Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations (Journal of Computational Mathematics, 2018)
The direct numerical simulation of Navier-Stokes equations in the turbulent regime is not computationally feasible either in the deterministic or (especially) in the stochastic case. Therefore, turbulent modeling must be employed. We consider the family of approximate deconvolution models (ADM) for the simulation of the turbulent stochastic Navier-Stokes equations (NSE). For moderate values of the Reynolds number, we investigate the effect stochastic forcing (through the boundary conditions) has on the accuracy of solutions of the ADM equations compared to direct numerical simulations. Although the existence, uniqueness and verifiability of the ADM solutions has already been proven in the deterministic setting, the analyticity of a solution of the stochastic NSE is difficult to prove. Hence, we approach the problem from the computational point of view. A Smolyak-type sparse grid stochastic collocation method is employed for the approximation of first two statistical moments of the solution – the expected value and variance. We show that for different test problems, the modeling error in the stochastic case is the same as predicted for the deterministic setting. Although the ADMs are arguably only applicable for certain boundary conditions (zero or periodic), we test the model on a problem with a boundary layer and recirculation region and demonstrate that the model correctly predicts the solution of the stochastic NSE with the noise in the boundary data.