PROFILING THE OPTIMAL CONTOUR OF A SUPERSONIC NOZZLE IN HIGHLY ROTATIONAL FLOW

The variational problem of constructing the wall of the supersonic part of a plane or axisymmetric nozzle of maximum thrust is solved for cases where the flow, in its smallest section, makes a large angle with the thrust vector, and a beam of rarefraction waves appears in the flow past the boundary of the second wall, which ends in the smallest cross-section. However, this scheme turns out to be unsuitable in the case where the restrictions on the size of the nozzle are comparatively rigorous (meaning, in the case of a case of a plane, that the length of the nozzle is not more than a few hundred times the ''height'' of the smallest cross-section). The optimal shape for such cases, proposed below, has no breaks and contains an initial sonic part of the boundary extremum. Optimality is proved in calculations which involve the use of the general method of Lagrange multipliers. Examples of optimal plane nozzles with uniform sonic flow, perpendicular to the component of the thrust vector to be optimized, demonstrate that the solution holds over a wide range of sizes.