A High-Order Spectral Element Solver for Steady-State Free Surface Flows

TL;DR

This paper introduces a high-order spectral element solver for steady free surface flows, leveraging the Firedrake framework, and demonstrates its accuracy and efficiency through benchmark tests.

Contribution

It develops a novel high-order spectral element method for free surface flows using an iterative pseudo-time approach within an open-source framework.

Findings

High-order convergence demonstrated in benchmark cases

Significant computational speed-up over low-order schemes

Effective handling of free surface and submerged bodies

Abstract

We present a spectral element solver for the steady incompressible Navier-Stokes equations subject to a free surface. Utilizing the kinematic behaviour of the free surface boundary, an iterative pseudo-time procedure is proposed to determine the a priori unknown free surface profile. The numerical model is implemented in the open-source finite element framework Firedrake, which enables the use of a high-order polynomial basis on unstructured meshes through weak formulations. Additionally, the curvature of the free surface and submerged bodies is incorporated through curvilinear elements obtained via transfinite linear blending, which conserves the high-order convergent properties of the overall scheme. The model is applied to several benchmark cases in two spatial dimensions. Initially, it addresses fixed-domain problems, including the lid-driven cavity flow and flows around bodies such as a cylinder and a NACA airfoil. Subsequently, with the presence of a free surface, it is extended to determine the flow around a bathymetry bump and a submerged NACA airfoil. The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.