AMS 691Special Topics in Applied

MathematicsLecture 8

James GlimmDepartment 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,• }

Useful Theories for Turbulence

• Large Eddy Simulation (LES) and dynamic Subgrid Scale Models (SGS)

• Kolmogorov 41• PDF convergence in the LES regime

– Observed numerically• K41 implies Lp convergence for LES• 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 / 32 3; 5 / 3

( )

a b a a bE k l t k l t la aa 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",

Subgrid models for turbulent flow

turbulent

turbulent

;

we want an approximate model ( )

(key modeling step)

SGS

SGS SGS

t SGS

SGS

t

F F U F U

F U F

U F U U F U

F U U

U F U U

The choice of is step 2 of the RNG expansion: selection of parameters for the essential variables.

The coefficient multiplies a “model” which is the current length scale observable. (More later)

turbulentturbulent

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%”

13

Convergence of PDFs/CDFs• CDF = Cumulative distribution function =

indefinite integral of PDF• PDF tends to be very noisy, distribution is

regularized• Apply conventional function space norms

to convergence of CDF (distribution functions)– L1, Loo, etc.

• PDF/CDF 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

15

W* convergenceX = 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. W* limit gives a probability measure at each space time point (random variable, or pdf). Actually a pdf is a further assumption, with a density continuous with respect to Lebesgue measure in the state space.

Classical (weak) solution: Young measure = delta function (in state space).

*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 – Convergence for Euler equations, not NS– 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

20

Variation of Re and mesh: About 10% effect for

L1 norm comparison of joint CDFs for concentration and temperature.

21

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 vv

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; continuous functions.– Multiply by an element of dual space of bounded continuous

functions. Dual space is space of measures.– 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 = renormalization group

Three steps:

1. Integrate from mesh level Mn+1 to level Mn

2. Reset coefficients of “essential terms” = coefficients of turbulent models3. Rescale so always at a fixed length scale

We carry out 1, 2 but not 3.

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

26

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

Provides accurate model for unclosed term, but fails to have a definite sign, and so is numerically unstable. This is why models (such as grad U) which are less accurate, but give a definite flux (div grad U) are preferred.

The RNG integration step

• To present the ideas in their simplest manner, we consider the incompressible constant density Euler equation. Consider a mesh division Mn of the periodic unit cube into n3 cells, n on a side.

• Conventionally, we consider finite differences, formed by cell averages of each term in the equation. The cell average of a spatial derivative becomes an average over cell faces, for the flux terms F(U). For nonlinear terms the F and the average do not commute, and the subgrid scale terms are just the differences, which are unclosed terms added to the equations. This term for the momentum equation is called the Reynolds stress.

• In its simplest form (incompressible constant density), the Reynolds stress is a 2-point velocity correlation function, defined relative to a local spatial average.

Reynolds stress in 2 space

V = <v> + v’ = veven + vodd

<vv> = <v> <v> + <v’ v’> (cross terms cancel)

R = <vv> - <v> <v> = <v’ v’>

= <v(1-Peven)v>

R = Reynolds stress

• R = <v v>-<v> <v>• where <…> is an average over a face of the grid Mn. We embed

Mn in the once refined grid Mn+1, and on faces, we introduce the projection operators

I = En+1 + Fn+1 onto Mn+1 functions which are constant on Mn cell faces and those

with mean zero on Mn cell faces.• In this notation,

R = <vFn+1v>.

We iterate this construction, starting from Mn+2, and substitute

I = En+2 + Fn+2

into the formula for R.

R = <vFn+1v> = <vFn+1En+2v> + <vFn+1Fn+2> The result is a term closed on Mn+2 and a remainder.

We continue and expand the remainder, obtaining a closed form expansion.

Assume convergence of the expansion.The remainder, tending to zero, is the Reynolds stress observed at a small (zero) scale.

The turbulent viscosity, observed at a small scale, is the magnitude of the Reynolds stress at this scale, and is vanishing as the scale length goes to zero (for the Euler equation).

Thus the bare coefficient of turbulent viscosity is zero.

Caveats and questions

• The convergence assumption does not apply to the Young measure solutions of DeLellis and Szelklyhidi. Thus we do not know if the D-S solutions have zero bare turbulent diffusion.

• Zero bare turbulent diffusion is related to K41. While there is little scientific doubt concerning the validity of K41 (except perhaps for a modification of the exponent to yield intermittency), we do not know if K41 is (only) a statement of physics, and describes a subclass of solutions of the NS equations, or (say as an upper bound with the -5/3 decay law as an upper bound, to give membership in a positive index Sobolev space) this property is a mathematical property of the NS solutions, and is universally valid as mathematics.

• Alternately stated, are the D-S solutions actually classical weak solutions (Young measures with delta function fluctuations)?

RNG expansion at leading order

33

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

34

Deviation of RT alpha for ILES simulations from experimental values

(100% effect)

Dependence of RT alpha on different ILES algorrithm (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

35

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