Analytical LS Model for Sheared Gaussian Homogeneous Turbulence

Abstract (of Wilson et al., 1993; Bound. Layer. Meteo. Vol. 62, 281-290)

Sir G.I. Taylor's (1921; Proc. London Math. Soc. Series 2, Vol. 20) exact Lagrangian solution for spread of a tracer in homogeneous turbulence is a classic tool to test more-complex computations. But it takes no account of mean velocity shear and shear stress (correlation among the components of the velocity-fluctuation vector), complications that are usual in environmental flow.

We introduce a two-dimensional Lagrangian stochastic trajectory model, which is consistent with Thomson's well-mixed condition, for the trajectory of a neutral particle released into linearly-sheared, Gaussian, homogeneous turbulence. By integrating the Fokker-Planck equation corresponding to the stochastic differential equation defining that LS model, we obtain an analytical solution for the joint probability density function p(x,u,t) for the position (x) and velocity (u) vectors at time t. The solution is compared with dispersion experiments conforming to the restrictions of the model and with a short-range experiment performed in highly inhomogeneous turbulence within and above a model crop canopy. When the turbulence intensity, wind shear and covariance are strong the present solution performs better than the simpler solutions (Taylor, 1921; and a one-dimensional solution of the form of ours, due to P.A. Durbin).

Comparison of the analytical LS model with the observed cross-flow profile of the turbulent along-wind heat flux downstream from a heated wire in the wind tunnel turbulence of a model plant canopy (strong mean wind shear and velocity fluctuation covariance): LS Model vs. Measured Turbulent Flux Above the source (which was at height z=hs), the turbulent flux is negative, ie. directed against the mean wind: this is because, upward (positive) vertical velocity fluctuations (w') are correlated negatively with alongwind velocity fluctuations (u').
The experimental data were obtained by B.J.Legg, M.R.Raupach & P.A.Coppin, CSIRO Environmental Mechanics, Australia.

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Analytical LS Model for Sheared Gaussian Homogeneous Turbulence

Abstract (of Wilson et al., 1993; Bound. Layer. Meteo. Vol. 62, 281-290)

Last Modified: 4 Sept 1996