# Alexandr Labovschii : High accuracy numerical methods for fluid flow problems and turbulence modeling

We present several high accuracy numerical methods for fluid flow problems and turbulence modeling.

First we consider a stabilized finite element method for the Navier-Stokes equations which has second order temporal accuracy. The method requires only the solution of one linear system (arising from an Oseen problem) per time step.

We proceed by introducing a family of defect correction methods for the time dependent Navier-Stokes equations, aiming at higher Reynolds' number. The method presented is unconditionally stable, computationally cheap and gives an accurate approximation to the quantities sought.

Next, we present a defect correction method with increased time accuracy. The method is applied to the evolutionary transport problem, it is proven to be unconditionally stable, and the desired time accuracy is attained with no extra computational cost.

We then turn to the turbulence modeling in coupled Navier-Stokes systems - namely, MagnetoHydroDynamics. We consider the mathematical properties of a model for the simulation of the large eddies in turbulent viscous, incompressible, electrically conducting flows. We prove existence, uniqueness and convergence of solutions for the simplest closed MHD model. Furthermore, we show that the model preserves the properties of the 3D MHD equations.

Lastly, we consider the family of approximate deconvolution models (ADM) for turbulent MHD flows. We prove existence, uniqueness and convergence of solutions, and derive a bound on the modeling error. We verify the physical properties of the models and provide the results of the computational tests.

**Category**: Applied Math and Analysis**Duration**: 01:34:47**Date**: April 2, 2009 at 4:25 PM**Views**: 109-
**Tags:**seminar, Applied Math And Analysis Seminar

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