Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

CSIC Building (#406), Seminar Room 4122.
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Accurate, Stable and Efficient Navier-Stokes Solvers Based on Explicit Treatment of the Pressure Term

Dr. Jian-Guo Liu

IPST at University of Maryland, College Park

Abstract:   In this talk, I will present numerical schemes for the incompressible Navier-Stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure Poisson equation. The pressure is treated explicitly in time, completely decoupling the computation of the momentum and kinematics equations. The result is a class of extremely efficient Navier-Stokes solvers. Full time accuracy achieved for all flow variables. The key to the schemes is a Neumann boundary condition for the pressure Poisson equation which enforces the incompressibility condition for the velocity field. Irrespective of explicit or implicit time discretization of the viscous term in the momentum equation, the explicit time discretization of the pressure term does not affect the time step constraint. Indeed, we show some unconditional stability properties of the this new formulation for the Stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term. Systematic numerical experiments for the full Navier-Stokes equations indicate that a second order implicit time discretization of the viscous term, with the pressure and convective terms treated explicitly, is stable under the standard CFL condition. Additionally, various numerical examples are presented, including both implicit and explicit time discretizations, using spectral and finite difference spatial discretizations, demonstrating the accuracy, flexibility and efficiency of this class of schemes. In particular, a Galerkin formulation is presented requiring only C0 elements to implement.