A Variational Formulation of the Compressible Throughflow Problem

Abstract

A variational formulation of the compressible throughflow problem is developed. The method is suitable for the calculation of throughflow flow fields in which large rotational effects, large compressibility effects and large variations in hub and tip radii may exist. The formulation requires the absence of viscous forces between the blade rows, though the effects of losses within the blade rows may be included through the variation of entropy across stream surfaces. The meridional Mach number is restricted to be less than unity, though the complete flow Mach number may be much in excess of unity. The variational formulation represents a complete statement of the problem in that the boundary conditions, far upstream and far downstream conditions, and matching conditions at all actuator disks are all natural conditions of the variational formulation. Furthermore, terms involving density variations vanish. The variational problem is posed in terms of the streamline position and the density. A finite element approximation produces a coupled nonlinear algebraic problem for numerical solution. Example calculations of flows with highly-loaded actuator disks, existing in annuli with large variations in hub and tip radii, are given.