1

V&V for Turbulent Mixing

and Combustion

James Glimm

1,3

Stony Brook University

With thanks to:

Wurigen

Bo

1

,

Gui-Qiang

Chen

4

,

Xiangmin

Jiao

1

,

Tulin

Kaman

1

, Hyun-

Kyung Li m

1

,

Xaolin

Li

1

, Roman Samulyak

1,3

, David H. Sharp

2

, Yan Yu

1

n

1. SUNY at Stony Brook

n

2. Los Alamos National Laboratory

n

3. Brookhaven National Laboratory

n

4. Oxford University

2

Turbulence and Mixing

n

Mixing generated by instabilities

n

Acceleration generated: Rayleigh-Taylor,

Couette

,

Richtmyer-Meshkov

n

Turbulent combustion

n

H

2

flame in engine of a scram jet

n

Large Eddy Simulation (LES)

n

Resolve some but not all turbulent scales; model the rest

n

Models generally largest source of error and uncertainty

n

Stochastic convergence of solutions

n

Probability distribution functions (PDFs) and Young measures

n

Numerical methods

n

Front Tracking +

Subgrid

scale (SGS) models

3

A mathematical theorem

(G-Q Chen, JG)

n

Incompressible Euler equations

n

Assume

Kolmogorov

1941 turbulence bounds

n

Fluctuations in velocity satisfy an integrable power law decay

n

Thus velocity belongs to a

Sobolev

space

n

Bounds and convergence (through a subsequence) to a classical

weak solution

n

With passive scalar (mixing)

n

Volume fractions w* convergent (subsequences), as a

pdf

to a

pdf

limit, i.e. a Young measure solution of the

concentration equation coupled incompressible Euler

solution

Weak vs.

pdf

(Young measure) solutions

n

Weak solution

n

No

subgrid

fluctuations

n

Nonlinear functions not preserved in the limit

n

Nonlinear processes (chemistry) require additional terms

(models) to account for the missing fluctuations

n

pdf

(Young measure) limit

n

Fluctuations and nonlinear functions preserved.

4

Classical vs. Young measure

(stochastic) convergence

o

Integral of numerical solution u

n

with test function g converges to limit

o

w* convergence in

o

Values of g multiply primitive variables: density, energy, … in

R

m

o

Argument of g = space, time

o

integation

over space time

o

g(

x,t

) multiplies density, energy, …

o

w* convergence in

o

Values of g multiply probabilities

o

Argument of g = space, time, density, energy, …

o

Integration over space, time, random density, energy, momentum, …

o

g(

x,t,random

density,…) multiplies probability

5

n

u g ug

→

∫ ∫

∞

4 4 *

1

( ) ( )

L R L R

∞

*

4 4

1 0

( ( )) (;( )

m m

L R R L R C R

∞

;M

Young Measure of a Single Simulation

n

Coarse grain and sample

n

Coarse grid = block of n

4

elementary space time grid

blocks. (coarse graining with a factor of n)

n

All state values within one coarse grid block define an

ensemble, i.e., a

pdf

n

Pdf

depends on the location of the coarse grid block, thus

is space time dependent, i.e. a numerically defined Young

measure

6

Turbulent Combustion

LES with finite rate chemistry

7

w* convergence with stochastic integration over random

variables extends to all nonlinear functions of the solution.

Chemical reaction source terms converge

Flame structure models not needed

Other nonlinear physical processes converge as well

For chemistry (H

2

flame in scram jet)

turbulence scale (

Kolmogorov

) = 5-10 microns

<< grid scale = 60-100 microns

<< chemistry scale = 300 microns

Scram Jet

8

Chemistry computed directly (without models)

in an LES simulation

Removes chemistry model

from turbulent combustion

Only turbulent fluid transport models needed

Model form uncertainty (epistemic : most difficult) eliminated

OH radical density indicating internal layer in H

2

flame

H

2

fuel density in center plane through flame

11

Rayleigh-Taylor Instable Mixing

n

Light fluid accelerates heavy

n

Across a density contrast

interface

n

Overall growth of mixing

region

n

Molecular mixing: second

moment of concentration

2

2 1

2 1

a c c e l e r a t i o n f o r c e

(1 )

1

h Agt

A

g

f f

f f

α

ρ ρ

ρ ρ

θ

−

〈 − 〉

〈 〉〈 − 〉

12

Simulation study of RT alpha for

Smeeton-Youngs experiment #112

n

Agreement with experiment (validation)

n

Agreement under mesh refinement (verification)

n

Agreement under statistical refinement (verification)

n

Agreement with Andrews-

Mueschke

-Schilling (code

comparison; different experiment)

n

Agreement within error bounds established for

uncertain initial conditions (uncertainty

quantification)

13

Experiment : V. S. Smeeton and D. L. Youngs, Experimental

inves;ga;on of turbulent mixing by

Rayleigh-‐Taylor instability (part 3). AWE Report Number 0

35/87, 1987

Simula;on : H. Lim, J. Iwerks, J. Glimm, and D. H. Sharp,

Nonideal Rayleigh-‐Taylor Mixing

The simula+ons reported here were performed on New York Blue, the BG/L computer operated jointly by Stony Brook University and BNL.

Simulation-Experiment

Comparison

14

Does simulation depend on unmeasured initial

data?

Transfer data from early time to initial time

n

Record all bubble minima

n

Fourier analyze these minima

n

Apply linear growth law dynamics to each

mode A(n) to infer initial amplitudes from

early time data

n

Compare results from different early times

for consistency

15

A(n) vs. wave number n at t = 0

A(k) ~ ka with

a = 0. Omit k = 0

mode as this is the

mean bubble position

and is a short wave

length signal.

T = 0

Early time

16

Uncertainty quantification regarding possible

long wave length initial perturbations

Reconstruction of long wave

length initial perturbations

simulated at +/- 100% to allow

for uncertainty in reconstruction,

simulations I, II.

Net effect: +/-5% for alpha.

Fine grid simulation III

fully resolves Weber scale.

Mesh convergence of normalized

second moment of concentrations

17

Medium and fine grid

simulations (red and

green) are in close

agreement.

18

Molecular level mixing:

second moment of concentration

2 experiments, 3 simulations (one DNS) compared

19

Thank You

Smiling Face: FronTier art simulation

Courtesy of Y. H. Zhao

Richtmyer

Meshkov

Instability

n

Circular domain, perturbed circular interface

n

Ingoing circular shock passes through interface, causes instability

n

Reaches origin, reflects there,

recrosses

(

reshocks

) perturbed

interface

n

High level of chaotic mixing

n

Convergence of

pdfs

of concentrations shown

n

L

1

norm of distribution functions

20

21

Circular

RM instability

Initial (left)

and after

reshock

(right) density

plots. Upper and

lower inserts

show enlarged

details of flow.

22

Re = 6000. Theta = normalized

second moment of concentrations

Theta(T) vs. T (left);

Pdf

for T (right)

23

Kolmogorov-Smirnov Metric for

comparison of PDFs

n

Sup norm of integral of PDF differences

n

Also L

1

norm of distribution function

1 2 1 2

| | | | | | ( ) ( ) | |

x

K S

p p p y p y dy

− ∞

−∞

− −

∫

24

Theoretical Model for PDFs

n

d = distance to computed interface

n

Use heat equation, d, elapsed time (since

reshock

), and the turbulent + laminar

diffusion constant in 1D diffusion equation;

predict the mixing PDF

25

Convergence, model comparison, intrinsic fluctuation

for reaction rate pdf

1 2

c o n s t. e x p (/)

n

AC

w f f T T T

−

Re

c to f

m to f

model to f

300

0.04

0.03

0.06

3K

0.49

0.04

0.07

600K

0.09

0.03

0.07

IV.

Uncertainty Quantification

26

27

Conclusions

n

Reacting, turbulent, mixing flows require

n

LES solutions

n

Concentrations converge as Young measures

n

Control over numerical mass diffusion (front tracking)

n

Subgrid

scale turbulence models

n

Pdf

convergence

n

V&V+UQ

n

Testing in realistic examples where “truth” is known

n

High Re flow is universal in Schmidt number

n

Relative to changes in laminar transport properties

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