The use of dynamical RG in the
development of spectral subgrid
models of turbulence
Khurom Kiyani, David McComb
Turbulence Theory group, School of Physics,
University of Edinburgh
Overview of this talk
• Brief phenomenology of the statistical theory of turbulence
• Large-eddy simulations (LES) & subgrid modeling
• Dynamical renormalization group (RG) method
• Results
- Homogeneous & isotropic turbulence
- Passive scalar advection (by above)
- Other LES comparisons
• Problems with the current schemes - introduction of slaved
modes to handle near-grid terms
Phenomenology
Incompressible spectral
Navier-Stokes eqn
We will be working with the divergence-free Fourier
transformed Navier-Stokes equation with no mean velocity
relatively arbitrary
Homogeneous, Isotropic &
stationary NSE for infinite fluid
The simplest non-trivial case -- shrink the monster to a
smaller monster. Makes the maths a bit easier.
• k-space allows us to deal directly with the many strongly
coupled degrees of freedom.
• Statistically steady state - the only reason why we have
included f in NSE.
• No mean velocity implicitly implies global isotropy
Leaves us with the most quintessential, unadulterated
turbulence -- but pretty artificial(ish)
Dimensionless NSE
Move to the dimensionless form of the NSE
Where the local Reynolds number is
work in shorthand notation
where
Richardson energy cascade
ew
e
Statistics e
d
(we’ll need
this later)
Characteristic
dissipation length scale
Scaling, self-similarity & K41
Generalized homogeneity
N=b1.585
QuickTime™ and a
GIF decompressor
are needed to see this picture.
f(x,y)=bf(baxx,bayy)
From dimension arguments,
Kolmogorov showed that for very
large Re there exists an intermediate
inertial range with scaling
Sierpinski gasket
independent of viscosity and forcing.
Turbulence ‘forgets it’s roots’
Animation from: http://classes.yale.edu/fractals/IntroToFrac/InitGen/InitGenGasket.html
McComb
(1990)
log E(k)
Inertial Range
-5/3 gradient
Dissipation
Range
Here be
dragons
End of known
NSE world
Coherent
structures , etc.
kL
kd
log k
Large Eddy Simulations (LES)
- subgrid modeling problem • Aim: To model the large scales of a turbulent flow whilst
accounting for the missing scales in an appropriate way.
• Using a sharp spectral filter (Heaviside unit-step fn)
Approx DNS
limitations go as
N~Re9/4
5123 -> ~4000 Re
Pipe flow
transition~2x103
=k0
Dynamical RG analysis
Renormalization Group (RG)
We can find what kc is and the form of the eddy viscosity
using Renormalization Group (RG) techniques.
What is RG?
RG in k-space
• RG is an iterative method
for reducing the number of
degrees of freedom (DoF’s)
in a problem involving
many DoF’s.
• Coarse-grain or average out
the effect of the high-k modes
and add it onto the kinematic
viscosity.
•In our context of fluid
turbulence, this can be
interpreted as the
elimination of Fourier
velocity fluctuation modes.
•Rescale the variables so that
the new renormalized NSE look
like the original one.
•Repeat until you get to a fixed
point - picture does not change.
DISCLAIMER
• Non-equillibrium phenomena different (nastier, richer) monster from
equillibrium physics -- analogies to ferromagnetism etc. quite hard;
Don’t quite know what the order parameter is* (ask me about this at
the end).
• Confining ourselves to LES - so no critical exponents etc. calculated
-- don’t think anyone has obtained K41 from NSE using RG.
• RG has to be formulated appropriately/delicately -- not a magic
black box -> exponents, renormalized quantities etc. You really have
to have an inclination of the ‘physics’ before you start RG’ing.
• Involves approximations (often) and blatant abuses.
However…
Very deep and profound ideas of the perceived physics of the system
and explanation of universality in physically distinct systems
D. Forster et al., Large-distance and long-time properties of a randomly stirred fluid, PRA 16 2, (1977)
* M. Nelkin, PRA 9,1 (1974); Zhou, McComb, Vahala -- icase 36 (1997)
Coarse-graining
Conditional average with asymptotic freedom
u(k) - conditional field;
w(k) - ensemble realisations
~ small
~ small
Partitioned equations & the
eddy viscosity
nPartition
NSE0
Coarse-grain
Rescale
NSE1
NSE2
RG
parameter
space
NSEFP
NSE3
Iterate
Re
Quantities being renormalized: n & local Reynolds #
*M. E. Fisher, Rev. Mod. Phys. 70 2 , (1998) [Nice picture of whats happening in RG]
• Slightly deceptive picture/map of the RG flow -but good to show validity of our approximations
k1=(1-h)k0
k2=(1-h)k1
RG iteration
E(k)
Where
0<h<1
Use for
LES
kc
k3 k2
k1
k0
k
RG recursive eqns and
approximations
‘Assymptotic freedom’
Results
RG map - Evolution of (scaled) eddy viscosity
with RG iteration
Eddy viscosity (unscaled)
What eddy viscosities should look like from
Direct Numerical Simulations
* A. Young, PhD Thesis, University of Edinburgh (1999)
Variation of the Kolmogorov constant a with
shell width h
E(k)=ae2/3k-5/3
*
* K. Sreenivasan, Phys. Fluids 7 11, (1995); P. Yeung, Y. Zhou, PRE 56 2, (1997)
323 LES using the RG subgrid model -- comparisons
Model
Chi2
RG
48.6
TFM TFM
DNS
77.3
88.21
K41 comparison
2563 DNS
RGModel
Chi2
RG
205.9
TFM
276.3
DNS
35.9
2563 comparison
Results from the work of C. Johnston, PhD Thesis, Edinburgh Uni (2000)
Passive scalar convection
* H. A. Rose, J. Fluid Mech. 81 4, (1977)
RG fixed point eddy diffusivity (scaled)
Prandtl number Independence
Pr*=n*/c*
Variation of the Kolmogorov (a) and ObukhovCorrsin (b) constants with shell width h
Ef(k)=befe-1/3k-5/3
Slaved modes & Near-grid
interactions
u+ = u~+ + u++
Problems -- pathological
divergence over here,
have to introduce cut-off
-> not desired
??? Questions ???
• The reason why we do not introduce extra couplings is due
to us not wanting to compute higher order terms like u-u-u- in
an LES -- it would be a poor subgrid model.
• Pessimistic - Possible existence of infinite number of
marginal scaling fields (your approximations are never good
enough)*.
• Optimistic - Apart from cusp behaviour results are not
doing too bad. Get pretty good values for ‘universal’
constants. Eddy viscosity performs just as well as other
leading brands**.
* G. Eyink, Phys. Fluids 6 9, (1994)
** McComb et. al -- (see next slide)
Parting thoughts
• RG of McComb et al. has been used in actual LES.
- W. D. McComb et al., Phys. Fluids 13 7 (2001)
- C. Johnston PhD Thesis, Edinburgh Uni (2000)
• Need more analysis on including near-grid cross terms.
Look at some way of ascertaining fixed point behaviour of
different terms/couplings (relevant scaling fields etc.)
• Maybe have a look at non-perturbative variational
approaches.
Thank you!
End