Transcript Document

Wave Hydrodynamics
Juan Carlos Ortiz Royero Ph.D.
From:
wavcis.csi.lsu.edu/ocs4024/ocs402403waveHydrodynamics.ppt
and the book: wind generated ocean wave by Ian R. Young 1999
Fields Related to Ocean Wave
•Ocean Engineering:
Ship, water borne transport,
offshore structures (fixed and
floating platforms).
• Navy:
Military activity, amphibious operation,
• Coastal Engineering: Harbor and ports, coastal structures,
beach erosion, sediment transport
The inner shelf is a friction-dominated zone where surface
and bottom boundary layers overlap.
(From Nitrouer, C.A. and Wright, L.D., Rev. Geophys., 32, 85, 1994. With permission.)
Conceptual diagram illustrating physical transport
processes on the inner shelf.
(From Nitrouer, C.A. and Wright, L.D., Rev. Geophys., 32, 85, 1994. With permission.)
Approximate distribution of ocean surface wave energy
illustrating the classification of surface waves by wave
band, primary disturbance force, and primary restoring
force.
SEAS
Waves under the influence of
winds in a generating area
SWELL
Waves moved away from the
generating area and no longer
influenced by winds
WAVE CHARACTERISTICS
T = WAVE PERIOD
Time taken for two successive crests to pass a given point
in space
Wave Pattern Combining Four Regular Waves
Linear Wave or small amplitude theory
• Assumptions:
– The water is of constant depth d
– The wave motion is two-dimensional
– The waves are of constant form (do not
change with time)
– The water is incompressible
– Effect of viscocity, turbulence and surface
tension are neglected.
– The wave height H: H / L  1 and H /d  1 ( L
is the wave length)
Regular Waves
1
f  ; f -- frequency (1/s) and T -- Wave period
T
a  H / 2 a -- amplitude and H -- Wave height
Governing equations
• Conservation of Mass:
u v w
 
0
x y z

u ( x, z , t )  
x

w( x , z , t )  
z

1 d
  u  0
 dt
Continuity equation,
for incompressible fluids
Velocity potential

u  0
Governing equations
• Laplace Equation:
 2  2
 2 0
2
x
z
• Navier- Stokes equation


du
1
2
  p  F  u
dt

p is pressure
is the water density
 is diffusion coefficient
Fluid is incompressible, no viscous, irrotational, etc..
• Euler equation:


du
1
  p  F
dt

• Unsteady Bernoulli equation:
 p
  gz  0
t 
Boundary conditions
• Dynamic boundary condition at the free surface:
In z = , p = 0

 g  0
t
• Kinematic boundary condition at the free surface:
In z = , there can be no transport of fluid through the
free surface (the vertical velocity must equal the vertical
of the free surface
d 

w

u
dt
t
x
Boundary conditions
• Kinematic boundary condition at the bed:
In z = - d, there can be no transport of fluid through the
free surface (the vertical velocity must equal zero)
Solution (Airy 1845, Stokes 1847) :
H g coshk (d  z )

sin(kx  wt )
2  cosh(kd )
 ( x, z, t )  a sin(kx  wt )
 2  gk tanh(kd )
C
C
Dispersion relationship

k
g
tanh(kd )
k
Deep water
C
g
k
g
C
L
2
Intermediate water
C
g
tanh(kd )
k
Shallow water
C  gd
1. Longer waves travel faster than shorter waves.
2. Small increases in T are associated with large increases in L.
Long waves (swell) move fast and lose little energy.
Short wave moves slower and loses most energy
before reaching a distant coast.
Example: What is the fase velocity of tsunami in
deep water?
Solution:
The typical wave length of a tsunami is thousand of
kilometers and periods of hours. Since the wave
length of tsunami is very large compared with the
depth, then tsunami is a shallow water wave.
C  gd  800 km / h
Velocity components of the fluid particles
(HORIZONTAL)
(VERTICAL)
Motions of the fluid particles
WAVE ENERGY AND POWER
Kinetic + Potential = Total Energy of Wave System
Kinetic: due to H2O particle velocity
Potential: due to part of fluid mass being above trough.
(i.e. wave crest)
WAVE ENERGY FLUX
(Wave Power)
•Rate at which
energy is
transmitted in the
direction of
progradation.
HIGHER ORDER THEORIES
1. Better agreement between theoretical and
observed wave behavior.
2. Useful in calculating mass transport.
HIGHER ORDER WAVES ARE:
• More peaked at the crest.
• Flatter at the trough.
• Distribution is skewed above SWL.
Comparison of second-order Stokes’ profile with linear
profile.
Stokes, 1847
H g coshk (d  z )

sin(kx  wt ) 
2  cosh(kd )
3 H cosh2k (d  z )
sin 2(kx  wt )
4
32 
sinh (kd )
2
H
 ( x, z, t )  sin(kx  wt ) 
L
H 2 k cosh kd
(2  cosh(2kd ) cos 2(kx  t )
3
16 sinh (kd )
Waves theories
Regions of validity for various wave theories.
Conclusions
•Linear Wave Theory: Simple, good approximation for
70-80 % engineering applications.
•Nonlinear Wave Theory: Complicated, necessary for about
20-30 % engineering applications.
•Both results are based on the assumption of non-viscous
flow.
Thanks!!