Artificial Viscosity

As usual in SPH ¹ implementations, viscosity is rather an inter-particles pressure than a bulk pressure. It was shown that the use of 式 1 and 式 2 generates a substantial amount of entropy in regions of strong shear even if there is no compression.(1)

π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}

$π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}$

with (2)

μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}}

$μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}}$

Where,

X_{i}

$X_{i}$ (resp.

X_{j}

$X_{j}$ ) indicates the position of particle I (resp.

j

$j$ ) and

c_{i}

$c_{i}$ (resp

c_{j}

$c_{j}$ ) is the sound speed at location

i

$i$ (resp.

j

$j$ ), and

q_{a}

$q_{a}$ and

q_{b}

$q_{b}$ are constants. This leads us to introduce 式 3 and 式 4. ² The artificial viscosity is decreased in regions where vorticity is high with respect to velocity divergence.(3)

π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}

$π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}$

with (4)

μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}} \frac{(f_{i} + f_{j})}{2}, f_{k} = \frac{‖ {\nabla \cdot v |}_{k} ‖}{‖ {\nabla \cdot v |}_{k} ‖ + ‖ {\nabla \times v |}_{k} ‖ + ε^{'} \frac{c_{k}}{d_{k}}}

$μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}} \frac{(f_{i} + f_{j})}{2}, f_{k} = \frac{‖ {\nabla \cdot v |}_{k} ‖}{‖ {\nabla \cdot v |}_{k} ‖ + ‖ {\nabla \times v |}_{k} ‖ + ε^{'} \frac{c_{k}}{d_{k}}}$

Default values for $q_{a}$ $q_{a}$ and $q_{b}$ $q_{b}$ are respectively set to 2 and 1.

Monaghan J.J., 「Smoothed Particle Hydrodynamics」, Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.

Balsara D.S., 「Von Neumann Stability Analysis of Smoothed Particle Hydrodynamics Suggestions for Optimal Algorithms」, Journal of Computational Physics, Vol. 121, pp. 357-372, 1995.