Probability density functions in steady-state Burgers turbulence

Probability density functions (PDFs) for steady-state Burgers turbulence supported by white-in-time random forcing at low wave numbers are studied by direct numerical simulation and compared to theoretical predictions. The velocity PDFs decay slightly faster than a Gaussian at large amplitudes. The putative power law exponent alpha of the PDF Q(xi) proportional to \xi\(-alpha) velocity gradient xi is examined at large Reynolds number and found to be approximately 3 or slightly greater. The tail of Q(xi) behaves like \R xi\(-1)exp(-c(\xi\/R xi(f))(theta 1)) at large negative xi, where xi(f) is a forcing parameter. The exponent theta(1) is near unity, which is smaller than predicted by theory. It decreases slowly with the Reynolds number R up to R = 14 000. The central parts of the PDFs of higher velocity space derivatives are found to be cusp-like, and the cusp exponents are measured. The PDF tails are stretched exponentials. (C) 1999 American Institute of Physics. [S1070-6631(99)00608-Xc].