Random walk models for particle displacements in inhomogeneous unsteady turbulent flows

Abstract

First a small time analysis is developed for the first and second moments of the velocity (W) and displacement (Z) in one direction of particles marked at a given point in an inhomogeneous unsteady turbulent flow, in terms of the local energy dissipation rate, and the local derivatives of the second and third moments of the vertical component of the velocity field, ∂∼(u²₃)/∂z and ∼(∂u³₃)/∂z. Then the appropriate form of a Langevin equation in inhomogeneous turbulence is suggested, namely, dW=(−W/T_L+a₁)dt+a^1/2₂ dω_t where a₁, a₂, and T_L are functions of the particle position and time, and dω_t is a random Gaussian velocity increment with ∼(dω_t)=0 and ∼((dω_t))²=dt. For simplicity, only one component of the particle motion, W(t), is considered. The functions a₁ and a₂ are determined by relating the random walk model to the Eulerian conservation equations for the mass of the contaminant and volume of the flow (i.e., the continuity equation), using the Fokker–Planck equation and the Eulerian equations for the moments of a vertical velocity. The coefficients a₁ and a₂ reduce to the same form as that obtained by the statistical analysis, namely a₁=∂∼(u²₃)/∂z, (≊dW̄/dt, when t → 0) and a₂=2∼(u²₃)/T_L +d∼(u²₃)/dt (≊2∼(W’²)/T_L+d∼(W’²)/dt, when t → 0). It is shown that the random walk model has the correct behavior as t/T_L → 0. The theory is shown to agree reasonably well with the measurements of mean height and mean vertical displacement of particles released in a convective boundary layer [Q. J. R. Meteorol. Soc. 1 0 2, 427 (1976)]. Yaglom’s [Isv. Atmos. Oceanic Phys. 8, 333 (1972)] surface similarity result is recovered as a special case. For t ≫ T_L, and in a zero‐skewness steady turbulence, the random walk model reduces to the familiar K‐diffusion equation. Some examples are presented to show that mean and mean square particle displacements from the random walk model are virtually identical to those obtained from the analytical solution of the corresponding Eulerian moment equations. Careful analysis is still required when concentration distributions in turbulent flows near a boundary are evaluated using random walk models.