The performance of Lagrangian stochastic models in uniform shear flow

Thomson’s theory (JFM, 180, 529-556, 1987) is now well accepted as the basis for constructing a Lagrangian stochastic model for turbulent dispersion from specified Eulerian velocity statistics such as the Eulerian velocity pdf. It is also well known that this theory does not yield a unique solution, and that in general there are many different Lagrangian models consistent with the specified Eulerian statistics. Several recent authors (Wilson and Flesch, Bound. Layer Meteorol., 84, 411-426, 1997; Borgas, Flesch and Sawford, JFM, 332, 141-156, 1997; Reynolds, Bound. Layer Meteorol., 88, 77-86, 1998; Sawford, Bound. Layer Meteorol., 93, 411-424, 1999) have associated this non-uniqueness with the extent to which trajectories rotate.

Uniform shear flow is perhaps the simplest flow for which this non-uniqueness is manifested. Recently (Sawford and Yeung, Phys. Fluids, Submitted, 2000) have compared some Eulerian acceleration statistics from DNS calculation for turbulence in uniform shear with two different Lagrangian stochastic models – Thomson’s (1987) model and an alternative solution due to Borgas (see Sawford and Guest, 8th Symposium on Turbulence and Diffusion, 96-99, 1988). These two models are representative of a whole class of models which are quadratic in the velocity. Sawford and Yeung found that the Eulerian acceleration statistics from DNS, which include an essentially direct measure of the drift term and a measure of the rotation of trajectories, agree very well with those from Thomson’s model, but that there are major discrepancies with those from Borgas’s model. They concluded that these Eulerian acceleration statistics provide a means of discriminating between alternative models and that on these grounds Thomson’s model is to be preferred.

Here we compare DNS results for Lagrangian statistics, the dispersion tensor and the velocity correlation function, against predictions of these two models. We find that for Borgas’s model the velocity correlation function exhibits oscillations, particularly in the correlation between the two velocity components in the plane of the shear. This behaviour is similar to that noted by Borgas, Flesch and Sawford (1997) in their model of axisymmetric turbulence with rotation and is not present in either Thomson’s model or the DNS data. It arises because Borgas’s model imparts a spurious mean rotation to the trajectories in the plane of the shear.

This rotation reduces the dispersion in the plane perpendicular to the axis of rotation, i.e. the shear plane. There is also some evidence in our results for Borgas’s model of oscillations in the dispersion curves, as also observed by Borgas, Flesch and Sawford (1997).

We conclude that the failure of Borgas’s model to correctly represent Eulerian acceleration statistics also degrades its prediction of Lagrangian statistics such as the correlation function and the dispersion. The Eulerian criteria advanced by Sawford and Yeung (2000) thus also serve as a useful measure of the performance of these models in their prediction of Lagrangian statistics.