Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported (Quirk JJ. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18: 555-574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd -even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd -even decoupling. This analysis is applied to Roe's scheme (Roe PL, Approximate Riemann solvers, parameters vectors, and difference schemes,
SUMMARYSince the development of shock-capturing methods, the carbuncle phenomenon has been reported to be a spurious solution produced by almost all currently available contact-preserving methods. The present analysis indicates that the onset of carbuncle phenomenon is actually strongly related to the shock wave numerical structure. A matrix-based stability analysis as well as Euler ÿnite volume computations are compared to illustrate the importance of the internal shock structure to trigger the carbuncle phenomenon.
The theoretical linear stability of a shock wave moving in an unlimited homogeneous environment has been widely studied during the last fifty years. Important results have been obtained by Dýakov (1954), Landau & Lifchitz (1959) and then by Swan & Fowles (1975) where the fluctuating quantities are written as normal modes. More recently, numerical studies on upwind finite difference schemes have shown some instabilities in the case of the motion of an inviscid perfect gas in a rectangular channel. The purpose of this paper is first to specify a mathematical formulation for the eigenmodes and to exhibit a new mode which was not found by the previous stability analysis of shock waves. Then, this mode is confirmed by numerical simulations which may lead to a new understanding of the so-called carbuncle phenomenon.
Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of them is the odd-even decoupling that occurs along planar shocks aligned with the mesh. Quirk proposed to test this shortcoming with the propagation of a planar shock in a duct. First, we give a few results on some failings. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd-even decoupling. This analysis is applied to Roe's scheme, EFM Pullin's scheme, EIM Macrossan's scheme and AUSM Liou's scheme. Strict stability is shown to be desirable to avoid must of these flaws.
This paper deals with the construction of a conservative method for coupling a fluid mechanics solver and a heat diffusion code. This method has been designed for unsteady applications. Fluid and solid computational domains are simultaneously integrated by dedicated solvers. A coupling procedure is periodically called to compute and update the boundary conditions at the solid/fluid interface. First, the issue of general constraints for coupling methods is addressed. The concept of interpolation scheme is introduced to define the way to compute the interface conditions. Then, the case of the Finite Volume Method is thoroughly studied. The properties of stability and accuracy have been optimized to define the best coupling boundary conditions: the most robust method consists in assigning a Dirichlet condition on the fluid side of the interface and a Robin condition on the solid side. The accuracy is very dependent on the interpolation scheme. Moreover, conservativity has been specifically addressed in our methodology. This numerical property is made possible by the use of both the Finite Volume Method and the corrective method proposed in the current paper. The corrective method allows the cancellation of the possible difference between heat fluxes on the two sides of the interface. This method significantly improves accuracy in transient phases. The corrective process has also been designed to be as robust as possible. The verification of our coupling method is extensively discussed in this article: the numerical results are compared with the analytical solution of an infinite thick plate in a suddenly accelerated flow (and with the results of other coupling approaches).
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