Free-surface Waves

by Jong-Chun Park

Class: Environmental Fluid Modeling, May 17, 2002

Water Waves• It is important to distinguish between the various types of water waves

that may be generated and propagated.

• One way to classify waves is by wave period T, or by the frequency f.

Fig.2 Approximate distribution of ocean surface wave energy illustrating the classification of surface waves by wave band, primary disturbing force, and primary restoring force.

Gravity Waves

• Seas when the waves are under the influence of wind in a generating area

– Usually made up of steeper waves with shorter periods and lengths, and the surface appears much more disturbed than for swell.

• Swell when the waves move out of the generating area and are no longer subjected to significant wind action

– Behaves much like a free wave (i.e., free from the disturbing force that caused it), while seas consist to some extent of forced waves (i.e., waves on which the disturbing force is applied continuously).

Small-Amplitude Wave Theory (1)

• Several Assumptions commonly made in developing a simple wave theory

a. The fluid is homogeneous and incompressible; therefore, the density ρ is a constant.

b. Surface tension can be neglected.c. Coriolis effect can be neglected.d. Pressure at the free surface is uniform and constant.e. The fluid is ideal or inviscid (lacks viscosity).f. The particular wave being considered does not interact with any

other water motions.g. The bed is a horizontal, fixed, impermeable boundary, which im

plies that the vertical velocity at the bed is zero.h. The wave amplitude is small and the waveform is invariant in ti

me and space.i. Waves are plane or long crested (two dimensional).

Small-Amplitude Wave Theory (2)

Wave length L, height H, period T, and depth d, the displacement of the water surface η relative to the SWL and is a function of x and time t.

Fig.3 Definition of terms-elementary, sinusoidal, progressive wave.

Small-Amplitude Wave Theory (3)

• Wave Celerity:

(1a)

(1b)

(1c)

• Wave length:

(2)

• Approximate wave length by Eckart (1952):

(3)

kdg

kdk

g

T

LC

tanh

tanh

kdgT

L tanh

g

dgTL 2tanh

Maximum error of 5% occurs when .1kd

Involving some difficulty since the unknown L appears on both sides of the equation.

Wave number:

Wave angular frequency:

Gravity acceleration: g

Lk

2

T

2

Small-Amplitude Wave Theory (4)• Gravity waves may be classified by the water depth in which they travel. Th

e classifications are made according to the magnitude of d/L and the resulting limiting values taken by the function tanh(kd):

• In deep water, tanh(kd) approaches unity, Eqs. (2) and (3) reduce to

(4)

• When the relative water depth becomes shallow, Eq.(2) can be simplified to

(5)

classification d/L kd tanh(kd)

Deep water

Transitional

Shallow water

> 1/2

1/25 to 1/2

< 1/25

> π

1/4 to π

< 1/4

1

tanh(kd)

kd

g

T

LgLC 00

0 2

gdC 0

Small-Amplitude Wave Theory (5)• The sinusoidal wave profile:

(6)• In wave force studies, or numerical wave generation, it is often desirable to know the

local fluid velocities and accelerations for various values z and t during the passage of a wave.

• The horizontal component u of the local fluid velocity:

(7)

• The vertical component w of the local fluid velocity:

(8)

• The local fluid particle acceleration in the horizontal:

(9)

• The local fluid particle acceleration in the vertical:

(10)

tkxkd

dzk

L

gTHu cos

cosh

cosh

2

tkxkd

dzk

L

gTHw sin

cosh

sinh

2

tkxH cos2

tkxkd

dzkgk

Hax

sincosh

cosh

2

tkxkd

dzkgk

Haz

coscosh

sinh

2

Small-Amplitude Wave Theory (6)• A sketch of the local fluid motion indicates that the fluid under the

crest moves in the direction of wave propagation and returns during passage of the trough.

Fig.4 Local fluid velocities and accelerations.

Small-Amplitude Wave Theory (7)• Water particles generally move in elliptical paths in shallow or

transitional water and in circular paths in deep water.

Fig.5 Water particle displacements from mean position for shallow-water deepwater waves.

Small-Amplitude Wave Theory (8)• Stokes’ Second-Order Wave Theory• Equation of the free-surface:

(11)

• Expressions for wave celerity and wave length are identical to those obtained by liner theory.

• Stokes (1880) found that a wave having a crest angle less that 120o would break (angle between two lines tangent to the surface profile at the wave crest).

• Michell (1893) found that in deep water the theoretical limit for wave steepness was;

(12)

tkdkdkd

kdkH

tkxH

22cos2cosh2sinh

cosh

16

cos2

3

2

7

1142.0

max0

0

L

H

Small-Amplitude Wave Theory (9)• Linear theory: A wave is symmetrical about the SWL and has water particles that

move in closed orbits.

• Second-order theory: A waveform is unsymmetrical about the SWL but still symmetrical about a vertical line through the crest and has water particle orbits that are open. The wave profile of second-order theory shows typical non-liner features, such as higher and narrower crest and smaller and flatter trough than the linear one.

Fig.6 Comparison of second-order Stokes’ profile with linear profile

Ocean Waves (1)• Physical values are varied randomly in time

• Stochastic process with multi-directionality

• Impossibility of Prediction for Physical Values

• Possibility of Prediction for Probability Distribution

• Wave Characteristics of Target Sea Environment

• Directional Spectrum in general;

(20)

where, S(f) is frequency spectrum and G(f;θ) directional spreading spectrum.

• Bretschneider-Mitsuyasu type Frequency Spectrum in coasts:

(21)

• Mitsuyasu type Directional Spreading Function in costs:(22)

);()(),( fGfSfS

43/1

53/13/1

23/1 75.0exp205.0)( fTfTTHfS

2cos);( 20 p

sGfG

12

0max

min 2cos

dG s

pp

pp

ffffS

ffffSs

:

:5.2

max

5max

where,

where, θ is the azimuth measured counterclockwise from the principle wave direction, θp, fp the

peak frequency (fp= T1/3/1.13), G0 a constant to normalize the directional function, s the directi

onal wave energy spreading determined by angular spreading parameter Smax (Goda & Suzuk

i, 1975).

Ocean Waves (2)• Offshore near cost

– Mono-peak directional spectrum

• Bretschneider-Mitsuyasu type Frequency Spectrum

• Mitsuyasu type Directional Spreading Function

– Double-peak directional spectrum

• Obtained from Large-Scale Field Observation in Off-Iwaki

• Wind wave: short period, Swell: long period

• North-Pacific Ocean

– ISSC Standard Directional Spectrum

(23)

(24)

0.25

0.25

0.50

0.50

0.75

1.00

2.00

FREQUENCY (1/s)

WAVEANGLE(o)

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

S(f:)[cm2sec]

H1/3 = 5.0cmT1/3 = 1.33sSmax = 5p = 0o

h = 90cm

2.24

2.24

4.49

4.49

6.73

6.73

8.98

FREQUENCY (1/s)

WAVEANGLE(o)

0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

S(f:)[cm2sec]

H1/3 = 5.0cmT1/3 = 1.33sSmax = 5p = 0o

h = 90cm

0.27

0.27

0.54

0.82

1.091.36

FREQUENCY (1/s)

WAVEANGLE(o)

0.1 0.3 0.5 0.7-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

S(f:)[m2sec]

H1/3 = 6.0mT1/3 = 4.5sSmax = 5p = 0o

41

51

11

2 44.0exp11.0 HS

T 21 where,

otherwise:0

22:cos

;2

pfG

Ocean Waves (2)Mono-Peak Double-Peak

North-Pacific Ocean

ANIMATIONANIMATIONANIMATIONANIMATION

ANIMATIONANIMATION

Reproduction of Random Seas in

Laboratory

b

wave direction

wave cre

st line

= -2bsin/= -kbsin

wavemaker panels

b

wave direction

wavecre

st line

= -2bsin/ = -kbsin

wavemaker panels

Reproduction of Random Seas (1)• For multi-directional wave generation, a snake-like wavemaker motion i

s used on the basis of linear wavemaker theory (Dean and Darlymple, 1991).

Fig.7 Principle of snake-type wavemaker

Reproduction of Random Seas (2)

• Equation of wave elevation

(17)

• Velocity components on panels of wavemaker

(18)

(19)

N

n n

n

n

nnn hk

hzkA

v

u

1 sin

cos

sinh

cosh

nnnynx tykxk cos

N

nnnnynx

n

nnn tykxk

hk

hzkAw

1

sinsinh

sinh

N

nnnnynxn tykxkA

1

cos

Reproduction of Random Seas (3)

Free-surface Conditions

• Kinematic condition

• Dynamic condition

implements the law of mass conservation

implements the law of momentum conservation

Free-surface Conditions-Kinematic Condition 1-

 In case the variables of density , velocity v, normal vector n and infinitesimally small segment of free-surface ds are defined as shown in Figure, the conservation of mass across ds becomes,

  (12)

 Suppose that two fluids are not mixed and then

(13)

 that is

(14)

 Eq.(3) is the kinematic condition on the free-surface boundary, which means that fluid particles on the free-surface remain on the same boundary.

ds 1

2

2v1vnvnv 2211

nvnv 21

021 nvv

Free-surface Conditions-Kinematic Condition 2-

When the free-surface is assumed to be a function of horizontal coordinate (x,y) and time t as

(15)

 and the kinematic condition in the Eulerian coordinate system becomes

(16)

 where substantial differential is used.The kinematic condition by use of Lagrangian coordinate system is

  (17)

 where the Lagrangian coordinate on the free-surface and is the components of velocity.

.const;,, 0htyxF

0,, tyxDt

DF

vDt

Du

Dt

D

,

Free-surface Conditions-Kinematic Condition 3-

To implement the kinematic condition of free-surface and to determine the free-surface configuration the MDF is used. Two-layer flow is considered and the density of the fluid in the lower and upper layers is denoted and . The MDF is governed by the following transportation equation.

  (18)

 where the MDF takes the value between 0 and 1 all over the computational domain and th

is scalar value has the meaning of porosity in each cell. Eq.(18) is calculated at each time step and the free-surface location is determined to be the position where the MDF takes the mean value as

 (19)

 The interface location is the same as the wave height function h in general unless overturning and breaking waves are considered. Thus, Eq.(18) is more general and solved for the movement of fluid interface.

0

z

Mw

y

Mv

x

Mu

t

M

5.0M

Free-surface Conditions-Dynamic Condition-

By the law of momentum conservation the following dynamic conditions are derived in the normal n and tangential t directions, respectively (Levich & Krylov, 1969).

  (13)

  (14)

 Here, is the surface tension, the radius of curvature, the dynamic viscosity and the pressure. The subscripts 1 and 2 denote the fluid 1 (lower layer) and fluid 2 (upper layer).

Assume that the viscous stress and surface tension on the free-surface are ignored and then the dynamic conditions are written in the following simple forms.

  (15)

  (16)

n

v

n

vpp

nn

22

1121 22

tn

v

t

v

n

v

t

v tntn

22

211

1

21 pp

n

v

n

v tt2

21

1

Some Applications of Free-surface FlowUsing the Simulation Techniques

• Nonlinear Free-surface Motions around Arctic Structure

• Wave Breaking

• Propagation of Solitary Wave

Non-linear Wave MotionsNon-linear Wave Motionsaround around Arctic StructureArctic Structure

• Various types of conical-shaped structures has been constructed in the Arctic in order to give rise to reduced ice loads and to protect the island top from wave attack.

• Needs to predict maximum wave run-up in order to determine the suitable deck elevation.

Non-linear Wave MotionsNon-linear Wave Motionsaround around Arctic StructureArctic Structure

Model to be simulated

Non-linear Wave MotionsNon-linear Wave Motionsaround around Arctic StructureArctic Structure

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