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In-Depth Information
Beyond boundaries, we need ghost velocity values to plug into these
formulas. For viscous solid walls with the no-slip condition, it's natural to
simply use the solid velocity itself. For free surfaces we have to be a little
more careful. The obvious first approach to try is to just extrapolate the
velocity into the air, as we have done before, effectively setting
u
at a point
in the air to the velocity at the closest point on the surface. Similarly, at
an inviscid solid boundary, we can extrapolate the tangential component
of fluid velocity into the solid while using the normal component of solid
velocity—in fact, Rasmussen et al. [Rasmussen et al. 04] suggest an inter-
esting intermediate condition between inviscid and fully viscous, where a
weighted average of the solid's tangential velocity and the fluid's tangential
velocity is used.
However, this simple extrapolation induces a non-negligible error. As
a thought experiment, imagine a blob of fluid moving rigidly in free flight:
it has zero deformation since its internal velocity field is rigid, therefore it
should experience no viscous forces—i.e.,
τ
should evaluate to zero, even
at the boundary. However, if a rigid rotation is present,
τ
only evaluates
to zero if the ghost velocities keep that same rotation: extrapolating as
a constant doesn't, and will induce erroneous viscous resistance at the
boundary. Ideally a more sophisticated extrapolation scheme such as linear
extrapolation should be used.
That said, at present our first-order time-splitting of advection from
pressure also will induce a similar erroneous drag on rotational motion,
which we'll discuss in Chapter 9. Reducing one error but not the other is
probably not worth the bother, and thus we'll leave this question open for
further research.
The chief problem with the method as presented is stability. Unfortu-
nately this method is liable to blow up if Δ
t
is too large. Let's examine a
simple 1D model problem to understand why,
∂t
=
k
∂
2
q
∂q
∂x
2
,
where
q
models a velocity component and
k
models
η/ρ
, the kinematic
viscosity. Our explicit discretization would give
+Δ
tk
q
i
+1
−
2
q
i
+
q
i−
1
q
n
+1
i
=
q
i
.
Δ
x
2
Consider the highest spatial-frequency component possible in the numerical
solution, say
q
i
=
Q
n
(
1)
i
.Here
Q
n
is the time
n
scalar coecient mul-
−
tiplying the
±
1 underlying basis function, with grid index
i
the exponent
