An approximate solution for a cone-cylinder in axially symmetric transonic flow

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Virginia Polytechnic Institute


In this thesis an approximate method is developed which predicts the aerodynamic force on a cone-cylinder body in axially symmetric transonic now. The method places more emphasis on the physics of the now than on the mathematical rigors of solving the typical reduced non-linear transonic equation of motion.

Under the assumption that the now is that of a steady, irrotational, inviscid, compressible gas, the body pressures are determined and the associated force defined. Recognizing that the transonic pressures are influenced by the character of the subsonic compressible pressures, which are obtained in this analysis through Gothert’s Rule, it is then mandatory that the incompressible case be defined with the best possible accuracy. Comparisons with experiments indicate that the classical method (axially distributed sources and sinks) does not provide this required accuracy. Thus the surface distributed vortex ring theory is used in the present analysis to obtain the incompressible body pressures.

Gothert’s Rule, which represents a linear solution for the subsonic case, is known to be applicable up to a limit value of tree stream Mach number. An investigation is carried out herein to determine both the correct form of the rule and its limits of applicability. As a result of this investigation, it is concluded that the upper limit is the lower free stream critical Mach number. Also, at this Mach number, a solution is immediately available tor the lower limit of the transonic range of Mach number.

In solving the transonic problem the law or stationarity of local Mach number is of fundamental importance. For an assumed isentropic flow over the body, and for sonic conditions being present at some point on the surface, the body pressures can be described in the ratio pL/p*. Here pL is the local surface pressure and p* is the sonic (body) pressure. Through the stationarity law, this ratio is recognized as an invariant for transonic speeds so long as the flow field remains essentially irrotational. Thus any change in local pressure is only a function of the free stream Mach number for any given body position. By this approach, the pressure distribution is defined for a range of Mach number from below to above the sonic stream value. The method is then capable of prediction for almost all of the transonic range of Mach number. It is only when the head shock baa significant curvature, causing the now adjacent to the body to be rotational, that the method fails. Though the procedure developed here is not capable of spanning the entire transonic range, it does provide a wider range of applicability than other known theories.

Finally, for this problem, a correlation of transonic pressure drag data is formulated. This correlation is founded on physical interpretation and is not limited to the usual transonic similarity restrictions. In fact, to the author's knowledge, this is the first known such correlation tor axially symmetric flow covering the range of body sizes and Mach numbers considered in this investigation.

In so far as is practicable the results obtained in this thesis have been compared to available experimental results. In particular, the drag data from this analysis compare closely with experimental transonic values. Experiment bears out the conclusion that the upper limit for linear theory is the lower critical tree stream Mach number. And, the pressures determined by the vortex ring theory agrees well with the low-speed experimental results obtained by the author.