Three dimensional free-surface flow over arbitrary bottom topography

TL;DR

This paper develops an efficient numerical method to simulate three-dimensional steady free-surface flows over arbitrary bottom topographies, revealing complex wave patterns and their relation to ship wakes.

Contribution

It introduces a Jacobian-free Newton Krylov method with a block-banded preconditioner for large-scale free-surface flow simulations over complex bottoms.

Findings

Significant reduction in computational time and memory usage.

Ability to simulate larger meshes and more complex geometries.

Numerical solutions illustrating diverse wave patterns in different regimes.

Abstract

We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.