Graphics Reference
In-Depth Information
1.5 Dropping Viscosity
In some situations, viscosity forces are extremely important: e.g., simulat-
ing honey or very small-scale fluid flows. But in most other cases that we
wish to animate, viscosity plays a minor role, and thus we often drop it:
the simpler the equations are, the better. In fact, most numerical methods
for simulating fluids unavoidably introduce errors that can be physically
reinterpreted as viscosity (more on this later)—so even if we drop viscosity
in the equations, we will still get something that looks like it. In fact,
one of the big challenges in computational fluid dynamics is avoiding this
viscous error as much as possible. Thus for the rest of this topic, apart
from Chapter 8 that focuses on high or even varying viscosity fluids, we
will assume viscosity has been dropped.
The Navier-Stokes equations without viscosity are called the
Euler equa-
tions
and such an ideal fluid with no viscosity is called
inviscid
.Justto
make it clear what has been dropped, here are the incompressible Euler
equations using the material derivative to emphasize the simplicity:
Du
Dt
+
1
ρ
∇
p
=
g,
∇·
u
=0
.
It is these equations that we'll mostly be using.
1.6 Boundary Conditions
Most of the, ahem, “fun” in numerically simulating fluids is in getting
the boundary conditions correct. So far, we've only talked about what's
happening in the interior of the fluid: so what goes on at the boundary?
In this topic we will only focus on two boundary conditions,
solid walls
and
free surfaces
. One important case we won't cover is the boundary
between two different fluids: most often this isn't needed in animation, but
if you are interested, see papers such as that by Hong and Kim [Hong and
Kim 05].
A solid wall boundary is where the fluid is in contact with a solid. It's
simplest to phrase this in terms of velocity: the fluid had better not be
flowing into the solid or out of it, thus the normal component of velocity
has to be zero:
u
·
n
=0
.
