Chapter 7 TURBULENCE MODELING FOR CFD - FEM

Chapter 7

TURBULENCE MODELING FOR CFD1

7-1 Introduction: What is Turbulence? Turbulence denotes a state of fluid in which all properties (velocity, P, T, ρ, ...) fluctuate in an irregular, disorderly, chaotic, non-repeating and unpredictable manner, but no discontinuity ... statistics are well behaved. Turbulence is a continuum phenomenon. The smallest scales are still much larger than molecular scales. Accordingly, the NS equations are still valid for every small part of the fluid. • “Turbulence is the last great unsolved problem of classical physics” (Feynman) • “Turbulent Motion. It remains to call attention to the chief outstanding difficulty of our subject” (Horace Lamb, 1932) • “Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another” (G. I. Taylor, according to T. von Karman, 1937) • “Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation with time and space coordinates, so that statistically distinct average values can be discerned” (J. O. Hinze, 1959) • “Turbulence is a dangerous topic, ... it represents extremely different points of view, all of which have in common its complexity, as well as an inability to solve the problem. It is even difficulty to agree on what exactly is the problem is be solved” (M. Lesieur, 1987) • “Turbulence is common, important and WEIRD” (P. Bradshaw, 1992) • “Shall I refuse my dinner because I do not fully understand the process of digestion?” (Oliver Heaviside)

1. Most of the material in this chapter is taken from the class notes of Professor K. Hanjalic and Dr. D. C. Wilcox

Page 7-1

7-2 Properties of Turbulence

Turbulence by its nature is an unsteady, three-dimensional, non-linear, irreversible, stochastic phenomenon, which can be characterized by serveral features:

• Randomness: meaning disorder and non-repeatability. (not to confused with “random flows”) • Continuity of eddy structure: a continuous spectrum of fluctuations over a range of frequencies, a spectrum of eddies of varying sizes (“eddy” - vaguely identifiable fluid structure in swirling motion, revealed by flow visualization, and associated with a Fourier component of a particular frequency or wave length; not to be confused with a “vortex” which has nothing to do with turbulence.) • Non-linearity and Three-dimensionality: basic feature of turbulence, originating in vorticity production due to vortex self-stretching, a 3D phenomenon which maintains the turbulence vorticity (2D vorticity can not be stretched!) • Diffusivity: fluctuations stretch and distort the fluid element containing a lump of property inhomogeneity (hot spot, species; it will enhance the mixing and transport of momentum, heat, and species by several orders of magnitude. • Dispersivity and Spreading: diffusion and mixing enhance (act as agents of) the dispersion and spreading of the convected scalar (heat and species) • Vorticity: high intensity of vorticity (rotation of fluid element, ω = ∇ × u ) the dynamics of which dominates the turbulence field (vortex stretching, elongation and spinning, breakup, colescing, etc.) • Energy Cascading and Irreversibility: one of the most useful hypothesis in turbulence theory, originated by Richardson (1929?) “Big whirls have little whirls that feed on their velocity; Little whirls have lesser whirls, and so on to viscosity” and later elaborated by Kolmogorov (no interaction of eddies of very different sizes!)

• Dissipativeness: turbulence is a dissipative phenomenon. Energy flows only in one direction, from large to small eddies and will ultimately be dissipated by viscosity. Hence, turbulent flow requires a continuous supply of energy. • Intermittency: turbulence can interact with non-turbulent flow or can occupy only parts of flow domain. Its appearance in time or space is characterized by “intermittency” factor. • The dominant processes are independent of viscosity - large eddies control turbulence dynamics • Etc.

Page 7-2

7-3 Turbulence Modeling History • Boussinesq (1877): Eddy-Viscosity Approximation • Reynolds (1895): Time-Averaging Procedures • Five classes of turbulence models have evolved: 1.

Algebraic (Zero-Equation) Models

2.

Half-Equation Models

3.

One-Equation Models

4.

Two-Equation Models

5.

Second Order (Moment) Closure Models

• Classes 1 ~ 4 all use the Boussinesq eddy-viscosity approximation.

7-4 Reynolds Averaging Reynolds averaging assumes a variety of forms involving either an integral or summation to define the mean value. • Time Averaging: appropriate for statistically stationary turbulence (turbulence that, on average, does not vary with time)

1 F T ( x ) = lim --T → ∞T

t+T

∫

f ( x, t ) dt

(7-1)

t

• Ensemble Averaging: appropriate for general turbulence (N samples should be employed for nonstationary flows)

F E ( x, t ) = lim

N→∞

1 ---N

N

∑ n=1

f n ( x, t )

• Phase Averaging: appropriate for flows with an imposed periodic mean flow with period τ

Page 7-3

(7-2)

< f ( x, t ) > = lim

N→∞

1 ---N

N–1

∑

f ( x, t + nτ )

(7-3)

n=0

• Mass (Favre) Averaging: appropriate for Compressible flows

1 1 ˜f = ------ lim --i ρ T → ∞T

t+T

∫

ρ ( x, t ) f i ( x, t ) dt

t

(7-4)

7-5 Time Averaging Operations u’

u U

t • Express the flow properties as the sum of mean and fluctuating parts

u i ( x, t ) = U i ( x ) + u'i ( x, t )

(7-5)

• The mean velocity is given by

1 U i ( x ) = lim --T → ∞T

t+T

∫

t

u i ( x, t ) dt

• The time average of the mean velocity is again the mean velocity

Page 7-4

(7-6)

1 U i ( x ) = lim --T → ∞T

∫

t+T

U i ( x ) dt = U i ( x )

(7-7)

[ u i ( x, t ) – U i ( x ) ] dt = U i ( x ) – U i ( x ) = 0

(7-8)

t

• The time average of the fluctuating part is zero

1 u'i ( x ) = lim --T → ∞T

∫

t+T t

• Time averaging commutes with spatial and temporal differentiations, e.g.,

∂p ∂( P + p' ) ∂P ∂ p' ∂P ∂P = ---------------------- = ------ + ------- = +0 = ∂x ∂x ∂x ∂x ∂x ∂x

(7-9)

• Time averaging is a linear operation

a+b = A+B

c1 a + c2 b = c1 A + c2 B

(7-10)

• However, there is no a priori reason for the time averaging of two (or more) fluctuating quantities to vanish! ab = ( A + a' ) ( B + b' ) = AB + Ab' + Ba' + a'b' = AB + Ab' + Ba' + a'b' = AB + Ab' + Ba' + a'b' = AB + a'b'

∴

(7-11)

ab = AB + a'b'

• Time averaged products of fluctuating properties are called correlations • Advantages: (1)

Averaged conservation equations reduced to a form manageable by the available numerical codes for laminar

Page 7-5

flow. (2)

Details of fluctuating motion are not needed for computing MEAN flow properties.

• Disadvantages: (1)

Statistical averaging brings about a “loss of information” (“closure problem” - “Turbulence Models”)

(2)

Inability to compute any effect associated with well-organized large-scale “coherent (sticking together, consistent)” motion. => promising prospects: “Large Eddy Simulation” (LES) which resolves large scale motion, but models small (smaller than the mesh) structure.

7-6 Reynolds-Averaged Equations for Incompressible Flow For incompressible flow, the equations for conservation of mass and momentum are: ∂u i ------- = 0 ∂x i ∂u i ∂u i ∂p ∂τ ij + ρ ------- + ρu j -------- = – ∂ xi ∂ x ∂t ∂x j j

τ ij = 2µs ij ≡ Viscous Stress Tensor

1  ∂u i ∂u j s ij = ---  -------- + -------- ≡ Strain-Rate Tensor 2  ∂x j ∂x i 

(7-12)

Time averaging these equations gives the Reynolds-averaged equations of motion: (Please see Wilcox’s book p.14 for the justification of the unsteady term)

∂U i --------- = 0 ∂x ∂U i ∂U i ∂P ∂ ρ --------- + ρU j --------- = – + ( 2µS ij – ρu'i u'j ) ∂t ∂x j ∂ xi ∂ x j

Page 7-6

S ij

1 ∂U ∂U = ---  ---------i + ---------j 2  ∂x j ∂x i 

(7-13)

where the velocity correction term is called the Reynolds stress tensor

R ij = – ρu'i u'j

(7-14)

and is a symmetric tensor (=> 6 independent components). • The fundamental problem of turbulence for the engineer is to find a way of computing Rij: Unknowns: Velocity (3) + P (1) + Rij (6) = 10 Equations: Mass (1) + Momentum (3) = 4 ➾

Closure Problem

• The Boussinesq Eddy-Viscosity approximation

R ij = 2µ T S ij

or

2 R ij = 2µ T S ij – --- ρkδ ij 3

(7-15)

• This symmetric tensor equation (6 independent components), in effect, defines the eddy viscosity, µT • This simply replaces the unknown Rij with the unknown µT • We hope that µT is better behaved than Rij

7-7 Formal Derivation of Rij Equations and the Problem of Closure Take moments of the Navier-Stokes equations and them do the time averaging, i.e.,

u'i ℵ ( u j ) + u'j ℵ ( u i ) = 0

(7-16)

where ℵ ( u i ) is the Navier-Stokes operator: 2

∂u i ∂u i ∂p ∂ ui -----------ℵ ( ui ) = ρ + ρu k + –µ ∂t ∂x k ∂ x i ∂ x k ∂x k

Page 7-7

(7-17)

gives the Reynolds stress equation as

∂R ij ∂R ij ∂R ij ∂ t + Dp --------- + U k --------- = P ij + ε ij – Π ij + - + D ijk ν ijk ∂x k ∂ x k -------∂t ∂x k ∂U j ∂U i --------------– R jk P ij ≡ Production (Generation) = – R ik ∂x k ∂x k ∂u i' ∂u j' ε ij ≡ Destruction ( Dissipation ) = 2µ -------- --------∂x k ∂x k

(7-18)

∂u' ∂u' Π ij ≡ Pressure Redistribution = p'  -------i- + --------j-  ∂x j ∂x i  t ≡ Turbulent Diffusion = ρu' u' u' D ijk i j k p ≡ Pressure Diffusion = p'u' δ + p'u' δ D ijk i jk j ik

Note that this equation for a double correlation ( R ij = – ρu'i u'j ) involves other unknown double and triple correlations, e.g., ρu'i u'j u'k → 10 unknowns

∂u i ' ∂u j ' 2µ -------- --------- → 6 unknowns ∂x k ∂x k

∂ p' ∂ p' u'i -------- + u'j ------- → 6 unknowns ∂x j ∂x i

(7-19)

This is the closure problem of turbulence. Because of the nonlinearity of the Navier-Stokes equation, we never achieve closure by taking higher moments. ➾ Turbulence modeling: To establish approximations for the unknown

Page 7-8

correlations in terms of known properties, so that unknowns and equations are balanced.

7-8 Algebraic Turbulence Models • Origin is with Prandtl’s Mixing-Length Hypothesis (1925). • Important contributions made by Van Driest (1956), Clauser(1956), Escudier (1966), Corrsin (1954), and Klebanoff (1956). • These models achieve closure by relating the Reynolds stresses directly to mean-flow properties by (1)

Boussinesq eddy-viscosity approximation, i.e., R ij = 2µ T S ij

(2)

Empirical representation for the eddy viscosity µ T

• No differential equations beyond the Reynolds-averaged NS and continuity equations are needed. • We sometimes call them Zero-Equation Models. • Cebeci-Smith Model (Cebeci, T. and Smith, A.M.O., 1974, “Analysis of Turbulent Boundary Layers,” Academic Press.) Two-layer model with µ T given by separate expressions in the inner and outer layers y µT

O

Eddy Viscosity:

µT

 µ Ti =   µ To

µT

y ≤ ym y > ym

y

m

→ smallest value of y for which µ

Page 7-9

Ti

= µ

To

i

ym

µT

Inner Layer:

µ Ti =

2 ρl mix

 ∂U +  ∂V  ∂y  ∂x  2

+

2

l mix = κy

1–e

y

y – ------+ A

+

uτ y = ------ν

uτ =

(7-20)

τw ----ρ

Outer Layer:

µ To = αρU e δ v* F Kleb ( y ;δ )

δ v* =

δ

U  1 – -----dy   U 0 e

∫

F Kleb ( y ;δ ) = 1 + 5.5  --y  δ

6

–1

(7-21)

Closure Coefficients (Karman, Clauser, and Van Driest’s constants):

κ = 0.40

α = 0.0168

A

+

( dP ⁄ dx ) = 26 1 + y --------------------ρu τ2

1 – --2

(7-22)

• Baldwin-Lomax Model (Baldwin B.S. and Lomax, H, 1978, “Thin-Layer Approximation and Algebraic Model for Separated Turbulent Flow,” AIAA Paper 78-257, 1978)

Another two-layer model with no dependence on difficult to determine quantities such as displacement thickness and edge velocity Inner Layer:

µ Ti =

2 ρl mix

ω

ω =

2 2 2  ∂V – ∂U +  ∂W – ∂V  +  ∂U – ∂W  ∂ y ∂z  ∂z ∂ x  ∂x ∂y 

(7-23)

Outer Layer:

µ To = αρC cp F wake F Kleb ( y ;y max ⁄ C Kleb )

Page 7-10

(7-24)

2 ⁄F F wake = min y max F max ; C wk y max U dif max

1 F max = --- max ( l mix ω ) κ

where y max is the value of y at which l mix ω achieves its max. value. Closure Coefficients:

κ = 0.40

α = 0.0168

1 – --( dP ⁄ dx ) 2 + --------------------1 + y A = 26 ρu τ2

C cp = 1.6

C Kleb = 0.3

C wk = 1

(7-25)

• Comments on the Algebraic Models and Range of Applicability - Accurate for free shear flows, although mixing length is different for each type - Accurate for boundary layers with mild to moderate adverse pressure gradient. Erratic predictions for strong adverse gradients - Completely unreliable for separated flows - Easy to implement, conceptually simple and very robust

7-9 The Half-Equation Model

(Johnson, D.A. and King, L.S., 1985, “A Mathematically Simple Turbulence Closure Model for Attached and Separated Turbulent

Boundary Layers,” AIAA J., Vol. 23, No. 11, pp. 1684-1692)

• The model has 7 closure coefficients, and shares algebraic strengths and weaknesses for simple flows • However, it is much better for separated flows! • The model is difficult to implement, even in 2D, and requires an iterative solution for the non-equilibrium parameter built in the model equation

7-10 Turbulence Energy Equation Models (1- and 2-equation Models) • To get some flow history into the modeling. • Prandtl (1945), Emmons (1954), and Glushko (1965) postulated writing the eddy

Page 7-11

viscosity as the product of k1/2 (a velocity scale), and a length scale l , where k is kinetic energy of the fluctuation velocity, i.e.,

1 1 2 2 2 k = --- u'i u'i = --- ( u' + v' + w' ) 2 2

(7-26)

• Correspondingly, the eddy viscosity becomes 1 --2

µ T = cons tan t ⋅ ρk l

(7-27)

• Kolmogorov (1942) and Chou (1945) made similar proposals using k and a second turbulence property (dissipation, dissipation per unit of k) • These proposals resulted in what we now call one- and two-equation models of turbulence • The turbulence kinetic energy equation can be derived directly from the NS equation, or by taking the trace of the Reynolds-stress equation, since

R ii = – ρu'i u'i = – 2ρk

(7-28)

• Accordingly, the turbulence kinetic energy equation is

∂U i ∂k ∂k ∂ ∂k 1 ρ ----- + ρU j ------- = R ij --------- – ρε + µ ------- – --- ρu'i u'i u'j – p'u'j ∂x j ∂x j ∂t ∂ x j ∂x j 2 •

∂k ∂k ρ ----- + ρU j ------∂t ∂x j

•

∂U i R ij --------∂x j

is the rate of change of k following a fluid particle

is called production and is the rate at which kinetic energy is transferred

Page 7-12

 ∂u i' ∂u i' ε = ν  -------- --------  ∂x j ∂x j 

(7-29)

from the mean flow to the turbulence. Rij term needs to be modelled. •

ε

•

∂k µ ------∂x j

•

1 --- ρu' i u' i u' j 2

is called dissipation and is the rate at which k is converted into thermal internal energy. This term must be modelled. is the molecular diffusion of k

is called turbulent transport and is the rate at which k is transported through the fluid by turbulent

fluctuations. This term must be modelled. •

p'u'j

is called pressure diffusion and is another form of turbulent transport. This term must be modelled

• Modelling of (1)

1 R ij , --- ρu' i u' i u' j , p'u'j 2

ε

Invoking the Boussinesq eddy-viscosity approximation for

2 R ij = 2µ T S ij – --- ρkδ ij 3 NOTE that (2)

, and

where

R ij

1 ∂U ∂U S ij = ---  ---------i + ---------j 2  ∂x j ∂x i 

(7-30)

R ii = – 2ρk

Using a gradient diffusion hypothesis (process) for

1 --- ρu' i u' i u' j 2

µ ∂k 1 --- ρu'i u'i u'j + p'u'j = – -----T2 σk ∂ x j

Page 7-13

and

p'u'j (7-31)

(3)

Assuming we know how to specify a turbulent length scale, based on the dimensional analysis (Taylor , 1935), we have

k3/2 ε ∼ --------l

(7-32)

• Accordingly, the turbulence kinetic energy becomes

∂U i µ T  ∂k ∂k ∂k ∂  ρ ----- + ρU j ------- = R ij --------- – ρε + - ------∂x j ∂x j ∂t ∂ x j  µ + ----σ k  ∂x j

(7-33)

• Most one- and two-equation models use this equation • One-Equation Models -

Most one-equation models don’t offer much improvement over algebraic models. The only exception is the Spalart-Allmaras model (“A one-Equation Turbulence Model for Aerodynamic Flows, AIAA Paper 92-439, 1992).

-

Most one equation models need recalibrate length scale for each new application.

-

The Baldwin-Barth Model (“A One-Equation Turbulence Transport Model for High Reynolds Number Wall-Bounded Flows,” NASA TM-102847, 1990) is extremely sensitive to freestream conditions

-

Closure coefficient/Function count: Baldwin-Barth (7/3); Spalart-Allmaras (12/5)

• Two-Equation Models ( k

–ε

and

k–ω

models)

-

The foundation for most turbulence model research during the past two decades

-

Virtually all use the equation for k

-

A second equation is postulated from which the turbulence length scale can be computed

-

These are the simplest models that are complete, i.e., just specific initial and/or boundary conditions and solve.

Page 7-14

7-11 The Transport Equation for ε and the k – ε model • Recall

 ∂u i' ∂u i' ε = ν  -------- --------  ∂x j ∂x j  • The exact equation for the dissipation rate,

(7-34)

ε , can be derived by taking the following moment of the NS equation

∂u i' ∂ 2ν -------ℵ ( ui ) = 0 ∂x j ∂ x j

(7-35)

After a considerable amount of algebra (several pages), it can be shown

ρ

∂ε ∂ε + ρU j = – 2µ u' u' + u' u' i, k j, k k, i k, j ∂t ∂x j

2

∂ Ui ∂U i --------- – 2µ u' k u'i, j ∂x j ∂ x k ∂x j

– 2µ u'i, k u'i, m u'k, m – 2µν u' i, km u' i, km +

(7-36)

∂ ∂ε µ ------- – µ u'j u'i , m u'i , m – 2ν p',m u'j , m ∂ x j ∂x j

• Note that all the terms (except the molecular diffusion of

ε

term) in the RHS must be modelled!!

• Although this is exact GE for ε , this equation can not serve much as a basis for modelling except to give some indication on the meanings and importance of various terms • Virtually all those terms need to be modelled can not be measured accurately

Page 7-15

• Direct Numerical Simulation (DNS) results provide a little guidance at very low Reynolds number. • Most (All?) of the modelled equations are established by dimensional analysis • The Launder and Sharma’s

k–ε

model (Launder, B.E. and Sharma, B.I., 1974, “Application of the Energy Dissipation Model of Turbulence to the

Calculation of Flow Near a Spinning Disc,” Letters in Heat and Mass Transfer, Vol. 1, No. 2, pp. 131-138)

=> usually referred to as the Standard k – ε model

∂U i µ T  ∂k ∂k ∂k ∂  ρ ----- + ρU j ------- = R ij --------- – ρε + + ------ ------µ ∂x j ∂x j ∂t ∂x j  σ k  ∂x j 2 ∂U i µ T  ∂ε ε ∂ε ε ∂ε ∂  ρ ----- + ρU j ------- = C ε1 ------ R ij --------- – C ε2 ρ ----- + + ----- ------µ k ∂t ∂x j ∂x j k ∂x j  σ ε  ∂x j

µT C ε1 = 1.44

C ε2 = 1.92

(7-37)

2

k = ρC µ -------ε C µ = 0.09

σ k = 1.0

σ ε = 1.3

• This model can not be integrated all the way down to the wall. Need wall function. ➾ LOW Re Models (Chien, K.-Y., 1982, “Predictions of channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbulence Model,” AIAA J., Vol. 20, No. 1, pp. 33-38)

• The most widely used model in today’s engineering computations

7-12 The k – ω model (Wilcox, D.C., 1993, Turbulence Modeling for CFD, DCW Industries, Inc.) • In 1942, Kolmogorov postulated using specific dissipation rate

Page 7-16

ω

• He combined dimensional analysis with physical reasoning • Fundamental physical processes: (1) Convection; (2) Unsteadiness; (3) Diffusion; (4) Production; (5) Dissipation; and (6) Dispersion 1⁄2 •

k l ∼ ---------ω

k µ T ∼ ρ ---ω

ε ∼ ωk

• Recognizing the key role of large eddies, and the unimportance of viscosity, Kolmogorov combined this knowledge to postulate an equation for ω . • He called processes

ω

the “rate of dissipation of energy in unit volume and time” and tied it to both dissipative and diffusive

• In 1970, with no knowledge of Kolmogorov’s work, Saffman formulated a k – ω model and called ω “the mean square vorticity of the energy containing eddies” • In recent years, using a combination of singular perturbation method and numerical computations, Wilcox fine-tuned the k – ω formulation and presented his baseline k – ω model in 1988

∂U i ∂k ∂k ∂ ∂k ρ ----- + ρU j ------- = R ij --------- – β * ρkω + ( µ + σ∗ µ T ) ------∂t ∂x j ∂x j ∂x j ∂x j ω ∂U i ∂ω ∂ω 2 ∂ ∂ε ρ ------- + ρU j ------- = α ------ R ij --------- – β ρω + ( µ + σµ T ) ------k ∂t ∂x j ∂x j ∂x j ∂x j

5 α = --9

3 β = -----40

k µ T = ρ --------ω 9 β∗ = --------100

• LOW Re Model

Page 7-17

1 σ = --2

1 σ∗ = --2

(7-38)

7-13 Relationship between ε and ω • If we use

ε = C µ ωk

as the definition of

ω , transforming the ε

equation (with

σk = σε

for simplicity) yields

        

µ + µ T ⁄ σ k ) ∂k ∂ω ω ∂U i ∂ω ∂ω 2 ∂ ∂ε + 2 (-----------------------------ρ ------- + ρU j ------- = α ------ R ij --------- – β ρω + ( µ + σµ T ) ------∂x j k ∂x j ∂ x j∂ x j k ∂t ∂x j ∂x j Cross Diffusion • A Typical turbulent boundary layer velocity profile 60 Sublayer

U+ = U/uτ 40

Log Layer

Viscous Buffer Turbulent Wall Sublayer Zone Zone

Defect Layer Outer Region

U+ = y+ 20 U+ = (1/κ)lny+ + B 0

1

10

102

103

104

y+=uτy/ν

• The cross-diffusion term does the following things: (1) (2) (3) (4) (5)

Gives an excellent prediction for the plane jet. Disrupts defect-layer response to pressure gradient. Distorts the compressible law of the wall Makes κ−ε model very stiff in the sublayer and dictates the need for viscous damping terms Makes κ−ω model more difficult to integrate even for relatively simple flows.

Page 7-18

(7-39)