Stochastic model error in the LANS-alpha and NS-alpha deconvolution models of turbulence

This paper reports on a computational study of the model error in the LANS-alpha and NS-alpha deconvolution models of homogeneous isotropic turbulence. The focus is on how well the model error may be characterized by a stochastic force. Computations are also performed for a new turbulence model obtained as a rescaled limit of the deconvolution model. The technique used is to plug a solution obtained from direct numerical simulation of the incompressible Navier--Stokes equations into the competing turbulence models and to then compute the time evolution of the resulting residual. All computations have been done in two dimensions rather than three for convenience and efficiency. When the effective averaging length scale in any of the models is $\alpha_0=0.01$ the time evolution of the root-mean-squared residual error grows as $\sqrt t$. This growth rate is consistent with the hypothesis that the model error may be characterized by a stochastic force. When $\alpha_0=0.20$ the residual error grows linearly. Linear growth suggests that the model error possesses a systematic bias which dominates the stochastic terms. Finally, for $\alpha_0=0.04$ the LANS-alpha model possessed systematic bias; however, for this value of $\alpha_0$ the higher-order alpha models that were tested did not.