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V&V for Turbulent Mixing and Combustion

James Glimm1,3

Stony Brook University With thanks to:

Wurigen Bo1, Gui-Qiang Chen4, Xiangmin Jiao1, Tulin Kaman1, Hyun-Kyung Lim1, Xaolin Li1, Roman Samulyak1,3, David H. Sharp2, Yan Yu1

n  1. SUNY at Stony Brook n  2. Los Alamos National Laboratory n  3. Brookhaven National Laboratory n  4. Oxford University

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Turbulence and Mixing n  Mixing generated by instabilities

n  Acceleration generated: Rayleigh-Taylor, Couette, Richtmyer-Meshkov

n  Turbulent combustion n  H2 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

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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.

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Classical vs. Young measure (stochastic) convergence

o  Integral of numerical solution un with test function g converges to limit

o  w* convergence in o  Values of g multiply primitive variables: density, energy, … in Rm

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

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nu g ug→∫ ∫

∞

4 4 *1( ) ( )L R L R∞ =

( )*4 41 0( ( )) ( ; ( )m mL R R L R C R∞ =;M

Young Measure of a Single Simulation

n  Coarse grain and sample n  Coarse grid = block of n4 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

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Turbulent Combustion LES with finite rate chemistry

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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 (H2 flame in scram jet)

turbulence scale (Kolmogorov) = 5-10 microns << grid scale = 60-100 microns << chemistry scale = 300 microns

Scram Jet

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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 H2 flame

H2 fuel density in center plane through flame

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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

acceleration force

(1 )1

h Agt

A

g

f ff f

α

ρ ρρ ρ

θ

=

−=

+

=

〈 − 〉=〈 〉〈 − 〉

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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)

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 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

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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

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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

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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

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Medium and fine grid simulations (red and green) are in close agreement.

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Molecular level mixing: second moment of concentration 2 experiments, 3 simulations (one DNS) compared

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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  L1 norm of distribution functions

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Circular RM instability Initial (left) and after reshock (right) density plots. Upper and lower inserts show enlarged details of flow.

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Re = 6000. Theta = normalized second moment of concentrations Theta(T) vs. T (left); Pdf for T (right)

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Kolmogorov-Smirnov Metric for comparison of PDFs

n  Sup norm of integral of PDF differences

n  Also L1 norm of distribution function

1 2 1 2|| || || ( ) ( ) ||x

K Sp p p y p y dy− ∞

−∞

− = −∫

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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

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Convergence, model comparison, intrinsic fluctuation

for reaction rate pdf 1 2 const. exp( / )nACw 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

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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