Turbulence and Multiscaling in the Randomly Forced Navier-Stokes Equation

Abstract

We present an extensive pseudospectral study of the randomly forced Navier-Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance ∼ k 4−d−y , where k is the wavevector and the dimension d = 3. We present the first evidence for multiscaling of velocity structure functions in this model for y ≥ 4. We extract the multiscaling exponent ratios ζp/ζ2 by using extended self similarity (ESS), examine their dependence on y, and show that, if y = 4, they are in agreement with those obtained for the deterministically forced Navier-Stokes equation (3dNSE). We also show that well-defined vortex filaments, which appear clearly in studies of the 3dNSE, are absent in the RFNSE. PACS : 47.27.Gs, 47.27.Eq, 05.45.+b, 05.70.Jk Kolmogorov's classic work (K41) on homogeneous, isotropic fluid turbulence focussed on the scaling behavior of velocity v structure functions S p (r) = |v i (x + r) − v i (x)| p , where the angular brackets denote an average over the statistical steady state [1]. He suggested that, for separations r ≡ |r| in the inertial range, which is substantial at large Reynolds numbers Re and lies between the forcing scale L and the dissipation scale η d , these structure functions scale as S p ∼ r ζp , with ζ p = p/3. Subsequent experiments [2] have suggested instead that multiscaling obtains with p/3 > ζ p , which turns out to be a nonlinear, monotonically increasing function of p; this has also been borne out by numerical studies of the three-dimensional Navier-Stokes equation forced deterministically (3dNSE) at large spatial scales [2,3]. The determination of the exponents ζ p has been one of the central, but elusive, goals of the theory of turbulence. One of the promising starting points for such a theory is the randomly forced Navier-Stokes equation (RFNSE) [4][5][6], driven by a Gaussian random force whose spatial Fourier transform f (k, t) has zero mean and a covariancehere k, k ′ are wave numbers, t, t ′ times, i, j Cartesian components in d dimensions, and P ij (k) the transverse projector which enforces the incompressiblity condition. One-loop renormalization-group (RG) studies of this RFNSE yield [4,5] a K41 energy spectrum, namely, E(k) ∼ k 2 S 2 (k) ≡ k 2 |v(k)| 2 ∼ k −5/3 , if we set d = 3 and y = 4; this has also been verified numerically [6]. Nevertheless, these RG studies have been criticised for a variety of reasons [7,8] such as using a large value for y in a small-y expansion and neglecting an infinity of marginal operators (if y = 4). These criticisms of the approxima-tions used in these studies might well be justified; but they clearly cannot be used to argue that the RFNSE is in itself inappropriate for a theory of turbulence. It is our purpose here to check if, indeed, the RFNSE is a good starting point for such a theory. Specifically we want to test whether structure functions in the RFNSE display the same multiscaling as in the 3dNSE for some value of y; if they do, then we can argue that both equations are in the same universality class and the RFNSE can, defensi...

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