Numerical solution of the direct-problem of mixed nozzle flow

A large part of the known results of Laval nozzle theory relates to the inverse problem, in which the velocity distribution on some line (usually the axis of symmetry) is given rather than the nozzle contour. Many important properties of transonic flows have been disclosed as a result of numerous studies, whose basic results were presented together with an extensive bibliography in Ryzhov's monograph [1]. The solution of the inverse problem has recently been used not only to analyze the qualitative characteristics but also to construct nozzles with rather marked variation of the slope of the generator, which are of practical interest. In this connection we note the work of Pirumov [2] and also the studies of Hopkins and Hill [3, 4]. The latter authors, in addition to the classical Laval nozzle, studied several nozzle schemes with a centerbody. Pirumov used a specially developed numerical method for the solution of the inverse problem (we note that in the subsonic part of the nozzle the corresponding Cauchy problem is incorrect), while Hopkins and Hill used a series expansion which was preceded by a change of variables.There are considerably fewer studies devoted to the solution of the direct problem of mixed nozzle flow. Numerical methods have been used by Alikhashkin, Favorskii, and Chushkin [5], Favorskii [6], and Danilov [7], with the method of integral relations being used in the first two studies. Finally, there has recently been extensive development of the method of expansion in powers of ^1/2, where is the ratio of the radius (or half-width of the nozzle to the radius of curvature of the wall, calculated at the throat section. Such expansions have been used by Hall [8] and Kliegel and Quan [9] to study flow in classical Laval nozzles, and by Moore [10] and Moore and Hall [11] to study flow in nozzles with a centerbody. We note that the ^1/2-expansion method is suitable only in those cases in which the wall radii of curvature are large.In the following the asymptotic method is used to solve the direct problem of mixed flow in nozzles. This reduces the very complex boundary value problem for an elliptic-hyperbolic system of equations with two unknown variables to the Cauchy problem (more precisely, to a mixed problem with initial conditions in a bounded two-dimensional region and boundary conditions which are independent of the third variable) for a hyperbolic system with three unknown variables. The integration of the equations describing the two-dimensional (plane of axisymmetric) nonsteady flow was accomplished with the aid of the Godunov-Zabrodin-Prokopov difference scheme [12]. Several types of nozzles with centerbody are calculated as well as the classical Laval nozzle. The contours of the subsonic parts of the nozzles were either closed (finite combustion chamber) or open (nozzle joins an infinite cylindrical tube). In the first case the flow is provided by three-dimensional mass and energy sources which are introduced at some fixed part of the combustion chamber. In the second case there are no mass and energy sources, but a boundary condition is established at a plane perpendicular to the nozzle axis and located at a finite distance from the throat section, and this condition becomes the flow uniformity condition as this plane moves away to infinity.The authors wish to thank I. Yu. Brailovskii for valuable advice in the selection of the difference scheme, U. G. Pirumov for the kind offer of the results of his calculations, and A. M. Konkina and L. P. Frolova for assistance in the calculations.