Implicit Finite-Difference Procedures for the Computation of Vortex Wakes

Abstract

Implicit finite-difference procedures for the primitive form of the incompressible Navier-Stokes and the compressible Euler equations are used to compute vortex wake flows. The partial differential equations in strong conservation-law form are transformed to cluster grid points in regions with large changes in vorticity. In addition to clustering, fourth-order accurate, spatial difference operators are used to help resolve the flowfield gradients. The use of implicit time-differencing permits large time steps to be taken since temporal variations are typically small. Computational efficiency is achieved by approximate factorization. Both two-dimensional and preliminary three-dimensional calculations are described. I. Introduction T HE concentrated vorticity in the near wake of a large aircraft can pose a destructive threat to smaller aircraft within the same airspace. Consequently, experimental and theoretical efforts have been under way to understand, predict (for use in avoidance systems), and possibly reduce the vortex wake hazard. Most theoretical models developed to treat this problem rely on tracing discrete vortices,13 but an alternate and potentially more powerful approach is to use finitedifference procedures.4'5 Computer programs based on such methods can ultimately account for flowfield nonlinear effects with few ad hoc assumptions. Implicit finite-difference procedures are developed here to solve the incompressible Navier-Stokes equations and compressible Euler equations for simplified two- and threedimensional, unsteady vortex wake flows. This paper is divided into five interdependent sections. The flowfield and its numerical implications are discussed in Sec. II. The incompressible equations are developed for simulation as a system of first-order partial-differential equations in Sec. Ill, and the finite-difference algorithms are described in Sec. IV. Simulation based on the compressible flow equations is discussed in Sec. V. and, finally, simple wake-vortex flow calculations using both kinds of modeling are presented in Sec. VI.