A Semi-Implicit Navier-Stokes Solver and Its Application to a Study of Separated Flow about Blunt Delta Wings

Abstract

A novel semi-implicit scheme for the Navier-Stokes equations is presented and evaluated. The semi-implicit scheme combines an implicit temporal integration in the bodynormal directions with explicit temporal integrations in the streamwise and cross stream directions. Thus, advantages of both explicit and implicit schemes are retained in the semi-implicit scheme. Numerical stiffness due to disparate physical scales in the normal direction is eliminated, since stability of the algorithm depends only on relatively coarse streamwise and cross stream grid spacing, not on the typically fine normal spacing. Approximate factorization is unnecessary and only one matrix inversion per multi-stage time step is required. Computations show that while a explicit scheme employing multigrid and residual smoothing and a fully implicit scheme are competitive for inviscid calculations, the semi-implicit scheme is superior for viscous flow calculations. Efficiency of the semi-implicit scheme is exploited in a study of flow separation around delta wings with blunt leading edges. Three-dimensional laminar vortical flows over two 65* swept semi-infinite elliptical wings of thickness to chord ratio 1 : 11.55 and 1 : 20 at Moo = 1.6, ReL = 106, and angles of attack of 40, and 8*, and a 60* swept elliptical wing with t/c = 1 : 11.55 at Moo = 1.4, ReL = 2 x 106 and a = 14* are considered. In these flow cases, separation line locations are fixed not by a particular geometric factor (eg. sharp leading edge), but by interaction of physical and geometric factors. Solutions with the semi-implicit scheme are shown to be significantly more efficient than solutions with a corresponding explicit scheme. Two distinct leading edge separation processes are identified: separation due to shock-less flow recompression leeward of the leading edge expansion in the t/c = 11.55, a = 40 case and separation involving a leading edge shock in the remaining cases.