Finite element solvers and preconditioners for non-rotational and rotational Navier-Stokes equations

Abstract

Navier-Stokes equations (NSE), the governing equations of incompressible ows, and rotational Navier-Stokes equations (RNSE), which model incompressible rotating ows, are of great importance in many industrial applications. In this thesis, several selected preconditioners for solving NSE are compared and analyzed. These preconditioners are then modified for applying to RNSE.

Understanding the physics behind NSE and RNSE is essential when studying these two equations. The derivation of NSE from the law of conservation of mass and law of conservation of momentum is described. RNSE is obtained by changing the frame of reference of NSE to a rotational frame.

The rotating effect leads to the extra Coriolis force term in RNSE. The equations are then scaled to dimensionless form to eliminate the effect of physical units.

In practice, numerical solution of NSE instead of analytic solution is considered. To apply numerical solvers in this thesis, NSE is discretized by backward differentiation formula in time and finite element method in space. The non-linear term is linearized by extrapolation. The existence and uniqueness of the finite element solutions to NSE are shown in this thesis. Discretization and linearization result in a system of linear equations which is of saddle point type.

Generalized minimum residual method (GMRES) is applied to solve the saddle point system so as to improve efficiency. GMRES is combined with preconditioning technique to enhance the convergence. In this thesis, three preconditioners, pressure convection-diffusion (PCD) [18], least squares commutator (LSC) [11] and relaxed dimensional factorization preconditioner (RDF) [4], for non-rotational problems are considered and investigated. The performance of preconditioners is compared in terms of time step dependency, mesh size dependency and Reynolds number (Re) dependency. It is found that PCD shows time step and mesh size independence for small Reynolds number (Re = 500). RDF is the most stable preconditioner among three preconditioners, but it costs slow convergence, which contrasts to the results in [4].

Preconditioners PCD, LSC and RDF are modi_ed to deal with the Coriolis force term in RNSE. Discrete projection method (DPM) [24], an algorithm designed for RNSE, is also considered. This algorithm can also be viewed as a preconditioned iterative method. The time step and Ekman number (Ek) dependency of modi_ed preconditioners and DPM are compared. The numerical results indicates that LSC is the best preconditioner against time step and Ek. DPM is only the second best although it is designed for RNSE. PCD is the worst preconditioner as it shows high Ek dependency.