Laminar boundary-layer flow near separation with and without suction

Abstract

Numerical solutions of the laminar boundary-layer equation for the mainstream velocity U = t/0(l — ^x) without suction have been obtained by Hartree and Leigh, and the solutions have suggested that a singularity is present at the separation point. Assuming the existence of this singularity, Goldstein developed an asymptotic solution in the upstream neighbourhood of separation, but his solution required that a certain integral condition must be satisfied. Stewartson extended this asymptotic solution so as to be independent of any integral condition. Jones and Leigh have compared the numerical and asymptotic solutions and have found satisfactory agreement between them. In part I of this paper the work of Goldstein and Stewartson has been extended to include suction through a porous surface. The stream function xjrl is expanded in a series of the type = 2 ^ 3 £ 6 4 (f) + 2 H *In g[F,(f) +fiF,(f)] + 0(*“ In©, r = 0 where £ = x , y =yl/2*x and (jq, y x) are non-dimensional distances measured from the separation point. Analytical solutions for the functions f r(v) = 0, 1, ...,5 ) have been obtained and the solutions for r = 0, 1, ..., 4 reduce to those given by Goldstein in the case of zero suction. The solution for f^yj) without suction was confirmed by comparison with the numerical work of Jones, and corrections were made to his values for two constants. The solution for without suction was next considered so as to show that Goldstein’s condition is not satisfied. This condition required the vanishing of a certain integral estimated by Jones at ( — 4 + 4) <x its value is now found to be ( — 8-62+ 0*01) af. Following Stewartson, solutions for the functions F5(y) and F6(r/) are given. Numerical expansions for the skin friction and the velocity distribution near to separation have been obtained. Numerical tables are given for the functions and AO/) and their derivatives which are required for the computation of the velocity distribution. In part II there is developed a numerical method, suitable for an automatic computer, by which the velocity distributions at all cross-sections to separation can be obtained from that at the leading edge. In this method Gortler’s transformation is applied to the boundary-layer equations and then, by means of the Hartree-Womersley approximation, derivatives are replaced by differences. The resulting simultaneous equations are solved by an iterative procedure which involves the inversion of matrices. The program has been written so that given a general external velocity distribution and velocity of suction only a few specified subroutines are required. By this method, the boundary-layer flow was computed for the mainstream velocity = sin x (corresponding to potential flow past a circular cylinder) and a certain constant velocity of suction. Tables have been included showing the velocity distributions at selected cross-sections and giving the skin friction, displacement thickness and momentum thickness. The position of separation obtained was 114-7° from the forward stagnation point, whereas for the same suction velocity, Bussman & Ulrich gave 120-9° using a series expansion. The difference between these values was discussed and the former shown to be accurate. Near separation similar behaviour to that found by Hartree and Leigh was experienced, thus confirming the existence of a singularity at separation. The numerical results were compared with the solution given in part I and excellent agreement was obtained. The functions fi(v) •••fiiv) depend on a parameter a x, which was determined by comparing the numerical results with the asymptotic expressions for the skin friction and the velocity distribution near to separation. Both methods gave a x — 0-555. The work was repeated for the same mainstream flow U = U0 sin x without suction. The position of separation in this case was 104-45° and a l — 0-676. (Leigh obtained a x 0-492 for the mainstream flow U = U0{ — ^x) without suction.) A range of solutions of the equation of similar profiles has also been obtained. In particular, the curve which divides the wholly forward flows from those with backflow is shown. The separation profiles for the two cases of potential flow past a circular cylinder have been compared with corresponding solutions of the equation of similar profiles. Fuller details of the numerical results, giving the velocity profiles at different cross-sections for both flows past a circular cylinder and the solutions of the equation of similar profiles, are contained in the author’s Ph.D. thesis at Manchester University,