Graphics Reference
In-Depth Information
1
4
:thisis
2
Take note of the factor of
from the quaternion integration
1
formula and another
2
to get angular velocity from vorticity. Advancing
orientations in this manner is useful not just for actual rigid bodies but also
for oriented particles that carry a local coordinate system—see Rasmussen
et al. [Rasmussen et al. 03] for an example in constructing highly detailed
smoke plumes from explosions, where each particle carries a volumetric
texture.
More generally, we might want solids to have some inertia, with the
effect of the fluid felt in terms of force, not velocity. As we know, there are
two forces in effect in a fluid: pressure and viscous stress. The second is
perhaps more important for very small objects.
The net force due to viscosity is the surface integral of viscous traction,
the viscous stress tensor times the surface normal:
F
=
−
τ n.
∂S
Here I take
S
to be the volume of the solid and
∂S
to be its boundary—this
is a slight change of notation from earlier chapters where
S
represented the
solid surface. The normal here points
out
of the solid and
into
the fluid,
leading to the negative sign. In one-way coupling, the viscous boundary
condition
u
=
u
solid
isn't present in the simulation and thus the fluid's vis-
cous stress tensor isn't directly usable. Indeed, the assumption underlying
one-way coupling is that the solid objects don't have an appreciable effect
on the fluid at the resolution of the simulation. However, we can imagine
that
if
thesolidwereintheflow,therewouldbeasmall
boundary layer
around it in which the velocity of the fluid rapidly alters to match the solid
velocity: the gradient of velocity in this region gives us the viscous stress
tensor. The actual determination of this boundary layer and exactly what
average force results is in general unsolved. We instead boil it down to
simple formulas, with tunable constants. For small particles in the flow,
we posit a simple drag force of the form:
F
i
=
D
(
u
−
u
i
)
.
Here
D
is proportional to the fluid's dynamic viscosity coecient, and
might be a per-particle constant, or involve the radius or cross-sectional
area of the object, or might even introduce a non-linearity such as being
proportional to
—in various engineering contexts all of these have
been found to be useful. For flatter objects, such as leaves or paper, we
might constrain the normal component of the velocity to match the fluid
and only apply a weak (if not zero) viscous force in the tangential direction.
u
−
u
i
