Design of multimode axisymmetric hypersonic nozzles with the use of optimization methods

Abstract

On the basis of the solution of direct problems with the use of various medium models and numerical meth- ods of integration of gas flow equations, methods for designing super- and hypersonic nozzles have been de- veloped. Multimode nozzles in the ranges of Mach numbers 8-14 and 14-20 satisfying given conditions have been designed. optimization. Introduction. It is known that good quality of flow in a wind tunnel is provided by the application of thor- oughly shaped nozzles accelerating the working gas to a given velocity. Such nozzles have a spatial structure, are made with a high precision, and are rather expensive. Practically in all existing wind tunnels single-mode nozzles de- signed for definite operating conditions are used. Because of the complex structure of single-mode nozzles and the high cost of their production, the number of modes of operation of a wind tunnel is always limited. There exist two approaches to the design of multimode shaped nozzles in which the main outlet section has a fixed geometry and the small separable part adjoining the minimal section makes it possible to vary the Mach number at the outlet from the nozzle. The first approach is based on constructing the nozzle contour with the use of the method of characteristics (1-4), and the second approach is based on methods of direct numerical optimization. For example, in (5) the func- tional equal to the sum of standard deviations of the Mach number from its average values in a given region of the flow was minimized. Both approaches did not take into account the presence of viscosity and, accordingly, of the boundary layer that radically changes the characteristics of hypersonic nozzles (6). Therefore, to solve the design prob- lem stated, we used a complex approach combining the method of characteristics that permits calculating fairly easily supersonic nozzles with a uniform outlet characteristic for a perfect nonviscous non-heat-conducting gas, direct meth- ods based on the calculation of viscous flows, and methods of numerical optimization. Construction of the Initial Approximation. The first stage of designing a nozzle consists of constructing by the method of characteristics the nozzle contour for the maximum value of the Mach number at the outlet and deter- mining the size of the acceleration part of the nozzle for the minimum Mach number. This makes it possible to divide the contour into a fixed equailizing part and a variable accelerating part (much smaller in size) whose shape is deter- mined for the chosen value of the Mach number at the outlet from the nozzle (7). This stage is an important element of the chosen strategy since the solution of optimization problems for viscous flows required large computational re- sources and is successful, as a rule, only given a good initial approximation. Design of Multimode Nozzles. Since the viscous flow in a hypersonic nozzle of a given geometry was sup- posed to be calculated by means of the Fluent package, it was necessary to test the accuracy of the solution of the Navier-Stokes equation with the k-ω-model of turbulence. The contour obtained by the method of (5) was used in creating a real axisymmetric nozzle with a nozzle outlet diameter of 0.40 m meant for obtaining Mach numbers 8, 10, 12, and 14 at the outlet. The comparison between the experimental and numerical distributions of the Mach number