Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Abstract

Few rigorous results are derived for fully developed turbulence. By applying the scaling properties of the Navier-Stokes equation we have derived a relation for the energy spectrum valid for unforced or decaying isotropic turbulence. We find the existence of a scaling function ψ. The energy spectrum can at any time by a suitable rescaling be mapped onto this function. This indicates that the initial (primordial) energy spectrum is in principle retained in the energy spectrum observed at any later time, and the principle of permanence of large eddies is derived. The result can be seen as a restoration of the determinism of the Navier-Stokes equation in the mean. We compare our results with a windtunnel experiment and find good agreement.The Navier-Stokes equation (NSE) for hydrodynamic flow has been known for many years, and several interesting numerical results have been found. However, very few analytic statements have been derived from these equations [1,2]. In this Letter we consider decaying isotropic turbulence, i.e. hydrodynamics without external forcing.One well known feature of the hydrodynamic equations is their invariance under re-scaling when the coordinates are scaled by an arbitrary quantity l. Any solution to the NSE can then be mapped onto another solution with a corresponding change in the velocity, the time and the diffusion coefficient ν. The same scaling argument applies to the energy density. In the following we shall show that this leads to a general scaling behavior of the energy density considered as a function of k, t, ν.The result resembles the k 4 backscattering at integral scales obtained in EDQNM closure calculations [3]. In studies of the decay of homogeneous and isotropic turbulence self-similarity in the shape of the energy spectra are often assumed, expressed as the principle of permanence of large eddies (PLE) [2]. In a recent interesting phenomenological study it was proposed that for initial spectra not as steep as k 4 there will be three ranges of self-similarity where PLE only applies to the very large scales [4].We shall now derive the scaling properties of the energy density directly from the scaling properties of the NSE. This has been done by one of the authors in the case where viscosity was ignored [5]. Here we shall generalize these results. Consider the energy density in D dimensions, given bywhere L and K are infrared and ultraviolet cutoffs, respectively, and Ω D is the solid angle. We assume isotropy such that the energy density only depends on the modulus k = |k| of the wave vector. The total energy density is given byUsing the invariance of the unforced NSE under the scalingwhere h is an arbitrary parameter, the relationtrivially follows.In the following we assume the cutoffs such that for the relevant physical range we have 1/L ≪ k ≪ K. We will therefore suppress the explicit dependence of the energy density on the cutoffs. The viscosity provides a physical cutoff for large k while the cutoff 1/L is determined by the boundary conditions at the integral scale.Let...

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