Graphics Reference
In-Depth Information
If we are further interested in solids with orientation, the net torque on
an object due to viscosity is likewise
T
=
−
(
x
−
x
i
)
×
(
τ n
)
,
S
where
x
i
is the center of mass of the object, and similarly we can't hope
to derive a perfect physical formula for it. Instead we can posit simple
formulas now based on the difference between the angular velocity of the
solid and half the vorticity of the fluid:
T
=
E
2
ω
Ω
.
−
The proportionality
E
can be tuned similar to
D
and may even be gener-
alized to a matrix incorporating the current rotation matrix of the object
if the solids are far from round.
The effect of pressure is a little simpler. The net force in this case is
S
∇
F
=
−
pn
=
−
p,
∂S
where we have used the divergence theorem to convert it into an integral
of the pressure gradient over the volume occupied by the solid. For small
objects, we can evaluate
∇p
from the simulation at the center of mass and
multiply by the object's volume to get the force. Note that for water sitting
still (and assuming a free surface pressure of zero), the hydrostatic pressure
is equal to
ρ
water
|
g
|
d
where
d
is depth below the surface, giving a gradient
of
ρ
water
g
. Multiplying this by the volume that the object displaces, we
get the mass of displaced water, leading to the usual buoyancy law.
The torque due to pressure is
−
T
=
−
(
x
−
x
i
)
×
(
pn
)
.
∂S
If
p
is smooth enough throughout the volume occupied by the solid—say it
is closely approximated as a constant or even linear function—this integral
vanishes, and there is no torque on the object; we needn't model it. Do
note that in the case of water sitting still, the pressure is
not
smooth across
the surface—it can be well approximated as a constant zero above the
surface, compared to the steep linear gradient below—and thus a partially
submerged object can experience considerable torque from pressure. In the
partially submerged case, the integral should be taken (or approximated)
over the part of the solid below the water surface.
