Calculations of viscous transonic flow over airfoils

Abstract

A simple and economic iterative scheme is presented for calculating compressible viscous boundary layers and wakes over airfoils and for matching the shear-layer calculations to the calculations of the transonic potential external flows. The iterative scheme is an extension and improvement of the scheme developed by Mahgoub and Bradshaw for calculating incompressible flow. A new iterative method in the shear-layer calculations has been designed and applied in the near-wake region. The computing time is only a little greater than in conventional displacement-surface calculations that ignore normal pressure gradients and consequently incur errors in the near wake. Some comparisons are made with full Navier-Stokes solutions and experimental data. ONSIDERABLE effort is currently being devoted to the solution of the full time-averaged Navier-Stokes equations, with appropriate turbulence model equation(s), using explicit time-dependent methods,1 implicit factorization methods,2 etc. However, the computing time is often too expensive for engineering purposes. It seems certain that methods involving separate calculations of the laminar and turbulent shear layers over airfoils and in the wake and of inviscid flow external to these layers, together with a matching process that enables the mutual interaction of these flows to be determined, will continue to have a practical advantage in both computing time and accuracy for a wide variety of engineering problems. In this paper we will present an iterative scheme of this kind, which can be regarded as an extension and improvement of a scheme developed for incompressible flow.3 The conventional displacement-surface methods may be inaccurate on highly curved surfaces or in the wake near the trailing edge where streamline curvature is extremely high, since they ignore the normal pressure gradient that may have a significant effect on the solution. With the new iterative method for the shear-layer calculations presented here, the effects of normal pressure gradient, even where they are relatively large and changing rapidly in the streamwise direction, can be included by iterative improvement of a marching calculation rather than by a fully elliptic calculation. Therefore, the whole iterative scheme remains simple and the cost of computing time is low. The pressure, but not the velocity, in the shear layer is stored from one iteration to the next: this means that only the pressure is allowed to transmit upstream influence, so that the method is nominally restricted to attached flow. Jameson's potential flow calculation method46 has been modified and applied in the present calculation of com- pressible flows external to the shear layers. It smears shock waves over at least two mesh widths, so discontinuiti es need not be included in the shear-layer calculation. The compressible boundary-layer calculation method described in Ref. 7 has been extended to an s, n coordinate system. A simple compressibility correction has been made to