An Asymptotic Solution for the Flow About an Ellipsoid Near a Plane Wall

The inherent difficulties in obtaining the solution for the flow about arbitrary bodies of revolution near a wall usually precludes an exact evaluation of the effect of wall proximity on the pressure distributions. However, many bodies of revolution may be replaced with good approximation by an ovary ellipsoid. For this purpose, an approximate solution for the velocity potential is obtained for the flov1 about an ellipsoid near a plane wall which approaches the exact solution in an infinite stream as the ellipsoid recedes from the wall. The evaluation of the image potentials and rectifying images is accomplished by an expansion in associated Legendre polynomials. A first approximation, which results in a symmetric distribution on the ellipsoid, is essentially an expansion in associated Legendre polynomials of zero order. A second approximation, which correctly predicts differences of pressure on opposite sides of the ellipsoid, is carried out by an exact evaluation of the effects of the image potentials while evaluating the rectifying images by the same method as followed for the first approximation. The solutions are obtained in closed form with resulting expressions for the velocity and pressure distributions that are especially convenient for application to specific cases. The solutions are compared with pressure distributions measured on two ellipsoid models placed near a plate, simulating a wall, in the free surface flume of the Hydrodynamics Laboratory. The first approximation shows good agreement along the meridian parallel to the wall but rather large deviations at other points of the ellipsoids. This approximation is probably most useful only for estimates of the change in pressure distribution for varying separations, and where a high degree of precision in actual values is not required. The second approximation, on the other hand, shows very good agreement for distances even as small as one diameter from the center of the ellipsoid to the wall. For smaller distances this approximation shows large deviations at the minimum pressure point of the half-meridian closest to the wall with increasing accuracy for points on the ellipsoid that are farther from the wall.