1. The use of dynamical RG in the development of spectral subgrid models of turbulence
The turbulence theory group at edinburgh has for the past 3 decades focused on the use of renormalization methods in the study of turbulence. This has been a two pronged approached; the first being the use of renromalized perturbation schemes simlilar to those in particle physics in the view of closing the moment heirachy of the Navier Stokes equations. The other of which this talk is concerned, is the use of RG in helping in removing degrees of freedom in the view of assisting simulations.
Khurom Kiyani, David McComb
Turbulence Theory group, School of Physics, University of Edinburgh

2. 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
We will start by talking about the subgrid modelling problem in Large eddy simulations and then go onto how the application of the renormalization group can help. We describe the basic RG algorithm and then focus in onto the obstacles needed in constructing a suitable scheme which leads us into the main construction of our method in adopting the RG scheme to the subgrid modelling problem - being the need for a conditional average. The results are then presented and in particular the application of our scheme to the advection of a passive scalar field by a turbulent velocity field. Finally if we have some time I will breifly mention outstanding problems with the present theory and some strategies for tackling them. There is a lot of stuff in the details here - so questions are thoroughly encouraged. In particular if you think I am saying something wrong - interject. Talk has been catered for this specific subject so no real space kadanoff blocking etc.

3. Phenomenology

4. Incompressible spectral Navier-Stokes eqn
We will be working with the divergence-free Fourier transformed Navier-Stokes equation with no mean velocity
relatively arbitrary
Simultaneous excitation of many different length and time scales; strongly coupled; dissipative non-equillibrium phenomena
Confine ourselves to HI iNSE with zero mean field (isotropy) and statistically stationary (driven by a ”stirring” force)

5. 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)

6. Dimensionless NSE Move to the dimensionless form of the NSE
Where the local Reynolds number is
work in shorthand notation
where

7. Richardson energy cascadeew ed e
Statistics (we’ll need this later)
Characteristic dissipation length scale

8. Scaling, self-similarity & K41
From dimension arguments, Kolmogorov showed that for very large Re there exists an intermediate inertial range with scaling
independent of viscosity and forcing.
Turbulence ‘forgets it’s roots’
McComb (1990)
Generalized homogeneity
N=b1.585
f(x,y)=bf(baxx,bayy)
Sierpinski gasket
Bad press due to obsession with power laws. Power laws and scaling are important because they are universal characteristic signals of physical systems -- a way to identify them independent of boundary/initial conditions. They are a deeper result of symmetries and covariance requirements of physical laws. They are also good tests for models and theories (e.g. closure problem in turbulence DIA)
Animation from:

9. Inertial Range -5/3 gradient
log E(k)
log k
Here be dragons
Coherent structures , etc.
Inertial Range -5/3 gradient
Dissipation Range
End of known NSE world kL kd

10. 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)
=k0
Approx DNS limitations go as N~Re9/4
5123 -> ~4000 Re
Pipe flow transition~2x103
Analogies with molecular properties I.e. modelling turbulence effects as a viscosity type term. Notice that while we want an eddy viscosity for our momentum equation to get some detailed dynamics, the subgrid eddy viscosity is usually calculated from the energy transfer equation I.e. from averaged quantities, so any phase information is lost.

11. Dynamical RG analysis

12. 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 is an iterative method for reducing the number of degrees of freedom (DoF’s) in a problem involving many DoF’s.
In our context of fluid turbulence, this can be interpreted as the elimination of Fourier velocity fluctuation modes.
RG in k-space
Coarse-grain or average out the effect of the high-k modes and add it onto the kinematic viscosity.
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.
Instead of solving the NSE over all the band -- do it in bits. MACRO from MICRO. Rescaling and covariance requirements needed for bootstrapping the RG. If you don’t have covariance then you have to add more terms to your RG eqns; if you don’t rescale then your approximations might not be correct at the next stage of your RG. You need to get your system at each stage of the RG to look as much like the original system as possible. When you have done this on everything covariance, scales, AND parameters then your RG can’t do anything more. Also the rescaling builds in the check for generalized homogeneity. So when you have a fixed point of the RG transformation, you have scale-invariance.

13. D I S C L A I M E R
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
The places where it has been done nearer to the Gaussian fixed point, they have used the invariance of the correlation function to get critical exponents by using the standard widom scaling type method.
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)

14. Coarse-graining

15. Conditional average with asymptotic freedom
u(k) - conditional field; w(k) - ensemble realisations

16. ~ small
~ small

17. Partitioned equations & the eddy viscosity
NSE0
NSE1
NSE2
NSE3
NSEFP n Re
RG parameter space
Coarse-grain
Rescale
Quantities being renormalized: n & local Reynolds #
Iterate
*M. E. Fisher, Rev. Mod. Phys , (1998) [Nice picture of whats happening in RG]

18. k1 k1=(1-h)k0 k2 k2=(1-h)k1 k3 kc RG iteration Use for LES E(k) Where
Slightly deceptive picture/map of the RG flow -- but good to show validity of our approximations k1
k1=(1-h)k0 k2
k2=(1-h)k1 k3 kc
RG iteration
Use for LES
E(k)
Where
0 < h < 1 k0 k

19. RG recursive eqns and approximations
‘Assymptotic freedom’
We use the E=4pik^2Q relation to substitute here. Then we approximate E in the band with the kolmogorov form. Why? Cos we have nothing else - no other known statistics. Also as the RG proceeds we expect this approximation to get better and better. Difference of FNS type approaches and current method of RG.

20. Results

21. RG map - Evolution of (scaled) eddy viscosity with RG iteration

22. RG fixed point eddy viscosity (scaled)
nN(k/kN)
k/kN

23. Eddy viscosity (unscaled)

24. What eddy viscosities should look like from Direct Numerical Simulations
* A. Young, PhD Thesis, University of Edinburgh (1999)

25. 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)

26. 323 LES using the RG subgrid model -- comparisons
Chi2 RG
48.6
TFM
77.3
DNS
88.21 RG
TFM
2563 DNS
K41 comparison
Model
Chi2 RG
205.9
TFM
276.3
DNS
35.9
2563 comparison
Results from the work of C. Johnston, PhD Thesis, Edinburgh Uni (2000)

27. Passive scalar convection
* H. A. Rose, J. Fluid Mech. 81 4, (1977)

28. RG fixed point eddy diffusivity (scaled)

29. Prandtl number Independence
Pr*=n*/c*

30. Variation of the Kolmogorov (a) and Obukhov-Corrsin (b) constants with shell width h
Ef(k)=befe-1/3k-5/3

31. Slaved modes & Near-grid interactions
u+ = u~+ + u++
Problems -- pathological divergence over here, have to introduce cut-off -> not desired

32. ??? 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)

33. Parting thoughts Thank you! End
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