Efficient computation of two-dimensional steady free-surface flows

TL;DR

This paper introduces an efficient numerical method for solving two-dimensional steady free-surface flow problems, significantly improving computational speed and allowing for higher resolution simulations.

Contribution

It presents a novel sparse Jacobian approximation and a preconditioning strategy for Jacobian-free Newton-Krylov methods in free-surface flow computations.

Findings

Enhanced computational efficiency enabling higher grid resolution.

Successful application to flows past various disturbances.

Demonstrated robustness of the numerical scheme.

Abstract

We consider a family of steady free-surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli's equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally employed for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian-free Newton-Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface-piercing object and bottom topographies.