AMS 691 Special Topics in Applied Mathematics Lecture 7

1. AMS 691Special Topics in Applied MathematicsLecture 7 James Glimm Department of Applied Mathematics and Statistics, Stony Brook University Brookhaven National Laboratory

2. Turbulence Theories • Many theories, many papers • Last major unsolved problem of classical physics • New development • Large scale computing • Computing in general allows solutions for nonlinear problems • Generally fails for multiscale problems

3. Multiscale Science • Problems which involve a span of interacting length scales • Easy case: fine scale theory defines coefficients and parameters used by coarse scale theory • Example: viscosity in Navier-Stokes equation, comes from Boltzmann equation, theory of interacting particles, or molecular dynamics, with Newton’s equation for particles and forces between particles

4. Multiscale • Hard case • Fine scale and coarse scales are coupled • Solution of each affects the other • Generally intractable for computation • Example: • Suppose a grid of 10003 is used for coarse scale part of the problem. • Suppose fine scales are 10 or 100 times smaller • Computational effort increases by factor of 104 or 108 • Cost not feasible • Turbulence is classical example of multiscale science

5. Origin of Multiscale Science as a Concept • @Article{GliSha97, • author = "J. Glimm and D. H. Sharp", • title = "Multiscale Science", • journal = "SIAM News", • year = "1997", • month = oct, • }

6. Four Useful Theories forTurbulence • Large Eddy Simulation (LES) and Subgrid Scale Models (SGS) • Kolmogorov 41 • PDF convergence in the LES regime • Renormalization group }

7. LES and SGS • Based on the idea that effect of small scales on the large ones can be estimated and compensated for. K41

8. Conceptual framework for convergence studies in turbulence • Stochastic convergence to a Young measure (stochastic PDE) • RNG expansion for unclosed SGS terms • Nonuniqueness of high Re limit and its dependence on numerical algorithms • Existence proofs assuming K41

9. PDF Convergence • @Article{CheGli10, • author = "G.-Q. Chen and J. Glimm", • title = "{K}olmogorov's Theory of Turbulence and Inviscid Limit of the • {N}avier-{S}tokes equations in ${R}^3$", • year = "2010", • journal = "Commun. Math. Phys.", • note = "Submitted for Publication",

10. Idea of PDF Convergence • “In 100 years the mean sea surface temperature will rise by xx degrees C” • “The number of major hurricanes for this season will lie between nnn and NNN” • “The probability of rain tomorrow is xx%”

11. Convergence of PDFs • Distribution function = indefinite integral of PDF • PDF tends to be very noisy, distribution is regularized • Apply conventional function space norms to convergence of distribution functions • L1, Loo, etc. • PDF is a microscale observable

12. Convergence • Strict (mathematical) convergence • Limit as Delta x -> 0 • This involves arbitrarily fine grids • And DNS simulations • Limit is (presumably) a smooth solution, and convergence proceeds to this limit in the usual manner

13. Young Measure of a Single Simulation • Coarse grain and sample • Coarse grid = block of n4 elementary space time grid blocks. (coarse graining with a factor of n) • All state values within one coarse grid block define an ensemble, i.e., a pdf • Pdf depends on the location of the coarse grid block, thus is space time dependent, i.e. a numerically defined Young measure

14. W* convergence X = Banach space X* = dual Banach space W* topology for X* is defined by all linear functional in X Closed bounded subsets of X* are w* compact Example: Lp and Lq are dual, 1/p + 1/q = 1 Thus Lp* = Lq Exception: Example: Dual of space of continuous functions is space of Radon measures

15. Young Measures Consider R4 x Rm = physical space x state space The space of Young measures is Closed bounded subsets are w* compact.

16. LES convergence • LES convergence describes the nature of the solution while the simulation is still in the LES regime • This means that dissipative forces play essentially no role • As in the K41 theory • As when using SGS models because turbulent SGS transport terms are much larger than the molecular ones • Accordingly the molecular ones can be ignored

17. LES convergence is a theory of convergence for solutions of the Euler, not the Navier Stokes equations • Mathematically Euler equation convegence is highly intractable, since even with viscosity (DNS convergence, for the Navier Stokes equation), this is one of the famous Millenium problems (worth $1M).

18. Compare Pdfs:As mesh is refined; as Re changes • w* convergence: multiply by a test function and integrate • Test function depends on x, t and on (random) state variables • L1 norm convergence • Integrate once, the indefinite integral (CDF) is the cumulative distribution function, L1 convergent

19. Variation of Re and mesh: About 10% effect forL1 norm comparison of joint CDFs for concentration and temperature. Re 3.5x104-6x106 (left); mesh refinement (right)

20. Two equations, Two TheoriesOne Hypothesis • Hypothesis: assume K41, and an inequality, an upper bound, for the kinetic energy • First equation • Incompressible Navier Stokes equation • Above with passive scalars • Main result • Convergence in Lp, some p to a weak solution (1st case) • Convergence (weak*) as Young’s measures (PDFs) (2nd case)

21. Incompressible Navier-Stokes Equation (3D)

22. Definitions • Weak solution • Multiply Navier Stokes equation by test function, integrate by parts, identity must hold. • Lp convergence: in Lp norm • w* convergence for passive scalars chi_i • Chi_i = mass fraction, thus in L_\infty. • Multiply by an element of dual space of L_\infty • Resulting inner product should converge after passing to a subsequence • Theorem: Limit is a PDF depending on space and time, ie a measure valued function of space and time. • Theorem: Limit PDF is a solution of NS + passive scalars equation.

23. RNG Fixed Point for LES(with B. Plohr and D. Sharp) Unclosed terms from mesh level n (Reynolds stress, etc.) can be written as mesh level quantities at level n+1, plus an unclosed remainder. This can be repeated at level n+2, etc. and defines the basic RNG map. Fixed point is the full (level n) unclosed term, written as a series, each term (j) of which is closed at mesh level n+j

24. RNG expansion at leading order Leading order term is Leonard stress, used in the derivation of dynamic SGS. Coeff x Model = Leonard stress Coefficient is defined by theoretical analysis from equation and from model. Choice of the SGS model is only allowed variation.

25. Nonuniqueness of limit • Deviation of RT alpha for ILES simulations from experimental values (100% effect) • Dependence of RT alpha on different ILES algorithms (50% effect) • Experimental variation in RT alpha (20% effect) • Dependence of RT alpha on experimental initial conditions (5-30% effect) • Dependence of RT alpha on transport coefficients (5% effect) • Quote from Honein-Moin (2005): • “results from MILES approach to LES are found to depend strongly on scheme parameters and mesh size”

26. Numerical truncation error as an SGS term Unclosed terms = O( )2 Model = X strain matrix = Smagorinsky applied locally in space time. Numerical truncation error = O( ) [Assume first order algorithm near steep gradients.] For large , O( ) = O( )2 So formally, truncation error contributes as a closure term. This is the conceptual basis of ILES algorithms. Large Re limit is sensitive to closure, hence to algorithm. FT/LES/SGS minimizes numerical diffusion, minimizes influence of algorithm on large Re limit.