AMS 691Special Topics in Applied

MathematicsLecture 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 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

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,• }

Four Useful Theories forTurbulence

• 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.

K413 2 2 3

2/3 5/3

[ ( )] [ ][ ] [ ][ ] [ ][ ][ ]

3 2; 2 / 3

2 3; 5 / 3

( )

a b a a bE k l t k l t l

a a

a b b

E k k

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 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",

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

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 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

13

W* convergence

X = Banach spaceX* = dual Banach spaceW* topology for X* is defined by all linear functional in XClosed bounded subsets of X* are w* compact

Example: Lp and Lq are dual, 1/p + 1/q = 1Thus Lp* = Lq

Exception:

Example: Dual of space of continuous functions is space ofRadon measures

*1

*1

L L

L L

Young Measures

Consider R4 x Rm = physical space x state space

The space of Young measures is

Closed bounded subsets are w* compact.

41( ( ) * ( )m mL R C R L M R

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

• L1 norm convergence– Integrate once, the indefinite integral (CDF) is the

cumulative distribution function, L1 convergent

18

Variation of Re and mesh: About 10% effect for

L1 norm comparison of joint CDFs for concentration and temperature.

19

Re 3.5x104-6x106 (left); mesh refinement (right)

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)

Incompressible Navier-Stokes Equation (3D)

( )

0

( )

mass fraction of species

is a passive scalar because its

equation decouples from the

velocity (Navier Stokes) equation

t

ii i i

i

i

v v v P v

v

vt

i

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.

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.

U

UU UU

U2| / |U U x x