C. Y. Loh’s research while affiliated with Hong Kong University of Science and Technology and other places

This paper presents a new Lagrangian approach for the two-dimensional (2-D) time-dependent Euler equations. It may be considered as a sequel to the authors' previous Lagrangian approaches for steady supersonic flow computations [C. Y. Loh and W. H. Hui, J. Comput. Phys., 89 (1990), pp. 207--240; W. H. Hui and C. Y. Loh, J. Comput. Phys., 103 (1992), pp. 450--464; W. H. Hui and C. Y. Loh, J. Comput. Phys., 103 (1992), pp. 465--471; C. Y. Loh and M. S. Liou, J. Comput. Phys., 104 (1993), pp. 150--161; C. Y. Loh and M. S. Liou, SIAM J. Sci. Comput., 15 (1994), pp. 1038--1058; C. Y. Loh and M. S. Liou, J. Comput. Phys., 113 (1994), pp. 224--248]. The theoretical background and the intrinsic flow coordinates as well as the Lagrangian conservation form are introduced based on the concept of material functions (or path functions). A TVD scheme of the Godunov type is chosen to describe the numerical procedure. Several examples are then given to justify the claimed advantages of the new methodology, namely, (a) any contact discontinuities are crisply solved and (b) grids are automatically and accurately generated following pathlines.

In this paper, an unsplit random choice method (RCM) is developed and applied to numerically solve three-dimensional supersonic steady flow problems. In order to keep the contacts (slip surfaces) crisply resolved, a new Lagrangian formulation is employed. Due to the lack of exact solutions to 3D Riemann problems, approximate Riemann solutions in the weak sense are adopted. The RCM is thus as efficient as the deterministic TVD schemes, and yields almost identical results in the model problems. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper, an unsplit random choice method (RCM) is developed and applied to numerically solve three-dimensional supersonic steady flow problems. In order to keep the contacts (slip surfaces) crisply resolved, a new Lagrangian formulation is employed. Due to the lack of exact solutions to 3D Riemann problems, approximate Riemann solutions in the weak sense are adopted. The RCM is thus as efficient as the deterministic TVD schemes, and yields almost identical results in the model problems. Copyright (C) 1999 John Wiley & Sons, Ltd.

The exact real gas Riemann solution and the random choice method (RCM) are briefly reviewed and the derivation of the Lagrangian geometrical quantities, which represent the deformation of fluid particles in the motion, are described in detail. Extensive calculations were made to test the accuracy against the exact solution and the robustness of the Lagrangian RCM approach for real gas supersonic flows, including complex wave interactions of different types. The real gas effect is also presented by comparison with the perfect gas solution. The inherent parallelism in the Lagrangian approach lends a natural application in the massively parallel computation.

In this paper, numerical simulation of three-dimensional supersonic flow in a duct is presented. The flow field in the duct is complex and can find its applications in the inlet of air-breathing engines. A unique streamwise marching Lagrangian method is employed for solving the steady Euler equations. The method was first initiated by Loh and Hui (1990) for 2-D steady supersonic flow computations and then extended to 3-D computation by the present authors Loh and Liou (1992). The new scheme is shown to be capable of accurately resolving complicated shock or contact discontinuities and their interactions. In all the computations, a free stream of Mach numberM=4 is considered.

In this report, the new Lagrangian method introduced by Loh and Hui is extended for three-dimensional, steady supersonic flow computation. The derivation of the conservation form and the solution of the local Riemann solver using the Godunov and the high-resolution TVD (total variation diminished) scheme is presented. This new approach is accurate and robust, capable of handling complicated geometry and interactions between discontinuous waves. Test problems show that the extended Lagrangian method retains all the advantages of the two-dimensional method (e.g., crisp resolution of a slip-surface (contact discontinuity) and automatic grid generation). In this report, we also suggest a novel three dimensional Riemann problem in which interesting and intricate flow features are present.

The present extention of a Lagrangian approach of the Riemann solution procedure, which was originally proposed for perfect gases, to real gases, is nontrivial and requires the development of an exact real-gas Riemann solver for the Lagrangian form of the conservation laws. Calculations including complex wave interactions of various types were conducted to test the accuracy and robustness of the approach. Attention is given to the case of 2D oblique waves' capture, where a slip line is clearly in evidence; the real gas effect is demonstrated in the case of a generic engine nozzle.

The new Lagrangian method introduced by Loh and Hui (1990) is extended for 3-D steady supersonic flow computation. Details of the conservation form, the implementation of the local Riemann solver, and the Godunov and the high resolution TVD schemes are presented. The new approach is robust yet accurate, capable of handling complicated geometry and reactions between discontinuous waves. It keeps all the advantages claimed in the 2-D method of Loh and Hui, e.g., crisp resolution for a slip surface (contact discontinuity) and automatic grid generation along the stream.

It is well known that high-order accurate shock-capturing schemes, e.g., second-order TVD and ENO schemes, based on Eulerian formulation are capable of resolving a shock discontinuity in two grid points, but they smear a slip-line (contact-line) discontinuity over several grid points. In this paper we show theoretically and numerically that the first-order Godunov scheme based on the new Lagrangian formulation of Hui and Van Roessel for steady supersonic flow always resolves an isolated slip-line discontinuity crisply, provided it is initially aligned with a grid line. Moreover, a simple extension of the second-order scalar TVD scheme of Sweby to the system of Euler equations based on the new Lagrangian formulation, with no special procedure for slip-line detection, resolves slip-line discontinuities in at most two grid points. Many examples are given, showing excellent agreement with known exact solutions.