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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

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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.

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Numerical example: Richtmyer-Meshkov (shock accelerated) turbulent mixing

Circular shock starts at outer edge of (1/2) circle.

Moves inward, crosses a perturbed circular density difference

Reaches origin, reflects, recrosses the density difference

Highly unstable flow Illustrates all previous points

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Numerical example: Primary breakup of jet (diesel fuel); Jet in cross flow (jet engine) Liquid leaves narrow nozzle at speed near Mach 1.

Breaks up rapidly into small droplets. Breakup is essential feature of flow to predict

combustion rates. Burning is mainly film burning, and so is rate limited by surface area of droplets.

For scram jet, fuel leaves combustion chamber in 1 millisec, and must complete combustion before that.

Flows are only stabilized by transport (viscosity, mass diffusion or surface tension and heat conduction.

Illustrates main points for chaotic mixing.

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Numerical example:Chemical fuel separation

in a mixer/contactor High shear rate in rotating device (Couette flow)

produces high levels of Kelvin-Helmholtz instability Interface between immiscible fluids (oil and water

based) produces fine scale droplets. Chemical reactions occur on the interface and are

thus interface area limited. Prediction of total droplet area and distribution of

droplet sizes and shapes is needed to predict the chemical reaction rates.

Flow is stabilized by surface tension (Weber number).

Illustrates main points for chaotic flow.

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Numerical Example: Rayleigh-Taylor Turbulent Mixing

Steady acceleration of a density difference

Unstable if acceleration points from heavy to light Water over air

Penetration distance h of light fluid into heavy (bubbles) is a simple measure of mixing rate. 2/h Agt

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2D Chaotic Solutions: Shock implosion of perturbed interface—4 grid levels

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CircularRM instabilityInitial (left)and afterreshock (right) densityplots. Upper andlower insertsshow enlargeddetails of flow.

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Definitions for Turbulent Mixing An observable of the flow is SENSITIVE if it

shows dependence on numerical algorithms and/or on physical modeling (transport).

MACRO observables: edges of mixing zones, trajectories of leading shock waves, mean densities, velocities of each fluid species

MICRO observables: probability density functions for temperature and for species concentration; properties dependent on atomic mixing

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Explanation Macro observables describe the

average mixing properties of the large scale flow

Micro observables describe the atomic level mixing properties

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Sensitive and Insensitive Observables

SensitiveInsensitiveRichtmyer-Meshkov

SensitiveSensitiveRayleigh-Taylor

MicroMacro

RM has a highly unstable interface, in theabsence of regularization, diverging as 1/delta x.

This divergent interface length or area causes sensitivity to atomic scale observables.

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Simplistic Error Analysis for Micro Observables in Chaotic Flow

1

12

11 2 1 2

Interface length (area)

Interface length (area) =

1

ERROR C x

C x

ERROR C x C x C C O

Error as results from numerical or physical modeling, e.g.numerical mass diffusion or ideal vs. physical transport coefficients

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Chaotic dependence of interface length on mesh: Unregularized simulation

Surface Length/Area

vs. time

Surface Length

Areax

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Verification (Macro Observables) A mesh convergence study Analyze heavy fluid density Errors are of two types

Position errors due to errors in locations of shock waves Density errors in the smooth regions between the shock

waves Large reduction in error variance results from

separating these types of errors Some statistical averages performed to achieve

convergence Convergence achieved in the sense of mean

quantities

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Heavy fluid relative density errors, at t = 60

Convergence of mean density of heavy fluid

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Convergence of wave trajectories

Kolmogorov length =

= size of smallest eddy

Kolmogorov length / x

1: DNSKmesh

Kmesh

<- change in physics ->

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Validation: Agreement betweensimulation and experiment Rayleigh-Taylor instability

Classical hydro instability, acceleration driven Three simulation studies based on Front

Tracking show agreement with experiment Immiscible fluids (surface tension is regularizer) Miscible fluids high Schmidt number (viscosity and

initially mass diffused layer are regularizers) Miscible fluids (helium-air) (mass diffusion is

regularizer) All three cases give essentially perfect agreement

to experiment on overall mixing rate.

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Simulation compared to experiment:Rayleigh-Taylor mixing with small levels of mass diffusion (large Schmidt number).

Plot of h = penetration distance of light fluid into heavy.

Physical values of viscosity and initialmass diffusion are used.

The two simulations arequarter and half of physical problem.

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Comparison of simulations and experiment (Validation): For Rayleigh-Taylor unstable mixingImmiscible fluids (surface tension)

growth rate

for mixing zone

dimensionless

surface tension

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Other RT Simulation Studies 50 year old problem Most simulations disagree with experiment

by factor of 2 on overall mixing rate

A few agree with experiment Front Tracking (control numerical mass

diffusion) Particle methods (control numerical mass

diffusion) Mueschke-Andrews-Schilling (control numerical

mass diffusion and use experimental initial conditions)

2/h Agt

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Comparison of FronTier and RAGEfor 2D RM instability

T. Masser: Macro variables not sensitive Micro: Temperature is sensitive

Stony Brook: Macro variables: not sensitive Interface length divergent Micro: Mixture concentrations sensitive

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Major Temperature Differences at Reshock

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.

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Thank YouSmiling Face: FronTier art simulation

Courtesy of Y. H. Zhao