Transcript Document
AMS 599
Special Topics in Applied
Mathematics
Lecture 8
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 for
Turbulence
• Large Eddy Simulation (LES) and Subgrid
Scale Models (SGS)
– Last week’s lecture
• Kolmogorov 41
• PDF convergence in the LES regime
• Renormalization group
– I will not discuss this topic
– Aside from its use for turbulence to date, its
potential
} is probably high
LES and SGS
• From last lecture
• Based on the idea that effect of small
scales on the large ones can be estimated
and compensated for.
Kolmogorov 41
• The opposite and also very successful idea
in turbulence is that the main coupling and
influence between length scales is that
large scale motions (eddies) influence small
scale motions (eddies) but not the opposite.
• Distinguish three ranges of length scales:
– Large scale motions, very problem
dependent and nonuniversal. Called the
energy containing eddies
– Intermediate scale motions, called the
inertial range, because governed by Euler,
not Navier-Stokes effects
– Dissipation range, in which viscosity plays a
role. Starts at a length scale for which the
flow is laminar (nonturbulent), called the
Kolmogorov scale
• K41 is a theory of the inertial range. It is a
theory for E. It is a theory for E(k) as a
function of k, for frequencies corresponding
to the inertial range
= spatial, temporal
or ensemble average
v' v v
E v 'v '
turbulent kinetic energy
K41 and E(k)
1
2
E (t ) 3 u ( x, t )d3 x
2 R
0
4 k
2
u
(
t
,
k
)
d
kdk
E
(
t
,
k
)
dk
2
E (t , k )
2
0
2/3
k
5/3
G. Batchelor, The theory of homogeneous Turbulence.
Cambridge University Press. 1955, Chapter 6
Dimensional Analysis
Re Re(v, l ) vl /
Re 2, 000 to 10,000 typical for transition to turbulence
3 du 2
=
2 k 2 E (k , t )dk
2 dt
0
= rate of removal of kinetic energy due to viscosity
Dimensionless (Kolmogorov) length and velocity scales
1/4
v3
1/4
v
[ ] [l 2t 1 ];[ ] [l 2t 3 ];[ 3 1 ] [l 4 ];[ ] [l 4t 4 ]
3/4 1/4
v
Re(v, )
1
Inertial range is r l
1/4
•
K41: Hypothesis of Universal
Tubulence in the Inertial Range
Dimension [] of E(k): [ E (k )] [d 2 k ][ E (k )]
2
[l ][l / t ][l ] [l / t ] [v l ]
• E(k) must have the
identical form for
every k in the inertial
range, by the
universal assumption.
• Thus E(k) can depend
only on k and epsilon
2
2
3
3
2
2
K41, concluded
[ E (k )] [l ][t ] [ ][k ] [l ][t
3a 2; a 2 / 3
2
3
2a b 3; b 5 / 3
E (k )
2/3
k
5/3
a
b
2a
3 a
b
][l ]
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%”
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
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).
Two equations, Two Theories
One Hypothesis
• Hypothesis: assume K41, and an indquality, 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)
t v (v v ) P v
v0
i
(v i ) i i
t
i mass fraction of species i
i is a passive scalar because its
equation decouples from the
velocity (Navier Stokes) equation
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.