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 $\ln P\left(v\right)\propto {-v}^4$ at $v\gg v_{\mathrm{rms}},L/\tau$. For a short-correlated forcing when $\tau \ll {L/v}_{\mathrm{rms}}$ there is an intermediate asymptotics $\ln P\left(v\right)\propto {-v}^3$ at $L/\tau \gg v\gg v_{\mathrm{rms}}$.