High-resolution numerical schemes for compressible flows and\ud compressible two-phase flows

Abstract

Several high-resolution numerical schemes based on the Constrained Interpolation Profile Conservative Semi-Lagrangian (CIP-CSL), Essentially Non-Oscillatory (ENO), Weighted ENO (WENO), Boundary Variation Diminishing (BVD), and Tangent of Hyperbola for INterface Capturing (THINC) schemes have been proposed for compressible flows and compressible two-phase flows. In the first part of the thesis, three high-resolution CIP-CSL schemes are proposed. (i) A fully conservative and less oscillatory multi-moment scheme (CIP-CSL3-ENO) is proposed based on two CIP-CSL3 schemes and the ENO scheme. An ENO indicator is designed to intentionally select non-smooth stencil but can efficiently minimise numerical oscillations. (ii) Motivated by the observation that combining two different types of reconstruction functions can effectively reduce numerical diffusion and oscillations, a better-suited scheme CIP-CSL-ENO5 is proposed based on hybrid-type CIP-CSL reconstruction functions and a newly designed ENO indicator. (iii) To further reduce the numerical diffusion in vicinity of discontinuities, the BVD and THINC schemes are implemented in the CIP-CSL framework. The resulting scheme accurately capture both smooth and discontinuous solutions simultaneously by selecting an appropriate reconstruction function. In the second part of the thesis, the TWENO (Target WENO) scheme is proposed to improve the accuracy of the fifth-order WENO scheme. Unlike conventional WENO schemes, the TWENO scheme is designed to restore the highest possible order interAbstract iv polation when three sub-stencils or two adjacent sub-stencils are smooth. To further minimise the numerical diffusion across discontinuities, the TWENO scheme is implemented with the THINC scheme and the Total Boundary Variation Diminishing (TBVD) algorithm. The resulting scheme TBVD-TWENO-THINC is also applied to solve the five-equation model for compressible two-phase flows. Verified through a wide range of benchmark tests, the proposed numerical schemes are able to obtain accurate and high-resolution numerical solutions for compressible flows and compressible two-phase flows