1. The earlier solutions of Ohrenberger and Baum were restricted to the region of the wakeupstream of the saddle point singularity in the wakeneck {often referred to as the Crocco-Lees critical point). This singularity plays a fundamental role in the fonnu 1ation since the only acceptable wake solution is the one which passes through the singularity. In practice, the singular solution can only be bracketed by diverging solutions of opposite family, thus restricting the solution to the region upstream of the singularity. However,manysignificant changes must occur in the flow downstreamof the singularity before conditions at the edse of the vlscous wakebecome ambient, and hence becameproper conditions for a far wake analysis. The solution d(Jl;tnstreamof the singularity poses no extraordinary mathematical difficulties and consequently the mJjor mathematfral problem lies with obtaining a solution in the diverging region of the bracketing solutions. A methodof approximating the singular soiution in the diverging region is outlined, and app 1ied to the case of the ablating cone. The downstreamwake to 20 base diameters is then determined, and a number of features of the downstream1akeare presented. It was, of course, necessary to extend the lateral domainof the calculation to the bo 1shock 1-1ave and the technique for including a discontinuous bo 11shock in conjunction with the ffni te difference solution is outlined in the paper.
2. The formulation is based on a set of equations obtained from the Navier-Stokes equations by assuming* (1) that streanMiSe second derivatives arising from streanMise normal stress and heat conduction terms are negligible comparedwith crossstream second derivatil'es of velocity and temperature, ard (2) that the cross-stream pressure variations in the subsonic regions of the flow are negligible. The resulting set of equations appear to be boundary 1ayer 1ike but differ in one fundamenta 1ly important way; pressure variations can occur across streamlines in the supersonic regions of the flm-1. Whenthese equations are solved for the boundary layer and adjacent inviscid flm on, for example, the rounded aft shoulder of a cone, the solution is fundamentally divergent. The iriteract ion between the subsonic and supersonic regions of the boundary layer produces an exponentiallike increase or decrease in pressure about an envelope solution 1-1hichrepresents the solution of neutra 1 stab i1ity between the two diverging fomi lies. These solutions are valid interaction solutions up to the point where stre 11mwisesecond derivatives becomeimportant. The solutions which diverge with an increastng prssure (and an accompanyingdecrease in 1-1a11shear) thus describe the interacting boundary layer flow approaching separation, and are used in the present analysis to establish initiai conditions for the wakesolution. This is depicted in Figure 1. Beginning 1-1ith ordinary boundary layer development upstream of the shoulder, a unique separation solution exists for each of the infinitude of possible separation points on the shoulder. The proper separation solution is determined by the downstream flow.
3. Whenthis same set of equations is solved for a viscous 1-1akeand adjacent inviscid flow by marching in the streamwise direction from the base, the solutions are found to exhibit tv 10types of behavior, neither of which is wake 1ike. Onefamily of solutions is characterized by a centerline velocity which reaches a peak downstreamof the wake stagnation point, then decreases until a second stagnation point is reached (the source solutions). The other is characterized by a rapidly accelerating rate of pressure decrease on the centerline, with the calculation terminating as tile rate approaches infinity (the sink solutions). This behavior is due to the presence of a saddle point singularity in the wake neck region and is depicted by the axis pressure distributions in the wake of the body shownin Figure 1. The singu 1cir solution which passes through the sadcle point bounds the two non-wa!te fdmilies, and produces the only wake-like solution, where the velocity continues to increase, and the pressure gradient remains finite downstreamof the
4. The integral method is applied to the inner region primarily for reasons of compatibility with the outer f101,·. The inner region equations are ordinary differential equations with the axial coordinate as the independent variable. These equatfons maybe integrated from the base in step with the finite differences marching solution for the outer flow, and a matching of essential flow properties can be maintained bet 1-1eenthe two regions at each computing step. Afinite differences solution of the inner region, while seemingly more accurate, is extremely difficult to match to the outer region, and is computationally time consuming. For practical computing times, the matching reso 1ution would be so poor that the entire near wake solution would be seriously degraded.