The laminar boundary-layer equations, I. Motion of an elliptic and circular cylinders

The main results obtained are as follows: (a) The problem is reduced, for the front part of a cylinder, to a non-linear differential equation of the same form as that studied by Falkner & Skan (1930), namely, ƒ´´´ +ƒƒ´´ = λ (1—ƒ´2). This equation has been numerically solved by Hartree (1937), so that the evaluation of velocities and surface friction requires only very simple and short computations. Near the separation point the problem leads to a more complicated differential equation (3·20). (b) Elliptic cylinder. The calculated and observed velocities agree up to and including the point x = 1·457 (§5), where x is the distance along the boundary of the ellipse from the forward stagnation point, expressed as a multiple of the minor axis. An approximate integration, by neglecting certain terms in the equation (3·20), gives the separation point at about 108° 30' from the stagnation point, as against 103° observed, where the angle corresponds to the elliptic co-ordinate of (4·1). (c) Circular cylinder. In all cases the calculated surface friction shows good agreement with the observed results, and the results previously obtained from experimental pressure distributions, for the front part of the cylinder. In the case of the cylinder of diameter d = 5*89 in. the calculated results (by neglecting certain terms in the equation (3·20)) show good agreement with observations almost up to separation point. (d) Pressure consists of two terms: one is equal to the pressure of the potential flow, the other one depends on the thickness of the boundary layer.