Shear-viscosity-independent bulk-viscosity term in smoothed particle hydrodynamics

An angular momentum conservative pure bulk viscosity term for smoothed particle hydrodynamics (SPH) is proposed in the present paper. This formulation permits independent modeling of shear and bulk viscosities, which is of paramount importance for fluids with large bulk viscosity in situations where sound waves or large Mach numbers are expected. With this aim a dissipative term proportional to the rate of change of the volume is considered at the particle level. The equations of motion are derived from the minimization of a Lagrangian combined with an appropriate dissipation function that depends on this rate of change of particle volume, in analogy with the corresponding entropy production contribution in fluids. Due to the Galilean invariance of the formulation, the new term is shown to exactly conserve linear momentum. Moreover, its invariance under solid-body rotations also ensures the conservation of angular momentum. Two verification cases are proposed: the one-dimensional propagation of a sound pulse and a two-dimensional case, modeling the time decay of an accelerating-decelerating pipe flow. The SPH solutions are compared to exact ones, showing that the newly proposed term behaves indeed as a viscosity associated only with the local expansion-compression of the fluid. In view of these considerations, we conclude that the method presented in this paper allows for setting up a bulk viscosity independently of the shear one and as large as any particular problem may require. At the same time, together with the prescribed momentum conservation to reproduce the Navier-Stokes equation, the new term also keeps the angular momentum conservation required to properly model free interfaces or overall rotations of the bulk fluid.

Figure 1

Sinusoidal pulse. Initial velocity field, $n=256.$

Figure 2

Sinusoidal pulse. Velocity field, $n=256, t/(L/c_0)\approx 1.22.$

Figure 3

One-dimensional sinusoidal pulse propagation. $A{\rho }_0{c}_{0}L/\kappa =0.05, \mathrm{\Delta }x/h=0.25$. Kinetic energy; time history.

Figure 4

Pipe flow: initial velocity field for $\mathrm{Re}=50$ and different values of the ratio $\sigma =\lambda /\mu$.

Figure 5

Pipe flow: time decay of the velocity at the center of the fluid domain and of the total kinetic energy for different values of the ratio $\sigma =\lambda /\mu$. The dash-dotted lines represent the analytical solutions, while the solid lines are the SPH solutions.