Experiments and model equations for generating waves on top
of currents
H.Margaretha 0 " E. van Groesen", R.H.M Huijsmans 6
a
University of Twente, Faculty
b
Maritime
Research Institute
of Mathematica!
Sciences, The
Netherlands.
Netherlands
(MARIN),
The
Netherlands.
S e p t e m b e r 25, 2000
Abstract
This paper deals with experiments and theoretical models for wave-current interaction in hydrodynamic laboratories. Different from ocean-engineering applications, the laboratory situation
corresponds to a closed system. A model is proposed to compare wave-only and current-only
quantities with properties of the waves and current in interaction. The model is based on conservation of mass-trans port and energy-flux, and is an extension of other models known in literature.
We compare the various models with experiments; for regular waves we exploit the experimental
results of Thomas ([9]), while for irregular waves we report and use experiments performed at
MARIN. The proposed model performs best of all models when both cases are considered. But
for irregular waves the simpler model that takes for the energy-flux the product of group velocity
with energy performs even better. More experimental results would be needed to improve and
justify the theoretical models.
Keywords: Wave-current interaction; Mass-transport; Energy-flux.
1
Introduction
This paper deals with wave-current intcraction with particular emphasis on situations as they appear
in hydrodynamic laboratories.
In general, theorics about wave-current interaction are approximations of the full Navier-Stokes equations obtained by spUtting the velocity field of the fluid in two parts, onc part referring to the current,
the other to the wave. This splitting is not unique, but is made to obtain simpler descriptions whcn
the intcraction between the two parts is simpÜned. In this paper wc present results of experiments,
derive a theoretical model and compare it with other models, for these, and other, experiments.
The experiments wcrc cxecutcd in a largc wave tank at MARIN ([8]), and show the effect on gencratcd
regular and irregular waves and on the initial current. These experiments confirm known results that,
with wave and current in the same direction, the current deercases as a consequence of the interaction.
For regular waves of given frequency, the effect on the wave is an increase of wave length according to
modified dispersion relation including the current, and a deercase of amplitude; for irregular waves,
the cffects on the waves is more complicated.
We will present a theoretical model to predict the actual current and waves during interaction and
compare with measurements reported in the literature for regular waves ([9]) and the MARIN experiments.
'Correspooding author. Tel.: +31-53-489-3430; fax: +31-53-489-4888.
E-mail address: h.margarethaOmath.utwente.nl
1
The model will give a practical method to determine the waves to be generated (as if there is no
current) in such a way that the interaction will produce desired waves in interaction with a given,
initially unpcrturbcd, uniform currcnt.
The model to be derived is based on relating two basic principles, mass-transport and energy-flux.
The application of these principles depends on the closedness of the system under consideration, and
is different for applications in an ocean environment than in hydrodynamic laboratories. In fact, a
hydrodynamic laboratory is a closed system, for which the total flow of the water has to be taken
into account, the water being recirculated through pipes outside the basin by pumps with a given
(constant) discharge that maintains a certain horizontal mass-transport (of whatcver origin) in the
basin. If the currcnt in the absence of wave is prepared in this way, say of strcngth Uo, the effect of
wave on top of the current will lead to a current of strength U, that differs from the initial current
by the mass-transport due to the wave:
Uo = U + wave mass-transport
(1)
In an ocean environment, mostly a local situation is investigatcd and the system is not closed; an
existing natural current and the effect of mass-transport by wind generated wave are not necessarily
restricted by such a relation.
The same applics for the energy flux. In the laboratory, the initial current will inducc an energy flux,
say F(Uo,0)> while generating waves tt/o in the absensc of current will lead to an energy flux F(0,WQ)
that is produced by the wave makers at one side of the basin. If the resulting current and wave when
in interaction are denoted by U and w respectively, and the corresponding energy flux by F(U,w),
the second relation that will be used reads
F(U>u,)=F(Uo,0)
+ F(Q,w0).
(2)
In general, the raomentum conservation equation is also needed bccause in an open system the mean
water level may also changc. But in the situation at hand, with a closed system and a 'known' mean
water level, only two relations are necessary. Combining these two relations, it is shown how to
calculate the initial wave wo when the initial current Uo and the measured wave w during interaction
are given, thereby solving the practical problem for the laboratory.
In this paper we derive expressions for the mass-transport and the energy flux. In principle, these
expressions are known in the litcrature for regular waves ([6j, [1], [11], [5]), but we will takc into
account that for the relatively short waves in the basin, the finite depth of the basin plays a role.
Then, using the (few) available mcasurements, the results obtaincd with our model will be compared
with results obtaincd when using several other models known in the litcrature.
The model derived here is also exploited for irregular waves, albeit in a rathcr primitive way, by
splitting the given energy spectrum in a finite nuraber of regular waves and taking a simplified version
of the mass-trans port for irregular waves. Comparison with the cxperiments of MARIN will be given.
The organisation of the paper is as follows. Observation from MARIN cxperiments on irregular waves
with currents are reported in §2. The laboratory set up is included there. Derivation of the interaction
model is given in §3, and comparison of various models with cxperiments is given in §4. Sensitivity
analysis of the model is given in §5. Finally somc conclusions are given.
2
Experiments on irregular waves with currents
The experiment to be reported hcre is one of a few laboratory cxperiments which is useful for relating
the waves before the interaction with the currcnt to those after the current has been allowed to interact
with the waves. The experiments were performed in a basin consisting of two parts with different
depth; the dimensions are given by ( 4 0 m x 4 0 m x l m ) + (20mx35m x2m). Waves are generated
2
by fiap-type wave makers, and a beach prcvcnts most of the reflection. A current is generatcd by
recirculating the water with a pump from one sidc to the other side, with inflow and outflow from the
floor. The produced current in the basin is assumed to be uniform over the depth. The free surface
elevation is mcasured by portable electric rcsistance type of wave probes. The horizontal fluid velocity
is measurcd by a propeller-type current meter, placed at 10 cm below the still water level.
In the experiment, current and wave propagate in the samc direction. The wave maker was controlled
by a random signal which produecs waves with significant wave height 15.6 cm and wave period of the
peak frequency of 1.5s. Initial currents werc prepared with strengths of l O c m / s and 2 0 c m / s . The
experiment was donc on both parts of the basin, but in this report, we only take the measurements
from the part of the basin with l m depth. The measurements were taken at four in-line positions
with distances 3.53m, 7.67m, 11.73m, 20.53m from the wave maker.
The records of the horizontal velocity show rapid fluctuations caused by the wave. For irregular wave,
the current is taken as the time-avcraging over an 'infinitc' long period; which is the statistical-mcan
of the time series.
Figure l(a) shows a horizontal velocity record (for initial current of 2 0 c m / s at distancc 11.73m).
For this measurement, the significant wave height turned out to be 13.59 cm. This means that the
velocity probc at 10 cm below the still-water level was in the air cvery, on average, oncc cvery 76
waves. This numbcr is obtained from a direct derivation of the Raylcigh's distribution for the wave
height, assuming narrow-band wave spectrum. This affects the mean value produced by the probe
with an crror of the order of one or two percent, of the same order as (or less than) the accuracy of
the instrument. This crror is thcrefore neglcctcd.
For this measurement, we find out that the current dccrcascs as a consequencc of intcraction with
waves. It decreases from 2 0 c m / s to 11.7cm/s. At this point, it seems neecssary to go into some
details of the actual measurements and the equipment, in particular the current meter.
In fact, in the measurement at the part of the basin with 2m depth, the current decreases from
20cm/s to 17.7cm/s. This measurement is shown in figure l(b). From this we conclude in the
following way that the decrease of the current is caused mainly by the waves and is not an effect
of the interna! friction of the current-meter. This internal friction could be substantial, sincc the
fiuctuating velocity causes the current meter to change the direction of rotation with the period of
the waves. In the experiments the Standard deviation of the velocity records {without averaging) is
practically independent of the depth. The significant wave height of the free surface elevation is also
nearly independent of the depth. Hence, if the internal friction of the current meter would causc the
reduction of mean velocity, the rcduction should have been equal for both depths. Since the reduction
depends vcry much on the depth, we arrive at the conclusion that the measurcd current reduction is
not duc to technical failure of the equipment.
Figures 2-5 show wave spectra mcasured at different positions from the wave maker. The change in
these spectra with distance occurs both in the presencc and the absence of current. From figure 5
with all spectra displayed, it is observed that in the presencc of current the wave spectrum does not
uniformly decrease for all frequencies.
3
Derivation of the interaction model
In this section we derive a model to describe the interaction of waves and currents; the dctailcd
derivation makes it possible to comparc the result with othcr models known in the literature. We
start to split the velocity u into two parts:
u = G + U,
(3)
in which ü(x, z,t) — 0. The overbar operator denotes time-averaging over a wave period. The term
u = («,w) represents the wave, and U —(t/,0) represents the current. In relation (3), U can be a
3
function of x and z. In the following discussion, the problem is simplified by assuming U is uniform
in the vertical and horizontal directions.
A regular wave on a layer of depth h in the presence of a constant current U, is represented by a
surface elevation
(,* (x, t) = a cos (kx - ut)
(4)
and linearized wave velocities and pressure which are given by
ii TJ
\ coshfc(z + /i)
,.
- (kU - u) a
. ' —- cos (kx - ut) ,
(5)
51X1 IJ rbr*
.,_.
. sinh/:(r + /t) . ,,
,
/a,
w = - (kU - u) a
. . , ,—- sm (kx - ut),x
(6)
suih kh
P ,, rr
^2 coshk(z + h)
.,
.
._.
p = -pgZ + f a (kU - uY
, ,V
' cos (kx - ut).
(7)
k
sinh kh
In thosc formulation, the observed frequency and wave number are relatcd by the linear dispersion
relation
u
=
u = n (k) + kU,
(8)
with fi (k) the relative (or intrinsic) frequency, i.e. in a frame of reference moving with the current:
_ ,,.
n(fc)
3.1
E x p r e s s i o n for t h e e n e r g y
, /otanhfc/i
._.
{9)
*—'
= *V
flux
As explained in the introduction, one of the basic equations is that of the energy flux, which we will
now derive in detail. The equation for the energy flux in the horizontal direction is
\u (\pu2
F = j_
+ pgz + p\\ dz.
(10)
By substituting (4)-(7) into (10) and taking tcrms upto and including O (a 2 ), we obtain
F = F(k,U,a)
= E{a) {Ro{k)-i-U + Ri(k)U
+ R2(k)U2}
+ lphU3,
(11)
where
F(a)
-pga2,
=
2
Rl{k)
=
fc
V
^h2kh
+
sinh2fc/i
r
2n(*)'
The term £ (a) is known as the mcchanical energy of a regular wave, the term Ri (k) E (a) is called
the radiation stress and Ri (k) E (a) U is interpreted as the work done by the current U against the
4
radiation stress of the waves ([6]). The term Ro (*0 + U is equal to dw/dk, i.e. the group velocity, in
which u is given by (8).
The energy flux induced by the initial current is obtained by substituting a = 0 and t/ = f/0 in (11):
FuQ = \ph{UQf.
(12)
In the absence of current, suppose that the wave has amplitude <2Q and that the frequency is u,
the same as in the presence of a current since the frequency is enforced by the wave maker. The
corresponding wavenumber fco is then related to the frequency by u = Q (ko), and the energy flux is
found to be
Fso =
=
F(ko,QtaQ)
R0(h^)E(fl0)1
(13)
Rewriting (2), we have
F{k,U1a) = \ph{Uo)3 + F{koiOla0).
(14)
This relation can bc rewritten like
'a0(u)\2
a(w)*/
_ {Ro (k) + U + Ri (k)U+R2
~
fifl(M
(k)U2} + K
(15)
in which
h.
K-M*-™*)ga
(16)
With an additional rclation for the current, to be derivcd next, the wave amplitudes can be related.
3.2
E x p r e s s i o n for t h e m a s s - t r a n s p o r t
To find the expression for U in the formula above, we usc rclation (1) with an approximation for the
wave mass-transport as follows.
Whitham ([11]) derivcd the mass-transport velocity for irrotational flow with uniform main stream U:
total mass-transport velocity
=
U + wave mass-transport velocity
=
U +
* * * \ M * *
( 1 7 )
h
in which the overbar denotes time-averaging over a (characteristic) wave period. Below, the wave
mass-transport velocity is denoted by Uw. We first consider a regular wave, and include terms of
second order in the ampütude a, that is given by
£
<f>
— a cos (kx — ut) + a?n1 cos 2 (kx - urt),
aCl(k) coshk(z + h) . . .
.
, „, /
• •, . „ / .
^
2
= —-*-*•
r-r-r,—- s i n (kx ~ w i ) + a Pï cosh 2k (z + h) s>n 2 (kx - ut),
k
smh kh
with
5
(18)
„n\
(19)
By substituting this in (17), the wave mass-transport vclocity is found to be
E (
Zo ocosh4jfc/i- ±\
Uw = - j - v - 1 + -a2k2
i
6s
c(k)h y
4
sinh kh
J
v(21)
J
where c(k) = Cl (k) fk is the phase velocity and E = -pga2; the direction of the wave mass-transport
is in the direction of the wave propagation. In figure 6, the dimensionless wave mass-transport velocity
is denoted by Uw ' and Uw * is plotted versus k*. We use nondimensional quantities denoted by an
asterix, i.e. k' = kh, a' = a/h, Uw~ = Uw/y/gh~, u' = u/^/g/h, and took a* = 0.05 to produce the
figure The region of shallow water is k' < 7r/10 and the region of deep water is k' > TT/2. In the deep
water approximation, i.e. for large kh or smail amplitude wave, the wave mass-transport velocity (21)
can be approximated by
u
-=*m-
(22)
This result is for a rcgular, monochromatic, wave. For an irregular wave, being a superposition of
various regular waves, the nonlinear terms makc it difficult to dcrive the second order expressions in
the wave mass-transport. Therefore, we approximatc the wave mass-transport velocity for an irregular
wave by taking expression (22), in which E is the total energy, i.e. the arca below the spectrum, and
replace the velocity by the phase velocity at the pcak frequency: c = Upeak/f^peak where /cpeaJt is the
corresponding wavcnumbcr.
3.3
Wave prediction
For a regular wave, substitution of (21) in (1) and then in (15) leads directly to a relation betwecn
the wave amplitudes a$ and a.
For irregular waves the situation is more difficult, and we simplify it in a crude way as a superposition
of regular waves in the following way. Supposc So (w) is the wave spectrum in the absence of currcnt,
and S (w) is the wave spectrum affected by the currcnt. In order to prodict So (w) from a given S (w),
we discretize the frequencies in bands with width Aw, and the spectrum S (u;) by superposition of
regular waves at frequencies Uj and with amplitudes of the same energy:
a {ujjf = 25 (uj) Au.
(23)
Then, relation (15) can bc used for each of the constituent waves when the simplified wave masstransport (22) is used. The the relation between the discretised spectra is obtained from
4
Comparison of models with experiments
In this section we comparc various models with rcsults of experiments.
We will considcr experiments by Thomas [10] for rcgular waves, and by MARJN [8] for irregular wavcs.
In these experiments the initial currcnt £/n in the absence of waves; and the amplitude of the waves o in
the presence of current, which of the amplitude an without current, were measured. This means that
in the formula we can assume f/n and a to be given, and then calculate the amplitudes an and compare
them with the experimental data. In doing so, it is clear that at the same time we have to exploit the
equation for energy-fiux and the equation for mass-transport. The way how these equations are used
or approximated will define several model equations that we will describe first.
6
4.1
Various model equations
The first model is the present model: this is defined by equation (15) together with (21) for regular
wavcs, and (22) for irregular waves.
The sccond model, denoted by LHS, is the model dcrived by Longuet Higgins and Stewart [6]. They
discussed waves in infinite channels or open seas with deep water approximations. The wave energy
density flux given by equation (5.5) in [6] is
E(a){Ro(k)
+U+
Ri(k)U}.
It relates the amplitudes in the following way:
(ao(u>)\2
\a(u)J
flo(fc)+£r+fli(fc)g
Ro{ko)
=
(24)
We use relation (1) with the approximation (22). Both using (21) and (22) in the present model imply
that the present model is different with the LHS-model in second order to the amplitude, a (w).
The third model, denoted by cgE, is often used in engineering practicc and is based on the assumption
that the energy flux is the wave energy times the group velocity. This implies that for a regular wave
/aoHV
Ro(k) + U
For the current, we use relation (1) with the approximation (22). This model is applied, for example,
by Wichers ([13])
Compared to the LHS-model, in the cgE-model the radiation stress is neglected. This stress is of the
same (second) order conparcd to the other terms.
Another model, denoted by CWA, is based on the conservation of wave action. This model was
used by Thomas ([9]) for his laboratory experiment. The CWA-model express conservation of waveaction between so-called rays (wave-group lines) is a different energy approach for waves on large-scale
currents. This concept was introduced by Bretherton and Garret ([2]), based on Whitham's averaged
Lagrangian ([12]). This conservation equation statcs that the total wave-action (or corresponding
wave energy) between any two group lines remains constant. A simplified explanation of this method
is presented by Thomas and Klopman ([10]), and an extensive discussion on its applications to wavecurrent interaction is given by Jonsson ([4]). In the linear problem, the conservation of wave action
relates the amplitudes in the following way:
(aoHX2
\a(u)J
=
n(fr)(flo(*)+tQ
n(fc)
Ro(ko)
(26)
Wc use relation (1) with the approximation (22).
Thomas's experiment modelled waves originating on still water meeting a uniform current, which
inercases in strength duc to upwelling from bclow. For MARIN experiment, current is also generated
in a similar way (by inflow and outflow from the floor). This justifies the use of relation (26); this
relation represents the situation of waves on slowly varying current, for which the current strength
gradually increases from zero.
In the final section we will comment about another, numerical, model that leads to completely different
predictions.
4.2
Regular waves
We use data from laboratory measurements by Thomas [9]. Thomas reported about regular wavcs
on uniform currents of different strength. In the experiment the water depth is 0.57 m, the wave
7
amplitude in the absence of current is 9.18 mm, so approximately 2% of the waterdepth, and the wave
length in the absence of current is 2.261 m, corresponding to a frequency of u = 5.0059 rad / s.
The current was mcasured using laser Doppler anemometry at some positions over the dcpth; from the
description of the experiment it becomes clear that the current is measured in the absence of waves,
so f/o- The uniform current is taken as the depth-average of the measured currents. As reported ) the
experimental errors are accurate within 1% of the measured quantity.
In table 1 we display the relative error in predicting the initial wave amplitude, aop and comparc it
with the experimental value, aom, assuming the initial current UQ and wave amplitude a in presence
of current to be given. The relative error is given by the following formula:
Relative error =
Q
° p " Q ° m x 100%.
Table 1 shows that the present and the LHS-model givc almost the same prediction, which could
be expected since higher order erTccts are neglegible for this experiment with small amplitude waves.
The error inercases with increasing current. The performance of the CWA-model is comparable; the
cgE-model produces much largcr errors.
4.3
Irregular waves
We use rcsults from the MARIN experiments {[8]) for which the laboratory set up has been describcd
in §2. Sincc the wave energy spectrum is different at different positions in the basin, obviously the
calculatcd wave mass-transport will also vary. Spatial variations of the calculated wave mass-transport
result in spatial variations of the calculated current U.
For cach position we predict the spectrum Sb(w) from the measured spectrum 5(w).
Figurc 7 shows the prediction of So (w) for UQ = 10 cm / s, and figure 8 for UQ = 20 cm / s using
the model presented in this paper. For comparison with the other models, the prediction using the
cgE-model is also shown in those figures.
From these graphs we observe that the present method gives a rather good prediction in the low
frequency range (below the peak frequency) but less so for the higher frequencies (above the peak
frequency). Except at distance 7.67 m from the wave maker, the method predicts quite well the value
of the spectrum at the peak frequency for an initial current 20 cm / s. This prediction is less well when
the initial current is smaller, lOcm/s. We do not have a good explanation for this, and more precise
measurements of the vertical sructure of the current before and during the interaction would be needed
to investigate this in more detail. For higher frequencies, with low energy content, the prediction is
rather poor, even leading to unrealistically negative values of the spectrum at one position. The reason
for this is that the related wave amplitudes are very small and frequencies and wave numbers rather
large, forcing the quotiënt (15) to become negative because of the presence of the additional term K.
To quantify the rcsults in more detail, and to compare the performance between the different models,
we define two norms for the diffcrencc between the predicted and the measured spectrum. The first
measurc is of integral-type, comparing the area of the wave spectra; while the second one is a pointwise
error which is calculated by taking a weighted average.
1. Relative wave energy error
_ (total estimated energy) — (total experimental energy)
x 100%.
(total experimental energy)
(27)
2. The local relative errors are large where the energy is low, i.e. in the lower and higher frequency.
This is due to noise in the measurements and the spectral method of analysis (cross-talk), making
smaller values of the spectral density less reliable. This can be taken into account by taking a
weighted average, with the weight at each frequency equal to
£O(WJ) measured
^ 5 o ( w j ) measured'
So, the relative pointwise error
=
E l ^ o ( " j ) predicted -SQ(UJ)
measured[ ^ 1 0 Q %
^ 5 O ( W J ) measured
The differences bctween So (w) from laboratory measurement and £o (w) predicted are given in table
2 and tablc 3, for the two relative errors.
From these results we conclude that the errors of the present model are slighly smaller than the LHS
and the CWA-model. The cgE-model performs even better in predicting the total energy. For higher
velocity of current, the present model perform slightly better than the other models in predicting the
shapc of the spectrum. This is shown by figure (8). The cgE model over-predicts the spectra at some
frequencies, but under-predicts at other frequencies. Doing the integration balances the over-predicted
and the under-predicted values.
5
5.1
Sensitivity analysis of the model
Regular waves
The experiments provide us with good information about the wave. We have two sets of wave spectra:
one effected by the current and the other which is current-free. Since our aim is to predict the currentfree wave spectra, dctailcd information about the current is needed. Ideally, an experiment should
measure the current before and during interaction with the wave. Since the experiments which we use
here do not provide us with detaüed information about the vertical structure of the current effected
by the wave, then we have to make a prediction. The actual current in the basin could be different
from this prediction. This would influence the error of the wave prediction.
The analysis, which will be discussed here, is made for investigating the sensitivity of our wave
prediction to error in the current prediction. One way of doing so, keeping in mind that the vertical
structure of the current was not measured in detail, can be done by taking another value for the
current, slightly different from the prediction. By comparing errors of the two wave predictions which
are obtained from those values of current, we can decide whethcr r not a precisc current prediction is
needed in the model.
The sensitivity analysis is donc in the foilowing way. The actual current in the basin is predicted by
rclation (1), and it is denoted by U. Then
U = Uo — wave mass-transport,
where Uo is the initial current and the wave mass-transport is given by (21). Wc takc another value
denoted by U', slighly different from U. Thcy are relatcd as follow:
U' = U+ At/.
(29)
The quotiënt (ao (w) / o (w)) is denoted by Q. By substituting (29) in (15), and taking Uo andfcoas
constants, we can express Q (U') as follow:
Q{U') = Q{U) + ±Qt
9
where approximately AQ is
• AC/.
(30)
The term AQ represents the effect to the wave prediction, if a perturbation AU is taken in the currcnt
prediction. In the previous section somc altemative models have been discussed. In (??) the term C\
represents AQ of the cgE-model (25), and C\ + Ci represents AQ of the LHS-model (24).
Different from those models, the predicted amplitude obtained from the present model depends
strongly on AU. In table 4 the relative error of the wave prediction, due to perturbation AC/, is
given. For the first row we take AU = 0, and for the second row AC/ = 0.75 mm / s. This last value is
precisely the wave mass-transport, so in that way we compare the relevance of taking this transport
into account, i.e. for AU = 0, or not, i.e. for AU = 0.75mm/s.
The experimental value is taken from the experiment by Thomas [9], with UQ = 203 mm / s .
The table shows that, compared to the othcr models, a precisc information of the currcnt is more
important for the present model.
For the example given by table 4, the calculatcd values for each term in (??) are
C1
+
C2
=
2.25,
*h{. . % = 43.89.
gRo (ko) a2
The last term of (??) dominates AQ of the present model. This term appears from the term K in
(15) and appears naturally from relation (2), which is a direct derivation of conservation of the energy
density flux. This calculation shows that the sensitivity of the present model is caused by this term.
5.2
Irregular w a v e s
In order to study the sensitivity of the model on the value of the currcnt, considcr as bcforc a
perturbation AU from Uc.
In table 5 an cxample is given, with AC/ = 0 and AU — 0.41 m m / s . The experimental data is taken
from experiment by MARIN at distancc 20.53m from the wave maker; with C/o = lOOmra/s. The
relative wave energy error is given by relation (27).
As for regular wave, the present model depends strongly on AU. This dependency is caused by the
presencc of K in the relation for the wave amplitudes. An irregular wave can be approximated by
a superposition of regular waves. The amplitude of each regular wave is related to the value of the
spectra by (23). If the spectra's valuc around a certain frequency is small, thcn the relatcd wave
amplitude is aïso small. Each band of the spectrum is affectcd by the same currcnt. This makes
the term K in (15) becomes O I
« I ^ or
eacn
J» w n i c h becomes large when the related wave
amplitude is small.
6
Discussions a n d Conclusions
In this paper we discussed the intcraction bctween waves and currents as these are gencratcd in hydrodynamic laboratories. We argued that the closedness of the systcm makes it different from real-ocean
10
situations, and derived two basic laws, namely energy flux and mass-trans port, that enable to establish
a relation between the waves and current before and after interaction. Based on the basic equations,
we derived this relation, and compared it to some other models known in literature. To compare their
predictive performance, we compared the result of the various models with laboratory experiments.
For regular waves, results of careful experiments by Thomas are available; the conclusion is that for
such waves the proposed model performs as good as some other known models, but differences with
for instance the LHS-model are rather small since the additional higher effetcs are neglegible for the
small amplitudes in the experiments. For regular waves the model based on an expression for the
flux as group velocity times energy performs rather poorly. The situation, however, is quite different
for irregular waves. Results are reported here for experiments performed at MARIN, showing the
correct qualitative interaction properties. The comparison of the models, now show that the proposed
model performs better than some comparably good models for regular waves, but that, somewhat
surprisingly, the model with the group velocity flux performs even better.
The accuracy of the prediction for wave spectra or wave amplitude could be affected by the accuracy of
the measurement for the initial current, since the relation for the current is coupled with the relation
for the wave amplitude. In the measurement by MARIN, current is measured at one position only,
and this value is taken as the value of the uniform current. A more precise measurement was done by
Thomas. The current was measured at some positions over the depth, and the value of the uniform
current takes the value of the depth-averaged current. All models, except the cgE-model, predict
Thomas's measurement better than that of MARIN.
By observing tables 4 and 5, several conclusions are obtained. Unlike the other models, both the
predicted amplitude and spectra obtained from the present model strongly depend on the value of the
current. A precise prediction of the current is important for the present model. In predicting irregular
wave by the present model, taking into consideration the wave mass-transport (i.e. AU = 0) gives
better result. On the contrary, as shown in table 4, different case happens to regular wave.
Unfortunately, too few experimental results are available to make a more thorough investigation and
to arrive at more definite conclusions. Further experiments are planned at MARIN in the new wavecurrent basin.
We conclude with a remark about another model that has not been taken into consideration above.
This model is numerical in nature, and has been studied by Baddour and Song [1]. The model is for
regular waves, and solves simultaneously the current, the wave height, wave length, and the water
depth during interaction. These four variables are calculated numerically by solving a set of four
nonlinear equations. These nonlinear equations are derived to satisfy, to second order, conservation
of mean mass, momentum, and energy flux, as well as a dispersion relation on the free surface. The
expressions in the conservation equations are simplified by taking the sum of contributions from
wave-only and from current-only quantities. For the model presented here, the water depth is kept
unchanged during the interaction, and we predict the wavelength by using the known (Doppler-shifted)
dispersion relation. Furthemore we also use the conservation equations for mass and for energy flux.
However, the results of the computation by Baddour and Song show that, contrary to our expectation
based on relation (1), the current increases when flowing in the direction of the wave and decreases
when flowing in the opposite direction. This should lead to the conclusion that the system calculated
by Baddour and Song is not closed, and is of no direct relevance for laboratory situations.
Acknowledgement 1 The authors thank Dr. Gert Klopman from Albatros Flow Research for his
comments and the Maritime Research Instüute Nctherlands (MARIN) for their support.
References
[1] Baddour, R.e-, & Song, S., 1990. On the interaction between waves and currents, Ocean
Engineering 17 no. 1/2, pp. 1-21.
11
[2] Bretherton, F.P. & Garrett, G.J.R., 1968. Wavetrains in inhomogeneous moving media.
Proc. Roy. Soc. A 302, pp. 529-554.
[3] Iskandarani, M., Liu, P.L.-F. 1993. Mass transport in wave tank, J. Waterway, Port, Coast,
Ocean Engng 119 No.1, pp. 88-104.
[4] Jonsson, I.G., 1990. Wave-current interactions. In: 'The Sea\ Ed. B. Le Mehaute & D.M.
Hanes, Ocean Eng. Sc, Vol 9(A), John WUey (New York), pp. 65-120.
[5] Longuet-Higgins, M.S. 1953. Mass transport in water waves, Phü. TYans., R. Soc. Lond. A
245, pp. 535-581.
[6] Longuet-Higgins, M.S. & Stewart, R.W., 1960. Changes in the form of short gravity waves
on long waves and tidal currents, J. Fluid Mechanica 8, pp. 565-583.
[7] Longuet-Higgins, M.S. & Stewart, R.W., 1960. The changes in amplitude of short gravity
waves on steady non-uniform currents, J. Fluid Mechanics 10, pp. 529-549.
[8] MARTN, 1994. Bepaling Golf-Stroom Interactie en Flapkracht, Report No. 820002-1-GT.
[9] Thomas, G.P., 1981. Wave-current interactions: an experimental and numerical study. Part 1.
Linear waves, J. Fluid Mech. 110, pp. 457-474.
[10] Thomas, G.P. & Klopman, G., 1996. Wave-current interactions in the nearshore region. In:
'Gravity waves in water of finite depth', Ed. J.N. Hunt, Advances in Fluid Mechanics, Vol. 10,
Computational Mechanics Publications, Southampton, U.K., Chapter 7, pp. 255-319.
[11] Whitham, G.B. 1962. Mass, momentura and energy flux in water waves, J. fluid mech. 12,
pp. 135-147.
[12] Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley-Interscience, New York.
[13] Wichers, J.E.W., 1988. A simulation mode! for a single point moored tanker, PhD thesis, TU
Delft.
a
mm
9.96
10-61
11.63
12.02
mm/s
-59.7
-116.2
-159.8
-203.0
U
mm/s
-60.18
-116.76
-160.49
-203.75
present model
0.82%
-0.86%
1.13%
-3.79%
Relative error
LHS-model cgE-model
0.82%
3.06%
-0.87%
3.87%
1.12%
8.44%
-3.80%
6.09%
CWA-model
1.29%
0.30%
3.16%
-0.67%
Table 1: The relative error between the predicted and measured wave amplitude
12
(a)
1
time (s)
(b)
"g
time (s)
Figure 1: Horizontal vclocitics, for initial current f/o = 2 0 c m / s . Shown are (a) at part of the basin
with 1 m depth, and {b) with 2 m depth. For both, the grey Unes are records of the horizontal velocity
and the black lines are the mean values.
Distance from
the wave maker
3.53 m
7.67m
11.73m
20.53 m
3.53 m
7.67 m
11.73m
20.53 m
Uo
cm/s
10
10
10
10
20
20
20
20
U
cm/s
9.940
9.944
9.949
9.945
19.956
19.953
19.958
19.959
present
model
8.127%
12.308%
11.495%
10.928%
-3.270%
14.063%
9.860%
0.986%
LHS
model
10.030%
14.337%
13.498%
12.951%
2.270%
20.800%
16.193%
7.030%
cgE
model
4.956%
9.037%
8.242%
7.756%
-6.186%
10.787%
6.535%
-1.814%
Table 2: Relative wave-energy error; irregular wave
13
CWA
model
9.770%
14.087%
13.228%
12.640%
2.260%
20.820%
16.287%
6.901%
3.53 m
7.67 m
11.73 m
20.53 m
Figure 2: Evolution of the wave spectra along the tank in the absence of currcnt, horizontal axcs
is frcquency u (rad/s) and vcrtical axcs is S(w) ( c m 2 s / r a d ) . The figure in cach plot is the total
energy, the arca below the spectra.
Distance from
the wave maker
3.53m
7.67m
11.73m
20.53 m
3.53m
7.67 m
11.73m
20.53 m
Uo
cm/s
10
10
10
10
20
20
20
20
U
cm/s
9.941
9.944
9.949
9.945
19.956
19.953
19.958
19.959
present
model
10.353%
14.969%
13.208%
13.265%
9.784%
21.951%
18.308%
8.413%
LHS
model
10.623%
15.602%
13.773%
13.691%
9.442%
23.383%
17.854%
9.090%
cgE
model
6.274%
11.257%
9.014%
8.749%
10.755%
15.491%
18.851%
8.951%
CWA
model
10.466%
15.478%
13.593%
13.443%
10.077%
23.686%
18.193%
9.229%
Table 3: Rclativc pointwise error; irregular wave
Aa
(mm / s)
0
0.75
Relative error a ° ^ 0 m x 100%
present model LHS-model cgE-model CWA-model
-3.79%
-3.80%
6.09%
-0.67%
-0.78%
-3.65%
6-20%
-0.54%
Table 4: The rclativc error for two different values of current; regular wave
14
3.53 ra
7.67 m
11.73 m
20.53 m
Figure 3: As figure 2 in the presence of a current U = 10 cm / s.
AU
mm/s
0
0.41
present model
0.986%
8.812%
Relative-wave energy error
LHS-model cgE-model
7.030%
-1.814%
7.094%
-1.770%
CWA-model
6.901%
6.966%
Table 5: Rclative wave-cncrgy error for two different values of current
15
3.53 m
7.67 m
11.73 m
20.53 m
Figurc 4: As figurc 2 for currcnt U = 2 0 c m / s .
16
3.53 m
7.67 m
11.73 m
20.53 m
Figure 5: Evoiution of the wave spectra along the tank with and without current: — U = 0,
£7 = 10cm/s,
[l = 2 0 c m / s .
a* = 0.05
,
ÜL
0.00450004
0.0035
^ ^ ^ ^
^ ^ ^
0QO300025
s^
0002
00015
L/-
0 0 5 1 1.5 2 2.5 3 3.5 4 4 5 5 5 5 G 6 5 7
k*
Figure 6: Plot of the nondimcnsional mass transport velocity (Uw *) vs the nondimcnsional wavenumber (/:*) for a' = 0.05, calculated by using expression (21).
17
3.53 m
7.67 m
11.73 m
20.53 m
S0 (W) from laboratory measuremcnt,
Figure 7: The prediction of £o(w) f° r #o = lOcm/s. —
SQ (w) from the present model,
SQ (W) from the cgE model.
18
7.67 m
3.53 m
3
4
5
20.53 m
11.73 m
Figurc 8: As figure 7 for UQ = 20 cm / s.
19
6
?