Analytically, the surface tension term in the momentum equation can be expressed
as:
(1)
dudt=−α(∇⋅n)n
With
u
being the velocity vector,
α
being the surface tension coefficient and the
n
being the two-fluid interface unit normal vector.
The divergence term is the curvature, as is well known from vector calculus. This
term is the most complex to acquire using SPH. According to the Adami model
#reference_jwp_11z_k2b__fn_lnr_2v5_s2b, the SPH formulation for this term
becomes:
(2)
∇⋅φi=d∑jφij⋅∇WijnVj∑jrij∇WijVj
Where the indices
i
and
j
stand for so called ‘owner’ and ‘neighboring’
particles respectively, and the
ij
index is a difference between the respective
variables of particle
i
and particle
j
. The
d
stands for the number of dimensions of the problem,
the
∇W
is the gradient of the kernel,
r is the position vector and
V
is the particle volume.
The surface tension model has only four options required for the setup. First, and
most important, is to turn the surface tension model on. In the Simulation
parameters, the option
surften_model specifies the selected
surface tension model. The current version (v
2021.2) has three
options:
NONE,
SINGLE_PHASE or
ADAMI. For the
SINGLE_PHASE surface tension
model, refer to the section on
Adhesion Model and Single Phase Surface Tension. The second
important parameter is the reference curvature
ref_curv[1/m] in
the Domain parameters, which is the largest expected surface curvature. Third, in
the Phase parameters, specify the surface tension coefficient
surf_ten [N/m] for the two-phase interaction, for example if
you have an oil phase and an air phase, you would specify the same surface tension
coefficient for both phases. If surface tension model is set to
ADAMI or
SINGLE_PHASE, the reference curvature
and surface tension coefficient definitions are mandatory.
Important: Enabling of surface tension models in the simulation and requirement to resolve
small droplets, for example, ref_curv set to 1000, can be
very computationally expensive. Unless it is of utter importance to accurately
resolve small droplets, for example, Rdroplet < 1 cm, it is
recommended to use a relatively high ref_curv value of ≈ 20.
This will make runs much faster, while still including surface tension effects
for surface fluid structures which are of the approximate size of 5
cm.
S. Adami, X. Hu und N. Adams, „A new surface-tension formulation for
multi-phase SPH using a reproducing divergence approximation,“ Journal of
Computational Physics, Nr. 229, pp. 5011-5021, 2010.
M. P. Allen und D. J. Tildesley, Computer simulation of liquids, New York:
Oxford University Press, 1989.