A new rotated upwind difference scheme for the Euler equations

A new explicit, rotated upwind algorithm has been developed to solve the Euler equations. The upwinding angle for the rotated algorithm is determined by using the local pressure gradient data and the upwind fluxes are computed using Roe’s method. A “refining” procedure is then applied to obtain the final solution which exhibits little or no smearing across a discontinuity. Three test cases were computed to validate the new rotated algorithm. These include l ) a simple oblique shock problem, 2) a simple shock reflection problem, and 3) supersonic flow into a twc-dimensional converging inlet. The computed results are in excellent agreement with the exact solutions and show significant improvement over other rotated upwind schemes. INTRODUCTION In the 1960’s and 1970’s, the Euler equations were typically solved using central-difference schemes [I]. Some examples of these central-difference schemes are those developed by MacCormack [2], and Beam and Warming [3]. These schemes are quite accurate a t modelling the flow field except near discontinuities such as shock waves and slip lines. At these discontinuities, central-difference schemes produce spurious oscillations. To overcome this problem, artificial viscosity terms are frequently added to central-difference schemes. Although these terms dampen the spurious oscillations, they in turn cause a decrease in the retlution of the discontinuities. These damping terms are constructed rather arbitrarily and rely on the experience of the user to specify the magnitude of the added numerical dissipation. To complicate things even further, the values of these coefficients are different for each particular flow problem. *Graduate Research Assistant. Student Member AIAA t Professor and Manager of Computational Fluid Dynamics . Center, Iowa State University, Associate Fellow AIAA Copyright @American Institute ol Aeronautics and Astronautics. Inc.. 1993. All rights reserved. U In the 1980’s, upwind-difference schemes were de veloped to overcome the above mentioned deficiencies of the central-difference schemes. These upwind schemes have the advantage of being naturally dissipative, thus providing a built-in mechanism to dampen the spurious oscillations across the discontinuities, thus eliminating the need to add artificial viscosity or smoothing terms. In other words, “smart” smoothing terms are incorporated into these upwind schemes to stabilize the solution. Upwind algorithms can be classified into three general categories according to the way the equations are “split”. These are 1) flux vector splitting, 2) split coefficient matrix (SCM), and 3) flux difference splitting methods. The flux vector splitting method splits the flux vector into two parts, one with the positive, and the other with the negative eigenvalues. Differencing is ’ then applied to both parts in an upwind manner according to the directional propagation of the carrier wav Examples of algorithms that uses the flux vector splitting method are those of Steger and Warming 141, and Van Leer [5]. The split coefficient matrix (SCM) scheme, which was introduced by Chakravarthy et al. [ G I , is a nonconservative form of the flux vector splitting algorithm of Steger and Warming. Since it is nonconservative, this method is not capable of accurately capturing shocks. Another example of an algorithm that makes use of the split coefficient matrix method is Moretti’s A scheme [7]. The flux difference splitting method splits the flux differences in terms of the signs of the eigenvalues for proper upwind differencing. An example of an algorithm that uses the flux difference splitting method is the conservative supra-characteristics method (CSCM) introduced by Lombard et al. [SI. More recently, other types of flux vector splitting and flux difference splitting algorithms have been developed as a result of the work of Godunov [9]. Godunov developed a first-order upwind scheme that modelled the discontinuities in fluid flows with remarkable accuracy 191. The key to his scheme is that it uses an exact solution of the Riemann problem [lo] to ! determine the fluxes at the cell interfaces. Following the introduction of Godunov’s scheme, a whole new family of Godunov-type schemes was born, each differing only by the way in which the fluxes a t the cell interfaces were determined. Osher and Soloman [l l] used an approximate solution to the exact Riemann problem to determine the fluxes while Roe [I21 used an exact solution to an approximate Riemann problem to determine the fluxes. Although these methods capture discontinuities with remarkable clarity, they are only first-order accurate. To improve the accuracy of these schemes, they have been extended to secondand third-order accuracy which unfortunately “triggers” new oscillations. Flux limiters [I31 have been introduced to prevent these new oscillations. W When dealing with one-dimensional flow problems, the current upwind schemes have the ability to predict shock structures accurately. This is accomplished by determining the waves that propagate in the grid direction (i.e. solving the Riemann problem). Typically, the extension to a multidimensional problem has been to implement the one-dimensional operator in each grid direction separately. Although this method has been shown to work suprisingly well [14], the difficulty with this approach is that the physical features of the flow are not modelled exactly. This becomes particularly evident when computing shocks that cross the computational grid at an oblique angle as smeared results are produced across the discontinuity. To overcome this problem, research has been performed to develop grid-independent upwind-difference schemes. The notion of the rotated Riemann problem was developed by Davis [15]. He used the velocity field to determine the desired upwinding direction at each grid face, determined the flux components normal and tangent to this direction and then employed one of the common one-dimensional algorithms to solve the resulting Riemann problem. This idea was further developed by Levy et al. [16]. They employed a first-order accurate, finite-volume, Euler code which used the local pressure gradient data to determine upwinding directions and then solved a local Riemann problem in the dominant direction. Similar work was performed by Dadone and Grossman [17] who showed second-order comparisons. Rumsey et al. [IS] developed a two-dimensional, grid-independent approximate Riemann solver that determines fluxes on grid faces via wave decomposition. Also Kontinos and McRae [I91 have developed an explicit, rotated upwind algorithm that uses the local pres-.J 2 sure gradient data to obtain the upwinding direction. Interpolation was employed to obtain the fluxes in the rotated frame with the upwinding performed tangent to this rotated direction. The MacCormack stencil was then used to solve the Riemann problem. The rotated algorithm developed in this study is an explicit, timemarching scheme which also uses the local pressure gradient data to obtain the rotation angle which in turn gives the desired upwinding direction. The fluxes are then calculated in this rotated frame by interpolating between the appropriate grid points. Up to this point, the present rotated scheme is quite similar to the scheme presented by Kontinos and McRae. The present scheme differs from Kontinos and McRae’s scheme in several ways. First of all, Roe’s upwind scheme is employed as opposed to the central-difference MacCormack stencil used in Kontinos and McRae’s algorithm. Secondly, upwinding is performed in directions normal and tangent to the rotated direction as compared to only the tangent direction in Kontinos and McRae’s scheme. Finally, the rotated scheme in this study applies a refining procedure to the converged solution to obtain the final results. In order to validate the new rotated upwind scheme, three test cases were computed. These include 1) a simple oblique shock problem, 2) a simple shock reflection problem, and 3) supersonic flow into a twodimensional converging inlet. The present results are compared with the results obtained from the Dadone and Grossman rotated upwind scheme, and the conventional non-rotated Roe’s scheme. GOVERNING EQUATIONS The governing equations used in this study are the two-dimensional Euler equations. In Cartesian coordinates and nondimensional form, the equations can be where U is a vector of the conserved variables, and E and F are the vectors of the inviscid fluxes: