The Mathematics and Numerical Principles for Turbulent Mixing

Transcription

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 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 1

2 Hyperbolic Conservation Laws U t F (U ) 0 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 be classical weak solutions 2D: special solutions only 2

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 3

4 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: tu F (U ) 0 all test functions 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 4

5 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) 5

6 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 6

7 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). 7

8 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 partially8

9 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: U F (U ) D U t 9

10 Parabolic Conservation Laws U t F (U ) D U D Even in 1D, and in the limit distinct solutions occur (Smoller, Conley) Ratios of diffusion terms at the root of issue 0 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. 10

11 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 11

12 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. 12

13 Numerical example: RichtmyerMeshkov (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 13

14 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. 14

15 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. 15

16 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 2 mixing rate. h / Agt 16

17 2D Chaotic Solutions: Shock implosion of perturbed interface 4 grid levels 17

18 Circular RM instability Initial (left) and after reshock (right) density plots. Upper and lower inserts show enlarged details of flow. 18

19 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 19

20 Explanation Macro observables describe the average mixing properties of the large scale flow Micro observables describe the atomic level mixing properties 20

21 Sensitive and Insensitive Observables Macro Micro Rayleigh-Taylor Sensitive Sensitive RichtmyerInsensitive Sensitive Meshkov RM has a highly unstable interface, in the absence of regularization, diverging as 1/delta x. This divergent interface length or area causes sensit to atomic scale observables. 21

22 Simplistic Error Analysis for Micro Observables in Chaotic Flow ERROR C1 x Interface length (area) Interface length (area) = C2 x 1 ERROR C1 x C2 x C1C2 O 1 1 Error as results from numerical or physical modeling, e.g. numerical mass diffusion or ideal vs. physical transport coefficients 22

23 Chaotic dependence of interface length on mesh: Unregularized simulation Surface Length/Area vs. time Surface Length x Area 23

24 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 24

25 Convergence of mean density of heavy fl Heavy fluid relative density errors, at t = 60 25

26 Convergence of wave trajectories Kolmogorov length = = size of smallest eddy Kmesh Kolmogorov length / x Kmesh 1: DNS <- change in physics -> 26

27 Validation: Agreement between simulation 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. 27

28 Rayleigh-Taylor mixing with small levels of mass diffusion (large Schmidt number). Plot of h = penetratio distance of light fluid into heavy. Physical values of viscosity and initial mass diffusion are us The two simulations a quarter and half of 28 physical problem.

29 Comparison of simulations and experiment (Validation): For Rayleigh-Taylor unstable mixing Immiscible fluids (surface tension) growth rate for mixing zone dimensionless surface tension 29

30 Other RT Simulation Studies 50 year old problem Most simulations disagree with experiment by factor of 2 on overall mixing rate h / Agt 2 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) 30

31 Comparison of FronTier and RAGE for 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 31

32 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. 32

33 DNS (regularized simulation) convergence of interface lambda_c = avg. distance to exit a phase lambda_cmesh = lambda_c/mesh size lambda_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 mesh for fixed physics, ie mesh convergence for DNS regime. 33

34 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. 34

35 Combine strengths from two numerical traditions Capturing likes steep gradients, rapid time scales Turbulence models often use smooth solutions, slow time scales with significant levels of physical mass diffusion Tracking is an extreme version of this idea Often, no subgrid model and so not physically accurate for under resolved (LES) simulations Often, too many zones to transit through a concentration gradient Best part of two ideas combined in present 35 study

36 Subgrid models for turbulence, etc. Typical equations have the form Averaged equations: U t F (U ) U U t F (U ) U F (U ) F (U ) F (U ) F (U ) FSGS (U ) FSGS is the subgrid scale model and corrects for grid e 36

37 Subgrid models for turbulent flow FSGS U F U F U U t F U U FSGS U FSGS U ; turbulent U (key modeling step) U t F U turbulent U 37

38 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 38

39 Typical atomic level observable Mean chemical reaction rate (no subgrid model needed for computation of w) w f1 f 2eT / TAC TAC Activation Temperature f i mass fraction of species i T f1 f 2 ; L defined at fixed T f1 f 2 d (T ) probability distribution for T f1 f 2 T et / TAC d (T ) mean reaction rate w TAC 39

40 Re = 300. Theta(T) vs. T (left); Pdf for T (right) f f 1 2 f1 f 2 40

41 Re = Theta(T) vs. T (left); Pdf for T (right) 41

42 Re = 300k. Theta(T) vs. T (left); Pdf for T (right) 42

43 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 x p1 p2 K S p1 ( y ) p2 ( y )dy 43

44 Convergence properties for w pdf const.for f1 f 2 exp(t / TAC ) reaction rate Errors for pdf for reaction rate w, compare coarse to fine, medium to fine and relative fluctuations in coarse grid Re c to f m to fluct. c f K K Conclusion: numerical convergence of chemical reaction rates, using LES SGS models for high Schmidt number flows 44

45 Convergence of w pdf Not just mean converges Moments to all order, and full distribution converges 45

46 Microphysics to study combustion: Primary breakup of fuel jet injection to engine Parameters from diesel engine Above: no cavitation bubbles. Below: with cavitation 46

47 Comparison to experiment: simulation with and without cavitation bubbles Reynolds numbe observation window as vs. time Velocity of jet tip Mass flux through a measure of jet spreading 47 Base case: specification of numerical parameters for insertion of cavitation

48 3D Simulation 48

49 Comparison of simulation and experiment (Argon National Laboratory) 49

50 Simplified engine geometry 50

51 Jet in Cross Flow Mach 1 cross flow with a 2D (planar) jet of diesel fuel. 51

52 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. 52

53 Thank You Smiling Face: FronTier art simulation Courtesy of Y. H. Zhao 53