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)
πij=−qbci+cj2μij+qαμ2ij(ρi+ρj)2πij=−qbci+cj2μij+qαμ2ij(ρi+ρj)2
with
(2)
μij=dij(vi−vj)•(Xi−Xj)‖Xi−Xj‖2+εd2ijμij=dij(vi−vj)∙(Xi−Xj)∥Xi−Xj∥2+εd2ij
Where,
XiXi
(resp.
XjXj
) indicates the position of particle I (resp.
jj
) and
cici
(resp
cjcj
) is the sound speed at location
ii
(resp.
jj
), and
qaqa
and
qbqb
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)
πij=−qbci+cj2μij+qαμ2ij(ρi+ρj)2πij=−qbci+cj2μij+qαμ2ij(ρi+ρj)2
with
(4)
μij=dij(vi−vj)•(Xi−Xj)‖Xi−Xj‖2+εd2ij(fi+fj)2,fk=‖∇⋅v|k‖‖∇⋅v|k‖+‖∇×v|k‖+ε′ckdk
Default values for
qa
and
qb
are respectively set to 2 and 1.
1
Monaghan J.J., 「Smoothed Particle Hydrodynamics」,
Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.
2
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.