Analysis of sponge zones for computational fluid mechanics

Journal Article · Wed Mar 01 00:00:00 EST 2006 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2005.07.014· OSTI ID:20767031

Bodony, Daniel J ^[1]

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1. Center for Turbulence Research, Stanford University, 488 Escondido Mall, Stanford, CA 94305-3035 (United States)

The use of sponge regions, or sponge zones, which add the forcing term -{sigma}(q - q {sub ref}) to the right-hand-side of the governing equations in computational fluid mechanics as an ad hoc boundary treatment is widespread. They are used to absorb and minimize reflections from computational boundaries and as forcing sponges to introduce prescribed disturbances into a calculation. A less common usage is as a means of extending a calculation from a smaller domain into a larger one, such as in computing the far-field sound generated in a localized region. By analogy to the penalty method of finite elements, the method is placed on a solid foundation, complete with estimates of convergence. The analysis generalizes the work of Israeli and Orszag [M. Israeli, S.A. Orszag, Approximation of radiation boundary conditions, J. Comp. Phys. 41 (1981) 115-135] and confirms their findings when applied as a special case to one-dimensional wave propagation in an absorbing sponge. It is found that the rate of convergence of the actual solution to the target solution, with an appropriate norm, is inversely proportional to the sponge strength. A detailed analysis for acoustic wave propagation in one-dimension verifies the convergence rate given by the general theory. The exponential point-wise convergence derived by Israeli and Orszag in the high-frequency limit is recovered and found to hold over all frequencies. A weakly nonlinear analysis of the method when applied to Burgers' equation shows similar convergence properties. Three numerical examples are given to confirm the analysis: the acoustic extension of a two-dimensional time-harmonic point source, the acoustic extension of a three-dimensional initial-value problem of a sound pulse, and the introduction of unstable eigenmodes from linear stability theory into a two-dimensional shear layer.

• OSTI ID: 20767031
• Journal Information: Journal of Computational Physics, Vol. 212, Issue 2; Other Information: DOI: 10.1016/j.jcp.2005.07.014; PII: S0021-9991(05)00351-7; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
BOUNDARY CONDITIONS
CONVERGENCE
DISTURBANCES
EQUATIONS
FLUID MECHANICS
MATHEMATICAL SOLUTIONS
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
POINT SOURCES
PULSES
REFLECTION
SOUND WAVES
THREE-DIMENSIONAL CALCULATIONS
TWO-DIMENSIONAL CALCULATIONS
WAVE PROPAGATION
ZONES