Viscosity-dependent inertial spectra of the Burgers and Korteweg–deVries–Burgers equations

Abstract

We show that the inertial range spectrum of the Burgers equation has a viscosity-dependent correction at any wave number when the viscosity is small but not zero. We also calculate the spectrum of the Korteweg-deVries-Burgers equation and show that it can be partially mapped onto the inertial spectrum of a Burgers equation with a suitable effective diffusion coefficient. These results are significant for the understanding of turbulence.I n a series of papers we, with G. I. Barenblatt and V. M.Prostokishin (1-7), presented a theory based on similarity considerations that derived Reynolds number-dependent scaling laws for the mean velocity profiles in turbulent boundary layers, as well as related Reynolds number-dependent anomalous corrections for the inertial range of turbulence. A mathematical model for the turbulent boundary layer that exhibits this scaling was presented in ref.8. This scaling theory agrees very well with the experimental data.It is a basic postulate of turbulence theory that fully developed turbulence far from walls has, on intermediate scales, small compared with the energy-containing scales but large compared with the dissipation scales, a universal ''inertial'' energy spectrum E(k) of the form E(k) ϭ CD 2/3 k Ϫ␥ , where k is the wave number and D is the rate of energy dissipation (9). In their original analysis Kolmogorov and Obukhov (see ref. 9) deduced that ␥ ϭ 5͞3; in recent years it has been claimed that this ␥ must be corrected by a positive ''intermittency correction'' that is independent of Reynolds number. The prefactor C has always been assumed to be independent of the Reynolds number. To the contrary, we have found from our theory and from analysis of the data that C was a function of the Reynolds number R, and that there was indeed a correction to the Kolmogorov-Obukhov exponent but that it was Reynolds number-dependent and tended to zero as the Reynolds number increased; thus the effects of viscosity are felt even in the range of intermediate, inertial scales.It would, of course, be desirable to deduce the scaling laws from the Navier-Stokes equations, but until this can be done it is of interest to examine the spectral properties of simpler model problems that exhibit an inertial range. This is done in the present article, where we rigorously show that the Burgers equation, one of the simplest models in fluid dynamics, exhibits the phenomena we see in the more complex situation. We also produce a closely related analysis of the spectrum of the Korteweg-deVries-Burgers (KdVB) equation that provides a useful glimpse of what has to be done in turbulence calculations. The Inertial Range of the Burgers EquationAs is well known, the Burgers equationwhere x is a physical space variable, t is the time, is a viscosity and subscripts denote differentiation, has in the limit 3 0 a Fourier transform û(k) with an inertial range spectrum E(k) ϳ k Ϫ2 , as can be readily seen from the fact that the solution u develops shocks whose spectral signature is k Ϫ2 . But what happens to that spectrum ...

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