The design of shock-free transonic slender bodies

The existence of shock-free flow around a slender body of revolution at near-sonic speeds is investigated using transonic small-disturbance theory. The governing partial differential equation, known as the Karman-Guderley equation, and the boundary conditions are transformed into the hodograph plane. In the hodograph plane the spatial variables depend on the velocity components. The problem is solved numerically, employing an algorithm that involves a combination of finite-difference and iterative methods. A condition dependent upon the Jacobian of the transformation is developed to determine when a shock-free solution has been computed. A number of bodies possessing shock-free flows are calculated at different values of the transonic similarity parameter, $K=(1-M\sbsp{\infty}{2})/(\delta M\sb\infty)\sp2$, where $M\sb\infty$ is the flow Mach number and $\delta$ is the body thickness. The body profiles computed are both fore-aft symmetric and fore-aft asymmetric. For moderate values of K (e.g. K = 3.5, corresponding to $M\sb\infty$ = 0.985 and $\delta$ = 0.1) there is little difficulty in finding shock-free solutions in the hodograph. Solutions have not been calculated for K less than about 3, corresponding to speeds very close to sonic.

Calculations in the physical plane of the flow around transonic slender bodies of revolution are performed at moderate values of K in order to confirm the shock-free nature of the body profiles obtained from the hodograph calculations and to explore off-design conditions. It is found that the flow field in the physical plane around the shock-free hodograph-designed bodies appears to be nearly shock-free, having at most a weak shock. Small perturbations to K or to the hodograph-designed body shape do not appear to create qualitatively different flow fields. The numerical evidence suggests that shock-free flows are isolated but that nearby flows possess, at most, weak shocks.