Article

# Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

NASA Ames Research Center, Moffett Field, California, USA; Tel Aviv University, Tel Aviv, Israel; New York University, New York, USA

Journal of Computational Physics (Impact Factor: 2.49). 03/1985; DOI: 10.1016/0021-9991(85)90183-4 Source: NTRS

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**ABSTRACT:**La propulsión eléctrica en vehículos espaciales es en nuestros días una alternativa competitiva frente a la propulsión convencional o química. La aceleración de gases para propulsión lograda por fuerzas electromagnéticas comprende a los propulsores magnetoplasmadinámicos (MPD) y a los propulsores de plasma pulsante (PPT). Un módulo propulsivo de plasma ablativo (APPT) se está desarrollando por investigadores de la Universidad Nacional de Río Cuarto (UNRC), del Instituto Universitario Aeronáutico (IUA) y de la Universidad Nacional de Córdoba (UNC). Estas investigaciones tienen como uno de sus objetivos específicos describir en forma física y numérica el desarrollo del flujo de plasma dentro de la cámara de reacción de un propulsor de plasma de los tipos APPT y AMPD. El estudio de los flujos en los cuales un gas eléctricamente conductor se mueve en un campo magnético es denominado Magnetogasdinámica (MGD). Aquí se presenta una metodología para simular numéricamente flujos magnetogasdinámicos difusivos, viscosos, bidimensionales e inestacionarios. Al ser planteado el sistema de ecuaciones en forma conservativa las contribuciones parabólicas son escritas en forma de flujos, lo que permite actualizar las variables de estado considerando los flujos numéricos hiperbólicos y parabólicos. La aproximación numérica se basa en la utilización de volúmenes finitos sobre mallas estructuradas. Para el cálculo de los flujos numéricos se ha implementado un esquema TVD conjuntamente con un seguidor de Riemann aproximado. Poder simular adecuadamente el flujo dentro del propulsor requiere conjuntamente con las ecuaciones de la MGD un adecuado modelo de las condiciones de borde. En este trabajo se presenta un estudio de las mismas considerando los aspectos físicos y numéricos asociados.IV Congreso Argentino de Tecnología Espacial, Buenos Aires, Argentina; 05/2007 - [Show abstract] [Hide abstract]

**ABSTRACT:**A two-phase flow at high Reynolds numbers in the subsonic, transonic and supersonic parts of the nozzle is considered within the framework of Prandtl model, i.e. the flow is divided into an inviscid core and a thin boundary layer. The mutual influence of gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The authors consider the effect of various boundary conditions for the flow of particles in the inviscid region. For an axisymmetric nozzle, the influence of the contour of subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied.Prikladnaya Mekhanika i Tekhnicheskaya Fizika. 10/2005; 46(6). - [Show abstract] [Hide abstract]

**ABSTRACT:**The present work compares the TVD schemes of Roe, of Van Leer, of Yee,Warming and Harten, of Harten, of Yee and Kutler and of Hughson and Beran applied to the solution of an aeronautical problem. Only the Van Leer scheme is a flux vector splitting one. The others are of flux difference splitting type. The Roe and Van Leer schemes reach second order accuracy and TVD properties by the use of a MUSCL approach, which employs five different types of nonlinear limiters, that assures TVD properties, being them: Van Leer limiter, Van Albada limiter, minmod limiter, Super Bee limiter and -limiter. The other schemes are based on the Harten's ideas of the construction of a modified flux function to obtain second order accuracy and TVD characteristics. The implicit schemes employ an ADI ("Alternating Direction Implicit") approximate factorization to solve implicitly the Euler equations, whereas in the explicit case a time splitting method is used. Explicit and implicit results are compared trying to emphasize the advantages and disadvantages of each formulation. The Euler equations in conservative form, employing a finite volume formulation and a structured spatial discretization, are solved in two-dimensions. The steady state physical problem of the supersonic flow along a compression corner is studied. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. This technique has proved an excellent behavior in terms of convergence gains, as shown in Maciel. The results have demonstrated that the most accurate solutions are provided by the Roe TVD scheme in its Super Bee variant. Key-Words: -Roe scheme, Van Leer scheme, Yee, Warming and Harten scheme; Harten scheme; Yee and Kutler scheme; Hughson and Beran scheme; Explicit and implicit formulations; TVD formulation; Euler and Navier-Stokes equations.

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