Three-dimensional solution of compressible potential flow in turbomachinery by a finite element method

Abstract

The present study is concerned with the analysis of three-dimensional, compressible flows in axial-, radial-, or mixed-flow turbomachinery under the limitations of steady and potential flow conditions. The basic approach is to isolate a single blade row (which may be stationary or rotating at a constant speed) and consider the flow in one of the blade passages as representative of the total flow through the entire row of blades. It is assumed that the fluid is inviscid and enters this "characteristic" passage with uniform entropy, uniform total enthalpy and zero vorticity. Under these conditions, Kelvin's circulation theorem ensures irrotationality of the absolute flow throughout. The analysis is restricted to subsonic flows which may have local suporsonic spots. The fluid is either incompressible or assumed to be accurately represented by the perfect gas law. The analysis begins with the development of the classical velocity potential formulation of the problem stated above. An equivalent variational formulation is then described. This formulation incorporates a quasilinearization concept which leads to iterative solution. Density distribution is assumed to be given by a previous solution and therefore has no variation. The problem of three dimensional, compressible, potentiai flow in turbomachinery is thus reduced to the determination of the absolute velocity potential distribution which minimizes an equivalent functional in the solution domain with appropriate boundary conditions. A computer code has been developed to solve this problem based on the finite element analysis presented in this thesis. The solution domain is discretized by using hexahedral superelements each composed of six ten-node tetrahedral elements enabling quadratic interpolation of velocity potential. The code offers the flexibility of using a com. bination of subparametric and isoparametric elements. which provides high accuracy at reasonable cost in the treatment of turbomachinery flows that exhibit complicated design features. Applications of the code to the Gostelow cascade, an experimental turbine stator, the first stage stator and rotor of an electric utility axial-flow turbine and finally to a mixed-flow turbine rotor are presented. The validity of the code is established by comparing the results with the exact solution, experimental data and calculations by other numerical methods.