Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence

TLDR

Two‑dimensional turbulence differs from three‑dimensional turbulence but shares randomness and convective nonlinearity; its mean‑square vorticity is conserved by convection, its dissipation vanishes with viscosity, while convective mixing raises the vorticity‑gradient, implying a nonzero limiting decay rate κ, a vorticity‑cascade at high Reynolds numbers, and an equilibrium spectrum governed solely by κ and ν, with energy and vorticity components approaching a similarity state determined by total energy. The study aims to test the validity of Kolmogoroff equilibrium ideas by numerically integrating the unaveraged two‑dimensional equations of motion with random initial conditions. The authors performed a spectral‑line numerical integration of the two‑dimensional equations, representing velocity by modes with |(p,q)|<10 and evolving until significant vorticity had cascaded to higher wavenumbers. Although inconclusive, the limited simulations were consistent with the anticipated k⁻¹ vorticity spectrum.

Abstract

Two‐dimensional and three‐dimensional turbulence have different properties, but both contain the two basic ingredients of randomness and convective nonlinearity, and some of the statistical hypotheses which have been proposed for three‐dimensional turbulence should be applicable to two‐dimensional motion. This justifies a numerical integration of the unaveraged equations of motion in two dimensions with random initial conditions as a means of testing the soundness of ideas such as those leading to the Kolmogoroff equilibrium theory. In spatially homogeneous two‐dimensional turbulence, the mean‐square vorticity is unaffected by convection and can only decrease under the action of viscosity. Consequently the rate of dissipation of energy tends to zero with the viscosity (v). On the other hand, the mean‐square vorticity gradient is increased by convective mixing, and it seems likely that the rate of decrease of mean‐square vorticity tends to a nonzero limit κ as ν → 0. This suggests the existence of a “cascade” of mean‐square vorticity at large Reynolds number, and an “equilibrium range” in the vorticity spectrum determined by the parameters κ and ν alone to which familiar dimensional arguments can be applied. Also, the energy and vorticity‐containing components presumably settle down to an approximate similarity state determined by the total energy alone. A numerical integration of the equation of motion to test such similarity relations was attempted at Cambridge some years ago by R. W. Bray. The velocity distribution was represented by spectral lines at vector wavenumbers of the form (p,q), where p and q are integers and (p2 + q2)1/2 < 10, and the integration was carried out to times at which an appreciable amount of vorticity had been transferred to the larger wavenumbers. The results obtained with the rather small computing machine available at the time were not decisive, but they were consistent with the development of the expected k−1 form of the vorticity spectrum.