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Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations

Event Type: 
Date and Time: 
Friday, September 1, 2017 - 16:30
CTR Conference Room 103
Event Sponsor: 
Parviz Moin, Director of Center for Turbulence Research
Jan Nordström

We derive boundary conditions for the nonlinear incompressible Navier-Stokes equations following the general recipie given in [1]. We present two formulations stemming from different techniques to diagonalize the boundary terms. Both formulations lead to an energy estimate.

In the first formulation, the boundary conditions are obtained through a suitable set of rotations. In the second formulation, the boundary conditions are derived directly by a standard eigenvalue decomposition [2, 3]. The two formulations differ in character and have different pro’s and con’s.

The rotational technique lead to more natural formulations, but the formulation must be changed depending on whether there is inflow or outflow. The rotation formulation does not always lead to a nonlinear bound.

The characteristic technique leads to more involved formulations, but the formulation retains the same form independent of whether there is inflow or outflow. The characteristic formulation provides a nonlinear bound for the velocity field for both solid wall and far field boundary conditions.

The continuous problem is approximated by using finite differences on Summation-By-Parts (SBP) form. The solid wall boundary conditions are weakly imposed with the Simultaneous-Approximation-Term (SAT) procedure [4]. It is shown that by mimicking the continuous analysis, the resulting nonlinear SBP-SAT scheme is provably energy stable, divergence free and high-order accurate.

[1]  J. Nordström, “A Roadmap to Well Posed and Stable Problems in Computational Physics,” Accepted in Journal of Scientific Computing.

[2]  J. Nordström, N. Nordin, and D. Henningson, “The Fringe Region Technique and the Fourier-method Used in the Direct Numerical Simulation of Spatially Evolving Viscous Flows,” SIAM Journal of Scientific Computing, Vol. 20, No. 4, pp.1365-1393, 1999.

[3]  J. Nordström, K. Mattsson, and C. Swanson, “Boundary Conditions for a Divergence Free Velocity-Pressure Formulation of the NavierStokes Equations, Journal of Computational Physics, Vol. 225, Issue 1, pp. 874-8901, 2007.

[4]  M. Svärd and J. Nordström, “Review of summation-by-parts schemes for initial-boundary-value problems,” Journal of Computational Physics, Vol. 268, pp. 17-38, 2014.

Since 2010 Jan Nordström is a Professor in Scientific Computing and since 2012 serves as the Head of the Division of Computational Mathematics, Department of Mathematics, Linköping University. He was a Research Scientist at the Aeronautical Research Institute of Sweden (FFA) from 1980 to 1995. Nordström serves on the board of Linköping Institute of Technology (LiTH) and the National Supercomputer Center (NSC). Education: 1980 MSc (civ ing) Master of Science in Aeronautics, Royal Institute of Technology (KTH) in Stockholm, Sweden; 1993 PhD (Tekn. Dr) in Numerical Analysis, Department of Scientific Computing, Uppsala University (UU), Sweden; 1999 Docent (Habilitation) in Numerical Analysis, UU.