Wave HydrodynamicsJuan 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; -- frequency (1/s) and -- Wave period

/ 2 a -- amplitude and -- Wave height

f f TT

a H H

Governing equations

• Conservation of Mass:

ztzxw

xtzxu

),,(

),,( Velocity potential

01

udt

d

Continuity equation, for incompressible fluids

0

z

w

y

v

x

u

0 u

Governing equations

• Laplace Equation:

• Navier- Stokes equation

02

2

2

2

zx

uFpdt

ud 21

p is pressureis the water density is diffusion coefficient

0

gzp

t

• Euler equation:

• Unsteady Bernoulli equation:

Fluid is incompressible, no viscous, irrotational, etc..

Fpdt

ud

1

Boundary conditions

• Dynamic boundary condition at the free surface:

In z = , p = 0

• 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

0

gt

xu

tdt

dw

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) :

)sin(),,( wtkxatzx

)tanh(2 kdgk Dispersion relationship

kC

)tanh(kdk

gC

)sin(

)cosh(

)(cosh

2wtkx

kd

zdkgH

Deep water

k

gC

Intermediate water

)tanh(kdk

gC

gdC

Shallow water

Lg

C2

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.

hkmgdC /800

Velocity components of the fluid particles

(VERTICAL)

(HORIZONTAL)

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 THEORIES1. 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

)(2cos)2cosh(2()(sinh

cosh

16

)sin(),,(

3

2

tkxkdkd

kdkH

wtkxL

Htzx

)(2sin

)(sinh

)(2cosh

32

3

)sin()cosh(

)(cosh

2

4

2

wtkxkd

zdkH

wtkxkd

zdkgH

Waves theories

Regions of validity for various wave theories.

Conclusions

•Linear Wave Theory: Simple, good approximation for70-80 % 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!!