# Numerical methods for surface Navier-Stokes equations in stream function formulation

• Numerische Methoden für Oberflächen-Navier-Stokes-Gleichungen in Stromfunktionsformulierung

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022

Abstract

In this thesis, surface Stokes and Navier-Stokes problems in stream function formulation on a simply connected oriented stationary manifold $\Gamma \subset \mathbb{R}^3$ without boundary are considered and stream function based finite element methods are developed, analyzed and numerically investigated. We present a variety of properties of surface differential operators and briefly discuss derivations of surface Navier-Stokes equations. For stationary problems, well-posed variational formulations in primitive velocity and pressure variables are recalled and well-posed stream function formulations are derived. The latter ones can be reformulated as coupled systems of two second order scalar surface partial differential equations for the stream function and the vorticity. Reconstruction approaches for velocity and pressure are introduced. We apply a parametric trace finite element discretization method to these systems. An error analysis of this discretization method is presented for the generalized surface Stokes equations. For the stream function and vorticity system, we make the simplifying assumption that there are no geometrical errors due to surface approximation. Geometrical errors are, however, included in a new error analysis for the reconstruction methods of velocity and pressure, resulting in optimal discretization error bounds. A pressure robustness property of the stream function based method is studied. A variety of numerical experiments is included to validate theoretical findings and to further study the stream function approach. Furthermore, we present results of numerical simulations to illustrate that the method combined with reasonable parameter choices maintains convergence orders also for other stationary problems treated in this thesis, for which no error analyses are available yet. We also study time-dependent problems on a stationary surface. New well-posed weak formulations in primitive variables, in stream function and in stream function - vorticity variables of instationary surface Stokes equations are derived. For instationary surface Navier-Stokes equations, we introduce variational formulations that are appropriate for a finite element discretization method. For both instationary problems, we present new discretization methods based on a parametric trace finite element approach for spatial discretization (as used for the stationary problems) and a time stepping procedure (in form of a second order backward differentiation formula) for discretization in time. For instationary surface Navier-Stokes equations, a second order extrapolation approach is applied for linearization. Results of numerical studies are presented to illustrate the performance of the stream function based method. Dynamics of these instationary (Navier-)Stokes problems are discussed and numerically investigated. All methods have been implemented in the ngsxfem package which is an add-on of the NGSolve software package.

Institutions

• Department of Mathematics [110000]
• Chair of Numerical Mathematics [111710]