Calculation of nozzle flows using Pade fractions.

Abstract

A method is given for the calculation of two-dimensional and axially symmetric nozzles in which Fade fractions formed from certain power series are used. Starting from a given centerline distribution of velocity or Mach number in an ideal gas, Taylor expansions for the stream function and density in the neighborhood of a point on the centerline are first obtained. Other expansions are then found, including power series for the equations of the sonic line and the limiting characteristic and for flow quantities along them. Fade fractions are obtained from these single power series and are used to calculate the flow and the nozzle contour. Calculations have been carried out on the CDC 6400 using 25 terms of the single power series. It is found that the use of Fade fractions leads to convergence when the power series diverge, and that they accelerate convergence when the latter converge. Accurate results are obtained both in the subsonic and supersonic regions and no difficulty is found at the sonic line. Good agreement with calculations by the method of complex characteristics is found. N the design of convergent-divergent nozzles of short length, an accurate calculation of the flow in the transonic region is necessary. Rapid changes then occur near the throat, and approximate methods in which it is assumed that the flow in the throat is almost uniform are not valid. Nozzles of short length with a given exit Mach number are of interest in connection with gas dynamic lasers,1 where two-dimensional nozzles are used, and with chemical lasers,2 where both two-dimensional and axially symmetric nozzles have application, The present paper is concerned with the inverse problem, in which a distribution of velocity or Mach number is given on the centerline, and the nozzle contour corresponding to a desired value of the stream function is calculated. A commonly used procedure for the calculation of wind tunnel nozzles is to solve the inverse problem by use of a power series solution in the subsonic and transonic regions and the method of characteristics in the supersonic region.3'4 No more than three terms of each series are used in these references, however, and the resulting accuracy is therefore limited. More accurate calculations of the subsonic region have been carried out in Refs. 5 and 6 in the axially symmetric case by means of Garabedian's method of complex characteristics (Ref. 7, Chap. 16). In the present paper, a method of calculation of two- dimensional and axially symmetric nozzles involving the use of Fade fractions is given. Fade fractions have been used in a similar way in the calculation of blunt body flows.8 The first step in the present procedure is to find the Taylor expansions of the stream function and density in the neighborhood of a given point on the centerline, starting from a given analytic expression for the centerline velocity or Mach number. As in Ref. 8, these calculations are carried out on a computer by use of subroutines for power series manipulations. Other power series in the neigh- borhood of the given point on the centerline are then obtained, including expansions in powers of the stream function along the curve of constant potential through the given point. Power series are also obtained for the equations of the sonic line and the limiting characteristics and for flow quantities along them. Finally, Fade fractions are formed from these power series and are used to calculate the flow. These Fade fractions are used in order to obtain convergence when the series are divergent and