Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes

We formulate hydrodynamic equations and spectrally accurate numerical methods for investigating the role of geometry in flows within two-dimensional fluid interfaces. To achieve numerical approximations having high precision and level of symmetry for radial manifold shapes, we develop spectral Galerkin methods based on hyperinterpolation with Lebedev quadratures for $L^2$-projection to spherical harmonics. We demonstrate our methods by investigating hydrodynamic responses as the surface geometry is varied. Relative to the case of a sphere, we find significant changes can occur in the observed hydrodynamic flow responses as exhibited by quantitative and topological transitions in the structure of the flow. We present numerical results based on the Rayleigh-Dissipation principle to gain further insights into these flow responses. We investigate the roles played by the geometry especially concerning the positive and negative Gaussian curvature of the interface. We provide general approaches for taking geometric effects into account for investigations of hydrodynamic phenomena within curved fluid interfaces.Comment: 14 figure

Numerical simulation of incompressible fluid flow by the spectral element method

Tato diplomová práce prezentuje metodu spektrálních prvků. Tato metoda je použita k řešení stacionárního 2-D laminárního proudění Newtonovské nestlačitelné tekutiny. Proudění je popsáno stacionarní Navier-Stokesovou rovnicí. Dohromady s okrajovou pod- mínkou tvoří Navier-Stokesův problém. Na slabou formulaci této úlohy je aplikována metoda spektrálních prvků. Touto discretizací se získá soustava nelineárních rovnic. K obrdžení lineární soustavy je použita Newtonova iterační metoda. Podorobný algorit- mus tvoří jádro Navier-Stokeseva solveru, který je naprogramován v Matlabu. Na závěr jsou pomocí tohoto solveru řešeny dva příklady: proudění v kavitě a obtékání válce. Přík- lady jsou řešeny pro různé Reynoldsovy čísla. První od 1 do 1000 a druhý od 1 do 100.The thesis presents the spectral element method and its application to a steady 2-D laminar flow of an incompressible Newtonian fluid. Main features of this method are presented in the thesis. The flow is governed by the steady Navier-Stokes equation. Together with boundary data they form the steady Navier-Stokes problem. Its weak form is a starting point for the method. A space discretization is applied and it results into a nonlinear system of equations. Due to this, the nonlinearity has to be treated. To obtain a linear system of equations is the Newton iteration method used. This algorithm forms the kernel of a Navier-Stokes solver that is implemented in Matlab. Finally, there are presented two examples: the lid driven cavity flow and the flow over a cylinder. The first one is solved for Reynolds numbers from 1 to 1000 and the second one for Reynolds numbers from 1 to 100.

A Trace Finite Element Method for Vector-Laplacians on Surfaces

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed