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

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

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

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 is used for coarse scale part of the problem. –Suppose fine scales are 10 or 100 times smaller –Computational effort increases by factor of 10 4 or 10 8 –Cost not feasible –Turbulence is classical example of multiscale science

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

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

PDF 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",

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 –L 1, L oo, etc. PDF is a microscale observable

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

Young Measure of a Single Simulation Coarse grain and sample –Coarse grid = block of n 4 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 13

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: L p and L q are dual, 1/p + 1/q = 1 Thus L p * = L q Exception: Example: Dual of space of continuous functions is space of Radon measures

Young Measures Consider R 4 x R m = physical space x state space The space of Young measures is Closed bounded subsets are w* compact.

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

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).

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 L 1 norm convergence –Integrate once, the indefinite integral (CDF) is the cumulative distribution function, L 1 convergent 18

Variation of Re and mesh: About 10% effect for L 1 norm comparison of joint CDFs for concentration and temperature. 19 Re 3.5x x10 6 (left); mesh refinement (right)

Two equations, Two Theories One 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 L p, some p to a weak solution (1 st case) –Convergence (weak*) as Young’s measures (PDFs) (2 nd case)

Incompressible Navier-Stokes Equation (3D)

Definitions Weak solution –Multiply Navier Stokes equation by test function, integrate by parts, identity must hold. L p convergence: in L p 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.

RNG Fixed Point for LES (with B. Plohr and D. Sharp) 23 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

RNG expansion at leading order 24 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.

Nonuniqueness of limit 25  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”

Numerical truncation error as an SGS term 26 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.