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 u,,,~ 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 + At) = aw(t) + bu,,i + (1 -a)Tr.d(&)/dz, where w(t) is the vertical velocity at time t, [ a random number from a Gaussian distribution with zero mean and unit variance, r, the Lagrangian integral time scale for vertical velocity, a = exp( -AI/T,), and b = (1 -a*)'/'. This equation can be used for inhomogeneous turbulence in which the mean wind speed, crWwE and T, vary with height. A two-dimensional numerical simulation shows that when this equation is used, an initially uniform distribution of tracer remains uniform.

Recipe for 1‐D Lagrangian particle tracking models in space‐varying diffusivity

“…This is necessary, as neutrally buoyant particles would otherwise accumulate in low-diffusivity areas as shown by Visser (1997). It should be noted that the existence of this deterministic term has been established previously in the atmospheric literature (e.g., Legg and Raupach 1982;Sawford 1985). The last term on the right-hand side of Eq.…”

Turbulence structure in a deciduous forest

“…Some Lagrangian models assume a homogeneous turbulence field inside the canopy (Raupach, 1987), yet field and wind tunnel measurements indicate that turbulence is actually inhomogeneous inside plant canopies (Wilson et al, 1982;Raupach et al, 1986). Other Lagrangian models account for inhomogeneous turbulence (Wilson et al, 1981;Legg and Raupach, 1982). However, information on the vertical profiles of Lagrangian time-scales and velocity statistics is required and further testing of these models is needed.…”

Deriving turbulence characteristics from the COSMO numerical weather prediction model for dispersion applications

“…In LPDM the trajectory of each particle is calculated using the actual wind speed at the position of the particle, which is decomposed into a mean and a turbulent component. The mean wind component is taken directly from the COSMO model, while the turbulent component is computed using the Langevin equation (Legg and Raupach, 1982). Evaluation of this equation requires the following turbulence variables at the particle's position: autocorrelation function and the Lagrangian timescale (parameterized according to Taylor (1921), and the standard deviation of the wind fluctuations (σ k ).…”