Ananda Himansu

Robust and simple non-reflecting boundary conditions for the space-time conservation element and solution element method

This paper reports on a significant advance in the area of non-reflecting boundary conditions for unsteady flow computations. Sets of new non-reflecting boundary conditions for ID Euler problems are developed without using any characteristics-based techniques. These conditions are much simpler than those commonly reported in the CFD literature, yet so robust that they are applicable to subsonic, transonic and supersonic flows even in the presence of discontinuities. The paper details the theoretical underpinning of the boundary conditions, and explains their unique robustness and accuracy, in terms of the conservation of space-time fluxes. Some numerical results for an extended Sods shock-tube problem, illustrating the effectiveness of the boundary condi* Senior Research Scientist, e-mail: vvscc@noboo.lerc.nasa.gov ^Member, AIAA; e-mail: fshiman@trout.lerc.nasa.gov ^Member, AIAA; e-mail: fsloh@nasp03.lerc.nasa.gov § Member, AIAA, and Research Associate, e-mail: wangxy@rintintin.colorado.edu ^Member, AIAA, and Senior Engineer, e-mail: styu@lerc.nasa.gov II Member, AIAA, and Aerospace Engineer, e-mail: jorgenson@lerc.nasa.gov Copyright ©1997 American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright owner. tions, are included, together with the simple Fortran computer program with which they were obtained. Since the properties of the numerical boundary conditions are closely linked to the previously developed interior schemes, a summary of the interior schemes is also provided.

The CE/SE Method for Navier-Stokes Equations Using Unstructured Meshes for Flows at All Speeds

In this paper, we report an extension of the Space-Time Conservation Element and Solution Element (CE/SE) Method for solving the Navier-Stokes equations. Numerical algorithms for both structured and unstructured meshes are developed. To calculate the viscous flux terms, a ‘midpoint rule’ is used. In the setting of space-time flux conservation, a new and unified boundary-condition treatment for solid wall is introduced. The Navier Stokes solvers retain all favorable features of the original CE/SE method for the Euler equations, including high fidelity resolution of unsteady flows, easy implementation of nonreflective boundary conditions, and simplicity of computational logic. In addition, numerical results show that the present Navier-Stokes solvers can be used for high-speed flows as well as low-Mach-number flows without preconditioning. The present Navier Stokes solvers are efficient, accurate, and very robust for flows at all speeds.

Computation of an Underexpanded 3-D Rectangular Jet by the CE/SE Method

Recently, an unstructured three-dimensional space-time conservation element and solution element (CE/SE) Euler solver was developed. Now it is also developed for parallel computation using METIS for domain decomposition and MPI (message passing interface). The method is employed here to numerically study the near-field of a typical 3-D rectangular under-expanded jet. For the computed case-a jet with Mach number Mj = 1.6. with a very modest grid of 1.7 million tetrahedrons, the flow features such as the shock-cell structures and the axis switching, are in good qualitative agreement with experimental results.

Parallel CE/SE Computations via Domain Decomposition

This paper describes the parallelization strategy and achieved parallel efficiency of an explicit time-marching algorithm for solving conservation laws. The Space- Time Conservation Element and Solution Element (CE/SE) algorithm for solving the 2D and 3D Euler equations is parallelized with the aid of domain decomposition. The parallel efficiency of the resultant algorithm on a Silicon Graphics Origin 2000 parallel computer is checked.

A Modified Space-Time Conservation Element and Solution Element Method for Euler and Navier-Stokes Equations

In this paper, we report a variation of Chang’s SpaceTime Conservation Element and Solution Element CE/SE Method for structured mesh. The algorithm of the present modified CE/SE method is even simpler than the original CE/SE method. Nevertheless, all advantageous features of the original CE/SE method have been retained, including a unified treatment of space and time, accurate calculation of the space-time flux, high-fidelity resolution of unsteady flow motions, and full compliance with the physics of initial valued problems. The present modified method is also extended to solve Navier Stokes equations. To calculate the viscous flux terms, a ‘midpoint rule’ is used. In the setting of space-time flux conservation, two boundary-condition treatments for solid wall are introduced. Numerical results show that the present Navier Stokes solver can be used for high-speed flows as well as low-Mach-number flows without preconditioning. Results of several standard flow problems show that the present method is efficient and accurate.

Computation on Wake/Stator Interaction in a 2D Cascade

Numerical simulations of rotor-stator interaction using the space-time conservation element and solution element (CE/SE) method are presented. The wake generated by the rotor is specified through the unsteady inlet boundary condition of the stator, over which the time-marching is performed. The Giles approach is incorporated with the CE/SE method such that numerical simulations involving rotor and stator rows with unequal blade counts can be carried out using a single or a few flow passages instead of using the full set of passages. The benchmark problems 3.2 of the 2nd and 4th CAA Workshops are used to validate the Giles approach with the CE/SE method. For Prob. 3.2 of the 2nd CAA Workshop, both numerical results obtained by using a single passage and the full set of passages are presented and compared with the analytical solution, while for the problem of the 4th CAA Workshop, the numerical solution obtained by using two passages is presented.

High-Resolution Genuinely Multidimensional Solution of Conservation Laws by the Space-Time Conservation Element and Solution Element Method

In this overview paper, we review the basic principles of the method of space-time conservation element and solution element for solving the conservation laws in one and two spatial dimensions. The present method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. In contrast to the modern upwind schemes, the approach here does not use the Riemann solver and the reconstruction procedure as the building blocks. The drawbacks of the upwind approach, such as the difficulty of rationally extending the 1D scalar approach to systems of equations and particularly to multiple dimensions is here contrasted with the uniformity and ease of generalization of the Conservation Element and Solution Element (CE/SE) 1D scalar schemes to systems of equations and to multiple spatial dimensions. The assured compatibility with the simplest type of unstructured meshes, and the uniquely simple nonreflecting boundary conditions of the present method are also discussed. The present approach has yielded high-resolution shocks, rarefaction waves, acoustic waves, vortices, ZND detonation waves, and shock/acoustic waves/vortices interactions. Moreover, since no directional splitting is employed, numerical resolution of two-dimensional calculations is comparable to that of the one-dimensional calculations. Some sample applications displaying the strengths and broad applicability of the CE/SE method are reviewed.