4. Modified bulk viscosity
A smoothed quantity can be calculated at any point in the fluid by Eq. (1). The smoothed velocity at is
If this point coincides with a position for a particle, the smoothed and the particle's individual velocity will be different in general. This difference is used to construct a modified bulk viscosity. Benz (1990) suggests that this quantity could be used when integrating the position to prevent interparticle penetration. It is true that it prevents interparticle penetration, but it unfortunately introduces conservation problems when the particles are not moved at their individal velocity. If the contribution from particle i subtracted, it does however say something about the fluid around the particle. The smoothed velocity at r can then be redefined as
where . The denominator scales the expression so that , if for all j .
In the standard formulation of the bulk artificial viscosity particle i interacts with each neighbour separately, where the acceleration and time derivative of the internal energy is added to the particle as described in Sect. 2. This introduces problems described in Sect. 3. The individual velocity of particle i can then be seen as a deviation from the smoothed value at the particle's position and the artificial viscosity as a correction to the individual velocity. I propose a modified bulk viscosity to replace the standard bulk viscosity, where the fluid around the particle is considered from a collective contribution from the neighbours in one single interaction. This modified bulk artificial viscosity is defined as
where the number of neighbours. The constant around unity, and used in the same way as the constants and in Eq. (6). This interaction can be seen as if the particle interacts with a virtual particle with the smoothed velocity , have a mass of and lies at a distance of h the direction of . This expression thus becomes similar to Eq. (6) and affects the same velocity regime. The difference is that in Eq. (16) the collective contribution from all neighbours is considered in one single interaction.
Now consider the integration of the velocitites and internal energy density from time to with the time step . From Eq. (8) the velocity for particle i at is
which gives the change in kinetic energy for particle i:
If is the change in internal energy for the particle it is possible to conserve the energy, that is require that . The time derivative of the internal energy for the particle is then consequently defined as
to conserve the total energy.
The artificial also must prevent particle penetration. The modified artificial viscosity is calculated with respect to an integrated mean of the neighbours. Therefore a particle will move less than from the definition of the time step, Eq. (7), regardless of the neighbours' individual sound speed and individual velocities. This should be compared with the distance to the closest which are approximately one h with 64 neighbours. Since the neighbours have no identity, this may lead to penetration with a few particles which have a sufficient deviation from the integrated mean.
If the particles i and j are each others neighbours and that their other neighbours give the same contribution to their respective velocities, one concludes from Eq. (15) and (16) that the impulse is conserved. Their other neighbours do, however, not give the same contribution, due to the limited resolution. Tests of self gravitating rotating disks show that the angular momentum and impulse are well conserved.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998