First- and Second-Order Theory of Supersonic Flow Past Bodies of Revolution

Abstract

Methods are studied for improving the existing perturbation theories of supersonic flow past bodies of revolution. Applicability of the theory at high Mach Numbers is emphasized. For axial flow, a second-order solution isfound which represents a considerable improvement over the first-order result. For inclined flow, a second-order solution is not feasible except for a cone. Comparison with the exact solutions for cones shows that the slender-body series expansion causes large inaccuracies in both axial and inclined flow. The conclusion that first-order theory predicts the flow no better than slender-body theory is shown to be erroneous. When first-order theory is properly used,making no unnecessaryapproximations, greatly improved agreement is found with exact solutions and with experiment. The order estimates used to justify the approximations are shown to be invalid in most practical cases. A hybrid theory, combining first-order cross flow and second-order axial flow, gives further improvement. A physical explanation is advanced for the marked superiority of first-order theory over the true linearized theory. Nonlinearity in lift is shown to result primarily from viscous separation of the cross flow along the after portions of a long body. The magnitude of the resulting normal force can be estimated with reasonable accuracy using two-dimensional viscous sweep-back theory.