740 I. Introduction A VARIETY of aerospace applications require three-di- .lA,mensional supersonic internal flow calculations. These include, among others, nozzles for propulsion engines having nonsymmetric area constraints and for high-speed aircraft, which increasingly require careful airframe/propulsion system integration. Predictive methods for nozzle flows of varying complexity may each play an important role in the design process from initial concept to detailed aerothermodynamic simulation and performance evaluation. Computational methods which solve such complex three-dimensional flows are becoming more widely accessible. However, special techniques are usually adopted to suit specific cases and there is inevitably a tendency toward the ever greater specialization of generalized methods. Such sophistication creates substantial difficulty for the non- specialist user in both understanding the program and prepar- ing the necessary input. Lengthy operating and turn-around times result, and consequently high costs are incurred. Fur- thermore, significant technological deficiencies are still associ- ated with such methods - one of the most important being in the area of geometric modeling. Computer codes based on the finite-difference method, for example, are generally tailored to a specific grid topology. Although body geometries that fit this particular topology may be analyzed accurately, a variety of problems accompany the severe loss of grid orthogonality, which often occurs as a grid is body fitted about a new geometry not suited to the particular topology. Grid refine- ment, by increasing the number of cells or redesigning the grid geometry, may reduce the effects of nonorthogonality, but these approaches are often costly and inconvenient. This must vitiate much of the attraction in computational flow simula- tion, which lies in the alternative presented to time-consuming and expensive hardware demonstration or rig testing, particu- larly in relation to initial design and often involving the inves- tigation of a range of competing geometries. The objective of the present research is the development of a simple and reliable method for calculation of supersonic internal flows having complex shapes and the preliminary de- Received Oct.20, 1988; revision received Aug. 23, 1989. Copyright O 1990 by the American lnstitute of Aeronautics and Astronautics, Inc. All rights reserved. *Research Student, School of Mechanical Engineering. tProfessor, School of Mechanical Engineering. vol-. 6, NO. 6 sign of representative three-dimensional supersonic nozzles based on such computations. II. Approach The method developed is simply based on three key features of inviscid axisymmetric flows. l) The streamlines of such flows lie in planes through the streamwise axis. 2) The flow in any one such plane is the same as that in any other. 3) The stream sheets (formed by the preceding streamlines) generate surfaces across which there is no flow and which may be replaced by solid boundaries to a first approximation. These characterictics will be exploited in order to calculate comparatively simply the nozzle or inlet having the desired shape. First, the axisymmetric nozzle having the desired length and Mach number is computed. Then, choosing the desired cross-section shape at the exit, the streamlines which pass through its periphery are located and traced back to the throat. The stream sheets formed by these streamlines then constitute the walls of the desired nozzle. Two three-dimensional nozzles, one of elliptical cross sec- tion and a two-dimensional wedge, have been designed using this approach. The detailed procedures and the validation of the aerodynamic design are described in the following sections. III. Nozzle Designs Axisymmetric Design Inviscid flow calculations were carried out using the method of characteristics (see Zucrow and Hoffman2) and based in particular on the scheme proposed by Sauer.3 The axisymmet- ric nozzle, from which the elliptical nozzle was developed sub- sequently, is contoured, comprising a throat formed by two circular arcs of different radii of curvature. The upstream throat radius of curvature was equal to two throat radii. The circular arc downstream of the throat was joined to it tangen- tially; this in turn was continuous in the first derivative at an attachment point to a simulated quadratic polynomial wall, responsible for further downstream expansion (see Fig, l). The attachment point (xo, yo) is readily established from the throat radius /r, the downstream radius of curvature R,a, and the attachment angle -4o, namely xo : R6 sirtAo lo=lt+Rd(l-cos,4d) J. PROPULSION Aerodynamic Design for Supersonic Nozzles of Arbitrary Cross Section A. Haddad*and J. B. Mosst Cranfield Institute of Technology, Bedford, Englond, United Kingdom A comparatively sirnple method for obtaining wall contours of supersonic nozzles of arbitrary exit cross sections from readily determined axisymmetric flows is presented. An initial axisymmetric flowfield is calculated using the method of characteristics in two dimensions from which the desired three-dimensional shape may be generated by specifying the appropriate cross section at the streomwise station giving the required overall nozzle lenglh and exit Mach number. The describing points on the perimeter of this section are trtced along corre' sponding streamlines back to the thrort. The stream sheets formed by these streamlines then define the new nozzle contour. Elliptical and two-dimensional wedge-shaped nozzles are designed using this approach, and comparisons are reported between detailed finite-difference flowfield predictions and experimental measure- ment.