Global marching technique for predicting separated flows over arbitrary airfoils

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Introduction T HE aerodynamic design of a flight vehicle accounts for the drag, and the estimation of the drag is greatly affected by viscous effects. For flows of practical interest, the Reynolds number is sufficiently large for the flowfield to be divided into viscous and inviscid zones, e.g., the problem of flow past a wing. Different approaches are available for solving such a problem. Inherently, the Navier-Stokes formulations lead to an extremely stiff nonlinear system. Using an explicit algorithm to solve such problems results in the requirement of very small time-steps in order to satisfy the stability bounds. Therefore, many iterations and large amounts of computer time are required to reach the steady state. To remove the time-step restriction, fully implicit methods have been investigated. The implicit methods, however, still require many iterations to reach the steady state and consequently, require large computational costs. The present work is a generalization and improvement of an earlier work developed for studying separated flows using boundary-layer-type equations. The improvements include extensions to a general coordinate system and use of a more general zonal technique for solving the coupled equations. In order to be able to consider arbitrary geometries, secondorder-accurate (in space) conservative differences are generated by considering the integral formulation of the governing equations in a general coordinate system. The general coordinate system is handled in as general a manner as possible in order to allow for the use of either analytically or numerically generated coordinate systems. The present work used a marching procedure for solving the approximate Navier-Stokes (ANS) equations in the viscous region coupled in a fully implicit manner with the elliptic inviscid equation. To demonstrate the independence of the scheme, as to the way the grid was generated, different grid generation techniques were employed (parabolic and algebric grid generation algorithms). Both orthogonal and nonorthogonal grids were used to predict the flow solutions for the JO 12 and NACA-0012 airfoils. New results were obtained for the J025 airfoil and a comparison with the experiment is shown.