LARGE-EDDY SIMULATION OF COMBUSTION

LARGE-EDDY SIMULATION OF COMBUSTION SYSTEMS WITH CONVECTIVE HEAT-LOSS

by L. Shunn & P. Moin

Prepared with support from the U.S. Department of Energy through the Advanced Simulation & Computing program and the Franklin P. and Caroline M. Johnson Fellowship

Report No. TF-113

Flow Physics and Computational Engineering Group Department of Mechanical Engineering Stanford University Stanford, CA 94305-3035

March 2009

ii

Abstract Computer simulations have the potential to viably address the design challenges of modern combustion applications, provided that adequate models for the prediction of multiphysics processes can be developed. Heat transfer has particular significance in modeling because it directly affects thermal efficiencies and pollutant formation in combustion systems. Convective heat transfer from flame-wall interaction has received increased attention in aeronautical propulsion and power-generation applications where modern designs have trended towards more compact combustors with higher surface-to-volume ratios, and in diesel engines where enclosed volumes and cool walls provide ample conditions for thermal quenching. As intense flame-wall interactions can induce extremely large heat fluxes, their inclusion is important in computational models used to predict performance and design cooling systems. In the present work, a flamelet method is proposed for modeling turbulence/chemistry interactions in large-eddy simulations (LES) of non-premixed combustion systems with convective heat-losses. The new method is based on the flamelet/progress variable approach of Pierce & Moin (2004) and extends that work to include the effects of thermal-losses on the combustion chemistry. In the new model, chemicalstate databases are constructed by solving one-dimensional diffusion/reaction equations which have been constrained by scaling the enthalpy of the system between the adiabatic state and a thermally-quenched reference state. The solutions are parameterized and tabulated as a function of the mapping variables: mixture fraction, reaction progress variable, and normalized enthalpy. The new model is applied to LES of non-premixed methane-air combustion in a coaxial-jet with isothermal wallconditions to describe heat transfer to the confinement. The resulting velocity, species iii

concentration, and temperature fields are compared to experimental measurements and to numerical results from the adiabatic model. The new method shows distinct improvement in the prediction of temperature, mixture composition, and heat flux in the near-wall regions of the combustor.

iv

Acknowledgement This work was supported by the NDSEG Fellowship and by the U.S. Department of Energy through the Advanced Simulation and Computing (ASC) program. Computer resources were provided by Sandia National Laboratory and Lawrence Livermore National Laboratory. Many of the key ideas in Chapters 4 and 5 were the result of collaborations with Dr. Frank Ham of the Center for Turbulence Research. His contributions are gratefully acknowledged. Likewise, early drafts of this report were strengthened by the insightful comments and suggestions of Professors Heinz Pitsch and Gianluca Iaccarino. We thank them for their careful reading and thoughtful feedback.

v

vi

Nomenclature Upper-case Roman symbols A

atomic weight

C

model coefficient

I

identity matrix

M

molecular weight

P Q˙

probability density function volume source term

S

deviatoric stress tensor

T

total stress tensor

U

reference velocity

X

mole fraction

X

vertices of tetrahedron

Y

mass fraction

Lower-case Roman symbols a

speed of sound, absorption coefficient, interpolation weight

c

reaction progress variable

cp

constant pressure heat capacity

e

internal energy

g

body force vector

h

enthalpy

j

diffusive flux

p

pressure vii

q

heat flux vector, subgrid-scale (residual) flux

r

radial coordinate

t

time

u

x1 velocity component

u

= (u, v, w)T , velocity vector

v

x2 velocity component

v

diffusion velocity

w

x3 velocity component, quadrature weight

x

spatial coordinate vector = (x1 , x2 , x3 )T for Cartesian coordinate systems = (x, r, θ)T for cylindrical coordinate systems

z

mixture fraction

Upper-case Greek symbols ∆

difference or change in a quantity

Φ

fuel equivalence ratio

Γ

gamma function, flamelet enthalpy parameter

Λ

flamelet reaction progress parameter

Θ

temperature

Lower-case Greek symbols α

diffusivity

β

beta function, least-squares coefficient

χ

scalar dissipation rate

δ

differential amount, edge length of tetrahedron

δ

Kronecker delta tensor

ϵ

error

φ

generic scalar

η

heat-loss parameter

κ

thermal conductivity

λ

bulk viscosity viii

µ

dynamic viscosity

ψ

generic scalar

θ

azimuthal coordinate

ρ

mass density

ϱ

non-dimensional radial coordinate

σ

Stefan-Boltzmann constant

σ

symmetric rate-of-strain tensor

τ

time scale, non-dimensional time

τ

viscous stress tensor

ω˙

reaction source

ξ

non-dimensional axial coordinate

ζ

element mass fraction

Script symbols C

transformed progress variable coordinate

F

tabulated function

D

diffusion coefficient

H

transformed enthalpy coordinate

L

Leonard stress

M

model term

Q

residual flux computed from test-filtered field

R

universal gas constant

V

transformed mixture fraction variance coordinate

X

predictor values

Y

response values

Z

transformed mixture fraction coordinate

Other symbols (·)

( · )′ (! ·)

Reynolds filtered quantity fluctuation (Reynolds filtered) Favre filtered quantity ix

( · )′′ (" · )′′2

variance (Favre filtered)

⟨( · )⟩

spatially or temporally averaged quantity

∂φ ( · )

=

O( · ) D( · ) Dt

fluctuation (Favre filtered)

order of accuracy ∂(·) ∂φ

, partial differentiation with respect to φ

·) = ∂t ( · ) + uj ∂( , substantial (material) derivative ∂xj

Le

Lewis number

Pr

Prandtl number

Sc

Schmidt number

Superscripts ad

adiabatic

fuel

fuel conditions

oxid

oxidizer conditions

ref

reference value

wall

wall value

Subscripts ∞

far-field or background quantity

α, β

molecular or elemental species (no summation)

D

Dufour heat flux, dimensionality of the system

e

extinction

eq

equilibrium

i

ignition

i, j, k

generic summation index

R

radiation

RF

reference frame

st

stoichiometric

t

turbulent

tree

binary tree approximation

x

Abbreviations BSP

binary space-partitioning

CCP

cubic close-packed

CDP

large-eddy simulation code developed at Stanford University (named in memory of Charles D. Pierce)

DNS

direct numerical simulation

EOS

equation-of-state

FPV

flamelet/progress variable

FPV-GHL

flamelet/progress variable with generalized heat-loss

FWI

flame/wall interaction

LES

large-eddy simulation

MMS

method of manufactured solutions

PDF

probability density function

RANS

Reynolds averaged Navier-Stokes

SGS

subgrid-scale

SLF

steady laminar flamelet

TI

tetrahedral integration

V&V

verification and validation

xi

xii

Contents Abstract

iii

Acknowledgement

v

Nomenclature

vii

1 Introduction

1

1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Literature discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Large-eddy simulation . . . . . . . . . . . . . . . . . . . . . .

5

1.2.2

Turbulent non-premixed combustion . . . . . . . . . . . . . .

7

1.2.3

Heat transfer and combustion . . . . . . . . . . . . . . . . . .

9

1.3 Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2 Mathematical model

13

2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2.1

Low-Mach approximation . . . . . . . . . . . . . . . . . . . .

15

2.2.2

Mass transport . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2.3

Thermal transport . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.4

Additional simplifications . . . . . . . . . . . . . . . . . . . .

20

2.3 Working equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4 Chemistry models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.1

Mixture fraction

. . . . . . . . . . . . . . . . . . . . . . . . . xiii

21

2.4.2

Steady laminar flamelet models . . . . . . . . . . . . . . . . .

24

2.4.3

Flamelet/progress variable model . . . . . . . . . . . . . . . .

28

2.4.4

Physical-space flamelet formulation . . . . . . . . . . . . . . .

30

3 Large-eddy simulation

33

3.1

Filtered LES equations . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.2

Subgrid-scale models . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.2.1

Turbulent stress and scalar flux . . . . . . . . . . . . . . . . .

35

3.2.2

Subgrid chemistry closure . . . . . . . . . . . . . . . . . . . .

37

3.2.3

Residual scalar variance . . . . . . . . . . . . . . . . . . . . .

40

Numerical algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.3

4 Equation-of-state evaluations 4.1 4.2

4.3

43

Stable and accurate state evaluations . . . . . . . . . . . . . . . . . .

44

4.1.1

EOS-induced instabilities . . . . . . . . . . . . . . . . . . . . .

44

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.2.1

1-D example problem . . . . . . . . . . . . . . . . . . . . . . .

47

4.2.2

2-D example problem . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.3

Adaptive tetrahedral integration . . . . . . . . . . . . . . . . .

52

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

5 Code verifcation using MMS 5.1

59

Method of manufactured solutions . . . . . . . . . . . . . . . . . . . .

61

5.1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.1.2

Example problems . . . . . . . . . . . . . . . . . . . . . . . .

62

5.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

6 A flamelet model for heat-loss 6.1

77

Non-adiabatic FPV model . . . . . . . . . . . . . . . . . . . . . . . .

77

6.1.1

Modeling approach . . . . . . . . . . . . . . . . . . . . . . . .

78

6.1.2

Application in LES . . . . . . . . . . . . . . . . . . . . . . . .

85

xiv

6.1.3

Transformed coordinate system . . . . . . . . . . . . . . . . .

88

6.2 Efficient tabulation using kd-trees . . . . . . . . . . . . . . . . . . . .

91

6.2.1

Introduction to binary trees . . . . . . . . . . . . . . . . . . .

92

6.2.2

Partitioning algorithm . . . . . . . . . . . . . . . . . . . . . .

93

6.2.3

Data storage and retrieval . . . . . . . . . . . . . . . . . . . .

97

6.2.4

Application to combustion chemistry . . . . . . . . . . . . . .

99

7 Application to coaxial-jet combustor

109

7.1 Experimental configuration . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 8 Conclusions and further work

131

8.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Recommendations for further work . . . . . . . . . . . . . . . . . . . 133 A Symmetric quadrature rules for tetrahedra

137

A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 A.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xv

xvi

List of Tables 4.1 Execution time of simulations. . . . . . . . . . . . . . . . . . . . . . .

55

5.1 Parameter values for 1-D manufactured solution problem, equation (5.4). 64 5.2 Parameter values for 2-D manufactured solution problem, equation (5.5). 64 5.3 1-D manufactured solution: L∞ - and L2 -error at t=1 versus spatial grid refinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.4 EOS lookup table: resolutions and errors. . . . . . . . . . . . . . . . .

69

5.5 1-D manufactured solution: maximum L∞ - and L2 -error versus EOS lookup table refinement. . . . . . . . . . . . . . . . . . . . . . . . . .

72

6.1 Parameters for adiabatic flamelets used in comparison test with enthalpyconstrained flamelets. . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

6.2 Storage requirements for Cartesian tables and kd-trees. . . . . . . . .

99

7.1 Combustor dimensions and flow conditions specified in the experiment of Spadaccini et al. (1976). . . . . . . . . . . . . . . . . . . . . . . . . 111 7.2 Summary of LES test cases. . . . . . . . . . . . . . . . . . . . . . . . 116 A.1 Tetrahedral integration quadrature point locations and weights . . . . 145 A.2 Integration of g30 (x, y, z) over the unit cube. g30 is the polynomial that includes all possible monomial terms xi y j z k of order i + j + k ≤ 30. . 150

xvii

xviii

List of Figures 1.1 World marketed energy use by fuel type from 1980 to 2030. Renewable energy sources include solar, geothermal, biomass, wind, and hydroelectricity (source: U. S. Department of Energy, 2008). . . . . . . . .

2

2.1 Solutions of the steady flamelet equations for methane-air combustion (p = 3.8 atm, Θfuel = 300K, Θoxid = 750K). S-shaped curve (left) and selected temperature profiles (right). The stoichiometric mixture fraction is zst = 0.055. . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.2 Solution of the steady flamelet equations for methane-air combustion (p = 3.8 atm, Θfuel = 300K, Θoxid = 750K). The solid line represents the accessible solutions in the SLF model. The dashed line represents the accessible solutions in the FPV model. The arrows indicate how intermediate states “a” and “b” are mapped onto the S-shaped curve in different flamelet models. . . . . . . . . . . . . . . . . . . . . . . .

29

2.3 Example physical-space flamelet configuration: stagnant diffusion-reaction layer between porous walls. . . . . . . . . . . . . . . . . . . . . . . . .

32

4.1 Scalars versus outer-iterations at a single location in a non-convergent combustion simulation. . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.2 Example spatial profiles of mixture fraction, density, and product source term in a combustion problem. . . . . . . . . . . . . . . . . . . . . . .

45

4.3 1-D combustion problem. . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.4 Model reactive EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

xix

4.5

1-D combustion problem with reference-frame velocity uRF = 0.0. (a) analytic EOS evaluation, (b) linear interpolation ∆z = 0.01, (c) linear interpolation ∆z = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

50

1-D combustion problem with reference-frame velocity uRF = 0.2. (a) analytic EOS evaluation, (b) linear interpolation ∆z = 0.01, (c) linear interpolation ∆z = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

50

2-D Rayleigh-Taylor instability. The left figures use node EOS-evaluations, and the right figures use TI 56-point EOS-evaluations. (top to bottom) t = 0, t = 1.56, t = 3.08, t = 4.64. (left to right) 25 × 75, 50 × 150,

4.8

100 × 300, 200 × 600 grids. . . . . . . . . . . . . . . . . . . . . . . . . 2-D Rayleigh-Taylor instability at t = 5.2. (left to right) Node evalu-

ations on 50 × 150 grid, TI 56-point evaluations on 50 × 150 grid, TI 4.9

53

56-point evaluations on 100 × 300 grid. . . . . . . . . . . . . . . . . .

54

2-D Rayleigh-Taylor instability on 100 × 300 grid using adaptive TI

evaluations. Density on top, concentration of quadrature points on

bottom. (left to right) t = 0, t = 1.56, t = 3.08, t = 4.64. . . . . . . . 4.10 2-D Rayleigh-Taylor instability on 100 × 300 grid at t = 5.2. TI adap-

56

tive (left), TI 56-point (right). . . . . . . . . . . . . . . . . . . . . . .

56

5.1

¯˙ (x, t). . . 1-D manufactured solution. (left to right) u(x, t), φ(x, t), Q φ

64

5.2

2-D manufactured solution: ρ(x, y, t) (top to bottom) t = 0, t = 1/3, t = 2/3, t = 1. (black: ρ = 1, white: ρ = 20). . . . . . . . . . . . . . .

66

5.3

1-D manufactured solution: L2 -error at t=1. . . . . . . . . . . . . . .

68

5.4

L2 -error in velocity u(x, t) versus time for 1-D manufactured solution.

68

5.5

L2 -error in scalar φ(x, t) versus time for 1-D manufactured solution. .

68

5.6

L2 -error in velocity u(x, t) versus time for 1-D manufactured solution on nx = 1024 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

L2 -error in scalar φ(x, t) versus time for 1-D manufactured solution on nx = 1024 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8

71 71

1-D manufactured solution: convergence of maximum L2 -error on nx = 1024 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

71

5.9 1-D manufactured solution: convective outlet velocity u(x = 2, t) on nx = 1024 grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 2-D manufactured solution: L2 -error at t = 1 using 20 outer iterations.

71 73

5.11 2-D manufactured solution: L2 -error of u(x, t) at t = 1 for different numbers of outer iterations. . . . . . . . . . . . . . . . . . . . . . . .

73

5.12 2-D manufactured solution: L2 -error at t=0.1 using 1 outer iteration.

74

6.1 Enthalpy profiles for flamelets with heat-loss. Methane-air combustion, p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, 0 ≤ η ≤ 1. . . . . . . . . . .

81

p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, 0 ≤ η ≤ 0.9. . . . . . . . . .

81

6.2 S-shaped curves for flamelets with heat-loss. Methane-air combustion,

6.3 Temperature (left) and progress variable (right) for the non-adiabatic, trans-critical flamelets indicated by “•” in Figure 6.2. Methane-air

combustion, p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, η = 0, 0.2, 0.4, 0.6, 0.8, 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

6.4 Temperature contours from a (z, c, h) flamelet library as a function of mixture fraction and enthalpy. These data correspond to the maximum value of the progress variable (Λ = 1). . . . . . . . . . . . . . . . . . .

82

6.5 Temperature contours from a (z, c, h) flamelet library as a function of mixture fraction and heat-loss parameter. These data correspond to the maximum value of the progress variable (Λ = 1). . . . . . . . . .

82

6.6 Mapping of enthalpy and heat-loss parameter from adiabatic flamelets solved at different air temperatures to regions defined by the enthalpyconstrained flamelets. . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

6.7 Solution profiles from the adiabatic flamelets listed in Table 6.1: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO . . . . . . . . . . . . . . . . . .

84

6.8 Comparison of thermochemical properties obtained from solving the adiabatic flamelet equations and the enthalpy-constrained flamelet equations: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO . . . . . . . . . . . . xxi

86

6.9

Flamelet solutions in the original coordinate system (left) and the transformed coordinate system (right). Top to bottom: temperature at C = 1 and H = 0, temperature at V = 0 and H = 0, temperature

at V = 0 and C = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

6.10 Auxiliary relations for chemistry table coordinate transforms. Refer-

ence enthalpy, href (Z, V) [kJ/kg] (left), and maximum progress variable, cmax (Z, V, H) (right). The cmax shown above corresponds to the H = 0 plane.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.11 Example kd-tree data structure (left) and spatial partitioning (right).

93

6.12 Error measures for the tree partitioning algorithm: (a) example nodal error distribution, equation (6.16); (b) cumulative sum of the centered error, equation (6.13); (c) left and right cumulative mean error, equation (6.14); (d) node split criterion, equation (6.15). . . . . . . . . . .

96

6.13 Binary tree data structure for density with error tolerance of 1 × 10−2 .

Branch node △, leaf node •. . . . . . . . . . . . . . . . . . . . . . . . 101

6.14 Density, ρ [kg/m3 ]. Original data (left) and binary tree approximation

with error tolerance of 1 × 10−3 (right). . . . . . . . . . . . . . . . . . 102

6.15 Product source, ρ ω˙ c [kg/m3 /s]. Binary tree approximation with error

tolerance of 1 × 10−3. Full domain (left) and zoom view (right). . . . 102

6.16 Binary tree approximation with error tolerance of 1 × 10−3 . Tempera-

ture, Θ [K], (left) and CO mass fraction, YCO , (right). . . . . . . . . . 102

6.17 Maximum normalized error (log10 scale) versus table size for linear interpolation from 2-D uniform Cartesian tables: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.18 Maximum error versus memory requirement for 2-D uniform Cartesian tables and binary trees: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO . . . . . 105 6.19 Retrieval time for 1,000,000 2-D lookups with linear interpolation using various search methods. Each point represents an average of ten 1,000,000-point tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 xxii

7.1 Schematic of the coaxial-jet combustor experiment of Spadaccini et al. (1976). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2 Least-squares fit of adiabatic flamelet data at zavg = 0.0498 to predict XH2 O using XCO2 and XCO . The linear coefficients are βCO2 = 1.8989 and βCO = 1.7929.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3 Computational domain (top) and computational grid (bottom) in nondimensional coordinates ξ = x/R3 and ϱ = r/R4 .

. . . . . . . . . . . 113

7.4 Computational grid: axial (left) and radial (right) grid spacing in nondimensional coordinates ξ = x/R3 and ϱ = r/R4 .

. . . . . . . . . . . 116

7.5 Instantaneous mixture fraction, z#: (a) meridional plane, (b) ξ = 1, (c)

ξ = 3, (d) ξ = 5. Adiabatic Case 2 (top), isothermal Case 3 (bottom). 118

7.6 Instantaneous progress variable, # c: (a) meridional plane, (b) ξ = 1, (c)


# (a) meridional plane, (b) ξ = 1, (c) 7.7 Instantaneous temperature, Θ:


7.8 Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature, CO mass fraction. Experiment (mean) •, experiment (RMS) ×, Case 0, and Case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.9 Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature, CO mole fraction. Experiment (mean) •, experiment (RMS) ×, Case 1, and Case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.10 Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature (full view), temperature (interior zoom), CO mole fraction, mixture fraction, progress variable. Experiment (mean) •, experiment (RMS) ×, Case 2, Case 3, and Case 4. . . . . . . . . . . . . . . . . . . 125 xxiii

7.11 Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: enthalpy, enthalpy boundary layer, temperature boundary layer. Case 2, Case 3, and Case 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.12 Instantaneous snapshot of wall heat flux: Case 3 (left) and Case 4 (right).127 7.13 Axial profiles of time-averaged wall variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Clockwise from upper left: heat-flux, normalized enthalpy, progress variable, mixture fraction. Case 3, and Case 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.14 Histogram of peak heat-flux values for Case 3 (left) and Case4 (right). 129 A.1 4-point product Gaussian rule. The rendered volume of the spheres indicates the relative weighting that each point receives in the quadrature rule. Note the asymmetric clustering of the points along the lower left edge of the tetrahedron. . . . . . . . . . . . . . . . . . . . . . . . 139 A.2 Cubic close-packed structures. Top row (left to right): 1-, 4-, and 10point structures . Bottom row (left to right): 20-, 35-, and 56-point structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 A.3 Tetrahedral integration quadrature point locations and weights. The rendered volume of the spheres indicates the relative weighting that each point receives in the quadrature rule. Top row (left to right): 1-, 4-, and 10-point rules. Bottom row (left to right): 20-, 35-, and 56-point rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.4 Unit tetrahedron used in the monomial integration test. . . . . . . . . 147 A.5 Error from integrating individual monomials over the unit tetrahedron. 148 A.6 Order-of-accuracy of tetrahedral integration rules. . . . . . . . . . . . 151 A.7 Exponential convergence of tetrahedral integration rules. . . . . . . . 152 A.8 Order-of-accuracy of tetrahedral integration rules. . . . . . . . . . . . 154

xxiv

Chapter 1 Introduction 1.1

Motivation and objectives

It is difficult to overstate the importance of energy research in today’s dynamic and globally competitive environment. Analysts expect worldwide energy consumption to increase by approximately 50% by the year 2030 compared to 2005 (see U. S. Department of Energy, 2008), increases fueled largely by growing demand in developing countries. Despite a projected increase from nuclear and renewable sources, the fractional contribution from fossil fuels (petroleum liquids, natural gas, and coal) is expected to change by less than 1% over the next 20 years (see Figure 1.1). Given their importance in the transportation sector, petroleum liquids are expected to remain the world’s primary energy source, while coal will continue to supply roughly half of the world’s electricity through 2030 and beyond. With these trends, it is obvious that combustion of hydrocarbon fuels will dominate the energy landscape for the foreseeable future. Despite the maturity of combustion-based energy, a host of technological challenges remains. The inherent scarcity of fossil fuel reserves, coupled with rising demand, highlights the urgency of developing more fuel-efficient combustion devices. In addition, environmental concerns and the prospect of global-warming has led governments to enact stringent restrictions on the emission of pollutants. As these requirements push devices into leaner combustion regimes, considerations of stability 1

2

CHAPTER 1. INTRODUCTION

energy consumption [Btu ×1015 ]

250

history

projections

200 150

liquids

100

coal natural gas

50

1980

renewables nuclear 1990 2000

year

2010

2020

2030

Figure 1.1: World marketed energy use by fuel type from 1980 to 2030. Renewable energy sources include solar, geothermal, biomass, wind, and hydroelectricity (source: U. S. Department of Energy, 2008).

and safety present practical limits to combustor designs. Addressing these issues, with their complex economic, political, and environmental implications, requires a better understanding of the fundamental physics of combustion and the development of tractable methods for modeling and analyzing combustion systems. To facilitate the design of efficient, low emission, and stably operating combustion devices it is essential to accurately predict combustion processes. Prediction in this sense, comprises both understanding the processes that drive key phenomena and translating those processes into mathematical statements that approximate (or model) real systems. Until recently, experimental observation and theory were the primary means of developing predictive models. However, with the advances in computer technology and algorithms of the last half-century, numerical simulation has emerged as a full complement to experiment (in understanding fundamental phenomena) and to theory (in predicting the behavior of physical systems). Computer simulations have the potential to viably address the design challenges of modern combustion applications, provided that adequate models for the prediction

1.1. MOTIVATION AND OBJECTIVES

3

of multiphysics processes encountered in combustion systems can be developed. In well-designed combustors, for example, flow unsteadiness, separation, swirl, and/or flame lift-off can all combine to ensure stable operation of the device. Additional complexity is introduced by secondary mixing and reaction, heat transfer, quenching and re-ignition, pollutant formation, and acoustic interactions. Models that effectively describe these systems must consider the vast range of scales and conditions dictated by all the component physics. In the present work, a flamelet-based combustion model is used to describe nonpremixed combustion. In the flamelet framework, multiscale analysis is used to describe a turbulent diffusion flame as an ensemble of laminar flamelets embedded in a turbulent flow (Peters, 1983, 1984). Thermochemical species are represented by mapping the flamelet information to low-dimensional manifolds parameterized by globally descriptive scalars. A salient advantage of flamelet methods is their simplicity and ability to efficiently account for complex chemical kinetics by pre-computing the thermochemical relationships and tabulating them for easy retrieval. The computational cost of the model is modest, since only the solution of the low-dimensional set of mapping scalars is required to represent all of the thermochemical properties. Recently, Pierce & Moin (2001, 2004) developed a flamelet-based methodology for large-eddy simulation (LES) called the flamelet/progress variable (FPV) model. The new approach was used to predict flame lift-off in a confined combustor, with favorable comparison to the experimentally measured velocity, mixture fraction, and product concentration fields. The FPV model, however, did not account for convective heat transfer to solid surfaces, radiative transfer effects, or slow chemical processes such as pollutant formation. Discrepancies between the simulation and experiment were attributed to these shortcomings. Heat transfer, in particular, can dramatically affect the thermal efficiency and pollutant formation in combustion systems. In diesel engines, for example, the enclosed volume and cool walls of the engine cylinders provide ample conditions for flame-wall interaction (FWI). Intense FWI can lead to local flame quenching and a corresponding decrease in engine performance. Although negative impacts on the thermal efficiency are generally small, FWI notably affects pollutant formation and

4


provides a dominant mechanism for the emission of unburnt hydrocarbons. Similar effects are observed in aeronautical propulsion and power-generation applications, particularly as designs trend towards more compact combustors with higher surfaceto-volume ratios. As flames interact with walls before quenching, extremely large heat fluxes can be induced. The maximum heat flux values are important to consider when designing structural components and cooling systems. The primary objective of this work is to extend the FPV model to include a general description of convective heat transfer for gas-phase combustion. In the expanded model, chemical-state databases are constructed by solving one-dimensional diffusion/reaction equations which have been constrained by scaling the enthalpy of the system between the adiabatic state and a thermally-quenched reference state. The solutions are parameterized and tabulated as a function of the mapping variables: mixture fraction, reaction progress variable, and normalized enthalpy. This model extension represents an essential movement towards a generalized combustion model for the prediction of non-premixed turbulent combustion. It is also recognized that to usefully leverage computer simulations, the predictive capability of the numerical methods and models must be demonstrated. The process of establishing this predictive capability is generally referred to as verification and validation, or V&V. The term verification describes the process of demonstrating that a code correctly solves the intended governing mathematical equations. A code that has been properly verified, therefore, is likely free of programming errors (bugs) that affect the theoretical order-of-accuracy of the numerical algorithm. Verification is a largely mathematical endeavor, and ordinarily carries clearly identifiable metrics of success or failure for a specific test. The term validation, on the other hand, is often used to imply favorable comparison of numerical simulation results with data from a controlled laboratory experiment. The goal of validation is to probe the applicability or suitability of a proposed numerical approach (or model) to solve a given real-world application. Validation efforts, in the end, require a judgement call about whether or not the right computational model is being used to describe the physics of the target application. As the demand for high-fidelity simulations in science and engineering grows and

1.2. LITERATURE DISCUSSION

5

simulation codes become correspondingly more complex and sophisticated, V&V practices must assume an early and integrated role in establishing the predictive capabilities of simulation software. Only after careful and proper V&V, can simulation results be used with confidence to predict the behavior of applications and provide the understanding necessary to improve them. In the present work, strong emphasis is placed towards the development of V&V tests amenable to the low-Mach number, variable-density combustion codes that are popular in propulsion and automotive applications.

1.2 1.2.1

Literature discussion Large-eddy simulation

In large-eddy simulation (LES), large energy-containing motions in a turbulent flow are explicitly represented, whereas the effects of smaller-scale motions are modeled. In the spectrum of computational expense, LES lies between Reynolds-averaged NavierStokes (RANS) methods, where all turbulent stresses are modeled, and direct numerical simulation (DNS) where all scales are resolved. RANS computations are relatively fast and inexpensive, but can be inaccurate for flows where unsteady dynamics and separation are important. DNS is the method of choice for detailed studies at low-Reynolds numbers, but its enormous cost makes it infeasible for complex, high-Reynolds number flows. LES is a promising compromise for many engineering applications because it directly captures the dynamics of large-scale motions, but replaces the vast computational expense of representing the smaller scales with simple models. Since large-eddy simulations are three-dimensional and time-dependent, they incur considerable computational costs relative to RANS methods. Despite this, high-fidelity LES is an attractive alternative to physical testing, and its utility is increasing with further advancements in computer technology. Much of the pioneering work on LES was motivated by meterological applications (Smagorinsky, 1963; Lilly, 1967; Deardorff, 1974). Later development and testing focused on simple flows such as isotropic turbulence (Reynolds, 1976; Kraichnan, 1976)

6


and fully-developed turbulent channel flow (Deardorff, 1970; Schumann, 1975; Moin & Kim, 1982; Piomelli, 1993). A driving objective in current research is the application of LES to complex geometries for engineering analysis (Ghosal & Moin, 1995; Mahesh et al., 2004; Ham & Iaccarino, 2004; Ham et al., 2006; Kassinos et al., 2007). Overviews of the developments and history of LES can be found in the collection of papers by Galperin & Orszag (1993) and the reviews of Mason (1994); Lesieur & Métais (1998); Moin (1997, 2002); Meneveau & Katz (2000).

Significant effort has been spent on the formulation of subgrid-scale (SGS) stress models for LES. Most SGS models are a variation on the constant coefficient eddyviscosity model proposed by Smagorinsky (1963). A significant contribution was the development of the dynamic Smagorinsky model where model coefficients are computed automatically from resolved quantities, thereby eliminating the uncertainty of tunable model-parameters (Germano et al., 1991; Moin et al., 1991; Ghosal et al., 1995). To further improve the dynamic method, Lilly (1992) proposed that leastsquares averaging of the dynamic coefficients be performed in statistically homogeneous directions to provide additional robustness. For complex flows where spatial averaging is problematic, Meneveau et al. (1996) applied time-averaging along Lagrangian pathlines with success. These and other variants of the dynamic Smagorinsky model have been applied in a variety of flows with generally good results (Piomelli & Liu, 1995; Akselvoll & Moin, 1996; Wang & Squires, 1996; Ghosal & Rogers, 1997; Wu & Squires, 1998; Haworth & Jansen, 2000). Alternatives to Smagorinsky-type models include scale-similarity approaches (Bardina et al., 1980), renormalization group techniques (Yakhot et al., 1989; Smith & Woodruff, 1998), structure function models (Métais & Lesieur, 1992), deconvolution methods (Domaradzki & Saiki, 1997; Stolz & Adams, 1999), fractal-based approaches (Benzi et al., 1993; Juneja et al., 1994), optimal-estimation methods (Berkooz, 1993; Chorin et al., 1998), stochastically synthesized velocity methods (Kerstein, 2002; Schmidt et al., 2003), and recent algebraic eddy-viscosity formulations (Vreman, 2004; You & Moin, 2007).


1.2.2

7

Turbulent non-premixed combustion

Several models have been proposed for predicting turbulent non-premixed combustion. Common approaches include flamelet models (Peters, 1983, 1984), probability density function (PDF) methods (Pope, 1985, 1990; Dopazo, 1994), conditional moment closure (CMC) models (Klimenko, 1990; Bilger, 1993; Klimenko & Bilger, 1999), linear-eddy models (LEM) (Kerstein, 1992a,b), and the related one-dimensional turbulence (ODT) approach (Kerstein, 1999; Kerstein et al., 2001; Echekki et al., 2001). The monograph by Peters (2000) offers a comprehensive review of turbulent combustion models. Many of these established modeling approaches have recently been applied to LES of chemically reacting flows. A review of developments in the field of LES applied to combustion was conducted by Pitsch (2006). The steady laminar flamelet model was tested in LES of homogeneous turbulence (Cook et al., 1997) and a priori studies of DNS data (Cook & Riley, 1998; de Bruyn Kops et al., 1998). Steady flamelet methods have additionally been used in LES of jet flames and bluff-body stabilized flames (Kempf et al., 2002, 2005; Renfro et al., 2004; Stein & Kempf, 2007; Stein et al., 2007). Lagrangian and Eulerian versions of unsteady flamelet models have likewise been applied to LES (Pitsch et al., 1998; Pitsch & Steiner, 2000; Pitsch, 2002). Pierce & Moin (2001, 2004) developed an approach for LES of non-premixed combustion termed the flamelet/progress variable (FPV) model. The FPV model is conceptually similar to the steady laminar flamelet method, but parameterizes the chemistry using the mixture fraction and a reaction progress variable rather than by the mixture fraction and scalar dissipation rate (as is common in traditional flamelet methods). Several other combustion models have employed similar parameterizations (Janicka & Kollmann, 1978; Bruel et al., 1990; Zhang et al., 1995), and the model is in some ways analogous to the intrinsic low-dimensional manifold (ILDM) (Maas & Pope, 1992) and flamelet-generated manifold (FGM) methods (van Oijen & de Goey, 2002). The FPV model has recently been recast in an unsteady version (Ihme & Pitsch, 2004) and expanded to better describe flame extinction and re-ignition (Ihme et al., 2005; Ihme & Pitsch, 2008a,b). The generality and utility of the model has been demonstrated in LES of engineering flows (Mahesh et al., 2006).

8


The CMC approach models turbulent combustion by solving transport equations for reactive scalars which have been conditioned on the mixture fraction (Klimenko, 1990; Bilger, 1993; Klimenko & Bilger, 1999). The model was originally formulated for RANS simulations, but has recently been modified for application to LES (Kim & Pitsch, 2005) and used to simulate a bluff-body stabilized diffusion flame (Kim & Pitsch, 2006). A variant of the CMC technique, called conditional source-term estimation, was proposed by Bushe & Steiner (1999) and applied to LES of a piloted methane/air diffusion flame (Steiner & Bushe, 2001). Transported PDF methods have long been applied in RANS simulations of turbulent combustion (Pope, 1985; Chen et al., 1989; Xu & Pope, 2000). One attractive aspect of these methods is that, contrary to most other combustion models, reactive source terms appear in closed form and require no closure modeling. A limitation of the model, however, is the inherent high dimensionality of the PDF-transport equation and the need for cumbersome modeling of molecular mixing terms. Transported PDF methods have recently been applied in LES of jet flame configurations (Raman et al., 2005; Raman & Pitsch, 2007). The method remains very costly due to its reliance on Monte-Carlo methods with large numbers of Lagrangian particles. Another approach to account for non-equilibrium chemistry in turbulent combustion is the linear-eddy model (LEM). LEM was originally formulated for nonreacting flows (Kerstein, 1988), but later extended to model reactive scalars (Kerstein, 1992a,b). LEM deterministically simulates one-dimensional diffusive mixing, but models turbulent advection as a series of stochastic “rearrangement events” whose size and frequency are chosen randomly from a PDF designed to represent inertialrange scaling. LEM can be used in “stand-alone” mode to represent simple combustion scenarios (McMurtry et al., 1993; DesJardin & Frankel, 1997) or can be coupled with LES to model residual scalar fluxes and chemistry (Menon & Calhoon, 1996; ElAsrag & Menon, 2007). The core ideas in LEM have recently been extended beyond scalar modeling to also include the velocity field. The resulting model, called onedimensional turbulence (ODT), is a self-contained formulation that does not require assumptions about the energy-cascade (Kerstein, 1999; Kerstein et al., 2001). The ODT framework has been used to successfully predict many properties of turbulent


9

jet diffusion flames (Echekki et al., 2001; Hewson & Kerstein, 2001). In addition to the computational models discussed here, a vast literature of experimental investigations of non-premixed combustion exists. An early example is the series of coaxial-jet combustor experiments conducted at the United Technology Research Center (Spadaccini et al., 1976; Owen et al., 1976; Johnson & Bennett, 1981, 1984; Roback & Johnson, 1983). The results consist of relatively complete measurements of velocity, species, and temperature for a variety of swirling and non-swirling configurations. Another notable example is the turbulent non-premixed flame (TNF) library established by Sandia National Laboratories to provide welldocumented flames for model validation and basic combustion research (Sandia National Laboratories, 1996). The library includes experimental data from simple jet flames, piloted jet flames, bluff-body flames, and swirl flames with a variety of reactants and flow conditions represented. The TNF workshop’s focus on geometrically simple, canonical flames is not only indicative of the challenges of producing highquality measurements of combustion phenomena, but also reflective of the relatively early stage of development that characterizes many computational combustion models. At present, most LES studies in complex geometries have not been subject to validation against comprehensive experiments. Future developments in this field would greatly benefit from quantitative experimental data in complex configurations.

1.2.3

Heat transfer and combustion

Flame-wall interactions (FWI) have been the subject of many investigations, with a prevalent focus on laminar and turbulent quenching of premixed flames near cold boundaries. Results from laminar flame experiments indicate that quenching events near cold surfaces correspond to maximum values of the gas-solid heat flux (Jarosinski, 1986; Lu et al., 1990; Ezekoye et al., 1992), a result confirmed numerically (Westbrook et al., 1981). Computational modeling has also been used to evaluate how quenching is affected by turbulent flow conditions (Poinsot et al., 1993; Bruneaux et al., 1996, 1997; Alshaalan & Rutland, 1998), detailed gas-phase kinetics (Popp & Baum, 1995, 1997; Dabireau et al., 2003), and heterogeneous gas-solid surface chemistry (Popp

10


et al., 1996; Aghalayam et al., 1998; Raimondeau et al., 2002). Bruneaux et al. (1996) report that peak gas-solid heat fluxes from turbulent premixed flames are comparable to values in laminar configurations, a result suggesting that similar dynamics control the quenching process in both flow regimes. In many numerical studies, radiation effects are neglected and simple one-step chemistry is used. This approximation appears to correctly describe FWI for the case of low-temperature walls, however, detailed chemical kinetics are required to accurately predict the gas-solid heat flux for wall-temperatures greater than 400K (Popp & Baum, 1995; Popp et al., 1996; Popp & Baum, 1997), indicating that FWI is a complex phenomenon with both thermal and kinetic contributions. Studies of FWI in non-premixed configurations are less common. Vance & Wichman (2000) studied gas-solid heat transfer in triple flames, comparing Burke-Schumann and finite-rate chemistry effects. Numerical simulations of diffusion flames in laminar stagnation-point flow, where the flame directly impinges on a cold wall, indicate that quenching strongly depends on the strain rate of the flow (de Lataillade et al., 2002; Dabireau et al., 2003). For the hydrocarbon-air flames studied by de Lataillade et al. (2002), peak heat flux values were larger than in corresponding premixed flames. However, the extremely large strain rates prescribed in these flames, may have contributed to unrealistically short quenching distances and excessive heat fluxes. More recently, Wang & Trouvé (2006) conducted a two-dimensional DNS of non-premixed FWI, reporting maximum heat flux values about one order of magnitude smaller than the de Lataillade et al. (2002) flames. Despite the simplifications in the DNS (i.e., two-dimensional domain, one-step chemistry, no radiation, etc.), the results are wellsuited for flamelet applications as the chemical and thermal structure of the flame is analyzed using classical variables such as mixture fraction and scalar dissipation rate. Future studies should target more quantitative results by removing the modeling limitations mentioned above. A significant body of work related to heat transfer and FWI has been motivated by fire safety and industrial combustion applications. In these disciplines, a common experimental configuration locates flames either adjacent to, or impinging on surfaces. Measurements of the time-averaged heat flux in enclosure fires have been obtained for

1.3. ACCOMPLISHMENTS

11

the case of flammable vertical walls (Ahmad & Faeth, 1978, 1979; Babrauskas, 1995; Lattimer, 2003), inert walls adjacent to a separate burner (Quintiere & Cleary, 1994; Back et al., 1994; Babrauskas, 1995; Lattimer, 2003), and horizontal walls exposed to impinging buoyancy-driven flames (You & Faeth, 1979; You, 1985; Kokkala, 1991; Babrauskas, 1995; Lattimer, 2003). In many fire applications the magnitude of the wall heat flux depends strongly on the radiative properties of the flame. In small non-sooty flames, the gas-solid heat flux is controlled by convective heat transfer, whereas radiation is the dominant contributor in large flames with significant smoke production. In industrial applications, measurements have been conducted for jet flames impinging normal to cylinders (Hargrave et al., 1987; Hustad & Sønju, 1991; Hustad et al., 1991, 1992) and plane surfaces (Milson & Chegier, 1973; Hargrave & Kilham, 1984; Hargrave et al., 1987; Popiel et al., 1980; van der Meer, 1991; Eibeck et al., 1993; Rigby & Webb, 1995; Mohr et al., 1996; Baukal & Gebhart, 1998). Useful reviews of experimental flame impingement studies are available (Baukal & Gebhart, 1995a,b; Baukal, 2000).

1.3

Accomplishments

Important contributions of this work include: • Demonstration that insufficient resolution of the equation-of-state (i.e., mass density, reactive source terms, etc.) in variable-density flow simulations can

lead to numerical instabilities that prevent solution convergence and/or lead to unphysical flow features. • Derivation of a family of symmetric quadrature rules for efficient numerical integration over tetrahedral volumes.

• Application of the quadrature formulae to evaluate the equation-of-state in

variable-density simulations. The new method was shown to mitigate errors and produce a more physical evolution of the flowfield at a modest computational cost.

12


• Development of 1-D and 2-D test problems for verification of low-Mach number

combustion codes using the method of manufactured solutions (MMS). The various MMS test problems were implemented in combustion codes at Stanford University and Sandia National Laboratories.

• Development of an unstructured, tree-based tabulation technique for efficient

storage and retrieval of thermochemical properties used in combustion simulations. The tree-based method displayed improved accuracy and reduced storage requirements when compared to conventional tabulation techniques.

• Extension of the flamelet/progress variable (FPV) model to include the effects of generalized heat-loss on combustion chemistry. The extended model was applied in a LES of a coaxial jet combustor with isothermal walls.

Chapter 2 Mathematical model In this chapter, the governing equations for gas-phase reacting flows are presented, simplifying assumptions are detailed, and a final “working set” of equations is outlined.

2.1

Governing equations

Gas-phase combustion is governed by the equations for mass, momentum, and energy conservation, species transport, and a thermodynamic equation-of-state. The following description is valid for a mixture of ideal gases in local thermodynamic equilibrium and chemical nonequilibrium (see Williams, 1985). Mass: Momentum: Energy:

∂t ρ + ∇ · (ρu) = 0

(2.1a)

∂t (ρu) + ∇ · (ρuu) = −∇p + ∇ · τ + ρ

nS $

Yα g α

(2.1b)

α=1

∂t [ρ(e + 12 (u · u))] + ∇ · [ρu(e + 12 (u · u))] = nS $ ∇ · [−pu + (τ · u) − q] + ρ Yα g α · (u + v α )

(2.1c)

α=1

Species transport: Equation-of-state:

∂t (ρYα ) + ∇ · (ρuYα ) = −∇ · (ρv α Yα ) + ρ ω˙ α nS $ p = ρRΘ Yα /Mα α=1

13

(2.1d) (2.1e)

14

CHAPTER 2. MATHEMATICAL MODEL

The viscous stress tensor appearing in equation (2.1b) can be written in terms of the symmetric rate-of-strain tensor σ as τ = 2µσ + λ(∇ · u)I

where

σ = 21 [∇u + (∇u)T ].

(2.2)

The heat flux vector in equation (2.1c) is given by q = −κ∇Θ + ρ

nS $ α=1

v α Yα hα + RΘ

nS $ nS $ Xβ DΘα α=1 β=1

Mα Dαβ

(v α − v β ) + q R .

(2.3)

Working from left to right, the terms in equation (2.3) describe heat flux due to: conduction, species diffusion, mass/density gradients (or the Dufour effect), and radiation. The temperature is implicitly related to the internal energy and enthalpy through e=

nS $

n

Yα eα (Θ)

α=1

and

S p $ h=e+ = Yα hα (Θ) ρ α=1

(2.4)

where eα and hα are the species internal energies and enthalpies per unit mass.

The species diffusion velocities v α are determined from the implicit vector equation nS $ Xα X β

∇p Dαβ p β=1 % & nS nS $ Xα Xβ DΘβ DΘα ∇Θ ρ$ Yα Yβ (g α − g β ) + − . + p α=1 ρD Y Y Θ αβ β α β=1

∇Xα =

(v β − v α ) + (Yα − Xα )

(2.5)

Here the terms from left to right indicate intermolecular multicomponent diffusion (or Stefan-Maxwell diffusion), pressure-gradient diffusion, diffusion due to body-forces on individual species, and temperature-driven mass diffusion (or the Soret effect). Conversion between species mole and mass fractions is accomplished using Xα =

Yα /Mα n S $

Yβ /Mβ

β=1

and

Yα =

Xα Mα nS $ β=1

Xβ Mβ

.

(2.6)

15

2.2. SIMPLIFYING ASSUMPTIONS

Summing the scalar transport equations (2.1d) over all nS species leads directly to the equation for total mass conservation (2.1a). A consistent description of these nS + 1 equations, therefore, requires that the diffusion velocities and chemical sources satisfy the conditions nS $

Yα v α = 0

and

α=1

nS $

ω˙ α = 0.

(2.7)

α=1

Molecular transport, thermochemical, and chemical kinetic properties are needed in order to complete the specification of the governing equations. Transport properties include the viscosity coefficients µ and λ, thermal conductivity κ, binary mass diffusion coefficients Dαβ , and thermal diffusion coefficients DΘα . Thermochemical

properties include the species enthalpies hα and molecular weights Mα . Chemical kinetics provide the species production terms ω˙ α as a function of temperature, pressure, and mixture composition.

2.2

Simplifying assumptions

For many combustion systems it is advantageous to simplify some of the physics embodied by the equations of Section 2.1. In the following, the various simplifying assumptions employed in this work are enumerated and justified.

2.2.1

Low-Mach approximation

A commonly invoked simplification for combustion systems is the so-called “lowMach number approximation.” This refers to two key assumptions, namely that compressibility effects and viscous heating are both negligible. It is appropriate to note that, even in flows at nominally low Mach number, either of these phenomena can play important roles. As an example, consider a lubricating flow where the Mach number is small, but viscous heating is critically important. Consequently, while a low Mach number is necessary for these assumptions to be valid, it is by no means a sufficient justification for their use.

16


The exclusion of viscous heating is a solidly defensible approximation in hightemperature reacting flows. In most of these systems, heat release from combustion patently overwhelms the relatively limited dissipation of energy due to viscous mechanisms. To a good approximation, therefore, the viscous work term in equation (2.1c), ∇ · (τ · u), can be dismissed. Neglecting compressibility effects (including acoustic interactions) is a far more restrictive approximation. This simplification assumes that the flow is characterized by a relatively constant background pressure p∞ with weak deviations δp. It is further presumed that thermodynamic quantities, such as density and temperature, are largely unaffected by the small variations in the total pressure p∞ + δp, and depend only on the base pressure. For a system of ideal gases this means ρ= RΘ

p∞ n S $

.

(2.8)

Yα /Mα

α=1

Note that although the density is formally decoupled from changes in the pressure, it is still allowed to vary with the local temperature Θ and mixture composition Yα . A well-known implication of the low-Mach number approximation is revealed by expressing the density as a function of the independent thermodynamic quantities pressure and entropy. In the low-Mach number framework this leads to %

& ∂ρ . ρ(p∞ + δp, s) = ρ(p∞ , s) + δp ∂p s ' () *

(2.9)

≈0

Since the speed of sound in the mixture, a, is related to the isentropic change in density with respect to pressure, equation (2.9) implies that sound waves propagate at nearly infinite rates in these flows a=

+%

∂p ∂ρ

&

s

→ ∞.

(2.10)

17


2.2.2

Mass transport

It is convenient to replace the rigorous, but unwieldy description of species diffusion in equation (2.5) with a simpler expression. A commonly used approximation is to assign mixture diffusivities αβ to each species and use Fick’s law to describe the corresponding diffusion velocities v β = −αβ

∇Yβ . Yβ

(2.11)

The mixture diffusivities αβ can be thought of as effective binary diffusion coefficients for species β in the mixture, and are calculated from the binary diffusivity matrix according to the formula of Bird et al. (2002) 1 − Xβ . αβ = $ Xα α̸=β

(2.12)

Dαβ

The above formula is rigorously true only for the special case when all of the species α ̸= β have the same velocities. Nevertheless, it is frequently used to approximate multicomponent systems.

Nondiscriminatory use of equation (2.11) can lead to numerical inconsistencies since the resulting diffusion velocities do not necessarily satisfy the mass-conservation criteria identified in equation (2.7). A simple remedy is to apply Fick’s law for the first nS − 1 species and assign the residual to the final component v nS

nS −1 1 $ =− v α Yα . YnS α=1

(2.13)

Kee et al. (1990) suggest that equation (2.13) be applied to the species in excess, which for many combustion systems is nitrogen N2 . Owing to these simplifications, there is no explicit representation of diffusion due to pressure gradients or body-force interactions. The non-dimensional Lewis number is defined as the ratio of the thermal diffusivity

18


to the mass diffusivity. For species α, therefore, the Lewis number is Leα =

κ Scα αΘ = = αα ρcp αα Pr

(2.14)

where Scα and Pr are the Schmidt and Prandtl numbers. Williams (1985) demonstrated by asymptotic analysis that differential diffusion (as characterized by nonunity Lewis numbers) influences the structure and stability of laminar flames. Differentialdiffusion effects in turbulent flames have likewise been observed in experiments (Drake et al., 1981, 1986; Bergmann et al., 1998; Meier et al., 2000) and numerical simulations (Yeung & Pope, 1993; Nilsen & Kosály, 1997, 1999; Kronenburg & Bilger, 1997; Pitsch, 2000). Generally, the importance of differential diffusion decreases with axial distance from the initial point of mixing and with increasing Reynolds number (Drake et al., 1986; Meier et al., 2000). Throughout this work, Lewis numbers are assumed to be unity in all of the resolved-scale transport equations (where turbulent advection dominates), while differential diffusion is included in the description of subgrid-scale mass transport (the flamelet model). The Soret term identified in equation (2.5) is likewise included in the flamelet model for completeness. This approach has been demonstrated to yield reasonable results in confined hydrocarbon jet flames (Pierce & Moin, 2001, 2004).

2.2.3

Thermal transport

For consideration of the thermal field, it is useful to express the energy equation (2.1c) in terms of the enthalpy. This conversion is accomplished by subtracting the transport equation for kinetic energy, 12 (u · u), from equation (2.1c) and combining the result with equation (2.4)

nS $ ∂(ρh) Dp + ∇ · (ρuh) = + τ : ∇u − ∇ · q + ρ Yα g α · v α . ∂t Dt α=1

(2.15)

For the systems under consideration here, the approximation that p∞ is spatially

19


and temporally uniform is valid and Dp∞ Dp ≈ = 0. Dt Dt

(2.16)

Substituting equation (2.3) into (2.15), and neglecting viscous heating, body-forces, and Dufour diffusion gives ∂(ρh) + ∇ · (ρuh) = ∇ · ∂t

,

κ∇Θ − ρ

nS $ α=1

-

v α Yα hα − q R ,

(2.17)

and recasting the diffusion term in light of equation (2.11) produces ∂(ρh) + ∇ · (ρuh) = ∇ · ∂t

,

κ∇Θ −

nS $ α=1

-

ραα hα ∇Yα − q R .

(2.18)

Detailed treatment of radiative transfer within a turbulent flame is computationally expensive. In order to include the effect of radiation in combustion models without significantly increasing computational cost, a simplified treatment of radiative losses is needed. Here, an optically-thin approximation is adopted. The utility of such an approach has been demonstrated for hydrogen (Barlow et al., 1999) and methane jet flames (Frank et al., 2000; Zhu et al., 2002; Coelho et al., 2002). Assuming that each radiating point has an unimpeded, isotropic view of the uniformly cold surroundings, the radiative loss rate per unit volume can be approximated as (Barlow et al., 2001) ∇ · q R = Q˙ R = 4σ(Θ4 − Θ4∞ )

nR $

pα aα

(2.19)

α=1

where σ is the Stefan-Boltzmann constant, Θ∞ is the background temperature, pα and aα are the partial pressure and Planck mean absorption coefficient of species α, and the summation is performed over all radiatively important species. Species typically included in the calculation for hydrocarbon fuels are CO2 , H2 O, CH4 , and CO. Incorporating the simplification from equation (2.19) into equation (2.18), and

20


using the following identity to eliminate ∇Θ ∇h = cp ∇Θ +

nS $ α=1

hα ∇Yα

(2.20)

leads to the enthalpy equation ∂(ρh) + ∇ · (ρuh) = ∇ · ∂t

,

ραΘ ∇h +

nS $ α=1

ρ(αα − αΘ )hα ∇Yα

-

− Q˙ R .

(2.21)

Under the assumption of unity Lewis numbers this reduces to ∂(ρh) + ∇ · (ρuh) = ∇ · (ραΘ ∇h) − Q˙ R . ∂t

2.2.4

(2.22)

Additional simplifications

The complete stress tensor (including the thermodynamic pressure p) is given by Tij = −pδij + 2µσij + λσkk δij . A mean, mechanical pressure can also be defined by

averaging the diagonal terms of the stress tensor, p¯ ≡ − 31 Tii . Equating these two

definitions of the normal stress component, p and p¯, leads to the condition λ = − 32 µ.

This assumption, which is analogous to the hypothesis of Stokes (1845), simplifies the viscous stress tensor to τ = 2µ[σ − 31 (∇ · u)I] = 2µS

(2.23)

where S is the deviatoric rate-of-strain tensor. For most of the simulations in this work body-forces are negligible, and therefore, not considered. In cases where body-forces are present, it is assumed that they act identically on all species. The body-force vectors in equations (2.1b) and (2.1c) are therefore equivalent, and g α = g β = g. In the general case, the equation-of-state (2.8) is a function of nS + 1 independent scalars, specifying the temperature, pressure, and composition of the mixture. In the context of flamelet modeling, however, the equation-of-state is ultimately expressed as

21

2.3. WORKING EQUATIONS

a function of a lower-dimensional set of mapping scalars and tabulated for convenience and efficiency. This is indicated generically by the notation ρ = f (φ1 , φ2 , ...).

2.3

(2.24)

Working equations

The simplifying assumptions of Section 2.2 result in the following system of equations which will used throughout this work: Mass: Momentum: Energy: Scalar transport: Equation-of-state:

∂t ρ + ∇ · (ρu) = 0

(2.25a)

∂t (ρu) + ∇ · (ρuu) = −∇p + ∇ · (2µS)

(2.25b)

∂t (ρh) + ∇ · (ρuh) = ∇ · (ραΘ ∇h) − Q˙ R

(2.25c)

∂t (ρφα ) + ∇ · (ρuφα ) = ∇ · (ραα ∇φα ) + ρ ω˙ α

(2.25d)

ρ = f (φ1 , φ2 , ...)

(2.25e)

Note that due to the simplifications applied to the energy equation, equation (2.25c) has been reduced to a special case of equation (2.25d). In this work, scalar transport equations are solved for three quantities: mixture fraction z, reaction progress variable c, and total enthalpy h. All other thermochemical quantities are interpolated from a property database as functions of these scalars.

2.4 2.4.1

Chemistry models Mixture fraction

In non-premixed combustion, initially separated fuel and oxidizer are combined in a common region where they mix and burn under continuous interdiffusion. Within the mixture, conditions suitable for combustion typically only exist in relatively narrow regions of space where the reactants have mixed in the proper proportions. In these “reaction zones,” heat release from exothermic combustion creates high temperatures

22


and supplies energy necessary for continued reactions. Simultaneously, however, reaction products and heat diffuse outward from the reaction zone and mix with reactants and other products. Competition between the relative rates of heat and mass diffusion, compared with the rate-limiting steps of the chemical reaction, largely determines the local behavior of the non-premixed flame. Since the time-scales associated with chemical reaction are often several orders of magnitude faster than characteristic convection and diffusion time-scales, heat release and chemical conversion rates are mainly determined by diffusion effects (Peters, 1983). In the limit of infinitely fast chemical reactions, the reaction zone is infinitesimally thick. For this case, Burke & Schumann (1928) defined a coupling function using reactant mass fractions which allows an analytical characterization of diffusion flames. Since their pioneering work, the “conserved scalar” variable (called the mixture fraction z) has been an essential component in predicting global properties of non-premixed jet flames such as the flame length. In the fast chemistry approach, all scalars such as temperature, species concentrations, and density are uniquely related to z. Since the mixture fraction variable is chemistry-independent, however, it is equally useful for describing finite-rate chemistry processes, and, in fact, remains a fundamental tool in diffusion flame theory. For a review of the conserved scalar approach see Bilger (1980). Roughly speaking, the mixture fraction is a conserved scalar quantity that indicates the fraction of mass at a given location that originated from a particular reactant stream. For a system that has n separate inlet streams, n − 1 mixture fraction variables are needed to completely describe the mixing processes. By convention,

a mixture fraction is assigned a value of one in the stream from which it emanates and zero in all others. Since z is a conserved scalar, its transport equation can be formally derived by taking linear combinations of other species transport equations such that the reaction source terms mutually cancel. The particular combination of species that comprise a mixture fraction depends on details about the fuel and oxidizer. Barring differences due to boundary conditions and how the diffusion velocity is specified, all conserved

23

2.4. CHEMISTRY MODELS

scalars obey the same generic advection-diffusion equation ∂ρz + ∇ · (ρuz) = ∇ · (ραz ∇z). ∂t

(2.26)

Although z can be defined using several different conventions (Burke & Schumann, 1928; Bilger, 1976, 1988; Pitsch & Peters, 1998), most definitions provide a fairly similar characterization of the flame structure with subtle distinctions. In this work, the definition of z proposed by Pierce & Moin (2001, 2004) is used. First, the mass fraction of each element is computed for all nE distinct atomic elements in the system. Denoting Nαβ as the number of atoms of element β in a molecule of species α and Aβ as the atomic weight of element β, this is accomplished using ζβ =

nS $ Nαβ Aβ α=1

Mα

Yα

for

β = 1, 2, ... , nE.

(2.27)

An average mixture fraction for a system with two inlet streams is then given by normalizing the result with the element mass fractions in the oxidizer and fuel streams (ζβ0 and ζβ1, respectively): z=

,n E $ α=1

|ζα − ζα0 |

-., n E $ β=1

-

|ζβ1 − ζβ0 | .

(2.28)

A consistent diffusivity for this definition of the mixture fraction can be computed by similarly combining the diffusive fluxes of elements jβ =

nS $ Nαβ Aβ α=1

Mα

ρYα v α

for

β = 1, 2, ... , nE

(2.29)

and equating the normalized result to the diffusive flux of z ραz |∇z| =

nE $ α=1

|j α |

., n E $ β=1

-

|ζβ1 − ζβ0 | .

(2.30)

Jointly solving equations (2.26) and (2.30) is then equivalent to using (2.28) to define

24


the mixture fraction z.

2.4.2

Steady laminar flamelet models

There are situations in turbulent reacting flows where local diffusion time-scales are comparable to the rate-determining combustion reactions. In this case, the fast chemistry assumption is not valid and non-equilibrium effects must be accounted for. The steady laminar flamelet (SLF) model offers a convenient framework to do so. The SLF model views a turbulent diffusion flame as an ensemble of thin, quasilaminar reaction zones embedded in a turbulent flow field. This concept was introduced by Williams (1975) and further developed by Peters (1983, 1984). At sufficiently high Damköhler number or sufficiently high activation energy, a flame’s reaction zone is characterized by length-scales that are smaller than the surrounding turbulence. In this regime, turbulent structures can deform and stretch the flame sheet, but are unable to penetrate the reaction zone and alter its structure. This supports the notion of a “flamelet,” or a thin diffusive-reactive layer in a turbulent flow, where the local chemical state resembles that of a laminar flame. In other words, in the flamelet regime, combustion chemistry is largely decoupled from the turbulence. Flamelets exhibit one-dimensional structure in the direction normal to the flame surface, and can be characterized by the iso-surface of stoichiometric mixture fraction, z(x, t) = zst . The flamelet equations are derived by transforming equations (2.25c) and (2.25d) into a coordinate system tied to the mixture fraction. For simplicity, unity Lewis numbers are adopted in the following derivation, implying that all species, temperature, and enthalpy share a common diffusivity α. A more general derivation including the effect of non-unity Lewis numbers is given by Pitsch & Peters (1998). The transport equations are transformed from (x, t)-space to (z, τ )-space using the

25


transformation ∂ψ ∂z ∂ψ ∂ψ = + ∂t ∂τ ∂t ∂z ∂ψ ∂z ∂ψ = ∂x1 ∂x1 ∂z ∂ψ ∂z ∂ψ ∂ψ = + , k = 2, 3. ∂xk ∂zk ∂xk ∂z

(2.31)

By definition z is a coordinate locally normal to the stoichiometric mixture surface, z2 and z3 are tangential to z, and τ is the transformed time coordinate. Applying (2.31) to equations (2.25c) and (2.25d) results in ∂ψ ∂z

/

0 ∂z ∂ψ ∂2ψ ˙ +R ρ + ρu · ∇z − ∇ · (ρα∇z) + ρ = ρα|∇z|2 2 + m ∂t ∂τ ∂z

(2.32)

˙ = (ρ ω˙ T , −Q˙ R )T . The quantity in braces on the left-hand where ψ = (Y T , h)T and m

side of equation (2.32) is merely a restatement of the z transport equation (2.26), and

is identically zero. The term R on the right-hand side of equation (2.32) is % & % & 0 3 / $ ∂ψ ∂ ∂ψ ∂ψ ∂z ∂ ρα + ρα − ρuk . R= ∂x ∂z ∂z ∂z ∂z k ∂z k k k k k=2

(2.33)

All of these terms involve derivatives with respect to z2 and z3 , and are of low-order when compared with derivatives in the normal direction. The unsteady flamelet equations are comprised of the remaining terms in equation (2.32) ˙ ∂ψ χ ∂2ψ m = + . 2 ∂τ 2 ∂z ρ

(2.34)

In this equation, χ = 2α|∇z|2 is the scalar dissipation rate, which has units of inverse time and characterizes typical diffusion times in the flamelet. An alternate derivation of equation (2.34) using two-scale asymptotic analysis is given by Peters (2000). By omitting the temporal derivative in equation (2.34), the SLF model is obtained ˙ χ d2 ψ m − = . 2 dz 2 ρ

(2.35)

26


2500

•

χe

b

middle branch

1000

χi •

c

500 lower branch -4

10

-2

10

0

10

temperature, Θ [K]

temperature, Θst [K]

2000 1500

2500

a

upper branch

a

2000 1500

b

1000

c 500

2

10

0

0.2

dissipation rate, χst [1/s]

0.4

0.6

0.8

1

mixture fraction, z

Figure 2.1: Solutions of the steady flamelet equations for methane-air combustion (p = 3.8 atm, Θfuel = 300K, Θoxid = 750K). S-shaped curve (left) and selected temperature profiles (right). The stoichiometric mixture fraction is zst = 0.055.

In order to solve equation (2.35) additional assumptions are needed to obtain χ as a function of z. Peters (1984) obtained an analytic expression for χ(z) from a similarity solution of mixing in a laminar counter-flow configuration, reducing the flamelet problem to solving a non-linear two-point boundary value problem along the stagnation streamline. Given a suitable chemical mechanism, equation (2.35) can be solved as a function of a single parameter, χst = χ(zst ). This provides a unique relationship between the scalars ψ and z (for a given value of χst ). Numerical solutions to the flamelet equation (2.35) are shown in Figure 2.1. The plot on the left shows the locus of stoichiometric temperatures, Θst = Θ(zst ), as a function of the mixture fraction dissipation rate. This curve is commonly referred to as the “S-shaped” curve for laminar diffusion flames. The plot on the right in Figure 2.1 shows the thermal structure of three selected flamelets along the S-shaped curve. As noted in the figure, the S-shaped curve can be divided into three general regions, describing different regimes of mixing and burning. The upper branch of the curve, extending from the upper left of the plot to the point marked “χe ,” represents stably burning solutions of the flamelet equations. The lower branch of the curve,

27


from the bottom right of the plot to the point marked “χi ,” describes non-burning or purely mixing solutions. The middle branch, located between the points “χi ” and “χe ,” contains intermediate or partially-burning solutions to the equations. It should be noted that although these intermediate states are mathematically admissible, they are also highly unstable and are not observed experimentally. The S-shaped curve illustrates some interesting aspects regarding ignition, burning, and extinction in diffusion flames. Recall that the scalar dissipation rate, χst , designates a characteristic inverse time-scale for diffusive mixing in flamelets. Large values of χst describe short mixing-times and rapid diffusion, while small values of χst translate to long mixing-times and slow diffusion. In diffusion flames, the mixing time-scale is in direct competition with the chemical time-scale. For example, in a reacting flow on the upper branch of the curve, if mixing is slow compared to the chemistry, the reactions proceed vigorously and have ample time to progress towards equilibrium. As the dissipation rate increases, diffusion accelerates until mixing and reaction occur on similar time-scales. Under these conditions, heat generation from combustion balances the heat transfer out of the reactive zone due to conduction and species diffusion. This state corresponds to the extinction condition indicated in Figure 2.1 by χe . Any additional increase in χst results in flame quenching and a rapid transition to the lower, non-burning branch. Similarly, for mixing flows on the lower branch, a decrease in the rate of dissipation will eventually bring the system to the ignition point, χi . Here, reaction time-scales again balance diffusion time-scales, however, χi ≪ χe due to strongly temperature-dependent reaction kinetics. Further reductions in χst induce ignition and a sudden shift to the burning (upper) branch.

As explained earlier, the SLF model parameterizes the flame state by the mixture fraction and its dissipation rate. The thermochemical states of the flamelet are then represented by a two-dimensional manifold of the form ψ = Fψ(z, χst ).

(2.36)

This model has proven useful for engineering calculations because the one-dimensional

28


steady flamelet profiles can be computed using arbitrarily complex chemical mechanisms and then tabulated as a function of z and χst for efficient use in applications. Note, however, that the state-relation given by equation (2.36) does not provide a unique parameterization of the S-shaped curve, since multiple solutions exist in the region bounded by χi ≤ χst ≤ χe . In practical implementations of the SLF model, flames with χst ≤ χe are projected onto the upper (burning) branch of the S-shaped

curve, and flames with χst > χe are mapped onto the lower branch. This is indi-

cated by the vertical arrows in Figure 2.2. For many combustion codes, large density fluctuations caused by the sharp discontinuity at χe can cause numerical instabilities. For this reason, the lower non-reactive branch is often omitted from consideration.

2.4.3

Flamelet/progress variable model

The flamelet/progress variable (FPV) model was developed as an alternative to the SLF model, and offers a different, more complete parameterization of the solutions to equation (2.35). In the FPV model, the flamelets’ dependence on the reaction-coordinate is described by a reaction progress parameter instead of the dissipation rate. Ideally, this flamelet parameter is chosen such that it uniquely identifies flame states along the entire S-shaped curve, including the upper, lower, and middle branches. Many definitions of the reaction progress variable are possible. Pierce & Moin (2004) suggest using a sum of the major-product mass fractions: c = YCO2 + YH2 O + YCO + YH2 . Ihme et al. (2005) formalize this definition by introducing a flamelet parameter that samples the functional c at the stoichiometric mixture fraction in each flamelet Λ = c |zst .

(2.37)

An advantage to this approach is that Λ is statistically independent of z, whereas c and z have an obvious functional relationship. This is a distinct convenience when formulating models for the joint PDF of the thermochemical variables. In practice, however, the difference between using c and Λ to parameterize the

29



2500

✻

SLF

2000

◦

a FPV 1500

✛

1000

✲

•

χe

FPV

◦

b SLF

❄

χi • 500 -4

10

-2

10

100

2

10

104

dissipation rate, χst [1/s] Figure 2.2: Solution of the steady flamelet equations for methane-air combustion (p = 3.8 atm, Θfuel = 300K, Θoxid = 750K). The solid line represents the accessible solutions in the SLF model. The dashed line represents the accessible solutions in the FPV model. The arrows indicate how intermediate states “a” and “b” are mapped onto the S-shaped curve in different flamelet models.

flamelet database is extremely subtle. One motivation in defining tabulation parameters is to provide quantities that are easily obtained from transport equations or modeling. This is consistent with choosing c, but not necessarily Λ, whose transport equation involves complicated model terms. In order to avoid this inconvenience, Ihme et al. (2005) propose a table inversion to eliminate Λ in favor of c c = Fc (z, Λ) −→ Λ = Fc−1 (z, c).

(2.38)

Thermochemical quantities of interest can then be obtained from the tabulated flamelet solutions ψ = Fψ (z, Λ) = Fψ (z, Fc−1 (z, c)).

(2.39)

The FPV tabulation provides a description of burning and quenched flame states that is identical to the SLF model, but offers a different pathway between the two by including the unstable branch of the S-shaped curve. This is shown in Figure 2.2, where the intermediate conditions “a” and “b” are projected horizontally onto the middle branch in the FPV approach.

30


2.4.4

Physical-space flamelet formulation

In this work, an alternate flamelet formulation is used in place of equations (2.35). This formulation solves a set of diffusion-reaction equations in physical coordinates (rather than mixture fraction space) to obtain the mixture composition, Yα , and temperature, Θ. Once the composition has been determined, a mixture fraction can be computed using equation (2.28), at which point processing and applying data from the present formulation becomes procedurally identical to the classical approach. The physical-space flamelet equations are derived by simplifying equations (2.1) for steady (∂t = 0), one-dimensional (∂x2 = ∂x3 = 0) combustion, with no external velocity (u = 0). This results in the following system of diffusion-reaction equations d(ρYα vα ) = ρ ω˙ α dx , nS $ d d(κΘ) −ρ vα Yα hα − qD = 0 dx dx α=1 , -−1 n S $ ρ = p∞ RΘ Yα /Mα

(2.40a) (2.40b)

(2.40c)

α=1

h=

nS $

Yα hα (Θ).

(2.40d)

α=1

Equations (2.40) can be solved once the domain length, xL , and reactant conditions, Θfuel , Yαfuel , Θoxid , and Yαoxid , are specified. In this work, the flamelet diffusion velocities, vα , are defined using Fick’s law with mixture-averaged diffusivities and Soret contributions (see Section 2.2.2). The heat flux, qD , in equation (2.40b) is the Dufour term from equation (2.3). The total enthalpy is determined from equation (2.40d). The reactant states are applied as Dirichlet boundary conditions at x = 0 and x = xL . Equations (2.40) were introduced by Pierce & Moin (2001, 2004) who argued that assuming a velocity field of zero may be a more general model for chemistry in turbulent flows than prescribing the velocity for the specific case of a laminar counter-flow flame. Whether this is actually the case is still a matter of debate.


31

Solving equations (2.40) leads to profiles of scalar dissipation rate that exhibit broad regions of nearly constant χ(z) and local maxima on each side of the reactive zone. Although the conditional scalar dissipation rate in most experiments tends to zero at the endpoints, flat doubly-peaked profiles have been observed in turbulent jets (Karpetis & Barlow, 2002, 2005) and turbulent counter-flow flames (Kelman & Masri, 1997; Geyer et al., 2005). In the work of Geyer et al. (2005), the measurements were complemented with a LES of the experiment, where they concluded that the correct behavior could not be reproduced using the conventional SLF model with unity Lewis numbers. Contradicting these findings, other investigators have experimentally measured turbulent scalar dissipation rates that closely resemble the profiles imposed in the laminar counter-flow SLF model (Mi et al., 1995; Pantano et al., 2003; Markides & Mastorakos, 2008). To complicate the picture, some researchers have observed statistical independence between z and χ (Mi et al., 1995; Chen & Mansour, 1997), while others note significant non-linear interdependence (Starner et al., 1997; Pantano et al., 2003; Mellado et al., 2008). The diversity of these results suggests that a comprehensive understanding of the scalar dissipation rate still eludes the combustion community. Indeed, Peters (2006) and Pitsch (2006) have both identified a need to revisit common approaches to modeling fine-scale mixing quantities such as scalar dissipation. For high Reynolds number flows, where strong turbulent fluctuations and intermittency complicate the formulation of such models, the zero-velocity assumption used to derive equations (2.40), with its implications for χ(z), seems to have at least limited support from experimental observations. In the absence of a realistic stochastic description of χ(z) and u(x), the zero-velocity diffusion-reaction equations (2.40) provide a reasonable approximation as a flamelet model. Although Pierce & Moin (2001, 2004) were the first to use equations (2.40) as a flamelet model in LES, application of these equations to other combustion problems is not without precedent. An identical system of one-dimensional equations and boundary conditions was used to model a stagnant diffusion layer between porous plates (Sohn et al., 1999). A schematic of this problem is shown in Figure 2.3. A similar formulation has been used to study fuel-side heat loss in diffusion flames,

32


porous wall

porous wall

✲

✛

✲

✛

✲

✛

✲

✛

✲ ✲ ✲ ✲

diffusion flame

Θoxid Yαoxid

reaction zone

✛ ✛ ✛ ✛

✲

✛

✲

✛

x=0

Θfuel Yαfuel

x = xL

Figure 2.3: Example physical-space flamelet configuration: reaction layer between porous walls.

stagnant diffusion-

although the final equations were transformed and solved in mixture fraction space (Ray & Wichman, 1998). A variant of the porous plate configuration with fuel-side convection has likewise been proposed and solved in physical coordinates to analyze thermal-diffusive instabilities in non-premixed flames (Kim et al., 1996; Vance et al., 2001). Recently an experimental apparatus that is conceptually similar to Figure 2.3 has been constructed to study one-dimensional unstrained diffusion flames (Lo Jacono et al., 2005).

Chapter 3 Large-eddy simulation In this work, the reaction flow equations are numerically solved using large-eddy simulation (LES). In this chapter, the basic premises and approximations inherent in LES are reviewed, the LES equations and standard dynamic closure models are presented, and the presumed PDF approach for flamelet models is discussed.

3.1

Filtered LES equations

The LES methodology is a multiscale approach for fluid dynamics based on the separation of large- and small-scale motions in a turbulent flow. In LES a filtering operation, denoted here by an overbar, is introduced to formally separate the flow into resolved and unresolved scales. The large scales of motion are directly simulated while the smaller, dissipative scales are modeled. All field variables are decomposed into resolved and unresolved (or subgrid-scale) components using either a Reynolds decomposition (for volume-specific quantities) ρ = ρ¯ + ρ′

(3.1)

or the density-weighted Favre decomposition (for mass-specific quantities) ψ = ψ# + ψ ′′ 33

(3.2)

34

CHAPTER 3. LARGE-EDDY SIMULATION

Favre-decomposed quantities are related to Reynolds-filtered quantities by ρ. ψ# = ρψ/¯

(3.3)

Filtering the continuum Navier-Stokes equations yields equations for the resolvedscale variables: density ρ¯, pressure p¯, velocity u #i, and transported scalars φ#α . The filtering process removes instantaneous small-scale fluctuations, but these subfilter

quantities still exert a statistical effect on the resolved scales via residual flux terms which require modeling. Applying this procedure to the working equations of Section 2.3 yields the filtered

LES equations for variable-density reacting flows: ∂ ρ¯ ∂ ρ¯u #j ¯˙ + =Q ρ¯ ∂t ∂xj

(3.4)

% & ∂ ρ¯u #i ∂ ρ¯u #j u #i ∂ p¯ ∂ ¯˙ # + =− + 2¯ µSij − qij + Q u ei ∂t ∂xj ∂xi ∂xj

(3.5)

% & ∂ ρ¯φ#α ∂ ρ¯u ∂ φ#α #j φ#α ∂ ¯˙ + = ρ¯α #α − qφeα j + Q eα φ ∂t ∂xj ∂xj ∂xj

(3.6)

ρ¯ = f (φ1 , φ2 , ...) .

(3.7)

The resolved-scale stress in equation (3.5) involves the rate-of-strain tensor #kk S#ij = σ #ij − 13 δij σ

where

1 σ #ij = 2

%

∂# ui ∂# uj + ∂xj ∂xi

&

.

(3.8)

¯˙ has been added to the rightFor generality, a generic volumetric source term Q X hand side of each transport equation in (3.4)–(3.6). For the reactive scalar equation ¯˙ = (3.6), the source term could be written using the more descriptive notation Q e φα

#˙ α , however, the less explicit form above is retained because of its utility in the ρ¯ ω

verification methods discussed in Chapter 5. In addition, this generic notation allows for the inclusion of the simplified energy equation (2.25c) in the uniform description

35

3.2. SUBGRID-SCALE MODELS

of scalar transport presented in equation (3.6). All of the unclosed transport terms are embodied in the residual stress qij and residual scalar flux qφeα j terms of equations (3.5) and (3.6). These residual terms, as well as all volumetric sources and the state equation (3.7), require some form of closure modeling.

3.2 3.2.1

Subgrid-scale models Turbulent stress and scalar flux

The subfilter residual stress qij in equation (3.5) is modeled using an eddy-viscosity assumption qij − 13 qkk δij = ρ¯u" ¯u #i u #j = −2µt S#ij . i uj − ρ

(3.9)

In the current work, the isotropic part of the residual stress, qkk , is not explicitly treated, but is instead absorbed into the pressure term of the momentum equation. This is admissible in low-Mach number formulations where the pressure is decoupled from the thermodynamic variables. However, in compressible simulations this term should be properly modeled (see Moin et al., 1991). The eddy-viscosity is related to the resolved rate-of-strain tensor by the model proposed by Smagorinsky (1963) # . µt = ρ¯ Cµ ∆2 |S|

(3.10)

Cµ is a dimensionless scaling coefficient and ∆ is the filter width, which is often taken to be a characteristic length-scale of the local computational grid. The residual scalar flux in equation (3.6) is modeled in a manner similar to the turbulent stress ∂ φ#α qφeα j = ρ¯u" ¯u #j φ#α = −¯ ραt , j φα − ρ ∂xj

(3.11)

and the eddy-diffusivity is given by

# . αt = Cα ∆2 |S|

(3.12)

36


The values of the coefficients Cµ and Cα in equations (3.10) and (3.12) depend on the local flow conditions and can be estimated dynamically from the resolved fields. The dynamic procedure is a general method for calculating the values of the dimensionless scaling coefficients in subgrid-scale models (Germano et al., 1991; Moin et al., 1991; Ghosal & Moin, 1995). In the dynamic procedure, information about the smallest resolved scales is used to estimate the properties of the residual scales via an implicit assumption of scale-similarity. The smallest resolved scales are isolated by 1 > ∆ to the flow quantities. On uniform applying a second, “test” filter of width ∆

grids, the test filter width is often taken to be twice the width of the grid filter, 1 = 2∆. Analogous to equation (3.3), a test-filtered quantity is defined as ∆ ˇ 2# 1 ψ# = ρ¯ ψ/ ρ¯ .

(3.13)

In the dynamic procedure, the test filter is applied to the resolved field to compute a stress tensor based on test-filtered quantities. It is assumed that this stress tensor can be represented by a model of the same form as the residual stress tensor ˇ# ˇ# 1 2 |S| Qij = ρ¯u" ρ¯ u #ˇi u #ˇj = −2 1 ρ¯ Cµ ∆ S ij . i uj − 1

(3.14)

The residual stress tensor is likewise test-filtered to produce

# S#ij . q1ij = ρ¯u" ¯u #iu #j = −2 ρ¯ Cµ ∆2 |S| i uj − ρ

(3.15)

Using Germano’s identity (Germano et al., 1991), the Leonard stresses Lij can be

expressed in terms of Qij and qij

ˇ# ˇ# # S#ij − 2 1 1 2 |S| #i u #j − 1 ρ¯ u #ˇi u #ˇj = 2 ρ¯ Cµ ∆2 |S| ρ¯ Cµ ∆ S ij . Lij = Qij − qij = ρ¯u

(3.16)

Remarkably, all of the terms in equation (3.16) can be computed from the resolved flow variables. The right-hand side of the equation is referred to as the “model” term and assigned the notation Mij . Under the assumption that Cµ varies weakly in space

37


and time and can be removed from the filtering operation, equation (3.16) can be re-written as Lij = 2 Cµ Mij .

(3.17)

This system of equations is over-determined, but can be solved in a least-squares sense (Lilly, 1992) to provide numerical values for the model coefficient Cµ =

1 ⟨Lij Mij ⟩ , 2 ⟨Mkl Mkl ⟩

(3.18)

where the angled brackets indicate a spatial-averaging operation. The dynamic procedure applied to the subgrid turbulent stress model in equation (3.9) can then be summarized as Cµ =

1 ⟨Lij Mij ⟩ ˇ# ˇ# # S#ij − 1 1 2 |S| , Lij = ρ¯u #iu #j − 1 ρ¯ u #ˇi u #ˇj , Mij = ρ¯ ∆2 |S| ρ¯ ∆ S ij . (3.19) 2 ⟨MklMkl ⟩

Similarly, dynamic treatment of the subgrid scalar flux model from equation (3.11) yields Cα =

⟨Li Mi ⟩ ˇ ˇ# #ˇ # i φ#α − 1 1 2 |S|∂ , Li = ρ¯u #i φ#α − 1 ρ¯ u #ˇi φ#α , Mi = ρ¯ ∆2 |S|∂ ρ¯ ∆ i φα . (3.20) ⟨Mj Mj ⟩

The ratio of the two model coefficients defines the subgrid turbulent Prandtl number,

Prt = Cµ /Cα . In the present work, a single eddy-diffusivity field is computed using the mixture fraction, and then this field is applied to all other transported scalars in the flow.

3.2.2

Subgrid chemistry closure

In LES of non-premixed combustion, many important mixing and reaction processes occur at scales that are unresolved on the computational grid. Combustion-related variables such as the mixture fraction and progress variable are fluctuating quantities and their statistical distribution needs to be considered. Subgrid fluctuations in the combustion variables can have a dramatic impact on the mean properties of the flow

38


due to the sensitivity and non-linearity of combustion chemistry. Tractable methods to accurately describe these subgrid fluctuations are necessary in order to predict the resolved-scale behavior of the system. The presumed-shape approach to model the joint probability density function (PDF) of subgrid-scale scalar fluctuations has been successful in a variety of flameletbased models (see Peters, 2000). In this approach the joint PDF is directly modeled by a priori selecting a distribution with a simple, analytical form. The success of the method depends on the ability of the presumed PDF to capture the true statistics of the subgrid fluctuations. In the case of the FPV model, all thermochemical quantities are a function of the random variables z and Λ, whose Favre-weighted joint PDF P#(z, Λ) must be modeled. Mean values are obtained by integrating with the PDF over the sample space #= ψ

3 3

ψ(z, Λ)P#(z, Λ)dzdΛ .

(3.21)

The Favre-weighted PDF is related to the conventional PDF by 1 P#(z, Λ) = ρ(z, Λ)P (Z, Λ) . ρ¯

(3.22)

The joint PDF can be rewritten as a product of the marginal and conditional PDFs #= ψ

3 3

ψ(z, Λ)P#(z)P#(Λ|z)dzdΛ .

(3.23)

For convenience, the conditional PDF P#(Λ|z) is simplified by assuming independence

between Λ and z. The validity of this assumption was investigated by Ihme et al.

(2005), and shown to be well-founded for the definition of Λ used here. This leads to the factorization #= ψ

3 3

ψ(z, Λ)P#(z)P#(Λ)dzdΛ .

(3.24)

The individual PDFs are then modeled using the presumed-shape PDF approach. The mixture fraction PDF, P#(z), is well-represented by a beta distribution (Cook &

39


Riley, 1994; Jiménez et al., 1997), parameterized by its first two moments Γ(a + b) ′′2 ) = z a−1 (1 − z)b−1 P#(z) = β(z; z#, z! Γ(a)Γ(b)

where

′′2 )/z! ′′2 a = z#(# z − z#2 − z!

and

′′2 )/z! ′′2 . b = (1 − z#)(# z − z#2 − z!

(3.25)

(3.26)

In the work of Pierce & Moin (2001, 2004), the reaction progress PDF, P#(Λ), is

assumed to be a delta-function that samples the distribution at its mean value # . P#(Λ) = δ(Λ − Λ)

(3.27)

The physical implication of this assumption is that subgrid-scale fluctuations in the reaction progress parameter do not alter the thermochemical state predicted by the model. While this assumption is not rigorously defensible, it can be argued that progress variable fluctuations are of secondary importance compared to the mixture fraction, due to the relative steepness of thermochemical gradients in each of those dimensions. With these assumptions, the thermochemical properties of the flow are determined by evaluating #= ψ

3 3

′′2 )δ(Λ − Λ)dzdΛ ′′2 , Λ) #ψ(# # # . ψ(z, Λ)β(z; z#, z! =F z , z!

(3.28)

#ψ can be manipulated during As discussed in Section 2.4.3, the resulting database F # from Λ # to # post-processing to map the dependence of the functions ψ c, resulting in a collection of flamelet states that can be easily used in the FPV model ′′2 , # # =F #ψ (# ψ z , z! c) .

(3.29)

# contains the filtered values of all of the thermochemical quantities The vector ψ ¯˙ , Θ, # Y#α , etc. #˙ c , Q necessary to run and analyze the LES, e.g., ρ¯, α #α , ω R

40


3.2.3

Residual scalar variance

As indicated in Section 3.2.2, flamelet-based combustion models for LES require the ′′2 in order to determine the subgrid, or residual, variance of the mixture fraction z! local thermochemical state of the mixture. Indeed, the accurate prediction of the residual variance has been shown to be a key factor in obtaining good results with

presumed PDF methods (Cook & Riley, 1994; Wall et al., 2000). The residual scalar variance of the mixture fraction is defined as ′′2 = z#2 − z z! #2 .

(3.30)

′′2 in LES. A classic approach that is Several methods have been proposed to model z! ′′2 (Poinsot & Veyused in both RANS and LES is to solve a transport equation for z!

nante, 2001). The resulting transport equation contains unclosed terms for turbulent

transport, production, and dissipation which must be modeled. Alternatively, Cook & Riley (1994) proposed a scale-similarity model for passive scalar variance. Pierce & Moin (1998) modeled the residual variance by assuming local equilibrium between production and dissipation and using the dynamic procedure to determine a suitable model coefficient. Raman et al. (2005) noted that numerical inaccuracies in comput-

ing gradients of the filtered mixture fraction can contribute to under-prediction of the residual variance with both the variance transport equation and dynamic models. As an alternative, they advocate solving transport equations for both the filtered ′′2 using equation mixture fraction, z#, and its second moment, z#2 , then computing z!

(3.30). The transport equation for z#2 is simpler to model than the variance equation in that it does not contain a production term

% & / 0 ∂ ρ¯z#2 ∂ ρ¯u ∂z ′′ ∂z ′′ #j z#2 ∂ ∂ z#2 ∂# z ∂# z + = ρ¯α # − qze2 j − 2¯ ρα # + 2ρα . ∂t ∂xj ∂xj ∂xj ∂xj ∂xj ∂xj ∂xj

(3.31)

The expression in braces on the right-hand side of the equation represents a dissipation term with both resolved and subgrid contributions. Raman et al. (2005) modeled

3.3. NUMERICAL ALGORITHM

41

these jointly as 2¯ ρα #

∂z ′′ ∂z ′′ ∂# z ∂# z (α + αt ) ′′2 + 2ρα = ρ¯ Cze2 z! , ∂xj ∂xj ∂xj ∂xj ∆2

(3.32)

with Cze2 = 2 and αt determined from the dynamic procedure.

A similar approach is followed in the present work, with the residual mixture fraction variance computed from (3.30) after solving transport equations for z# and z#2 . Here, however, the resolved and subgrid portions in the dissipation term are handled separately. The resolved term is computed directly as 2¯ ρα #|∇# z |2 , and the

subfilter term is modeled as

2ρα

αt ∂z ′′ ∂z ′′ ′′2 = ρ¯ Cze2 z! . ∂xj ∂xj ∆2

(3.33)

The non-dimensional coefficient Cze2 is set statically to 40 for the current simulations. This value comes from estimates of the turbulent and scalar-mixing time-scales derived from model energy and scalar-energy spectra (Ihme, 2007). The eddy-diffusivity αt is determined from the dynamic procedure.

3.3

Numerical algorithm

The simulations throughout this work are performed using the unstructured largeeddy simulation code CDP. CDP is a set of massively parallel unstructured flow solvers developed specifically for LES by Stanford’s Center for Integrated Turbulence Simulations as part of the U. S. Department of Energy’s Advanced Simulation and Computing (ASC) Alliance Program. CDP uses a collocated, unstructured version of the algorithm of Pierce & Moin (2001, 2004). This algorithm employs a temporally-staggered variable arrangement in which velocity components are staggered in time with respect to density and other scalar variables. The equations are spatially discretized using low-dissipation, nodebased finite-volume operators developed by Ham et al. (2006). The variables are implicitly advanced in time using a fractional-step method, and an iterative approach

42


is used at each time level to repair splitting errors and enhance stability. The major features of the iteration process at each time step are listed below. Here the superscript m is used to denote the outer-iteration number. 1. The scalar equation(s) (3.6) are advanced in time. This yields (ρφ)m+1 , from which a provisional estimate for φ is obtained by φ1 = (ρφ)m+1 /ρm . 2. The momentum equations (3.5) are advanced to obtain provisional velocities: u1i .

3. The provisional scalar values are used to evaluate the density from the equation1 of-state: ρm+1 = f (φ). 4. The updated density is used to correct the scalar(s) to ensure primary conservation: φm+1 = (ρφ)m+1 /ρm+1 . 5. A Poisson equation is solved for pressure, and the result is used to correct the velocity field to discretely conserve mass. 6. The process is repeated from step 1 and continued until convergence. The linearized scalar and momentum equations (steps 1-2) are solved using a Jacobi method, and the Poisson solve (step 5) is accomplished using the HYPRE algebraic multigrid solver (Falgout & Yang, 2002; Henson & Yang, 2002). Linear analysis indicates that the iterative approach outlined above is second-order accurate when at least two outer iterations are employed (Pierce & Moin, 2001). Additional iterations may improve the stability of the scheme, but do not increase the order of accuracy. Formal verification of the second-order behavior of the algorithm requires convergence of the system at each time step.

Chapter 4 Equation-of-state evaluations For the case of low-Mach number combustion, the variable-density equations for reacting flows can realize substantial efficiency gains relative to fully compressible formulations. In variable-density formulations, pressure and density are formally decoupled by defining the density through an equation-of-state (EOS) in terms of one or more transported scalars: ρ = ρ(φ1 , φ2 , ...). The EOS may be given by an analytic expression, or as is common for complex reactive systems, it may be precomputed and tabulated as a function of the scalars. When the variable-density equations are discretized and solved numerically, it is common to use a fractional-step formulation where a constant-coefficient Poisson equation for pressure is derived through a constraint on the divergence of ρui coming from the continuity equation. The resulting equation for pressure has the time derivative of the density as a source term, and is solved and used to correct the velocity field and enforce mass conservation discretely. For large density variations (characterized by density ratios of approximately three or greater), instabilities are commonly encountered and resolved by largely ad hoc techniques whose effect on the solution cannot easily be quantified. For example, Pierce & Moin (2001, 2004) characterized the problem as “spurious heat release” related to inconsistencies between the mass and scalar transport, and resolved the instabilities by spatially filtering the computed ∂ρ/∂t source term several times. While this does not alter the mass in the simulation (assuming a conservative filter), it does spatially redistribute mass in a way not called 43

44

CHAPTER 4. EQUATION-OF-STATE EVALUATIONS

for by the governing equations. Forkel & Janicka (2000) performed temporal filtering of the density to stabilize their calculations. This introduces an additional complexity and hysteresis to the state equation whose effect is difficult to quantify. Other authors have reformulated the numerical method to solve a variable-coefficient Poisson system by constraining the divergence of velocity with the energy equation. While these approaches stably support higher density ratios, it appears necessary to relax the conservation properties of the scheme (Nicoud, 2000), or allow the state to wander somewhat from the state equation (Pember et al., 1998). In the present study the instability problem is resolved by recognizing that nonlinearities in the state equation can introduce a multi-scale resolution problem that is not supported by the grid. This problem can be largely resolved by simply evaluating the density (and other tabulated source terms) using an accurate integral over the computational control volume.

4.1 4.1.1

Stable and accurate state evaluations EOS-induced instabilities

When the time-stepping algorithm presented in Section 3.3 is applied to real combustion problems, it is generally not iterated to convergence. Normally, a fixed number of outer-iterations are performed (typically 3–5) and then the solution at a time step is considered to be “converged.” For many of the combustion problems investigated, however, rigorous convergence of the scheme was not possible. For example, Figure 4.1 shows the values of three scalar quantities sampled from one spatial location in a combustion simulation at successive outer-iterations of a single time step. In this example the density is obtained as a function of two transported scalars: the mixture fraction, z, and a reaction progress variable, c. Clearly, the system is non-convergent irrespective of the number of outer-iterations that are used, presenting a serious impediment to verification. Much simpler problems can be studied to try to reproduce and understand the sources of these instabilities. Consider, for example, the 1-D problem depicted in

45

4.1. STABLE AND ACCURATE STATE EVALUATIONS

mixture fraction, z density, ρ progress variable, c

1

1.5

0.8

mixture fraction, z density, ρ progress variable source, ω˙ c

φm

1

0.6 0.5

0.4 0.2 0 0

10

20

30

40

50

0 0

0.5

1

1.5

2

x

outer-iteration, m

Figure 4.1: Scalars versus outer- Figure 4.2: Example spatial profiles of mixture iterations at a single location in a fraction, density, and product source term in a non-convergent combustion simula- combustion problem. tion.

Figure 4.2, which shows the mixture fraction, z, as a function of the spatial coordinate, x. Also shown are the density and product source term, ω˙ c , from an EOS for non-premixed methane chemistry at slightly elevated pressures. An example computational grid is denoted by the solid symbols along the mixture fraction curve. Note that although the mixture fraction transition in the figure is reasonably resolved, the highly non-linear behavior of the density and source term would be grossly underrepresented on this grid. Instabilities can develop as the under-resolved, non-linear features of the EOS are transported through the grid. Consider the case of convective scalar transport (no diffusion) with the density given as a continuous function of one scalar, i.e., ∂ρφ ∂ρui φ + = 0, ∂t ∂xi Together with continuity,

ρ = ρ(φ).

∂ρ ∂ρui + = 0, ∂t ∂xi

(4.1)

(4.2)

46


these equations are equivalent to passive scalar advection and a divergence-free velocity field, i.e.,

∂φ ∂φ + ui = 0, ∂t ∂xi

∂ui = 0. ∂xi

(4.3)

This continuous equivalence is exploited in some numerical approaches for multiphase flow, such as the level set method, where equation (4.3) is discretized and solved. For the case of turbulent combustion, however, discrete conservation is considered important, so it is preferable to discretize and solve the conservative form of both scalar transport and continuity. Due to discretization errors, this equivalence is not realized discretely and manifests itself in some non-physical divergence in the velocity field. One way to reduce this non-physical divergence can be seen by considering the leading truncation error associated with the second-order approximations normally used in discretizing equations (4.1) and (4.2). Consider the 1-D case inside a control volume of size ∆x. The volume integration of density associated with the time derivative in the continuity equation is 1 ∆x

∆x/2 3

∆x2 ρ(φ)dx = ρ(φ0 ) + 24

−∆x/2

4

5 5 5 % 5 &2 6 ∂ρ 55 ∂ 2 φ 55 ∂ 2 ρ 55 ∂φ 55 + + O(∆x4 ) (4.4) ∂φ 5φ0 ∂x2 50 ∂φ2 5φ0 ∂x 50

and the volume integration of density times the scalar associated with the time derivative in the scalar transport equation is 1 ∆x

∆x/2 3

−∆x/2

ρ(φ)φ dx = ρ(φ0 )φ0 + 4 5 5 % 5 &2 ∆x2 ∂ρ 55 ∂φ 55 ∂ 2 φ 55 ρ(φ0 ) 2 5 + 2 5 + 24 ∂x 0 ∂φ φ0 ∂x 50 5 5 5 % 5 &2 6 ∂ 2 ρ 55 ∂φ 55 ∂ρ 55 ∂ 2 φ 55 + φ0 2 5 + O(∆x4 ). φ0 5 5 2 ∂φ φ0 ∂x 0 ∂φ φ0 ∂x 50

(4.5)

By introducing a higher-order reconstruction of density in the control volume, it

47

4.2. RESULTS

is clear that the truncation errors associated with both of these approximations can be reduced. For example, introducing a more accurate approximation for the density in the control volume, ρ1:

∆x2 ρ1 = ρ(φ0 ) + 24

4

5 5 5 % 5 &2 6 ∂ 2 ρ 55 ∂ρ 55 ∂ 2 φ 55 ∂φ 55 + ∂φ 5φ0 ∂x2 50 ∂φ2 5φ0 ∂x 50

(4.6)

removes all of the second-order error in equation (4.4) and two of the four secondorder terms in equation (4.5). Numerical experiments indicate that eliminating the curvature term involving ∂ 2 ρ/∂φ2 is particularly important, as this term is normally dominant. The optimized quadrature rules presented in Appendix A can be used to accurately approximate the density in flow computations on grids of arbitrary polyhedra in threedimensions. Example problems using this technique and a simple cost comparison of the method are presented in the following section.

4.2 4.2.1

Results 1-D example problem

The 1-D combustion problem shown schematically in Figure 4.3 has been designed to demonstrate the impact of EOS evaluations on a flow simulation. Here an initial transition from fuel (z = 1) to oxidizer (z = 0) is imposed and allowed to evolve subject to diffusive mixing. A constant reference-frame velocity uRF can be supplied to translate the entire system through the computational grid. For this problem, a polynomial state-relationship of the form ρ = 2z + (1 − z)18

(4.7)

is used to describe the density. This EOS exhibits strong non-linear behavior in the region z ≤ 0.2 (see Figure 4.4), and is indicative of the general behavior expected

in non-premixed combustion systems. In particular, the density minimum around

48


2

density, ρ

fuel

uout

z air

uRF

1.5 1

0.5

x

L

00

0.2

0.4

0.6

0.8

1

mixture fraction, z Figure 4.3: 1-D combustion problem.

Figure 4.4: Model reactive EOS.

z ≈ 0.12 suggests a region of reaction and heat release caused by the mixing of

relatively dense fuel at z = 1 with moderately dense (preheated) air at z = 0.

As the fuel front in Figure 4.3 diffuses, the low-density reactive region broadens and forces mass out of the right-hand side of the domain. The velocity at the exit, uout , can be used to monitor the global rate-of-change of mass in the system: −1 d (uout − uRF) = ρx=L dt

3L

ρ dx.

(4.8)

0

In the following, the outlet velocity is used to gauge the effectiveness of different density-evaluation methods. The 1-D diffusion problem was solved numerically using two different EOS-evaluation methods. In the first method, the mean density at a node is computed by simply evaluating the EOS using the mean mixture fraction value for that node. This method is referred to as “node evaluations.” In the second approach, the independent function (z in this case) is reconstructed linearly and continuously in the polyhedral region around each grid node. Each polyhedral control volume is then tessellated into tetrahedral subvolumes involving a combination of nodes, vertices, edges, and faces. A quadrature rule is applied within each subvolume to accurately integrate the state equation in space and construct an appropriate approximation to the density. This evaluation technique is referred to as “tetrahedral integration” (TI). 128 points in x were used in all of the simulations, and the most accurate (56-point) quadrature rule

4.2. RESULTS

49

was applied in the TI evaluations (see Appendix A). Figures 4.5 and 4.6 show numerical solutions to the 1-D diffusion problem with reference-frame velocities of uRF = 0 and uRF = 0.2, respectively. Solutions using node evaluations and the TI method are compared in each figure. Three cases, denoted as (a), (b), and (c), were run for each EOS-evaluation technique. In case (a), the EOS (equation (4.7)) was evaluated analytically to compute the density. In cases (b) and (c), the EOS was interpolated linearly from a grid of uniformly-spaced points in mixture-fraction space. Case (b) used 101 points in z (∆z = 0.01), and case (c) used 51 points in z (∆z = 0.02). Cases (b) and (c) are of practical interest due to the widespread use of tabulated state-relationships in industrial computations. Interpolation of the EOS at these z-resolutions is not unreasonable. The maximum and average errors in the density for case (b) are ϵmax = 3.5 × 10−3 and ϵavg = 1.5 × 10−4 .

For case (c) the errors are approximately four times larger: ϵmax = 1.3 × 10−2 and ϵavg = 6.0 × 10−4 .

At the beginning of the simulation, the outlet velocity starts from some initial value dictated by the rate of diffusion and decays smoothly as the mixture fraction gradient decreases. This behavior is captured very well with both EOS-evaluation techniques when the analytic EOS is used to compute the density, as indicated in case (a) shown in Figures 4.5 and 4.6. When linear interpolation is used to evaluate the EOS, however, interpolation errors cause unphysical oscillations in the velocity. These errors are clearly shown in cases (b) and (c) of Figures 4.5 and 4.6. The node-based EOS evaluations are particularly susceptible to these errors. To examine the cause of the unphysical oscillations, consider the evolution of the stoichiometric mixture fraction value, zst . This value indicates the ratio of fuel to air that leads to chemical reactions, and can be loosely identified in this example as the density minimum in Figure 4.4. As the flow evolves, the location of zst (and the corresponding non-linear density region) moves through the computational grid under the effects of diffusion and reference-frame translation. When the EOS is underresolved on the computational grid, this can lead to subtle errors in the instantaneous density field. These errors, when summed globally, lead to fluctuations with respect to time in the total amount of mass contained within the computational domain. In

50

uout − uRF uout − uRF uout − uRF


0.1

node evaluations TI method

(a)

0.05 0 0

1

2

3

4

5

7

6

0.1

1

2

3

4

5

7

6

0.1

0.05 1

2

3

4

5

7

6

10

8

9

10


(c) 0 0

9


(b)

0.05 0 0

8

8

9

10

time

uout − uRF uout − uRF uout − uRF

Figure 4.5: 1-D combustion problem with reference-frame velocity uRF = 0.0. (a) analytic EOS evaluation, (b) linear interpolation ∆z = 0.01, (c) linear interpolation ∆z = 0.02. 0.1


(a)

0.05 0 0

1

2

3

4

5

7

6

0.1

1

2

3

4

5

7

6

0.1

0.05 1

2

3

4

5

6

10

8

9

10


(c) 0 0

9


(b)

0.05 0 0

8

7

8

9

10

time Figure 4.6: 1-D combustion problem with reference-frame velocity uRF = 0.2. (a) analytic EOS evaluation, (b) linear interpolation ∆z = 0.01, (c) linear interpolation ∆z = 0.02.

4.2. RESULTS

51

pressure projection methods, any erroneous mass is instantaneously removed from the system by adjusting the pressure and velocity to “correct” the global mass-balance of the system. As a consequence, spurious density fluctuations give rise to spurious velocity and pressure fluctuations. The velocity and density in turn influence the evolution of the scalar field in a highly non-linear manner, further compounding the errors. The end result is that small errors in the EOS evaluation can amplify and lead to very large errors in the velocity and scalar fields. Another way to explain these errors is to consider the implications of the piecewiselinear EOS used in cases (b) and (c). This representation is characterized by discontinuous first derivatives and undefined second derivatives of the density in mixturefraction space. Examination of equations (4.4) and (4.5) shows that these derivative terms are important for accurate approximation of the density. Neglecting these terms, as is done with node evaluations, can lead to large errors in the numerical results. The TI method is more resistant to this type of error because the TI-evaluated density reflects a subgrid average over the control volume, including features of the EOS that are not resolved on the computational grid. Small oscillations are still observed in the TI results, but the errors remain small and do not overwhelm the physics as in the case of the node evaluations.

4.2.2

2-D example problem

In more complex systems, EOS-evaluation errors can lead to significant errors in the time evolution of the flow, including the development of spurious flow structures and unphysical mixing. This is demonstrated through the 2-D problem shown in Figure 4.7. The initial configuration is similar to the classical Rayleigh-Taylor mixing problem, however, the density is given by the “reactive” EOS used in the previous example (equation (4.7)). As the initial disturbance evolves under the influence of gravity, the low-density region around zst is confined to a thin region around the fuel-oxidizer interface. Discretely representing these highly complex mixing patterns is a challenging computational problem.

52


Figure 4.7 compares the time evolution of the 2-D mixing problem using nodebased density evaluations and the TI method. Simulation results on four different grids are presented to show the effects of grid refinement on the solution and to give a point of comparison to judge the physical realism of the coarse-grid solutions. For each EOS-evaluation method, the coarsest solution is shown in the left-most column of the figure matrix, and time increases from top to bottom. The most accurate (56-point) quadrature rule was applied for all evaluations in the TI computations. A non-dimensional viscosity of µ = 8.0 × 10−4 was used in all cases, while the mixture

fraction diffusivity was set to zero.

After only a short integration time, distinct differences emerge between the two EOS-evaluation methods. These differences are particularly pronounced on the coarser grids shown in Figure 4.7. In the case of the node-based evaluations, the interface develops unphysical corrugations that numerically amplify as the solution progresses. After some time, the flowfield is contaminated with spurious flow structures evolving from an unphysical mixing-history. The coarse-grid results for the TI method, while still unresolved on the computational grid, tend to better preserve the characteristics and appearance of the resolved solutions. The difference between the two methods is clearly noticeable in Figure 4.8, which shows the long-time evolution of the flowfield on a 50 × 150 grid. The different evaluation techniques are shown side-by-side with a

more resolved solution for comparison purposes.

4.2.3

Adaptive tetrahedral integration

The hierarchical nature of the quadrature rules in Appendix A lends itself naturally to an adaptive EOS-evaluation procedure. Such an adaptive approach should efficiently focus additional quadrature work only where it is needed and apply lower-order methods in regions that do not demand special treatment. One implementation strategy would be to adaptively select the quadrature rule based on the local spatial gradient of the scalar field. A computational cell that spans a large region of scalar-space would, for example, select a high-order, multi-point quadrature rule, whereas a cell with relatively homogeneous composition might only require a one-point evaluation.

4.2. RESULTS

53

Figure 4.7: 2-D Rayleigh-Taylor instability. The left figures use node EOSevaluations, and the right figures use TI 56-point EOS-evaluations. (top to bottom) t = 0, t = 1.56, t = 3.08, t = 4.64. (left to right) 25 × 75, 50 × 150, 100 × 300, 200 × 600 grids.

54


Figure 4.8: 2-D Rayleigh-Taylor instability at t = 5.2. (left to right) Node evaluations on 50 × 150 grid, TI 56-point evaluations on 50 × 150 grid, TI 56-point evaluations on 100 × 300 grid.

55

4.2. RESULTS

Table 4.1: Execution time of simulations.

method node TI 56-point TI adaptive

time steps 1120 1120 1120

total time (min) 180 211 178

average no. of MG cycles outer-iterations per iteration 40.1 12.3 34.1 9.05 34.5 9.00

The minimum and maximum values of the scalars at the vertices of tetrahedral subvolumes is a natural choice for gradient estimation in such a method. The adaptive method described above has been applied to the 2-D Rayleigh-Taylor mixing problem (see Figure 4.9). The criterion for rule selection in this example required at least one quadrature point per ∆z of 0.005. The time evolution of the solution is shown in Figure 4.9. The upper figures show the evolution of the density, and the lower figures show the concentration of quadrature points used in the simulation. It is clearly seen that the bulk of the EOS-evaluation work is concentrated in the thin interface region of the solution. Figure 4.10 compares the solution using the adaptive procedure with that obtained by universally applying the 56-point quadrature rule. The solutions are virtually indistinguishable. The total execution time of the simulations was monitored for the 100 × 300 grid

case, and the results are presented in Table 4.1. In all of the simulations, a sufficient

number of outer-iterations was performed at each time level to converge the maximum density difference |ρm+1 − ρm | to less than 1.0 × 10−6. Using TI with the 56-point quadrature rule incurred an increase of approximately 20% in the total execution

time compared to node-based density evaluations. The adaptive TI method was the least computationally expensive approach. In this case, the extra EOS-evaluation work was offset by a decrease in the number of outer-iterations per time step and multi-grid cycles per Poisson-solve required to achieve the given levels of density and pressure convergence, respectively.

56


Figure 4.9: 2-D Rayleigh-Taylor insta- Figure 4.10: 2-D Rayleigh-Taylor instabilbility on 100 × 300 grid using adaptive ity on 100×300 grid at t = 5.2. TI adaptive TI evaluations. Density on top, concen- (left), TI 56-point (right). tration of quadrature points on bottom. (left to right) t = 0, t = 1.56, t = 3.08, t = 4.64.

4.3. SUMMARY

4.3

57

Summary

A method for consistent and accurate EOS evaluations in variable-density flow simulations has been developed and implemented. Several example problems were studied demonstrating that under-resolving the EOS can lead to numerical instabilities and unphysical flow features. A hierarchy of symmetric quadrature rules for tetrahedral volume integration was developed and used to efficiently integrate the EOS in fluid dynamics simulations. The new method, termed tetrahedral integration (TI), was shown to reduce EOS-evaluation errors, mitigate many of the undesirable numerical artifacts that result from other techniques, and produce a more physical evolution of the flowfield. The extra cost of the TI method is offset by better convergence and stability properties of the numerical solution.

58


Chapter 5 Code verifcation using MMS The term verification, when applied to a flow solver, describes the process of demonstrating that the code correctly solves its governing mathematical equations. A code that has been properly verified, therefore, is in likelihood free of programming errors that affect the theoretical order-of-accuracy of the numerical algorithm. As such, code verification is an early and integral step in building confidence in the predictive capabilities of simulation software. Over the past several decades, the complexity of computational algorithms in simulation codes has grown in response to demands for high-fidelity simulations in science and engineering. State-of-the-art simulation codes often involve complex exchanges of information amongst various physics modules, each of which may solve different equations using different algorithms on different grid topologies. As simulation codes become more sophisticated, thorough verification becomes increasingly challenging and time consuming, yet also more essential. In this work, attention is focused on hydrodynamics codes amenable to low-Mach number combustion where acoustic effects are unimportant. In this framework, a variable-density formulation of the Navier-Stokes equations is often used due to its computational efficiency relative to fully compressible formulations. In the variabledensity equations, the pressure and density are formally decoupled by defining the density through an equation-of-state (EOS) expressed in terms of transported scalars. The EOS may be given by an analytical expression, or as is common for complex 59

60

CHAPTER 5. CODE VERIFCATION USING MMS

reactive systems, it may be precomputed and tabulated as a function of the scalars. Tabulated state-equations are heavily used in many popular combustion models. Examples include laminar flamelet models (Peters, 1983, 1984), conditional moment closure (CMC) methods (Klimenko, 1990; Bilger, 1993; Klimenko & Bilger, 1999), and some transported PDF methods (Pope, 1985, 1990; Dopazo, 1994). These combustion models are used in a variety of codes, targeting applications that include the design and optimization of engines and power systems, prediction of pollutant formation in combustion devices, and modeling and prediction of fires. While validation studies of combustion codes are routinely performed, the application of systematic verification studies is less common. In particular, the ramifications of tabulated state-relationships on the convergence and accuracy of combustion codes has not been widely investigated. As the EOS in typical combustion systems is multi-dimensional and highly non-linear, its implications on code performance are not straightforward. Owing to the non-linear character of the governing equations, it is common to use iterative algorithms to solve the discretized system of equations. With these methods, solutions are iteratively refined from an initial guess until all equations are approximately satisfied at a given point in time. Iterative approaches, by definition, do not exactly satisfy the discrete equations, and therefore, unavoidably involve certain residual errors. The matter of how small residuals must become for the numerical solution to be a valid approximation of the governing mathematical equations is of obvious and practical relevance. In large-scale computations it may be prohibitively expensive to converge all solvers to machine precision. A popular execution mode, therefore, is for a solver to perform a fixed number of iterations and then proceed to subsequent steps in the algorithm. The ramifications of such an approach on the evolution of the solution are difficult to conjecture. A powerful technique that can unravel this complexity and ultimately help to verify these solvers is the method of manufactured solutions (MMS). Manufactured solutions are exact solutions to a set of governing equations that have been modified with forcing terms. This concept has been around since the early days of computer codes (see, for example, Burggraf, 1966) and has been more recently formalized under the name MMS in a series of papers (Roache, 1998a,b, 2002; Knupp & Salari, 2003;

61

5.1. METHOD OF MANUFACTURED SOLUTIONS

Oberkampf et al., 2004; Roy, 2005). The objective of this work is to use the method of manufactured solutions to explore the effects of tabulated constitutive relationships and iteration errors on the computational performance of low-Mach number combustion codes.

5.1 5.1.1

Method of manufactured solutions Background

The method of manufactured solutions (MMS) is a general procedure that can be used to construct analytical solutions to the differential equations that form the basis of a simulation code. The resulting solutions, while not necessarily physically relevant, can be used as benchmark solutions for verification tests. The accuracy of the code is gauged by running the test problems on systematically refined grids and comparing the output with the analytical manufactured solution. The behavior of the error is examined against the theoretical order-of-accuracy inherent in the code’s numerical discretizations. Thus, a verification test using MMS provides an unambiguous result as to whether or not the algorithm is implemented correctly. MMS has been successfully applied in a variety of applications including fluid dynamics (Roy et al., 2004; Bond et al., 2006), heat transfer (Brunner, 2006; Domino et al., 2007), fluid-structure interaction (Tremblay et al., 2006), even turbulence modeling (Eca et al., 2007). Application of MMS is conceptually straightforward. Consider a generic system of differential equations D(ψ) = 0

(5.1)

where ψ is a vector of unknown variables and D(·) is a differential operator whose specific form depends on the governing partial differential equations. In MMS, the 1 to describe the desired evolution analyst selects a sufficiently differentiable function ψ

1 does not necessarily satisfy the original of the variables in space and time. Since ψ governing equations (5.1), a corresponding set of source terms Q˙ ψ are “manufactured”

1 in order to balance the system by simply applying the differential operator to ψ

62


1 = Q˙ ψ. D(ψ)

(5.2)

The new set of equations given by (5.2) constitutes an exact analytical solution that 1 is exercises all of the same differential terms as equation (5.1) (provided that ψ

sufficiently differentiable). Consequently, equation (5.2) can be used to test numerical codes designed to solve equation (5.1) with minimal additional coding.

As the generality of the current discussion demonstrates, there is near limitless flexibility in constructing manufactured solutions. A useful set of guidelines for the effective design and application of MMS are contained in the monograph by Knupp & Salari (2003).

5.1.2

Example problems

In this section we introduce example MMS problems which attempt to illustrate “canonical” phenomena in variable-density flows. It has been argued that because code verification is a purely mathematical exercise, manufactured solutions need not be “realistic” (Roache, 2002). While this statement is unquestionably true, it does not fully acknowledge the utility of well-crafted manufactured solutions in identifying the vulnerabilities and strengths of a computational algorithm. For instance, a manufactured solution that is suggestive of some elementary physics, provides not only a statement about the code’s order-of-accuracy, but also gives a preview of how the code might perform in more complex problems where the mimicked physics are pervasive. In this spirit, the current examples are constructed such that they identically obey a subset of the governing physics without extra source terms (for example, they analytically conserve mass), and apply manufactured sources to satisfy the remaining conservation laws. Mass conservation is afforded preferential treatment in these examples due to its central role in the solution algorithm presented in Section 3.3. The resulting manufactured solutions attempt to balance simplicity with realism in an effort to understand how the code performs in “representative” scenarios. The verification problems in this report are based on the EOS for isothermal

63

5.1. METHOD OF MANUFACTURED SOLUTIONS

binary mixing between miscible fluids:

ρ(φ) =

%

1−φ φ + ρ1 ρ0

&−1

.

(5.3)

Although this EOS is simple, it is deceptively nontrivial. Large density ratios result in extremely non-linear behavior that can challenge variable-density solvers in a manner similar to the reactive state-equations associated with combustion chemistry. The scalar variable φ in equation (5.3) is known as the mixture fraction and assumes values ranging from 0 to 1. A similar mixture fraction variable is ubiquitously used in combustion modeling to describe the “mixedness” between fuel and oxidizer. The quantities ρ0 and ρ1 are the pure component densities, i.e. ρ0 = ρ(φ = 0) and ρ1 = ρ(φ = 1). The first example problem is a one-dimensional manufactured solution reflective of binary diffusive mixing:

φ(x, t) =

ρ(x, t) =

exp(−k1 t) − cosh(w0 x exp(−k2 t)) 7 8 exp(−k1 t) 1 − ρρ01 − cosh(w0 x exp(−k2 t))

%

φ(x, t) 1 − φ(x, t) + ρ1 ρ0

u(x, t) = 2k2 exp(−k1 t)

ρ0 − ρ1 ρ(x, t)

&−1

,

(5.4)

u 1x + +1

u 12

1 − π4 ) ( kk21 − 1)(arctan u w0 exp(−k2 t)

-

where u 1 = exp(w0 x exp(−k2 t)) and w0 , k1 , and k2 are constant parameters. Note ¯˙ = 0, that equation (5.4) satisfies the continuous continuity equation (3.4) with Q ρ¯ but produces a non-zero source term in the scalar transport equation (3.6). No source term is specified in the momentum equation, instead the pressure is allowed ¯˙ = 0. If interested, one could solve to compensate to satisfy equation (3.5) with Q u e i

for the analytical pressure distribution by integrating equation (3.5) with respect to x. The relevant manufactured scalar source term is computed by substituting ¯˙ . The spatio-temporal evolution equation (5.4) into equation (3.6) and solving for Q e φα

1.2 0.9 0.6 0.3 0 0

time ❈ ❈ ❈ ❈ ❈❲ 0.2

0.4

x

0.6

0.8

1

1 0.8 0.6 0.4 0.2 0 0

❖❈ ❈

❈

❈

time 0.2

0.4

x

0.6

0.8

1

¯˙ (x, t) scalar source, Q φ


scalar, φ(x, t)

velocity, u(x, t)

64

20 15 10 5 0 0

✄ ✄✎

✄

✄

time

0.2

0.4

x

0.6

0.8

1

¯˙ (x, t). Figure 5.1: 1-D manufactured solution. (left to right) u(x, t), φ(x, t), Q φ

Table 5.1: Parameter values for 1-D manufactured solution problem, equation (5.4). parameter ρ0 ρ1 k1 k2 w0 ρ¯α #φ = µ ¯

value 20 1 4 2 50 0.03

Table 5.2: Parameter values for 2-D manufactured solution problem, equation (5.5). parameter ρ0 ρ1 uRF vRF ρ¯α #φ

value parameter 20 a 1 b 1 k 1/2 ω 0.001 µ ¯

value 1/5 20 4π 3/2 0.001

of equation (5.4) is shown in Figure 5.1 for the parameter values in Table 5.1. The computational domain for this problem is 0 ≤ x ≤ 2 and 0 ≤ t ≤ 1. A similar problem was investigated in Chapter 4, although not within the framework of MMS.

Figure 5.1 shows that the initial, steep profile of φ(x) broadens, or “diffuses,” over time. This pseudo-diffusion is actually controlled by the relative values of the parameters k1 and k2 , and is independent of the physical scalar diffusivity, α #α . The

diffusivity still appears in the MMS problem, but only affects the shape and magnitude ¯˙ . If k = k in this problem, then the integrated mass of the of the source term, Q eα 1 2 φ 9∞ system 0 ρ(x) dx is constant. A value of k1 > k2 is selected in order to provide a

positive velocity at the outflow boundary, and avoid the possibility of inflows due to (small) numerical errors. A second MMS problem involves a two-dimensional corrugated front with advection and diffusion:

65

5.2. RESULTS

φ(x, y, t) =

ρ(x, y, t) =

(1 + %

1 + tanh(b1 x exp(−ωt)) ρ0 + (1 − ρ1 ) tanh(b1 x exp(−ωt))

ρ0 ) ρ1

φ(x, y, t) 1 − φ(x, y, t) + ρ1 ρ0

ρ1 − ρ0 u(x, y, t) = ρ(x, y, t)

v(x, y, t) = vRF

%

&−1

ω1 x − uRF exp(2b1 x exp(−ωt)) + 1 & ω log(exp(2b1 x exp(−ωt)) + 1) + 2b exp(−ωt)

− ω1 x+

(5.5)

p(x, y, t) = 0 where x 1(x, y, t) = uRF t − x + a cos(k(vRF t − y)) and a, b, k, ω, uRF, and vRF are con-

stant parameters. Equation (5.5) satisfies the continuous continuity equation (3.4) ¯˙ = 0. Non-zero source terms appear in the x and y momentum equations (3.5) with Q ρ¯ and the scalar transport equation (3.6). The evolution of the density field described by (5.5) is shown in Figure 5.2 given the parameter values in Table 5.2. The computational domain for this problem is −1 ≤ x ≤ 3, −1/2 ≤ y ≤ 1/2, and 0 ≤ t ≤ 1.

In order to focus on the effects of density, the viscosity µ ¯ and the “dynamic”

diffusivity ρ¯α #α in equations (3.5) and (3.6) are assumed constant in the MMS examples presented here.

5.2

Results

The MMS problems of Section 5.1.2 were implemented in the unstructured LES code CDP. In all of simulations reported in this chapter, the code was run in so-called “DNS” mode, wherein all subgrid models were disabled. A spatial grid-refinement study using the one-dimensional example problem equation (5.4) was conducted to assess the convergence properties of CDP’s numerics.

66


Figure 5.2: 2-D manufactured solution: ρ(x, y, t) (top to bottom) t = 0, t = 1/3, t = 2/3, t = 1. (black: ρ = 1, white: ρ = 20).

67

5.2. RESULTS

Computational grids consisting of 64, 128, 256, 512, 1024, and 2048 control volumes were used. A time step of ∆t = 0.00125 was applied in all cases, leading to maximum CFL numbers in the range 0.048 to 1.53. The boundary conditions at x = 0 were u = 0 and ∂φ/∂x = 0. At the domain exit, an “outflow” boundary condition was used

∂ψ ∂ψ + uc = 0, ∂t ∂n

(5.6)

where ψ is any scalar variable or velocity component, uc is the convective velocity, and n is the outward normal at the boundary. This condition was applied at x = 2, a location sufficiently removed from the problem dynamics so as not to introduce significant errors in the solution. The velocity, pressure, and scalar values were solved from equations (3.4)–(3.6), and the density was evaluated using the analytical function equation (5.3) and the instantaneous scalar field. The convergence tolerance for solving transport equations and the pressure Poisson equation was 1 × 10−8 . Outer iterations at each time step were continued until the maximum density difference between iterations |ρm+1 − ρm | was less than 1 × 10−8 . This was typically achieved after 20–25 outer iterations. Additional problem parameters are listed in Table 5.1.

The maximum error (L∞ -error) and volume-averaged (L2 -error) for u(x, t), φ(x, t), and ρ(x, t) were monitored throughout the simulation. The error trends toward second-order convergence with respect to ∆x for all variables, as seen in Figure 5.3. Detailed convergence results are tabulated in Table 5.3. Plots of the L2 -error versus time for u(x, t) and φ(x, t) are shown in Figures 5.4 and 5.5. Note that the error smoothly decays with time in each simulation as the flow features diffuse and become more easily resolved. The L∞ -error behaves similarly on all grids except the nx = 2048 grid, where iteration and time errors begin to contaminate the solution after t ≥ 0.5. As part of a separate temporal-refinement study (not shown here) the time step was

halved to ∆t = 0.000625 and the simulations were repeated on the nx = 1024 grid. The results were almost indistinguishable, with a maximum difference on the order of 10−6 for all variables. This suggests that the results are “converged” in a temporal sense, and that time errors are subservient to spatial errors on all of the grids with nx ≤ 1024.

68


0

10

L2 -error

-2

1st order

10

-4

10

-6

10

-8

10

u, velocity φ, scalar ρ, density

2nd order

2

3

10

10

no. of control volumes, nx

10

-2

10

-3

10

-4

10

-5

nx = 64

L2 -error: φ(x, t)

L2 -error: u(x, t)

Figure 5.3: 1-D manufactured solution: L2 -error at t=1.

nx = 128 nx = 256 nx = 512 nx = 1024 nx = 2048

10

-2

nx = 64

10

-4

nx nx nx nx nx

10

-6

10

= 128 = 256 = 512 = 1024 = 2048

-6

0

0.2

0.4

time

0.6

0.8

1

Figure 5.4: L2 -error in velocity u(x, t) versus time for 1-D manufactured solution.

0

0.2

0.4

time

0.6

0.8

1

Figure 5.5: L2 -error in scalar φ(x, t) versus time for 1-D manufactured solution.

69

5.2. RESULTS

Table 5.3: 1-D manufactured solution: L∞ - and L2 -error at t=1 versus spatial grid refinement. no. of points 64 128 256 512 1024 2048

L∞ -error u(x, t) 1.4587e-02 2.5261e-03 6.5204e-04 1.7155e-04 4.4133e-05 1.1976e-05

observed order 2.53 1.95 1.93 1.96 1.88

L2 -error u(x, t) 1.4409e-03 2.4485e-04 7.6850e-05 2.2545e-05 6.3385e-06 1.6845e-06

observed order 2.56 1.67 1.77 1.83 1.91

L∞ -error φ(x, t) 2.6407e-02 3.2633e-03 1.1724e-03 3.5933e-04 9.9106e-05 2.6574e-05

observed order 3.02 1.48 1.71 1.86 1.90

L2 -error φ(x, t) 6.7197e-03 9.5846e-04 3.5391e-04 1.1075e-04 3.0840e-05 8.2872e-06

observed order

Table 5.4: EOS lookup table: resolutions and errors. no. of max error avg error points ρ(φ)/ρ0 ρ(φ)/ρ0 21 8.0591e-02 3.6803e-03 31 4.7323e-02 1.6942e-03 51 2.2125e-02 6.2316e-04 101 6.9391e-03 1.5738e-04 201 1.9681e-03 3.9449e-05 401 5.2606e-04 9.8689e-06 801 1.3613e-04 2.4676e-06

In order to evaluate the effect of EOS tabulation on code performance, a refinement study was conducted in which the EOS equation (5.3) was interpolated linearly from successively-refined tables of uniformly-spaced points in φ-space. A summary of the tabulation resolutions and their associated errors is found in Table 5.4. Interpolation of the EOS at the coarsest resolutions in Table 5.4 would not be uncommon in many engineering calculations where property tables are multi-dimensional (typically 3–4) and memory is limited. The simulations were effected on a grid of 1024 control volumes with a time step of ∆t = 0.00125. The boundary conditions and solver convergence limits were identical to the spatial grid-refinement study above. The full convergence results are tabulated

2.81 1.44 1.68 1.84 1.90

70


in Table 5.5, and plots of the temporal evolution of the L2 -error for u(x, t) and φ(x, t) are shown in Figures 5.6 and 5.7. The “nφ = ∞” label indicates results using the

analytical or non-interpolated EOS.

The data clearly indicate a degradation of accuracy when using a tabulated EOS. Velocity convergence inclines towards first-order behavior, while scalar and density convergence appears to be closer to second-order (with respect to ∆φ). These trends, however, are speculative at best as the data are not well converged, even with 801 interpolation points in the EOS. It is likely that convergence of the scalar outperforms velocity because of the manufactured source term in equation (3.6). In the simulations, the scalar source was evaluated as a function of x and t, rather than u(x, t), φ(x, t), and ρ(x, t). The source term, therefore, implicitly used the analytical EOS and was partially shielded from the influence of tabulation errors. It is not surprising, therefore, that scalar convergence was less affected than velocity, especially when considering the relative strength of the scalar source term in this example (see Figure 5.1). In addition to poor convergence rates, it is clear that EOS interpolation dramatically affects the character of the error in the field variables. The smooth error decay exhibited in Figure 5.4 is replaced by the unsteadiness apparent in Figure 5.6. These numerical fluctuations result from the tight coupling between density, velocity, and pressure in low-Mach number projection methods. Density errors arising from the tabulation are readily translated into velocity errors as the pressure acts to “correct” changes in the global mass-balance. The velocity and density in turn influence the evolution of the scalar field in a non-linear manner, adding further complexity. The end result is that small errors in the EOS evaluation can amplify and produce relatively large errors in the velocity and scalar fields. The numerical fluctuations induced by EOS interpolation errors undoubtedly find expression in the flow variables on a macro-scale. This is readily visible in Figure 5.9, which shows the convective outlet velocity u(x = 2, t) for different EOS resolutions. Here, interpolation errors cause dramatic fluctuations about the exact MMS value. The presence of these fluctuations, whose genesis is entirely numerical, holds serious implications for subgrid modeling of combustion and turbulence.

71

10

10

10

nφ nφ nφ nφ nφ nφ nφ nφ

-2

-4

= 21 = 31 = 51 = 101 = 201 = 401 = 801 =∞

L2 -error: φ(x, t)

L2 -error: u(x, t)

5.2. RESULTS

10

10

10

nφ nφ nφ nφ nφ nφ nφ nφ

-2

-4

-6

-6

0

0.2

0.4

0.6

0.8

1

0

0.2

time

10

1

nφ nφ nφ nφ nφ nφ nφ

0.02

-2

-4

-6

0.8

0.03

uout − uMMS out

max L2 -error

10

0.6

Figure 5.7: L2 -error in scalar φ(x, t) versus time for 1-D manufactured solution on nx = 1024 grid.

0

1st order 10

0.4

time

Figure 5.6: L2 -error in velocity u(x, t) versus time for 1-D manufactured solution on nx = 1024 grid.

10

= 21 = 31 = 51 = 101 = 201 = 401 = 801 =∞

0.01 0

= 21 = 31 = 51 = 101 = 201 = 401 = 801

-0.01

u, velocity φ, scalar ρ, density

2nd order 10

2

-0.02 10

3

no. of interpolation points, nφ Figure 5.8: 1-D manufactured solution: convergence of maximum L2 -error on nx = 1024 grid.

-0.03

0

0.2

0.4

0.6

0.8

1

time Figure 5.9: 1-D manufactured solution: convective outlet velocity u(x = 2, t) on nx = 1024 grid.

72


Table 5.5: 1-D manufactured solution: maximum L∞ - and L2 -error versus EOS lookup table refinement. no. of points 21 31 51 101 201 401 801

L∞ -error u(x, t) 3.0450e-02 1.8274e-02 9.2087e-03 3.5173e-03 2.0185e-03 9.2466e-04 5.2006e-04

observed order 1.31 1.38 1.41 0.81 1.13 0.83

L2 -error u(x, t) 2.3991e-02 1.4032e-02 8.5121e-03 3.2208e-03 1.9794e-03 8.6672e-04 4.3319e-04

observed order 1.38 1.00 1.42 0.71 1.20 1.00

L∞ -error φ(x, t) 5.4061e-02 2.8370e-02 1.1613e-02 3.2998e-03 9.6504e-04 3.4493e-04 1.5400e-04

observed order 1.66 1.79 1.84 1.79 1.49 1.17

L2 -error φ(x, t) 1.8100e-02 9.5723e-03 3.9251e-03 1.1030e-03 3.0885e-04 9.9923e-05 4.1577e-05

observed order

Issues relating to time-accuracy and the error-contribution from iteration residuals have also been studied using MMS. Here the two-dimensional problem (equation (5.5)) was simulated on computational grids of 200 × 50, 400 × 100, 800 × 200, and

1600×400 hexahedral control volumes with a uniform time step of ∆t = 0.00125. This

time step produced maximum CFL numbers ranging from 0.15 to 1.18 on the various grids. Dirichlet boundary conditions were imposed at x = −1, an “outflow” boundary condition (equation (5.6)) was applied at x = 3, and periodic boundary conditions were used at y = ±1/2. Tolerances for the scalar, momentum, and pressure solvers

were set to 1 × 10−10 in all simulations. Additional problem parameters are supplied in Table 5.2.

A major focus of this study was to investigate the effect of applying multiple non-linear iterations to “fully” converge the system of equations at each time step. Figure 5.10 shows the spatial-convergence of the L2 -error of the flow variables when 20 outer iterations are employed. At this level of iteration, the bulk of the time error is eliminated and all variables tend toward second-order convergence with grid refinement — confirming the expected accuracy of CDP’s spatial operators. Figure 5.11 shows how the convergence is affected when fewer outer iterations are used at each time level. The observed convergence is approximately first-order when 10 iterations are applied, and even less accurate with fewer iterations. The errors depicted in Figure 5.11 are a combination of spatial and temporal

1.64 1.79 1.86 1.85 1.63 1.27

73

5.3. SUMMARY

10

0

10

-1

10

-2

10

-3

10

-4

u, velocity v, velocity φ, scalar ρ, density 10

10

L2 -error

L2 -error

1st order

2nd order

2

no. of control volumes,

√

nx ny

103

Figure 5.10: 2-D manufactured solution: L2 -error at t = 1 using 20 outer iterations.

-1

10

-2

10

-3

1st order

niter = 5 niter = 10 niter = 20 10

2nd order

2

no. of control volumes,

√

nx ny

103

Figure 5.11: 2-D manufactured solution: L2 -error of u(x, t) at t = 1 for different numbers of outer iterations.

contributions that combine in a complex manner on different grids and time steps. A closer examination of the temporal component of the error is instructive. Figure 5.12 shows the convergence of the L2 -error with respect to ∆t for a simulation on the 800 × 200 grid and applying only one outer iteration per time step. Under these

conditions, the dominant error derives from the time advancement which is clearly first-order accurate. Additional outer iterations help to reduce the time-errors by increasing their accuracy towards second-order and/or reducing their magnitude. In a related study, the simulations of Figure 5.11 were repeated using a more moderate density ratio of ρ0 /ρ1 = 5. In this case, second-order convergence was observed after 5–10 outer iterations versus the 15–20 iterations for the ρ0 /ρ1 = 20 case. This accelerated convergence is encouraging for applications with modest density ratios, but also indicates an unappealing dependence of the algorithm’s convergence properties on the EOS.

5.3

Summary

In this study, the method of manufactured solutions (MMS) was used to investigate the effects of tabulated state-equations and temporal iteration errors on the convergence and accuracy of the multi-physics hydrodynamics code CDP. Two MMS

PSfrag 74


10

L2 -error

10 10 10 10

0

-1

1st order

-2

-3

-4

10−4

2nd order time step, ∆t

u, velocity v, velocity φ, scalar ρ, density 10

-3

Figure 5.12: 2-D manufactured solution: L2 -error at t=0.1 using 1 outer iteration.

problems were constructed whose evolution is reflective of some of the basic physics germane to combustion problems, namely: diffusive mixing of species and convection of density fronts. Both of the MMS examples analytically satisfy the source-free continuity equation, and use manufactured source terms to balance other transport equations in the system. Grid refinement studies performed using the MMS problems confirm the spatial convergence rate of CDP to be second-order when an analytical EOS is used. Convergence of the flow variables to the exact solution was markedly impaired when the EOS was linearly interpolated in φ-space. EOS interpolation errors introduce spurious numerical fluctuations in the flow variables, with velocity and pressure being particularly vulnerable. In some problems these errors can accumulate with time and potentially alter the temporal evolution of the flow. The variable density algorithm in CDP has first-order temporal accuracy when a single outer iteration is applied. Temporal errors were generally not dominant when multiple outer iterations were performed, making it difficult to confirm the temporal accuracy of the method with multiple outer iterations. The present results suggest that, for a given problem, a balance exists between factors such as the size of the time step and the number of outer iterations that need to be performed. Determining the optimal operating conditions (i.e. grid size, time step, number of outer iterations, etc.) is a nontrivial and problem-dependent task

5.3. SUMMARY

that deserves more attention than is currently afforded.

75

76


Chapter 6 A flamelet model for heat-loss Heat transfer is a complex phenomenon that can include contributions from conduction, convection, and radiation. Each of these processes is driven by different physical mechanisms and therefore requires different modeling approaches. Developing integrated models that account for the combined effects of these different modes of heat transfer can be challenging. The problem is compounded in reactive systems, where additional complexity results from the interaction of heat transfer processes with chemical kinetics and mass transport. In this chapter, a general framework for incorporating the effects of heat transfer on combustion chemistry is developed. The framework is discussed as an extension to the flamelet/progress variable (FPV) model for turbulent combustion, but it could be applied to other models as well.

6.1

Non-adiabatic FPV model

The usual pattern adopted in constructing computational models is to develop a mechanistic description of the process of interest, and then translate that mechanism into a system of algebraic or differential equations that govern the behavior of the process. In the case of the FPV model, the physical mechanism is embodied at the highest-level by equations (2.40), which describe conservation of chemical species and energy. In practice, the solutions to these equations are obtained, and the results are mapped to a tabulated parameter space for easy retrieval. The final outcome of 77

78

CHAPTER 6. A FLAMELET MODEL FOR HEAT-LOSS

the FPV model, then, is summarized by the simple functional representation which recovers thermochemical properties from the table ψ = Fψ (z, c).

(6.1)

In extending the FPV model to describe heat transfer, the objective is to expand the property map expressed by equation (6.1) to describe a more diverse range of chemical states. Since the goal is to include heat transfer effects, addition of an energy-related parameter such as temperature or enthalpy is a logical step. In this work, the total enthalpy is selected to supplement the parameterization of the flamelet solutions. The enthalpy is adopted over temperature due to the relative simplicity of its transport equation (the temperature equation includes cumbersome reaction terms), and the smoothness of its behavior (profiles of h(z) in adiabatic flamelets are nominally linear). The desired end-product of the extended flamelet model is a tabulated description of the resulting thermochemical states using this additional parameter ψ = Fψ (z, c, h).

6.1.1

(6.2)

Modeling approach

In devising a non-adiabatic flamelet model, an alternative to directly modeling the physics of heat transfer, is to use an ad hoc parameterization of the enthalpy-coordinate and solve the remaining flamelet equations subject to constraints imposed by the chosen parameterization. This strategy implicitly assumes that the resulting thermochemical state is a unique function of the mapping parameters z, c, and h, and is independent of details about how the enthalpy is specified. This general approach is particularly useful for dealing with convective heat transfer, which is highly dependent on an application’s geometry. Any considerations of “geometry,” or the introduction of solid boundaries to the one-dimensional flamelet equations (2.40), tends to spoil many of the assumptions and symmetries that are assumed in their derivation. A useful way to parameterize the enthalpy is to scale it linearly between the

79

6.1. NON-ADIABATIC FPV MODEL

adiabatic state, had , and a reference value, href , η=

had − h had − href

href = h (Yαad , Θref ).

where

(6.3)

Although the choice of the adiabatic and reference states in this expression is somewhat arbitrary, some options offer practical advantages over others. In this work, the adiabatic state, (Yαad , had ), is chosen as the equilibrium chemistry solution. Note that (Yαad , had ) also defines the adiabatic temperature, Θad , through equation (2.40d). The equilibrium solution offers a convenient characterization of the adiabatic state because it is physically and mathematically well-defined, relatively easy to compute, and depends only on the mixture fraction, i.e., ψ = ψ(z). As indicated in equation (6.3), the reference state, href , is chosen to be the adiabatic species composition, Yαad , cooled to the reference temperature, Θref . For convenience, the value of Θref = 300K is used throughout this study. With this choice for had and href , the parameter η can be viewed as a measure of the change in sensible enthalpy that is required to cool the adiabatic composition to the reference temperature, Θref . As defined by equation (6.3), positive values of the parameter η indicate an enthalpy defect, or loss of heat. Consequently, η is referred to as a “heat-loss parameter.” With the enthalpy constrained by equation (6.3), h is no longer an unknown in the flamelet equations (2.40), and the energy transport equation need not be explicitly solved. The modified system of flamelet equations, d(ρYα vα ) = ρ ω˙ α dx , -−1 nS $ ρ = p∞ RΘ Yα /Mα

(6.4a) (6.4b)

α=1

ad

h=h

ref

+ η(h

ad

−h )=

nS $

Yα hα (Θ) ,

(6.4c)

α=1

is solved for the remaining unknowns Yα and Θ. Although the energy equation is not directly solved in this approach, the non-linear summation that defines the total enthalpy in equation (6.4c) ensures a tight coupling between the energy and the

80


production and transport of species in the system. These equations must be mutually satisfied in order for the formulation to be consistent. In the current approach, the species equations (6.4a) are solved using a pseudotime stepping method to march the solution to a steady state. At each time step, the enthalpy is constrained to obey equation (6.4c) for a given (constant) value of the heat-loss parameter, η. The temperature is determined at each time step from the summation term using a Newton-Raphson method with the current estimate for Yα . All of the thermodynamic and transport properties are updated, and the process is continued until the steady solution is obtained. Enthalpy profiles corresponding to different values of the heat-loss parameter are shown in Figure 6.1. The uppermost, linear profile represents the adiabatic flamelet state, while the lower curves with η > 0 describe flamelets that are experiencing heatloss. The effects of sustained heat-loss on the flamelets are depicted in Figures 6.2 and 6.3. Figure 6.2 shows the non-extinguished branches of the S-shaped curves from flamelet solutions with different values of η. Note that heat-losses tend to weaken the flames, leading to lower overall temperatures and extinction at smaller dissipation rates. Several flamelet solutions that are close to extinction are identified in Figure 6.2 by solid symbols at the critical points (maximum value of χst ) along their respective curves. These nearly quenched flamelets are shown in greater detail in Figure 6.3, which depicts the temperature and progress variable profiles as a function of the mixture fraction. As noted earlier, larger values of η correspond to decreased temperatures, slower reaction kinetics, and a greater propensity for strain-induced quenching. On the upper (burning) branch of the S-shaped curve, moving towards lower dissipation rates brings one closer to chemical equilibrium and complete “reacted-ness.” Since non-adiabatic flamelets quench at lower dissipation rates, they generally have formed higher concentrations of combustion products upon extinction than their adiabatic counterparts. This trend is visible in the progress variable curves shown in Figure 6.3, where the mass fraction of major products (i.e., the progress variable) increases with η. The low-temperature, well-reacted solutions that result are important in describing the thermochemical state of combustion gases near cold surfaces. Flamelet solutions computed using different values of η are assembled to form a

81


enthalpy, h [kJ/kg]


2500

0

η = 0, h = h

-1000

ad

1500

η = 1, h = href

0

0.2

0.4

0.6

0.8

1

500

increasing η , , , ✠

500

0.2

0.4

0.6

102

104

Figure 6.2: S-shaped curves for flamelets with heat-loss. Methane-air combustion, p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, 0 ≤ η ≤ 0.9.

progress variable, c

2000

0 0

100

dissipation rate, χst [1/s]

Figure 6.1: Enthalpy profiles for flamelets with heat-loss. Methane-air combustion, p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, 0 ≤ η ≤ 1.

1000

•

•

10-10 10-8 10-6 10-4 10-2

mixture fraction, z

1500

•

•

1000

-4000

temperature, Θ [K]

✟ ✟✟ ✟ • • ✙✟ ✟

2000

-2000 -3000

increasing η

0.8

mixture fraction, z

1

0.3 0.25 0.2 0.15

✒ , , , increasing η

0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

mixture fraction, z

Figure 6.3: Temperature (left) and progress variable (right) for the non-adiabatic, trans-critical flamelets indicated by “•” in Figure 6.2. Methane-air combustion, p = 3.8 atm, Θfuel = 300K, Θoxid = 750K, η = 0, 0.2, 0.4, 0.6, 0.8, 0.9.


0 -1000 -2000 -3000 -4000

0

0.2

0.4

0.6

0.8

mixture fraction, z

1

Figure 6.4: Temperature contours from a (z, c, h) flamelet library as a function of mixture fraction and enthalpy. These data correspond to the maximum value of the progress variable (Λ = 1).

heat-loss parameter, η

enthalpy, h [kJ/kg]

82

0.8 2000 0.6 1500 0.4 1000 0.2 500 0

0

0.2

0.4

0.6

0.8

mixture fraction, z

1

Figure 6.5: Temperature contours from a (z, c, h) flamelet library as a function of mixture fraction and heat-loss parameter. These data correspond to the maximum value of the progress variable (Λ = 1).

library of flame states. The flamelet library supplies information about the thermochemical state of the flame as a function of the scalars z, c, and h. A section from such a library is shown in Figures 6.4 and 6.5. The contours of temperature are taken at the maximum value of the progress variable, cmax = f (z, h), or more appropriately, Λ = 1. This can be thought of as the temperature that the flamelet assumes as it approaches equilibrium. From Figure 6.5 it is also apparent that the heat-loss parameter, η, offers a convenient framework for the tabulation of properties, as it maps the quantities cleanly into a well-defined, orthogonal space. Note the thin peninsula of high temperatures that extends along the line of stoichiometric mixture fraction, zst = 0.055. Evidence of this reactive zone is also seen in Figure 6.3, where localized peaks are visible in the temperature profiles. The approach outlined above assumes that a thermochemical database constructed from the solution of the full, adiabatic flamelet equations (2.40), and a database from solutions of the heat-constrained equations (6.4), will produce the same output for a given set of inputs (e.g., z, c, h, etc.). This assumption was tested for flamelets with weak heat-loss by comparing solutions of the adiabatic flamelet equations (2.40) with solutions of the heat-loss equations (6.4). The adiabatic solutions can be made to cover a portion of the non-adiabatic parameter space by specifying reactant temperatures that are lower than those used to define the reference state (Y ad , Θad )


83

Table 6.1: Parameters for adiabatic flamelets used in comparison test with enthalpyconstrained flamelets. Θoxid = 400K Θoxid = 525K Θoxid = 650K xL [cm] Θst [K] xL [cm] Θst [K] xL [cm] Θst [K] 100.0 2293.1 100.0 2350.4 100.0 2406.0 2.000 2182.2 2.000 2146.9 0.220 2103.7 0.180 1943.4 0.120 1933.0 0.090 1926.6 0.144 1740.5 0.113 1788.1 0.100 1793.8 0.254 1597.8 0.240 1595.1 0.240 1582.3 0.699 1315.7 0.660 1302.5 0.585 1324.2 1.580 888.8 1.390 937.0 1.200 999.4

in the non-adiabatic formulation. In this validation test, a fuel-side temperture of Θfuel = 300K was used for all of the flamelet solutions. The air-side temperature was Θoxid = 750K for the non-adiabatic solutions, and between 400K and 650K for the adiabatic flamelets. The full test matrix of air-side temperatures and domain lengths is specified in Table 6.1. These parameters were chosen so as to produce adiabatic solutions on the burning and intermediate branches of the S-shaped curve, with maximum temperatures spaced at roughly 200K intervals from 1200K to 2400K. Results from the test are shown in Figures 6.6–6.8. The solutions to the adiabatic equations correspond to the dotted-line profiles of the enthalpy and heat-loss parameter shown in Figure 6.6. Although these are adiabatic solutions, the heat-loss parameter is non-zero in this example because η is defined at Θoxid = 750K. The one-dimensional solution profiles from the adiabatic flamelets were interpolated to 200 test points spaced quadratically along the z-coordinate. Quadratic spacing was used in order to focus points near the reactive region around zst . The solution values at the test points for some important combustion quantities are shown in Figure 6.7. Since each test point is also associated with a value for the progress variable and enthalpy, the results can be directly compared to values retrieved from a nonadiabatic flamelet database in order to validate the assumptions and solution method used for the non-adiabatic formulation. The non-adiabatic flamelet database pictured

84

0

-1000

increasing Θoxid ✡ ✣ η = 0, h = had ✡

heat-loss parameter, η

enthalpy, h [kJ/kg]


1

0.8

❅ ❅ η = 1, h = href

0.6

-2000

0.4

-3000 -4000 0

η = 1, h = h 0.2

ref

0.4

increasing Θoxid η = 0, h = had ✓ ✓ ❅ ✴ ✓ ❅

0.2

0.6

0.8

1

0 0

mixture fraction, z

0.2

0.4

0.6

0.8

1

mixture fraction, z

Figure 6.6: Mapping of enthalpy and heat-loss parameter from adiabatic flamelets solved at different air temperatures to regions defined by the enthalpy-constrained flamelets.

2500

3.5

(a)

3

(b)

2000

2.5

1500

2 1000

1.5

500

1 0.50

0.2

0.4

0.6

1

0.8

00

0.2

0.4

0.6

1

0.8

4500

(c)

3500

(d)

0.15

2500

0.1

1500 0.05 500 0

0.2

0.4

0.6

0.8

mixture fraction, z

1

00

0.2

0.4

0.6

0.8

mixture fraction, z

1

Figure 6.7: Solution profiles from the adiabatic flamelets listed in Table 6.1: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO .


85

in Figures 6.1–6.5 was used for the comparison. This database was constructed for use in LES and is a four-dimensional, Cartesian table with axes for the mean mixture fraction, mixture fraction variance, mean progress variable, and mean enthalpy. The table size is 100 × 30 × 50 × 40 for mixture fraction, scalar variance, progress variable, and enthalpy, respectively. All of the lookups for the comparison use a mixture fraction variance of zero. Results of the comparison are summarized in Figure 6.8, which plots the values retrieved from the non-adiabatic flamelet database versus the values computed using the adiabatic flamelet equations. The results from 4200 individual test points are plotted in each figure. Although small deviations are observed, the agreement between the two formulations is excellent. The largest discrepencies are observed in the most non-linear regions of the table, and are likely due to interpolation errors committed at the time of retrieval from the database, or accrued during the many post-processing steps required to construct the flamelet library. All-in-all, differences between the two flamelet formulations are relatively minor, and both produce nearly the same outputs for a given set of inputs. This demonstrates that the proposed parameterization of the enthalpy, with its corresponding constraints on the flamelet energy equation, leads to a consistent framing of the problem and that the solution methodology used for the non-adiabatic system (6.4) is appropriate. The approach outlined here for constructing flamelet databases is termed the flamelet/progress variable model with generalized heat-loss, or FPV-GHL model.

6.1.2

Application in LES

In Section 6.1.1, the FPV-GHL model was discussed in terms of fully resolved, instantaneous parameters without consideration of turbulence/chemistry interactions. When applied to LES, the model must account for the effects of subgrid-scale turbulent fluctuations of the chemistry variables. These interactions are approximated by a presumed-shape PDF in a manner similar to Section 3.2.2. The filtered thermochemical quantities are obtained by integrating equation (6.2) with the joint PDF of

86


3.5

(a)

(b) 2000

non-adiabatic

non-adiabatic

3

2500

2.5 2 1.5

1

1.5

2

2.5

3

0 0

3.5

5000

0.15

non-adiabatic

4000

non-adiabatic

500

1000

1500

2000

2500

0.18

(c)

3000 2000 1000 0 0

1000 500

1 0.5 0.5

1500

(d)

0.12 0.09 0.06 0.03

1000

2000

3000

adiabatic

4000

5000

0 0

0.03 0.06 0.09 0.12 0.15 0.18

adiabatic

Figure 6.8: Comparison of thermochemical properties obtained from solving the adiabatic flamelet equations and the enthalpy-constrained flamelet equations: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO .


z, Λ, and Γ #= ψ

3 3 3

ψ(z, Λ, Γ)P#(z, Λ, Γ)dz dΛ dΓ .

87

(6.5)

Here Λ is the reaction progress parameter, and Γ is a quantity that uniquely parameterizes the enthalpy coordinate. The heat-loss parameter, η, is a logical choice for Γ. The assumption of statistical independence amongst the arguments is used to factor the joint PDF into the product of marginal PDFs P#(z, Λ, Γ) = P#(z)P#(Λ)P#(Γ) .

(6.6)

The individual PDFs are then modeled with simple distributions ′′2 ) P#(z) = β(z; z#, z!

# P#(Λ) = δ(Λ − Λ)

(6.7)

# . P#(Γ) = δ(Γ − Γ)

The FPV-GHL model could benefit from more sophisticated models for P#(Λ) and P#(Γ). A balance must be struck, however, between augmenting the model with additional inputs and parameters, and keeping the resulting chemistry database a manageable size and dimensionality. The final expression for the filtered variables is #= ψ

3 3 3

′′2 )δ(Λ − Λ)δ(Γ # # ψ(z, Λ, Γ)β(z; z#, z! − Γ)dz dΛ dΓ .

(6.8)

The information is ultimately tabulated as a function of quantities that are readily produced in a LES, for example, ′′2 , # # =F #ψ (# ψ z , z! c, # h) .

(6.9)

The model is implemented by solving LES transport equations for the filtered

88


mixture fraction, progress variable, and enthalpy fields % & #j z# ∂ ∂# z ∂ ρ¯z# ∂ ρ¯u + = ρ¯(# αz + αt ) ∂t ∂xj ∂xj ∂xj % & ∂# c ∂ ρ¯# c ∂ ρ¯u #j # c ∂ #˙ c + = ρ¯(# αc + αt ) + ρ¯ ω ∂t ∂xj ∂xj ∂xj % & ∂ ρ¯# h ∂ ρ¯u ∂# h #j # h ∂ ¯˙ . + = ρ¯(# αh + αt ) −Q R ∂t ∂xj ∂xj ∂xj

(6.10)

These scalars serve as inputs to the chemistry database, along with the subgrid scalar variance which is modeled as discussed in Section 3.2.3. All of the transport and ¯˙ , etc.) #˙ c , Q thermochemical properties that appear in equation (6.10) (i.e., ρ¯, α #α , ρ¯ ω R

are retrieved from the database as needed.

6.1.3

Transformed coordinate system

In actuality, the table axes in the non-adiabatic database presented in Sections 6.1.1 and 6.1.2 do not directly correspond to mixture fraction, scalar variance, progress variable, and enthalpy. The coordinates are, rather, a set of transformed variables specifically designed to span a mutually orthogonal parameter space. The transformation is performed in order to maximize the storage of useful information in the database, and reduce redundancies and unphysical extrapolations. Consider, for example, the temperature contours shown in Figure 6.4. Tabulating the data in a Cartesian table as a direct function of enthalpy requires some kind of an approximation for the temperature in the white, inaccessible regions above and below the available data. Extrapolating valid data points to fill these areas can lead to unphysical estimates of the thermochemical quantities with unpredictable results when applied to real problems. A conservative, zeroth-order extrapolation could be applied, filling the space with a “nearest neighbor” approximation of the data. However, this approach unnecessarily populates the table with redundant values, taking up valuable space in memory that could alternatively be used to increase the accuracy of the representation. This is likewise an unattractive solution. In this work, auxiliary relations are used to map the table inputs from their


89

′′2 , # physical values (# z , z! c, # h) to a transformed description (Z, V, C, H). The variable

transformations are:

z# 1→ Z = z#

′′2 z! z#(1 − z#) # c # c 1→ C = cmax had − # h # h 1→ H = η# = ad . h − href

′′2 1→ V = z!

(6.11a) (6.11b) (6.11c) (6.11d)

As shown in equation (6.11) above, an identity map is applied to the mixture fraction, simply returning z#. The mixture fraction variance is scaled by its theoretical

upper-limit, z#(1 − z#). The progress variable is likewise scaled by its local maximum

value, cmax = f (Z, V, H), i.e., its maximum possible value given the other abscissae.

Enthalpy is mapped to the heat-loss parameter. In general, had and href in equation (6.11d) could be a function of all of the other input scalars (Z, V, C). Due to the defi-

nitions adopted in this work, however, their specification is much simpler: had = f (Z) and href = f (Z, V). Illustrative examples are shown in Figure 6.9. The coordinate transformations in equation (6.11) ensure that all of the information in the final database is directly traceable to a computed flamelet solution – not the result of an extrapolation or approximation outside of the known values. It also encourages a more faithful representation of information in the neighborhood of “pinch” regions that occur when properties converge near domain boundaries and endpoints. The subtle property variations that occur in these vicinities are often under-represented on table grids that span the global range of the un-transformed variable, where only one or two points may cover the region. In addition, the transformations proposed here organize the information into a space that is more natural for flamelet solutions, where gradients tend to align in the (Z, C, H)-direction, rather

than the (z, c, h)-direction. This further alleviates storage requirements by providing a coordinate system that is more attuned to the data, leading to increased accuracy

when interpolating on grids with a fixed number of samples. Although the auxiliary

90

′′2 mixture fraction variance, z!


1

0.2

2000 0.8

2000

0.15

1500 0.6

1500

normalized variance, V

0.25

0.1

0.4

0.2

0.05

0.2

0.4

0.6

0.8

0 0

1

mean progress variable, # c

0.35

0.3

0.25

0.2

0.15

0.2

0.4

0.6

0.8

1

1

2000 0.8

2000

1500 0.6

1500

0.4 1000

1000

0.1

0.2

0.05

0 0

500

500

normalized prog. variable, C

0 0

1000

1000

500

500 0.2

0.4

0.6

0.8

1

0 0

0.2

0.4

0.6

0.8

1

mean enthalpy, # h [kJ/kg]

normalized enthalpy, H = η#

0.9

0

2000

2000

-1000

0.6

1500

1500

-3000

1000 0.3

1000

-4000

500

500

-2000

0

0.2

0.4

0.6

0.8

mean mixture fraction, z#

1

0 0

0.2

0.4

0.6

0.8


1

Figure 6.9: Flamelet solutions in the original coordinate system (left) and the transformed coordinate system (right). Top to bottom: temperature at C = 1 and H = 0, temperature at V = 0 and H = 0, temperature at V = 0 and C = 1.

91

6.2. EFFICIENT TABULATION USING KD-TREES

1 -1000

0.8 0.6

-2000

0.4

-3000

0.2

0

-4000 0.2

0.4

0.6

0.8


1



1 0.3

0.8 0.25

0.6

0.2 0.15

0.4

0.1

0.2 0 0

0.05 0.2

0.4

0.6

0.8

1


Figure 6.10: Auxiliary relations for chemistry table coordinate transforms. Reference enthalpy, href (Z, V) [kJ/kg] (left), and maximum progress variable, cmax (Z, V, H) (right). The cmax shown above corresponds to the H = 0 plane. quantities cmax , had , and href must be stored in the database, this extra cost is more than compensated for by the other reductions that it affords.

6.2

Efficient tabulation using kd-trees

Detailed kinetic mechanisms for combustion can comprise thousands of chemical reactions amongst hundreds of individual species. In numerical simulations of these systems, it is generally not feasible to directly integrate transport equations for each chemical component due to the large number of species and the disparate range of important chemical and flow scales. In order to decrease the dimensionality of the problem, automatic reduction techniques have been developed which project the thermochemical state vector onto a lower-dimensional manifold. For example, in the SLF and FPV models discussed in Chapter 2, all chemical and thermodynamic quantities are represented by two scalars. As models are adapted to incorporate more complex physics such as turbulence, heat-loss, pollutant formation, etc., accurate predictions might require the use of five or more independent scalars to describe the system (Peters, 2000). The thermochemical state variables are typically stored in conventional tables as a function of the independent parameters. The size of the chemistry database

92


increases with the dimension of the parameter space, placing severe limitations on the resolution that can be afforded in multi-dimensional lookup tables. Several methods have been proposed to address this challenge, including the use of orthogonal polynomials (Turányi, 1994), in situ adaptive tabulation (ISAT) (Pope, 1997), artificial neural networks (Christo et al., 1996; Blasco et al., 1998; Ihme et al., 2008), high-dimensional model representation (Rabitz & Alis, 1999), self-similarity properties (Ribert et al., 2006), and adaptive Cartesian meshes (Xia et al., 2007). In this section, a multi-dimensional tabulation method that utilizes binary tree data structures and recursive partitioning to store and retrieve information is outlined and applied to chemically-reacting systems.

6.2.1

Introduction to binary trees

A kd-tree is a space-partitioning data structure that can be used to organize and search a k-dimensional region of space. In a kd-tree, each branch in the structure represents a (k-1)-dimensional hyperplane located at an isovalue of one of the coordinate dimensions. These axis-aligned hyperplanes are used to divide a given region (the parent) into two distinct sub-regions (the children). The “root” node of a kd-tree refers to the upper-most node in the structure and encompasses the entire region under consideration. Nodes on lower levels of the tree are designated as either “branch” nodes if they are a parent to child-regions, or “leaf” nodes if they are the bottom-most node in a sub-tree. Figure 6.11 shows an example of a two-dimensional tree structure and the corresponding partitioning of the region. Traversing a kd-tree requires a binary test at each branch of the tree (i.e., is the target point located left or right of this node’s hyperplane?), hence, kd-trees are a special case of binary space partitioning (BSP) trees. Many applications have utilized kd-trees for efficient organization of spatial data, including geographic information systems (GIS), computer graphics, and robotics control. Tree structures such as kd-trees are also a mainstay in computational geometry and form the basis of many nearest neighbor and range query algorithms. In this work, kd-trees are used to efficiently search multi-dimensional chemistry databases to

93


1.0 ✻

y

root node

0.5

❤ = branch node ✐ = leaf node 0.0 0.0

0.5

x

✲ 1.0

Figure 6.11: Example kd-tree data structure (left) and spatial partitioning (right).

locate appropriate interpolation intervals.

6.2.2

Partitioning algorithm

The ultimate objective of the tree-construction algorithm is to build a decision tree for predicting the response, Y, as a function of the predictors, X. The response values are the nP input data points that will be approximated by the final tree. For each response value there is a vector of nD predictor values, or parameters that classify that data point. The predictors and response values associated with the learning sample (input data set) are denoted as XP and YP . In the case of the FPV model, the predictors are the values of the independent scalars: Z, V, C, and the response # Y#α , etc. #˙ α , Θ, values are the thermochemical quantities that need to be tabulated: ρ¯, ω

Each quantity for which a tree is to be constructed is considered independently of the others in order to customize the tree-structure and adaptively fit it to the function of interest. Tree-structured regression is necessarily a recursive and iterative procedure. A

partitioning algorithm requires two core elements: a criterion for deciding when a node should be split, and a rule for selecting the “best” split at any node. The specification of these criteria depend on the requirements of the application. For

94


example, for approximating thermochemical properties in combustion simulations, a driving motivation is to maximize the accuracy of the retrieved data while minimizing the storage requirements of the tree. Other factors such as the overall computational performance of the algorithm during the construction of the tree, and the creation of a “balanced” tree, where all of the leaf nodes are approximately the same distance from the root node, are of secondary importance here. During the construction of the tree, information about each node is managed in a linked list. Six basic data fields are associated with each node: the split dimension icut , the split value Xcut , a link to the parent node idad , a link to each child node ikids , the spatial limits of the node’s bounding-box Xbox , and the value of the response function at the bounding-box corners Ybox = Y(Xbox ). Not all of this information is needed in the final tree, but it is convenient to have while building and rearranging the tree’s structure. The algorithm begins by initializing the root node and populating its six required data fields. Fields without a currently applicable value are left null. The values of the response function at the corners of the node’s bounding-box, Ybox , must be interpolated from the input data points, YP . The current code can interpolate from either structured input data, or from scattered data if a simplicial tessellation of the points can be constructed. The interpolated Y-values at each node are used to approximate the response function within the bounding-box by multi-linear interpolation. The initialization is completed by evaluating the tree approximation at the given predictor locations, XP , and computing the normalized error ϵtree

5 5 5 Ytree − YP 5 5, = 55 Ymax − Ymin 5

(6.12)

where Ytree is the current approximation provided by the kd-tree. max The code then conducts up to Nlevel refinement passes to divide nodes and create

sub-trees. At the beginning of each refinement pass, all nodes with errors larger than some user-specified tolerance are flagged to be split. Each of these nodes is then statistically analyzed to determine which dimension should be split. The overall strategy for refinement is to design a split that confines the error into

95


ever smaller nodal bins. In this way, non-linear regions of the parameter space can be adaptively refined to eliminate errors, while nearly-linear regions can be represented by coarsely-spaced nodes. During the analysis of a node, each of the nD dimensions are considered in turn. For each dimension j, the data points within the node are 1 j . The cumulative sum of the sorted in order of increasing values of their predictor X centered error in the node is calculated using (j)

(j)

ϵC = cumsum( 1 ϵsort − ⟨ 1 ϵ ⟩) ,

(6.13)

where the superscript (j) signifies the jth dimension, the hat accent denotes the subset of data points that are located in the current node, and the angle brackets indicate a simple average. The centered error is used to compute the left and right cumulative means of the error (j)

(j)

µLi =

ϵC i i

(j)

and

(j)

µRi =

(j)

ϵC m − ϵC i , m−i+1

(6.14)

where m denotes the total number of data points in the node and i indicates the ith element in the vector. The left and right cumulative means are combined to form the following error measure (j)

(j)

(j)

σi = i (µLi )2 + (m − i + 1) (µRi )2 .

(6.15)

Equation (6.15) can exhibit strong, bimodal peaks when the error distribution in the node is biased in a given direction. Consequently, it is a good indicator for differentiating amongst candidate split dimensions. The dimension that produces the largest value of σ (j) is selected for splitting. This is similar to a split criterion proposed by Breiman et al. (1984) for general classification trees. To illustrate application of the splitting rule, consider the following example. Suppose that the error in a node is described by the Gaussian function % & (X1 − X01 )2 (X2 − X02 )2 ϵ(X1 , X2 ) = exp − − . 2∆21 2∆22

(6.16)

96


1

300

(a)

0.8

250

0.7

0.8

200

0.6

X2

0.5 0.4

0.4

0.2

0.4

0.6

0.8

1

X1 0.08

0.2

0

0.1

-50

0

-100 0

0.2

0.4

X1 X1 X2 X2

(c)

dimension, dimension, dimension, dimension,

0.04

µL µR µL µR

0.6

0.8

1

Xj

.

0.06

35 30

X1 dimension X2 dimension

(d)

25 20

0.02

σ

µL , µR

100 50

0.3

0.2 0 0

150

ϵC

0.6

X1 dimension X2 dimension

(b)

15

0

10 -0.02

5

-0.04 2000

4000

6000

i

8000

2000

4000

6000

8000

i

Figure 6.12: Error measures for the tree partitioning algorithm: (a) example nodal error distribution, equation (6.16); (b) cumulative sum of the centered error, equation (6.13); (c) left and right cumulative mean error, equation (6.14); (d) node split criterion, equation (6.15).

This error distribution is shown in Figure 6.12a for X01 =0.25, ∆1 =0.05, X02 =0.40, and ∆2 =0.15 on a 100 × 100 grid. The various error measures from equations (6.13)–

(6.15) are plotted in Figure 6.12b-d. Since the example error function is biased most strongly in the X1 -dimension, σ (1) exhibits a pronounced peak, signifying that X1 is a good candidate for splitting. After selecting the split dimension, the nodal hypervolumes are divided, and a new hyperplane is inserted at 12 (Xmin + Xmax ). The simple bisection proposed here is j j not necessarily the optimal split location for the node. Experience with the present application, however, indicates that more complicated splitting rules offer only small improvements (if any), and have very little impact on the final size of the tree. Bisection, therefore, is used in the interest of simplicity. In the current implementation,


97

a user-defined limit on the minimum allowable spacing in each dimension is also imposed. If a proposed cut would violate this limit, then the largest scaled-dimension is split instead. Upon splitting the node, two new child leaves are appended to the tree, parent/child links are updated, and the response function is interpolated to the new bounding-box locations introduced by the splitting plane. The error estimate from equation (6.12) is then updated for all of the points in the recently split node. Refinement passes are continued until either the supplied error tolerances are met, or max the maximum allowable number of levels, Nlevel , is reached.

6.2.3

Data storage and retrieval

Not all of the data fields used during the construction of the tree are retained in the final data file. Branch and leaf nodes have different functions and, therefore, require different information. Branch nodes direct queries down the tree until they terminate at a leaf. To accomplish this, branch nodes must carry their split dimension icut , split location Xcut , and references to each of their child nodes ikids . The storage requirements per branch node are, therefore, 3 integers and 1 float. Leaf nodes, on the other hand, must contain all of the information necessary to reconstruct the response function within their hypervolume. For the case of multi-linear interpolation, this includes the minimum and maximum extents of the nD bounding-box Xbox , and the values of the function at the bounding-box vertices Ybox . This requires 2nD + 2nD floats for each leaf node. The tree data file must also contain the dimension of the parameter space, nD , and the number of branches in the tree, ntree . Since a properly constructed tree will contain ntree + 1 leaves, there is no need to explicitly save the number of leaves. The storage requirements for a kd-tree with linear interpolation can be summarized as follows: dimensions :

nD , ntree → 2 I

branches : icut , ikids , Xcut → ntree (3 I + R) leaves :

Xbox , Ybox → (ntree + 1)(2nD + 2nD )R

98


where “I” and “R” denote the memory required for a single integer and float, respectively. Note that this is the most compact (low-memory) storage scheme for the case of single-precision floats, or when I = R. For higher-precision floats, it can be advantageous to store all of the unique X and Y values in a separate floating-point array, and reference them with integer maps stored at the node level. The break-even point for this strategy depends on the dimension of the problem nD , the size of the binary tree ntree , and the fraction of redundancy amongst the float values. Given the other errors incurred during tree construction and data retrieval, however, double-precision arithmetic seems unlikely to provide any practical benefit. For comparison purposes, the memory requirements for uniform and non-uniform Cartesian tables are also presented. For a uniform Cartesian table, the storage requirements are: max grid : nD , nj , Xmin → I + nD (I + 2R) j , Xj nD : data array : Y → R nj . j=1

Likewise, a non-uniform Cartesian table requires: grid : nD , nj , Xj → I + nD I + R data array :

Y → R

nD :

nD $

nj

j=1

nj .

j=1

The total storage requirements for Cartesian tables and for the kd-trees described in this report are summarized in Table 6.2. Table 6.2 also indicates the scaling trend that each method exhibits with respect to the number of dimensions, nD . The size of Cartesian tables increases as ∼ nnD , while kd-trees scale as ∼ 2nD . This makes kd-

trees an attractive alternative to Cartesian tables for functions of moderate dimension, say nD ≤ 10.

Data retrieval is accomplished via a recursive navigation of the kd-tree. Each query point, XQ , starts at the root node, where it is passed from parent to child

99


Table 6.2: Storage requirements for Cartesian tables and kd-trees. table type

integers, I

uniform Cartesian

1 + nD

non-uniform Cartesian

1 + nD

floats, R nD : 2nD + nj

nD $ j=1

kd-tree

2 + 3ntree

j=1 nD :

nj +

nj

j=1

ntree + (ntree + 1)(2nD + 2nD )

nD scaling ∼ nnD ∼ nnD ∼ 2nD

down the tree until it arrives at one of the many leaf nodes. At every branch that it encounters along the way, the point is tested against that node’s splitting plane. If (i

)

XQcut < Xcut , the point is passed to the left, otherwise it is directed right. When the query reaches a leaf node, the function is interpolated using the information stored in Xbox and Ybox , and the result is returned to the calling function.

6.2.4

Application to combustion chemistry

In this section, the tree construction method outlined in Sections 6.2.2–6.2.3 is applied to tabulate flamelet data for use in the FPV combustion model. Two-dimensional examples are presented which demonstrate the method’s accuracy, memory requirements, and timing properties. The learning sample for the trees was prepared by transforming flamelet profiles resulting from the solution of equations (2.40) into (Z, C)-coordinates. The flamelets

were solved on adaptively-refined grids with 200 mesh points in the mixture fraction dimension. Also, roughly 200 solution profiles were obtained to span the reaction progress dimension from near-equilibrium burning to pure binary mixing (no reaction). In order to produce a very high-quality learning sample for this illustrative example, each flamelet profile was interpolated onto a Z-grid of 2000 non-uniformlyspaced points using one-dimensional cubic interpolation. The grid spacing was deter-

mined by averaging the grids from all of the flamelet solutions and scaling the result

100


from 200 to 2000 points. A similar procedure was then used to interpolate the data to a C-grid of 1000 points, creating a finely resolved collection of samples to use as the starting point for the analysis.

Binary trees were fit to the flamelet data using error tolerances ranging from 1 ×10−2 to 2 ×10−4 . A representative data structure from the tree-building algorithm

is shown in Figure 6.13. This tree was produced from the density data with a scaled

error tolerance of 1 × 10−2 . More stringent tolerances result in trees with more branching and deeper sub-trees.

Tree-based approximations of the flamelet data are shown in Figures 6.14–6.16 for trees using an error tolerance of 1 × 10−3. Figure 6.14 compares density from the original flamelet data with the binary tree approximation. The tree’s spatial partitions are overlaid on the plot to convey the complexity of the nodal structures and the adaptive nature of the tree approximation. Figure 6.15 shows the mass source term for the progress variable, c. In the expanded view, note the intense refinement that results from the sharp non-linear features near the stoichiometric mixture fraction zst = 0.055. Tree approximations for the temperature and mass fraction of carbon monoxide are shown in Figure 6.16. In all of the examples presented here, observe that each flamelet quantity produces a tree with substantially different structure than the others. This underscores the potential value of fitting approximations for each individual property rather than lumping them together in a single database as is often done. In order to appraise the effectiveness of the tree-based approach, the approximations from the binary trees are compared with tabulations on uniform Cartesian grids. It is acknowledged that a uniform orthogonal mesh is usually not the most cost-efficient way to store data. Nevertheless, equi-spaced Cartesian tables offer a useful and repeatable standard by which to benchmark alternative methods. The Cartesian tables were constructed by interpolating the learning sample data from the very fine, non-uniform grid onto a series of uniform Cartesian grids using bi-cubic interpolating splines. All of the variables were then reverse-interpolated from the uniform Cartesian tables, back to the original non-uniform mesh, using bilinear interpolation. The error incurred by this transaction was assessed in a manner


101

Figure 6.13: Binary tree data structure for density with error tolerance of 1 × 10−2 . Branch node △, leaf node •.

0.62 102


1

0.8

2

0.6

0.8

2

0.6

1.5

0.4

1.5

0.4

1

0.2 00

progress variable, C


1

0.2

0.4

0.6

0.8

00

1

1

0.2

mixture fraction, z

0.2

0.4

0.6

0.8

1

mixture fraction, z

Figure 6.14: Density, ρ [kg/m3 ]. Original data (left) and binary tree approximation with error tolerance of 1 × 10−3 (right). . progress variable, C

0.8

8000

0.6

6000

0.4

4000

0.2

2000

00

0.2

0.4

0.6

0.8

1

0


1

1

0.9

8000

0.8

6000

0.7

4000

0.6

2000

0.50

mixture fraction, z

0.02

0.04

0.06

0.08

0.1

0.12 0

mixture fraction, z

Figure 6.15: Product source, ρ ω˙ c [kg/m3 /s]. Binary tree approximation with error tolerance of 1 × 10−3 . Full domain (left) and zoom view (right). . 1

0.8

2000

0.6

1500

0.4

1000

0.2 00

500 0.2

0.4

0.6

0.8

mixture fraction, z

1



1

0.8

0.20

0.6

0.15

0.4

0.1

0.2

0.05

00

0.2

0.4

0.6

0.8

1

0

mixture fraction, z

Figure 6.16: Binary tree approximation with error tolerance of 1×10−3 . Temperature, Θ [K], (left) and CO mass fraction, YCO , (right).


103

analogous to equation (6.12). Uniform Cartesian tables of size 100×100 to 1000×1000 were investigated, and the maximum errors are shown in Figure 6.17 as a function of the number of points in each dimension. As expected, as the Cartesian grids become more refined, the maximum errors decrease. However, the directional dependence of the convergence rate is different for each of the quantities that was tested. The maximum errors in the density and the progress variable mass source, for instance, are almost independent of the resolution of the C-dimension over the ranges investigated. This is due to the steep non-linearities

in the z-dimension in these functions. Temperature and CO mass fraction, on the other hand, show fairly steady convergence with both z and C, at least in some regions of (nz , nc )-space. For all cases, it is only on the largest grids that were considered

that the maximum error levels in the Cartesian tables approach those of the binary trees pictured in Figures 6.14–6.16. From Figure 6.17 it is also apparent that for each of the flamelet quantites there are “optimal” values of (nz , nc ) that give the lowest errors given the total number of elements in the Cartesian table. These “best” combinations of (nz , nc ) have been extracted and plotted as the black symbols in Figure 6.18, showing the maximum error in the table versus the memory requirements. Also pictured in Figure 6.18 are the values for the kd-tree approximations of the flamelet data. Comparing the tree approximations with the uniform Cartesian tables reveals almost an order of magnitude difference in the memory required to obtain a given error tolerance. Additionally, the maximum error in several of the Cartesian tables stagnates as the table gets large. By contrast, the binary trees, with their capability for local anisotropic refinement, leads to a smooth and monotonic reduction of the error. Given the scaling relations shown in Table 6.2, this comparison should only improve as the number of table dimensions increases. Accuracy and memory constraints are important factors in gauging the utility of tabulated chemistry approximations. Data retrieval must also come at a reasonable cost in order for the method to be viable. To assess the cost of data retrieval, the time required to perform 1,000,000 two-dimensional table lookups on a single-processor

-0.8 104


1000

-1.5

(a) 800

1000

(b)

-2

800 -2.2

nc

-2 600

600

400

-2.5

-2.4 -2.6

400

-2.8 200

200 200

400

600

800

1000

1000

-3

-0.8

(c)

200

-1.2

600

800

1000

1000

-2

800

-2.2

600

-2.6

nc

-1.4 600

-1.6

400

-2

-2.4

-1.8

-2.8 400

-3

-2.2 200

-2.4 200

400

600

nz

800

1000

-3

-1.8

(d)

-1

800

400

-3.2 200

-3.4 200

400

600

nz

800

1000

Figure 6.17: Maximum normalized error (log10 scale) versus table size for linear interpolation from 2-D uniform Cartesian tables: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO .

105


-1

-1

10

(a) log10 ϵmax

10

Cartesian kd-tree

-2

-2

10

-3

10

10

-3

10

-4

-4

10 0 10 10

log10 ϵmax

10

1

10

2

10

3

10

4

10 1 10

10

2

10

3

10

4

-1

0

-1

Cartesian kd-tree

(b)

10

Cartesian kd-tree

(c)

Cartesian kd-tree

(d)

10

-2

10

-2

10

-3

10

-3

10

-4

10 1 10

-4

10

2

10

3

memory [KB]

10

4

10 1 10

10

2

10

3

10

4

memory [KB]

Figure 6.18: Maximum error versus memory requirement for 2-D uniform Cartesian tables and binary trees: (a) density, ρ [kg/m3 ]; (b) temperature, Θ [K]; (c) product source, ρ ω˙ c [kg/m3 /s]; (d) CO mass fraction, YCO .

106


workstation was measured. The lookup points were uniformly distributed pseudorandom numbers that covered the full range of (Z, C)-values.

The kd-trees are compared with several other methods that are used to search

conventional tables: a uniform Cartesian lookup, a “dumb” search loop that incrementally cycles through each table dimension until it finds the appropriate interval, and a bisection search suitable for non-uniform table coordinates. With each lookup a bi-linear interpolation was conducted as part of the test. The Cartesian lookups were tested on tables of size: 30 × 30, 50 × 50, 80 × 80, 130 × 130, 220 × 220, 360 × 360, 600 × 600, and 1000 × 1000. The kd-trees used in the test are the ones shown in Figure 6.18. Ten separate timings were conducted for each method and grid, with the average lookup times reported in Figure 6.19. The uniform Cartesian lookups are difficult to beat in terms of efficiency. For small- and moderately-sized tables they show no appreciable scaling with table size. When arrays become large and are spread over many memory blocks, however, inefficiencies accumulate and the scaling behavior is altered. The “dumb” loop search strategy is comparable with the other methods on small grids, but scales poorly as the size of the table grows. The bisection search maintains respectable scaling over the range of table-sizes that were tested. Data retrieval in the kd-trees exhibits similar scaling properties as the bisection search. These results suggests that kd-trees are a competitive alternative to conventional tabulation methods on uniform and non-uniform Cartesian grids. The techniques outlined here have been applied to construct a three-dimensional chemistry database for methane-air combustion at the conditions described in Section 7.1. The coordinate transformations defined in equations (6.11a)–(6.11c) were applied to the flamelet data before the binary trees were constructed. In terms of the coordinate transformation, the only difference between this three-dimensional problem and the four-dimensional example in Section 6.1.3 is the functional dependence of the auxiliary quantity, cmax . For the three-dimensional database cmax = f (Z, V).

107

time [sec]


10

1

10

0

10

-1

10

-2

10

uniform Cartesian “dumb” loop bisection kd-tree

0

10

1

10

2

10

3

10

4

memory [KB] Figure 6.19: Retrieval time for 1,000,000 2-D lookups with linear interpolation using various search methods. Each point represents an average of ten 1,000,000-point tests.

108


Chapter 7 Application to coaxial-jet combustor 7.1

Experimental configuration

The FPV-GHL model has been applied to simulate the combustion experiments performed by Owen et al. (1976). The full experimental study examined turbulent flow and combustion in a coaxial-jet combustor operated under various conditions and geometric modifications. In all, eight separate test cases were documented, exploring the effects of swirling inflow conditions, the air/fuel velocity ratio, and pressure. The present validation of numerical results focuses on the conditions referred to as “Test 1” in the laboratory report (Spadaccini et al., 1976). This experiment was selected for its simple geometry and boundary condtions, as well as its relatively detailed set of measurements, including velocity, temperature, and several reactant and product species. Additionally, this experiment has been the subject of several previous computational investigations, particularly the original LES implementation of the FPV model (Pierce & Moin, 2001, 2004). In that work, disagreement with the experimental temperature measurements was attributed to the neglect of convective and radiative heat-transfer effects. An investigation of this conjecture partially motivates the current study and the selection of this experiment for validation. Although, this 109

110

CHAPTER 7. APPLICATION TO COAXIAL-JET COMBUSTOR

water-cooled outer wall, Θwall ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲

✲ ✲ ✲

✻ ✻ ✻ R3 R4 R 2 R1

fuel

✲ ✲ ✲

air

✻

✟ ✟✟ ✟ ✟✟

combustion region

❅ ❅ ❅

recirculation zone

Figure 7.1: Schematic of the coaxial-jet combustor experiment of Spadaccini et al. (1976).

combustor is not geometrically sophisticated, its complex recirculation and flow patterns provide a reasonable approximation of conditions in a gas turbine combustor. An illustration of the experimental configuration is provided in Figure 7.1. In the experiment, relatively low-speed fuel enters the combustion chamber through the central pipe, and non-swirling, higher-speed air enters through the outer annulus. The high velocity ratio between the streams results in dramatic mixing directly downstream of the fuel inlet and the formation of a large recirculation zone. As hot combustion products circulate in this region, they ignite incoming reactants and stabilize the burning. Highly unsteady dynamics were observed in the experiment near the nozzle, with intermittent lift-off and reattachment of the flame. The region of active combustion persisted beyond the 1-meter-long test section of the experiment. The combustor was fueled with natural gas at 300K and preheated air at 750K. The entire system was pressurized to 3.8 atm. The walls of the combustor were water-cooled to maintain a constant wall temperature of approximately 500K. Porous metal discs were installed upstream of the fuel and air inlets to provide uniform inflow conditions. Some of the pertinent dimensions and flow conditions of the experiment are summarized in Table 7.1. Spadaccini et al. (1976) report radial profiles of velocity, temperature, and selected species concentrations measured at four axial stations. The velocity was measured using laser Doppler velocimetry, temperature was measured with a calibrated-heat-loss thermocouple, and species concentrations were measured

111

7.1. EXPERIMENTAL CONFIGURATION

Table 7.1: Combustor dimensions and flow conditions specified in the experiment of Spadaccini et al. (1976). quantity central pipe radius (R1 ) annulus inner radius (R2 ) annulus outer radius (R3 ) combustor radius (R4 ) combustor length overall equivalence ratio (Φ) combustor pressure

: : : : : : :

value quantity 0.03157 m fuel mass flow rate 0.03175 m air mass flow rate 0.04685 m fuel bulk velocity 0.06115 m air bulk velocity 1.0 m fuel temperature (Θfuel ) 0.9 air temperature (Θoxid ) 3.8 atm wall temperature (Θwall )

: : : : : : :

value 0.0072 kg/s 0.137 kg/s 0.9287 m/s 20.63 m/s 300K 750K 500K

using a traversing gas-sampling probe. Mole fraction data were reported for CH4 , O2 , CO2 , CO, NO2 , and NO. Experimental mixture fraction and progress variable profiles were estimated from these species concentrations according to the following procedure. First, the mole fraction of H2 O was estimated from a linear least-squares fit to the adiabatic flamelet data for CO2 and CO at the globally-averaged value of the mixture fraction zavg =

zst Φ . 1 + zst (Φ − 1)

(7.1)

For this system, zst = 0.055 and Φ = 0.9, thus, zavg = 0.0498. The resulting fit, for flamelets ranging from pure-mixing conditions (XH2 O = 0) to near equilibrium (XH2 O ≈ 0.18), is shown in Figure 7.2 for the least-squares coefficients βCO2 = 1.8989

and βCO = 1.7929. Next, the mole fraction of H2 was computed from a balance on C

and H atoms: XH2 = 2XCO2 + 2XCO − XH2 O . N2 results from atom balances on N and

O: XN2 = 1.881(2XO2 +2XCO2 +XCO +XH2 O ). If the sum of the mole fractions at this

point totaled greater than one, the estimated concentrations, XH2 O , XH2 , and XN2 , were proportionally adjusted so as to preserve the normalization constraint. With these estimates it was possible to calculate the mass fractions of all of the species, assuming that only major products were formed. An experimental mixture fraction was then constructed from the total mass fraction of C and H atoms in the system.

112


βCO2 XCO2 + βCO XCO

0.15

0.1

0.05

0 0

0.05

0.1 XH 2 O

0.15

Figure 7.2: Least-squares fit of adiabatic flamelet data at zavg = 0.0498 to predict XH2 O using XCO2 and XCO . The linear coefficients are βCO2 = 1.8989 and βCO = 1.7929.

The progress variable was likewise computed from c = YCO2 + YH2 O + YCO + YH2 . Six of the data points for XCH4 at the first measurement station were not provided because concentrations were out of the calibrated range of the gas analyzer. These missing data were estimated using the above assumptions and the requirement that mole fractions should sum to one.

7.2

Computational details

A diagram of the computational setup for the LES is presented in Figure 7.3. Throughout the following discussion, all references to geometrical dimensions are reported in the non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . The reference velocity oxid scale is taken to be the bulk air velocity Ubulk , leading to a non-dimensional time of oxid τ = t Ubulk /R3 . Air is introduced in the outer annulus at an axial location of ξ = −8,

and fuel enters the system at ξ = −3. A recycling strategy is used to establish

fully-developed inflow conditions in the air and fuel streams. This is accomplished

by sampling the flow profiles at ξ = −2 and recycling the information to the inlet

113

✲ air

✲ air

ξ=8

ξ=8 outflow b.c.

ξ=2

ξ=1

ξ=0

ξ = −2 recycle sample plane

ξ = −3 recycle b.c. (pipe)

ξ = −8 recycle b.c. (annulus)

7.2. COMPUTATIONAL DETAILS

✲ ✲ ✲ ✲ fuel (CH4 ) ✲ ✲ ✲

ξ = −8

ξ=2

ξ = −2

ξ=0

ξ=1 Figure 7.3: Computational domain (top) and computational grid (bottom) in nondimensional coordinates ξ = x/R3 and ϱ = r/R4 .

114


planes of the reactant streams. The inflow velocities are rescaled during the recycle procedure to maintain the desired bulk velocities. A convective “outflow” boundary condition is applied at ξ = 8 (see equation (5.6) for details). The computational mesh is composed of 6.7 million high-quality hexahedral control volumes, or approximately 6.9 million grid nodes. The mesh was generated using Tommie, a Cartesian-based meshing tool developed at Stanford University that is capable of anisotropic mesh refinement (Ham et al., 2002; Iaccarino & Ham, 2007). Although the grid is largely structured, 2-to-1 volume-transitions are used in the azimuthal direction to eliminate the high-concentration of grid points near the centerline that is typical in conventional cylindrical grids. This allows improved azimuthal resolution at large radii, while maintaining manageably-sized grids. For example, in the current grid, 512 intervals were used to azimuthally resolve the outer walls of the combustor and annulus. Given the grid dimensions in the other directions (nξ = 256 and nϱ = 128), if this azimuthal resolution were continued radially to the centerline, the grid would contain almost 17 million control volumes. Volume transitions are not used in the axial and radial directions in the current mesh. Instead, the mesh spacing is stretched to provide resolution in critical regions such as the mixing zone immediately downstream of the fuel nozzle and in boundary layers near solid surfaces. A depiction of the computational grid is provided in the lower portion of Figure 7.3. Each 60-degree sector in the figure displays a cut through the grid at a different axial location. The downstream progression of grid slices proceeds clockwise from the top center sector. Note how the fine spacing near the pipe/annulus boundary is relaxed after the step expansion at ξ = 0 to provide more radial resolution near the combustor wall and nearly uniform spacing in the interior of the flow. Additional quantitative details about the grid spacing are offered in Figure 7.4. The flamelet libraries used in this study were computed using GRI-Mech 2.11, a detailed kinetic mechanism containing 277 elementary chemical reactions and 49 component species (Bowman et al., 1995). Multicomponent mass diffusion including Soret and Dufour effects was used. In the FPV-GHL model, radiation effects were included by post-processing the flamelet data to include a radiative source term in accordance with equation (2.19). A summary of the final table-sizes and tabulation

7.3. RESULTS

115

methods is available in Table 7.2. The simulations were initialized on a relatively coarse mesh of 1.7 million nodes. An incompressible, constant density simulation with a passive mixture fraction was run until approximately τ = 25 to establish a steady base flow. Combustion was then ignited on the coarse mesh by applying the equilibrium value of the progress variable, # ceq (# z ), according to the instantaneous mixture fraction field. Full variable-

density effects were gradually applied to the flow over the course of approximately

1,000 time steps, after which the burning flow was advanced until τ = 70. During this initialization period, the code was run with a constant CFL number of 1.5 on 64 processors of a generic Linux cluster. The flow variables were then interpolated to the refined mesh pictured in Figure 7.3. The flow was advanced using a CFL number of 1 for another ∆τ = 20, and then the sampling of flow statistics was initiated. Statistics were collected for a total sampling period of ∆τstats = 100 for each LES case. To obtain the statistics, the resolved fields were averaged in the azimuthal direction and in time. The average time step during the sampling period was about ∆τ = 0.0034 for all of the cases. The simulations were performed on the Red Storm computing platform at Sandia National Laboratories. Red Storm is a massively parallel Cray XT3/4 supercomputer with distributed memory and more than 38,000 processing cores. Each simulation was run on 320 processors, where an individual time step took approximately 14.9 seconds for the FPV model and 16.3 seconds for the FPV-GHL model. This represents a cost-increase of about 9%, due to the additional scalar equation and four-dimensional chemistry lookups in the FPV-GHL model.

7.3

Results

Four separate large-eddy simulations of the coaxial-jet combustor experiment described in Section 7.1 were performed. They differ in their choice of flamelet model, chemistry tabulation technique, use of coordinate transformations in the chemistry table, specification of energy boundary conditions, and inclusion of radiation. Outside of these differences, identical execution parameters, grids, and starting conditions are


0.2 0.15 0.1 0.05 0 -8

-6

-4

-2

0

2

4

6

8

radial grid spacing, ∆ϱ

axial grid spacing, ∆ξ

0.007 116

axial coordinate, ξ

0.013 0.011 0.009 0.007 0.005 0.003 0.001

0

ξ ξ ξ ξ ξ

= = = = =

0 1 2 3 8 0.2

0.4

0.6

0.8

1

radial coordinate, ϱ

Figure 7.4: Computational grid: axial (left) and radial (right) grid spacing in nondimensional coordinates ξ = x/R3 and ϱ = r/R4 .

Table 7.2: Summary of LES test cases. case 0 1 2 3 4

color magenta gold red green blue

code structured CDP CDP CDP CDP

flamelet model adiab. FPV adiab. FPV adiab. FPV FPV-GHL FPV-GHL

chemistry table type and size Cartesian (200 × 50 × 50) Cartesian (200 × 50 × 100) binary tree (tol = 1 × 10−3 ) Cartesian (100 × 30 × 50 × 40) Cartesian (100 × 30 × 50 × 40)

coord. transf. N N Y Y Y

enthalpy wall b.c. no-flux no-flux no-flux isothermal isothermal

radiation N N N N Y

used for all of the cases. Specific details of the different test cases are enumerated in Table 7.2. Also listed in the table are details of the LES conducted by Pierce & Moin (2004), which is used as a preliminary point of comparison for the current results. There are many differences between the current simulations and the work of Pierce & Moin (2004). The Pierce & Moin (2004) simulation used a different code, grid, and inflow conditions, as well as an alternate definition of the reaction progress variable, namely c′ = YCO2 + YH2 O . A direct comparison with the new simulations facilitates a precursory quantification of these contrasts. The various models lead to appreciably different flame behavior in the combustor. Some of the similarities and differences can be seen in the instantaneous snapshots of the flow shown in Figures 7.5–7.7. In all of the cases, an unsteady mixing region develops in the core of the flow between 1 < ξ < 4. The transient mixing leads to fuel-rich pockets of mixture fraction that precess azimuthally around the shear layer.

7.3. RESULTS

117

The non-uniformities in the mixture fraction field are apparent in the slices shown in Figure 7.5a-b. As the fuel-rich fluid mixes further downstream, different behavior is observed in the cases with and without convective heat transfer. In the adiabatic simulations, the excess fuel produces a flame which becomes stabilized near the walls of the combustor at around ξ = 3. This can be seen in the upper panels of Figures 7.6c-d and 7.7c-d, where there are pervasive regions of hot combustion products in close proximity to the outer wall. In the adiabatic cases, the flame can persist near the wall for long periods of time before eventually migrating back to the core of the combustor. In the non-adiabatic cases, however, flame-wall interactions weaken flames that come near the wall and prevent the kind of near-wall flame stabilization described above. The cold walls lead to increased flame quenching and the formation of a thin thermal boundary layer. This boundary layer is evident in the lower panels of Figures 7.6c-d and 7.7c-d as a dark region near the combustor wall. Although combustion is inhibited in this region, the steep temperature gradients can produce large heat fluxes. A comparison of the time-averaged statistics from Case 0 (Pierce & Moin, 2004) with Case 1 from the present work is shown in Figure 7.8. Modest differences are observed in the velocity and temperature profiles, but pronounced improvements are seen in the prediction of the CO mass fraction in Case 1. These differences are likely due to the improved definition of the progress variable in the current work, which produces a more monotonic parameterization of the flamelet space. Also, much finer resolution is used in the chemistry table for Case 1 compared to Case 0. For instance, Pierce & Moin (2004) used 200 uniformly-spaced points to resolve the z-dimension in the chemistry table, while the flamelet database used in Case 1 has non-uniform table-spacing with aggressive refinement around the stoichiometric region. Case 1 additionally provides twice the resolution in the progress variable dimension of the flamelet library. Both simulations do a poor job of representing the temperature at the first two measurement stations and over-predict the thermal field in that region by 300-450K. In many aspects, the behavior of the temperature field qualitatively resembles that of the product mass fraction. As an example, compare the overall characteristics of

118


(a)

(b)

(c)

(d) .

(a)

(b)

(c)

(d)

Figure 7.5: Instantaneous mixture fraction, z#: (a) meridional plane, (b) ξ = 1, (c) ξ = 3, (d) ξ = 5. Adiabatic Case 2 (top), isothermal Case 3 (bottom).

119

7.3. RESULTS

(a)

(b)

(c)

(d) .

(a)

(b)

(c)

(d)

Figure 7.6: Instantaneous progress variable, # c: (a) meridional plane, (b) ξ = 1, (c) ξ = 3, (d) ξ = 5. Adiabatic Case 2 (top), isothermal Case 3 (bottom).

120


(a)

(b)

(c)

(d) .

(a)

(b)

(c)

(d)

# (a) meridional plane, (b) ξ = 1, (c) ξ = 3, Figure 7.7: Instantaneous temperature, Θ: (d) ξ = 5. Adiabatic Case 2 (top), isothermal Case 3 (bottom).

121

7.3. RESULTS

1.2

g ′′2 ⟩ 2 ⟨e u⟩, ⟨u

1

1

1.2

ξ = 0.1357

1

1

ξ = 0.3811

0.8

ξ = 4.6727 0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.6

0.4

0.4

0.2

0.4

0.2

0.2

0

0

0

−0.2

−0.2 0 2000

0.2 0.4 0.6 0.8

1

ξ = 0.8876

−0.2 0 2000

1500

1000

1000

0.2 0.4 0.6 0.8

1

ξ = 1.5663

e ⟨Θ⟩

1500

500 0 0.08

0.2 0.4 0.6 0.8

1

500 0 0.12

ξ = 0.2088 ⟨Yg CO ⟩

1

ξ = 1.2713

0.2 0.4 0.6 0.8

1

−0.4 0 2500

1000

1000

500 0 0.12

0.06

0.06

0.02

0.03

0.03

0.2 0.4 0.6 0.8

ϱ

0.2 0.4 0.6 0.8

1

500 0 0.15

ξ = 3.8374

0.04

0 0

ξ = 4.5161

1500

0.09

1

0 0 2500

1500

ξ = 3.1587

ϱ

1

2000

0.09

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

2000

0.06

0 0

0.2

1

0 0

0.2 0.4 0.6 0.8

1

ξ = 5.1948

0.2 0.4 0.6 0.8

1

ξ = 7.4137 0.1

0.05

0.2 0.4 0.6 0.8

ϱ

1

0 0

0.2 0.4 0.6 0.8

1

ϱ

Figure 7.8: Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature, CO mass fraction. Experiment (mean) •, experiment (RMS) ×, Case 0, and Case 1.

Figure 7.6 with Figure 7.7. In an adiabatic flow, over-predicting the temperature suggests a corresponding over-prediction of combustion products. For this flow, it appears that the FPV model incorrectly predicts the onset of combustion and the lift-off height of the flame. Note that the axial measurement stations for temperature are different from those for species concentrations. Since species measurements are not available in this region, it is difficult to draw any definitive conclusions about the source of the discrepancy. Figure 7.9 shows how the tabulation method and the use of the coordinate transformations outlined in equation (6.11) affect the simulations. There is little change

122


1.2

1

g ′′2 ⟩ 2 ⟨e u⟩, ⟨u

1

1.2

ξ = 0.1357

1

1

ξ = 0.3811

0.8

ξ = 4.6727 0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.6

0.4

0.4

0.2

0.4

0.2

0.2

0

0

0

−0.2

−0.2 0 2000

0.2 0.4 0.6 0.8

1

ξ = 0.8876

−0.2 0 2000

1500

1000

1000

0.2 0.4 0.6 0.8

1

0.2 0.4 0.6 0.8

1

0.06

500 0

ξ = 1.5663

0.2 0.4 0.6 0.8

−0.4 0 2500

e ⟨Θ⟩

1500

500 0

1

0.1

0.2 0.4 0.6 0.8

1

0 0 2500

ξ = 4.5161

2000

2000

1500

1500

1000

1000

500 0

ξ = 3.1587

0.04

0.2

0.2 0.4 0.6 0.8

1

0.1

ξ = 0.2088 ⟨Xg CO ⟩

1

ξ = 1.2713

0.08

0.06

0.06

0.04

0.04

0.02

0.02

1

ξ = 5.1948

0.2 0.4 0.6 0.8

1

0.12

ξ = 3.8374

0.08

500 0

0.2 0.4 0.6 0.8

0.1

ξ = 7.4137

0.08 0.06

0.02

0 0

0.2 0.4 0.6 0.8

1

0 0

0.2 0.4 0.6 0.8

ϱ

ϱ

1

0 0

0.04 0.02 0.2 0.4 0.6 0.8

ϱ

1

0 0

0.2 0.4 0.6 0.8

1

ϱ

Figure 7.9: Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature, CO mole fraction. Experiment (mean) •, experiment (RMS) ×, Case 1, and Case 2.

in the velocity and temperature within the first few radii downstream of the expansion, however, predictions of temperature and CO mole fraction are significantly improved in the latter half of the domain. Temperatures increase by 50-100K in the core of the combustor, producing better agreement with the experiment. The shift in temperature also leads to better predictions of the concentration of CO at the last three measurement stations. The results suggest that the increased accuracy and fidelity afforded by the binary tree representation of the chemistry, combined with the advantages of using the coordinate transformations, greatly enhance the predictive capability of the FPV model. The effects of isothermal wall conditions and heat-loss on the time-averaged flow

7.3. RESULTS

123

variables is presented in Figures 7.10 and 7.11. Near the inlet nozzle, there is very little difference in the velocity profiles obtained from Cases 2, 3, and 4. This is not surprising considering that the recycling inlet condition was forced to provide the same bulk velocity in all three of the cases. Subtle differences in the temperature, however, begin to emerge less than one radius downstream of the expansion. This is much sooner than was observed for Cases 1 and 2 in Figure 7.9, suggesting a delicate sensitivity of the near-field mixing processes to the heat transfer and adiabaticity of the flow. Similar velocities, yet different temperatures, imply changes in the local density and mass fluxes, which have consequences on the mixing patterns in this region. Figure 7.10 supports this notion, showing large changes in the time-averaged mixture fraction and progress variable near the fuel nozzle. The differences in the scalar fields are mainly confined to the mixing zone in the fore-region of the combustor, and diminish as the flow stabilizes downstream. The unsteady dynamics of the mixing region present a significant challenge to predicting this flow, and dramatically affect the performance of the different models near the nozzle. This explains the erratic temperature predictions from Case 3 at the first measurement station. In the region after the recirculation zone (roughly ξ > 4), temperature profiles settle into the hierarchy one would expect, i.e., Case 2, then Case 3, then Case 4 as heat-losses increase and temperatures decrease. This is easily seen in the expanded view of the interior temperature in Figure 7.10. In the core of the flow, the peak temperature drops by approximately 100K from Case 2 to Case 4. This leads to a 20% change in the predicted CO concentration near the exit of the domain. The adiabatic model (Case 2) seems to do the best job of capturing the evolution of CO in this problem, but the isothermal model without radiation (Case 3) attains nearly as good results. This is impressive given the relatively coarse tabulation in the fourdimensional chemistry table compared to the binary tree. The temperature changes due to radiation in Case 4 have a pronounced negative impact on the prediction of CO, suggesting that radiation is not a significant factor in the experiment. The temperature and enthalpy in the cases with and without heat transfer exhibit markedly different behavior in the regions near the walls of the combustor. This is clearly visible at the latter measurement stations in Figure 7.11, where a difference

124


of almost 800K is observed in the near-wall temperatures between Case 2 (adiabatic) and Cases 3 and 4 (non-adiabatic). The zoom view of the near-wall temperature and enthalpy in Figure 7.11 shows the thermal boundary layer with solid symbols to denote the grid spacing in the LES. The enthalpy profiles have been normalized using the adiabatic values at z = 0 and z = 1. It is apparent from the figure that the developing boundary layer is somewhat under-resolved at the first two measurement stations, with only a few points to represent the steep thermal gradients. The mesh resolution is better downstream, where the boundary layer is thicker and the grid has been gradually stretched to provide more points near the wall. The temperature profiles for the two non-adiabatic cases are very similar in the near-wall region, but the inclusion of radiation in Case 4 leads to a pronounced enthalpy defect and correspondingly lower temperatures in the interior of the combustor. By contrast, heat-transfer effects in Case 3 remain fairly localized and primarily affect the region near the cold wall. The FPV-GHL model enables the determination of heat fluxes applied to the combustor confinement. Depictions of the instantaneous wall heat flux are shown in Figure 7.12 for Case 3 (isothermal wall) and Case 4 (isothermal wall with radiation). When the air stream encounters the step expansion, the flow separates and creates a recirculation bubble in the region immediately behind the step. The recirculating fluid pulls hot gases away from the center of the combustor towards the wall, where they impinge on the solid surface. Large heat fluxes are produced in the region just downstream of the stagnation line located around ξ = 1. Pockets of large heat fluxes tend to meander azimuthally along the wall and migrate slowly downstream until they exit or dissipate. Figure 7.13 shows the time-averaged profiles of heat flux and several scalars variables at the wall. The maximum time-averaged heat flux was 75 kW/m2 for Case 3, and 83 kW/m2 for Case 4, thus radiation increased the maximum time-averaged heat flux by approximately 10%. The location of the maximum was ξ = 3.16 for Case 3 and ξ = 3.66 for Case 4. The ensemble-averaged maximum heat flux values were about an order of magnitude larger, namely 690 kW/m2 for Case 3 and 720 kW/m2 for Case 4, a difference of 5%. Histograms of the maximum heat flux are available in Figure 7.14. The peak values of the instantaneous heat flux were roughly

125

7.3. RESULTS

1.2 1 g ′′2 ⟩ 2 ⟨e u⟩, ⟨u

1

1.2

ξ = 0.1357

1

1

ξ = 0.3811

0.8

0.8

0.6

0.6

0.6

0.4

0.6

0.4

0.4

0.2

0.4

0.2

0.2

0

0

−0.2

0

−0.2 1 0 2000

0.2 0.4 0.6 0.8

ξ = 0.8876

1500

1000

1000

0.2 0.4 0.6 0.8

ξ = 1.5663

e ⟨Θ⟩

1500

−0.4 1 0 2500

e ⟨Θ⟩

500 0 1800

0.2 0.4 0.6 0.8

1

ξ = 0.8876

500 0 1800

0.2 0.4 0.6 0.8

1

0.2 0.2 0.4 0.6 0.8

1

ξ = 4.5161

0 0 2500

2000

2000

1500

1500

1000

1000

500 0 2200

0.2 0.4 0.6 0.8

1

500 0 2200

1600

1600

2000

2000

1400

1400

1800

1800

1200

1200

1000 0

0.06

0.2 0.4 0.6 0.8

1

1000 0

0.1

ξ = 0.2088

0.04

1600

ξ = 1.5663 0.2 0.4 0.6 0.8

1

1400 0

0.1

ξ = 3.1587

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

ξ = 4.6727

0.8

0.8

−0.2 0 2000

⟨Xg CO ⟩

1

ξ = 1.2713

0.2 0.4 0.6 0.8

1

ξ = 5.1948

0.2 0.4 0.6 0.8

1

1600

ξ = 4.5161 0.2 0.4 0.6 0.8

1

1400 0

0.12

ξ = 3.8374

0.1

ξ = 5.1948 0.2 0.4 0.6 0.8

1

ξ = 7.4137

0.08 0.06

0.02

0 0 1

0 1 0 0.15

0.2 0.4 0.6 0.8

ξ = 0.2088

0 1 0 0.15

0.2 0.4 0.6 0.8

ξ = 3.1587

0.04 0.02 0.2 0.4 0.6 0.8

1

0 0 0.15

ξ = 3.8374

0.2 0.4 0.6 0.8

1

ξ = 7.4137

⟨e z⟩

0.8 0.6 0.4

0.1

0.1

0.1

0.05

0.05

0.05

0.2 0 0 0.3

0.2 0.4 0.6 0.8

1

0 0 0.4

ξ = 0.2088

⟨e c⟩

0.2

0.1

0 0

0.2 0.4 0.6 0.8

ϱ

1

0.2 0.4 0.6 0.8

1

0 0 0.4

ξ = 3.1587

0.2 0.4 0.6 0.8

1

0 0 0.4

ξ = 3.8374 0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.2 0.4 0.6 0.8

ϱ

1

0 0

0.2 0.4 0.6 0.8

ϱ

1

ξ = 7.4137

0.3

0 0

0.2 0.4 0.6 0.8

1

0 0

0.2 0.4 0.6 0.8

1

ϱ

Figure 7.10: Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: velocity, temperature (full view), temperature (interior zoom), CO mole fraction, mixture fraction, progress variable. Experiment (mean) •, experiment (RMS) ×, Case 2, Case 3, and Case 4.

126


⟨(e h − h0 )/(h1 − h0 )⟩

⟨(e h − h0 )/(h1 − h0 )⟩

0.3

0.2

0.1

0.25

0.25

0.25

0.2

0.2

0.2

0.15

0.15

0.15

0.1

0.1

0.1

0.05 0 0 0.08

ξ = 0.8876 0.2 0.4 0.6 0.8

1

0 0 0.08

ξ = 0.8876 0.06

0.04

0.04

0.02

0.02

0.95

0.2 0.4 0.6 0.8

1

0 0 0.25

ξ = 1.5663

0.06

0 0.9 800

0.05

ξ = 1.5663

1

0.95

0.2 0.4 0.6 0.8

1

1

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 0.9 1500

0.95

ξ = 5.1948 0.2 0.4 0.6 0.8

1

ξ = 5.1948

0.2

1

0 0.9 1500

0.95

1

700 1000

600

500 0.9

0 0 0.25

ξ = 4.5161

e ⟨Θ⟩

700

0 0.9 800

0.05

ξ = 4.5161

1000

600

ξ = 0.8876 0.95

ϱ

1

500 0.9

ξ = 1.5663 0.95

ϱ

ξ = 4.5161

1

500 0.9

0.95

ϱ

ξ = 5.1948

1

500 0.9

0.95

1

ϱ

Figure 7.11: Radial profiles of time-averaged flow variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Top-to-bottom: enthalpy, enthalpy boundary layer, temperature boundary layer. Case 2, Case 3, and Case 4.

127

azimuthal coordinate, θ/π

7.3. RESULTS



Figure 7.12: Instantaneous snapshot of wall heat flux: Case 3 (left) and Case 4 (right).

the same for both Case 3 and 4, reaching values as high as 1.6 MW/m2 . This is an order of magnitude larger than the maximum value of 90 kW/m2 reported from the two-dimensionsal DNS of Wang & Trouvé (2006), but it is comparable to the peak heat fluxes of 1-3 MW/m2 reported for laminar diffusion flames in stagnation flow by de Lataillade et al. (2002).

128


0.25

80

enthalpy, ⟨(# h − h0 )/(h1 − h0 )⟩wall

heat-flux, ⟨# q ⟩wall [kW/m2 ]

100

0.2

60

0.15

40

max at ξ = 3.16

max at ξ = 3.66

0.1

20 0

0

2

4

6

8

0.05

2

4

6

8

2

4

6

8

0.1

progress variable, ⟨# c⟩wall

mixture fraction, ⟨# z ⟩wall

0.02

0

0.08

0.015

0.06

0.01

0.04

0.005

0

0.02

0

2

4

6


8

0

0


Figure 7.13: Axial profiles of time-averaged wall variables in non-dimensional coordinates ξ = x/R3 and ϱ = r/R4 . Clockwise from upper left: heat-flux, normalized enthalpy, progress variable, mixture fraction. Case 3, and Case 4.

129

7.3. RESULTS

0.2

mean=690 at ξ = 2.48

kW/m2

0.15

0.1

0.05

0

500

1000

maximum heat-flux,

1500 wall q#max

[kW/m2 ]

relative frequency per interval

relative frequency per interval

0.2

mean=720 kW/m2 at ξ = 2.59 0.15

0.1

0.05

0

500

1000

maximum heat-flux,

1500 wall q#max

[kW/m2 ]

Figure 7.14: Histogram of peak heat-flux values for Case 3 (left) and Case4 (right).

130


Chapter 8 Conclusions and further work 8.1

Summary and conclusions

Accurate predictions of chemically reactive flows are needed in order to design and optimize combustion devices to meet increasingly demanding performance and emission requirements. Recently, large-eddy simulation (LES) has moved beyond showing promise in this area, and has begun demonstrating real utility in predicting fluid flow in complex multi-physics applications. In this work, several aspects relating to the application of LES to non-premixed combustion problems were studied. Issues relating to tabulation and interpolation of non-linear equations-of-state (EOS) were addressed in Chapters 4 and 5. Many popular combustion models utilize some form of property tabulation to efficiently couple hydrodynamics codes with complex equations-of-state. It was shown that errors introduced by tabulated-EOS lookups can lead to numerical instabilities that affect the accuracy and convergence of low-Mach number combustion codes. The impact of these errors can be reduced by improving the accuracy of property lookups by implementing key state-functions as volume-integrated quantities in the flow solver. Optimized quadrature formula were developed to efficiently integrate the EOS on arbitrary 3-D mesh elements. The method of manufactured solutions (MMS) was used to formally verify the performance of the low-Mach number, variable-density flow solver used throughout 131

132

CHAPTER 8. CONCLUSIONS AND FURTHER WORK

this work. The MMS problems developed here mark the first application of this technique to quantify errors introduced by tabulated state-relations in combustion simulations. The verification tests were also used to examine the iterative, fractional-step algorithm applied in the code. The results underscore the importance of providing adequate resolution for tabulated chemistry and sufficient iterations to converge each time step. A tree-based tabulation strategy for combustion quantities was shown to provide increased accuracy and improved storage requirements compared to conventional Cartesian tables. The benefits demonstrated in the 2-D example implementation should be amplified in tables of higher-dimensionality. The cost of data retrieval is competitive with efficient (bisection) methods on non-uniform Cartesian grids. The accuracy of table representations was further increased by introducing coordinate transformations for the parameters in common flamelet models. An extension to the flamelet/progress variable (FPV) model was proposed as a way to include the effects of convective heat-loss on combustion chemistry. In addition to the mixture fraction and reaction progress variable, the new model uses the total enthalpy to provide a three-parameter representation of the thermochemistry. Predictions from the model were compared with the original FPV approach in a LES of a coaxial-jet combustor with isothermal walls. The new model was used to estimate the maximum and average values of the wall heat flux – quantities that are useful to make design decisions for combustion engineers. The cost of the new method relative to the original model, is essentially the solution of an additional scalar transport equation and higher-dimensional table lookups. The flamelet model used in this study obtained favorable comparison with the experiment by including differential diffusion effects characterized by non-unity Lewis numbers. This result is contrary to a body of established literature that suggests that differential diffusion is important in laminar and transitional flames, but is largely negligible in high-Reynolds number turbulent flames (see, for example, Drake et al., 1986; Meier et al., 2000; Pitsch, 2000). This contradiction likely stems from implications regarding the scalar dissipation rate in the physical-space flamelet formulation that was used in this work (see Section 2.4.4. In conventional flamelet models, the

8.2. RECOMMENDATIONS FOR FURTHER WORK

133

scalar dissipation rate appears as a diffusivity in mixture fraction space, thus, it affects mass transfer in the flamelet relative to the mixture fraction. It is possible that the current formulation achieves good results here in large measure because the application was a methane-fueled burner, and methane has a Lewis number that is close to unity (LeCH4 = 0.955). The applicability of the physical-space flamelet model may be in question for reactants whose Lewis number deviates substantially from one.

8.2

Recommendations for further work

The work presented here touches many aspects of turbulent combustion simulations. Much work remains, however, before the full potential of LES can be realized when applied to turbulent reacting flows. Some suggestions for model extensions and future research are discussed below. • Method of manufactured solutions: The method of manufactured solutions

(MMS) problems explored here provide a documented example of verification for a low-Mach number, variable-density flow solver. The results indicate that the accuracy and performance of the code is affected by factors such as: the size of the time step (or equivalently the CFL number), the number of outer iterations performed at each time step, and the residual tolerance for each component solver. The development of automatic techniques to determine the problemspecific, run-time parameters that optimize accuracy for a given computational cost would be a significant contribution. For example, the relative economics of taking a large time step with many outer iterations, versus a smaller time step and only a few outer iterations could be explored. New and existing MMS problems could be used to find practical solutions to these questions, and improve the performance of segregated schemes. Alternative solution techniques, such as coupled solvers, should also be considered. For example, it may be advantageous to initially use a segregated solver with a few outer iterations to produce an approximation that is within the “neighborhood” of the true solution, then switch to a coupled, Newton-type

134


method to drive the residuals to acceptable tolerances. MMS problems could be used to provide a benchmark to compare the accuracy-weighted cost of the solution. As interest increases in hypersonic combustion and re-entry problems, codeverification using the MMS can provide important assurance and direction during code-development. There are several challenges, however, to verification of compressible codes that admit discontinuous solutions. MMS problems have traditionally been constructed from smooth analytic functions, and the method has not been demonstrated in problems with shocks or other discontinuities. The applicability of MMS for these situations should be investigated. • Equation-of-state representation: Tabulation and interpolation of the equationof-state (EOS) in combustion simulations was shown to introduce spurious nu-

merical fluctuations into the solution. This could have potentially grave implications for subgrid-scale modeling in LES of combustion. For example, many non-premixed combustion models use the subgrid scalar variance and dissipation rate as input parameters. Both of these quantities provide information about mixing at the smallest scales of the flow. Given the numerical noise that can result from current implementations of tabulated chemistry, it may be difficult to ensure that subgrid-scale models for these quantities are not overwhelmed by unidentified and un-quantified errors. The impact of these errors should be further studied to develop guidelines for model development and implementation. As more complex physical processes are incorporated into combustion models, higher-dimensional sets of parameters are needed to describe them. The models discussed in this work employ two to four scalar parameters to specify the thermochemical state of the flow. Conventional Cartesian-based tabulation techniques are quickly become obsolete due to the competing needs of accuracy and memory. Novel storage and retrieval methods are needed to overcome these issues. The binary tree approximations introduced in Chapter 6 are one solution, but alternate methods should also be considered.

8.2. RECOMMENDATIONS FOR FURTHER WORK

135

The current implementation of tree-based chemistry could be improved in several ways. More accuracy could be provided by using higher-order reconstruction methods or least-squares regression during data retrieval. The added costs of these options, should be justified by considering the overall tree size and the complexity of retrievals. Tree structures could also be optimized to improve balancing, or to reduce retrieval time in frequently accessed parts of the table. For example, in well-mixed regions of the combustion system, most of the fluid parcels are characterized by a mixture fraction that is reflective of the overall equivalence ratio. Tree structures could be optimized to reduce lookup times in this region of the parameter space, at the expense of less-frequently accessed regions. Improved search methods could also be implemented that use a previous value as an initial guess and search up the tree until finding a relevant parent node, then navigating back down the tree to the correct leaf node. • LES of combustion with heat-loss: The flamelet/progress variable model with

generalized heat-loss (FPV-GHL) that was introduced here represents a critical starting point for the inclusion of complex heat-transfer and its effect on combustion chemistry. The model is by no means complete in its present form, however, and could be improved in several ways. In the FPV-GHL model, heat-losses are parameterized using an ad hoc specification of the enthalpy. The reference conditions used to scale the enthalpy, had and href , are reasonable choices, but others are possible and alternatives should be examined. The FPV-GHL model was implemented in a LES of the coaxial-jet flames of Spadaccini et al. (1976) as a demonstration of its feasibility. This experiment provided a useful preliminary test, but it is not an ideal case for quantifying heat-transfer effects. The model should be applied in an experiment that has direct measurements of the heat flux, and where convective heat transfer has a more pronounced role. Configurations such as an impinging flame, a recirculating burner, or an object embedded in a fire are all good candidates, but detailed measurements in these configurations with non-premixed reactants are difficult to find.

136


The FPV-GHL model could also be augmented with a more realistic model for radiation. In this work, radiative losses were included in the optically thin limit. A more detailed model, including the effects of absorption and geometric view-factors, is necessary to capture complex radiation effects. This would be necessary for applications in sooting flames, such as pool fires or diesel engines. Supplementary models could also be included to describe other slow chemical processes including NOx formation. Application of flamelet models to LES currently requires an assumed PDF to account for subgrid fluctuations of chemistry variables. The beta-distribution provides a good approximation for conserved scalars (Cook & Riley, 1994; Jiménez et al., 1997), but there has been little progress in PDF-modeling for reactive scalars. A notable exception is the contribution by Ihme & Pitsch (2008a,b) using the statistically most likely distribution (SMLD) to model the PDFs of reactive scalars. Work in this vein should be continued and expanded to consider its feasibility for higher-dimensional models.

Appendix A Symmetric quadrature rules for tetrahedra High-accuracy numerical integration is essential to many areas of applied mechanics including fluid dynamics, combustion, and aero-acoustics. Many of these applications are trending towards a heavier reliance on unstructured grids for the solution of the relevant conservation laws. Tetrahedral or tet-dominant grids are common in engineering calculations due to their relative ease of construction and flexibility. In addition, arbitrary polyhedral elements can always be tessellated into tetrahedra, lending additional utility to tet-based quadrature schemes for complex elements. This appendix details the derivation of a family of optimized quadrature rules for integration over tetrahedral volumes.

A.1

Background

Determining quadrature points for volume integration has been the subject of a vast body of literature. Much of the research has considered quadrature rules for the triangle (Lyness & Jespersen, 1975; Dunavant, 1985; Lyness & Cools, 1994; Hesthaven, 1998; Taylor et al., 2001; Wandzura & Xiao, 2003). There has been more limited work on rule development for the tetrahedron (Stroud, 1969; Keast, 1986). Integration rules for higher-dimensional simplices have been also been considered (Silvester, 137

138 APPENDIX A. SYMMETRIC QUADRATURE RULES FOR TETRAHEDRA

1970; Grundmann & Möller, 1978; Lyness & Genz, 1980). Many key studies are summarized in the classic works of Stroud (1971) and Engles (1980), the reviews of Cools & Rabinowitz (1993) and Cools (1999), and the recently updated text of Davis & Rabinowitz (2007).

In a general approach, accurate quadrature points can be generated for many volumes by the recursive application of 1-D Gaussian quadrature, referred to as the product Gaussian rule (see Stroud, 1971). When the product Gaussian rule is applied to tetrahedra, however, it results in an unpleasantly asymmetric distribution of quadrature points with inefficient clustering near one of the vertices (see Figure A.1). The product Gaussian rule implicitly assumes an underlying structured cubic grid for the point locations. This grid samples a hexahedral space more or less uniformly, but maps awkwardly to tetrahedral volumes. A more natural choice for tetrahedra is the cubic close-packed (CCP) arrangement, alternatively referred to as the face-centered cubic (FCC) structure. Close-packing is the configuration obtained by arranging spheres of equal size so as to achieve the maximum possible number-density. In a close-packed structure, each sphere in the interior of the lattice symmetrically contacts its 12 nearest neighbors.

In the present study we describe a method for developing quadrature points for tetrahedra based on parameterizing the point locations in terms of an underlying CCP grid, and then optimizing the precise point locations and weights to reduce the truncation error in the quadrature approximation. Such an approach results in a family of symmetric rules for tetrahedral integration where the number of integration points progresses as Np = 1, 4, 10, 20, 35, 56, etc. (see Figure A.2). For bookkeeping purposes, it is useful to express Np algebraically as a function of the number of stacked-layers in the CCP structure, Nq . Np and Nq are related by:

Np = Nq (Nq + 1)(Nq + 2)/6.

(A.1)

139

A.2. METHOD

Figure A.1: 4-point product Gaussian rule. The rendered volume of the spheres indicates the relative weighting that each point receives in the quadrature rule. Note the asymmetric clustering of the points along the lower left edge of the tetrahedron.

A.2

Method

The integration of the function f (x) over the tetrahedron Ω with volume VΩ is defined as

3

f (x) dV = VΩ

Ω

, Np $

wi f (xi ) + O(δ n )

i=1

-

(A.2)

where δ is a length scale associated with the tetrahedron (e.g., the longest edge length) and n is the order of the leading error term of the integration. The integration points (or quadrature points) xi are defined in terms of the four vertices of the tetrahedron X 1 , X 2 , X 3 , X 4 as xi =

4 $

ai,j X j .

(A.3)

j=1

An integration scheme of this sort should have the following desirable properties (adapted from Vavasis, 1998): 1. The scheme is optimal in some sense, e.g., minimizing truncation error. 2. It is symmetric, i.e., the scheme is independent of vertex ordering or the rule is invariant under affine maps of the tetrahedron to itself.


Figure A.2: Cubic close-packed structures. Top row (left to right): 1-, 4-, and 10-point structures . Bottom row (left to right): 20-, 35-, and 56-point structures.

141

A.2. METHOD

3. The weights are all positive: wi ≥ 0. 4. The quadrature points are all interior to the tetrahedron: ai,j ≥ 0. The integration schemes are developed by first parameterizing the integration point locations in terms of the vectors of the unit equilateral tetrahedron centered at the origin and scaled by δ X1 = X2 = X3 = X4 =

,

√ √ 1 3 6 δ − ,− ,− 2 6 12 , √ √ 3 6 1 ,− ,− δ 2 6 12 , √ √ 3 6 δ 0, ,− 3 12 , √ 6 . δ 0, 0, 4

(A.4)

In general, the parameters used to describe the system indicate the quadrature weight that a particular point will receive in the final formula and the specific location of that point as a linear combination of the vertex vectors in equation (A.4). The number of free-variables in the parameterization depends on the number of points in each CCP configuration, but it is reduced somewhat by exploiting the natural symmetry of the tetrahedron. In other words, all points that have corresponding spatial symmetries are assigned the same weighting and positioning relationships. A trivial example is the 1-point quadrature formula, which is entirely specified by symmetry considera; tions and the normalization constraint that wi = 1. By inspection, the optimal

solution for the 1-point rule is that w1 = 1 and all of the a1,j = 1/4. There are no other possible combinations that honor the symmetry of the system. Similarly, the optimal 4-point rule is solved by applying the normalization condition and determining the value of a single parameter that specifies the “best” location of the quadrature points along the vector from the centroid of the tetrahedron to one of its vertices. Specifying this parameter fixes the relative distribution of interpolationweighting that each point receives from the vertex closest to it versus the other three


vertices in the tetrahedron. The number of parameters that need to be optimized for each rule grows as the quadrature formulae become more complex. For the rules presented here, the degrees-of-freedom in the parameterizations (after accounting for the normalization requirement) are: 1-point rule – 0, 4-point rule – 1, 10-point rule – 3, 20-point rule – 6, 35-point rule – 10, and 56-point rule – 15 degrees-of-freedom. The free-variables in the parameterization are determined by considering the integral in equation (A.2) over the tetrahedron given in equation (A.4). This amounts to F (x, y, z) =

3

z1

z0

3

y1

y0

3

x1

f (x, y, z) dx dy dz

(A.5)

x0

where the limits of integration are x0 = x1 = y0 = y1 = z0 = z1 =

6 √ 4 √ , √ 3 2 δ 6 − −z −y 3 2 4 6 √ 4 √ , √ 3 2 δ 6 −z −y 3 2 4 √ , √ 2 δ 6 −z − 4 4 √ , √ 2 δ 6 −z 2 4 √ δ 6 − 12 √ δ 6 . 4

(A.6)

The nth-order Taylor expansion of f (x, y, z) about the centroid of the tetrahedron is then formed as: 0 0 0 0 0 0 fT (x, y, z; f 0, fx0 , fy0 , fz0 , fxx , fxy , fxz , fyy , fyz , fzz , ...) = f (x, y, z) − O(δ n )

(A.7)

where f 0 , fx0 , fy0 , fz0 , etc. are partial derivatives of f evaluated at the centroid of the tetrahedron. Equation (A.7) can be integrated exactly up to order n + 3 by replacing

143

A.2. METHOD

f with fT in equation (A.5): FT (δ; f

0

0 0 0 0 0 0 , fx0, fy0 , fz0 , fxx , fxy , fxz , fyy , fyz , fzz , ...)

=

3

fT (x, y, z) dx dy dz

(A.8)

Ω

The error ϵ associated with a parameterized quadrature approximation to F (x, y, z) is

Np

FT (δ) $ ϵ(δ, φ) = − wi(δ, φ)fT (xi (δ, φ), yi (δ, φ), zi (δ, φ)) ∼ O(δ n ) VΩ i=1

(A.9)

where the φ are the yet-to-be-determined factors in the parameterization of the quadrature rule’s weights and point-locations. Specific values of φ are determined by a constrained optimization of the quadrature error ϵ in equation (A.9). Optimization constraints are defined by collecting all of the terms that multiply a given power of δ and extracting independent relationships from amongst the coefficients. The resulting expressions, given in terms of φ, are set equal to zero to produce a series of nonlinear equations that can be solved for the weights and positions of the quadrature points. When there is not a sufficient number of parameters φ to completely eliminate the truncation error at the highest level, the error of the highest-order term is minimized in a least-squares sense by assuming that the partial derivatives are all of equal magnitude. In this work, a sequential quadratic programming (SQP) method was used to determine the optimal values for φ. The optimization routine solves a quadratic programming subproblem at each iteration with Hessian updates using the BroydenFletcher-Goldfarb-Shanno (BFGS) method (Fletcher & Powell, 1963; Goldfarb, 1970). A line search algorithm with a merit function similar to that proposed by Han (1977) and Powell (1978a,b) was used in the optimization. The quadratic programming subproblem was solved using an active-set strategy as described in Gill et al. (1981). Once the optimal solutions were obtained, the point locations and weights were assembled and stored in a computer library for efficient implementation in numerical applications. The optimized rules are shown in Figure A.3, with numerical values for the interpolation and quadrature weights given in Table A.1. Inspection of the ai,j in Table A.1 reveals that all of the points are located strictly inside the tetrahedron,


Figure A.3: Tetrahedral integration quadrature point locations and weights. The rendered volume of the spheres indicates the relative weighting that each point receives in the quadrature rule. Top row (left to right): 1-, 4-, and 10-point rules. Bottom row (left to right): 20-, 35-, and 56-point rules.

with no points on edges or faces. The theoretical orders-of-accuracy of the formulae are: 1-point rule – O(δ 2), 4-point rule – O(δ 3), 10-point rule – O(δ 4), 20-point rule – O(δ 6 ), 35-point rule – O(δ 7 ), and 56-point rule – O(δ 9 ).

A.3

Results

The order-of-accuracy of each of the quadrature rules was checked by individually integrating all possible monomial terms xi y j z k of order i + j + k ≤ m over the unit tetrahedron (see Figure A.4). For each rule the quadrature approximation was

145

A.3. RESULTS

Table A.1: Tetrahedral integration quadrature point locations and weights Nq 1 2

3

4

5

point i 1 1 2 3 4 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

ai,1 0.2500000000000000 0.5854101966249680 0.1381966011250110 0.1381966011250110 0.1381966011250110 0.7784952948213300 0.0738349017262234 0.0738349017262234 0.0738349017262234 0.4062443438840510 0.4062443438840510 0.4062443438840510 0.0937556561159491 0.0937556561159491 0.0937556561159491 0.9029422158182680 0.0323525947272439 0.0323525947272439 0.0323525947272439 0.2626825838877790 0.6165965330619370 0.2626825838877790 0.6165965330619370 0.2626825838877790 0.6165965330619370 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.3097693042728620 0.3097693042728620 0.3097693042728620 0.0706920871814129 0.9197896733368800 0.0267367755543735 0.0267367755543735 0.0267367755543735 0.1740356302468940 0.7477598884818090 0.1740356302468940 0.7477598884818090 0.1740356302468940 0.7477598884818090 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.4547545999844830 0.4547545999844830 0.4547545999844830 0.0452454000155172 0.0452454000155172 0.0452454000155172 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.0504792790607720 0.0504792790607720 0.0504792790607720 0.2500000000000000

ai,2 0.2500000000000000 0.1381966011250110 0.5854101966249680 0.1381966011250110 0.1381966011250110 0.0738349017262234 0.7784952948213300 0.0738349017262234 0.0738349017262234 0.4062443438840510 0.0937556561159491 0.0937556561159491 0.4062443438840510 0.4062443438840510 0.0937556561159491 0.0323525947272439 0.9029422158182680 0.0323525947272439 0.0323525947272439 0.6165965330619370 0.2626825838877790 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.2626825838877790 0.6165965330619370 0.2626825838877790 0.6165965330619370 0.0603604415251421 0.0603604415251421 0.3097693042728620 0.3097693042728620 0.0706920871814129 0.3097693042728620 0.0267367755543735 0.9197896733368800 0.0267367755543735 0.0267367755543735 0.7477598884818090 0.1740356302468940 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.1740356302468940 0.7477598884818090 0.1740356302468940 0.7477598884818090 0.0391022406356488 0.0391022406356488 0.4547545999844830 0.0452454000155172 0.0452454000155172 0.4547545999844830 0.4547545999844830 0.0452454000155172 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.0504792790607720 0.0504792790607720 0.0504792790607720 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.2500000000000000

ai,3 0.2500000000000000 0.1381966011250110 0.1381966011250110 0.5854101966249680 0.1381966011250110 0.0738349017262234 0.0738349017262234 0.7784952948213300 0.0738349017262234 0.0937556561159491 0.4062443438840510 0.0937556561159491 0.4062443438840510 0.0937556561159491 0.4062443438840510 0.0323525947272439 0.0323525947272439 0.9029422158182680 0.0323525947272439 0.0603604415251421 0.0603604415251421 0.6165965330619370 0.2626825838877790 0.0603604415251421 0.0603604415251421 0.6165965330619370 0.2626825838877790 0.0603604415251421 0.0603604415251421 0.2626825838877790 0.6165965330619370 0.3097693042728620 0.0706920871814129 0.3097693042728620 0.3097693042728620 0.0267367755543735 0.0267367755543735 0.9197896733368800 0.0267367755543735 0.0391022406356488 0.0391022406356488 0.7477598884818090 0.1740356302468940 0.0391022406356488 0.0391022406356488 0.7477598884818090 0.1740356302468940 0.0391022406356488 0.0391022406356488 0.1740356302468940 0.7477598884818090 0.0452454000155172 0.4547545999844830 0.0452454000155172 0.4547545999844830 0.0452454000155172 0.4547545999844830 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.0504792790607720 0.0504792790607720 0.0504792790607720 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2500000000000000

ai,4 0.2500000000000000 0.1381966011250110 0.1381966011250110 0.1381966011250110 0.5854101966249680 0.0738349017262234 0.0738349017262234 0.0738349017262234 0.7784952948213300 0.0937556561159491 0.0937556561159491 0.4062443438840510 0.0937556561159491 0.4062443438840510 0.4062443438840510 0.0323525947272439 0.0323525947272439 0.0323525947272439 0.9029422158182680 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.0603604415251421 0.6165965330619370 0.2626825838877790 0.0603604415251421 0.0603604415251421 0.6165965330619370 0.2626825838877790 0.6165965330619370 0.2626825838877790 0.0706920871814129 0.3097693042728620 0.3097693042728620 0.3097693042728620 0.0267367755543735 0.0267367755543735 0.0267367755543735 0.9197896733368800 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.0391022406356488 0.7477598884818090 0.1740356302468940 0.0391022406356488 0.0391022406356488 0.7477598884818090 0.1740356302468940 0.7477598884818090 0.1740356302468940 0.0452454000155172 0.0452454000155172 0.4547545999844830 0.0452454000155172 0.4547545999844830 0.4547545999844830 0.0504792790607720 0.0504792790607720 0.0504792790607720 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2232010379623150 0.2232010379623150 0.5031186450145980 0.2500000000000000

wi 1.0000000000000000 0.2500000000000000 0.2500000000000000 0.2500000000000000 0.2500000000000000 0.0476331348432089 0.0476331348432089 0.0476331348432089 0.0476331348432089 0.1349112434378610 0.1349112434378610 0.1349112434378610 0.1349112434378610 0.1349112434378610 0.1349112434378610 0.0070670747944695 0.0070670747944695 0.0070670747944695 0.0070670747944695 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.0469986689718877 0.1019369182898680 0.1019369182898680 0.1019369182898680 0.1019369182898680 0.0021900463965388 0.0021900463965388 0.0021900463965388 0.0021900463965388 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0143395670177665 0.0250305395686746 0.0250305395686746 0.0250305395686746 0.0250305395686746 0.0250305395686746 0.0250305395686746 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0479839333057554 0.0931745731195340


Table A.1: Tetrahedral integration quadrature point locations and weights (continued ) Nq 6

point i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

ai,1 0.9551438045408220 0.0149520651530592 0.0149520651530592 0.0149520651530592 0.7799760084415400 0.1518319491659370 0.7799760084415400 0.1518319491659370 0.7799760084415400 0.1518319491659370 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.3549340560639790 0.5526556431060170 0.3549340560639790 0.5526556431060170 0.3549340560639790 0.5526556431060170 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.0055147549744775 0.0055147549744775 0.0055147549744775 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.0992057202494530 0.0992057202494530 0.0992057202494530 0.5965649956210170 0.1344783347929940 0.1344783347929940 0.1344783347929940

ai,2 0.0149520651530592 0.9551438045408220 0.0149520651530592 0.0149520651530592 0.1518319491659370 0.7799760084415400 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.7799760084415400 0.1518319491659370 0.7799760084415400 0.1518319491659370 0.0340960211962615 0.0340960211962615 0.5526556431060170 0.3549340560639790 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.3549340560639790 0.5526556431060170 0.3549340560639790 0.5526556431060170 0.0462051504150017 0.0462051504150017 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.0055147549744775 0.0055147549744775 0.0055147549744775 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.0992057202494530 0.0992057202494530 0.0992057202494530 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1344783347929940 0.5965649956210170 0.1344783347929940 0.1344783347929940

ai,3 0.0149520651530592 0.0149520651530592 0.9551438045408220 0.0149520651530592 0.0340960211962615 0.0340960211962615 0.1518319491659370 0.7799760084415400 0.0340960211962615 0.0340960211962615 0.1518319491659370 0.7799760084415400 0.0340960211962615 0.0340960211962615 0.7799760084415400 0.1518319491659370 0.0462051504150017 0.0462051504150017 0.5526556431060170 0.3549340560639790 0.0462051504150017 0.0462051504150017 0.5526556431060170 0.3549340560639790 0.0462051504150017 0.0462051504150017 0.3549340560639790 0.5526556431060170 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.0055147549744775 0.0055147549744775 0.0055147549744775 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.0992057202494530 0.0992057202494530 0.0992057202494530 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.1344783347929940 0.1344783347929940 0.5965649956210170 0.1344783347929940

ai,4 0.0149520651530592 0.0149520651530592 0.0149520651530592 0.9551438045408220 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.0340960211962615 0.1518319491659370 0.7799760084415400 0.0340960211962615 0.0340960211962615 0.1518319491659370 0.7799760084415400 0.1518319491659370 0.7799760084415400 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.0462051504150017 0.5526556431060170 0.3549340560639790 0.0462051504150017 0.0462051504150017 0.5526556431060170 0.3549340560639790 0.5526556431060170 0.3549340560639790 0.0055147549744775 0.0055147549744775 0.0055147549744775 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.2281904610687610 0.2281904610687610 0.5381043228880020 0.0992057202494530 0.0992057202494530 0.0992057202494530 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.3523052600879940 0.3523052600879940 0.1961837595745600 0.1344783347929940 0.1344783347929940 0.1344783347929940 0.5965649956210170

wi 0.0010373112336140 0.0010373112336140 0.0010373112336140 0.0010373112336140 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0096016645399480 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0164493976798232 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0153747766513310 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0293520118375230 0.0366291366405108 0.0366291366405108 0.0366291366405108 0.0366291366405108

147

A.3. RESULTS

Figure A.4: Unit tetrahedron used in the monomial integration test.

compared to the exact value, and m was increased until a non-zero (to machineprecision) result was obtained. The first non-zero result indicates the order of the leading error term in the quadrature rule, i.e., the O(δ p ) term from equation (A.2). The results are presented in Figure A.5 which shows the error magnitude scaled by the exact value of the integral 3

1 0

3

0

1−x

3

1−x−y

xi y j z k dz dy dx = 0

i! j! k! . (i + j + k + 3)!

(A.10)

Figure A.5 confirms the validity of the theoretical orders-of-accuracy stated in Section A.2. The quadrature rules were additionally verified by integrating a test function over the unit cube on a series of grids comprising successively smaller tetrahedra. The grids were constructed by first creating a uniform cartesian grid and decomposing each hexahedral volume into 24 equally-sized tetrahedra. Each tetrahedron includes the centroid, one face-center, and two corner-vertices of its parent hexahedron. Grids consisting of 24, 192, 648, 3000, 8232, 24000, and 81000 tetrahedra were used for the


error magnitude

10

10

0

-6

-12

10

1-point rule 4-point rule 10-point rule 20-point rule 35-point rule 56-point rule

-18

10

12 3

4

5

6

7 8 monomial order, m

9

Figure A.5: Error from integrating individual monomials over the unit tetrahedron.

149

A.3. RESULTS

test. The function that was investigated is the polynomial composed of all possible monomials with order i + j + k ≤ m: gm (x, y, z) =

;m ;m−i ;m−i−j i=0

(i + 1)(j + 1)(k + 1) xi y j z z . ;m ;i+1 i=0 j=0 j

k=0

j=0

(A.11)

The multiplicative coefficients of the monomials in equation (A.11) have been chosen such that gm has the property Gm =

3

1 0

3

0

1

3

1

gm (x, y, z) dx dy dz = 1.

(A.12)

0

A value of m = 30 was selected for a highly nonlinear and challenging test problem. The corresponding test problem g30 contains 5456 individual monomial terms. Values for the quadrature approximations are presented in Table A.2 and Figure A.6. The results again verify the theoretical convergence properties of the quadrature rules. Several of the rules even exhibit convergence at rates greater than their theoretical order-of-accuracy. This is possibly the result of advantageous error-cancellation on the nearly-uniform grids used in this study. The quadrature rules also exhibit “exponential” or “spectral” convergence, where the error scales with the total number of approximation points as ϵ ∼ O[exp(−qNpr )]

(A.13)

where q is a constant and r > 0 (see Boyd, 2001). This property is demonstrated in Figure A.7, where each dotted line connects quadrature results on a given grid using rules of increasing order-of-accuracy. The exponential factor r in equation (A.13) asymptotically approaches r ≈ 0.41 as the grid is refined. The quadrature formulae can also be used to approximate non-polynomial functions. Consider the “Genz product peak,” a problem designed to test multi-dimensional


Table A.2: Integration of g30 (x, y, z) over the unit cube. g30 is the polynomial that includes all possible monomial terms xi y j z k of order i + j + k ≤ 30. Nq 1

2

3

4

5

6

ntet 24 192 648 3000 8232 24000 81000 24 192 648 3000 8232 24000 81000 24 192 648 3000 8232 24000 81000 24 192 648 3000 8232 24000 81000 24 192 648 3000 8232 24000 81000 24 192 648 3000 8232 24000 81000

δ 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02 3.4668e-01 1.7334e-01 1.1556e-01 6.9336e-02 4.9526e-02 3.4668e-02 2.3112e-02

9 INp ≈ g(x) dx 0.172300573672941 0.575481829109966 0.774464088656016 0.910441641402659 0.952934024342237 0.976545166092715 0.989476215733917 0.668365165445412 0.954690978097844 0.990156897657781 0.998732743497481 0.999674385708818 0.999922494864099 0.999984765990626 0.978987383577130 1.001332510244897 1.000364525672625 1.000039223544953 1.000008391376955 1.000001697624973 1.000000296143472 1.012787093156747 1.000287729675973 1.000020417884059 1.000000599350408 1.000000058559376 1.000000005322641 1.000000000386502 0.997796612987978 0.999968766219781 0.999998269401933 0.999999962633169 0.999999997233612 0.999999999831991 0.999999999993257 1.000441298283147 1.000002034599156 1.000000060762914 1.000000000536598 1.000000000020979 1.000000000000637 1.000000000000003

error -8.2770e-01 -4.2452e-01 -2.2554e-01 -8.9558e-02 -4.7066e-02 -2.3455e-02 -1.0524e-02 -3.3163e-01 -4.5309e-02 -9.8431e-03 -1.2673e-03 -3.2561e-04 -7.7505e-05 -1.5234e-05 -2.1013e-02 1.3325e-03 3.6453e-04 3.9224e-05 8.3914e-06 1.6976e-06 2.9614e-07 1.2787e-02 2.8773e-04 2.0418e-05 5.9935e-07 5.8559e-08 5.3226e-09 3.8650e-10 -2.2034e-03 -3.1234e-05 -1.7306e-06 -3.7367e-08 -2.7664e-09 -1.6801e-10 -6.7434e-12 4.4130e-04 2.0346e-06 6.0763e-08 5.3660e-10 2.0979e-11 6.3660e-13 3.1086e-15

order — 0.963 1.560 1.808 1.912 1.953 1.977 — 2.872 3.765 4.013 4.039 4.024 4.012 — 3.979 3.197 4.364 4.583 4.480 4.307 — 5.474 6.525 6.907 6.912 6.723 6.468 — 6.140 7.135 7.508 7.737 7.854 7.930 — 7.761 8.659 9.259 9.634 9.799 13.13

151

A.3. RESULTS

10

0

error magnitude

2nd order 10

-5

3rd order 4th order

10

-10


6th order 7th order

10

-15

9th order 10

-2

-1

10 tetrahedron length scale, δ

Figure A.6: Order-of-accuracy of tetrahedral integration rules.


error magnitude

10

10

0

24-tet grid

-5

192-tet grid 648-tet grid

10

10


-10

-15

10

0

3000-tet grid 8232-tet grid 24000-tet grid 81000-tet grid 2

4

10 10 10 total number of points, Np × ntet

6

10

Figure A.7: Exponential convergence of tetrahedral integration rules.

8

153

A.4. SUMMARY

integration routines (Genz, 1987)

gprod

,n -−1 D : = (c2j + (xj − x0j )2 ) .

(A.14)

j=1

Integration over the unit cube in three dimensions gives Gprod =

3

1

0 3 :

3

0

1

3

1

gprod (x1 , x2 , x3 ) dx1 dx2 dx3

(A.15)

0

< % & % 0 &= 1 − x0j xj 1 arctan + arctan . = c cj cj j=1 j The convergence of the quadrature rules applied to equation (A.14) is shown in Figure A.8. In this problem c1 = c2 = c3 = 0.2 and x01 = x02 = x03 = 0.5. The performance of the numerical integration is more sporadic compared with the polynomial quadrature in Figure A.6. Even so, as the grid is refined, most of the quadrature rules display orders-of-accuracy that exceed their expected theoretical values.

A.4

Summary

A family of symmetric quadrature formulae was developed for efficient integration over tetrahedral volumes. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement in order to preserve the innate symmetry of tetrahedra. Optimal point configurations were determined for 1-, 4-, 10-, 20-, 35-, and 56-point quadrature rules. This progression of points produces quadrature formula with leading truncation error ranging from O(δ 2 ) to O(δ 9 ). Higher-order rules are theoretically possible, however, the polynomial systems that must be optimized grow increasingly complex and ill-conditioned as the number of quadrature points increases. Monomial and polynomial test functions were integrated in order to verify the validity of the new quadrature rules in numerical applications. The rules exhibit their expected orders-of-accuracy for the test problems considered here, and displayed exponential convergence properties over the range of conditions tested.


10

0

error magnitude

2nd order 10

-5

3rd order 4th order

10

-10


6th order 7th order

10

-15

9th order 10

-2

-1

10 tetrahedron length scale, δ

Figure A.8: Order-of-accuracy of tetrahedral integration rules.

References Aghalayam, P., Bui, P.-A. & Vlachos, D. G. 1998 The role of radical wall quenching in flame stability and wall heat flux: hydrogen-air mixtures. Combust. Theory Modelling 2, 515–530. Ahmad, T. & Faeth, G. M. 1978 An investigation of the laminar overfire region along upright surfaces. J. Heat Trans. 200, 112–119. Ahmad, T. & Faeth, G. M. 1979 Turbulent wall fires. Proc. Combust. Inst. 17, 1149– 1160. Akselvoll, K. & Moin, P. 1996 Large-eddy simulation of turbulent confined coannular jets. J. Fluid Mech. 315, 387–411. Alshaalan, T. M. & Rutland, C. J. 1998 Turbulence, scalar transport, and reaction rates in flame-wall interactions. Proc. Combust. Inst. 27, 793–799. Babrauskas, V. 1995 Specimen heat fluxes for bench-scale heat release rate testing. Fire & Materials 19, 243–252. Back, G., Beyler, C., DiNenno, P. & Tatem, P. 1994 Wall incident heat flux distributions resulting from an adjacent fire. In Fourth Int. Symp. Proc. (ed. T. Kashiwagi), pp. 241–252. International Association for Fire Safety Science, Ottawa. Bardina, J., Ferziger, J. H. & Reynolds, W. C. 1980 Improved subgrid scale models for large-eddy simulation. Paper 80-1357. AIAA. Barlow, R. S., Karpetis, A. N., Frank, J. H. & Chen, J.-Y. 2001 Scalar profiles and NO formation in laminar opposed-flow partially premixed methane/air flames. Combust. Flame 127 (3), 2102–2118.

155

156

REFERENCES

Barlow, R. S., Smith, N. S. A., Chen, J.-Y. & Bilger, R. W. 1999 Nitric oxide formation in dilute hydrogen jet flames: isolation of the effects of radiation and turbulence-chemistry submodels. Combust. Flame 117 (1-2), 4–31. Baukal, C. E. 2000 Heat Transfer in Industrial Combustion. Boca Raton, FL: CRC Press. Baukal, C. E. & Gebhart, B. 1995a A review of flame impingement heat transfer studies Part 1: Experimental conditions. Combust. Sci. Tech. 104 (4-6), 339–357. Baukal, C. E. & Gebhart, B. 1995b A review of flame impingement heat transfer studies Part 2: Measurements. Combust. Sci. Tech. 104 (4-6), 359–385. Baukal, C. E. & Gebhart, B. 1998 Heat transfer from oxygen-enhanced/natural gas flames impinging normal to a plane surface. Exp. Therm. Fluid Sci. 16 (3), 247–259. Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–R32. Bergmann, V., Meier, W., Wolff, D. & Stricker, W. 1998 Application of spontaneous Raman and Rayleigh scattering and 2D LIF for the characterization of a turbulent CH4 /H2 /N2 jet diffusion flame. Appl. Phys. B 66 (4), 489–502. Berkooz, G. 1993 An observation on probability density equations, or, when do simulations reproduce statistics? Nonlinearity 7, 313–328. Bilger, R. W. 1976 The structure of diffusion flames. Combust. Sci. Tech. 13 (1-6), 155– 170. Bilger, R. W. 1980 Turbulent flows with nonpremixed reactants. In Turbulent Reacting Flows (ed. P. A. Libby & F. A. Williams), pp. 65–113. Berlin: Springer-Verlag. Bilger, R. W. 1988 The structure of nonpremixed flames. Proc. Combust. Inst. 22, 475– 488. Bilger, R. W. 1993 Conditional moment closure for turbulent reacting flow. Phys. Fluids A 5 (2), 436–444. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena, 2nd edn. New York, NY: John Wiley & Sons.

REFERENCES

157

Blasco, J. A., Fueyo, N., Dopazo, C. & Ballester, J. 1998 Modelling the temporal evolution of a reduced combustion chemical system with an artificial neural network. Combust. Flame 113, 38–52. Bond, R. B., Ober, C. C. & Knupp, P. M. 2006 A manufactured solution for verifying CFD boundary conditions, part III. In 36th AIAA Fluid Dynamics Conference, , vol. 3, pp. 1966–1982. San Francisco, CA. Bowman, C. T., Hanson, R. K., Davidson, D. F., Gardiner, W. C., Lissianski, V., Smith, G. P., Golden, D. M., Frenklach, M. & Goldenberg, M. 1995 GRIMech 2.11. Available from http://www.me.berkeley.edu/gri_mech/, accessed March 2006. Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods, 2nd edn. Mineola, NY: Dover Publications. Breiman, L., Friedman, J. H., Olshen, R. A. & Stone, C. J. 1984 Classification and Regression Trees. Belmont, CA: Wadsworth International. Bruel, P., Rogg, B. & Bray, K. N. C. 1990 On auto-ignition in laminar and turbulent non-premixed systems. Proc. Combust. Inst. 23, 759–766. Bruneaux, G., Akselvoll, K., Poinsot, T. & Ferziger, J. H. 1996 Flame-wall interaction simulation in a turbulent channel flow. Combust. Flame 107, 27–44. Bruneaux, G., Poinsot, T. & Ferziger, J. H. 1997 Premixed flame-wall interaction in a turbulent channel flow: budget for the flame surface density evolution equation and modeling. J. Fluid Mech. 349, 191–219. Brunner, T. A. 2006 Development of a grey nonlinear thermal radiation diffusion verification problem. Transactions of the American Nuclear Society 95, 876–878. ´ly, G. & Cook, A. W. 1998 Investigation of de Bruyn Kops, S. M., Riley, J. J., Kosa modeling for non-premixed turbulent combustion. Flow, Turb. Combust. 60 (1), 105–122. Burggraf, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113–151.

158

REFERENCES

Burke, S. P. & Schumann, T. E. W. 1928 Diffusion flames. Ind. Eng. Chem. 20 (10), 998–1004. Bushe, W. K. & Steiner, H. 1999 Conditional moment closure for large eddy simulation of nonpremixed turbulent reacting flows. Phys. Fluids 11 (7), 1896–1906. Chen, J.-Y., Kollmann, W. & Dibble, R. W. 1989 PDF modeling of turbulent nonpremixed methane jet flames. Combust. Sci. Tech. 64 (4-6), 315–346. Chen, Y.-C. & Mansour, M. S. 1997 Measurements of scalar dissipation in turbulent hydrogen diffusion flames and some implications on combustion modeling. Combust. Sci. Tech. 126, 291–313. Chorin, A. J., Kast, A. P. & Kupferman, R. 1998 Optimal prediction of underresolved dynamics. Proc. Natl. Acad. Sci. USA 95, 4094–4098. Christo, F. C., Masri, A. R. & Nebot, E. M. 1996 Artificial neural network implementation of chemistry with PDF simulation of H2 /CO2 flames. Combust. Flame 106, 406–427. Coelho, P. J., Teerling, O. J. & Roekaerts, D. 2002 Spectral radiative effects and turbulence-radiation-interaction in Sandia flame D. In TNF6: Sixth International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames (ed. R. S. Barlow, S. B. Pope, A. R. Masri & J. C. Oefelein), pp. 31–32. Sapporo, Japan. Cook, A. W. & Riley, J. J. 1994 A subgrid model for equilibrium chemistry in turbulent flows. Phys. Fluids 6 (8), 2868–2870. Cook, A. W. & Riley, J. J. 1998 Subgrid-scale modeling for turbulent reacting flows. Combust. Flame 112 (4), 593–606. ´ly, G. 1997 A laminar flamelet approach to subgridCook, A. W., Riley, J. J. & Kosa scale chemistry in turbulent flows. Combust. Flame 109 (3), 332–341. Cools, R. 1999 Monomial cubature rules since “Stroud”: a compilation — part 2. J. Comput. Appl. Math. 112, 21–27. Cools, R. & Rabinowitz, P. 1993 Monomial cubature rules since “Stroud”: a compilation. J. Comput. Appl. Math. 48, 309–326.

REFERENCES

159

Dabireau, F., Cuenot, B., Vermorel, O. & Poinsot, T. 2003 Interaction of H2 + O2 flames with inert walls. Combust. Flame 135, 123–133. Davis, P. J. & Rabinowitz, P. 2007 Methods of Numerical Integration, 2nd edn. Mineola, NY: Dover Publications. Deardorff, J. W. 1970 A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453–480. Deardorff, J. W. 1974 Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer. Boundary-Layer Meteorol. 7, 81–106. DesJardin, P. E. & Frankel, S. H. 1997 Linear eddy modeling of nonequilibrium turbulent reacting flows with nonpremixed reactants. Combust. Flame 109 (3), 471–487. Domaradzki, J. A. & Saiki, E. M. 1997 A subgrid-scale model based on the estimation of unresolved scales of turbulence. Phys. Fluids 9, 1–17. Domino, S. P., Wagner, G., Luketa-Hanlin, A., Black, A. & Sutherland, J. 2007 Verification for multi-mechanics applications. In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Honolulu, HI. Dopazo, C. 1994 Recent developments in PDF methods. In Turbulent Reacting Flows (ed. P. A. Libby & F. A. Williams), pp. 375–474. London: Academic Press. Drake, M. C., Lapp, M., Penney, C. M., Warshaw, S. & Gerhold, B. W. 1981 Measurements of temperature and concentration fluctuations in turbulent diffusion flames using pulsed Raman spectroscopy. Proc. Combust. Inst. 18, 1521–1531. Drake, M. C., Pitz, R. W. & Lapp, M. 1986 Laser measurements on nonpremixed H2 -air flames for assessment of turbulent combustion models. AIAA J. 24 (6), 905–917. Dunavant, D. A. 1985 High degree efficient symmetrical gaussian quadrature rules for the triangle. Int. J. Num. Meth. Eng. 21, 1129–1148. Eca, L., Hoekstra, M., Hay, A. & Pelletier, D. 2007 On the construction of manufactured solutions for one- and two-equation eddy-viscosity models. Int. J. Num. Meth. Fluids 54 (2), 119–154.

160

REFERENCES

Echekki, T., Kerstein, A. R., Dreeben, T. D. & Chen, J.-Y. 2001 ‘One-dimensional turbulence’ simulation of turbulent jet diffusion flames: model formulation and illustrative applications. Combust. Flame 125 (3), 1083–1105. Eibeck, P. A., Keller, J. O., Bramlette, T. T. & Sailor, D. J. 1993 Pulse combustion: impinging jet heat transfer enhancement. Combust. Sci. Tech. 94, 147–165. El-Asrag, H. & Menon, S. 2007 Large eddy simulation of bluff-body stabilized swirling non-premixed flames. Proc. Combust. Inst. 31, 1747–1754. Engles, H. 1980 Numerical Quadrature and Cubature. London: Academic Press. Ezekoye, O., Greif, R. & Sawyer, R. F. 1992 Increased surface temperature effects on wall heat transfer during unsteady flame quenching. Proc. Combust. Inst. 24, 1465–1472. Falgout, R. D. & Yang, U. M. 2002 HYPRE: a library of high performance preconditioners. In Computational Science - ICCS 2002 Part III (ed. P. M. A. Sloot, C. J. K. Tan, J. J. Dongarra & A. G. Hoekstra), Lecture Notes in Computer Science, vol. 2331, pp. 632–641. New York, NY: Springer-Verlag. Fletcher, R. & Powell, M. J. D. 1963 A rapidly convergent descent method for minimization. Computer Journal 6, 163–168. Forkel, H. & Janicka, J. 2000 Large-eddy simulation of a turbulent hydrogen diffusion flame. Flow, Turb. Combust. 65, 163–175. Frank, J. H., Barlow, R. S. & Lundquist, C. 2000 Radiation and nitric oxide formation in turbulent non-premixed jet flames. Proc. Combust. Inst. 28, 447–454. Galperin, B. & Orszag, S. A., ed. 1993 Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge: Cambridge University Press. Genz, A. 1987 A package for testing multiple integration subroutines. In Numerical Integration: Recent Developments, Software and Applications (ed. P. Keast & G. Fairweather), pp. 337–340. New York, NY: Springer. Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760–1765.

REFERENCES

161

Geyer, D., Kempf, A., Dreizler, A. & Janicka, J. 2005 Scalar dissipation rates in isothermal and reactive turbulent opposed-jets: 1-D-Raman/Rayleigh experiments supported by les. Proc. Combust. Inst. 30 (1), 681–689. Ghosal, S., Lund, T. S., Moin, P. & Akselvoll, K. 1995 A dynamic localization model for large-eddy simulation of turbulent flows. J. Fluid Mech. 286, 229–255. Ghosal, S. & Moin, P. 1995 The basic equations for the large eddy simulation of turbulent flows in complex geometry. J. Comput. Phys. 118 (1), 24–37. Ghosal, S. & Rogers, M. M. 1997 A numerical study of self-similarity in a turbulent plane wake using large-eddy simulation. Phys. Fluids 9, 1729–1739. Gill, P. E., Murray, W. & Wright, M. H. 1981 Practical Optimization. London: Academic Press. Goldfarb, D. 1970 A family of variable-metric methods derived by variational means. Mathematics of Computing 24, 23–26. ¨ ller, H. M. 1978 Invariant integration formulas for the n-simplex Grundmann, A. & Mo by combinatorial methods. SIAM J. Numer. Anal. 15 (2), 282–290. Ham, F. & Iaccarino, G. 2004 Energy conservation in collocated discretization schemes on unstructured meshes. Annual Research Briefs 2004 Center for Turbulence Research, Stanford University/NASA Ames, 3–14. Ham, F., Mattsson, K. & Iaccarino, G. 2006 Accurate and stable finite volume operators for unstructured flow solvers. Annual Research Briefs 2006 Center for Turbulence Research, Stanford University/NASA Ames, 243–261. Ham, F. E., Lien, F. S. & Strong, A. B. 2002 A Cartesian grid method with transient anisotropic adaptation. J. Comput. Phys. 179, 469–494. Han, S. P. 1977 A globally convergent method for nonlinear programming. J. Optim. Theory & Appl. 22, 297–309. Hargrave, G. K., Fairweather, M. & Kilham, J. K. 1987 Forced convective heat transfer from premixed flames. 2. Impingement heat transfer. Int. J. Heat Fluid Flow 8 (2), 132–138.

162

REFERENCES

Hargrave, G. K. & Kilham, J. K. 1984 The effect of turbulence intensity on convective heat transfer from premixed methane-air flames. Inst. Chem. Eng. Symp. Series 2 (86), 1025–1034. Haworth, D. C. & Jansen, K. 2000 LES on unstructured deforming meshes: towards reciprocating IC engines. Comput. Fluids 29 (5), 493–524. Henson, V. E. & Yang, U. M. 2002 BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Num. Math. 41, 155–177. Hesthaven, J. S. 1998 From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35 (2), 655–676. Hewson, J. C. & Kerstein, A. R. 2001 Stochastic simulation of transport and chemical kinetics in turbulent CO/H2 /N2 flames. Combust. Theory Modelling 5, 669–697. Hustad, J. E., Jacobsen, M. & Sønju, O. K. 1992 Radiation and heat transfer in oil/propane jet diffusion flames. Inst. Chem. Eng. Symp. Series 10 (129), 657–663. Hustad, J. E., Røkke, N. A. & Sønju, O. K. 1991 Heat transfer to pipes submerged in lifted buoyant diffusion flames. In Second International Conference on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics (ed. J. F. Keffer, R. K. Shah & E. N. Ganic), pp. 567–574. New York, NY: Elsevier. Hustad, J. E. & Sønju, O. K. 1991 Heat transfer to pipes submerged in turbulent jet diffusion flames. In Heat Transfer in Radiating and Combusting Systems (ed. M. G. Carvalho, F. Lockwood & J. Taine), pp. 474–490. Heidelberg: Springer-Verlag. Iaccarino, G. & Ham, F. 2007 LES on Cartesian grids with anisotropic refinement. In Complex Effects in Large Eddy Simulations (ed. S. C. Kassinos, C. A. Langer, G. Iaccarino & P. Moin), Lecture Notes in Computational Science and Engineering, vol. 56, pp. 219– 234. New York, NY: Springer. Ihme, M. 2007 Pollutant formation and noise emission in turbulent non-premixed flames. PhD thesis, Stanford University.

REFERENCES

163

Ihme, M., Cha, C. M. & Pitsch, H. 2005 Prediction of local extinction and re-ignition effects in non-premixed turbulent combustion using a flamelet/progress variable approach. Proc. Combust. Inst. 30, 793–800. Ihme, M., Marsden, A. L. & Pitsch, H. 2008 Generation of optimal artificial neural networks using a pattern search algorithm: Application to approximation of chemical systems. Neural Computation 20 (2), 573–601. Ihme, M. & Pitsch, H. 2004 An unsteady/flamelet progress variable method for an unsteady/flamelet progress variable method for LES of nonpremixed turbulent combustion. Paper 2004-557. AIAA. Ihme, M. & Pitsch, H. 2008a Prediction of extinction and reignition in nonpremixed turbulent flames using a flamelet/progress variable model: 1. a priori study and presumed PDF closure. Combust. Flame 155 (1-2), 70–89. Ihme, M. & Pitsch, H. 2008b Prediction of extinction and reignition in nonpremixed turbulent flames using a flamelet/progress variable model: 2. application in LES of Sandia flames D and E. Combust. Flame 155 (1-2), 90–107. Janicka, J. & Kollmann, W. 1978 A two-variable formalism for the treatment of chemical reactions in turbulent H2 -air diffusion flames. Proc. Combust. Inst. 17, 421–430. Jarosinski, J. 1986 A survey of recent studies on flame extinction. Combust. Sci. Tech. 12, 81–116. ã ń, J., Rogers, M. M. & Higuera, F. J. 1997 A priori testing of Jim´ enez, J., Lin subgrid models for chemically reacting non-premixed turbulent shear flows. J. Fluid Mech. 349, 149–171. Johnson, B. V. & Bennett, J. C. 1981 Mass and momentum turbulent transport experiments with confined coaxial jets. Tech. Rep. CR-165574. NASA. Johnson, B. V. & Bennett, J. C. 1984 Statistical characteristics of velocity, concentration, mass transport, and momentum transport for coaxial jet mixing in a confined duct. J. Eng. Gas Turbines & Power 106, 121–127.

164

REFERENCES

Juneja, A., Lathrop, D. P., Sreenivasan, K. R. & Stolovitzky, G. 1994 Synthetic turbulence. Phys. Rev. E 49, 5179–5194. Karpetis, A. N. & Barlow, R. S. 2002 Measurements of scalar dissipation in a turbulent piloted methane/air jet flame. Proc. Combust. Inst. 29, 1929–1936. Karpetis, A. N. & Barlow, R. S. 2005 Measurements of flame orientation and scalar dissipation in turbulent partially premixed methane flames. Proc. Combust. Inst. 30, 665–672. Kassinos, S. C., Langer, C. A., Iaccarino, G. & Moin, P., ed. 2007 Complex Effects in Large Eddy Simulations, Lecture Notes in Computational Science and Engineering, vol. 56. New York, NY: Springer. Keast, P. 1986 Moderate-degree tetrahedral quadrature formulas. Comp. Meth. Appl. Mech. Eng. 55, 339–348. Kee, R. J., Dixon-Lewis, G., Warnatz, J., Coltrin, M. E. & Miller, J. A. 1990 A Fortran computer code package for the evaluation of gas-phase multicomponent transport properties. Sandia National Laboratories Report SAND86-8246. Kelman, J. B. & Masri, A. R. 1997 Reaction zone structure and scalar dissipation rates in turbulent diffusion flames. Combust. Sci. Tech. 129, 17–55. Kempf, A., Flemming, F. & Janicka, J. 2005 Investigation of lengthscales, scalar dissipation, and flame orientation in a piloted diffusion flame by LES. Proc. Combust. Inst. 30, 557–565. Kempf, A., Sadiki, A. & Janicka, J. 2002 Prediction of finite chemistry effects using large eddy simulation. Proc. Combust. Inst. 29, 1979–1985. Kerstein, A. R. 1988 A linear- eddy model of turbulent scalar transport and mixing. Combust. Sci. Tech. 60 (4-6), 391–421. Kerstein, A. R. 1992a Linear-eddy modeling of turbulent transport. Part 4. Structure of diffusion flames. Combust. Sci. Tech. 81 (1-3), 75–96. Kerstein, A. R. 1992b Linear-eddy modeling of turbulent transport. Part 7. Finite-rate chemistry and multistream mixing. J. Fluid Mech. 240, 289–313.

REFERENCES

165

Kerstein, A. R. 1999 One-dimensional turbulence: model formulation and application to homogeneous turbulence, shear flows, and buoyant stratified flows. J. Fluid Mech. 392, 277–334. Kerstein, A. R. 2002 One-dimensional turbulence: a new approach to high-fidelity subgrid closure of turbulent flow simulations. Comput. Phys. Comm. 148 (1), 1–16. Kerstein, A. R., Ashurst, W. T., Wunsch, S. & Nilsen, V. 2001 One-dimensional turbulence: vector formulation and application to free shear flows. J. Fluid Mech. 447, 85–109. Kim, J. S., Williams, F. A. & Ronney, P. D. 1996 Diffusional-thermal instability of diffusion flames. J. Fluid Mech. 327, 273–301. Kim, S. H. & Pitsch, H. 2005 Conditional filtering method for large-eddy simulation of turbulent nonpremixed combustion. Phys. Fluids 17 (10), 105103. Kim, S. H. & Pitsch, H. 2006 Mixing characteristics and structure of a turbulent jet diffusion flame stabilized on a bluff-body. Phys. Fluids 18 (7), 075103. Klimenko, A. Y. 1990 Multicomponent diffusion of various admixtures in turbulent flow. Fluid Dynamics 25 (3), 327–334. Klimenko, A. Y. & Bilger, R. W. 1999 Conditional moment closure for turbulent combustion. Prog. Energy Combust. Sci. 25, 595–687. Knupp, P. & Salari, K. 2003 Verification of Computer Codes in Computational Science and Engineering. Boca Raton, FL: Chapman & Hall/CRC. Kokkala, M. A. 1991 Experimental study of heat transfer to ceiling from an impinging diffusion flame. In Third Int. Symp. Proc. (ed. G. Cox & B. Langford), pp. 261–270. International Association for Fire Safety Science, Edinburgh. Kraichnan, R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521–1536. Kronenburg, A. & Bilger, R. W. 1997 Modelling of differential diffusion effects in nonpremixed nonreacting turbulent flow. Phys. Fluids 9 (5), 1435–1447.

166

REFERENCES

de Lataillade, A., Dabireau, F., Cuenot, B. & Poinsot, T. 2002 Flame/wall interaction and maximum wall heat fluxes in diffusion burners. Proc. Combust. Inst. 29, 775–779. Lattimer, B. Y. 2003 Heat fluxes from fires to surfaces. In SFPE Handbook of Fire Protection Engineering, 3rd edn. (ed. P. J. DiNenno). Quincy, MA: National Fire Protection Association. Lesieur, M. & M´ etais, O. 1998 New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid Mech. 28 (45-82). Lilly, D. K. 1967 The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of IBM Scientific Computing Symposium on Environmental Sciences (ed. H. H. Goldstine), pp. 195–210. Yorktown Heights, NY: IBM. Lilly, D. K. 1992 A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A 4, 633–635. Lo Jacono, D., Papas, P., Matalon, M. & Monkewitz, P. A. 2005 An experimental realization of an unstratined planar diffusion flame. Proc. Combust. Inst. 30, 501–509. Lu, J. H., Ezekoye, O., Greif, R. & Sawyer, R. F. 1990 Unsteady heat transfer during side wall quenching of a laminar flame. Proc. Combust. Inst. 23, 441–446. Lyness, J. N. & Cools, R. 1994 A survey of numerical cubature over triangles. In Mathematics of computation 1943-1993: A half-century of computational mathematics (ed. W. Gautschi), Proceedings of Symposia in Applied Mathematics, vol. 48, pp. 127–150. American Mathematical Society, Providence, RI. Lyness, J. N. & Genz, A. C. 1980 On simplex trapezoidal rule families. SIAM J. Numer. Anal. 17 (1), 126–147. Lyness, J. N. & Jespersen, D. 1975 Moderate degree symmetric quadrature rules for the triangle. J. Inst. Maths Applics 15, 19–32. Maas, U. & Pope, S. B. 1992 Simplifying chemical kinetics: Intrinsic, low-dimensional manifolds in composition space. Combust. Flame 88 (3-4), 239–264.

REFERENCES

167

Mahesh, K., Constantinescu, G., Apte, S., Iaccarino, G., Ham, F. & Moin, P. 2006 Large-eddy simulation of reacting turbulent flows in complex geometries. J. Appl. Mech. 73 (3), 374–381. Mahesh, K., Constantinescu, G. & Moin, P. 2004 A numerical method for large-eddy simulation in complex geometries. J. Comput. Phys. 197 (1), 215–240. Markides, C. N. & Mastorakos, E. 2008 Measurements of the statistical distribution of the scalar dissipation rate in turbulent axisymmetric plumes. Flow, Turb. Combust. 81, 221–234. Mason, P. J. 1994 Large-eddy simulation: A critical review of the technique. Quart. J. Royal Meteorol. Soc. 120, 1–26. McMurtry, P. A., Menon, S. & Kerstein, A. R. 1993 Linear eddy modeling of turbulent combustion. Energy & Fuels 7 (6), 817–826. van der Meer, T. H. 1991 Stagnation point heat transfer from turbulent low Reynolds number jets and flame jets. Exp. Therm. Fluid Sci. 4, 115–126. Meier, W., Barlow, R. S., Chen, Y. L. & Chen, J.-Y. 2000 Raman/Rayleigh/LIF measurements in a turbulent CH4 /H2 /N2 jet diffusion flame: Experimental techniques and turbulence-chemistry interaction. Combust. Flame 123 (3), 326–343. ¨ bbert, J. H. 2008 Variation of the Mellado, J. P., Peters, N., Wang, L. & Go conditional dissipation rate across a shear layer. In 2a Reuni´ on de la Secci´ on Espa˜ nola del Instituto de Combusti´ on. Valencia, Spain. Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32, 1–32. Meneveau, C., Lund, T. S. & Cabot, W. H. 1996 A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech. 319, 353–385. Menon, S. & Calhoon, W. H. J. 1996 Subgrid mixing and molecular transport modeling in a reacting shear layer. Proc. Combust. Inst. 26, 59–66. M´ etais, O. & Lesieur, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194.

168

REFERENCES

Mi, J., Antonia, R. A. & Anselmet, F. 1995 Joint statistics between temperature and its dissipation rate components in a round jet. Phys. Fluids 7 (7), 1665–1673. Milson, A. & Chegier, N. A. 1973 Studies of methane-air flames impinging on a cold plate. Combust. Flame 21, 295–305. Mohr, J. W., Seyed-Yagoobi, J. & Page, R. H. 1996 Combustion measurements from an impinging Radial Jet Reattachment flame. Combust. Flame 106 (1-2), 69–80. Moin, P. 1997 Progress in large eddy simulation of turbulent flows. Paper 97-749. AIAA. Moin, P. 2002 Advances in large eddy simulation methodology for complex flows. Int. J. Heat Fluid Flow 23 (5), 710–720. Moin, P. & Kim, J. 1982 Numerical investigation of turbulent channel flow. J. Fluid Mech. 118, 341–377. Moin, P., Squires, K., Cabot, W. & Lee, S. 1991 A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids A 3, 2746–2757. Nicoud, F. 2000 Conservative high-order finite-difference schemes for low-Mach number flows. J. Comput. Phys. 158, 71–97. ´ly, G. 1997 Differentially diffusing scalars in turbulence. Phys. Fluids Nilsen, V. & Kosa 9 (11), 3386–3397. ´ly, G. 1999 Differential diffusion in turbulent reacting flows. Combust. Nilsen, V. & Kosa Flame 117 (3), 493–513. Oberkampf, W. L., Trucano, T. G. & Hirsch, C. 2004 Verification, validation, and predictive capability in computational engineering and physics. Appl. Mech. Rev. 57 (5), 345–384. van Oijen, J. A. & de Goey, L. P. H. 2002 Modelling of premixed counterflow flames using the flamelet-generated manifold method. Combust. Theory Modelling 6, 463–478. Owen, F. K., Spadaccini, L. J. & Bowman, C. T. 1976 Pollutant formation and energy release in confined turbulent diffusion flames. Proc. Combust. Inst. 16, 105–117.

REFERENCES

169

Pantano, C., Sarkar, S. & Williams, F. A. 2003 Mixing of a conserved scalar in a turbulent reacting shear layer. J. Fluid Mech. 481, 291–328. Pember, R. B., Howell, L. H., Bell, J. B., Colella, P., Crutchfield, C. Y., Fiveland, W. A. & Jessee, J. P. 1998 An adaptive projection method for unsteady, low-Mach number combustion. Combust. Sci. Tech. 140 (1-6), 123–168. Peters, N. 1983 Local quenching due to flame stretch and non-premixed turbulent combustion. Combust. Sci. Tech. 30, 1–17. Peters, N. 1984 Laminar diffusion flamelet models in non-premixed turbulent combustion. Prog. Energy Combust. Sci. 10 (3), 319–339. Peters, N. 2000 Turbulent Combustion. Cambridge: Cambridge University Press. Peters, N. 2006 Concepts and key parameters in turbulent combustion modeling. In Turbulence, Heat and Mass Transfer , Proceeding of the International Symposium on Turbulence, Heat and Mass Transfer , vol. 1, pp. 47–58. Dubrovnik, Croatia. Pierce, C. D. & Moin, P. 1998 A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar. Phys. Fluids 10, 3041–3044. Pierce, C. D. & Moin, P. 2001 Progress-variable approach for large-eddy simulation of turbulent combustion. PhD thesis, Stanford University. Pierce, C. D. & Moin, P. 2004 Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. J. Fluid Mech. 504, 73–97. Piomelli, U. 1993 High Reynolds number calculations using the dynamic subgrid-scale stress model. Phys. Fluids A 5, 1484–1490. Piomelli, U. & Liu, J. 1995 Large-eddy simulation of rotating channel flow using a localized dynamic model. Phys. Fluids 7, 839–848. Pitsch, H. 2000 Unsteady flamelet modeling of differential diffusion in turbulent jet diffusion flames. Combust. Flame 123 (3), 358–374.

170

REFERENCES

Pitsch, H. 2002 Improved pollutant predictions in large-eddy simulations of turbulent nonpremixed combustion by considering scalar dissipation rate fluctuations. Proc. Combust. Inst. 29, 1971–1978. Pitsch, H. 2006 Large-eddy simulation of turbulent combustion. Annu. Rev. Fluid Mech. 38, 453–482. Pitsch, H., Chen, M. & Peters, N. 1998 Unsteady flamelet modeling of turbulent hydrogen/air diffusion flames. Proc. Combust. Inst. 27, 1057–1064. Pitsch, H. & Peters, N. 1998 A consistent flamelet formulation for non-premixed combustion considering differential diffusion effects. Combust. Flame 114, 26–40. Pitsch, H. & Steiner, H. 2000 Large-eddy simulation of a turbulent piloted methane/air diffusion flame (Sandia flame D). Phys. Fluids 12, 2541–2554. Poinsot, T., Haworth, D. C. & Bruneaux, G. 1993 Direct simulation and modeling of flame-wall interaction for premixed turbulent combustion. Combust. Flame 95, 118–132. Poinsot, T. & Veynante, D. 2001 Theoretical and Numerical Combustion. Philadelphia, PA: R. T. Edwards. Pope, S. B. 1985 PDF methods in turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192. Pope, S. B. 1990 Computations of turbulent combustion: progress and challenges. Proc. Combust. Inst. 23, 591–612. Pope, S. B. 1997 Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation. Combust. Theory Modelling 1, 41–63. Popiel, C. O., van der Meer, T. H. & Hoogendoorn, C. J. 1980 Convective heat transfer on a plate in an impinging round hot gas jet of low Reynolds number. Int. J. Heat Mass Trans. 23, 1055–1068. Popp, P. & Baum, M. 1995 Heat transfer and pollutant formation mechanisms in insulated combustion chambers. Tech. Paper 952387. SAE.

REFERENCES

171

Popp, P. & Baum, M. 1997 Analysis of wall heat fluxes, reaction mechanisms, and unburnt hydrocarbons during the head-on quenching of a laminar methane flame. Combust. Flame 108, 327–348. Popp, P., Smooke, M. & Baum, M. 1996 Heterogeneous/homogeneous reaction and transport coupling during flame-wall interaction. Proc. Combust. Inst. 26, 2693–2700. Powell, M. J. D. 1978a The convergence of variable metric methods for nonlinearly constrained optimization calculations. In Nonlinear Programming 3 (ed. O. L. Mangasarian, R. R. Meyer & S. M. Robinson). London: Academic Press. Powell, M. J. D. 1978b A fast algorithm for nonlinearly constrained optimization calculations. In Numerical Analysis (ed. G. A. Watson), Lecture Notes in Mathematics, vol. 630. New York, NY: Springer-Verlag. Quintiere, J. G. & Cleary, T. G. 1994 Heat flux from flames to vertical surfaces. Fire Technol. 30, 209–231. Rabitz, H. & Alis, O. 1999 General foundations of high-dimensional model representations. J. Math. Chem. 25, 197–233. Raimondeau, S., Norton, D., Vlachos, D. G. & Masel, R. I. 2002 Modeling of high-temperature microburners. Proc. Combust. Inst. 29, 901–907. Raman, V. & Pitsch, H. 2007 A consistent LES/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry. Proc. Combust. Inst. 31, 1711–1719. Raman, V., Pitsch, H. & Fox, R. O. 2005 Hybrid large-eddy simulation/Lagrangian filtered-density-function approach for simulating turbulent combustion. Combust. Flame 143, 56–78. Ray, A. & Wichman, I. S. 1998 Influence of fuel-side heat loss on diffusion flame extinction. Int. J. Heat Mass Trans. 41, 3075–3085. Renfro, M. W., Chaturved, A., King, G. B., Laurendeau, N. M., Kempf, A., Dreizler, A. & Janicka, J. 2004 Comparison of OH time-series measurements and large-eddy simulations in hydrogen jet flames. Combust. Flame 139 (1-2), 142–151.

172

REFERENCES

Reynolds, W. C. 1976 Computation of turbulent flows. Annu. Rev. Fluid Mech. 8, 183– 208. Ribert, G., Gicquel, O., Darabiha, N. & Veynante, D. 2006 Tabulation of complex chemistry based on self-similar behavior of laminar premixed flames. Combust. Flame 146, 649–664. Rigby, J. R. & Webb, B. W. 1995 An experimental investigation of diffusion flame jet impingement heat transfer. In Proceedings of the Fourth ASME/JSME Thermal Engineering Joint Conference, pp. 117–126. ASME, Maui, HI. Roache, P. J. 1998a Verification and Validation in Computational Science and Engineering. Albuquerque, NM: Hermosa Publishers. Roache, P. J. 1998b Verification of codes and calculations. AIAA Journal 36 (5), 696–702. Roache, P. J. 2002 Code verification by the method of manufactured solutions. J. Fluids Eng. 124 (1), 4–10. Roback, R. & Johnson, B. V. 1983 Mass and momentum turbulent transport experiments with confined swirling coaxial jets. Tech. Rep. CR-168252. NASA. Roy, C. J. 2005 Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205 (1), 131–156. Roy, C. J., Nelson, C. C., Smith, T. M. & Ober, C. C. 2004 Verification of Euler/Navier-Stokes codes using the method of manufactured solutions. Int. J. Num. Meth. Fluids 44 (6), 599–620. Sandia National Laboratories 1996 International workshop on measurement and computation of turbulent nonpremixed flames. Available from http://public.ca.sandia. gov/TNF/abstract.html, accessed October 2008. Schmidt, R. C., Kerstein, A. R., Wunsch, S. & Nilsen, V. 2003 Near-wall LES closure based on one-dimensional turbulence modeling. J. Comput. Phys. 186 (1), 317– 355. Schumann, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys. 18, 376–404.

REFERENCES

173

Silvester, P. 1970 Symmetric quadrature formulae for simplexes. Mathematics of Computing 24 (109), 95–100. Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weather Rev. 91 (3), 99–164. Smith, L. M. & Woodruff, S. L. 1998 Renormalization-group analysis of turbulence. Annu. Rev. Fluid Mech. 30, 275–310. Sohn, C. H., Chung, S. H. & Kim, J. S. 1999 Instability-induced extinction of diffusion flames established in the stagnant mixing layer. Combust. Flame 117 (1-2), 404–412. Spadaccini, L. J., Owen, F. K. & Bowman, C. T. 1976 Influence of aerodynamic phenomena on pollutant formation in combustion (Phase I. Gaseous fuels). Tech. Rep. EPA-600/2-76-247a. U. S. Environmental Protection Agency. Starner, S. H., Bilger, R. W., Long, M. B., Frank, J. H. & Marran, D. F. 1997 Scalar dissipation measurements in turbulent jet diffusion flames of air diluted methane and hydrogen. Combust. Sci. Tech. 129, 141–163. Stein, O. & Kempf, A. 2007 LES of the Sydney swirl flame series: a study of vortex breakdown in isothermal and reacting flows. Proc. Combust. Inst. 31, 1755–1763. Stein, O., Kempf, A. & Janicka, J. 2007 LES of the Sydney swirl flame series: an initial investigation of the fluid dynamics. Combust. Sci. Tech. 179 (1-2), 173–189. Steiner, H. & Bushe, W. K. 2001 Large eddy simulation of a turbulent reacting jet with conditional source-term estimation. Phys. Fluids 13 (3), 754–769. Stokes, G. G. 1845 On the theories of the internal friction of fluids in motion and of the equilibrium and motion of elastic solids. Trans. Cambr. Phil. Soc. 8, 287–319. Stolz, S. & Adams, N. A. 1999 An approximate deconvolution procedure for large-eddy simulation. Phys. Fluids 11, 1699–1701. Stroud, A. H. 1969 A fifth degree integration formula for the n-simplex. SIAM J. Numer. Anal. 6 (1), 90–98.

174

REFERENCES

Stroud, A. H. 1971 Approximate Calculation of Multiple Integrals. Englewood Cliffs, NJ: Prentice-Hall. Taylor, M. A., Wingate, B. A. & Vincent, R. E. 2001 An algorithm for computing Fekete points in the triangle. SIAM J. Numer. Anal. 38 (5), 1707–1720. Tremblay, D., Etienne, S. & Pelletier, D. 2006 Code verification and the method of manufactured solutions for fluid-structure interaction problems. In 36th AIAA Fluid Dynamics Conference, , vol. 2, pp. 882–892. San Francisco, CA. ńyi, T. 1994 Parameterization of reaction mechanisms using orthogonal polynomials. Tura Comput. Chem. 18, 45–54. U. S. Department of Energy 2008 International Energy Outlook 2008, DOE/EIA0383(2008). Available from the Energy Information Administration http://www.eia. doe.gov/oiaf/ieo/, accessed August 2008. Vance, R., Miklavcic, M. & Wichman, I. S. 2001 On the stability of one-dimensional diffusion flames established between plane, parallel, porous walls. Combust. Theory Modelling 5, 147–161. Vance, R. & Wichman, I. S. 2000 Heat transfer analysis of a diffusion flame leading edge near a cold, chemically inert surface. Int. J. Heat Mass Trans. 43, 921–933. Vavasis, S. A. 1998 Quadrature over tetrahedra. Available from http://www.cs.cornell. edu/home/vavasis/quad.html, accessed July 2008. Vreman, A. W. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. Phys. Fluids 16 (10), 3670–3681. Wall, C., Boersma, B. & Moin, P. 2000 An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release. Phys. Fluids 12, 2522–2529. Wandzura, S. & Xiao, H. 2003 Symmetric quadrature rules on a triangle. Computers and Mathematics with Applications 45, 1829–1840. Wang, Q. & Squires, K. D. 1996 Large eddy simulation of particle-laden turbulent channel flow. Phys. Fluids 8, 1207–1223.

REFERENCES

175

Wang, Y. & Trouv´ e, A. 2006 Direct numerical simulation of nonpremixed flame-wall interactions. Combust. Flame 144, 461–475. Westbrook, C. K., Adamczyk, A. A. & Lavoie, G. A. 1981 A numerical study of laminar flame wall quenching. Combust. Flame 40, 81–99. Williams, F. A. 1975 Recent advances in theoretical descriptions of turbulent diffusion flames. In Turbulent Mixing in Nonreactive and Reactive Flows (ed. S. N. B. Murthy), pp. 189–208. New York, NY: Plenum Press. Williams, F. A. 1985 Combustion Theory, 2nd edn. Boulder, CO: Westview Press. Wu, X. & Squires, K. D. 1998 Numerical investigation of the turbulent boundary layer over a bump. J. Fluid Mech. 362, 229–271. Xia, G., Li, D. & Merkle, C. L. 2007 Consistent properties reconstruction on adaptive cartesian meshes for complex fluids computations. J. Comput. Phys. 225, 1175–1197. Xu, J. & Pope, S. B. 2000 PDF calculations of turbulent nonpremixed flames with local extinction. Combust. Flame 123 (3), 281–307. Yakhot, A., Orszag, S. A. & Yakhot, Y. 1989 Renormalization group formulation of large-eddy simulations. J. Sci. Comput. 4 (2), 139–158. Yeung, P. K. & Pope, S. B. 1993 Differential diffusion of passive scalars in isotropic turbulence. Phys. Fluids A 5 (10), 2467–2478. You, D. & Moin, P. 2007 A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries. Phys. Fluids 19 (6), 065110. You, H.-Z. 1985 Investigation of fire impingement on a horizontal ceiling. 2. Impingement and ceiling-jet regions. Fire & Materials 9, 46–56. You, H.-Z. & Faeth, G. M. 1979 Ceiling heat transfer during fire plume and fire impingement. Fire & Materials 3, 140–147. Zhang, Y., Rogg, B. & Bray, K. N. C. 1995 2-D simulation of turbulent autoignition with transient laminar flamelet source term closure. Combust. Sci. Tech. 105, 211–227.

176

REFERENCES

Zhu, X. L., Gore, J. P., Karpetis, A. N. & Barlow, R. S. 2002 The effects of selfabsorption of radiation on an opposed flow partially premixed flame. Combust. Flame 129 (3), 342–345.