Markov-chain simulation of particle dispersion in inhomogeneous flows: The mean drift velocity induced by a gradient in Eulerian velocity variance

Abstract

The Langevin equation is used to derive the Markov equation for the vertical velocity of a fluid particle moving in turbulent flow. It is shown that if the Eulerian velocity variance Σ wE is not constant with height, there is an associated vertical pressure gradient which appears as a force-like term in the Markov equation. The correct form of the Markov equation is: w(t + δt) = aw(t) + bΣ wEζ + (1 − a)T L ∂(Σ wE 2)/∂z, where w(t) is the vertical velocity at time t, ζ a random number from a Gaussian distribution with zero mean and unit variance, T L the Lagrangian integral time scale for vertical velocity, a = exp(−δt/T L), and b = (1 − a 2)1/2. This equation can be used for inhomogeneous turbulence in which the mean wind speed, Σ wE and T L vary with height. A two-dimensional numerical simulation shows that when this equation is used, an initially uniform distribution of tracer remains uniform.