5Water waves
IB Fluid Dynamics
5.2 Equation and boundary conditions
We now try to solve for the actual solution.
z = 0
z = −H
h(x, y, t)
H
We assume the fluid is inviscid, and the motion starts from rest. Thus the
vorticity
∇ × u
is initially zero, and hence always zero. Together with the
incompressibility condition ∇ · u = 0, we end up with Laplace’s equation
∇
2
φ = 0.
We have some kinematic boundary conditions. First of all, there can be no flow
through the bottom. So we have
u
z
=
∂φ
∂z
= 0
when z = −H. At the free surface, we have
u
z
=
∂φ
∂z
=
Dh
Dt
=
∂h
∂t
+ u
∂h
∂x
+ v
∂h
∂y
when z = h.
We then have the dynamic boundary condition that the pressure at the
surface is the atmospheric pressure, i.e. at z = h, we have
p = p
0
= constant.
We need to relate this to the flow. So we apply the time-dependent Bernoulli
equation
ρ
∂φ
∂t
+
1
2
ρ|∇φ|
2
+ gρh + p
0
= f(t) on z = h.
The equation is not hard, but the boundary conditions are. Apart from them
being non-linear, there is this surface h that we know nothing about.
It is impossible to solve these equations just as they are. So we want to make
some approximations. We assume that the waves amplitudes are small, i.e. that
h H.
Moreover, we assume that the waves are relatively flat, so that
∂h
∂x
,
∂h
∂y
1,
We then ignore quadratic terms in small quantities. For example, since the waves
are small, the velocities
u
and
v
also are. So we ignore
u
∂h
∂x
and
v
∂h
∂y
. Similarly,
we ignore the whole of |∇φ|
2
in Bernoulli’s equations since it is small.
Next, we use Taylor series to write
∂φ
∂z
z=h
=
∂φ
∂z
z=0
+ h
∂
2
φ
∂z
2
z=0
+ ··· .
Again, we ignore all quadratic terms. So we just approximate
∂φ
∂z
z=h
=
∂φ
∂z
z=0
.
We are then left with linear water waves. The equations are then
∇
2
φ = 0 −H < z ≤ 0
∂φ
∂z
= 0 z = −H
∂φ
∂z
=
∂h
∂t
z = 0
∂φ
∂t
+ gh = f(t) z = h.
Note that the last equation is just Bernoulli equations, after removing the small
terms and throwing our constants and factors in to the function f.
We now have a nice, straightforward problem. We have a linear equation
with linear boundary conditions, which we can solve.