report - water waves

8/21/2019 Report - Water Waves

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Contents

Introduction ######################################################################################################################################## $

Nomenclature and main relations ############################################################################################### $

Question 1. ######################################################################################################################################### %

Question 2. ######################################################################################################################################### &

Question 3. ####################################################################################################################################### '%

Question 4. ####################################################################################################################################### '&

Question 5. ####################################################################################################################################### "(

Question 6. ####################################################################################################################################### ""


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Introduction

The objective of this practical work is to have deep insight in the water waves

theory with respect to the lectures Water Waves and Sea State Models for

Ship Design . We intend to introduce the most important relations and to plot

some graphs as a result of our work of sea waves.

Nomenclature and main relations

! – angular frequency (rad/s)

g – gravity acceleration (9,81 m/s2)

k – wave number

h – water depth (m)

" – wavelength (m)

T – wave period (1/s)

# – free surface ?


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Question 1.

Study the dispersion relation and extract the main features that are explained

by this elation (phase velocity, group velocity. . . ). The influence of the main parameters has to be studied.

The dispersive relation is given by following relation:

!!! ! ! ! ! !"#$ !! ! !!

The phase velocity is given by relations:

! ! !!

! ! !!

!

The group velocity is given by relation:

!! !!

!! !!

!!!

!"#$ !!!!!

!! !!

! !!

!!!

!"#$ !!!!!

In this question we studied dispersion relation and we try to show how it

behave for different depth and we plot the graphs for deep, intermediate and

shallow water. Also we show the phase and group velocity for these three

types of depth. We defined types of water depth with respect to the following

relations:

!

! !

!

! !""# !"#$%

!

! !

!

!"

!!!""#$

!"!"#

!

!"!!

! !

!

! !"#$%&$'()#$ !"#$!

For the deep water depth was chosen 100m, for intermediate depth 5m and

for shallow water 0.4m. For these water depth criteria will work for all

frequencies from 0.4 to 2 [rad/s].


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Results obtained for the calculations:

Figure 1. Dispersion relation for different water depth

Figure 2. Phase and group velocity as f(!) for deep water

Figure 3. Phase and group velocity as f(!) for intermediate water


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Figure 4. Phase and group velocity as f(!) for shallow water

We can see that speed related with frequency of wave and decrease by

increasing the frequency. We can se that group velocity is always smaller

than phase and also we can prove relations, which is visible from the plots,

that for deep water group velocity is half of the phase velocity and for the

shallow water Cg=(gh)^0.5. So that relation are:

!!! ! !!! ! ! !!

!!! ! !!! !

!

Figure 5. Phase and group velocity as function of kh


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Question 2.

In order to characterize the behavior of linear dispersive waves, study the free

surface elevation as well as kinematics and pressure fields. The influence ofthe different wave parameters has to be studied in details.

The solution for free surface elevation " of airy waves is given by following

relation: ada

! !! ! ! ! ! !"# ! ! ! !!"

! !!!! ! ! ! ! !"# ! !"#$ ! ! !"#$ ! !!"

And solution of the velocity potential problem is:

!!!! !! ! ! ! ! ! ! ! !"#! ! ! ! !

! ! !"#$ ! ! ! ! !"# !! ! ! ! ! ! !!

These equations are valid for intermediate and deep water and for regular

(periodic) waves only.

We will present the free surface elevation in time domain and in frequency

domain.

Figure 6. Free-surface elevation at given position x


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Figure 7. Free-surface elevation at given time position t

The velocity field and pressure under the free surface for linear problem are

presented in following relations: asdad

! !!!

!" !

!"# ! !"#! !!! ! !!

! ! !"#$ !!!! !"# !!" ! !"!

! !!!

!" !

!"# ! !"#! !!! ! !!

! ! !"#$ !!!! !"# !!" !!"!

! ! !!"#! !!!

!" !

!

!! !!

!

! ! !!"#! !!!

!"

At the following graphs we present the velocity and the pressure field under

the free surface with different water depth with same amplitude and also by

changing the wave height.


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Figure 8. Velocity and presure field for h=10m and A=1m




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Figure 11. Velocity and presure field for h=1m and A=0.5m

The velocity of the water particles is presented in the following graphs for

different water depth (3m, 5m and 10m). It is showed how the velocity is

changing with respect to water depth.

Figure 12. Velocity profile under free-surface for vertical

and horizontal velocity (h=10m)


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The amplitude and period is keeped the same and we can see that velocity is

the same at z=0, only decrease how deeper you going.

For smaller depth slip on the bottom for horizontal velocity is bigger.


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At the next graphs is presented horizontal velocity on the crest of wave.

Figure 15. Velocity profile at the crest of wave for h=5m and A=1m

Now we know how it velocity profile looks, and we study the particlest

movement under the free surface. The trajectories has shape of elipse, and

each particles start movement from one point and finish at the same point

after one wave period. The trajectories of water particules are given by

following relations:

! ! ! !!"#! ! !! !

!"#$ !! !"# !!" !!"!

! ! ! !!"#! ! ! ! !

!"#$ !! !"# !!" ! !"!

So now we will show what is the particle movement for the different water

depth and amplitude.


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Figure 16. Trajectories of water particles for h=10m and A=1m




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Question 3.

The study performed with linear solution has to be pursued with the Stokes

second order solution (see lecture notes for different expressions). Demonstratethe possible differences experienced between linear and second order solution.

The presentation of correction associated to total solution may be useful. The

main physical features explained by second order theory should be brought

into light.

The second order waves studied by Stokes solutions where elevation and

potential are perturbed in wave steepness (e=kA) which is a small parameter:

!!! !!!!! ! !!!!!! ! !!!!!!!! !! ! !!!!

!!! !!!!! ! !!!!!! ! !!!!!!!!!! !!! ! !!!!

Where the second order term for eta and potential are given by following

relation:

!!!!! ! !!

!! !!

!!! !"# !!!" !!"!

!!!! ! ! !

!

!

!!"#!!!!!!!"#! !!!! ! !! !"# !!!"

!!"!

! ! !"#$ !!!!

Then the final elevation eta will be sum of these two term:

! ! !!!!! !

!!!

So to the firs order wave we add second order term with the same parameter

and result will be second order wave.

So now on the next graph we present first order wave and second order term

and then final wave which is result of sum previous two. Also we will plot for

different parameter (water depth and wave amplitude).


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Figure 19. Second order Stoces solution for h=2m and A=0.5m

Figure 20. Second order Stoces solution for h=5m and A=1.5m

Figure 21. Second order Stoces solution for h=10m and A=4m


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Now we plot second order velocity and presure field which is shown below:

Figure 22. Second order velocity and presure field


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Question 4.

In this study, JONSWAP spectrum is studied. It is a function of Hs, Tp and !.

Fix a value to Hs and Tp (reasonable value!).

a.) Look at the influence of ! parameter on the shape of the spectrum.

The JONSWAP spectrum is given by following relations:

!! ! !!

!!!!!!!

!! !"# !

!

!

!

!!

!

! !"#!! ! ! !!!

!

!!! !!!

!"#! ! ! !!!" ! ! !!

! ! !!!" ! ! !!

So now for JONSWAP spectra we fix parameters Hs=3 and Tp=10 and plot

the spectrum for three different value of gamma (1,3.3 and 10), and we

obtain following results:

Figure 23. JONSWAP spectra for # =1, 3.3 and 10

We can see that higher # cause more narrow spectrum with higher pick value

and for lower value of # we obtain very different spectrum of frequencies. The

mostly used value of # for sea state is 3.3.


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b.) Give wave profile at a given location as a function of time for different

values of !. Make sure to use the same phases for each of the test.

On the next graph we plot three wave profiles for different three # value (thesame from a. ) we show influence of coefficient to free surface elevation.

Wave profiles are plotted in function of time with fixed X.

Figure 24. Wave profile for # =1, 3.3 and 10

c.) Once the surface elevation is known (choose one value for !), it is also

possible to have a look to the corresponding velocity and pressure field.

To define the velocity and pressure field we used the relations for regular

waves. Every irregular waves decomposed by some number of regular waves.So we will calculate for each frequency of spectrum pressure and velocity

field, and make summation of it. In the next plot we present pressure and

velocity field for irregular waves.


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Figure 25. Velocity and presure field for irregular waves for # =3.3

Figure 26. Wave profil for irregular waves for # =3.3


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Question 5.

Compare regular and irregular waves. This has to be done in terms of free

surface levation as well as kinematics/pressure (you may have to fix thelocation for comprehensive comparison).

To compare regular and irregular waves, first we will plot free surface

elevation of amplitude A=0.5 m and than present irregular wave for the

similar characteristic.

Figure 27. Velocity and presure field for irregular waves for # =4

Figure 28. Velocity and presure field for irregular waves


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Now we compare pressure field of regular waves and irregular one.

Figure 29. Velocity and presure field for irregular waves for # =4

Figure 30. Velocity and presure field for regular waves for A=1.7

We can see pressure and velocity field for both cases, we can see that

pressure and velocity are well distributed with regular waves, but with

irregular waves are not. But even waves with irregular waves are little bigger

(around 3m) pressure is generally smaller than with regular one where high

is 1.7m and pressure is going until the bottom (mostly constant), and bothcases are with the same water depth of 7m.


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Question 6.

Compar After the choice of an adequate sea state, determine the response

spectrum (in heave) of the ship of interest:

• Without forward speed

• With forward speed in head sea

For the large dimensional body the Keulegan-Carpenter number Kc should

be relatively small:

!" !!!!!!

!! !

A - wave amplitude;

L - length of the body (ship)

Inferences from small Kc number:

• flow is attached to the body;

• large perturbation of the incident flow;

•

diffraction / radiation forces are most important;

• effect of drag forces important only at resonance

For determing the Responce Spectrum, following computational methods

and concepts are used:

• Linear Time-Invariant (LTI) System, where causality (relation betweeninput and output parameters) is time-independent;

• Response Amplitude Operator (RAO) - amplitude of response per unit

of input of a linear system for a given period / frequency;

• encounter frequency "e - frequency (and derivable parameters,including equations of ship motions), which is written on the base ofreference frame moving along the body (ship);


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For the deep water:

!! ! ! !

!!!!

!! !"# !

! - frequency with fixed reference frame;! - velocity vector;

g - gravity acceleration;

! - angle for determining the waves direction (for the head sea: 90° < $ < 270°)

• observed sea response spectrum S #( "e ) :

!! !! !!! !

!!!!!!!

!!

!"# !

For the heave motion we have:

!!!

!! ! !!" !!

!

! !!! !!

Previous formula is dependant on forward speed through Hz#(!e), and at each

speed and angle the RAO is different.

Now we are going to use the data file provided to us, we will load the file and

use only column for heav motion (the fourth column). The first column is in

period so we will transform to frequency (1/T).

After that we obtain spectrum S by JONSWAP and our output is now given

by relation:

!"#$"# ! !! !! ! !"# !

So now we can plot our respons spectrum of the ship (for heave) without

forrward speed.

Figure 31. Response spectrum for ship without forward speed


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To find response with forward speed we cannot use the same spectrum S

obtained from JONSWAP, because the frequency will be changed, so we have

to modify the spectrum with encounter frequency, which is related with

ship’s speed and given by relation:

!! ! ! !

!!!!

!! !"# !

And then we obtain the new spectra (for 180°) by using relation:

!! !! !!! !

!!!!!!!

!!

!"# !

And than we have our output:

!"#$"#!"#$%#& ! !!! !! ! !"# !

And when we plot we obtain following spectrum:

Figure 32. Response spectrum for ship with forward speed fn=0.22

report - water waves

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