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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 wayfrom data contained in the simulation itself. The dynamic model is initiallytested in forced and decaying isotropic turbulent flows where $\alpha$ isconstant in space but it is allowed to vary in time. It is observed that byusing the dynamic LANS-$\alpha$ procedure a more accurate simulation of theisotropic homogeneous turbulence is achieved. The energy spectra and the totalkinetic energy decay are captured more accurately as compared with theLANS-$\alpha$ simulations using a fixed $\alpha$. In order to evaluate theapplicability of the dynamic LANS-$\alpha$ model in anisotropic turbulence, apriori test of a turbulent channel flow is performed. It is found that theparameter $\alpha$ changes in the wall normal direction. Near a solid wall, thelength scale $\alpha$ is seen to depend on the distance from the wall with avanishing value at the wall. On the other hand, away from the wall, where theturbulence is more isotropic, $\alpha$ approaches an almost constant value.Furthermore, the behavior of the subgrid scale stresses in the near wall regionis captured accurately by the dynamic LANS-$\alpha$ model. The dynamicLANS-$\alpha$ model has the potential to extend the applicability of theLANS-$\alpha$ equations to more complicated anisotropic flows.Comment: 17 pages, 17 figure

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