SPH compressible turbulence

In this paper a smoothed particle hydrodynamics (SPH) version of the alpha turbulence model devised by Holm and his colleagues is formulated for compressible flow with a resolution that varies in space and time. The alpha model involves two velocity fields. One velocity field is obtained from the momentum equation, the other by averaging this velocity field as in the version of SPH called XSPH. The particles (fluid elements) are moved with the averaged velocity. In analogy to the continuum alpha model we obtain a particle Lagrangian from which the SPH alpha equations can be derived. The system satisfies a discrete Kelvin circulation theorem identical to that obtained with no velocity averaging. In addition, the energy, linear and angular momentum are conserved. We show that the continuum equivalent of the SPH equations are identical to the continuum alpha model, and we conjecture that they will have the same desirable features of the continuum model including the reduction of energy in the high-wavenumber modes even when the dissipation is zero. Regardless of issues concerning turbulence modelling, the SPH alpha model is a powerful extension of the XSPH algorithm, which reduces disorder at short length-scales and retains the constants of the motion. The SPH alpha model is simple to implement.