Analysis of a propeller in compressible, steady flow

Abstract

THIS Synoptic describes an analytical method for solving the equation governing the inviscid, irrotational, compressible, potential flow about a propeller. The equation and boundary conditions are transferred to a noninertial system of coordinates rotating with the propeller, in which the basic problem becomes a steady one. The solution method takes advantage of the linearity of the model by superposing a compressible solution to the potential equation on an incompressible wake solution. In addition, the boundary conditions are satisfied by dividing the flowfield at the propeller plane, solving the equations separately ahead of and behind this plane, and enforcing continuity matching conditions. Applying the final boundary condition yields an infinite-series integral equation for the unknown circulation distribution. A lifting-line method is used to produce numerical results. Presented results establish the effect of compressibility on the induced field. Contents Except in the vicinity of the propeller and its wake, the flowfield caused by an advancing propeller in an inertial coordinate system may be considered irrotational and isentropic. This approach treats the viscous drag on the blade as inconsequential to the general induction theory. Consider a transformation of the linearized convective wave equation to a bladefixed coordinate system that rotates with the propeller at angular velocity w. This noninertial, cylindrical system (r, 6, z, t) is defined by r = r, 6 = 0 + ui, z = z, and t = t. Tildes represent quantities in the inertial, translating coordinate system. Performing these coordinate transformations gives the unsteady, linearized potential equation in the rotating system. Although the general methodology works for unsteady flows, this Synoptic focuses on the aerodynamics of the steady propeller, that is, a rigid propeller experiencing a uniform freestream. This allows the time-dependent terms to be excluded, since the flow now appears steady in the rotating, translating coordinates. Converting the dimensional spatial variables to dimensionless form (p = a>/-/£/and z = uz/U) and dropping the bar over the dimensionless z variable, the steady, potential equation becomes - M2p2 (1)