Viscous-inviscid interactive procedure for rotational flow in cascades of airfoils

Abstract

A VISCOUS-INVISCID interactive calculation procedure for application to flow in cascades of two-dimension al airfoils is described. This procedure has four components. The first is a grid generation method that relies in part on a succession of conformal mappings to produce a nonor- thogonal curvilinear grid mesh fitted to the geometry of the cascade. This method can accommodate an arbitrarily specified cascade geometry. Second, a numerical solution of the Euler equations is carried out on this grid. Third, a viscous solution for use in boundary-layer and wake regions has been programmed. Finally, an interactive scheme which takes the form of a source-sink distribution along the blade surface and wake centerline is employed. Contents method relying in part on a succession of conformal map- pings. An implicit time-marching solution of the Euler equations is then carried out on this grid in the manner described in Refs. 6 and 7 except for certain differences in the treatment of boundary conditions. We have accounted for the effect of viscosity on the flow by coupling the inviscid calculation with a separate viscous shear layer calculation in an interactive procedure. The various components of this viscous-inviscid interactive calculation are now discussed separately; additional details may be obtained by consulting Ref. 8. The inviscid computations of the present work are per- formed on a C-type body-fitted grid in which one family of lines form open loops (Cs) around the blade and wake. The grid is periodic and nearly orthogonal. This choice permits accurate resolution of the leading-edge region and provides an appropriate location for the interactive wake boundary conditions. Typical grids for a turbine cascade and a com- pressor cascade are shown in Figs. 1 and 2. The grid generation employs two analytical mappings that take the multiply connected exterior of a cascade of airfoils to the interior of a simply connected domain. A numerically con- structed mapping is then used to take this region into a rec- tangular computational space. During this process, a small amount of coordinate straining is introduced in the vicinity of the blade trailing edge to insure grid continuity across the wake. Consequently, this grid continuity is obtained at the expense of a small amount of nonorthogonality. The inviscid component of this interactive procedure consists of a time-marching solution of the Euler equations using an approach described in detail in Ref. 6. The successful application of this method to cascade flows, however, seems to require a more careful treatment of certain computational boundaries than is generally the case with isolated airfoils.