Stability calculations of the two-dimensional boundary layer along a concave wall: Part 1: A numerical method of solution for the simultaneous ordinary differential equations governing disturbances

Linear and weakly non-linear developments of Goertler vortices induced by instability of the boundary layer along a concave wall are governed by simultaneous ordinary differential equations of second and fourth order. The eigenvalue problems in linear analysis are posed by homogeneous equations and boundary conditions, but in weakly non-linear theories several boundary-value problems of inhomogeneous equations have to be solved, some of which are singular in the sense that there exist solutions only if the forcing terms satisfy a kind of solvability condition. A numerical method of solution is presented for three kinds of such equation systems and is shown to give very accurate solutions. Also, a very fast method of iteration appropriate for calculation of eigenvalues is given. Tentative calculations are made to obtain the neutral stability curve of the Blasius velocity profile to Goertler vortices as well as the maximum-growth-rate curve, the former of which is in very good agreement with the most reliable result of earlier investigations.