1 The Mathematics and Numerical Principles for Turbulent Mixing James Glimm 1,3 With thanks to: Wurigen Bo 1 , Baolian Cheng 2 , Jian Du 7 , Bryan Fix 1 , Erwin George 4 , John Grove 2 , Xicheng Jia 1 , Hyeonseong Jin 5 , T. Kaman 1 , Dongyung Kim 1 , Hyun-Kyung Lim 1 , Xaolin Li 1 , Yuanhua Li 1 , Xinfeng Liu 6 , Xingtao Liu 1 , Thomas Masser 1,2 , Roman Samulyak 3 , David H. Sharp 2 , Justin Iwerks 1 ,Yan Yu 1 1. SUNY at Stony Brook 2. Los Alamos National Laboratory 3. Brookhaven National Laboratory 4. Warwick University, UK 5. Cheju University, Korea 6. University of Southern California 7. University of South Carolina
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The Mathematics and Numerical Principles for Turbulent Mixing · Turbulent mixing and turbulent combustion Atomic or molecular level mixing requires a new length scale (the atoms
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1
The Mathematics and Numerical Principles for Turbulent Mixing
James Glimm1,3
With thanks to:
Wurigen Bo1, Baolian Cheng2, Jian Du7, Bryan Fix1,Erwin George4, John Grove2, Xicheng Jia1, Hyeonseong Jin5, T. Kaman1, Dongyung Kim1, Hyun-Kyung Lim1, Xaolin Li1, Yuanhua Li1, Xinfeng Liu6, Xingtao Liu1, Thomas Masser1,2, Roman Samulyak3, David H. Sharp2, Justin Iwerks1,Yan Yu1
1. SUNY at Stony Brook2. Los Alamos National Laboratory3. Brookhaven National Laboratory4. Warwick University, UK5. Cheju University, Korea6. University of Southern California7. University of South Carolina
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Hyperbolic Conservation Laws
( ) 0tU F U Status of existence theory:
1D and small data: Perturbation expansion in wave interactions
existence, uniqueness,smooth dependence on data(G., Liu, Brezan, others)
1D and 2x2 system: Existence (Diperna, Ding, Chen, others)Measure valued solutions, then shown to beclassical weak solutions
2D: special solutions only
3
Nature of Solutions (1D) Solutions are typically discontinuous Shock waves form Solution space is functions of bounded
variation and L_infty Solutions are interpreted in the weak
sense, as distributions
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Compensated Compactness Proofs (for 1D, large data, 2x2 systems)
First establish existence of solutions in a very weak space of measure valued distributions
Then (more difficult) show that such a weak solution is a classical weak solution, as a distribution:
Theorem (Gangbo and Westdickenberg): For 2D, 3D: only first step (measured valued solution) Isentropic equations Comm. PDEs (In press) Limit may not proved to satisfy the original equation
( ) 0 all test functions tU F U
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Obstacles to extensions Some 3x3 and larger systems are unstable in
BV norm (but perhaps these are not physically motivated)
In 2D, even for gas dynamics, special solutions can be unbounded. Also BV is unstable.
Nonuniqueness, 2D: Scheffer (1989), Snirelman DeLellis and Szelkelyhidi (Archives) Volker Elling (numerical) Lopez, Lopez, Lowengrub and Zheng (numerical)
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Numerical implications: Standard view Typically numerical solutions appear to
be convergent Problems with existence seems to be a
strictly mathematical concern This point of view is incorrect as
mathematics and as physics This lecture: to identify and cure
problems as mathematics and as physics
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Implications, continued For many problems, physical regularization (viscosity, mass
diffusion, etc.) is very small. Even if included in the simulation, the effects are under resolved.
Thus effects of physical regularization are dominated by numerical effects such as numerical mass diffusion.
Large effort in V&V = Verification and Validation Verification is proof that the numerical solutions are bona fida
solutions of the mathematical equations. This step fails if mathematical solutions are nonunique or discontinuous
in dependence on initial conditions (loss of well posedness) or fail to exist or if the nonuniqueness etc. is resolved numerically by artifacts of algorithm
Validation is agreement with physical experiments In practice, validation fails dramatically for turbulent mixing
(Rayleigh-Taylor instability).
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Chaotic Mixing: A challenge to the standard view Solutions are unstable on all length
scales Under mesh refinement, new structures
emerge In this sense there is no convergence Optimistically, we hope that the large scale
structures converge and the new small scale ones that emerge under mesh refinement will not influence the large scale ones
This hope is partially correct and partially not
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Failure of the standard view Turbulent mixing and turbulent
combustion Atomic or molecular level mixing requires a
new length scale (the atoms or molecules) And a change in the laws of physics at
these length scales or above. New terms in the conservation laws, to
express mass diffusion, viscosity, heat conduction:( )tU F U D U
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( )tU F U D U Even in 1D, and in the limitdistinct solutions occur (Smoller, Conley)
Ratios of diffusion terms at the root of issue
For combustion, dimensionless ratios in D influence mixing properties and combustion rates, thus the global dynamics (Schmidt and Prandtl numbers). In this sense the small scales can easily affect the large ones.
For turbulent mixing, parabolic transport terms can affect solution, especially the ratios of transport terms Density differences drive instabilities, and so mass diffusion
diminishes the instability. Viscosity limits the complexity of the flow and hence the amount of the interfacial mass diffusion. In the limit D 0, the Schmidt number (viscosity/mass diffusion) determines the net amount of mass diffusion and hence the RT instability growth rate.
0D
Parabolic Conservation Laws
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Requirements for simulation of turbulent combustion Averaged flow velocities and pressures as a
function of x,y,z,t Joint probability distributions of the
concentrations of species (fuel, oxidizer) and temperature, as a function of x,y,z,t
This depends on the diffusion matrix D and if the calculation is under resolved, it depends on the numerical code and the numerical D, not the physical one
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Mathematical implications for hyperbolic conservation laws The ultra weak solutions as measure valued
distributions (PDFs) depending on x,y,z,t is exactly the framework of compensated compactness
Existence proofs in this framework should be easier than for classical (weak) solutions
Uniqueness, which typically fails for such weak solution methods, should fail, and is not correct (in the hyperbolic setting), neither as mathematics nor as physics.
On numerical grounds, appears to correctly express the physics of unregularized solutions.
At reshock the fingers of tin are heated to a much higher temperature in the FronTier simulation than the corresponding fingers in the RAGE simulation.
There are at least three possible mechanisms that might be responsible.
Velocity shear in FronTier missing in RAGE
Thermal and Mass diffusion at the interface in RAGE
Differences in the hyperbolic solver After reshock FronTier continues to have
a significantly higher maximum temperature.
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DNS (regularized simulation) convergence of interfacelambda_C = avg. distance to exit a phaselambda_Cmesh = lambda_C/mesh sizelambda_K = Kolmogorov length (smallest eddies)
LHS of left frame shows uniform behavior in mesh units, independent of mesh and physics, for LES regime.RHS of right frame shows uniform behavior relative to meshfor fixed physics, ie mesh convergence for DNS regime.
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Atomic Scale Mixing Properties Atomic scale mixing properties are sensitive
to physical modeling and to numerical methods unless fully resolved (direct numerical simulation = DNS)
Correct simulation for both micro and macro variables: use sub grid models (large eddy simulation = LES)
LES modify equations to compensate for physics occurring on small scales (below the grid size) but not present in the computation.
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Combine strengths from two numerical traditions Capturing likes steep gradients, rapid time
scales Tracking is an extreme version of this idea Often, no subgrid model and so not physically
accurate for under resolved (LES) simulations Turbulence models often use smooth
solutions, slow time scales with significant levels of physical mass diffusion Often, too many zones to transit through a
concentration gradient Best part of two ideas combined in present
study
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Subgrid models for turbulence, etc. Typical equations have the
form Averaged equations:
( )tU F U U
( )
( ) ( )
( ) ( ) ( )
t
SGS
U F U U
F U F U
F U F U F U
is the subgrid scale model and corrects for grid errorsSGSF
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Subgrid models for turbulent flow
turbulent
turbulent
(key modeling step)
SGS
t SGS
SGS
t
F U F U F U
U F U U F U
F U U
U F U U
;
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Subgrid Scale Models (Moin et al.) No free (adjustable) parameters in the SGS terms Parameters are found dynamically from the
simulation itself After computing at level Delta x, average solution
onto coarser mesh. On coarse mesh, the SGS terms are computed two ways: Directly as on the fine mesh with a formula Indirectly, by averaging the closure terms onto the coarse
grid. Identity of two determinations for SGS terms becomes an
equation for the coefficient, otherwise missing. Assume: coefficient has a known relation to Delta x and
otherwise is determined by an asymptotic coefficient. Thus on a fine LES grid the coefficient is known by above algorithm.
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Typical atomic level observable Mean chemical reaction rate (no
subgrid model needed for computation of w)
/1 2
1 2
1 2
/1 2
Activation Temperature
mass fraction of species
; defined at fixed
( ) probability distribution for
( ) mean reaction rate
AC
AC
AC
T T
AC
i
T
T TT
T
w f f e
T
f i
f fT
f f
d T T
w f f e d T
L
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Re = 300. Theta(T) vs. T (left); Pdf for T (right) 1 2
1 2
f f
f f
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Re = 3000. Theta(T) vs. T (left); Pdf for T (right)
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Re = 300k. Theta(T) vs. T (left); Pdf for T (right)
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Kolmogorov-Smirnov Metric for comparison of PDFs Sup norm of integral of PDF
differences PDF data is noisy and the smoothing
in the K-S metric is needed to obtain coherent results
1 2 1 2|| || || ( ) ( ) ||x
K Sp p p y p y dy
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Convergence properties for reaction rate pdf for
Conclusion: numerical convergence of chemical reaction rates, using LES SGS models for high Schmidt number flows
1 2 const. exp( / )ACw f f T T
Errors for pdf for reaction rate w, compare coarse to fine, medium to fine and relative fluctuations in coarse grid
0.250.030.09300K
0.490.040.493K
0.240.030.04300
fluct. cm to f
c to fRe
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Convergence of w pdf Not just mean converges Moments to all order, and full
distribution converges
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Microphysics to study combustion:Primary breakup of fuel jet injection to engineParameters from diesel engine
Above: no cavitation bubbles. Below: with cavitation
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Comparison to experiment:simulation with and without cavitation bubbles
Velocity of jet tip Mass flux through observation window asa measure of jet spreading
Reynolds numbervs. time
Base case: specification of numerical parameters for insertion of cavitation bubbles
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3D Simulation
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Comparison of simulation and experiment (Argon National Laboratory)
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Simplified engine geometry
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Jet in Cross Flow
Mach 1 cross flow with a2D (planar) jet of dieselfuel.
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Conclusions Chaotic flow simulations are sensitive to numerical and physical
modeling Several examples of natural physical problems RM mixing: Temperature, concentration and chemical reaction
rate PDFs are sensitive to transport, to numerical algorithms (under resolved) and are convergent with use of SGS model (no adjustable parameters).
RM mixing: mesh convergence including microphysical variables (verification)
RT mixing: agreement of simulation with experiment (validation)
Solutions as measure valued distributions from the compensated compactness theory provides a useful framework for interpretation of simulations.
Simulations suggest that this framework might be a basis for 2D/3D existence proofs.