report - water waves
TRANSCRIPT
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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
Figure 9. Velocity and presure field for h=5m and A=1m
Figure 10. Velocity and presure field for h=3m 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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Figure 13. Velocity profile under free-surface for vertical
and horizontal velocity (h=5m)
Figure 14. Velocity profile under free-surface for vertical
and horizontal velocity (h=2m)
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
Figure 17. Trajectories of water particles for h=5m and A=1m
Figure 18. Trajectories of water particles for h=1m 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