## 2. The SPH methodologyIn SPH one models a number of particles that carry the physical
quantities, where the particles' distribution in space describes the
density distribution. To simulate a fluid each particle's mass is
smoothed over a radius where , ,
and . The smoothing kernel,
The kernel used here is the spline kernel from Monaghan and Lattanzio (1985): This is a smooth kernel with compact support over the radius
around the particle. The smoothing length
In the SPH formulation there are different forms of the discretization of the Navier Stokes equations. Here the expressions of Hernquist & Katz (1989) are used, i.e. where and . The continuity equation is automatically satisfied due to the Lagrangian formulation, and the density calculated from Eq. (1) becomes The standard artificial viscosity term, , is defined where . The first and second term in the
expression of in Eq. (6) represents the bulk
and the von Neumann Richtmeyer artificial viscosity respectively. The
constant is a fudge parameter to prevent the
artificial viscosity to become too large. The artificial viscosity is
used only when , that is when two particles are
approaching each other. To close the system, the pressure is defined
as , where where is the Courant factor to stabilize the
integration, and the maximum The viscous acceleration terms in Eq. (4) scales with as The artificial viscosity can therefore lead to undesirable effects, because the velocity differences are smoothed on a time scale of roughly . The velocities in the model will therefore be smoothed out unless it expands or if there is some driving mechanism such as gravitation. To conserve the energy the particles are heated, which may be unphysical. The heated gas may reach an equilibrium state earlier than expected. A way of preventing interparticle penetration without unnecessary heating of the gas could therefore be useful, and this implies that there is a need to restrict the artificial viscosity to the shocks as much as possible. © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |