Simulations of viscous transonic flows over lifting airfoils and wings Mohamed Hafez * , Essam Wahba Department of Mechanical and Aerospace Engineering, University of California at Davis, CA 95616, USA Received 15 June 2005; accepted 12 July 2005 Available online 2 December 2005 Abstract In this paper, the hierarchical formulation for steady viscous transonic flow simulations of Refs. [Hafez M, Wahba E. Numerical Simulations of Transonic Aerodynamic Flows. AIAA paper 03-3564, 2003] and [Hafez M, Wahba E. Viscous/Inviscid Interaction Procedures for Compressible Aerodynamic Flow Simulations (in press)] is reviewed and a simplified version for the calculation of the vortical velocity components is presented. The results are compared to available solutions of standard Navier–Stokes equations for laminar flows. Ó 2005 Elsevier Ltd. All rights reserved. 1. Introduction In Refs. [1,2], the authors present a hierarchical for- mulation for steady inviscid and viscous transonic flows over airfoils and wings. The main idea is to use a Helm- holtz-type velocity decomposition, where the velocity vector is split into a gradient of a potential function plus vortical components. The latter are only calculated in limited regions, where potential flow is not an adequate model. The advantages of this approach have been dis- cussed in the above references and can be summarized as follows. The formulation can be viewed as a viscous/inviscid interaction procedure [3–6] where the outer flow field is governed by a potential equation. This way, the calcu- lations of the vortical flow components in the outer region, where the grid is stretched to the far field, are eliminated. The artificial entropy and vorticity, usually generated in the standard methods in the outer region, are therefore avoided. The potential flow calculations provide proper boundary conditions for the limited regions of rotational flows, typical of external aerody- namic applications. The present natural decomposition does not suffer from the difficulties associated with the interface boundaries of heterogeneous zonal methods and at the same time it is not limited to boundary layer approximations and the coupling between the viscous and inviscid regions is built-in automatically. Another advantage is the simple implementation of upwind schemes for the convection/diffusion scalar equations of the entropy and vorticity components. Moreover, the separation of the acoustic mode repre- sented by the augmented potential equation facilitates the application of multigrid acceleration techniques. Finally, the formulation has a built-in low Mach number preconditioning on the differential level. Indeed, incompressible flow (zero Mach number) is recovered without loss of accuracy or efficiency. The present paper deals with the evaluation of the vortical velocity components in the viscous flow region for the airfoil and wing problems. A simple and flexible method is discussed and the results are compared to those obtained in Refs. [1,2] as well as available data in the literature. In the following, the governing 0045-7930/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2005.07.002 * Corresponding author. Tel.: +1 530 7520212; fax: +1 530 7524158. E-mail address: mhafez@ucdavis.edu (M. Hafez). Computers & Fluids 36 (2007) 39–52 www.elsevier.com/locate/compfluid