Hyperviscosity, Galerkin Truncation, and Bottlenecks in Turbulence

Abstract

It is shown that the use of a high power α of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid conservative dynamics with a finite range of spatial Fourier modes. Those at large wavenumbers thermalize, whereas modes at small wavenumbers obey ordinary viscous dynamics [C. Cichowlas et al. Phys. Rev. Lett. 95, 264502 (2005)]. The energy bottleneck observed for finite α may be interpreted as incomplete thermalization. Artifacts arising from models with α > 1 are discussed. PACS numbers: 47.27 Gs, 05.20.Jj A single Maxwell daemon embedded in a turbulent flow would hardly notice that the fluid is not exactly in thermal equilibrium because incompressible turbulence, even at very high Reynolds numbers, constitutes a tiny perturbation on thermal molecular motion. Dissipation in real fluids is just the transfer of macroscopically organized (hydrodynamic) energy to molecular thermal energy. Artificial microscopic systems can act just like the real one as far as the emergence of hydrodynamics is concerned; for instance, in lattice gases the "molecules" are discrete Boolean entities [1] and thermalization is easily observed at high wavenumbers [2]. Another example has been found recently by Cichowlas et al. [3] wherein the Euler equations of ideal non-dissipative flow are (Galerkin) truncated by keeping only a finite -but largenumber of spatial Fourier harmonics. The modes with the highest wavenumbers k then rapidly thermalize through a mechanism discovered by T.D. Lee [4] and studied further by R.H. Kraichnan [5], leading in three dimensions (3D) to an equipartition energy spectrum ∝ k 2 . The thermalized modes act as a fictitious microworld on modes with smaller wavenumbers in such a way that the usual dissipative NavierStokes dynamics is recovered at large scales [25].All the known systems presenting thermalization are conservative. As we shall show themalization may be present in dissipative hydrodynamic systems when the dissipation rate increases so fast with the wavenumber that it mimics ideal hydrodynamics with a Galerkin truncation. This is best understood by considering hydrodynamics with hyperviscosity: the usual momentum diffusion operator (a Laplacian) is replaced by the αth power of the Laplacian, where α > 1 is the dissipativity. Hyperviscosity is frequently used in turbulence modeling to avoid wasting numerical resolution by reducing the range of scales over which dissipation is effective [6].The unforced hyperviscous 1D Burgers and multidimensional incompressible Navier-Stokes (NS) equations are:The equations must be supplemented with suitable initial and boundary conditions. We employ 2π-periodic boundary conditions in space, so that we can use Fourier decompositions such as v(x) = kv k e i k·x . Note that minus the Laplacian is a positive operator, with Fourier transform k 2 , which can be raised to an arbitrary power α. The coefficient µ is taken positive to make the hyperviscous operator dissipative. The Galerkin wavenumber k G > 0 is chos...

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

“…But scientists doing numerical simulations of the NSE, say, for engineering, astrophysical or geophysical applications, have also been using hyperviscosity because it is often believed to allow effectively higher Reynolds numbers without the need to increase spatial resolution. Recently, three of us (UF, WP, SSR) and other coauthors have shown that when using a high power α of the Laplacian in the dissipative term for 3D NSE or 1D Burgers, one comes very close to a Galerkin truncation of Euler or inviscid Burgers, respectively [38]. This produces a range of nearly thermalized modes which shows up in large-Reynolds number spectral simulations as a huge bottleneck in the Fourier amplitudes between the inertial range and the far dissipation range.…”

Statistical properties of turbulence: An overview

“…13 for the case α = 200]. Such thermalisation effects in the Galerkin-truncated Euler equation have also attracted a lot of attention [137]; and the link between bottlenecks and thermalisation has been explored in our recent work [134] to which we refer the interested reader. …”

“…It turns out that such a bottleneck does not occur in the conventional Burgers equation. However, it does [134] occur in the hyperviscous one, in which usual Laplacian dissipation operator is replaced by its α th power; this is known as hyperviscosity for α > 1. We show a representative compensated energy spectrum for the case α = 4 in the left panel of Fig.…”

Generation of non-Gaussian statistics and coherent structures in ideal magnetohydrodynamics

“…These ideas may also be relevant to the relationship between bottleneck effects, statistical mechanics, and effective viscosity caused by eddy noise. 21,22,32 Finally, the breakdown of the correspondence between ideal and dissipative cases is associated with interaction between spectral transfer and the k-space cutoff due to numerical discretization. Understanding this interaction is of importance in assessing the adequacy of resolution in a numerical model.…”

“…The connection between ideal and dissipative dynamics was suggested decades ago 7 and has been of recent interest. 21,22 By t = 1, however, the plots are radically different with small scale coherent structures much less prominent in the ideal run. The correlation coefficient between the current density in ideal and dissipative cases ͑not shown͒ remains very high ͑Ͼ0.98͒ until t Ϸ 0.3, and it declines smoothly toward zero for t Ͼ 1.…”