Proceedings of OMAE’05:
24th International Conference on Offshore and Arctic Engineering
12-17 June, 2005, Halkidiki, Greece
OMAE2005-67436
VALIDATION OF AN ANALYTICAL METHOD TO CALCULATE WAVE SETDOWN ON CURRENT
Arjan Voogt, Tim Bunnik and René Huijsmans
Maritime Research Institute The Netherlands (MARIN)
ABSTRACT
This paper describes an analytical model to calculate the
setdown in waves with and without current. This model is
based on a second order quadratic transfer function and a
Lagrangian transformation for the effect of current. The method
is validated against basin experiments with a large number of
wave probes through the basin. The wave signals are separated
in incoming and reflected bound and free waves based on the
phase differences between the wave probes. The optimum
probe spacing and transient phenomena in the restricted sized
basin are discussed. The resulting separated bound wave shows
good comparison to the setdown calculated with the analytical
model.
As reported in earlier studies, the effect of the second order
bound wave in shallow water plays an important role in the
second order wave drift forces. Apart from these bound
setdown waves, long free waves can occur in the coastal
regions and basins due to the bottom geometry or finite
boundaries.
This paper starts with a description of the analytical model
to calculate the setdown in waves with and without current.
After that a separation scheme is discussed to separate the
different components in a measured wave. The paper closes
with discussion and conclusions on the comparison between the
separated measured setdown and the analytically calculated
setdown.
INTRODUCTION
The increase in demand for LNG and the associated safety
requirements has resulted in a large number of offshore LNG
terminals under construction or preparation. Most of these LNG
terminals are located near shore in relative shallow water. In
shallow water the contribution to the low frequency excitation
of the wave setdown increases. Combined with the low
damping on the streamlined LNG carrier hulls, this can result in
significant resonant motions and related mooring loads.
Already in the early days of offshore engineering shallow
water hydrodynamics played an important role in the design of
jetty systems and the design of the vertical dimensions of
harbors and waterways. The first studies on shallow water
related bound waves have been reported by Longuet Higgins et
al (1977), bowers (1977), Pinkster (1980) and Huijsmans and
Pinkster (1992). The applicability of second order theory of
water waves to harbor design have been investigated by
Huijsmans Dallinga (1983).
The renewed attention to the shallow water hydrodynamics
is based on the fact that the new design concepts for LNG
offloading (see Naciri (2004)) posses a very low damping as
opposed to the mooring systems as seen in the earlier years.
This low damping ratio leads to a large variation of the low
frequency rms values of the horizontal motions. In order to
reduce these large variations in rms values, long simulation
times in either experiments or computations are required.
ANALYTICAL MODEL TO CALCULATE SETDOWN
The wave setdown calculation is based on the well known
analysis of Sharma and Dean (1981). The wave elevation is
described by the dynamic and kinematic conditions on the free
surface, which state that atmospheric pressure holds at the free
surface and that fluid particles remain at the free surface.
Applying the assumption of an irrotational and non-viscous
flow, the fluid velocity field is described by a velocity potential
that satisfies the equation of Laplace (mass conservation). This
set of equations is non-linear and analytical solutions do not
exist. Perturbation theory can be applied to find the solution to
these equations up to a certain order.
In this paper, long-crested waves are considered moving on
the current. The first-order wave elevation of a long-crested
wave can be written as a sum of harmonic components:
N
ζ (1) = a cos(ω t + ε − k x )
∑
i=1
1
i
i
i
i
Where:
ai =
wave amplitude [m]
ωi =
wave frequency [rad/s]
εi =
random phase
[rad]
ki =
wave number
[1/m]
Copyright © 2005 by ASME
The dispersion relation follows from inserting this
formulation into the free-surface condition and gives the
relation between wave frequency, wave number, current speed
and water depth:
(
)
D ωi , ω j , k i , k j , h = A
1
A=
2 ωi − ω j
(
B=
C=
ki
2g
−
(
( ))
k ik j g 1 + tanh(k ih) tanh k jh
4ωi ω j
+
(
( ))
1
k i tanh(k ih) + k j tanh k jh
4
g2 (B + C)
2
−
)(
kj
2
( )
ω j cosh 2 k jh
-2
-1
-0 .4
-0 .7
-5
-20
-1
-2
0.5
0
-2
-2
-5
-5
-20
2.5
-5
-20
1
1.5
2
wavenumber times water depth (ki.h)
-20
0.5
0
3
In case of a situation with current, the wave setdown is
evaluated in a system of axis moving with the current. Using
the transformation x*=x-Ut, the following formulation for the
wave elevation moving with the current is obtained:
N
( ))
2k ik j ωi − ω j 1 + tanh(k ih) tanh k jh
ωi ω j
1
ki − k j
4
The transfer function is made non-dimensional with the
water depth and plotted for all combinations of the wave
numbers in Figure 1. It should be noted that the wave numbers
themselves are also normalized with the water depth to ensure a
direct relationship with the wave frequencies wi and wj through
the dispersion relation. Therefore both axes from Figure 1 show
the relative water depth (2πh/λ) of the two wave components
that contribute to the setdown.
In deep water, the transfer function becomes D = −
From the Figure the following can be concluded:
• The setdown increases significantly for shallow water
(dotted arrow)
• The setdown increases for larger difference
frequencies (solid arrow)
(
(
))
(
N
ζ = ∑ a i cos ωi t + ε i − k i x * + Ut = ∑ a i cos ωi* t + ε i − k i x *
i =1
)2 − (k i − k j )g tanh((k i − k j )h)
ωi cosh 2 (k ih)
(
ωi − ω j
-0 .7
-0 .7
-1
Figure 1 Non dimensional transfer functions
As can be seen, the setdown is in phase with the wave group.
The transfer function is given by:
5
-0 .
))
-0 .7
(
-0 .5
-1
)
-0 .4
-0 .5
-5
) ((
1
-2
i =1 j =1
1.5
-2 0
(
N N
ζ (2 ) = ∑ ∑ aia jD ωi, ωj, ki, k j, h cos ωi − ωj t + εi − ε j − k i − k j x
2
-20
Since the free surface condition for the second order
potential contains the products of first-order quantities, the
solution for the second order wave elevation contains sum and
difference frequency components. The difference frequency
components are generally referred to as wave setdown.
Physically it can be observed the mean water level drops below
a group of waves. In the situation without current (U=0), the
quadratic transfer functions between the first-order wave
elevation and the wave setdown has been derived by for
example by Longuet Higgins. The setdown is described by the
following formulation:
2.5
-5
= gk tanh kh
wavenumber times water depth (kj.h)
(ω − Uk )2
Non dimensional transfer function (D.h)
3
i=1
)
With the frequency of the waves in the system of axis moving
with the current (encounter frequency). In this system of axis,
the apparent current speed is zero, and the standard procedure
for the analysis of wave setdown can be applied, thus:
(
N N
) ((
)
(
) )
ζ (2 )current fixed = ∑ ∑ aia jD ωi*,ω*j,k i,k j, h cos ωi* - ω*j t + εi − ε j − k i − k j x *
i =1 j =1
Where D is the quadratic transfer function of the wave setdown,
evaluated at the encounter frequencies and at the wave
numbers, corrected for the effect of current.
In order to obtain the wave setdown in the earth-fixed
system of axis, the back substitution is made. This results in:
N N
(
) ((
)
(
))
ζ (2 )earth fixed = ∑ ∑ aia jD ωi* , ω*j ,k i ,k j ,h cos ωi - ω j t + εi − ε j − k i − k j x
i =1 j =1
The encounter frequencies in the moving reference frame
are coupled to the wave numbers through the dispersion
relation without current. Therefore Figure 1 can be used to
assess the values of the transfer function. Due to the current the
wave numbers decrease, resulting in a shift towards origin of
the graph (more shallow water). In deep water this will have
little effect on the setdown, but for shallow water the setdown
will increase significantly.
WAVE SPLITTING THEORY
Model tests in ocean laboratories are more and more used
to calibrate and validate numerical models. Therefore, accurate
knowledge of what is happening in ocean basins is required in
order to provide correct input to these numerical models. One
of these things is the presence and quantification of free low
frequency waves. These free waves occur due to several types
of basin effects:
1. Generation of waves with a flat wave flap
2. Reflections due to the finite size of the basin
3. Shoaling effects due to the bathymetry of the basin
bottom
2
Copyright © 2005 by ASME
Due to the first effect, there is a mismatch between
generated water velocities and the water velocities in an ocean
wave at the wave flap. To minimize this effect a second order
correction is applied to the wave generation. The second effect
is minimized by using parabolic beaches, but especially the
long wave components are insufficiently damped. The third
effect is important in shallow water situations where the wave
setdown reflects as a free wave from the parabolic beaches.
These low frequency waves must be separated from the
bound wave components to provide a direct comparison with
the analytical calculated setdown. For this separation it is
assumed that the waves are long-crested and moving parallel to
the current. The wave elevation can be described as follows:
ζ = ζ inc,free + ζ inc,bound + ζ ref ,free + ζ ref ,bound
Within the wave spectrum there is a large amount of firstorder frequency pairs that result in the difference frequency ∆ω
. The incident bound wave therefore contains not only one
wavelength ∆k , but an infinite number of wavelengths.
Because the information to split the wave components (wave
probe measurements) is often limited, it is assumed that the
bound wave propagates with the group speed corresponding to
the peak period in the wave spectrum:
c bound =
∂ω
at ω = ωp
∂k
And thus ∆k =
ω
c bound
ζ inc,free is the incident free wave, propagating from the
wave flaps to the beach.
ζ inc,bound is the wave bound to the incident free wave, also
referred to as wave setdown.
ζ ref ,free is the reflected free wave, propagating from the
beach towards the wave flap.
ζ ref ,bound is the wave bound to the reflected free wave.
Since the reflections are generally small, it is assumed that
the reflected bound waves are negligible due to the quadratic
nature of the setdown response to first order waves.
ω
The free waves propagate with the wave speed c free = .
k
∂ω
The bound waves propagate with the group speed c bound =
∂k
These different speeds and the different direction of
propagation of the incident and reflected waves, provide the
opportunity of finding the different wave components. This
splitting up has to be carried out for each wave frequency and
requires knowledge of the wave elevation at at least three
different spatial positions. The incident and bound free waves
at one particular (difference) frequency can be written as
follows:
ζ inc,free = ζ if cos(ωt − k inc x + ε if )
ζ ref ,free = ζ rf cos(ωt + k ref x + ε rf )
ζ inc,bound = ζ ib cos(ωt − ∆kx + ε ib )
Where the wave frequency and wave number satisfy the
dispersion relation:
(ω − Uk inc )2
(ω + Uk ref )2
= gk inc tanh kh
= gk ref tanh Kh
Figure 2 Wave spectrum
This is a reasonable assumption for narrow-banded spectra
and limits the number of bound wavelengths to one at each
frequency. In order to split the resulting three wave
components, the wave elevation must be known at at least three
different points in space. If more than three, a least squares
approximation will give the best fit for the three wave
components. Suppose the wave elevation is known at location
x = x n . A Fourier Transform will give the individual frequency
components of the wave. Each frequency component can then
be written as:
ζ x = x n , ω = ω j = A cos ω j t + B sin ω j t
(
)
( )
( )
with known coefficients A and B.
Inserting this into the formulation for the total wave
elevation gives:
ζ if cos ω j t − k j x n + ε if + ζ rf cos ω j t + K j x n + ε rf +
(
)
(
)
ζ ib cos(∆ω j t − ∆k j x n + ε ib ) = A cos(ω j t ) + B sin(ω j t )
The difference frequency and difference wave number do
not satisfy this dispersion relation:
Defining the following solution vector with unknowns:
(ω − U∆k )2 ≠ g∆k tanh ∆kh
3
Copyright © 2005 by ASME
phase difference only occurs for a large distance between the
probes.
1
ωp 0.5 rad/s
bound relative to free wave length
⎛ ζ if cos ε if ⎞
⎜
⎟
⎜ ζ if sin ε if ⎟
⎜ ζ cos ε ⎟
rf ⎟
y = ⎜ rif
⎜ ζ rif sin ε rf ⎟
⎜
⎟
⎜ ζ ib cos ε ib ⎟
⎜ ζ sin ε ⎟
ib ⎠
⎝ ib
And collecting the cosine and sine terms, the following two
equations are found:
( ( ) sin(k j x n ) cos(K j x n ) − sin(K j x n ) cos(∂k j x n ) sin(∂k j x n )) • y
B = (sin(k j x n ) − cos(k j x n ) − sin(K j x n ) − cos(K j x n ) sin(∂k j x n ) − cos(∂k j x n )) • y
A = cos k j x n
In case of three wave probes, 6 equations are found for the
six unknowns. Provided that this system of equations is well
conditioned, the solution can be found by matrix inversion. The
conditioning of the matrix and thus the quality of the separation
depends on the distance between the wave probes relative to the
wavelength of the free waves and on the difference in
wavelength for the free and bound waves.
Figure 3 shows the conditioning of the matrix system in
case of a 2-component split up (incident free and reflected free
waves). It shows that the conditioning becomes large when the
distance between the two wave probes is an integer multiple of
half the wavelength. These are the wavelength probe spacing
ratio’s were aliasing occurs. The optimum separation of the
incident and reflected waves can be achieved with a probe
distance of 0.25 times the wavelength.
Figure 3: Condition number of matrix for wave separation
For large distances between the probes aliasing effects can
occur. On the other hand large distances are needed to separate
the free and bound waves traveling in the same direction.
Figure 4 shows the ratio between the wavelengths of free and
bound waves as function of the water depth at a frequency of
0.05 rad/s in waves without current. The different lines show
the effect of the mean frequencies (ωp) of the underlying first
order waves. The figure shows that for decreasing water depth
the wavelength of the bound components approaches the
wavelength of the free components. Therefore a significant
ωp 0.6 rad/s
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
water depth [m]
70
80
90
100
Figure 4: Wave length ratio between bound and free waves
For a water depth of 20 meters the wave and a mean
frequency of the first order waves at 0.6 rad/s, the bound
wavelength equals 66% of the free wavelength. As the
optimum separation can be achieved with a phase shift of 180
degrees between two components this results in optimum probe
spacing of one free wavelength.
As discussed before the optimum probe spacing for
separation of incident and reflected free waves equals one
quarter of the (free) wavelength. For the present case the
optimum array of wave probes to separate the different wave
systems consist of a combination of these two distances.
Additional probes can be added to minimize the effect of
noise on the measurement signals with a least square fit through
the data. These additional probes are positioned such that the
optimum probe spacing becomes available for a whole range of
frequencies with different wavelengths.
To use both the distances between two neighboring probes
and the probe further away a setup is chosen in which the
distances between probe 1 and 2 (d1) and probe 2 and 3 (d2)
are related as follows:
d2=d1*q
d2+d1=d1*q^2
These equations hold for q = 0.5+sqrt(1.25) = 1.618, which is
exactly the 'Golden Section' Coefficient known from classical
mathematics and the generation of the fibonacci numbers.
With this factor (1.618) between two successive probes
distances, the ratio between each wavelength and its
corresponding optimum distance for the free wave separation
varies between 0.2 and 0.3. This system can be further extended
with more wave probes as follows
Wave probe number
Distance preceding probe
Distance to first probe
2
d1
3
q. d1
q2. d1
4
q3.d1
q4.d1
5
q5. d1
q6. d1
6
q7. d1
q8. d1
RESULTS AND DISCUSSION
The wave measurements were carried out in the Offshore
Basin. This basin offers a number of unique possibilities for the
modeling of current, waves and wind. Figure 5 shows a cross
section of the Offshore Basin.
4
Copyright © 2005 by ASME
The bound component is compared to the theoretical
setdown in Figure 7. The theoretical setdown was calculated for
frequencies above 0.02 rad/s as larger wavelengths do not fit in
the basin. The comparison shows small deviations on the
energy spectrum but the comparison in time domain and
statistics is rather good.
sum wave components versus measured wave
x = 1610 m
x = 800 m
x = 491 m
x = 300 m
0.5
0.5
0
-0.5
1000
0.5
1100
1200
1300
1400
1500
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.5
0
-0.5
1000
0.5
1100
1200
1300
1400
1500
0
0.5
0
-0.5
1000
0.5
1100
1200
1300
1400
1500
0
-0.5
1000
0.5
0.1
1100
1200
1300
1400
1500
0
1100
1200
1300
1400
1500
0
0.2
0
0.1
1100
1200
1300
1400
1500
0
0.5
0.2
0
0.1
-0.5
1000
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1
0.5
-0.5
1000
0
0.2
0
-0.5
1000
0.2
1100
1200
1300
Time [s]
1400
1500
0
analytical
measured
0
0.05
0.1
0.15
0.2
Frequency [rad/s]
0.25
0.3
Figure 7: Bound wave versus analytical calculated set down
REFERENCES
Bowers E.C.: Long Period Oscillation of moored ships
subject to short wave seas. RINA proceedings 1975 London.
Pinkster J.A.: Low Frequency Second Order Wave
Exciting Forces on Floating Structures. PhD thesis Delft 1980.
Huijsmans R.H.M.,Pinkster J.A.. Wave Drift Forces in
Shallow Water. Proceedings of the ISOPE 1992 Conference
London.
Huijsmans R.H.M.,Dallinga R.P. Non-Linear Ship Motions
in Shallow Water. Ship and platform motion symposium 1983
Berkely.
Naciri, M. Bunnik, T. Buchner B. Huijsmans R.H.M.:
Low frequency motions of LNG carriers moored in shallow
water. Proceedings of the OMAE 2004 Conference ,
OMAE2004-51169 Vancouver
Kostense, J.C. Study to long periodic waves at Sines,
Portugal and its application in port design. International
Conference on Numerical and Hydraulic Modelling of Ports
and Harbours, Birmingham, United Kingdom, 23-25 April
1985.
Sharma S. Dean P. Simulation of Wave systems due to
non-linear spectra. Proc of the Conf on Hydrodynamics in
Offshore Engineering 1981 Trondheim Norway.
Longuet Higgins, M.S. Steward R.W : Radiation stress
and mass transport in gravity waves with application to surfbeats. J of Fluid Mechanics 13, 1977 p481-504.
Ottnes Hansen N.E. et al: Correct reproduction of group
induced long waves, Proc 17th Coastal Eng. Conf. Sydney
1980.
1
0
-0.5
1000
0
x = 800 m
The basin measures 46 m * 36 m and has a movable floor,
which is used to adjust the water depth. The current generation
system consists of 6 separate layers, each equipped with its own
pump. The water is re-circulated through a system of channels
outside the basin, in order to avoid re-circulation in the basin
itself.
To allow application of the discussed method to the
mooring design of a vessel all results are scaled to prototype
values using a scale factor of 50. At this scale the water depth
in the basin was adjusted to 25.8m. On this water depth a
Jonswap wave spectrum is adjusted with a peak period of 12s
and significant wave height of 5m. The current of 1.2 m/s is
collinear to the waves.
Figure 6 shows a time trace sample and the low frequency
energy content at four different distances from the start of the
flat bottom. Six probes were used to separate this low
frequency wave energy into free and bound components. A
least square method is used to estimate the 3 unknowns from
the 6 measurements. The sum of these 3 components is added
in the solid (blue) line in Figure 6. For this separation the
following assumptions were made:
• The measured wave system is an ergodic system over
the length of the measurement array. This means that
the energy of the individual components is constant
over the length of the array.
• The wavelength of the bound components can be
described with the group velocity of the peak period of
the first order waves.
• The wave system is limited to these three components,
i.e. no other sloshing modes exist.
Comparison shows that the assumed wave system does fit
reasonably to the low frequency content of the signal.
x = 1610 m
Figure 5: Offshore Basin
x = 491 m
x = 300 m
Analytical setdown versus bound wave component
0.5
sum comp.
measured
0.5
1100
1200
1300
Time [s]
1400
1500
0
0
0.05
0.1
0.15
0.2
Frequency [rad/s]
0.25
0.3
Figure 6: sum wave components versus measured wave
5
Copyright © 2005 by ASME