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**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

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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.

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Phenomenology

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**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)

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**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)

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**Dimensionless NSE Move to the dimensionless form of the NSE**

Where the local Reynolds number is work in shorthand notation where

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**Richardson energy cascade**

ew ed e Statistics (we’ll need this later) Characteristic dissipation length scale

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**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:

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**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

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**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.

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Dynamical RG analysis

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**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.

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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)

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Coarse-graining

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**Conditional average with asymptotic freedom**

u(k) - conditional field; w(k) - ensemble realisations

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~ small ~ small

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**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]

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**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

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**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.

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Results

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

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**RG fixed point eddy viscosity (scaled)**

nN(k/kN) k/kN

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**Eddy viscosity (unscaled)**

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**What eddy viscosities should look like from Direct Numerical Simulations**

* A. Young, PhD Thesis, University of Edinburgh (1999)

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**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)

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**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)

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**Passive scalar convection**

* H. A. Rose, J. Fluid Mech. 81 4, (1977)

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**RG fixed point eddy diffusivity (scaled)**

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**Prandtl number Independence**

Pr*=n*/c*

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**Variation of the Kolmogorov (a) and Obukhov-Corrsin (b) constants with shell width h**

Ef(k)=befe-1/3k-5/3

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**Slaved modes & Near-grid interactions**

u+ = u~+ + u++ Problems -- pathological divergence over here, have to introduce cut-off -> not desired

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??? 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)

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**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

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