Numerical Proof of Self-Similarity in Burgers' Turbulence

Abstract

We study the statistical properties of solutions to Burgers' equation, $v_t+v{v}_x=\nu v_{xx}$, for large times, when the initial velocity and its potential are stationary Gaussian processes. The initial power spectral density at small wave numbers follows a steep power-law $E_0(k)\sim |k{|}^n$ where the exponent $n$ is greater than two. We compare results of numerical simulations with dimensional predictions, and with asymptotic analytical theory. The theory predicts self-similarity of statistical characteristics of the turbulence, and also leads to a logarithmic correction to the law of energy decay in comparison with dimensional analysis. We confirm numerically the existence of self-similarity for the power spectral density, and the existence of a logarithmic correction to the dimensional predictions.

arXiv commentLaTeX, 18 pages (no figures) Hardcopy with figures available from [email protected], Physics of Fluids A (submitted)