Numerical analysis of cavitational flow-theory
When trying to obtain numerical approximations of ideal two dimensional cavitational flows, two approaches come to mind. The first one is to give a tentative free boundary (the region separating the fluid and the air), construct a mesh from it, solve the potential problem in the fluid region, correct the free boundary so that Bernoulli's law is satisfied and start again. This approach has two problems: the mesh has to be redefined at each step and oscillations could arise if an unsuitable way of correcting the free boundary is given, these two can be severe under certain geometries. A more palatable way, from the mathematician's point of view, is to consider a fixed mesh and use a variational principle. \textit{P. R. Garabedian} and \textit{D. C. Spencer} [J. Rat. Mech. Anal. 1, 359-409 (1952; Zbl 0046.185)] considered the functional \(J(v)=\int_{\Omega}| \nabla v|^ 2dx+m(\{x\in \Omega:\) \(v(x)>0\})\) so that a minimum for it represents either the velocity potential or the stream line function of the fluid flow, with appropriate boundary conditions. Unfortunately, the functional J is not convex, and it is not even continuous on \(H^ 1(\Omega).\) In this paper we will follow the fixed mesh-variational approach to show what the notions of local minima are in the discrete case, and then show convergence to a continuous function.
ideal two dimensional cavitational flows
scientific article; zbMATH DE number 4114225