Optimal two-dimensional models for wake flows

Abstract

In the case of nominally two-dimensional (2D) cylinders of arbitrary cross section in cross flow, the three-dimensionality of the wake manifests in the form of quasi-streamwise vortices. These three-dimensional (3D) features profoundly influence lift and drag forces. However, a two-dimensional projection of such a flow, where the effects of three-dimensionality are modeled, will be computationally very attractive. One can consider the two-dimensional projection as the limiting case of large eddy simulation, where the spanwise direction has been completely averaged out. The transport equation for the span-averaged spanwise component of vorticity, ω̄z, is considered; the 3D effects to be modeled appear as a subgrid scale flux of torque. It is shown that simple minded eddy viscosity type models that assume the flux vector to be proportional to the spatial gradient of ω̄z are inadequate. Here we extend the optimal modeling formalism [Moser, Balachandar, and Adrian, Turbulence and Internal Flow/Unsteady Aerodynamics and Hypersonics Conference, Annapolis, MD, pp. 269–274 (1998); Langford and Moser, J. Fluid Mech. 398, 321 (1999)] to address issues pertaining to complex flows with multiple directions of inhomogeneity. We present optimal closures for subgrid flux modeled in terms of ω̄z distribution, based on linear and quadratic stochastic approximations. These ideas are tested using the database of flow over a flat plate held normal to a cross flow. It is observed that even the optimal model has about 70% normalized error, indicating that the subgrid flux is only about 30% deterministic. Furthermore, it is observed that local models are inadequate, but there exists a region of nonlocality for model dependence, expanding beyond which does not improve the estimate. Higher order nonlinearities however do not seem to improve the model’s predictability.