Free-surface Waves

1
Free-surface Waves
Class Environmental Fluid Modeling, May 17, 2002

• by Jong-Chun Park

2
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.

3
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).

4
Small-Amplitude Wave Theory (1)

• Several Assumptions commonly made in developing a
  simple wave theory
• The fluid is homogeneous and incompressible
  therefore, the density ? is a constant.
• Surface tension can be neglected.
• Coriolis effect can be neglected.
• Pressure at the free surface is uniform and
  constant.
• The fluid is ideal or inviscid (lacks viscosity).
• The particular wave being considered does not
  interact with any other water motions.
• The bed is a horizontal, fixed, impermeable
  boundary, which implies that the vertical
  velocity at the bed is zero.
• The wave amplitude is small and the waveform is
  invariant in time and space.
• Waves are plane or long crested (two dimensional).

5
Small-Amplitude Wave Theory (2)
Fig.3 Definition of terms-elementary, sinusoidal,
progressive wave.

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

6
Small-Amplitude Wave Theory (3)

• Wave Celerity
• (1a)
• (1b)
• (1c)
• Wave length
• (2)
• Approximate wave length by Eckart (1952)
• (3)

Involving some difficulty since the unknown L
appears on both sides of the equation.
7
Small-Amplitude Wave Theory (4)

• Gravity waves may be classified by the water
  depth in which they travel. The 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 gt 1/2 1/25 to 1/2 lt 1/25 gt p 1/4 to p lt 1/4 ?1 tanh(kd) ?kd
8
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)

9
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.

10
Small-Amplitude Wave Theory (7)

• Water particles generally move in elliptical
  paths in shallow or transitional water and in
  circular paths in deep water.

11
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)

12
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.

13
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?)
  directional spreading spectrum.
• Bretschneider-Mitsuyasu type Frequency Spectrum
  in coasts
• (21)
• Mitsuyasu type Directional Spreading Function in
  costs
• (22)

where,
where, ? is the azimuth measured counterclockwise
from the principle wave direction, ?p, fp the
peak frequency (fp T1/3/1.13), G0 a constant to
normalize the directional function, s the
directional wave energy spreading determined by
angular spreading parameter Smax (Goda Suzuki,
1975).
14
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)

where,
15
Ocean Waves (2)
ANIMATION
ANIMATION
ANIMATION
16
Reproduction of Random Seas in Laboratory
17
Reproduction of Random Seas (1)

• For multi-directional wave generation, a
  snake-like wavemaker motion is used on the basis
  of linear wavemaker theory (Dean and Darlymple,
  1991).

Fig.7 Principle of snake-type wavemaker
18
Reproduction of Random Seas (2)

• Equation of wave elevation
• (17)
• Velocity components on panels of wavemaker
• (18)
• (19)

19
Reproduction of Random Seas (3)
20
Free-surface Conditions

• Kinematic condition

implements the law of mass conservation

• Dynamic condition

implements the law of momentum conservation
21
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.

22
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.

23
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 this 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.

24
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)

25
Some Applications of Free-surface FlowUsing the
Simulation Techniques

• Nonlinear Free-surface Motions around Arctic
  Structure
• Wave Breaking
• Propagation of Solitary Wave

26
Non-linear Wave Motionsaround Arctic 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.

27
Non-linear Wave Motionsaround Arctic Structure
Model to be simulated
28
Non-linear Wave Motionsaround Arctic Structure
Front View
Back View
29
(No Transcript)