Balsara

Finally, we include the AV form developed by Balsara (1995). He suggests

$$\Pi_{ij}=\left({p_i\over\rho_i^2}+{p_j\over\rho_j^2}\right) \left(-\alpha \mu_{ij} + \beta \mu_{ij}^2\right),$$ (23)

where

$$\mu_{ij}=\cases{ {(\vec{v}_i-\vec{v}_j)\cdot(\vec{r}_i-\vec{... ... if $(\vec{v}_i-\vec{v}_j)\cdot(\vec{r}_i-\vec{r}_j)\ge0$.\cr}$$ (24)

Here $f_i$ is the form function for particle $i$, defined by

$$f_i={\vert\vec{\nabla}\cdot \vec{v}\vert _i \over \vert\vec{... ..._i +\vert\vec{\nabla}\times \vec{v}\vert _i + \eta' c_i/h_i },$$ (25)

where the factor $\eta'=10^{-5}$ prevents numerical divergences, and

$$(\vec{\nabla}\times \vec{v})_i={1 \over \rho_i}\sum_j m_j (\vec{v}_i-\vec{v}_j)\times\vec{\nabla} W_{ij}.$$ (26)

The function $f_i$ acts as a switch, approaching unity in regions of strong compression ( $\vert\vec{\nabla}\cdot \vec{v}\vert _i »\vert \vec{\nabla}\times \vec{v}\vert _i$) and vanishing in regions of large vorticity ( $\vert\vec{\nabla}\times \vec{v}\vert _i »\vert\vec{\nabla}\cdot \vec{v}\vert _i$). Consequently, this AV has the advantage that it is suppressed in shear layers. In our code, we set $\eta'=10^{-5}$, a choice which does not significantly affect our results. Note that since $(p_i/\rho_i^2+p_j/\rho_j^2)\approx 2c_{ij}^2/(\Gamma\rho_{ij})$, Balsara's AV resembles Monaghan's AV when $\vert\vec{\nabla}\cdot \vec{v}\vert _i »\vert \vec{\nabla}\times \vec{v}\vert _i$, provided one rescales Balsara's $\alpha$ and $\beta$ in to be a factor of $\Gamma/2$ times the $\alpha$ and $\beta$ in Monaghan's. We have taken this into account in the code, multiplying the AV term by a factor of $\Gamma/2$, so we recommend setting $\alpha\approx\beta\approx 1.0$ in the input files.