Axisymmetric transonic flow past slender bodies, including perforated wall interference effects

Abstract

Solutions of the transonic small-perturbation equation for flow past slender bodies of revolution at subsonic freestream Mach numbers are presented in free air as well as in the presence of perforated walls. An artificial viscosity term which enables shock capture is added explicitly to the small-perturbation equation. The modified equation is then converted into an integral equation by the use of Green's theorem. Numerical solution to the integral equation is obtained by discretizing the region of integration into rectangular panels wherein the flow quantities can be considered uniform. Type-dependent operators are introduced in the calculation of the nonlinear source term. The resulting system of algebraic equations is then iteratively solved either by a single- step direct iteration scheme or a quasi-Newton scheme. HE finite difference relaxation procedure has been ap- plied to axisymmetric flow around bodies of revolution by Bailey 1 and Krupp and Mmrman2 and compared with the experimental data of McDevitt and Taylor.3'4 Using the method of split potential Sedin5 has devised a stable iterative scheme for the same problem. Both Bailey and Sedin have included tunnel wall interference effects. They have used the classical homogeneous wall boundary condition developed by Baldwin et al. 6 to model the presence of a perforated wall. Bailey's theoretical calculations seem to confirm the validity of this formulation insofar as the theoretical pressure distribution shows good agreement with the measured values and the shock location is correctly predicted for the ex- perimentally determined value of the porosity parameter. Thus, the theoretical model on which the computation of axisymmetric transonic flow with tunnel wall boundaries is based appears to be reasonably effective in predicting the flow details. It is well known that the axisymmetric transonic small- perturbation equation (TSPE), together with the linearized boundary condition on the body surface and a vanishing far- field disturbance, can be converted into an integral equation by the application of Green's theorem to the flow domain exterior to the body.7 In this formulation, the body boundary condition appears as a line integral on the body contour. Because the line integral for the axisymmetric flow is in- dependent of the unknown nonlinear field source term, iterative readjustment of the boundary condition is not necessary. Since the accuracy with which the tangency con- dition is satisfied depends only on the accuracy of the evaluation of the line integral, choice of the grid details in the vicinity of the body is unimportant to the overall com- putation. Further, since the far-field conditions are implicitly satisfied by the integral equation, no matching is necessary at the far-field boundary as is generally the case in finite difference calculations. The grid boundary is chosen after carrying out numerical experiments to assess its influence on the surface pressure distribution; the number of grid points can thus be minimized. The presence of the wall gives rise to an additional line integral on the wall, with appropriate changes in the domain of integration for the field integral.