4.2.1 Reynolds-Averaged Approach vs. LES

Time-dependent solutions of the Navier-Stokes equations for high Reynolds-number turbulent flows in complex geometries which set out to resolve all the way down to the smallest scales of the motions are unlikely to be attainable for some time to come. Two alternative methods can be employed to render the Navier-Stokes equations tractable so that the small-scale turbulent fluctuations do not have to be directly simulated: Reynolds-averaging (or ensemble-averaging) and filtering. Both methods introduce additional terms in the governing equations that need to be modeled in order to achieve a "closure'' for the unknowns.

The Reynolds-averaged Navier-Stokes (RANS) equations govern the transport of the averaged flow quantities, with the whole range of the scales of turbulence being modeled. The RANS-based modeling approach therefore greatly reduces the required computational effort and resources, and is widely adopted for practical engineering applications. An entire hierarchy of closure models are available in ANSYS FLUENT including Spalart-Allmaras, $k$- $\epsilon$ and its variants, $k$- $\omega$ and its variants, and the RSM. The RANS equations are often used to compute time-dependent flows, whose unsteadiness may be externally imposed (e.g., time-dependent boundary conditions or sources) or self-sustained (e.g., vortex-shedding, flow instabilities).

LES provides an alternative approach in which large eddies are explicitly computed (resolved) in a time-dependent simulation using the "filtered'' Navier-Stokes equations. The rationale behind LES is that by modeling less of turbulence (and resolving more), the error introduced by turbulence modeling can be reduced. It is also believed to be easier to find a "universal'' model for the small scales, since they tend to be more isotropic and less affected by the macroscopic features like boundary conditions, than the large eddies. Filtering is essentially a mathematical manipulation of the exact Navier-Stokes equations to remove the eddies that are smaller than the size of the filter, which is usually taken as the mesh size when spatial filtering is employed as in ANSYS FLUENT. Like Reynolds-averaging, the filtering process creates additional unknown terms that must be modeled to achieve closure. Statistics of the time-varying flow-fields such as time-averages and r.m.s. values of the solution variables, which are generally of most engineering interest, can be collected during the time-dependent simulation.

LES for high Reynolds number industrial flows requires a significant amount of computational resources. This is mainly because of the need to accurately resolve the energy-containing turbulent eddies in both space and time domains, which becomes most acute in near-wall regions where the scales to be resolved become much smaller. Wall functions in combination with a coarse near wall mesh can be employed, often with some success, to reduce the cost of LES for wall-bounded flows. However, one needs to carefully consider the ramification of using wall functions for the flow in question. For the same reason (to accurately resolve the eddies), LES also requires highly accurate spatial and temporal discretizations.