Subfilter-scale scalar transport for large-eddy simulation

Large-eddy simulation (LES) is an important tool for studying micro- and meso-scale atmospheric flow fields, where horizontal scales are on the order of ten kilometers and practical grid sizes are much larger than what is required to resolve all of the turbulent motions. The quality of the subfilter-scale (SFS) model used to represent the unresolved motions is thus very important for accurate calculations. At the last BLT Symposium, Cederwall and Street (1999) showed that use of an improved SFS model revealed turbulent episodes known to occur in the stable boundary layer that were not found in previous simulations.

The scalar transport of pollutants or suspended matter is difficult to model but is of particular interest in the atmospheric boundary layer. Most simulations use an eddy diffusivity model in which the SFS transport terms are aligned with the resolved-scale strain rate and are entirely dissipative, leading to an inadequate representation of turbulent quantities. Here, a Taylor series expansion model is proposed for the unclosed terms of the scalar transport equation, analogous to the model presented by Street (1999) and Katopodes et al. (2000). This model has no free parameters, is straightforward to derive, and correlates very well with the exact transport term from the direct numerical simulation (DNS) data in a priori tests. The model is of scale-similar form, and thus allows for backscatter, or scalar flux from the small to the large scales.

To model the unclosed SFS terms in the scalar transport equation, we use a Taylor series expansion to express the unfiltered velocity and scalar concentration in terms of their filtered (resolved) counterparts. We can then derive SFS models of arbitrary order of accuracy in the filter width. Furthermore, the SFS model satisfies the evolution equations for the SFS scalar transport to the specified order of accuracy.

A priori tests for several SFS models (including general scale-similarity models) are performed using a DNS dataset for stably-stratified shear flow. For the 6th-order Taylor series model and a LES-DNS grid ratio of 2, the correlation for the SFS scalar dissipation is 0.999; the ratio of rms exact to modeled values is 1.021, indicating that the SFS dissipation is captured to within 2%. On the other hand, the eddy diffusivity model for this case exhibits a correlation of -0.010, and a ratio of 1.904. In meso-scale simulations of the atmosphere, the grid size will be considerably larger than the DNS grid size, making it harder to accurately construct the SFS motions using the knowledge of the resolved scales only. However, even for a LES-DNS grid ratio of 8, the 6th-order model gives a ratio for the SFS scalar dissipation of 1.273, with a correlation much higher than any other model tested, at 0.973.

The Taylor series model is easy to implement and provides an estimate to any order of accuracy desired. While the model will be tested further in actual LES (a posteriori tests), the a priori tests presented here indicate that the Taylor series expansion model is considerably better than the eddy diffusivity closure models traditionally used to represent SFS scalar transport.