| Abstract: | The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry.The inverse problem of Laval nozzle theory was considered in 1908 by Meyer 1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor 2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl 3] and Fal'kovich 4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle.A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in 3–6], has been studied in detail for the plane and axisymmetric cases. In 7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with large gradients of the gasdynamic parameters along the normal to the axis of symmetry.Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii 9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases.The present paper provides a numerical solution of the inverse problem of Laval nozzle theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzles is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution.The author wishes to thank G. D. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer. |
