Work for my Ph.D. thesis involved the development of a new flow-solving approach where the three-dimensional (3D) shape of a shock surface was specified as input, and the flow solver computed the post-shock flowfield by marching away from the shock surface in a roughly normal direction. The general cross-stream marching problem in supersonic flow is ill-posed, however, if the streamwise shock curvature is not too great, and if the proper marching direction is used, it was shown that accurate and stable solutions could be obtained. The new approach was applied to the inverse design of hypersonic waverider aircraft. The inverse marching algorithm is graphically illustrated in the following series of images.

In the figure to the right an arbitrary, 3D shock surface is shown by the shaded grey surface. This shock surface along with the freestream flow conditions are the input to the flow solver. The origin of the coordinate axis indicates the upstream centerline of the symmetric (about y) surface, with the freestream flow aligned with the x-axis. The black line across the surface indicates the downstream cross-section for the flow region we wish to define. Note, however, that the shock must be defined much further downstream from that line in order to provide the necessary boundary data for the problem.

Next a grid of discrete points is defined on the shock surface in such a way as to promote stability in the marching scheme. The post-shock flow variables are shown in the figure to the right. These are computed directly at each surface grid point using the Rankine-Hugoniot shock-jump relations, and these values form the initial conditions for the cross-stream marching scheme.

As shown in the figure to the right, the post-shock conditions are used to compute the optimal marching direction at each grid point, and a new grid layer is computed. While the general, three-dimensional, cross-stream marching problem is mathematically ill-posed, it can be shown that the instabilities are minimized or eliminated by marching within the osculating-plane. The marching distance is limited based on the domain of influence of the boundary data.

Once the new grid layer is formed, the Euler equations, written in gradient form, are used to compute the flow variables on the new grid layer. Using the solution on the new grid layer as initial conditions, a new grid layer is generated, and the solution on this layer is computed, and so forth. This technique for advancing a solution is refered to as a "marching" scheme. In the end, the solution is marched away from the initial shock surface, bounded by the upstream and downstream characteristics, until the region shown in the figure to the right is defined. The perimeter grid topology and pressure isolines on this surface are shown in the figure.

For more information regarding the application of this method, you might be interested in my waverider design and analysis page.