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AMS 691Special Topics in Applied
MathematicsLecture 7
James Glimm
Department of Applied Mathematics and Statistics,
Stony Brook University
Brookhaven National Laboratory
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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
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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
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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
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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,• }
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Four Useful Theories forTurbulence
• Large Eddy Simulation (LES) and Subgrid Scale Models (SGS)
• Kolmogorov 41• PDF convergence in the LES regime• Renormalization group
}
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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
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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
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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",
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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%”
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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
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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
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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
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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
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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
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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
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• 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).
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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
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Variation of Re and mesh: About 10% effect for
L1 norm comparison of joint CDFs for concentration and temperature.
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Re 3.5x104-6x106 (left); mesh refinement (right)
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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)
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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
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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.
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RNG Fixed Point for LES (with B. Plohr and D. Sharp)
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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
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RNG expansion at leading order
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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.
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Nonuniqueness of limit
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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”
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Numerical truncation error as an SGS term
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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