Numerical experiments with the split-flux-vector form of the Euler equations

An explicit, finite-volume algorithm based on the flux-vectorsplitting concept has been developed for the two-dimensional Euler cquations. The method is second-order accurate in space and time, uses body-conforming coordinate systems to map complex geometries to a uniform computational space, and is stablc for Courant numbers as high as two. A variable time step, determined by the local Courant number, is used to accelerate convergence of steady-state calculations. This code has been used to calculate lifting and nonlifting transonic flow about airfoils, diffuser flows with shocks, and flows about a cylinder in the supersonic to hypersonic range. Characteristics of this type of split-flux-vector algorithm, as applied to two-dimensional flows with shocks, are discussed. Introduclion The concept of using upwind finite-difference schemes for solving the inviscid fluid flow equations has generated considerable recent interest. A number of algorithms have attempted to incorporate ideas from the mathematical theory of charactcristics into finite-difference method^. "^ Initial computational results indicate that the new methods can lead to improvcd algorithms which are more robust and have a greater range of application than do standard central-difference schemes. In particular, they can handle stronger shocks and high-gradient regions where central methods fail, and do not need specially designed artificial viscosity terms for each problem. However, upwind methods are more complex than central schemes and require more computational effort pcr time step. Consequently, at this early stage of thcir development, upwind schemes have been applied to a relatively small number of problems. The upwind schemes can be divided into three basic classes. The first class consists of the flux-vector-splitting technique as applied to thc conservation law form of thc Euler equations; the second is the flux-difference-splitting methods, based on Riemann solution soIvers, also applied to thc conservation law form of the Euler equation; and the third class contains the up wind mcthods as applied to the primitive-variable form of the Euler equations. A brief description of the schemes in each class and the results presented to date follow. Interest in split-flux-vector methods as applied to the conscrvation law form of the Euler equations was triggered by the work of Steger and Warming.l Both implicit and explicit algorithms presented in Ref. 1 were applied to one-dimensional shock tube flow and two-dimensional airfoil flow using a uniform grid. The algorithms encounter difficulties when applied to nonuniform grid systems because transformation of the governing equations introduces metrics inside the differential operators. It is then difficult to numerically satisfy the metric conservation law since both forward and backward spatial difference operators may be applied at the same point. Instead of satisfy.c/ 'This work was supported by the McDonnell Doualus Indenendent Reseurch und Development program. Luborutories; member AIAA. "Research Scientist, McDonnell Douglus Research ing the metric conservation law exactly, Buning and Steger2 proposed a first-order correction that subtracts the freestream metric terms so that for a uniform freestream, no numerical error is introduced. As local conditions dcviate from freestream, the grid must be sufficiently refined to minimize any errors that result from improper metric differentiation. Using this improvement, two-dimensional flows about cylinders and airfoils were calculated on nonuniform grids. To alleviate the metric conservation law problem, Reklis and Thomas3 proposed that a finite-volume method be used with the split-flux-vector conservation law form of the Euler equations. Under the finite-volume formulation, the flowfield, which has directional components, is uncoupled from the metrics which are not directional. At a particular point, the same metrics are used for the fluxes computed from both forward and backward extrapolations, so that the metric conservation law is satisfied. Reklis and Thomas applied their implicit algorithm to onedimensional shock tube problems and to two-dimensional viscous flow over a flat plate. The earliest algorithm in the flux-difference-splitting class of methods based on Riemann solvers is the Godunov4 method, first introduced in 1959. Recent variations developed by Roe,S Engquist and Osher,6 Chakravarthy and Osher,7 and Lombard et appear to be simpler to implement than the Godunov method. Sellsio applied Roe's method to two-dimensional flow about an airfoil using a MacCormack predictor-corrector time intcgration. An explicit version of Osher's scheme has been used to calculate quasi-one-dimensional nozzle flow and two-dimensional supersonic flow past wedges and cylinders for Mach numbers as large as eight. ' The conservative supra-characteristics method, which was introduced recently, has been used to calculate quasi-one-dimensional nozzle flow for both real and perfect gases.9 Application of upwind-difference schemes to the nonconservative form of the Euler equations can be traced to the work of Courant, Isaacson, and Rees. " However, the leading proponent of using the characteristic formulation of the nonconservative primitive variable form of the equation has been Moretti with the "A" scheme.i2 This method and the closely related splitcoefficient-matrix (SCM) method of Chakravarthy et aLi3 have been applied to a number of one-dimensional and two-dimensional flowfields, generally with good results. Daywitt14 has used the SCM method to calculate three-dimensional flow about blunt bodies at supersonic speeds. Certainly this calculation is the most complex one performed using an upwind Euler scheme. However, with the nonconservative primitive variable form of the equations, shocks cannot be captured accurately. Signals cannot propagate across the shock, so that shocks cannot move upstream against the flow unless a shock tracking and fitting scheme is used, greatly increasing the algorithm complexity. The approach taken in the present work evolves from the ideas developed by Reklis and tho ma^. ^ This method was chosen because of its shock capturing capability, its simplicity as compared with flux-difference-splitting, and the ability of the finite-volume method to satisfy the metric conservation law. As a first step toward applying the upwind split-flux-vector finitevolume method to more complex problems, an explicit method has been derived on the basis of the Beam and Warming upwind algorithm.IS Although an implicit scheme might haw been more efficient, the simpler derivation and coding involved in the explicit scheme make it attractive as a development tool for the study of such algorithm characteristics as shock capturing capability, boundary condition implementation, robustness, accuracy, dissipative properties, and range of application. Two-Dimensional Euler Equations The set of equations to be solved consists of the twodimensional Euler equations. Letting 0, u, v, E, p and t denote the density, Cartesian velocity components, total energy, pressure, and time, respectively, this set of equations can be written in Cartesian coordinates x, y as