| Abstract: | One of the possible methods is considered for profiling short plane nozzles for aerodynamic tubes. The nozzle has a straight sonic line, which allows the subsonic and supersonic sections to be constructed separately. The problem is solved numerically in the plane of a hodograph. In the subsonic region, Dirichlet's problem is formulated for Chaplygin's equation in a rectangle, one side of which is the sonic line. At the present time, two approaches have been defined in papers on calculations of a Laval nozzle, associated with the solution of the so-called “direct” and “inverse” problems (one has in mind a study of the flow in the interconnected region of sub- and supersonic flow). The direct problem determines the flow field in the case of a previously specified contour of the channel wall, the shape of which from technical considerations is obtained with certain geometry conditions. The direct problem can be applied in the construction of the Laval nozzle, if the contour of the inlet section of the channel (generally speaking, quite arbitrary) is chosen so successfully that neither shock compressions nor breakaway zones result in the flow. Although a strictly mathematical theory of the direct problem of the Laval nozzle is only being developed at present, there are still very effective numerical methods for its solution 1, 2]. In the inverse problem (which, by definition, is a problem of profiling), the contour of the nozzle is found with respect to a specified velocity distribution on the axis of symmetry. It is assumed that this quite arbitrary dependence can be selected from the condition of the absence of breakaway zones and shock compressions in the nozzle. By its formulation, the inverse problem is Cauchy's problem which, as is well-known, is incorrect in the classical sense in the ellipticity region — the subsonic section of the nozzle. At present, there are also efficient methods of solving the inverse nozzle problem 3], by interpreting it as an arbitrarily correct problem. Difficulties can arise in the inverse problem, in the provision of short (and, consequently, steep) nozzles because of the sharp increase of the error in the calculation. Together with the stated problems, a procedure can be evolved which is associated with the solution of the correctly posed problem for Chaplygin's equation in the plane of the hodograph. This approach is convenient in that it succeeds a priori in fulfilling the important condition of monotonicity of the velocity at the wall, ensuring (in the absence of shock compressions) nonseparability of the streamline flow at any Reynold's numbers. |
