A Theory of the Direct and Inverse Problems of Compressible Flow Past Cascade of Arbitrary Airfoils

Abstract

A simple unified approach to the inverse and direct problems of compressible flow past cascade of arbitrary airfoils is presented. In this approach, the close relation between the shapes of the mean line of the blade and mean stream line in the channel and that between the variations in channel width and specific mass flow on the mean stream line are utilized. The customary assumption of a linear relation between pressure and specific volume is not required in this approach, and the computations involved are also relatively simpler and quicker. The method is particularly useful for high-solidity blades. In the inverse problem, with the inlet and exit angles, a desirable blade thickness distribution, and either a desirable mean line of blade or a certain mean stream-line shape chosen, the flow along the mean stream line can be properly determined and easily extended out along the pitch direction by the equations of continuity and motion. The blade boundaries are then determined from the inlet mass flow and best velocity distribution over the blade. For a quick approximate solution of the direct problem, the same process is used with one or more adjustments on the mean stream line to get the correct blade shape. The solution obtained in this manner can also be used to give a good starting value for a more accurate solution by either relaxation method for hand computation or matrix method for automatic machine computation; these are also briefly described. The theory is illustrated with solutions of both inverse and direct problems of compressible flow past typical turbine cascades of highly cambered thick blades in which accurate results are obtained by using three terms in the series. The results compared well with experimental data. The method is directly applicable to two-dimensional flows on cylindrical surfaces in axial turbomachines and can be extended to two-dimensional flows on arbitrary surfaces of revolution in radial and mixed-flow turbomachines. Families of blade sections for these machines can be built up by this method in a relatively simple manner. INTRODUCTION A BASIC AERODYNAMIC PROBLEM of turbojet and turbopropeller engines is the flow of compressible fluid past a series of blades in circular arrangement. In axial-flow type turbomachines, if the blades are relatively short in the radial dimension and are bounded by cylindrical walls, the theoretical flow passing through the blades is usually computed on the basis of two-dimensional flow on cylindrical surfaces, which are developed into planes for convenience of calculation. Presented at the Propulsion Session, Nineteenth Annual Meeting, I.A.S., New York, January 29-February 1, 1951. * Aeronautical Research Scientist. Now, Professor of Mechanical Engineering, Polytechnic Institute of Brooklyn. t Aeronautical Research Scientist. Now Aero Research Engineer, Minneapolis-Honeywell Regulator Company. A number of methods have been proposed to obtain the theoretical flow through such a given cascade of airfoils or to obtain a desirable airfoil shape for the blade section, including analytical methods using conformal mapping, interference technique, and series expansion; graphical procedure; and other mechanical and electrical devices. In the present paper a simple unified approach to the direct and inverse problems of compressible flow past cascade of arbitrary airfoils is presented. In both problems, calculation is first made for the flow along a particular stream line in the channel formed by two neighboring blades, preferably the mean stream line that divides the mass flow in the channel into two equal parts. In this calculation, the close relation between the shapes of the blade mean line and the mean stream line and that between the variations in channel width and specific mass flow on the mean stream line are employed. The flow is then extended in the pitch direction by the use of Taylor series, whose successive terms are obtained by the use of the equations of continuity and motion. In the inverse problem, the blade boundaries are determined by the given mass flow at the inlet, and, by interpreting the starting mean stream line as dividing the mass flow in the channel into two slightly different portions, a number of blade profiles with different velocity distributions are obtained and the best one is chosen. In the direct problem, successive correction of the shape of, and the flow on, the mean stream line is necessary to match the given blade shape. Because the computation involved is relatively short (in the turbine example given herein, less than 16 hours were needed for one complete calculation), iteration in the inverse problem for most desirable blade thickness distribution or velocity distribution and iteration in the direct problem to fit the given blade shape are practical. This iteration, however, will be reduced to a minimum if a number of families of cascades are built up by this method. A check of the accuracy of the solution in either problem and its improvement, if found desirable, can be readily made by finite difference methods using the available flow variations obtained in the solution. Several numerical examples are given at the end of the paper to illustrate these methods. Results obtained are compared with available experimental data. 183 184 J O U R N A L O F T H E A E R O N A U T I C A L S C I E N C E S — M A R C H , 1 9 5 2