II. Kinematics of Fluid Motion

1
An Application of Bernoulli Irrotational Flows
Surface Gravity Waves
EFML flume ca. 1994
2
Basic wave properties
Wavelength l
Wave height h0
Wave-induced current u, w
Period T Frequency w2p/T Speed c
Depth H
3
Linear vs. nonlinear waves

• Linear Speed c only a function of g, l, H.
• Nonlinearity c is a function of wave amplitude.

Linear wave
Nonlinear wave
Depth H ? H h0
Depth H ? H- h0
4
Linear
Inviscid
Incompressible
Irrotational flow Barotropic, inviscid,
non-rotating
Irrotational flow ?? Potential flow
5
The math problem we consider
Crest
h0
x3
x30
x1
Trough
l
Suppose the wave propagates across an infinitely
deep ocean, What velocities are associated with
this wave and how fast does it travel? Assume
Linear, irrotational, incompressible.
6
The group is known as
the wave phase if
we stay with same phase (say the crest).
This gives
We need to impose boundary conditions.
7
(a) A particle that starts on the surface remains
there
Velocity of particle on surface is same as that
of surface if
or
Surface is a streamline
8
In general
(b) Pressure on surface is constant 0 This
is the dynamic condition. It couples the velocity
and surface elevation through the Bernoulli Eqn
9
Assume small-amplitude motions. This means
10
To see how (2) works, lets consider the
linearized kinematic condition
Suppose we write
Scaling
11
So we must satisfy the approximate conditions
we assumed that
It must take the form
(in order to satisfy free surface bcs!)
12
Using this guess for f we find that
Recall that the solution to
So
But for
But on x30 the kinematic B.C. gives
But, we still dont know !
13
We now use the dynamic B.C. that tells us that
14
Finally, we can calculate the velocities
The group wave steepness (max
0.42 for deep water)
How big is the neglected term in Bernoulli
applied on surface?
So as we said above if we expect we
have an accurate solution.
15
Velocity field at t0 (red line is free surface)
16
An improvement Plotted in deformed coordinates
17
Deep water wave orbitals (to lowest order)
Initial point
18
Stokes (1847) It appears that the forward
motion of particles is not altogether compensated
by their backwards motion.. ? Stokes Drift
Trajectory for 20 wave periods calculated by
numerical integration of deep water wave velocity
field (e0.1). All dimensions scaled by k-1.
Initial position is (x1, x3) (0,-0.5).
19
Duck, North Carolina
Wave driven flow
Shoaling
20
Effects of finite depth
see Dean Dalrymple Water wave mechanics for
engineers scientists
x30
x3-H
Solid bottom (e.g coral reef)
We would re-work the problem imposing
rather than
. This makes the following changes.
Finite depth effect e.g. kH1 tanh0.76
21
Some limits (see next slide)
Practically is a long wave (to
within 3)
Practically is a deep water wave
(within 0.5) General velocities
22
Effect of kH on velocity field
(note difference of scales)
23
Deep limit
Shallow limit
Tides, tsunamis
24
Note in the case where we get
is the velocity field for a long wave Note that
the vertical velocity, while small, is not zero,
but instead varies linearly from zero at the
bottom to at the surface.