Adiabatic One-Dimensional Flow of a Perfect Gas through a Rotating Tube of Uniform Cross Section

Abstract

SummaryThe paper contains an analysis of the flow of a perfect gas with constant specific heats through a rotating channel of constant cross-sectional area, as used in certain helicopter propulsion systems and wind-driven gas turbines. The analysis is restricted to the adiabatic one-dimensional treatment, the Coriolis accelerations acting across a section being disregarded.The equations of motion and energy are deduced and, together with the equation of continuity, reduced to an ordinary non-linear differential equation of the first order, involving a dimensionless form of velocity and distance.The patterns of the integral curves of the differential equation are discussed and sketched by examining their asymptotic behaviour as well as that in the neighbourhood of singular points. It is shown that there exists a critical value of the angular velocity below which the flow remains subsonic in the pipe, if the entrance velocity is subsonic; it may, however, become sonic at the exit of the pipe. For supercritical angular velocities the flow may become sonic or supersonic in the pipe if the entrance velocity attains a given “ correct” but subsonic value. A method of examining for the possibility of shock formation is indicated.The initial conditions for the differential equation are deduced for the design and for the performance problem, two new flow functions being introduced and tabulated to facilitate practical calculations. Formulae are also deduced for the calculation of the pressure and Mach number variation from the previously calculated velocity variation along the pipe.Finally an approximate solution in closed terms is given for the case of small entrance velocities.