The Complex Variable Boundary Element Method For Potential Flow Problems
Free (open access)
The Cauchy type integral, used to represent the complex velocity, is converted to a line distribution of sources and vortices in the complex plane. The specification of the normal velocity on the bounding contour leads to a Riemann-Hilbert problem, which provides the theoretical foundation of the method. The boundary element discretization results in a simple algorithm for calculating potential flows in multiple-connected domains. Flow problems with periodic or homogeneous outer boundary conditions are treated using the concept of the Green’s function in the complex plane. 1 Introduction The Complex Variable Boundary ElementMethod (CVBEM) evolved as a numerical procedure for solving boundary value problems for analytic functions in terms of discretized Cauchy type integrals . The method described in this paper is based on the same principle. However, instead of linking the integral to the complex potential, as is commonly done, it is linked to the complex velocity. The main advantages of this approach are: 1) the complex velocity is single-valued and hence no cuts in the computational domain are necessary and 2) the representing Cauchy type integral is equivalent to the contour distribution of source and vortex singularities. By selecting the Cauchy density as the boundary value of a function analytic in the external flow region, it is possible to specify the far field condition such that there is no flow in the complementary interior domain. In this particular case the normal and tangential velocities become decoupled, corresponding to the source and vortex densities respectively. The imposition of the normal-velocity boundary condition and the subsequent discretization by boundary elements leads to the vortex panel method in the complex plane, reported earlier .