Ch2

1

Ocean EnvironmentProf. Dr Kurian V. John

Ocean Environment (I)

Lecture Outline

Linear Airy wave TheoryDesign wave EnvironmentWave StatisticsWave Forces on Small StructuresWave Forces on Large Structures

2

Linear Airy Wave Theory


Definition Sketch for a progressive wave train



k=wave number = 2π/L where L is the wave length

ω = wave frequency = 2π/T where T is the wave period

Wave surface elevation η is given by

where

3



The linear dispersion relationship is given by

The velocity potential Φ is given by

Wave Celerity C=L/T is given by

, which is another form of

dispersion relationship.



The wave length L is obtained as

=

where =

The water-particle velocities in the x and ydirections are obtained from the expressions

and v

4



The horizontal water-particle velocity

The vertical water-particle velocity



The water particle accelerations in x and ydirections are given by

Horizontal:

Vertical:

5



The horizontal and vertical displacementsare given by

and

The dynamic pressure p is given by

PMO Metocean Data


6



0

10

20

30

40

50

60

70

-2.5000 -2.0000 -1.5000 -1.0000 -0.5000 0.0000 0.5000 1.0000 1.5000 2.0000

s (m

)

Particle Velocity (m/s)

Velocity Distribution

u

v



0

10

20

30

40

50

60

70

-2.0000 -1.5000 -1.0000 -0.5000 0.0000 0.5000 1.0000 1.5000

s (m

)

Particle Acceleration (m/s²)

Acceleration Distribution

u dot

v dot

7



0

10

20

30

40

50

60

70

-4.0000 -3.5000 -3.0000 -2.5000 -2.0000 -1.5000 -1.0000 -0.5000 0.0000

s (m

)

Particle Displacement (m)

Displacement Distribution

ξ

η



0

10

20

30

40

50

60

70

-40000 -35000 -30000 -25000 -20000 -15000 -10000 -5000 0

s (m

)

Pressure (N/m²)

Pressure

P

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Design Wave Environment


For a single sinsoidal wave of frequency, ,the wave profile may be given as

Choosing the origin at x = 0,where =H/2


9



The total energy of a wave, E(per unit surfacearea) in the wave record between infinite timelimits is given by the integral

The total energy is obtained from the areacovered by the energy density curve as afunction of frequency.



-10

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

S(f

) (m

²/s)

f (1/s)

Wave Spectrum

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Pierson-Moskowitz Spectrum

The peak frequency is related to significant wave height by

. ⁄

An equivalent expression for P-M spectrum in terms of cyclic frequency

.



Bretschneider Spectrum

. . ⁄

ISCC Spectrum

. . ⁄

The relationship between the peak frequency, and for ISCC spectrum is

.

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ITTC Spectrum

where .



JONSWAP Spectrum

. ⁄

in which γ = peakedness parameterand τ = shape parameter

τa for ω≤ω0 and τb for ω≥ω0 .Considering a prevailing

wind field with a velocity of UwandafetchofX,theaveragevaluesofthesequantitiesaregivenby

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3.30 1 7τa 0.07andτb 0.09 consideredfixed

0.076 . 0.0081 2 ⁄ . X gX U usuallynotused⁄



Goda (1979) derived an approximateexpression for the JONSWAP spectrum interrms of and as folows:

∗ . ⁄

where

∗ .. . . .

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-10

0

10

20

30

40

50

60

70

0.000 0.100 0.200 0.300 0.400 0.500

S(f

)

frequency

Various spectra

JS(Goda)Spectrum

PM Spectrum



Simulation of Wave Profile from Spectra

The wave height frequency is obtained asfollows

2 2 ∆

and the corresponding wave period is given by1⁄

14



The phase angle associated with each pair ofheight and period is chosen uniformlydisributed in the range (0, 2 ) by a randomnumber generator, as

The wave profile is computed from

,



-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300 350 400 450 500η(m

)

t (sec)

Time Series

15

Wave Statistics


Wave Statistics


The zeroth moment, , represents the areaunder the energy density spectrum curve

From the above equality, the relationshipbetween and is obtained

16

Wave Statistics


Significant Wave HeightThe average of highest one-third waveheights gives the significant wave height.

⁄

⁄

If is the total area under the waveenergy density spectrum, then

Wave Statistics


RMS Wave HeightThe root -mean square (rms) wave heightis calculated from the wave formula

⁄

In the frequency domain, is given by

The rms value of water elevation is

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Wave Statistics


Maximum Wave Height

.

The most probable maxima, , in 1000 waves can be obtained by letting

so that

.

Or . ⁄

Wave Forces on Small Structures


Morison Equation

The Morison Equation was developed byMorison, O’Brien, Johnson and Shaaf in theirpaper ‘The force exerted by surface waves onpiles’ published in the Journal PetroleumTransactions at AIME 1950.

They proposed that the force exerted byunbroken surface waves on a vertical cylindricalpile which extends from the bottom through thefree surface is composed of two components,inertia and drag.

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Inertia Force

The principle involved in the concept of theinertia force is that a water particle moving in awave carries a momentum with it.

As the water particle passes around the circularcylinder, it accelerates and then decelerates.

This requires that work be done through theapplication of a force on the cylinder to increasethis momentum.



The incremental force on a small segment of thecylinder, ds, needed to accomplish this is proportionalto the water particle acceleration at the center of thecylinder (in the absence of the cylinder)

In which dfI = inertia force on the segment ds of the vertical cylinder,

is the mass density of sea water,

D = cylinder diameter,

= = local water particle acceleration at the center line of the cylinder & CM = inertia coefficient

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Drag Force

The principle cause of the drag force component is thepresence of a wake region on the downstream side ofthe cylinder.

The wake is a region of low pressure compared to thepressure on the upstream side and thus a pressuredifferential is created by the wake between the upstreamand downstream of the cylinder at a given instant oftime.

As the water particle motion under a wave is oscillatorywithin a given wave period, the downstream side of thecylinder reverses every half cycle and a mirror image iscreated after half a cycle.



The pressure differential causes a force to be exertedin the direction of the instantaneous water particlevelocity.

In a steady flow, downstream side is fixed and thedrag force is proportional to the square of the waterparticle velocity.

In an oscillatory flow, the absolute value of the waterparticle velocity is inserted to ensure that the dragforce is in the same direction as the velocity.

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Where dfD = drag force on the segment ds of the vertical cylinder,

U = instantaneous water particle velocity &

CD = drag coefficient



Combining the inertia and drag components of force,the Morison Equation is written as

f = fD + fIwhere

f = total wave force per unit length

fD = drag force per unit length =

fI = inertia force per unit length =

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Example



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24



25



26



27



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Wave Forces on Large Structures

• If the incident flow separates from the surface of thestructure forming a wake alternatively in front of andbehind the structure: Morison Equation

• If the incident wave experiences scattering from thesurface of the structure in the form of reflected wave thatis of the order of magnitude of the incident wave:Diffraction theory

• If neither separation (structure not too small compared toL), nor large reflection is evident (structure not too largecompared to L) Froude-Krylov theory


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• The forces are calculated by a pressure-area method inwhich the expression of the pressure due to the incidentwaves is used on the surface of the structure.

• It assumes the structures is not there and hence limitedpractical applications.

• Closed-form expressions for wave forces on a fewsubmerged basic structures of symmetry may beobtained using linear wave theory.

• The forces derived by this theory are to be correctedbecause of the change of the oscillatory flow around thestructure, like a force coefficient.


• The expressions for force components can be written as:

• ∬

• ∬

• In which

• and are horizontal and vertical force coefficients(assigned values)

• and are direction normals in the x and y directions

• is an elemental surface area of the submergedstructure

• The use of these coefficients limit the applicability of thismethod in general.



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• When the structure spans a significant portion of a wavelength, the incident waves upon arriving at the structureundergo significant scattering or diffraction.

• In this case, the diffraction of waves from the surface ofthe structure should be considered in the wave forcecalculation.

• The total velocity potential, Φ, is obtained as a sum ofthe incident and a scattered potential. It satisfies theLaplace equation as follows.

•



Definition of Boundary Conditions for the Linear Diffraction Problem


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• The dynamic boundary condition is given by

•

• The kinematic boundary condition is given by

•

• Velocities are defined as

• ; ;

• Bottom boundary condition is given by

•



• The body surface boundary condition is given by

•

• where is the velocity potential due to incident waves and is the velocity potential due to diffracted waves.

• Closed form solutions are available for standardproblems and many commercial software areavailable for numerical solutions.



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• When D/L ratio is less than 0.2, Morison equation isstill applicable and has proven to give good results.

• Many research papers have used Morison Equationfor studying the wave responses because of theease of programming. This use should be carefullydone in high frequency region, where scattering ordiffraction effects will dominate.




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