Graphics Reference
In-Depth Information
3.2 Boundary Conditions
If the starting point of the imaginary particle is in the interior of the fluid,
then doing the interpolation is no problem. What happens though if the
estimated starting point happens to end up outside of the fluid boundaries?
This could happen because fluid is flowing in from outside the domain (and
the particle is “new” fluid), or it could happen due to numerical error (the
true trajectory of the particle actually stayed inside the fluid, but our
forward Euler or Runge-Kutta step introduced error that put us outside).
This is really the question of boundary conditions. In the first case,
where we have fluid flowing in from the outside, we should know what the
quantity is that's flowing in: that's part of stating the problem correctly.
For example, if we say that fluid is flowing in through a grating on one
side of the domain at a particular velocity
U
and temperature
T
,thenany
particle whose starting point ends up past that side of the domain should
get velocity
U
and temperature
T
.
In the second case, where we simply have a particle trajectory that
strayed outside the fluid boundaries due to numerical error, the appropri-
ate strategy is to extrapolate the quantity from the nearest point on the
boundary—this is our best bet as to the quantity that the true trajectory
(which should have stayed inside the fluid) would pick up. Sometimes that
extrapolation can be easy: if the boundary we're closest to has a specified
velocity we simply use that. For example, for simulating smoke in the open
air we could assume a constant wind velocity
U
(perhaps zero) outside of
the simulation domain.
The trickier case is when the quantity isn't known
apriori
but has to be
numerically extrapolated from the fluid region where it is known. We will
go into more detail on this extrapolation in Chapter 6 on water. For now,
let's just stick with finding the closest point that is on the boundary of the
fluid region and interpolating the quantity from the fluid values stored on
the grid near there. In particular, this is what we will need to do for finding
velocity values when our starting point ends up inside a solid object, or for
free surface flows (water) if we end up in the free space.
Note that taking the fluid velocity at a solid boundary is
not
the same
as the solid's velocity in general. As we discussed earlier, the normal com-
ponent of the fluid velocity had better be equal to the normal component
of the solid's velocity, but apart from in viscous flows, the tangential com-
ponent can be completely different. Thus we usually interpolate the fluid
velocity at the boundary and don't simply take the solid velocity. However,
for the particular case of viscous flows (or at least, a fluid-solid interaction
