Contouring the nozzles producing a uniform supersonic flow or a thrust maximum in the presence of a curvilinear sonic line

Abstract

Within the framework of the ideal, i.e., inviscid and non-heat conducting, gas model we consider the problem of designing the supersonic section of a two-dimensional or axisymmetric nozzle realizing a uniform supersonic flow limitingly similar with a sonic flow when the choked flow involves a curvilinear sonic line. Emphasis is placed on nozzles with abruptly or steeply converging subsonic sections and a strongly curved sonic line formed by the C−-characteristics of the expansion fan with the focus at the lower bend point of the vertical section of the subsonic contour. In the two-dimensional case, the least possible greater-than-unity Mach number Mem at the nozzle exit corresponds to the flow in which the first intersection of the C+-characteristics originated at the closing C−-characteristic of the expansion fan falls on the unknown contour of its supersonic part. For a uniform flow with Me 1 ensures a limitingly rapid flow acceleration and forms the initial region of the supersonic generator of a maximum-thrust nozzle. For this reason, in the case of a curvilinear sonic line the supersonic generators of these nozzles have two, rather than one, bends, which, however, is interesting only for the theory. At least, in the calculated examples the thrusts of the nozzles with one and two bends differ only by a hundredth or even thousandth fractions of per cent.