A dynamic subfilter-scale model for plane parallel flows
Résumé
We present a dynamic model of the subfiltered scales in plane parallel geometry using a generalized, stochastic rapid distortion theory (RDT). This new model provides expressions for the turbulent Reynolds subfilter-scale stresses via estimates of the subfilter velocities rather than velocity correlations. Subfilter-scale velocities are computed using an auxiliary equation which is derived from the Navier–Stokes equations using a simple model of the subfilter energy transfers. It takes the shape of a RDT equation for the subfilter velocities, with a stochastic forcing. An analytical test of our model is provided by assuming delta-correlation in time for the supergrid energy transfers. It leads to expressions for the Reynolds stresses as a function of the mean flow gradient in the plane parallel geometry and can be used to derive mean equilibrium profiles both in the near-wall and core regions. In the near-wall region we derive a general expression for the velocity profile which is linear in the viscous layer and logarithmic outside. This expression involves two physical parameters: the von Karman constant and the size of the viscous layer (which can be computed via a numerical implementation of our model). Fits of experimental profiles using our general formula provides reasonable values of these parameters (kappa= 0.4 to kappa= 0.45, the size of the viscous layer is about 15 wall units). In the core region, we find that the shape of the profile depends on the geometry of the flow; it ranges from algebraic in channel flow, to exponential in the bulk of boundary layers, and linear in plane Couette flow. This classification is consistent with Oberlack's system, which is based on symmetry arguments. Fits of boundary layer flows or channel flows at different Reynolds number over the whole flow region are performed using our results, and are found to be in very good agreement with available data.