Numerical analysis of cavitational flow-theory
type: Item
instance of: scholarly article
title: Numerical analysis of cavitational flow-theory
zbMATH Open document ID: 0679.76019
DOI: 10.1007/BF01395870
author: Néstor E. Aguilera
published in: Numerische Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133342
review text: 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.
Mathematics Subject Classification ID: 76B10
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4114225
zbMATH Keywords: numerical approximations
zbMATH Keywords: ideal two dimensional cavitational flows
zbMATH Keywords: free boundary
MaRDI profile type: Publication
cites work: Regularity results for discrete solutions of second order elliptic problems in the finite element method
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description: scientific article; zbMATH DE number 4114225