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STABILIZED, MULTISCALE, AND MULTIPHYSICS METHODS IN FLUID MECHANICS

Time-Derivative Preconditioning Methods for Multicomponent Flows—Part I: Riemann Problems

[+] Author and Article Information
Jeffrey A. Housman, Mohamed M. Hafez

 University of California Davis, 2132 Bainer Hall, One Shields Avenue, Davis, CA 95616

Cetin C. Kiris

NASA Advanced Supercomputing (NAS) Division, NASA Ames Research Center, Moffett Field, CA 94035

J. Appl. Mech 76(2), 021210 (Feb 04, 2009) (13 pages) doi:10.1115/1.3072905 History: Received January 31, 2008; Revised July 10, 2008; Published February 04, 2009

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid multicomponent and multiphase flows at all speeds is described. The system is shown to be hyperbolic in time and remains well conditioned in the incompressible limit, allowing time marching numerical methods to remain an efficient solution strategy. It is well known that the application of conservative numerical methods to multicomponent flows containing sharp fluid interfaces will generate nonphysical pressure and velocity oscillations across the component interface. These oscillations may lead to stability problems when the interface separates fluids with large density ratio, such as water and air. The effect of which may lead to the requirement of small physical time steps and slow subiteration convergence for implicit time marching numerical methods. At low speeds the use of nonconservative methods may be considered. In this paper a characteristic-based preconditioned nonconservative method is described. This method preserves pressure and velocity equilibrium across fluid interfaces, obtains density ratio independent stability and convergence, and remains well conditioned in the incompressible limit of the equations. To extend the method to transonic and supersonic flows containing shocks, a hybrid formulation is described, which combines a conservative preconditioned Roe method with the nonconservative preconditioned characteristic-based method. The hybrid method retains the pressure and velocity equilibrium at component interfaces and converges to the physically correct weak solution. To demonstrate the effectiveness of the nonconservative and hybrid approaches, a series of one-dimensional multicomponent Riemann problems is solved with each of the methods. The solutions are compared with the exact solution to the Riemann problem, and stability of the numerical methods are discussed.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Results for Riemann problem I (top to bottom): pressure, velocity, temperature, and mass fraction

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Figure 2

Results for Riemann problem II (top to bottom): pressure, velocity, temperature, and mass fraction

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Figure 3

Results for Riemann problem III (top to bottom): pressure, velocity, temperature, and mass fraction

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Figure 4

Grid convergence of pressure (upper) and velocity (lower) for Riemann problem III using PROE

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Figure 5

Results for Riemann problem IV (top to bottom): pressure, velocity, temperature, and mass fraction

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Figure 6

Top two: grid convergence of pressure (upper) and velocity (lower) for Riemann problem IV using PROE. Bottom two: grid convergence of pressure (upper) and velocity (lower) for Riemann problem IV using HYBR.

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Figure 7

Results for Riemann problem V (top to bottom): pressure, velocity, temperature, and mass fraction

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Figure 8

Grid convergence of pressure (upper) and velocity (lower) for Riemann problem V using HYBR

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Figure 9

Results for Riemann problem VI (top to bottom): pressure, velocity, temperature, and mass fraction

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