Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach

The stability and accuracy of three methods which enforce either a divergence-free velocity field, density invariance, or their combination are tested here through the standard Taylor–Green and spin-down vortex problems. While various approaches to incompressible SPH (ISPH) have been proposed in the...

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Bibliographic Details
Published in:Journal of computational physics 2009-10, Vol.228 (18), p.6703-6725
Main Authors: Xu, Rui, Stansby, Peter, Laurence, Dominique
Format: Article
Language:English
Subjects:
ACCURACY
ANISOTROPY
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Computational techniques
DENSITY
Density invariance
DISTURBANCES
Divergence-free velocity field
Exact sciences and technology
HYDRODYNAMICS
Incompressible smoothed particle hydrodynamics (ISPH)
INSTABILITY
INTERPOLATION
Mathematical methods in physics
NOISE
OSCILLATIONS
Physics
RANDOMNESS
REYNOLDS NUMBER
SPIN
STABILITY
VELOCITY
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Summary:The stability and accuracy of three methods which enforce either a divergence-free velocity field, density invariance, or their combination are tested here through the standard Taylor–Green and spin-down vortex problems. While various approaches to incompressible SPH (ISPH) have been proposed in the past decade, the present paper is restricted to the projection method for the pressure and velocity coupling. It is shown that the divergence-free ISPH method cannot maintain stability in certain situations although it is accurate before instability sets in. The density-invariant ISPH method is stable but inaccurate with random-noise like disturbances. The combined ISPH, combining advantages in divergence-free ISPH and density-invariant ISPH, can maintain accuracy and stability although at a higher computational cost. Redistribution of particles on a fixed uniform mesh is also shown to be effective but the attraction of a mesh-free method is lost. A new divergence-free ISPH approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines, although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative. This avoids the highly anisotropic particle spacing which eventually triggers instability. Importantly pressure fields are free from spurious oscillations, up to the highest Reynolds numbers tested.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2009.05.032