Contouring spike nozzles and determining the optimal direction of their primary flows

Problems related with the optimal contouring of two-dimensional and axisymmetric spike nozzles providing maximum thrust for given dimensions and external pressure are studied. The nozzles under consideration are self-adjustable which is ensured by the non-zero inclination of their primary supersonic flow to the plane (axis) of symmetry. Along with the optimal contouring of the spike, the optimal orientation of the “primary” nozzle producing the primary flow is obtained. In the exact formulation, its optimal inclination is determined by an exhaustive search for the configurations providing maximum thrust for a given spike length and various fixed inclinations of the primary nozzle. The spike and primary nozzle contours of these configurations are generally joined through a bend with the formation of an expansion fan in the flow around the bend. The efficiency of a simpler approximate method for contouring the spike and determining a near-optimal inclination of the primary nozzle is demonstrated. The method is based on passage to a modified formulation of the problem differing from the original one in that it is only the part of the spike extending beyond the primary nozzle edge that is preassigned rather than its entire length. In the modified formulation, there is no a bend in the flow around which an expansion fan could be formed, while the inclination of the primary nozzle is determined in the process of designing the unique optimal configuration.