Numerical Proof of Self-Similarity in Burgers' Turbulence

Details Numerical Proof of Self-Similarity in Burgers' Turbulence Numerical Proof of Self-Similarity in Burgers' Turbulence

1996-02-23

arxiv.org/abs/patt-sol/9602005v1

Nonlinear Sciences

Pattern Formation and Solitons

LaTeX, 18 pages (no figures) Hardcopy with figures available from britta@pdc.kth.se, Physics of Fluids A (submitted)

Scientific paper

We study the statistical properties of solutions to Burgers' equation, $v_t + vv_x = \nu v_{xx}$, 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.

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