Single-Point Velocity Distribution in Turbulence

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 randomly forced 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 {τ} produces velocity PDF tails lnP(v)∝-v⁴ at vvᵣₘₛ,L/τ. For a short-correlated forcing when τL/vᵣₘₛ there is an intermediate asymptotics lnP(v)∝-v³ at L/τvvᵣₘₛ.