Three-Dimensional Internal Flow Calculations

Abstract

A marching in time finite-volume numerical procedure is presented for three-dimentional Euler analysis of internal configuration flows using general curvilinear coordinates. The results of the performed experiments could be used not only to back up the predictions of the proposed numerical technique, but the same time to shed more light on the threedimensional transonic and supersonic internal flow behaviour. Numerical solution results have been presented and compared with measurements as well as with predictions using two other three-dimensional flow solvers. 1Introduction The non working flow passages constitute transitional pieces with sharp bends, rapid flow area changes and irregular cross sections. Nowdays, 3D Euler solvers are well developed and are available for routine turbomachinery calculations. Several of these dealing with internal and external flows are described by Hirsch*. Stuart and Hetherington as early as in 1974 determined the solution for the full 3D inviscid flow field in curved ducts using finite difference technique to solve continuity and momentum equations. Euler solvers for 3D tyrbomachinery (cascade) flows have been reported by Denton and who developed an explicit time marching method. Shieh and Delaney applied the hopscotch scheme written in general curvilinear coordinates using an O type grid system. Weber et al presented a 3D Euler analysis on a C type grid using the well known Beam and Warming implicit algorithm. Results for compressor cascade and rotor flows are presented. Arts presented an invisvig flow solution for transonic axial turbine stage. Holmes and Tong developed an algorithm based on the explicit four step Runge Kutta finite volume method advocated by Jameson. The purpose of the present study is to calculate the 3-D flow in ducts with arbitrary varying cross sections as well as for annular passage configurations. At the same time measurements were Transactions on Modelling and Simulation vol 16, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 572 Computer Methods and Experimental Measurements performed on an internal flow passage in order not only to back up the predictions but the same time to shed more light on the three dimensional flow behaviour. The proposed scheme has several advantages. 2. Problem Definition It is convenient to write down the three-dimensional Euler equations in a cylindrical polar coordinate system (z,e,r). These equations are expressed in conservation form as a(rp) a(rpu) a(pv) a(rpw) az ae ar ^ a(rpu) a[r(pu + p)] a(pvu) at az ae a(rpv)_a(rpuv) a(p at ~ az ae a(rpe) _afr(pe+p)u] 8t az jKgvw) ^ a(rpvw) ar