Sin-Chung Chang

Solving the MHD equations by the space-time conservation element and solution element method

We apply the space-time conservation element and solution element (CESE) method to solve the ideal MHD equations with special emphasis on satisfying the divergence free constraint of magnetic field, i.e., @?.B=0. In the setting of the CESE method, four approaches are employed: (i) the original CESE method without any additional treatment, (ii) a simple corrector procedure to update the spatial derivatives of magnetic field B after each time marching step to enforce @?.B=0 at all mesh nodes, (iii) a constraint-transport method by using a special staggered mesh to calculate magnetic field B, and (iv) the projection method by solving a Poisson solver after each time marching step. To demonstrate the capabilities of these methods, two benchmark MHD flows are calculated: (i) a rotated one-dimensional MHD shock tube problem and (ii) a MHD vortex problem. The results show no differences between different approaches and all results compare favorably with previously reported data.

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.

Local time-stepping procedures for the space-time conservation element and solution element method

A local time-stepping procedure for the space-time conservation element and solution element (CESE) method has been developed. This new procedure allows for variation of time-step size in both space and time, and can also be extended to become multi-dimensional solvers with structured/unstructured spatial grids. Moreover, it differs substantially in concept and methodology from the existing approaches. By taking full advantage of key concepts of the CESE method, in a simple and efficient manner it can enforce flux conservation across an interface separating grid zones of different time-step sizes. In particular, no correction pass is needed. Numerical experiments show that, for a variety of flow problems involving moving shock and flame discontinuities, accurate and robust numerical simulations can be achieved even with a reduction in time-step size on the order of 10 or higher for grids across a single interface.

Solving magnetohydrodynamic equations without special treatment for divergence-free magnetic field

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ACCURACY STUDY OF THE SPACE-TIME CE/SE METHOD FOR COMPUTATIONAL AEROACOUSTICS PROBLEMS INVOLVING SHOCK WAVES

The space-time conservation element and solution element(CE/SE) method is used to study the sound-shock interaction problem. The order of accuracy of numerical schemes is investigated. The linear model problem.govemed by the 1-D scalar convection equation, sound-shock interaction problem governed by the 1-D Euler equations, and the 1-D shock-tube problem which involves moving shock waves and contact surfaces are solved to investigate the order of accuracy of numerical schemes. It is concluded that the accuracy of the CE/SE numerical scheme with designed 2nd-order accuracy becomes 1st order when a moving shock wave exists. However, the absolute error in the CE/SE solution downstream of the shock wave is on the same order as that obtained using a fourth-order accurate essentially nonoscillatory (ENO) scheme. No special techniques are used for either high-frequency low-amplitude waves or shock waves.

Noise Computation of a Shock-Containing Supersonic Axisymmetric Jet by the CE/SE Method

The space-time conservation element solu- tion element (CE/SE) method (l) is em- ployed to numerically study the near-field of a typical under-expanded jet. For the computed case-a circular jet with Mach number Mj = 1.19-the shock-cell struc- ture is in good agreement with experimen- tal results (2, 31. The computed noise field is in general agreement with the experi- ment, although further work is needed to properly close the screech feedback loop.

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.

Prediction of Sound Waves Propagating Through a Nozzle Without/With a Shock Wave Using the Space-Time CE/SE Method

AbstractThe benchmark problems in Category l(InternalPropagation) of the third Computational Aeroacous-tics (CAA) Workshop sponsored by NASA GlennResearch Center are solved using the space-timeconservation element and solution element (CE/SE)method. The first problem addresses the propaga-tion of sound waves through a nearly choked tran-sonic nozzle. The second one concerns shock-soundinteraction in a supersonic nozzle. A quasi 1-DCE/SE Euler solver for a nonuniform mesh is de-veloped and employed to solve both problems. Nu-merical solutions are compared with the analyticalsolution for both problems. It is demonstrated thatthe CE/SE method is capable of soMng aeroacous-tic problems with/without shock waves in a simpleway. Furthermore, the simple non-reflecting bound-ary condition used in the CE/SE method which isnot. based on the characteristic theory works verywell.1. IntroductionThe method of space-time conservation elementand solution element (abbreviated as the CE/SEmethod) is an innovative numerical method for solv-ing conservation laws. It is different in both con-cept and methodology from the well-established tra-ditional methods such as the finite difference, finitevolume, finite element and spectral methods. It isdesigned from a physicists perspective to overcomeseveral key limitations of the traditional numericalmethods.Simplicity, generality and accuracy are weightedin the development of this method while the funda:mental requirements are satisfied by the scheme. Its

Computation of Feedback Aeroacoustic System by the CE/SE Method

It is well known that due to vortex shedding in high speed flow over cutouts, cavities, and gaps, intense noise may be generated. Strong tonal oscillations occur in a feedback cycle in which the vortices shed from the upstream edge of the cavity convect downstream and impinge on the cavity lip, generating acoustic waves that propagate upstream to excite new vortices. Numerical simulation of such a complicated process requires a scheme that can : (a) resolve acoustic waves with low dispersion and numerical dissipation, (b) handle nonlinear and discontinuous waves (e.g. shocks), and (c) have an effective (near field) non-reflecting boundary condition (NRBC). The new space time conservation element and solution element method, or CE/SE for short, is a numerical method that meets the above requirements [1–4]. A detailed description of the 2-D CE/SE Euler scheme can be found in [1, 2], only a brief sketch is given here.

Local Mesh Refinement in the Space-Time CE/SE Method

A local mesh refinement procedure for the CE/SE method which does not use an iterative procedure in the treatments of grid-to-grid communications is described. It is shown that a refinement ratio in the order of 10 can be applied successfully across a single coarse grid/fine grid interface.