A dynamic model for the Lagrangian-averaged Navier-Stokes-α equations

Abstract

A dynamic procedure for the Lagrangian-averaged Navier-Stokes-α (LANS-α) equations is developed where the variation in the parameter α in the direction of anisotropy is determined in a self-consistent way from the data contained in the simulation itself. In order to derive this model, the incompressible Navier-Stokes equations are Helmholtz filtered at the grid and test filter levels. A Germano-type identity is derived by comparing the filtered subgrid-scale stress terms with those given in the LANS-α equations. Assuming constant α in homogenous directions of the flow and averaging in these directions result in a nonlinear equation for the parameter α, which determines the variation of α in the nonhomogeneous directions or in time. Consequently, the parameter α is calculated during the simulation instead of a predefined value. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where α is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-α procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic-energy decay are captured more accurately as compared with the LANS-α simulations using a fixed α. In order to evaluate the applicability of the dynamic LANS-α model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter α changes in the wall normal direction. Near a solid wall, the length scale α is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, α approaches an almost constant value. Furthermore, the behavior of the subgrid-scale stresses in the near-wall region is captured accurately by the dynamic LANS-α model. The dynamic LANS-α model has the potential to extend the applicability of the LANS-α equations to more complicated anisotropic flows.