EULER SOLUTIONS FOR FLOW PAST AN AIRFOIL FROM SUBSONIC TO HIGH SUPERSONIC MACH NUMBERS

Abstract

Inviscid flowfields around an NACA 0012 airfoil are computed up to Mach 5 covering all the three flow regimes subsonic, transonic, and supersonic by solving Euler equations on body-conforming curvilinear grids. For the calculation of these flowfields, two numerical codes, designated as FLO-53 and SFVZD, are employed. The numerical algorithm employed in FLO-53 is based on an explicit Runge-Kutta time-stepping finite-volume procedure wherein the spatial terms are centraldifferenced and a combination of secondand fourth-order artificial dissipation is added to stabilize the algorithm. The numerical algorithm employed in SFVZD is an explicit MacCormack predictor-connector time-marching finite-volume method with split-flux-vector splitting of the spatial terms, which introduces upwind bias in regions of supersonic flow. Both codes employ variable time-steps for faster convergence to steady-state. The influence of the general characteristics of the numerical algorithms, such as numerical dissipation, the treatment of farfield arid body boundary conditions on the solution accuracy, and convergence, is investigated for 0.3 5 M, 5 5. Numerical experiments indicate that SFVZD computes the flowfield for the entire Mach number range without any difficulty, while convergence problems are encountered with FLO-53 for M, > 2. Furthermore, the boundary treatment of the artificial dissipation terms in FLO-53 has considerable influence on solution accuracy, especially at high angles of attack. At the present stage of development, SFV2D appears to be a more versatile and reliable code than FLO-53; however, it is about four times slower than FLO-53 per computational time-step. Introduction In recent years, there has been considerable emphasis on the development of reliable numerical codes for the solution of Euler equations for calculating transonic and supersonic flowfields in and around complex aerodynamic configurations. This emphasis has evolved as a natural next step to the progress made toward numerical solution of the full potential equation during the past decade. While potential solutions have proved useful for predicting transonic flows with shock waves of moderate strength, the validity of the approximation of ignoring entropy changes and vorticity becomes questionable in the presence o f strong shocks. A number of numerical algorithms have been proposed to integrate the unsteady Euler equations in time until the steady state is reached.'-'' For time integration, both explicit and implicit time integration methods have been employed, and for *This work was supported by the McDonnell Douglas Independent Research and Deuelopment program Scienlisl, McDonrrell Doirglas Research Laboraiories, Associate Fellow, AIAA. tResearcli Scientist, McDonnell Doirglas Research Laboratories, Member AIAA. ,, spatial terms, central as well as upwind finite-difference/finitevolume methods have been used. However, the vast majority of these algorithms have been tested on model problems such as one-dimensional shock tubes or nozzle flows; there has been only limited development toward solution of 2-D/3-D Euler equations on nonuniform curvilinear grids.12-15 For an Euler code to become an alternative to potential flow calculations in aerodynamic design, it should have the following properties: (a) it should be robust, that is accurate and stable on arbitrary curvilinear grids for a wide range of flow parameters, @) it should have better shock-capturing ability in both the transonic and supersonic flow regimes, (c) it should have rapid convergence to steady state so that the computational cost is reasonable, and (d) it should be easily vectorizable. From the viewpoint of better shock capturing ability in both the transonic and supersonic flow regimes, upwind schemes have been claimed to be more robust and versatile than the central-difference schemes because they do not need the artificial viscosity terms specially designed for each problem that are often required by central-difference schemes. From the viewpoint of computational efficiency, upwind methods require more computational effort per time step than the central-difference schemes, Although computational efficiency remains an important consideration in design work, at this early stage of developmen! the most stringent requirement on the numerical algorithm for solution of the Euler equations is its ability to compute accurate solutions on arbitrary curvilinear grids for the entire Mach number range from subsonic to supersonic. The main purpose of the present paper is to evaluate the performance of two representative Euler codes one employing the central differencing of the spatial terms with added artificial dissipation'2 (Jameson's code FLO-53) and the other employing an upwind treatment (split-flux-vector type) of the spatial terms13 (Deese's code SFVZD). Both codes employ finite-volume formulation and explicit time integration with variable time step. FLO-53 uses a fourth-order Runge-Kutta time-stepping scheme, which is stable for Courant numbers up to 2 8 . SFVZD employs a MacCormack predictor-corrector time-marching method, which is stable for Courant numbers < 2. Performance of the two codes is evaluated by calculating the flow past an NACA 0012 airfoil up to Mach 5, covering all the three flow regimes subsonic, transonic, and supersonic. The influence of the general characteristics of the numerical algorithms, such as artificial dissipation, treatment of farfield, and body boundary conditions on the solution accuracy and convergence, is investigated. Numerical Solution Algorithms Details of the numerical solution algorithms employed in FLO-53 and SFV2D are given in Refs. 12 and 13, respectively. Table 1 summarizes salient features of the numerical methods employed in these two codes.