Mousavi, Roozbeh (2014)
Level Set Method for Simulating the Dynamics of the Fluid-Fluid Interfaces: Application of a Discontinuous Galerkin Method.
Technische Universität Darmstadt
Dissertation, Erstveröffentlichung

Kurzbeschreibung (Abstract)

A discontinuous Galerkin (DG) method was applied for simulating the dynamics of the fluid-fluid interfaces. The numerical implementations were performed in the context of an available in-house code, BoSSS. The flow field was assumed to be governed by a single-set of the Navier-Stokes equation in terms of the phase-dependent density and viscosity fields. As in the discontinuous Galerkin method the variables in each cell are expressed in terms of a polynomial space, the solution may exhibit spurious oscillations in the presence of the steep variations such as the density jumps across the interface. In order to overcome this problem, a diffuse interface assumption was made, according to which a jump is approximated by a continuous variation employing a regularized heaviside function. The interface diffusion is supposed to take place in a region with a reasonable width. Therefore, in order to properly express the smoothed jumps in terms of a polynomial space of a certain degree, only jumps with limited hight could be considered. Otherwise, the grid needs to be highly refined in the interface diffusion region for preventing the non-physical spatial oscillation of the solution. Surface tension effects as well as gravity were also involved in the simulations by adding the corresponding source terms to the Navier-Stokes equation. The interface kinematics was simulated using the level set method. Taking the advantage of the discontinuous Galerkin method, a precise solution to the level set advection equation was achieved. As the regularized Heaviside and delta functions are commonly expressed in terms of the level set function, the level set function needs to remain signed distance in order to keep a uniform diffusion width. The signed distance property of a level set functions was recovered by solving the re-initialization equation. A Godunov's scheme was applied for approximating the Hamiltonian of the re-initialization equation, in order to obtain a solution with a monotonicity preserving behavior. A notable stability improvement was achieved by adding an artificial diffusion along the characteristic lines of the re-initialization equation. The solution showed an appropriate hp-convergence behavior and almost no spurious movement of the interface was detected. For solving the Navier-Stokes equation, an explicit-implicit stiffly stable time integration method was employed combined by a splitting method for decoupling the velocity and pressure fields within the DG framework. This solver, which had been priorly implemented for the single phase formulation of the equation, was used as a basis for implementing a new solver for the multiphase formulation. The multiphase flow solver was verified by considering a number of the test cases, such as a rising bubble.

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