Volume 29, Issue 2
Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations

M. Gunzburger & A. Labovsky

J. Comp. Math., 29 (2011), pp. 131-140

Published online: 2011-04

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  • Abstract

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.

  • Keywords

turbulence modeling stochastic Navier-Stokes equations deconvolution

  • AMS Subject Headings

65M70 76F65.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-29-131, author = {M. Gunzburger and A. Labovsky}, title = {Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {2}, pages = {131--140}, abstract = { 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.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1006-m3244}, url = {http://global-sci.org/intro/article_detail/jcm/8469.html} }
TY - JOUR T1 - Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations AU - M. Gunzburger & A. Labovsky JO - Journal of Computational Mathematics VL - 2 SP - 131 EP - 140 PY - 2011 DA - 2011/04 SN - 29 DO - http://doi.org/10.4208/jcm.1006-m3244 UR - https://global-sci.org/intro/article_detail/jcm/8469.html KW - turbulence modeling KW - stochastic Navier-Stokes equations KW - deconvolution AB - 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.
M. Gunzburger & A. Labovsky. (1970). Effects of Approximate Deconvolution Models on the Solution of the Stochastic Navier-Stokes Equations. Journal of Computational Mathematics. 29 (2). 131-140. doi:10.4208/jcm.1006-m3244
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