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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

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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

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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

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

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Richardson energy cascade Statistics (we’ll need this later) ww dd Characteristic dissipation length scale

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Scaling, self-similarity & K41 Sierpinski gasket Animation from: N=b f(x,y)=bf(b ax x,b ay y) Generalized homogeneity 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)

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

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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) =k 0 Approx DNS limitations go as N~Re 9/ > ~4000 Re Pipe flow transition~2x10 3

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Dynamical RG analysis

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Renormalization Group (RG) We can find what k c 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.

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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 I S C L A I M E R 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

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Partitioned equations & the eddy viscosity Partition Coarse-grain Rescale Quantities being renormalized: & local Reynolds # Iterate NSE 0 NSE 1 NSE 2 NSE 3 NSE FP Re RG parameter space *M. E. Fisher, Rev. Mod. Phys. 70 2, (1998) [Nice picture of whats happening in RG]

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k E(k) k0k0 k1k1 k 1 =(1- )k 0 Where 0 < < 1 k3k3 kckc RG iteration Use for LES k2k2 k 2 =(1- )k 1 Slightly deceptive picture/map of the RG flow -- but good to show validity of our approximations

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RG recursive eqns and approximations ‘Assymptotic freedom’

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Results

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

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k/k N (k/k N ) RG fixed point eddy viscosity (scaled)

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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 with shell width E(k)= k -5/3 * * K. Sreenivasan, Phys. Fluids 7 11, (1995); P. Yeung, Y. Zhou, PRE 56 2, (1997)

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32 3 LES using the RG subgrid model -- comparisons RG TFM DNS ModelChi 2 RG48.6 TFM77.3 DNS88.21 ModelChi 2 RG205.9 TFM276.3 DNS35.9 K41 comparison 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*=

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Variation of the Kolmogorov ( and Obukhov- Corrsin ( ) constants with shell width E (k)= k -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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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) ??? Questions ???

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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. Parting thoughts Thank you! End

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