Applied Ocean Research 33 (2011) 88-99
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Applied Ocean Research
journal homepage: www.elsevier.conn/locate/aporo ' c k A N R E S E A R C H
Forces and moment on a horizontal plate due to regular and irregular waves in
the presence of current
V. Rey.J.Touboul*
Laboratoire de Sondages Électromagnétique de I'Environnement Terrestre, Institut des Sciences de Ingénieur Toulon - Var, Avenue Georges Pompidou, BP 56, 83162 La Valette Cedex, France
A R T I C L E I N F O A B S T R A C T
Article history: Received 1 October 2010 Received in revised form 23 December 2010 Accepted 1 February 2011 Available online 1 March 2011
Keywords: Coastal structure Submerged plate Wave scattering Current
Partially reflective wave absorber
This worl< presents an experimental study of a submerged plate used as a breakwater for coastal areas protection. Questions addressed concern the influence of current on the reflective power of the plate, and its influence on the hydrodynamic loads exerted on it. Results concern both monochromatic and irregular waves. Generally speaking, an influence of the current is found, changing the reflecting power of the structure up to 50%. A homogenized behavior of the loads and moments is found in the presence of currents, meaning that the load values become less sensitive to the frequency. Furthermore, the influence of waves reflected by the wave absorber, representing partially reflective conditions at the shore, is found to be of same order in the absence of current. In any case, the linear behavior of the breakwater is emphasized through the irregular waves approach.
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1. Introduction
In coastal areas, rapid changes in the seafloor are known to have a strong impact on tiie dynamics of water waves. St:rong variations in bathymetry, or submerged obstacles are responsible for a pardal reflection of incident wave energy [ 1 ]. Several studies took advan-tage of this phenomenon to design coastal structures protecring coastal areas (see e.g. [2-4] and references herein). Among coastal structures, the submerged plate is especially indicated for steeply sloping shorelines [5]. It provides an economical solution for pro-tecring areas without a significant tidal range.
A large number of studies have been carried out in the last thirty years to analyze the behavior of such a submerged breakwater. Pioneering works were conducted by Durgin and Shiau [6]. They estimated experimentally and theoretically the loads exerted on the structure by regular waves. However, they did not provide data concerning reflecrion of waves due to the structure. This aspect was initially introduced in the framework of shallow water and long waves theory [7]. Later works tried to extend analytically and experimentally these results in a more general case. Among them, one may cite Grue and Palm [8) who studied wave reflection by deeply submerged bodies, including the flat plate, by means of integral equations. Patarapanich [9] and Sturova [10] described the power of reflection of the plate as a funcrion of the wavelength.
* Corresponding author.
E-mail address; julien.toubouKaiuniv-tln.fr {]• Touboul).
0141-1187/$ - see front matter © 2 0 1 1 Elsevier Ltd. All rights reserved, doi; 10.1016/j.apor.2011.02.002
They showed analytically the presence of maxima and minima in reflection, due to interference processes above the plate. Patarapanich [11] provided analytically a detailed description of the loads exerted on the structure, extending results of [6].
The description of the maxima and minima of reflecrion was detailed analyrically and experimentally by Le-Thi-Minh [12], who showed the necessity to take evanescent modes into account to describe the phenomena.
Other studies focused on the mechanisms responsible for en-ergy dissipation i n the vicinity of the plate. Among these mecha-nisms, one may cite the generation of harmonics downstream of the plate. Dattari et al. [13] showed experimentally that an energy transfer from the fundamental mode to higher harmonics occurred while crossing the plate. This aspect was invesrigated experimen-tally by Brossard and Chagdali [14] who introduced a method based on a moving probe, amenable to dissociate free modes f r o m bound modes. A recent study concluded that the generation of free har-monics downstream of the plate was due to the resonant interac-don of bound harmonics on top of the plate, where the water is shallow [15]. Quantitative results were confirmed experimentally and numerically by Liu et al. [16].
Another mechanism responsible for energy dissipation is the generation of vortices at both edges of the plate, observed experi-mentally by Boulier and Belorgey [17]. This phenomenon cannot be studied i n the framework of potendal flow theory, and was ignored in the studies mentioned above. However, Huang and Dong [18] and more recently [19] conducted numerical simulations based on the f i i l l Navier-Stokes equations, and concluded these vortices had
little role on the free surface dynamics, even i f their interaction w i t h the seabed could be strong.
None of these studies took the presence of currents into ac-count. However, in coastal zones, currents of order 10~^-10° m/s are often generated, due to tides, wave breaking or local or global circulation. Their interaction w i t h water waves is known to be i m -portant [20]. The impact of currents on the efficiency of wave bar-riers for reflecdng and damping wave energy can thus be really important.
Several authors studied analytically and numerically the inter-action of currents w i t h submerged bodies. Among them, one may cite Grue and Palm [21 ] who studied the influence of the current on the problems of diffraction and radiation of waves by submerged circular cylinders. Mo and Palm [22] extended these results for el-lipdc cylinders. These authors found a strong influence of the cur-rent on waves amplitudes, and loads exerted on the submerged body. The equivalent problem of a body moving in calm water was addressed by Grue et al. [23], who studied the efficiency of a ver-dcally oscillating submerged foil w i t h forward speed, used as a propeller. IVIore complex geometries were considered by Nossen et al. [24]. I n the framework of a fixed submerged plate, former w o r k by Rey et al. [25] invesdgated experimentally the impact of currents on the efficiency of the reflective power of the device for both monochromatic and bichromatic waves. They emphasized that a Doppler shift was responsible for an attenuation of reflected wave energy inversely proportional to the current velocity. How-ever, the range of efficiency of the plate remained unchanged. The main conclusion concerned the linear behavior of the breakwater, and its relevance to model the phenomenon.
The present work, conducted in the Ocean Engineering Basin FIRST (La Seyne sur Mer, France), aims to extend previous results to spectral wave signals, for both wave scattering and pressure efforts on the structure. This approach is helpful to conclude on the relevance of using linear theory to describe the efficiency of the wave breaker i n the presence of current. Furthermore, an effort is made to estimate the loads exerted on the structure, extending previous work by Patarapanich [11] to more general wave conditions, and under the influence of current.
2. Experimental setup and measurements
2.1. Experimental set-up
The Ocean Engineering Basin (BGO) FIRST is designed to conduct ocean and coastal engineering model studies. Its useful length is 24 m, while its effecrive w i d t h is 16 m. The bottom of the basin is mobile, allowing to adapt the bathymetry. It can be inclined up to ± 7 % for coastal engineering studies w i t h variable bottom topography. The maximum water depth is 5 m, although a 10 m depth pit, of diameter 5 m, can be used to study structures in deep water. Water waves can be generated by means of a surface wavemaker, covering the entire w i d t h of the basin. The wavemaker is composed of horizontally oscillating cylinders, allowing to produce regular and irregular waves in the presence o f currents. The waves' frequency extends f r o m 0.3 to 1.4 Hz, while the maximum crest to trough wave height is 0.8 m . At the other end of the basin, a 7 m parabolic permeable wave absorber dissipates wave energy. A carriage can be moved over the whole basin's useful length. It allows the quick installation and repositioning of the instrumentation. The current generation is provided by water pumps, calibrated by means of velocity measurements i n the water.
In the framework of this study, the model is constituted by a 1.53 m long plate, 0.1 m thick. Its topside immersion depth is 0.5 m. To avoid three dimensional effects, the plate extends over the complete w i d t h of the basin. The mobile bottom has been raised
to keep a constant water depth of 3 m. The coordinate system is Cartesian. The (Ox) axis is parallel to the waves' propagation direction, along the basin, the origin being the upstream edge of the plate. The (Oy) axis is parallel to the basin w i d t h , its origin being the axis of symmetry of the basin. The (Oz) axis is vertical, oriented upwards. Its origin is the still water level.
Waves propagating across the plate are considered w i t h o u t cur-rent, and in the presence of a current flowing in the wave prop-agation direction. Several wave conditions are considered. Both monochromatic and irregular waves are considered. Monochro-matic waves present various amplitudes for periods extending from 1.1 to 3.2 s. Irregular waves have peak periods Tp = 1.4,1.8, 2.1 and 2.3 s, w i t h an enhancement factor y = 1 cor-responding to a Pierson-Moskowitz spectrum. For every case, the propagation is considered without any current, and w i t h a current velocity equal to 0.3 m/s. A list of the experiments conducted is detailed in Table 1.
The synchronous instrumentation is composed of 18 wave probes ( W „ ) „ = i i s , 3 velocity propeller sensors (Cn)n=\ 3. and 12 pressure sensors (Pn)n=i 12. Resolution o f t h e pressure sensors is 0.2 mbar. Locations of the sensors are detailed in Table 2, while the sketch of the experimental set-up is summarized in Fig. 1. 2.2. Reflected waves measurement
The characterization of the breakwater efficiency is based on the estimate of the reflection coefficient R, corresponding to the ratio of reflected wave amplitude to incident wave amplitude. A method to measure this coefficient was initially introduced by Goda and Suzuki [26]. This method was based on the measurement o f t h e phase shift between incident and reflected waves propagating between two probes. However, this method presented some d i -vergences for specific values of the ratio between wavelength and distance between the probes. It has been significantly improved by Mansard and Funke [27], who used a larger number of probes (at least three), and a least square method to minimize the error. In the presence of current, the Doppler shift induced by the non homo-geneous media imposes a correction in the method, as mentioned by Suh et al. [28]. The method is based on linear theory. Far from the structure, where local modes (evanescent modes) can be ne-glected, the water elevation ;;(x, t) can be understood as two Airy waves of frequency co = 2n/T propagating in opposite directions along the (Ox) axis. Thus,
?7(x, f ) = 5?e[(a,e'''"''' + are"'^''+*')e-''"'], (1) where a, and are real numbers corresponding respectively to
incident and reflected wave amplitude, and where /<+ and i<~ are algebraic values of incident and reflected wave numbers respec-tively, obtained by solving the dispersion relation
(w + ii^Uf = gii^ tanh(l;*/i). (2)
Let x„ be the location of probe W„. The theoretical expression of water elevation at probe Wn reads
rj(x, t) = m(aie"-''^"+''"^ + a,e'<''"^'+^"+^')e-""], (3)
where A+ = i<'^(Xn - X i ) , A~ = ii~(Xn - x O and ip is the phase lag of the reflected wave. By introducing complex amplitude, Eq. (3) can be rewritten
nix, t) = 3ie[(biê'^" + bré'^")e-"'"'\. (4)
Each probe Wn delivers an elevation signal o f t h e f o r m
,,„(f) = ^^e[/l„e'<«'"-"")] = J?e[B„e-'"^], (5) where amplitude A„ and phase <pn can be obtained by means of
90 V. ReyJ. Touboul /Applied Ocean Research 33 (2011) 88-99
Wl W2 W3 W4 Ws Wa W? Ws W9 Wio Wii Wi2 Wi3 W n Wis Wj6 W„ Wis
P2 • Cl P J P I P5 Pe* 0,5 m P7 I Pl ' 0.1 m 3 m Ps P9 PlO Pll Pl2 •Cl • Cs 1.53 m
Fig. 1. Side view of the experimental set-up.
Let E be the error committed by understanding the signal as two plane progressive counter propagating Airy waves. It follows as
bre'- (6)
The solution o f t h e system dE^/dhi = dE^/dbr = 0 w i l l minimize the error, and provide the value of complex amplitude of incident and reflected wave. After some algebra, we have
where 52S3 - S4S12 S1S4 — S3S12 ( 7 ) (8) n=1 n=1 S3 = X : B n e ' " " , S4 = ^ B „ e ^ ^ " , S12 and Ss = S1S2 - s,
The reflection coefficient is then defined by R = | ör | / 1 I , while the phase lag a t x i iscp = arg(br)-arg(fa,)-t-(/<+-/<:")Xi.The limitation of the method due to wave probes spacing is studied in [25], and is found to be negligible while considenng probes I V ] , W2 and W3. While considenng probes W15, Wis, W n and Wis, the same result is obtained in the absence of current. However, in the presence of the current, the coefficient S5 is nil for several wave periods lower than T = 1.1s, leading to the divergence of the method for these periods.
When considering spectral waves, the same approach can be used. The elevation signal is considered being the superimposition of incident and reflected Airy waves of frequencies a)„. Fourier analysis of the elevation signal provides the values An{co„,) and (pn((»m), and the approach described above can be directly applied, providing a set of incident and reflected amplitudes
(OiCwm), ar(wm))- Following Drevard et al. [29], we introduce a discrete energy density spectra for both incident and reflected waves as e,-r(tt>m) = al^ia>m)/2. The spectral density function can be defined as
S i , r ( « m ) = ^^^^ E ^i-r(<^m+j).
(9)
where p is a smoothing parameter. The reflection coefficient can now be defined as
SriOm)
(10)
In the following, the smoothing parameter w i l l be taken equal to p = 5, when the typical number of modes considered, corresponding to half the number of samples in the time series considered, is close to 6500.
2.3. Validation of tlie hydrodynamical model
While performing experiments involving water waves and current in a closed basin, several possible errors might arise.
The first source of error is the excitation of eigenmodes of the basin. Energy transfer f r o m fundamental mode to eigenmodes is a well k n o w n problem in hydrodynamics. This phenomenon might be responsible for a significant deviation f r o m results obtained in open seas. To check this issue, we present in Fig. 2 the spectrum obtained w i t h monochromatic waves of various frequencies in the presence of current. The choice of the frequencies corresponds to spectral peak of irregular waves. From this figure, it appears that the l o w frequency mode's energy is o f a n order between 10"^ and 10"'^ times the incident energy. Special attention should be given to the frequencies co = 0.42 rad s~' and a) = 1.01 rad s"'. These t w o frequencies correspond to the first eigenmodes of the basin, longitudinally and transversally respectively. It has to be mentioned that the length used to compute the longitudinal eigenmode is the total length, equal to 4 0 m, and not the useful length of 26 m, as mentioned in Section 2.2. In the present study, all the probes are located i n the center of the basin, allowing no disturbance by the transverse mode. However, the result
V. Rey.J. Touboul/Applied Ocean Research 33 (2011) 88-99
Table 1
List of experiments conducted for monochromatic and irregular waves. N° T ( s ) a (mm) U ( m / s ) N" Tp (s) Hs (mm) » ( m / s ) 59 1.4 93 0 60 1.8 91 0 61 2.1 90 0 62 2.3 91 0 63 1.4 86 0.3 64 1.8 77 0.3 65 2.1 87 0.3 66 2.3 84 0.3
emphasizes that the long waves' energies remain weak regarding the incident waves energy. The same analyses were conducted for waves generated in the absence of current. Long waves' energies were of an order between 1 0 " ' ° and 10~^ the incident energy.
The second source of error is the uniformity of the current. In the hydrodynamic model, we assume being i n the presence of waves propagating i n a uniform current, and this issue might be checked. Fig. 3 shows the current velocity distribution as a function of depth, obtained at velocity propeller sensors (Cn)„=i 3. Current velocities are obtained by means of time average of velocities recorded at the propellers. Dashed lines correspond to velocities obtained before the generation of waves, while solid lines correspond to an average obtained in the presence of waves. From this figure, currents appear to be slightly affected by the presence of waves. This is correlated to the physics of the propeller sensors, which are not able to measure negative values of the current. Thus, the mean velocity component due to the waves isn't nil, as it should be. Furthermore, the current appears to be slightly lower for the peak period Tp = 1.4 s. However, the most important information appearing in this figure concerns the uniformity of the current along depth. Deviations f r o m the mean depth value of the current velocity remain lower than 4%. Currents appear to be almost uniform over depth.
3. Wave scattering due to tlie plate 3.Ï. In the absence of current
In this section, we present results of the experiments involving both monochromatic and irregular waves in the absence of current. Fig. 4 shows the normalized wave envelope a/a, = |Bnl/a/ as a function of space, for monochromatic incident waves of period T = 2.1 s and T = 2.2 s. Experimental data are plotted together w i t h the wave envelope obtained through analytical models 1 and 2, described in Appendix. Analytical model 1 corresponds to the solution i n the absence of reflection at the wave absorber, while model 2 takes account of the reflection due to the wave absorber. In the latter case, two propagating components are considered downstream of the plate. These components correspond to the wave transmitted by the plate, and the counter propagating wave, reflected by the absorber. This approach allows the plate and the wave absorber to interact together. From this figure, it appears that reflection at the end of the basin plays a significant role. For the case T = 2.1 s, differences of wave envelope of order 10% are observed upstream f r o m the plate, while this value can reach 20% on its top. For the case T = 2.2 s, a similar difference i n envelope amplitude is observed together w i t h an important phase lag on top of the plate. This phase lag could significandy affect the hydrodynamic loads and moments exerted on the structure. It is interesting to notice that the presence of reflection at the wave absorber modifies the wave properties, even upstream f r o m the plate. This is explained through the role played by the counter propagating component downstream of the plate i n the interference process.
This result raises the question of the role played by the presence of partially reflected waves downstream of the plate. First of all, i t is common that shorelines, or coastal structures being protected by such brealcwaters are partially reflective. The role played by the coastal structures on the brealoA/ater's efficiency is of major interest by itself Secondly, since the purpose of this work is to evaluate the importance of current on reflective behavior and loads exerted on the structure, an effect of 20% cannot be neglected. In this framework, it is essential to have a good characterization of the wave absorber at hand.
Fig. 5 presents the reflection coefficients due to the wave ab-sorber, and the equivalent location of the wave absorber corre-sponding to the phase lag measured. Reflection coefficients and phase lags are obtained using the probes W15, W16, W n and Wis, assuming the weight of evanescent modes to be negligible at this distance f r o m the plate. According to the method presented in
1 1.1 60 0 2 1.1 120 0 3 1.2 72 0 4 1.3 70 0 5 1.4 94 0 6 1.4 120 0 7 1.5 80 0 8 1.6 73 0 9 1.7 77 0 10 1.8 88 0 11 1.9 105 0 12 1.9 125 0 13 1.9 210 0 14 2 38 0 15 2 78 0 16 2 120 0 17 2.1 112 0 18 2.1 220 0 19 2.2 30 0 20 2.2 80 0 21 2.3 110 0 22 2.4 88 0 23 2.4 180 0 24 2.5 80 0 25 2.6 70 0 26 2.7 104 0 27 2.7 145 0 28 2.8 88 0 29 2,8 120 0 30 2.9 134 0 31 3 72 0 32 3,1 55 0 33 3.2 165 0 34 1.1 60 0,3 35 1.2 70 0,3 36 1.3 75 0,3 37 1.4 70 0,3 38 1.5 95 0,3 39 1.6 90 0,3 40 1.7 110 0,3 41 1.8 85 0,3 42 1.8 155 0.3 43 1,9 42 0.3 44 1,9 80 0.3 45 2 90 0,3 46 2,1 50 0,3 47 2,1 78 0,3 48 2,2 95 0,3 49 2,3 85 0,3 50 2,4 80 0,3 51 2,5 87 0,3 52 2,6 75 0,3 53 2,7 70 0,3 54 2,8 87 0,3 55 2,9 105 0,3 56 3 80 0,3 57 3,1 105 0,3 58 3,2 68 0.3
92 v. Rey.J. Touboul/Applied Ocean Research 33 (2011)88-99
Table 2
Location of the wave probes (W„)„=i ig, pressure sensors (P„)„=i n . and velocity propeller sensors (C„)„=i 3.
Sensor VV, IV2 W3 W4 Ws We W , Ws Wg
x ( m ) - 4 - 3 . 2 - 2 . 7 - 0 . 9 - 0 . 3 0 0.165 0.465 0.765
z ( m ) 0 0 0 0 0 0 0 0 0
Sensor W,o W „ W12 W,3 Wu Wis W,6 Wl7 W,8
x ( n i ) 1.065 1.365 1.53 2.13 2.73 3.33 3.93 4.53 5.13 zCm) 0 0 0 0 0 0 0 0 0 Sensor Pl Pl P3 PA Ps Pe P7 Ps Pg x ( m ) 1.53 0 0.165 0.465 0.765 1.065 1.365 1.365 1.065 z ( m ) - 0 . 5 5 - 0 . 5 5 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 5 - 0 . 6 - 0 . 6 Sensor PK Pll P12 Cl C2 C3 x ( m ) 0.765 0.465 0.165 0.765 0.765 0.765 z ( m ) - 0 . 6 - 0 . 6 - 0 . 6 - 0 . 4 - 0 . 7 - 1 . 2 cofrad.s' 10' 10' (£i(rad.s- •'j io' 10" 10' (£i(rad.s- ^) (ü(rad.s-'') Fig.2. Energy spectrum recorded at probe W15 for monochromatic waves of periods T = 1.4 s, T = 1.8 s, T = 2.1 s and T = 2.3 s (From Top-Left to Bottom-Right).
Section 2, the phase lag of reflected wave to incident wave (corre-sponding to the waves transmitted by the brealcwater) is obtained w i t h reference to probe W15. This phase lag can be interpreted as the phase lag induced by a pardally reflecting vertical wall, located at position Labsorber- Results for monochromatic waves and irregu-lar waves of peak periods Tp = 1.4 s, Tp = 1.8 s, Tp = 2.1 s and Tp = 2.3 s (y = 1) are plotted together.
An equivalent location of the wave absorber presents a surpris-ingly linear behavior. Locations vary continuously from Labsorber = 19 m for the shortest waves to Labsorber = 14 m for the longest waves. For sake of reference, the parabolic wave absorber i n the basin extends f r o m 13 to 20 m. It seems understandable that the
shortest waves interact w i t h the top of the absorber, while the longest waves interact w i t h a deeper (and closer) part of the ab-sorber. A solid line corresponds to a polynomial interpoladon of the data, using a polynomial function of degree 5, Labsorber = Q(T). This function will provide the phase parametrization of the wave absorber in the following.
Reflecdon coefficients present a more complicated behavior, as it was expected. The efficiency of the wave absorber depends on strongly nonlinear phenomena. If the dissipative process is wave breaking, the efficiency increases w i t h the steepness of the waves. As a result, the dispersion of the data is more important than it was for the phase. However, the use of a polynomial function of
Or -0.2 • -0,4 • f f » •1.4 • •'•» -4- rp=i.4s,i=i -o- Tp=1.Bs,y=1 -l.B „„- Tp=2.1s,-f=1 1 1 — Tp=2.3s,r1 I '^0 0.05 a ( o.rs 0.2 0.25 0.3 0.35 0.4
U(m.s-Fig. 3. Current velocity magnitude as a function of deptii, obtained in the presence of irregular waves of various peal< periods at sensors Ci to C3. Solid lines correspond to current in the presence of waves, while dashed lines correspond to the current alone. The horizontal thick lines symbolize the location of the plate.
degree 15 of the form ^absorber = P(T) provides an acceptable representation of the data. This function w i l l be considered as the reflecdon coefficients parametrization of the wave absorber in the following.
However, it has to be emphasized that an important discrep-ancy appears at high wave periods while computing i?absorber and
2 r 1.8 0.2 0 I . . . - 6 - 4 - 2 0 2 4 6 X(m)
iabsorber from the lowcst peak period signals (Tp = 1.4 s). This dis-crepancy can be interpreted through several phenomena. First of all, one may cite the deterioration o f t h e signal to noise ratio. High period waves in these conditions suffer from a lack of energy while noise signals, such as transversal fundamental period of the basin, become predominant. Secondly, the wave absorber suffers f r o m a lack of efficiency for really low amplitude components. For low am-plitude long waves, no breaking occurs, and these waves are almost totally reflected. Finally, the highly nonlinear behavior of the wave absorber might be responsible of an energy transfer from a high period to lower periods. For these reasons, the lowest peak periods of irregular wave signals were not considered in the polynomial interpolation.
Figs. 6 and 7 presents respectively the reflection coefficients R and transmission coefficient T induced by the plate as a function of the non dimensional incident wave number kh. Experimental data for monochromatic waves and irregular waves of periods Tp = 1.4 s, Tp = 1.8 s, Tp = 2.1 s, and Tp = 2.3 s are plotted together w i t h theoretical models 1 and 2. Model 2, including the effect of reflection due to the wave absorber, is based on the polynomial interpolation obtained above (l^absorber, ^-absorber) = (P(T), Q ( r ) ) . These figures illustrate a good agreement between experimental data and theoretical model 2, except for the longest waves in irregular seas. The analytical model predicts no scattering in both asymptotic cases corresponding to short waves and long waves, as it is well known f r o m previous works. A t the same
2 • l.e • 0.2 • 01 1 . 1 1 1 1 - 6 - 4 - 2 0 2 4 6 X(m)
Fig. 4 . Spatial evolution of the normalized wave envelope for a monochromatic wave of period T = 2.1 s (Left) and T = 2.2 s (Right). ( — ) Theoretical model without reflection at the wave absorber (Model 1), ( - ) Theoretical model with reflection at the wave absorber (Model 2) ((Ratsorber. iabsorber)(T = 2.1 s) = (0.25, 17.95) and (Rabsorber. iabso,ber)(T = 2.2 s) = ( 0 . 4 5 , 1 7 . 2 5 ) ) and (*) experimental data.
Fig. 5. Charactenzadon of the reflection induced by the wave absorber. (Left) Phase equivalent wave absorber location Labsorber as a function ofthe incident wave penod T. (Right) Reflection coefficient Rabsorber as a function of the incident wave period T.(—) Polynomial interpolation of the experimental data ( • , O, - f , * ) .
94 V. Rey,}. Touboul I Applied Ocean Research 33 (2011)88-99
OC 0.5
Fig. 6 . Reflection coefficient induced by the breakwater versus non dimensional incident wave number. ( — ) Theoretical model without reflection at the wave absorber reflection (Model 1), ( - ) Theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x, + , * ) experimental data.
123 I.s 1.4 1.3 1.2 1.1 h- ( 0.9 0 8 0 7 0.6 0.5 T(s) — Theoretical model 1 — Ttieoretical model 2 m Mortochromatic waves + 7p=r.4s, y=1 o Tp=1.8s,-/=1 « Tp=2.1s,y=1 • Tp=2.3s, y=1 1 2 3 4 5 6 7 B 9 10 kh
Fig. 7 . Transmission coefficient induced by the breakwater versus non dimensional incident wave number. ( — ) Theoretical model without reflection at the wave absorber reflection (Model 1), ( - ) Theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x, + , * ) experimental data.
time, the longest waves in irregular seas seem to be reflected. The explanation might be the same as in the framework of the
reflection at the wave absorber. A deterioradon of the signal to noise ratio might be responsible for this discrepancy, together w i t h a possible generation of long waves at the wave absorber.
However, the general agreement between model 2 and ex-perimental results emphasizes the significant role of a reflective structure behind the brealcwater. Because of the partially reflective absorber, the efficiency of the brealcwater exhibits a strongly oscil-lating behavior around the frequencies corresponding to the most reflecdve values, i.e. around its iFunctioning point. The reflection co-efficient R presents values oscillating f r o m 0.3 to 0.7 where model 1 predicts a value of R = 0.5. The presence of the reflective struc-ture behind the breakwater is responsible for a change of behavior of about 40%, and of the disappearance of the zero reflection point predicted by Patarapanich [9]. Every experimental data agree w i t h model 2 to predict a m i n i m u m reflection coefficient of R = 0.05 around the wave period T = 1.2 s. According to Patarapanich [9], and accordingly to model 1, a non reflected period should be found around this value. Linear theory predicts that resonant interaction of waves on top of the structure should result on a total transmis-sion across the brealcwater. In the presence of a partially reflective structure, this resonance is clearly affected. It seems that the phase lag of waves reflected on the wave absorber destroys perfect reso-nant interaction of waves on top of the breakwater.
3.2. In the presence of current
In this section, we question the role of the current on the efficiency of the breakwater. W i t h i n sight of answering this question, and given the results obtained previously, the role of the wave absorber in the presence of current should be addressed in detail. Fig. 8 presents the phase equivalent wave absorber location together w i t h the wave absorber reflection coefficient, both plotted as a function of incident wave period in the presence of current. Results concern monochromatic and spectral waves data. On this figure aiso appears a polynomial interpolation, as defined in Section 3. It appears that reflection from the wave absorber i n the presence of current is significantly less important than it was without any current. One can notice a peak for the results concerning reflection coefficients f o r the shortest waves and a strong increase of the reflection at the low frequency band of the spectral waves. The discrepancy at high frequency is due to the divergence ( i n fact the peak is smoothed according to Eq. (9)) of the method for these wave periods in the presence of current, as mentioned i n Section 2.2. The increase at low frequency is due to the very small amplitude of the wave components as already
13 12 11 10 — Polynomial Interpolation • Monociiromalic waves • rp=1.4s, -pi o 7p=(.8s, Y=* . Tp=2,1s,-i=1 • Tp=2.3s, -pi (.5 as O.i 0 / CO 0.4 0.i 0.2 Polynomiai interpolation t^onochn:>matlc waves Tp'1.4s.t=1 Tp=1.es, t=1 Tp=2.1s,-f=1 Tp=2.3s, -pi T(s) T(s)
Fig. 8. Characterization of the reflection induced by the wave absorber in the presence of current. (Left) Phase equivalent wave absorber location Labsorber as a function of the incident wave penod T. (Right) Reflection coefficient Rabsorber as a funcdon of the incident wave penod T. ( - ) Polynomial interpolation of the experimental data ( • , 0 , - 1 - , X , * ) .
Fig. 9. Reflection coefficient induced by the breal<water versus non dimensional incident wave number. ( — ) Theoretical model without reflection at the wave absorber reflection (Model 1), ( - ) Theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x , + , *) experimental data in the presence of current. 2.50 1.74 1.23 1.4 1.3 1.2 1.1 1 0.9 O.S 0.7 o.e 0.5 T(s) Theoretical model 1 Theoretical model 2 Monochromatic waves Tp=1.4s,-i=1 Tp=1.Ss,y=1 Tp=2.1s,y=1 Tp=2.3s,-f=1 1 2 3 4 5 e 7 8 9 10 kh
Fig. 10. Transmission coefficient induced by the breakwater versus non dimensional incident wave number. ( — ) Theoredcal model without reflection at the wave absorber reflection (Model 1), ( - ) Theoretical model with reflecdon at the wave absorber (Model 2) and ( • , O, x, + , * ) experimental data in the presence of current.
discussed i n Section 3.1 in the absence of current. Excluding these data, reflection is then found to be, most of the time, of order 5%, and never exceeds 15%. The phase lag computed suffers a lack of precision. A large number of phase shifts are observed, and the polynomial interpolation associated is then questionable. However, given the weak amplitude of waves reflected by the wave absorber, the phase lag associated is expected to play a little role in the results. Furthermore, one can nodce a discrepancy of the results concerning reflection coefficients for the shortest waves. This discrepancy is due to the divergence of the method for these wave periods in the presence of current, as mentioned in Section 2.2.
Figs. 9 and 10 present respectively the reflection coefficients R and transmission coefficient T induced by the plate as a function of the incident non dimensional wave number kh, in the presence of current. Experimental data for monochromatic waves and irregular waves of periods Tp = 1.4 s, Tp = 1.8s,Tp = 2.1 s, and Tp = 2.3 s are plotted together w i t h theoretical models 1 and 2. Model 2, including the effect of reflection due to the wave absorber, is based on the polynomial interpolation obtained f r o m Fig. 8 (Rabsorber, ^absorber) = (P(T), Q ( T ) ) . However, It has to be mentioned that this model does not correspond to a solution in
the presence of current, since no current is taken into account in the dispersion relation (A.2). From this figure, it appears that the wave absorber plays a weak role on the reflection and transmission coefficients, as suggested by Fig. 8. Thus, the difference between theoretical models (1 and 2) and experimental data can be interpreted as the impact of the current on brealcwater efficiency. It appears that the amplitude of reflected waves is increased in the presence of current. The maximum value of R is shifted from 0.5 to 0.75, corresponding to an increase of 50%. However, the reflected wave amplification is partially due to wave amplification while propagating against the current. It does not necessarily correspond to an increase of reflected energy, as emphasized by Grue and Palm [21], and more recently by Rey et al. [25]. These authors introduced a Doppler coefficient defined as
U U
k-U\ al
k+U (11)
This coefficient corresponds to the ratio of reflected wave action to incident wave action, and is smaller than 1. It emphasizes that reflected waves in the presence of current can both present a higher amplitude while propagating less energy, comparatively to the values obtained in the absence of current.
Introducing this Doppler coefficient for transmitted waves has no impact, since incident and transmitted waves propagate in the same direction. Transmission coefficients should remain lower than unity i n the absence of beach reflection. But Fig. 9 presents transmission coefficients much higher than unity. This result is surprising, since it corresponds to an increase of the energy flux. This question should be addressed properly. To achieve this goal, we present i n Fig. 11 the energy flux conservation w i t h and without any current. The quantity introduced corresponds to
tf'R + (PT + ^ W A Ï ' I
(12) where * R , * T and cPyvA are algebraic values corresponding to the energy flux correlated respectively to incident waves, reflected waves, transmitted waves and waves reflected at the wave absorber. These quantities are defined as
U
(1
<1>R =
u
(1 - U / C p - ) 2 2
(1 u/c+y 2 and <PwA
U
(13)
(14) ( l - ( ; / C p - ) 2 2
In the absence of current, it appears that energy flux is globally conserved. A weak damping is observed, and is not surprising. A peak exceeding unity is observed for high wave numbers in irregular waves, but can be explained by the generation of free harmonics due to nonlinear interaction.
In the presence of current, the curve exhibits values often greater than 1, for both irregular and monochromatic waves. Thus, the generation of free modes due to second order terms cannot explain this phenomenon for the lowest periods. It really seems that energy is produced. A possible explanation is that waves and current present a nonlinear interaction. The presence of waves might affect the current distribution beyond the plate, leading it to become non uniform. In this case, the current might act like a source term i n the conservation equation. However, we observed that the vertical profile of the current remains almost constant as shown in Fig. 3.
However, coming back to the R and T coefficients, the agree-ment between monochromatic data, and irregular waves is strik-ing. Whatever the amplitude of incident waves, the reflection coefficient remains unchanged. The process at hand exhibits a l i n -ear behavior, which has to be emphasized. This result is of major interest for the design of coastal structures and sea state predic-tions around the brealcwater.
96 V. Rey.J. Touboul/Applied Ocean Research 33 (201 J j 88-99 1.23 1.6 1.4 1.3 1.2 1.1 e ' 0.9 as 0.7 0,6 0.5 T(s) • Theoretical model 1 + Theoretical model 2 o Monochromatic waves < rp=f.4s, - p J . Tp=1.8s, 1=1 Tp=2.1s,-f=1 Tp'2.3s, 1=1 Theoretical model 1 Theoretical model 2 Monochromatic waves Tr^1.4s,i=1 Tp%1.8s, 1=1 Tp^.1s,i=1 V o 5 kh
Fig, 11. Energy flux conservation measured experimentally in the absence of current (Left), and in the presence of current (Right),
0.90 *inf 1.55 1.10 0.90 1.10 T(s) - - Theoretical model 1 — Theoretical model 2 • Monochromatic waves + Tp'1.4s,i'1 o rp=t,ss, Y=r K Tp=2.1s,i=1 Tp=2.3s, 1=1
Fig. 12. Horizontal hydrodynamical loads exerted on the plate versus non-dimensional incident wave number. (Left) In the absence of current and (Ilight) in the presence of current. ( — ) T h e o r e t l c a l model without reflection at the wave absorber (Model 1), ( - ) theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x, + , . ) experimental data.
4. Forces and moments exerted on the plate
This section presents experimental results concerning the hydrodynamical loads exerted on the plate. As mentioned in Secdon 2, 12 pressure sensors were distributed on the plate. Analytical model 1 has been used to demonstrate numerically that the error of integration due to this discretization while computing forces and moments was acceptable. Simulations for the vertical efforts and moments have been conducted using 10 collocation points (located as the pressure sensors P3 to Pu). or 200 collocation points, corresponding to an almost continuous description of the plate. Results obtained were compared, and divergences observed never exceeded 2% in the computation of loads, and 8% in the computation of moments. It emphasizes that the error committed by using only 12 pressure sensors is negligible.
Fig. 12 presents the normalized horizontal loads exerted on the plate due to the action of waves. Following Patarapanich [11], normalization is chosen such as Fx(, = pgab, a being the incoming wave component amplitude, and b the thickness of the plate.
The left part of the figure shows experimental data obtained in the absence of current for both monochromatic and irregular waves. They are plotted together w i t h analytical models 1 and 2. Wave absorber reflection coefficients used for model 2 are the values obtained i n Section 3.1. From this figure, reflection due to the wave absorber is found to play a significant role on the horizontal loads. Divergences between models 1 and 2 can
reach 40%. However, experimental data exhibit a good agreement w i t h analytical model 2, and retrospectively, w i t h results obtained analytically by Patarapanich [11]. Furthermore, it appears that normalized horizontal loads due to wave action still exhibit a strongly Unear behavior, depending weakly on the incident waves' amplitude.
The right-hand part of Fig. 12 shows experimental data ob-tained i n the presence of current for both monochromatic and ir-regular waves. They are plotted together w i t h analytical models 1 and 2, obtained w i t h no current. Wave absorber reflection coeffi-cients used for model 2 are the values obtained i n Section 3.2. In terms of non-dimensional current velocity, T = coU/g, values are ranging f r o m T = 0.05 to r = 0.2. From this figure, a significant increase on the horizontal loads is found, for current velocity of or-der T = 0.1. This increase can reach 50% w i t h reference to model 1. Furthermore, the role played by wave absorber reflection is weaker than it was i n the absence of current. Thus, the divergence between data and model can be interpreted as the influence o f t h e current. This can be understood by the form drag induced by the current on the plate. The pressure in the presence of current is increased w i t h the term Uodtp/dx.
Fig. 13 presents the normalized vertical loads exerted on the plate due to the action of waves. Following Patarapanich [11], normalization is chosen such as F^^ = pgaB, a still being the incoming wave component amplitude, and B the length of the plate.
current. ( — ) Theoretical model without reflection at the wave absorber (IVlodel 1), ( - ) theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x, + , *) experimental data.
Fig. 14. Hydrodynamical moments exerted on the plate versus non dimensional incident wave number. (Left) In the absence of current and (Right) in the presence of current. ( — ) Theoretical model without reflection at the wave absorber (Model 1), ( - ) theoretical model with reflection at the wave absorber (Model 2) and ( • , O, x, + . *) experimental data.
As previously, the left-hand part of the figure shows experi-mental data obtained i n the absence of current for both mono-chromatic and irregular waves. They are plotted together w i t h analytical models 1 and 2 and here again, wave absorber reflection coefficients of model 2 are those obtained in Secdon 3.1. Reflection due to the wave absorber is also found to play a significant role on the verrical loads. Divergences between models 1 and 2 are also of order 40%. However, experimental data exhibit a good agreement w i t h analytical model 2. Furthermore, it appears that normalized vertical loads due to wave action still exhibit a strongly linear behavior, depending weakly on the incident waves' amplitude.
The right-hand part of the figure presents experimental data obtained i n the presence of current, including both monochromatic and irregular waves. Models 1 and 2, obtained i n the absence of current, are plotted for sake of reference. Reflection coefficients for the wave absorber used i n model 2 result f r o m the polynomial interpoladon presented in Section 3.2. From this figure, vertical loads i n the presence of current appear to be more homogeneous than without any current, meaning that their values less depend on frequency. Vertical loads present a mean value 10%-15% lower under the action of current. This result is surprising. It might be interpreted as the redistribution of the f l o w due to the current around the plate, modifying the vertical profile. Waves at hand tend to be longer, and pressure on top and beneath the plate are of same order However, the Unear behavior of experimental data is still striking.
In Fig. 14, the normalized hydrodynamical moments exerted on the plate are presented. After [11], the moment is calculated w i t h respect to the center o f t h e plate (x = 0.765 m ) . Normalization is chosen such as My^ = pgaB^.
The left-hand part of the figure still corresponds to the exper-imental data including both monochromatic and irregular waves without any current. Results for analytical models 1 and 2 are also presented in the figure. Reflection coefficients relative to the wave absorber in model 2 still correspond to the polynomial interpo-lation of Section 3.1. Moments are found to be significantly less important than loads (about an order of magnitude), confirming previous theoretical results of [ 11 ]. Reflection due to the wave ab-sorber is also found to have a weak influence. Divergences be-tween model 1, model 2, and experimental data never exceed 15% if we do not consider the low frequency band already discussed in Section 3.1.
The right-hand part of Fig. 14 presents experimental results for both monochromatic and irregular waves in the presence of current. A comparison to analytical models 1 and 2, obtained without any current, using polynomial interpolation of the wave absorber's reflection coefficients f r o m Section 3.2 is also presented. It appears that high incident wave numbers are weakly affected by the presence of current. At the same time, moments due to the lowest wave numbers present a faster decrease than they did in the absence of current. This result confirms observations made for the vertical loads. The presence of current tends to redistribute
98 V. ReyJ. Touboul/Applied Ocean Research 33 (201 I j 88-99
the velocity around the plate, providing an homogenization of the pressure on top, and under the plate. As a result, the moment is decreased for longer waves. However, the linear behavior of the plate remains remarkable, since the moments due to waves o f various amplitude, for a given frequency, remain equal.
5. Conclusions
Wave scattering f r o m a submerged horizontal plate and the resulting hydrodynamical loads and moment have been studied experimentally. Results obtained i n the absence of current, and i n the presence of current were confronted to the analytical model based on integral matching of boundary conditions at the vertical discontinuities for linear waves (see e.g. [3] and references herein). To take care of partial reflection at the wave absorber, the analytical model has been modified and compared to the results.
Because o f this partial reflection, which can be due to reflective beaches or structures at the shore, an interference process is observed betw/een the plate and the shore. As a consequence, the location of the plate might have a significant influence on its efficiency.
In the absence of current, reflection coefficients have been found to be influenced by up to 40% due to the presence of partial reflection at the end of the basin. The same amount of influence is found for the horizontal and vertical hydrodynamical loads, while the moment seems to be weakly affected.
In the presence of current, the role played by the wave ab-sorber was found to be weaker. However, reflection coefficients were found to be affected by the presence of the current, up to 50%. Results concerning the horizontal force were similar, confirming results obtained by Mo and Palm [22] in the framework of sub-merged horizontal cylinders. However, the vertical force exerted on the plate seemed to be affected by the current in a different way. Loads were more constant, and a little weaker, especially for low incident wave numbers. Similar results were found for the moment.
In every case, the linear behavior observed has to be empha-sized, experimental data for both regular and irregular waves pre-sented a surprising superposition. These results were i n agreement w i t h the analytical model, as soon as they were modified to take care of partially reflected waves at the wave absorber. This re-sult emphasizes that the model is w e l l adapted for studying wave-structure interactions for engineering applications.
Appendix. Analytical model
The analytical approach used here is the model initially intro-duced by Takano [30] i n the framework of water waves propagat-ing over a parallelepipedic submerged obstacle i n the absence of current. This approach was later extended to waves propagating over a submerged trench by Kirby and Dalrymple [31]. More re-cently, it was extended by Rey [3] to arbitrary topographies and submerged structures.
The approach is based on a linear description of local and propagating modes, solutions of the problem in each domain m of constant depth h^-As soon as a free surface belongs to the domain, the velocity potential writes
4>(x,z, 0 = 3le (z)e
(A.1)
to local (or evanescent) modes, /<* and a : * „ being the related wavenumbers. One should notice that > 0 correspond to modes decaying in the ( - O x ) direction, while K„^m < 0 correspond to modes decaying in the (Ox) direction. These wavenumbers are obtained by solving the dispersion relation
= -gktanikhm) (A.2)
w i t h k = ikm for propagating modes, and k = Km for evanescent modes. The functions Xm(z) and 1'n,m(z) are given by
Xm(z) = COSh(lim(.Z + h„)) COShilimilm) _ COS(;Cn,m(z -|- / l m ) ) COS(Kn,mf!m) (A.3) (A.4) In the cases involving no free surface, Eq. ( A . l ) has to be replaced w i t h
<^(x, z, t) = Sie
(A.5)
where the first term corresponds to propagating modes, i n both directions (Ox) and (—Ox), and the second term corresponds
Once a form o f t h e solutions known in every domain, the continuity conditions of pressure and velocity on the domain interface are w r i t t e n in an integral form, allowing to take advantage of the orthogonality of Xm and ^'n^m in a given domain m. This formulation leads to the solution of a linear system.
The problem at hand (see Fig. 1) defines 4 domains of solutions. Domain 1 corresponds to the upstream part of the plate. In this domain, both propagating modes are considered, while the pos-itive wavenumbers corresponding to local modes decaying i n the (—Ox) direction are only considered. Domain 2s corresponds to the area above the plate, where all modes are considered. Area 2F is the area under the plate. It is slightly different f r o m other areas, since it involves no free surface. Solutions in this domain are of the f o r m given by Eq. (A.5). Finally, domain 3 is the area downstream of the plate. To study the far field due to wave transmission across the brealcwater, we should consider the only propagating modes i n the (Ox) direction, together w i t h the only evanescent modes w i t h negative wavenumbers, corresponding to modes decaying i n the (Ox) direction. This model is referred to as model 1 in the body of the manuscript.
However, the wave absorber turns out to play a significant role in the problem, as discussed i n Section 3. To take account of the reflected waves due to the presence of the wave absorber, we suggest to modify model 1. In domain 3, the only waves considered were the propagating modes i n the (Ox) direction, of amplitude a^, and the local modes decaying in the (Ox) direction, of amplitude y6„"3. The influence of the wave absorber can be introduced by considering the presence of another propagating mode, i n the ( - O x ) direction. Thus, the wave absorber is understood as a partially reflecting vertical wall, of reflective power J?absorber(ü^). located at x = Labsorber(«)- Thus, the amplitude of the reflected mode corresponds to a fraction of incident energy, namely = RabsorberaJ. Furthermore, the phase lag of this mode w i t h respect to incident mode o;^ is known f r o m the location of the wave absorber (p = (/<+ - /<:J)Labsorber- It appears that the wave absorber can be taken into account through a parametrization of (Rabsorber(ft*)> Labsorber(w)). In the dlscussiou of the manuscript, this model, together w i t h the parametrization of the wave absorber, is referred to as model 2.
For both models 1 and 2, convergence of the results is rapid and the number of evanescent modes considered is taken equal to 1 1 = 10. Higher order local modes are neglected.
V. Rey.J. Touboul/AppUed Ocean Research 33 (2011) 88-99
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