1
V&V for Turbulent Mixing
and Combustion
James Glimm
1,3
Stony Brook University
With thanks to:
Wurigen
Bo
1
,
GuiQiang
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: RayleighTaylor,
Couette
,
RichtmyerMeshkov
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
(GQ 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
) = 510 microns
<< grid scale = 60100 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
RayleighTaylor 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
SmeetonYoungs 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.
SimulationExperiment
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
KolmogorovSmirnov 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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