High-order finite difference scheme for compressible multi-component flow computations

Abstract

This paper presents a high-order weighted essentially non-oscillatory (WENO) finite difference scheme for compressible multi-fluid and multi-phase dynamics. The scheme overcomes the difficulty of applying the common flux-based WENO finite difference scheme to multi-fluid problems by applying the reconstruction on primitive variables and, thus avoiding the spurious oscillations inherent in the standard methods. Schemes of orders up to nine are introduced and analyzed. The proposed finite difference schemes are significantly more efficient than the available high-order finite volume schemes in both storage requirement, operation counts, and inter-processor message passing in parallel computations with efficiency gains being higher at higher orders and higher spatial dimensions. For the same level of accuracy, in three-dimensional calculations, a fourfold speedup or higher at fifth-order accuracy or higher over the finite volume scheme is expected. A comparison of the proposed scheme with the standard flux-based finite difference scheme in solving a single-fluid shock small entropy wave interaction is presented, demonstrating its excellent performance. In a challenging, two-fluid problem of a strong shock interacting with a helium-air interface, the accuracy, non-oscillatory, conservation, and convergence of the scheme are illustrated. Moreover, in a liquid water-air shock tube problem, the scheme effectively captures compression and expansion waves and contact discontinuity, and outperforms low-order schemes. In computations of a two-dimensional shock bubble interaction, good agreements with experimental data are obtained and competitive performance to the high-order finite volume scheme is shown.