Ideal Planar Fluid Flow Over a Submerged Obstacle: Review and Extension

Abstract

A classical problem in free-surface hydrodynamics concerns flow in a channel, when an obstacle is placed on the bottom. Steady-state flows exist and may adopt one of three possible configurations, depending on the fluid speed and the obstacle height; perhaps the best known has an apparently uniform flow upstream of the obstacle, followed by a semiinfinite train of downstream gravity waves. When time-dependent behaviour is taken into account, it is found that conditions upstream of the obstacle are more complicated, however, and can include a train of upstream-advancing solitons. This paper gives a critical overview of these concepts, and also presents a new semianalytical spectral method for the numerical description of unsteady behaviour.

A comparison of explicit Runge-Kutta methods

“…The 14th-order method appears to be on course to exceed all lower methods if very high accuracy is desired, but caution should be used, as it may under-perform all of the methods used here for large stepsizes, as may be seen in most of the figures. We note that in a very different setting, fluid flow over a bump [15], we tested these ERK integrators, and found at least an order of magnitude decrease in run-time by choosing Zhang's 10th-order method [37] over the classic RK4 method, for the same level of accuracy. This suggests that modern higher-order integrators may be worth considering as an alternative to the classic RK4 method in a wide range of problems.…”

“…This present paper is concerned with higher-order ERK methods, which will be examined in the solution of n-body systems, and their practical implementation. These methods are equally applicable to the solution of PDEs as described above, and Forbes et al [15] employed a 10th-order, 16-stage ERK recently to study a problem in free-surface hydrodynamics.…”

A Comparison of Explicit Runge–kutta Methods

“…The th-order method appears to be on course to exceed all lower methods if very high accuracy is desired, but caution should be used, as it may under-perform all of the methods used here for large stepsizes, as may be seen in most of the figures. We note that in a very different setting, fluid flow over a bump [15], we tested these ERK integrators, and found at least an order of magnitude decrease in run-time by choosing Zhang’s th-order method [37] over the classic RK4 method, for the same level of accuracy. This suggests that modern higher-order integrators may be worth considering as an alternative to the classic RK4 method in a wide range of problems.…”

“…This present paper is concerned with higher-order ERK methods, which will be examined in the solution of n -body systems, and their practical implementation. These methods are equally applicable to the solution of PDEs as described above, and Forbes et al [15] employed a th-order, 16-stage ERK recently to study a problem in free-surface hydrodynamics.…”

Editorial: Special Issue in Honour of Professor Graeme Hocking

“…This special issue contains nine research papers which reflect various aspects of Graeme’s interests. We have a paper discussing a classical applied mathematics problem concerning flow over a heat island by Forbes and Walters [1]. Larry Forbes is probably Graeme’s longest standing collaborator; they first published together over 30 years ago and have established many important results since then.…”