Application of an Eulerian One Dimensional

Application of an Eulerian One Dimensional Turbulence model to Simulation of Turbulent Jets Naveen K. Punati, James C. Sutherland

Department of Chemical Engineering, University of Utah, Salt Lake City, UT, USA

Abstract

An alternative formulation for One-dimensional turbulence model (ODT), an out-growth of LEM model, a method for stochastic simulation of turbulent ows is proposed and applied for non-reacting jets. The formulation proposed herein solves the governing equations in an Eulerian frame of reference, rather than a Lagrangian frame as is traditionally done. Furthermore, equations for conserved variables, rather than primitive variables, are solved. After discussing the primary dierences between the present formulation and prior formulations, we compare predictions of the model to data from a non- reacting propane-air jet. Ensemble averages for velocity and mixture fraction from multiple realizations of the ow eld are compared with experimental data. The predictions of the proposed approach for both temporal and spatial formulation shows good agreement with experimental data.

1 Introduction Turbulence is, inherently, three dimensional. The vortex stretching that causes the cascade of energy from large to small scales is not present in the NavierStokes equations in fewer than three dimensions. The One-Dimensional Turbulence (ODT) model developed by Kerstein, is formulated on a one-dimensional (1D) spatial domain simulates turbulent ows to avoid the expense of computations in three dimensions by introducing a model for the nonlinear advective terms that drive turbulence through vortex stretching [1].

The ODT model has been successfully applied to sta-

tionary and decaying homogeneous turbulence, shear ow, and various buoyant stratied ows by evolving a single velocity component without accounting for interactions among velocity components [1]. ODT is an outgrowth of the Linear Eddy Model (LEM) developed by Kerstein [2]. LEM is a modeling strategy of maintaining a distinction between turbulent stirring and molecular diusion on a 1D domain.

ODT generalizes LEM by using ideas from mixing length theory to allow the

scalar elds to self-consistently determine the rate of turbulent mixing in the model, diusive mixing is also explicitly represented on the small length scales. Hence turbulent advection and physical diusion are algorithmically distinct processes within the ODT model. ODT model is capable of reproducing the turbulent cascade process, transferring uctuations to successively smaller scales. Pressure-scrambling eects are incorporated in to the model by introducing a vector velocity eld whose three components exchange energy in a manner that emulates pressure velocity interactions and successfully applied to free shear ows by Ashurst [3]. Ashurst [4] generalized ODT to incorporate the kinematic and dynamical eects of density variations, examined performance of the model and the underlying physics of variable-density turbulence with reference to the measured structure and growth rate of variable-density planar mixing layers. Hewson [5] used ODT model to turbulent reacting ows and demonstrated its capability to reproduce dierential diusion eects by studying the conditional means and uctuations of temperature

1

and the species mass fractions.

Hewson and Kerstein [6] applied ODT model to study local ex-

tinction and reignition phenomena in non-premixed turbulent jet ames. Echekki [7] discussed the dierential diusion eects in the near ow region by applying ODT model to simulate turbulent diusion ames. By using ODT-based closure approach to predict scalar statistics to couple with the Reynolds-averaged NavierStokes (RANS), Ranganath [8] predicted the extinction and reignition in piloted methaneair jet diusion ames. ODT model is also used as a subgrid scale modeling technique for LES [9]. In the present paper we discuss the derivation of governing equations solved as part of ODT model in an Eulerian reference frame for temporal and spatial formulation, how the solution variables instead of primitive variables are solved, how the new approach enforces momentum conservation implicitly and calculation of kernel coecients from energy conservation principle. The discussion will be followed by code validation study.

2 Model Formulation In ODT we restrict the computational domain to a one-dimensional line of sight. For the present work we consider a domain aligned with the

x-direction.

The elds dened on the one-dimensional

domain evolve by two mechanisms: molecular evolution and a stochastic process representing advection. For temporal ows (T-ows), each ODT realization represents a time history of the transverse proles of velocities, and/or time histories of scalar proles. For spatially developing ows (S-ows), ODT generates realizations parametrized by

(x; y)

instead of

(x; t),

where

y

is the streamwise coor-

dinate.

2.1 T-ows The integral form of governing equations for conservation of mass,momentum, and energy along one-dimensional domain aligned with

ˆ

∂ρ dV V (t) ∂t ˆ ∂ρux dV V (t) ∂t ˆ ∂ρuy dV V (t) ∂t ˆ ∂ρYi dV V (t) ∂t ˆ ∂ρe0 dV V (t) ∂t where

ux , uy ,

and

usx

x-direction

may be written as

ˆ = −

ρ (ux − usx ) ax ds,

(1)

S(t)

ˆ

ˆ

= −

[ρux (ux − usx ) ax + τxx ax + pax ] dS + ρgx dV, V (t) ˆ ˆ = − [ρuy (ux − usx ) ax + τyx ax ] dS + ρgy dV, S(t) V (t) ˆ ˆ = − [ρYi (ux − usx ) ax + jix ax ] dS + Si dV, S(t) V (t) ˆ = − [ρe0 (ux − usx ) ax + qx ax + τxx ux ax + P ux ax ] dS,

(2)

S(t)

(3)

(4)

(5)

S(t) are the lateral, streamwise, control volume surface velocities respectively.

If the surfaces move with the same velocity as

ux , usx = ux ,

then the above governing equations

reduce to a Lagrangian formulation. Existing ODT implementations solve the governing equations in Lagrangian form. In the traditional approach the assumption

usx = ux

is equivalent to imposing

constant density, and mass conservation is enforced by modifying control volume sizes. The

y -momentum equation is solved in ODT to approximate local shear stress.

The shear stress,

in turn, aects the model for turbulence by inuencing the rate at which model eddies occur.

2

In the current approach we set

usx = 0,

leading to an Eulerian formulation where the control

volumes are xed in space and time. The advection mechanism represented by the stochastic process in ODT denes eddy motions based on variations in streamwise velocity otherwise to say the shear in streamwise direction, so even though the domain is represented in 1-D

y−momentum

equation

is also solved. Setting

usx = 0,

we arrive at the governing equations for the Eulerian form of the ODT model

proposed here,

∂ρ ∂t ∂ρux ∂t ∂ρuy ∂t ∂ρYi ∂t ∂ρe0 ∂t P The temperature

T

∂ρux , ∂x ∂ρux ux ∂P ∂ − − − (τxx ) , ∂x ∂x ∂x ∂ρuy ux ∂ − − (τxy ) , ∂x ∂x ∂ρYi ux ∂ − − (jix ) , ∂x ∂x ∂ρe0 ux ∂ ∂ ∂ − − (qx ) − (τxx ux ) − (P ux ) , ∂x ∂x ∂x ∂x ρRT . M

= −

(6)

=

(7)

= = = =

(8) (9) (10) (11)

is calculated from total internal energy and is subsequently used in the equation

of state (11) to nd the pressure.

Dening the constitutive relationships for

τxx ,τxy , jix

and

qx

completes the formulation.

2.2 S-ows Occasionally we want to represent a spatially evolving ow (S-ow). Equations (6)-(10) are best suited for temporally evolving ows (T-ows). The equations derived for T-ows are readily transformed to S-ows by applying the chain rule,

∂φ ∂t

=

∂φ ∂y ∂y ∂t and noting that

uy ≡

∂y ∂t . Using this, the

governing equations for S-ows represented as

∂ρ ∂y ∂ρux ∂y ∂ρuy ∂y ∂ρYi ∂y ∂ρe0 ∂y

1 ∂ρux , uy ∂x   1 ∂ρux ux ∂P ∂ − + + (τxx ) , uy ∂x ∂x ∂x   1 ∂ρuy ux ∂ − + (τxy ) , uy ∂x ∂x   1 ∂ρYi ux ∂ − + (jix ) , uy ∂x ∂x   1 ∂ρe0 ux ∂ ∂ ∂ − + (qx ) + (τxx ux ) + (P ux ) . uy ∂x ∂x ∂x ∂x

= −

(12)

=

(13)

= = =

(14)

(15)

(16)

2.3 Stochastic simulation method The stochastic process used to represent the eect of turbulence, propsed by Kerstein [3], consists of a sequence of events each of which involves an instantaneous transformation of the velocity and

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scalar elds. These events may be interpreted as the model analogue of individual turbulent eddies which are referred to as eddy events or simply eddies. Each eddy event is characterized by three properties: a length scale, a time scale, and a measure of kinetic energy, and a key physical input to the model is a postulated relationship among these quantities. The advection submodel is specied by dening the mathematical operations performed during an eddy event and by formulating the rules that govern the random selection of events.

Because advection is implemented as an event

sequence rather than a continuous process, the velocity eld does not directly prescribe the uid motions. Motions and velocities are nevertheless closely linked through the dynamics embodied in the event selection rules. An eddy event consists of two mathematical operations. One is a measurepreserving map representing the uid motions associated with a notional turbulent eddy(triplet mapping).

The other is a modication of the velocity proles(kernel transformation) in order to

implement energy transfers prescribed by the dynamical rules.

These operations are represented

symbolically as

(ρu)i [x] = (ρu)i [f (x)] + ci K(x) According to this representation, uid at location

f (x)

is moved to location

(17)

x

by the mapping

operation, which is the model analogue of the advection operator of the Navier-Stokes equations, is applied to all solution variables. The additive term

ci K(x),

aecting only the velocity components,

represents velocity changes due to pressure gradients or body forces and in vector formulation capture pressure-induced energy redistribution among velocity components.

2.3.1 Triplet map The functional form chosen for

f (x),

triplet map, is the simplest of a class of mappings that

satisfy the physical requirements of measure preservation, continuity and scale locality over the eddy interval can be represented in the following form

 x0 ≤ x ≤ x0 + 13 l  3 (x − x0 ) 2l − 3 (x − x0 ) x0 + 13 l ≤ x ≤ x0 + 23 l . f (x) ≡ x0 +  3 (x − x0 ) − 2l x0 + 23 l ≤ x ≤ x0 + l This mapping takes a line segment

[x0 , x0 + l]

(18)

shrinks it to a third of its original length, and

then places three copies on the original domain. The middle copy is reversed, which ensures that property elds remain continuous and introduces the rotational folding eect of turbulent eddy motion. Property elds outside the size-l segment are unaected.

2.3.2 Kernel Transformation K(x) is conveniently dened as K(x) = x−f (x) is continuous and integrates to zero over the eddy interval. The amplitudes ci are chosen to transfer the amount of kinetic energy specied 00 by the model. Change in i-component momentum (ρu)i can be represented as, The kernel

00

0

(ρu)i = (ρu)i + ci K(x), where

0

(ρu)i

(19)

represents triplet mapped momentum eld. Solving for momentum deterministically

instead of velocities along with continuity equation enforces the momentum conservation implicitly avoiding the computation of extra kernel to account for variable density eects.

4

An eddy event causes the kinetic energy of velocity component

1 2

= ∆Ei

ˆ h

(20)

(21)

can be represented as

4Ei =

 1 2 ci A + 2ci Bi , 2

(22)

´ (ρu)0i K Bi = dx. For a three velocity component system conservation of ρ0 kinetic energy, 4Ei = 0, imposes only one constraint on (22) in calculating the amplitudes of ci . Further modeling to determine the three amplitudes is based on the following observation. ci for given i can be chosen so as to add an arbitrarily large amount of kinetic energy to component i,

where

A=

´

to change by an amount

i 00 0 00 0 (ρui ) ui − (ρu)i ui dx # 0 0 ˆ " 00 00 (ρu) (ρu)i (ρu)i 1 i (ρu)i − dx 2 ρ0 ρ0

4Ei =

From (19) and (21),

i

K2 0 dx and ρP

but the maximum amount that can be removed is a nite value, which is evaluated by maximizing the kinetic energy change with respect to

ci

d(4Ei ) 1 = [2ci A + 2Bi ], dci 2 so that

ci = − Dene

Qi

Bi . A

(23)

as the maximum energy that can be removed from velocity component

Qi = −

i,

Bi2 2A

(24)

Motivated by the phenomenological interpretation of pressure scrambling as a tendency to restore isotropy, the amplitudes are further constrained by requiring invariance under exchange of indices. The kinetic energy changes imposed on the velocity components must be of the form,

4Ei where

α

is a model parameter and

The value

α=1

  1 = α −Qi + (Qj + Qk ) , 2

{i, j, k}

is any permutation of the component indices

(25)

{1, 2, 3}.

maximizes the inter-component transfer.

From (22), (24) and (25), expressions for amplitudes of

ci

ci

v u 2 Bi u Bi α = − (1 − α) + ±t A A 2

can be formulated as

! Bk2 + . A A

Bj2

(26)

2.3.3 Eddy rate distribution The physical model, developed by Ashurst [4], used to decide whether to accept a selected eddy is described in this section. The quantity

ρl3 l τ is interpreted as an eddy velocity and τ 2 is interpreted as 5

a measure of the energy of eddy motion. To determine eddy turn over time

τ

this energy is equated

to an appropriate measure of the eddy energy based on the current ow state, which is, available energy of the

i=2

velocity component upon completion of eddy implementation, minus an energy

penalty that reects viscous dissipation eects.

 2 l ∼ τ where

C

2 νavg (1 − α)Q2 + α2 (Q1 + Q3 ) −Z 2 . ρavg l l

(27)

is an adjustable parameter that controls the overall event frequency and

penalty coecient. disallowed.

Z

is a viscous

If the eddy energy cannot overcome the viscous dissipation eects eddy is

The energy penalty introduces a threshold Reynolds number that must be exceeded

for eddy turnover to be allowable. Eddy rate distribution governing the occurrence of eddy events during simulated realizations is dened from scaling analysis and 27 as

λ = =

C , l2 τs   2 νavg C (1 − α) Q2 + α (Q1 + Q3 ) − Z . l3 ρavg l l2

(28)

(29)

2.3.4 Large eddy suppression The eddy selection procedure can occasionally allow the occurrence of large unphysical eddies. This is more commonly seen in the case of free shear ows than wall bounded ows.

A method for

suppressing these highly energetic eddies, developed by Ashurst [4], is implemented into the ODT formulation.

This procedure termed as the scale-reduction method involves auxiliary eddy-rate

computations for each of three equal sub-intervals of the eddy interval[x0 , x0

  λ is evaluated as if the eddy interval were x0 + (j − 1) 3l , x0 + j 3l ,

+ l]. In other words, for j = 1, 2, and 3, respectively.

If any of these three candidate eddies are disallowed due to dominance of the viscous penalty, then the acceptance probability is set to zero. Otherwise it is unchanged from its value computed for the complete eddy interval

[x0 , x0 + l].

In the implementation of this method, the viscous penalty

employed in the sub-interval computations is based on the size of the complete eddy interval, not the sub-interval size.

2.3.5 Eddy selection For each eddy event, eddy length(l) and location(x0 ) selected from randomly generated numbers and ow properties. From scaling analysis eddy length and shear stress value eddy starting location will be calculated. For the selected

x0 , l ,

eddy rate distribution will be calculated using the procedure

outlined so far. Based on the probability density functions dened for of

x0 , l

and

λ,

probability

will be calculated using

probability = where

λdt f (x0 )g(l)

(30)

dt is integration step size and f (x0 ), g(l) are probability density functions for x0 , l respectively

dened as

6

f (x0 ) = g(l) = where

Lp , eddymax ,

and

eddymin

ceddy exp l2



−2Lp l

 ,

1 , eddymax − eddymin

are the most probable eddy length, maximum and minimum eddy

size respectively. If the probability calculated from (30) is greater than a randomly selected number on the interval

[0, 1]

then eddy will be accepted.

3 Results & Discussion 3.1 Non-reacting Propane-Air jet In this work, the radial and axial proles of the mean streamwise velocity and mixture fraction that are predicted by the model (after ensemble averaging) are compared with those of the experiment. The non-reacting propane jet measurements taken as a part of the International-Workshop on Measurement and Computation of Turbulent Non-premixed ames [10] are used for this validation study.

3.1.1 Initial Conditions The initial conditions used for simulating non-reacting propane jet are given in Table 1. The velocity

Table 1: Initial conditions. Reynolds number

68000

Inner Diameter

0.0052 m

Outer Diameter

0.0090 m

Propane Jet velocity

53 m/s

Propane Jet Temperature

294 K

Co-ow air velocity

9.2 m/s

Co-ow Air Temperature

294 K

and scalars proles at the inlet are dened by

       A x − L1 1 x − L2 ϕ(x) = 1 + tanh 1− 1 + tanh , 2 w 2 w where

A

is the amplitude of the change

w

is the width of the transition and

midpoints of the transition. This is illustrated in Figure 1.

7

L1

and

L2

are the

Figure 1: Schematic of tanh prole used to specify scalar and mean velocity proles

3.1.2 T-ows An indirect approach has been adopted to compare the temporal ows simulation results with experimental data. Based on the average streamwise velocity at each time step an ordinary dierential equation is solved to determine the axial location of ODT line,

dy =u ¯y , dt where

u ¯y

is a suitable average of the

y -velocity.

The variation in the mean velocity along the jet centerline is compared with experimental data

4

in Figure 2.

The radial proles of the mean axial velocity at

4, 15, 30, 50

are compared with experimental data in Figure 3.

dierent axial locations,

y/D =

In this work, the experimental

data obtained by seeding the jet stream are used for the comparison with ODT prediction. Good agreement between the ODT results and the experiments is seen at all axial locations except at

y/D = 50.

The spreading rate predicted by the model matches with the experimental data at all

axial locations, but the over prediction in the velocity can be seen at

y/D = 50.

is also predicted by mean centerline velocity decay in Figure 2.

70 Expt ODT

Mean Centerline Velocity

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

x/D

Figure 2: Centerline prole of mean streamwise velocity

8

The same behavior

50 Expt ODT

60

Mean Streamwise Velocity

Mean Streamwise Velocity

70

50 40 30 y/D = 4 20 10 0 −6

−4

−2

0 x/D

2

4

20

y/D = 15

10

−4

−2

0 x/D

2

4

6

25 Expt ODT

Mean Streamwise Velocity

Mean Streamwise Velocity

30 25 20 y/D = 30

15 10 5 −6

30

0 −6

6

Expt ODT

40

−4

−2

0 x/D

2

4

Expt ODT

20

10

5 −6

6

y/D = 50

15

−4

−2

0 x/D

2

4

6

Figure 3: Mean radial proles of streamwise velocity at dierent axial locations

The variation in the mean mixture fraction along the jet centerline is compared with experimental data in Figure 4. As seen from the gure, the mean centerline mixture fraction remains constant for few jet diameters downstream of the jet exit before it starts to decay. This region is called the potential core of the jet. In the potential core region, the eects of viscous shear and diusion are not very signicant and hence the mixture fraction remains unchanged from its nozzle exit value. Beyond this region, the mean mixture fraction decreases rapidly as co-owing air is entrained by the jet. The decay of the centerline mixture fraction predicted by the ODT model is rapid between

10 − 25

jet diameters and slows down after that.

The radial proles of the mean mixture fraction at four dierent axial locations are also compared with experimental data in Figure 5. The proles at

y/D = 4, 15, 30

and

50

show good agreement

between the simulations and the experiment although a slightly higher value of the mixture fraction towards the centerline is predicted by the ODT model at

y/D = 50.

As seen in Figure 4, this region

was over-predicted by the ODT model.

1 Expt ODT

Mean Centerline Mixture Fraction

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40 x/D

50

60

70

80

Figure 4: Centerline prole of mean mixture fraction

9

0.6

1 0.8

Mean Mixture Fraction

Mean Mixture Fraction

Expt ODT

0.6 0.4

y/D = 4

0.2 0 −6

−4

−2

0 x/D

2

4

Expt ODT

0.5 0.4 0.3 y/D = 15

0.2 0.1 0 −6

6

−4

−2

0 x/D

2

4

6

0.4 Mean Mixture Fraction

Mean Mixture Fraction

Expt ODT

0.3

0.2 y/D = 30 0.1

0 −6

−4

−2

0 x/D

2

4

Expt ODT

0.25 0.2 0.15

y/D = 50 0.1 0.05 0 −6

6

−4

−2

0 x/D

2

4

6

Figure 5: Mean radial proles of mixture fraction at dierent axial locations

3.1.3 S-ows Figure 6 shows the comparison between the experimental data and model prediction for centerline

4

velocity decay.

The radial proles of the mean axial velocity at

= 4, 15, 30, 50

are compared with experimental data in Figure 7.

reasonable agreement with data for

y/D ≤ 15,

dierent axial locations,

y/D

Though the prediction is in

signicant dierence is observed for

y/D ≥ 25 which

is also clearly evident in center line velocity decay in Figure 6. The spreading rate predicted by the model matches with the experimental data at all axial locations, but the over prediction in the velocity can be seen for

y/D ≥ 30. 70 Expt ODT

Mean Centerline Velocity

60

50

40

30

20

10

0

0

10

20

30

40

50

60

70

80

90

x/D

Figure 6: Centerline prole of mean streamwise velocity

10

50 Expt ODT

Mean Streamwise Velocity

Mean Streamwise Velocity

80

60 y/D = 4 40

20

0 −6

−4

−2

0 x/D

2

4

y/D = 15

20 10

−4

−2

0 x/D

2

4

6

30 Expt ODT

30

Mean Streamwise Velocity

Mean Streamwise Velocity

30

0 −6

6

35

25 y/D = 30 20 15 10 5 −6

Expt ODT

40

−4

−2

0 x/D

2

4

Expt ODT

25 20

y/D = 50

15 10 5 0 −6

6

−4

−2

0 x/D

2

4

6

Figure 7: Mean radial proles of streamwise velocity at dierent axial locations

The variation in the mean mixture fraction along the jet centerline predicted by the model shows signicant deviation from the experimental data in Figure 8. The decay is over-predicted for

y/D < 15

and under-predicted downstream. Figure 9 shows the comparison between radial proles

predicted by the model and experimental data at dierent axial locations. High Spreading rate and higher mixture fraction values are predicted by the model in the region

y/D ≥ 15

evident from Figure 8 where slow decay is observed in the same region.

1 Expt ODT

Mean Centerline Mixture Fraction

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40

50

60

70

x/D

Figure 8: Centerline prole of mean mixture fraction

11

which is also

0.6 Expt ODT

0.8 0.6

Mean Mixture Fraction

Mean Mixture Fraction

1

y/D = 4

0.4 0.2 0 −6

−4

−2

0 x/D

2

4

0.4

y/D = 15 0.3 0.2 0.1 −4

−2

0 x/D

2

4

6

0.5 Expt ODT

Mean Mixture Fraction

Mean Mixture Fraction

0.4

0 −6

6

Expt ODT

0.5

0.3 y/D = 30 0.2

0.1

0 −6

−4

−2

0 x/D

2

4

y/D = 15

0.3 0.2 0.1 0 −6

6

Expt ODT

0.4

−4

−2

0 x/D

2

4

6

Figure 9: Mean radial proles of mixture fraction at dierent axial locations

4 Conclusions The ODT model of Kerstein [1] was reformulated using an Eulerian approach. Simulation results from both temporal and spatial formulation are compared with experimental data of a spatially developing non-reacting propane jet.

The validation study has shown that the ODT formula-

tion can provide qualitative agreement with experimental data.

Entrainment and mixing eects

are represented better by the temporal formulation than for spatial formulation. Further adjustment/optimization of model parameters, as well as modifying initial conditions and changing mesh resolution may result in better agreement between model prediction and experimental data. The approach proposed herein (Eulerian) is quite dierent from the traditional ODT approaches in the following respects:

•

It solves the momentum equation in the tuations in the

x-direction.

x-direction

and directly incorporates pressure uc-

The implementation described herein treats this in a fully com-

pressible manner.

•

It uses a static mesh whereas other formulations for variable density ow using ODT rely on a Lagrangian mesh where cell volumes change to maintain zero advective uxes through cell boundaries.

•

The eddy events are implemented dierently than traditional ODT approaches.

We apply

(18) directly, interpolating values as needed. In the traditional approach, a discrete analog of (18) is used, where Lagrangian volumes are relocated in space. Thus, while our mesh is unaected by triplet mapping, traditional ODT approaches must remesh when a triplet map occurs.

•

We solve the governing equations for conserved quantities

ρux , ρuy , ρ, ρe0 , ρYi ,

whereas

previous formulations solve the primitive equations. Application of triplet maps to the primitive variables requires additional kernel transformations to ensure conservation; we apply the

12

triplet map to the conserved variables and thus do not require additional kernel transformations. The only kernel transformation required in this formulation is on the momentum to guarantee conservation of kinetic energy. In the event that a nonuniform mesh is employed, an additional kernel transformation is required to maintain conservation. Both temporal and spatial formulations are directly applicable for reacting ows as well. Comparison with turbulent reacting jet ames (Sandia ames D and E) is currently underway. Future work will focus on evaluating ODT against DNS data to determine if statistics from DNS can be reasonably represented by ODT so that ODT can form the basis for subgrid modeling of turbulent reacting ow.

5 References 1. Kerstein, A. R., J. Fluid Mech 392:277 (1999). 2. Kerstein, A. R., J. Fluid Mech 216:411 (1990). 3. Kerstein, A. R., Ashurst, Wm. T., Wunch, S., Nilsen, V., J. Fluid Mech 447:85 (2001). 4. Ashurst, Wm. T., Kerstein, A. R., Phys. Fluids 17:26 (2005). 5. Hewson, J. C., Kerstein, A. R., Combust. Theo. Modelling 5:669 (2001). 6. Hewson, J. C., Kerstein, A. R., Combust. Sci. and Tech 174:35 (2002). 7. Echekki, T., Kerstein, A. R., Dreeben, T. D., Combust. Flame 125:1083 (2001). 8. Ranganath, B., Echekki, T., Combust. Flame 154:23 (2008). 9. McDermott, R. J. (2001). Ph.D. thesis, University of Utah. 10. International workshop on measurement and computation of turbulent nonpremixed ames. http://public.ca.sandia.gov/TNF/abstract.html.

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