Computational Fluid Dynamics for Free Surface Flows using the Spectral Element Method

This work aims to develop novel and spectrally accurate numerical solvers for the efficient simulation of nonlinear waves. This is done by mainly solving the incompressible Navier-Stokes (INS) equations with a free surface; however, we have also considered the fully nonlinear potential flow (FNPF) equations, with both modeling approaches facing similar challenges. Both wave models consist of complex coupled systems of partial differential equations that deal with nonlinear properties and time-varying fluid domains. Additionally, solving the models at scale implies designing efficient solver strategies to address the computational bottleneck problem of solving Poisson-type problems during all steps of temporal integration procedures.

In this thesis, we present work done on two types of spectrally accurate numerical methods. Firstly, a single-domain pseudo-spectral method that have a mixed Fourier collocation and Chebyshev collocation method. This method has been used to solve the FNPF and INS equations with a free surface. The model allows for exponential (spectral) error convergence and takes advantage of the inherent periodic nature of the Fourier method to efficiently solve problems in periodic domains. Moreover, the use of the Fast Fourier Transform (FFT) and Fast Co-sine Transform (FCT) allows for efficient O(n log n) computation of derivatives. A novel method for Fourier continuation based on relaxation zones has also been developed for a new numerical wave tank concept, allowing for reproducing measurement campaigns obtained from physical wave tank experiments.

The method allows for non-reflective finite-domain simulations despite the inherent periodicity of the method. Secondly, we propose a multi-domain spectral element method (SEM) based on nodal Lagrange basis functions, mass matrix-based integration, and gradient recovery using global L² projections. This method has solely been used to solve the INS equations with a free surface. Like the pseudo-spectral model, this model also allows for spectral error convergence. Compared to the pseudo-spectral method, SEM allows for greater geometric flexibility and the ability to do reflective finite- domain simulations. However, this comes at the cost of much higher complexity in the implementation of such models. Due to aliasing from nonlinear terms, all models have used mild, but effective anti-aliasing techniques to ensure robust and stable simulations. The spectral methods have been used for spatial discretization and combined with classical Runge-Kutta methods for the temporal discretization using the standard method-of-lines. The new free surface INS model is conceptually demonstrated to work for water wave propagation across varying bathymetry in 2D; however, it remains to be utilized to incorporate structures placed in the fluid domain, that would make it possible to use such simulator in design loops, e.g. for applications that require assessing the wave-induced loads on such structures.

To efficiently solve the occurring linear systems of equations that stems from the involved Poisson-type problems and with a view to enabling massively parallel implementations on modern many-core computing systems, we develop scalable geometric p-multigrid solvers for both numerical models since such solvers have to date not found widespread use despite the potential for accelerating free surface solvers. Multigrid solvers employ a hierarchy of discretization levels, can achieve grid size independent convergence rates, and are therefore attractive choices for larger scale wave simulations. The geometric p-multigrid method takes advantage of the inherent higher order of spectral methods to construct grids through p-coarsening, thus maintaining the high order properties of the numerical discretization and potentially avoiding refining mesh topologies when dealing with wave-structure interaction applications. Moreover, we also combine the proposed multigrid methods with other iterative solvers, such as the preconditioned generalized minimal residual method (GMRES) and the preconditioned defect-correction (PDC) method, to improve robustness and overall performance.

Through numerical experiments for non-trivial standard benchmarks, we show that the developed solvers can achieve spectral error convergence for both free surface wave models. With the use of numerical stabilization techniques, the models can accurately do long time wave propagation, both in periodic domains, as well as in wave-tank setups. This includes benchmarks such as harmonic wave generation over a submerged bar. Moreover, the developed geometric p-multigrid solvers are shown to obtain O(n log n) and O(n) computational scaling for the pseudo-spectral method and SEM, respectively. From this, we conclude that both spectral methods can be used for phase-resolving nonlinear water wave simulation by efficiently solving the FNPF and INS equations with a free surface.