A Dynamic model for the Lagrangian Averaged Navier-Stokes-$\alpha$ Equations

A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$\alpha$ (LANS-$\alpha$) equations is developed where the variation in the parameter $\alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $\alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$\alpha$ 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-$\alpha$ simulations using a fixed $\alpha$. In order to evaluate the applicability of the dynamic LANS-$\alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $\alpha$ changes in the wall normal direction. Near a solid wall, the length scale $\alpha$ 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, $\alpha$ 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-$\alpha$ model. The dynamic LANS-$\alpha$ model has the potential to extend the applicability of the LANS-$\alpha$ equations to more complicated anisotropic flows.