Single-point velocity distribution in turbulence

Abstract

We show that the tails of the single-point velocity probability distribution function (PDF) are generally non-Gaussian in developed turbulence. By using instanton formalism for the Navier-Stokes equation, we establish the relation between the PDF tails of the velocity and those of the external forcing. In particular, we show that a Gaussian random force having correlation scale $L$ and correlation time $\tau$ produces velocity PDF tails $\ln{\cal P}(v)\propto-v^4$ at $v\gg v_{rms}, L/\tau$. For a short-correlated forcing when $\tau\ll L/v_{rms}$ there is an intermediate asymptotics $\ln {\cal P}(v)\propto-v^3$ at $L/\tau\gg v\gg v_{rms}$.