Numerical analysis of jet flows based on the variational inequality formulation

Abstract

The free boundary problems of jet flows of an incompressible inviscid fluid, either plane and axially symmetric or impinging on an infinite plane wall, are considered and are studied from the numerical point of view. The jet problem has been formulated as the variational problem of minimizing a functional which contains an indicator function defined on a variable domain and the unknown parameters to be determined as part of the solution due to Alt, Caffarelli and Friedman [33-35]. In order to construct the efficient method for solving the jet problem numerically, the method of reducing the variational problem to a variational inequality is developed. By using the variational inequality formulation and the mathematical results established by variational principles, a new numerical algorithm for solving the jet problem is proposed. In the proposed algorithm, the convergence is assured and the difficulty of dealing with the variable domain is overcomed. Also, the proposed algorithm has the advantage of being able to overcomed. Also, the proposed algorithm has the advantage of being able to implement simply by using the conventional finite element method instead of the method of variable finite elements, though the domain of integration is varing at each iteration. The numerical calculations which have been performed for several test problems are also presented. So far as comparison with other results as possible, the present numerical results agree well with those obtained by other methods either theoretically or numerically. In applying the present algorithm, the adjustment of the free boundary position can be carried out very simply and there do not appear any difficulties in attaining the converged solutions of test problems.