A precise conformally mapped method for water waves in complex transient environments

TL;DR

This paper introduces an exact, efficient conformal mapping-based model for simulating water waves in complex, transient environments, avoiding numerical diffusion and supporting various boundary conditions.

Contribution

It presents a novel conformal mapping approach that is exact, computationally efficient, and capable of handling complex transient boundaries without iterative procedures.

Findings

Model accurately simulates shallow water waves.

Supports complex geometries and transient boundaries.

Validated through multiple wave simulation examples.

Abstract

A two-dimensional water wave model based on conformal mapping is presented. The model is exact in the sense that it does not rely on truncated series expansions, nor suffer any numerical diffusion. Additionally, it is computationally highly efficient as it numerically evaluates only the surface line while using a fixed number of FFT operations per time step. A double layered mapping enforces prescribed outer boundaries without iteration. The model also supports transient boundaries, including walls. Mapping models are presented that support smooth bathymetries and angled overhanging geometries. An exact piston-type wavemaker model demonstrates the method's potential as a numerical wave tank. The model is tested and validated through a number of examples covering shallow water waves, wavemaker generation, rising bathymetry shelves, and wave reflection from slanting structures. A paddle-type wavemaker model, developed from the present theory, will be detailed in a forthcoming paper.