Numerical solution of Bernoulli-type free boundary value problems by variable domain method

The variable domain method for a free surface problem is considered as a weighted residual method. This considerably simplifies discretization of the problem and at the same time allows implementation of any standard discretization technique. Discretization of the problem results in a non-linear system of equations for discrete values of a field variable and co-ordinates of the nodes on the free boundary. Variables corresponding to a field variable are eliminated and the resulting non-linear system is of considerable smaller size. Some new update strategies of the Jacobian together with discussion of the cost-effectiveness and covergence rate are introduced and compared to Broyden's method. Numerical results which confirm validity of the presented discussion are given for three potential free surface problems, (i) two-dimensional free jet impingement against an infinite wall, (ii) axisymmetric free jet impingement subjected to gravity and surface tension forces, and (iii) computation of critical flow over a weir. In case (i) comparison with known analytical results is given, while in case (iii) a good agreement with previously published results is established.