Numerical solution of a free surface flow problem over an obstacle

Samira Beyoud, Dahbia Boukari-Hernane

We consider a free surface flow problem of an incompressible and inviscid fluid, perturbed by a topography placed on the bottom of a channel. We suppose that the flow is steady, bidimensional and irrotational. We neglect the effects of the superficial tension but we take into account the gravity acceleration $g$. The main unknown of our problem is the equilibrium free surface of the flow; its determination is based on the Bernoulli equation which is transformed as the forced Korteweg--de Vries equation. The problem is solved numerically via the fourth-order Runge--Kutta method for the subcritical case, and the finite difference method for the supercritical case. The results obtained are illustrated by several figures, where the height $h$ of the obstacle, and the value of the Froude number $F$ of the flow, are varied. Note that different shapes of the obstacle have been considered.