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 $α$ of the PDF $Q(ξ)∝|ξ|−α$ of velocity gradient $ξ$ is examined at large Reynolds number and found to be approximately 3 or slightly greater. The tail of $Q(ξ)$ behaves like $|Rξ|−1exp(−c(|ξ|/Rξf)θ1)$ at large negative $ξ,$ where $ξf$ is a forcing parameter. The exponent $θ1$ 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.