- Email: [email protected]
Wave transmission in harbors through flushing culverts
ARTICLE IN PRESS Ocean Engineering 36 (2009) 434–445
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Wave transmission in harbors through flushing culverts V.K. Tsoukala , C.I. Moutzouris National Technical University of Athens, School of Civil Engineering, Laboratory of Harbor Works, 5 Iroon Polytechniou, 15780 Zografou, Athens, Greece
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 August 2008 Accepted 8 January 2009 Available online 20 January 2009
An important aspect that must be considered in harbor design and construction is the water quality within the basin, which depends on the water exchange between the harbor and the surrounding water body. The most internationally distinguished methods to counteract diminished flushing and insufficient renewal occurring leeward of coastal structures include mechanical shakers, overflows, pumps, permeable breakwaters, and flushing culverts. Among these methods, the construction of flushing culverts is favorable due to the low costs of construction and operation. In the present paper, wave transmission through flushing culverts was investigated experimentally in two physical models. Incident wave and transmitted wave heights were determined using wave gauges for various combinations of wave characteristics and geometric characteristics of the flushing culverts. Wave height transformation through the flushing culvert was processed and analyzed for all experimental conditions. The sensitivity of the wave transmission coefficient with respect to other dimensional and non-dimensional parameters was comprehensively investigated in order to define which parameters could most effectively predict the wave transmission coefficient. The wave transmission coefficient increased when the incident wave period, the width, and the height of the flushing culvert increased, the incident wave became steeper, the length of the flushing culvert decreased, and the incident wave angle approached 90o. An empirical equation that correlates the wave transmission coefficient with the wave characteristics and the geometrical characteristics of the flushing culvert was derived using nonlinear regression analysis. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Breakwater Harbor Physical model Wave transmission Water renewal
1. Introduction An important aspect that must be considered in harbor design and construction is the water quality within the basin, which depends on the water exchange between the harbor and the surrounding water body (US Army Corps of Engineers, 2002). Water exchange produces a flushing action in the harbor, which is important, because it reduces the level of pollution by chemical, biological, and floating solids (US EPA, 1993). Studies have shown that adequate flushing improves water quality, reduces or eliminates water stagnation, and helps maintain biological productivity and aesthetic appeal (Schwartz and Imberger, 1988; Schwartz, 1989). Flushing may reduce pollutant concentrations in a harbor basin by 70–90% over a 24-h period (Cardwell and Koons, 1981). Generally, the small dimension and the location of the entrance of the harbor and specially the marinas do not permit the inflow or outflow of significant flow rates, thus resulting in high values of flushing time. Several methods have been developed to counteract diminished flushing and insufficient
Corresponding author. Fax: +30 210 772 2368.
E-mail address: [email protected] (V.K. Tsoukala). 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.01.005
renewal occurring leeward of coastal structures (US EPA, 2001). The most internationally distinguished methods include mechanical shakers, overflows, pumps, permeable breakwaters, and flushing culverts. Among these methods, the construction of flushing culverts is favorable due to the low costs of construction and operation. The use of flushing culvert enhance wave transmission which results to significant increase of the flow rate passing through the harbor and to increase of the magnitude of water particles velocity. Flushing culverts therefore enhance water circulation through the amplification of the velocity field inside the harbor basin and consequently contribute to the reduction of the flushing times. Previous studies (Stamou et al., 2004) in specific harbor have already shown that the flushing time decreases by 10–32%, depending on the wave characteristics, with the construction of flushing culvert. Underwater placement of culverts is internationally standardized; in this case, tidal hydrodynamics is the main mechanism of enhanced flushing. In Greece, where there are more than 600 harbors of different sizes and operations, construction of flushing culverts for the improvement of water quality is common. Flushing culverts are mainly constructed in protective structures, i.e., breakwaters, to significantly improve water circulation and flushing (Papaioannou et al., 1999). As the ranges of the tides are low, it is preferred that culverts are constructed with their longitudinal axis at sea
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
Nomenclature
L0
b d g h hs Hi Ht Kt l
L R SWL tan j T
width of the flushing culvert water depth gravitational acceleration, g ¼ 9.81 m/s2 height of the flushing culvert submergence depth incident wave height transmitted wave height wave transmission coefficient length of the flushing culvert
water level (Fig. 1). Additionally, as the culvert is placed away from the seabed, sediment transport into the harbor basin is restricted. Although the design of flushing culverts, including determination of their shape, location, and dimensions has been used to improve water circulation and flushing without excessive wave penetration into the harbor basin (Fountoulis and Memos, 2005), the design process has thus far depended on empirical rules. Consequently, the resulting flushing is occasionally inadequate. Additionally, problems can sometimes arise in the functionality of
a x
435
offshore length of the incident wave or deep water wavelength wave length correlation coefficient of the empirical equation surface water level breakwater slope wave period incident wave angle surf similarity parameter or Iribarren number, x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan j= 2pHi =gT 2
the harbor and craft anchorage due to excessive wave penetration into the harbor basin (Tsoukala and Moutzouris, 2009). The effect of the wave penetration is quantified using the wave transmission coefficient, which is defined as the ratio of the transmitted wave height (Ht) to the incident wave height (Hi), i.e., Kt ¼ Ht/Hi. Several efforts have been made to demonstrate the beneficial effect of designing flushing culverts using numerical models (Stamou et al., 2004) or finite difference models of an idealized harbor (Fountoulis and Memos, 2005). However, there is no
Flushing culverts
Marina Mytilini
Flushing culverts Marina Kalamata
Flushing culverts
Marina Mati Fig. 1. Characteristic views of flushing culverts in three marinas in Greece.
ARTICLE IN PRESS 436
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
experimental work in the literature dealing with the influence of the flushing culvert on the characteristics of the transmitted waves or on the flushing time of the harbor. The majority of existing studies have mainly analyzed wave transmission through various types of structures. In all of these studies, Kt was used as the main non-dimensional parameter, and it was correlated with the wave and geometrical characteristics of the structures using dimensional, parametric, and regression analyses. In this paper, wave transmission through flushing culverts was investigated experimentally in two physical models. A constitutive set of experiments was carried out to observe the effect of the geometrical characteristics of flushing culverts on the wave transmission coefficient in order to define which parameters could most effectively predict the wave transmission coefficient. Furthermore, the experimental data were processed using dimensional, parametric, and regression analyses. As a result a preliminary empirical equation that can be used in the design of flushing culverts was derived. The proposed equation correlates the wave transmission coefficient (Kt) to the characteristics of flushing culverts and to the wave characteristics.
intra-gravity waves. In the present study, only regular waves were produced (Fig. 2). 2.2. Test layouts and structures Two physical models of two different harbors were used in order to examine wave transmission through flushing culverts. Model 1 is a physical model of Ikaria Harbor (Fig. 3), and Model 2 is a physical model of Kolybari Harbor (Fig. 4). These models were constructed using Froude similarity at scales of 1:100 and 1:60, respectively. The breakwater of each model was composed of two layers of quarry rocks having densities of 2.65 kg/m3. In Model 1, the breakwater slopes were 1:2.5 and 1:1.5 for the seaward and leeward sides, respectively, while for Model 2, the slopes were 1:1.5 for both sides of the breakwater. Three flushing culverts in Model 1 and one in Model 2 were placed at the windward breakwaters of the harbors, with their axes at the mean water surface level as shown in Fig. 3. The different dimensions of the flushing culverts that were tested for each model are shown in Table 1.
2. Experimental setup and data sampling
2.3. Instrumentation
In order to give a better description of wave transmission through a flushing culvert in the lee of a breakwater, a comprehensive set of laboratory experiments were preformed in one of the three wave basins of the Laboratory of Harbor Works (LHW), National Technical University of Athens (NTUA). The wave basin has the dimensions 26.8 m 24.3 m 1.0 m (length width depth).
2.3.1. Wave gauges Resistance-type wave probes were used (see Fig. 5) to record and determine wave characteristics for each test. A quoted precision of 72% can be achieved with these wave gauges. The resistance from the gauges is converted to a voltage. The voltage outputs of the wave probes are converted to water levels using calibration factors for each wave gauge. The HR Wave Data Calibration Module (Bersford et al., 2005) was used to calibrate the wave gauges; every day prior the experiments, several different water levels (3 or 5 depending on the length of the wave gauge) were measured against the corresponding output voltages from the gauges. The voltage conversion factor was then determined as the slope of the linear curve best fitting the height–voltage diagram.
2.1. Wave generation The hydraulic wave generation system is composed of two paddles side by side, each 6.0 m wide, and can generates waves based on the signal-generator system. Produced waves must be in the range of 0.3–3 Hz and should not be higher than 0.5 m. No active absorption on the wave paddles was used, but passive absorption was placed at the rear end of the basin and at both sides of the basin. Waves are absorbed on sidewalls by rubble mound structures made of stones having a characteristic diameter equal to 0.08 m, and they are also absorbed on wiredrawn cylinders. The input signal to the wave generator is produced by a random wave synthesizer that uses HR Wave-Maker software (Bersford, 2007). HR Wave-Maker is an advanced software package produced by HR Wallingford and it is designed to simulate a variety of sea states for both single and multi-element wavemakers. It can generate regular and irregular waves and can also produce compensation for set down or very long-period
2.3.2. Monitoring All tests were monitored using a video camera, with observations being recorded on video tape and with a digital camera for further analysis.
3. Test conditions and data sampling Preliminary experiments were conducted to understand the flow characteristics better as well as to select the final wave conditions. Waves that generated agitation due to entering the harbors and interdigited flows between the three flushing
Fig. 2. Schematic representation of the flushing culvert placed in the breakwater.
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
437
Fig. 3. Model 1: (a) top view of Ikaria Harbor and (b) physical model of Ikaria Harbor in LHW.
Fig. 4. Model 2: (a) top view of Kolybari Harbor and (b) physical model of Kolybari Harbor in LHW.
culverts in Model 1 were discarded. Based on these tests, it was decided that the results from the middle flushing culvert would be utilized. Based on the preliminary observations and in order to avoid the diffraction observed in Model 2, the seabed on the seaward side of the breakwater was also demolished. A total of 105 tests performed under different combinations for 24 monochromatic wave climates were finally separated for further analysis. As no active absorption on the wave paddles was used, the constructed Models may reflect from 20% to 80% of the incident wave amplitude back towards the wave generator. To overcome this problem all the measurements of incident wave height used in the following analysis were made by repeating the test sequence without the structure in place. The experiments were organized into 13 series: five in Model 1 and eight in Model 2. The ranges of tested variations in water depth (d), wave characteristics including height (Hi), wave period (T), and incident wave angle (a), as well as the geometrical characteristics of the flushing culverts including length (l), height (h), and width (b), are presented in Table 1. The list of wave conditions is presented in Table 2. To avoid surface tension effects and the production of
capillary waves, the wave periods were selected to be higher than 0.35 s (Hughes, 1993). The effect of flushing culvert width in wave transmission coefficient was investigated through different wave conditions for four different flushing culvert widths (see Table 1) through the experiments conducted in Model 1 (series I2–I6). In Model 2 the influence of the length, the height, and the angle of wave attack on transmission coefficient was also investigated. Experimental series I1, K1, and K5 represent the experiments conducted without the harbor Models in the wave basin in order to estimate the incident wave characteristics. Water level time series were measured using resistance-type wave probes. The location of the probes and their distances from each other depended on the length of the flushing culvert employed in the test. Two wave gauges (named 1 and 2 in Fig. 5) were used to monitor generated wave conditions. Generated wave conditions were calibrated at those wave gauges for all experiments. Additional wave gauges were placed at other points inside the wave basin to obtain data for additional numerical analysis; those data will not be reported here. Fig. 5 shows the locations of the probes in each test condition (with and without the harbor model).
ARTICLE IN PRESS 438
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
Table 1 Characteristics of the experimental series. Series
Harbor
I1 I2 I3 I4 I5
Model 1
K1 K2 K3 K4
Model 2
K5 K6 K7 K8
Model 2
Geometrical characteristics of the flushing culverts
Water depth (m)
Incident wave angle (1)
0.23 0.23
901 901
Height (m)
Length (m)
Width (m)
No flushing culvert 0.025
0.375
0.025 0.050 0.075 0.100
No flushing culvert 0.025 0.025 0.030
0.160 0.320 0.160
0.050
0.25 0.25
901 901
No flushing culvert 0.025 0.025 0.030
0.160 0.320 0.160
0.050
0.25 0.25
601 601
A sampling rate for water surface elevation measurement using wave probes was adopted such that a minimum of 15 samples were collected during one wave period. Data were sampled at a rate of 20 Hz. Test models each consisted of 1000 regular waves. HR Wave Data (Bersford et al., 2005) was used for data acquisition, and the output signals of wave probes (in volts) were stored in different data files for each test.
incident wave. All the acquired time series show strongly nonlinear wave forms that begin to split into a series of smaller waves at the lee of the flushing culvert and further down inside the harbor basin. This includes considerable growth of the second and the third harmonics which are then drastically reduced as the wave continues on the leeward side of the structure. 4.3. Incident wave height
4. Experimental data processing 4.1. Harmonic generation in the wave basin without the harbor models Monitoring of water surface elevation in the water basin without the harbor models was carried out as discussed in 2.4. Recorded signals from the probes were spectrally analyzed using fast Fourier transformation (FFT) method in order to investigate the behavior of water surface elevation and waves traveling along the wave basin. The water-level spectra revealed that higher harmonic wave generation took place under certain test conditions. These tests were discarded from further analysis.
Water surface elevation data collected by wave gauges along the wave basin were used to calculate incident, reflected, and transmitted wave heights. The incident wave height was calculated for each test from data obtained without the harbor models.
5. Analysis and discussion of experimental results Wave height transformations occurring through the flushing culvert under all experimental conditions were processed and analyzed. The effectiveness of several parameters (dimensional and dimensionless) was examined to improve the understanding of the hydrodynamic processes.
4.2. Harmonic generation caused by the flushing culvert
5.1. Transmitted wave height
Harmonic generation due to abrupt changes in water depth is well known, and some experimental and numerical approaches have been developed (e.g., Mei and Black, 1969; Massel, 1983; Goda et al., 1999) to investigate this process. When waves approach the breakwater containing the flushing culvert, part of the wave energy is reflected by the barrier, but, at the bottom of the flushing culvert, harmonic generation with energy transfer to higher harmonics is observed. The higher harmonics generated inside the flushing culvert are transmitted to deeper water behind the breakwater as free waves. The generation and growth of harmonics and their progression leeward were obtained by analyzing all the probe records. The shape of the wave spectra offshore from the breakwater, showed similar trends of harmonic generation to those noted in the flume without the breakwater with the flushing culverts. In the probes in front of the flushing culvert, an increase in wave spectra is observed, which is due to the overlapping of the reflected to the
The height of transmitted waves passing through the experimental flushing culverts was measured using two- and threeresistance wave gauges in Models 1 and 2, respectively, located as shown in Figs. 4 and 5. Waves transmitted onshore included higher harmonic-free waves that were generated over the bottom of the flushing culvert and transmitted to the deeper water behind the breakwater as free waves. Hence, the wave height behind the flushing culvert varied spatially and depended on the number of significant harmonics generated and transmitted inside the culvert. This phenomenon was clearly observed throughout testing. The amplitude of the second harmonic wave was unknown until recently, although this subject has widely been examined in the past for the case of wave transmission over submerged breakwaters (Longuet-Higgins, 1977; Massel, 1983; Driscoll et al., 1992; Losada et al., 1997). An interesting discussion about the observed physics for wave transmission through flushing culverts is given in Tsoukala et al. (2009).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
439
Table 2 Characteristics of the waves.
Model 1
Model 2
a/a
Harbor
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
T (s)
Hi (m)
L0 ¼ gT2/2p (m)
Hi/L0
0.52 0.52 0.52 0.52 0.75 0.75 0.75 1.00 1.00 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.50 0.50 0.50 0.50 0.50 0.67 0.67 0.67
0.060 0.014 0.018 0.023 0.012 0.017 0.023 0.022 0.030 0.004 0.007 0.010 0.014 0.018 0.023 0.025 0.005 0.008 0.011 0.015 0.020 0.012 0.017 0.025
0.42 0.42 0.42 0.42 0.88 0.88 0.88 1.56 1.56 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.39 0.39 0.39 0.39 0.39 0.70 0.70 0.70
0.014 0.033 0.043 0.054 0.014 0.019 0.026 0.014 0.019 0.014 0.028 0.040 0.056 0.072 0.092 0.100 0.013 0.020 0.028 0.038 0.051 0.017 0.024 0.036
ˆ rms) calculated from the wave probes at squared wave height (H the leeward side of the flushing culvert was calculated with the following three methods:
Average Hˆrms at each probe calculated from the simple statistic method pffiffiffi ^ rms ¼ 2 2 H
Pn
Average Hˆrms
s
i (1) n calculated from the zero moment of each
i¼1
spectrum Pn pffiffiffiffiffiffiffiffi i¼1 m0i n Square root of the mean zero moment of the spectra sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ rms ¼ 2 2 i¼1 m0i H n
pffiffiffi ^ rms ¼ 2 2 H
Model 2
(2)
(3)
where si is the standard deviation of the water surface level enhanced by the ith wave gauge behind the flushing culvert, n is the number of wave gauges, and m0i is the zero moment of the spectrum of the ith wave gauge.
Fig. 5. Wave basin layout and wave gauge positions: (a) Model 1, (b) Model 2 with 16 cm flushing culvert, and (c) Model 2 with 32 cm flushing culvert.
Comparison of the three above-mentioned methods indicated no significant difference for the calculation of transmitted wave ˆ rms obtained from Eq. (1) was adopted as heights. In this paper, H representative of the transmitted wave height. It should be noted that the calculated transmitted wave height is an average and includes the fundamental and higher generated harmonic waves since the total spectrum (wave energy) was taken into account. This is consistent with the total energy of wave approaches to the breakwater that was considered in the calculation of the incident wave height (see Section 5.1). 5.2. Transmission coefficient
In order to obtain more accurate measurements of the height of waves transmitted through the flushing culvert, different approaches were applied and compared. The average root-mean-
The transmission coefficient was calculated as the ratio of the average transmitted wave height obtained from Eq. (1) to the
ARTICLE IN PRESS 440
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
incident wave height
0.4
^ rms H Kt ¼ Hi
(4)
0.3
Hi ~ 0.004m Hi~ = 0.026m
Analysis of the transmission coefficient and the effects of different variables (dimensional and non-dimensional) on Kt were examined both graphically and statistically. The purpose of this analysis was to define which parameters could most effectively predict the transmission coefficient. The variation of the transmission coefficient is plotted in Figs. 6–11 as a function of the incident wave height Hi, wave period T, flushing culvert length l, flushing culvert width b, submergence depth hs (water depth over the bottom of the flushing culvert, hs ¼ h/2 for all the experiments reported in the present study), and incident wave angle a. Graphical representation demonstrated that the transmission coefficient increased with submergence depth, flushing culvert width, and wave period, whereas culvert length and incident wave height had inverse effects on Kt. The above results provide valuable guidance for the geometric design of flushing culverts. However, non-dimensional parameters
Kt
5.3. Effective parameters for prediction of Kt
0.2
0.1
0.0 0.00
0.16
0.08
0.24
0.32
0.40
l (m) Fig. 8. Plot of transmission coefficient against flushing culvert length (T ¼ 0.40 s and hs ¼ 0.013 m).
0.3 Hi ~ 0.014m
0.4
Hi~ = 0.019m
hs = 0.0125m 0.2
hs = 0.015m
Kt
Kt
0.3
0.2
0.1
0.1 0.0 0.01 0.0 0.00
0.01
0.01
0.02 Hi (m)
0.02
0.03
0.03
0.01
0.01
0.01 hs (m)
0.01
0.02
0.02
Fig. 9. Plot of transmission coefficient against flushing culvert submergence depth (T ¼ 0.40 s and l ¼ 0.16 m).
Fig. 6. Plot of transmission coefficient against incident wave height (T ¼ 0.40 s and l ¼ 0.16 m).
0.3
0.3 Hi ~ 0.014m
Hi ~ 0.013m
Hi~ = 0.019m
Hi ~ 0.024m 0.2
Kt
Kt
0.2
0.1
0.1
0.0 0.00
0.20
0.40 T (sec)
0.60
0.80
Fig. 7. Plot of transmission coefficient against wave period (hs ¼ 0.13 s and l ¼ 0.16 m).
0.0 0.000
0.025
0.050
0.075
0.100
0.125
b (m) Fig. 10. Plot of transmission coefficient against flushing culvert width (T ¼ 0.40 s, l ¼ 0.16 m and hs ¼ 0.0013 m).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
can provide a more effective correlation for the above-dependent parameters. Based on dimensional analysis and a review of the single parameters above, the following dimensionless parameters were considered for further analysis: Kt ¼ f
l hs Hi H H ; ; or x; i ; i ; sin a L0 Hi L0 l b
(5)
0.3
Hi~0.007m Hi~0.014m
0.2
441
in which the deep water wave length L0 is employed instead of the wave period T. The dimensionless parameter, Iribarren number x ðx ¼ tan f= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pHi =gT 2 Þ, can be used to represent wave steepness since it is commonly used for breaking wave analysis in the surf zone, where tan j is the breakwater’s slope. Various plots of the variation of the transmission coefficient as a function of the above non-dimensional parameters are presented in Fig. 12. Graphical representation of the dimensionless parameters demonstrates that the transmission coefficient increases as the length of the flushing culvert decreases. In particular, based on the graphical interpretation of Fig. 12, the effect of the dimensionless parameters on the wave transmission coefficient is described as follows:
Kt
Dimensionless flushing culvert length l/L0 has an inverse effect 0.1
0.0
0
20
60
40
80
100
α (°) Fig. 11. Plot of transmission coefficient against incident wave angle (T ¼ 0.40 s, l ¼ 0.16 m, and hs ¼ 0.0013 m).
on Kt. For the tests with shorter culvert lengths (l/L0o1.0), the transmission coefficient is more sensitive to l/L0; thus, the effect is less for longer culverts. It is therefore likely that, for flushing culverts longer than the incident wave length, variation of the crest width will have a small effect on the transmission coefficient. The submergence ratio hs/Hi has a direct influence on the transmission coefficient. Increasing the submergence ratio allows waves to pass through the flushing culvert with less energy losses, such more energy transfers onshore. Variation of the transmission coefficient with respect to the submergence ratio for different flushing culvert lengths (Fig. 12b) revealed
0.4
0.4
l = 0.16m
0.3
l = 0.32m
0.3
Kt
Kt
l = 0.38m
0.2
0.1
0.1 0.0 0.0
0.2
0.5
1.0
0.0 0.0
1.5
0.5
1.0
0.4
0.4
0.3
0.3 Kt
Kt
l/L0
0.2 0.1 0.0 0.00
1.5 2.0 hs/Hi
2.5
3.0
3.5
0.2 0.1
0.05
0.10
0.15
0.0 0.00
0.05
0.10 Hi/l
Hi/L0
0.15
0.20
0.4 b = 0.025m b = 0.050m b = 0.075m b = 0.010m
Kt
0.3 0.2 0.1 0.0 0.0
0.3
0.6
0.9
1.2
1.5
Hi/b Fig. 12. Plots of transmission coefficient against different dimensionless variables. (a) flushing culvert length ratio, (b) submergence ratio, (c) wave steepness, (d) wave height to flushing culvert length ratio, and (e) wave height to flushing culvert width ratio.
ARTICLE IN PRESS 442
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
greater effects for the submergence ratio in tests with longer flushing culverts. Increasing the incident wave steepness (Hi/L0) causes the transmission coefficient to decrease due to stronger wave breaking. The transmission coefficient decrease as the ratio of incident wave height to flushing culvert length (wave height ratio) increases. The wave height to culvert width ratio has an inverse effect on Kt. Fig. 12(e) shows that, in tests with wider flushing culvert widths (Hi/bo0.3), the transmission coefficient is more sensitive to Hi/b; this effect is decreased for narrower flushing culverts.
The Pearson (linear) correlation test is used to investigate the relative dependency of the transmission coefficient on the above-independent variables. Although the linear correlation coefficient does not conclusively indicate a non-linear relationship, it provides a general estimation of the effectiveness of the variables. The correlation coefficients between Kt and other dimensionless variables were calculated for all the conducted experiments, and the values are provided in Table 3. The correlation coefficients indicate that the wave height to flushing culvert width ratio (Hi/b), the submergence ratio (hs/Hi), the wave steepness (Hi/L0), the ratio of flushing culvert length to wave height (Hi/l), and the flushing culvert ratio (l/L0) are the parameters that most affect the transmission coefficient and are used in the next section for the formation of a design equation for the transmission coefficient. 5.4. Proposed empirical model Based on the graphical and statistical analyses, six dimensionless variables were finally adopted for generation of a design
Correlation coefficient
l/L0 hs/Hi Hi/L0 Hi/l Hi/b
0.28 0.58 0.57 0.31 0.62
(6)
Four functions were considered for developing an appropriate design equation for Kt. The first function was a non-linear equation that included the submergence ratio, wave height to breakwater crest ratio, dimensionless crest width, and inverse ratio of wave steepness K t ¼ b1
hs Hi
b2
þ b3
l Hi
b4
þ b5
L0 Hi
b6
þ b7
b8 b10 l b þ b9 L0 Hi (7)
where the coefficients b1–b10 are defined by non-linear regression analysis. The second form of the equation is similar to Eq. (7) except that the inverse ratio of the wave steepness was replaced by the Iribarren number K t ¼ b1
hs Hi
b2
þ b3
l Hi
b4
þ b5 ðxÞb6 þ b7
b8 b10 l b þ b9 L0 Hi
(8)
The third form of the equation is similar in form to that proposed by d’Angremond et al. (1996) and van der Meer et al. (2004) for wave transmission over submerged breakwaters K t ¼ b1
hs Hi
b2
þ b3
l Hi
b5
ð1 expðb6 xÞÞ þ b7
b Hi
b8
(9)
where the coefficients b1–b8 are determined by non-linear regression analysis. Finally, the fourth form of the equation is similar to Eq. (9) except that the impact of the Iribarren parameter is incorporated into the dimensionless ratios of flushing culvert length and width as follows: b2 " b4 b8 # hs l b K t ¼ b1 þ b3 þ b7 (10) ð1 expðb6 xÞÞ Hi Hi Hi An easy way to include the effect of oblique waves is to add a sin a2/3 (van der Meer et al., 2004) as function to Eq. (10). This leads to the final prediction formulas for wave transmission through flushing culverts, including obliquity. Therefore, the influence of incident wave angle a was also investigated in all the above equations by multiplying the equations by sin a2/3. Non-linear regression analysis was carried out using SPSS15 (SPSS Inc, 2004) software. Several scripts were developed to determine parameters based on non-linear least-squares regressions.
Table 3 Correlation coefficient between Kt and other dimensionless variables. Dimensionless parameter
equation for the transmission coefficient Ht hs Hi l b L0 Kt ¼ ; ; ¼f ; ; x or ; sin a Hi Hi l L0 Hi Hi
Table 4 Calibration coefficients calibrated by non-linear regression analysis for best fitting of data and the corresponding statistical parameters for evaluation of the models. Equation number
Model no.
Calibration coefficients b1
b2
b3
b4
b5
R2
b
RMSE
Iw
e
b6
b7
b8
b9
b10
0.001 0.000 0.000 0.001 0.001 0.000
0.20 0.21 11.34 0.19 0.20 0.19
0.45 0.44 0.01 0.46 0.44 0.46
0.71 0.73 0.64 0.71 0.73 0.73
1.04 1.04 1.05 1.04 1.04 1.04
0.30 0.29 0.34 0.30 0.29 0.29
0.93 0.94 0.92 0.93 0.94 0.94
0.20 0.20 0.22 0.20 0.20 0.20
1 2 3 4 5 6
162.05 342.52 164.79 382.35 374.71 611.97
0.00 0.00 0.00 0.00 0.00 0.00
33.11 286.04 158.35 253.42 256.25 612.15
0.00 0.00 0.00 0.00 0.00 0.00
0.20 0.01 4.82 92.04 71.10 0.19
1.08 0.94 0.01 0.00 0.00 0.79
128.39 56.34 0.00 36.77 47.22 0.00
9
7
0.47
0.04
0.39
0.26
8277.30
1.09
0.00
–
–
–
0.65
1.05
0.32
0.92
0.22
10
8 9
0.00 0.10
3.31 0.76
0.16 2.64
0.58 0.03
0.01 3.00
0.88 0.05
0.53 0.21
– –
– –
– –
0.73 0.79
1.06 1.06
0.27 0.27
0.94 0.98
0.20 0.19
8
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
A total of 10 different design equations for Kt were investigated based on Eqs. (7)–(10). The calculated values for calibration coefficients b1–b10 were derived from regression analysis and are provided in Table 4. Models 1–3 in Table 4 correspond to Eq. (7), and Models 4 and 6 correspond to Eq. (8). In Models 2 and 4, the effect on the predicted equations was examined by incorporating the sin a function. The term l/L0 on the right-hand side of Eqs. (7) and (8) was ignored in Models 3 and 6 to assess the sensitivity of Kt, to the culvert length ratio l/L0; coefficients b7 and b8 were considered to be zero in regression
0.4 Model 1
Kt predicted
0.3
Correlation coefficient R2 P ð N ðX m X¯ m Þi ðX c X¯ c Þi Þ2 R2 ¼ PN i¼1 P ¯ 2 N ¯ 2 i¼1 ðX m X m Þi i¼1 ðX c X c Þi
0.1
0.0
analysis of Models 3 and 6. Models 7 and 8 were extracted from the formulae of d’Angremond et al. (1996) and van der Meer et al. (2004) (Eq. (9)), while Model 9 was extracted from Eq. (10). Graphical comparison of the measured transmission coefficient with the calculated values using the proposed equations is displayed in Fig. 13. Comparison of the figures for Models 1 and 3, as well as Models 4 and 5 in Fig. 13, indicates that the flushing culvert length ratio (l/L0) has little influence on the transmission coefficient. The dimensional parameters l and L0 have both been employed to define other non-dimensional parameters, and their combination can be ignored from the corresponding equations. Statistical evaluation of the models was carried out by calculating the following statistical indicators: Bias or distortion (b): PN ðX =X Þ (11) b ¼ i¼1 c m i N Root-mean-squared error (RMSE) "P #1=2 N 2 i¼1 ððX c X m Þ=X m Þi RMSE ¼ N
0.2
0.2 Kt measured
0
0.4
0.4 Model 2
Kt predicted
¼
2 0.1
0.0
0.2 Kt measured
0
0.4
0.4 Model 3
Kt predicted
0.3
0.2
0.1
0.0
0
0.2 Kt measured
0.4
Fig. 13. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (7).
(12)
(13)
The Wilmott number Iw (Wilmott, 1981) and error function e (Haller et al., 2002), were also implemented for additional discussion of the validity of the various models PN 2 i¼1 ðX c X m Þi (14) Iw ¼ 1 P 2 ¯ ¯ i¼1 ðjX c X m j þ jX m X m jÞ
0.3
0.2
443
"P N
2 i¼1 ðX c X m Þi PN 2 i¼1 X m
#1=2 (15)
where Xc are the calculated values, Xm are the measured (available data) values, and the overbars indicate the average value of the parameter. Perfect agreement is indicated when R2, bias b, and Wilmott index Iw are 1.0, and the error function e and RMSE are zero. R2 is considered unsatisfactory for measurement of fit quality in a multivariate regression-based relationship (Draper, 1984); thus, the validity of the model was mostly assessed using other parameters. The values of the five statistical parameters are provided in Table 4. Graphical and statistical comparison of the measured transmission coefficient with the calculated values from the proposed model revealed that Models 1 through 8, with the exception of Model 3, provided reasonable approximations of the transmission coefficient with respect to laboratory measurements. Iw and b were greater than 0.92, and the RMSE and e error values were less than 0.30 and 0.20, respectively. Models 6 and 9 present marginally more accurate results when compared with the other models hs l K t ¼ 374:7 þ 256:3 þ 71:1ðxÞ0:001 Hi Hi 0:001 0:44 # l b þ 0:204 (16) sin a2=3 L0 Hi " " 0:032 0:76 hs l þ 2:64 K t ¼ 0:1 Hi Hi # 0:05 # b 3 ð1 expð0:21 xÞÞ sin a2=3 Hi
(17)
ARTICLE IN PRESS 444
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
culvert geometrical characteristics. Therefore, Eq. (17) should be adopted for calculating the transmission coefficient through flushing culverts with axes at the mean sea water level.
0.4 Model 4
Kt predicted
0.3 6. Conclusions
0.2
Wave transmission through flushing culverts placed across a breakwater was investigated in the present study for improvement of harbor basin quality. An extended set of experiments was
0.1 0.4 Model 7
0.2 Kt measured
0
0.4
0.4 Model 5 0.3 Kt predicted
0.3 Kt predicted
0.0
0.2
0.1 0.2
0.0
0
0.2 Kt measured
0.1
0.4
0.5 Model 8 0
0.2 Kt measured
0.4 Model 6 0.3 Kt predicted
0.4
0.4
Kt predicted
0.0
0.3
0.2
0.1 0.2 0.0 0.0
0.1
0.1
0.2 0.3 Kt measured
0.4
0.5
0.5 Model 9
0
0.2 Kt measured
0.4
0.4
Fig. 14. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (8).
The structure of Eq. (17) (Model 9) is less complicated than that of Eq. (16) (Model 6). Moreover, it is more common in the literature, as its predictive form was reported by d’Angremond et al. (1996), and it has subsequently been used by other investigators for predicting the transmission coefficient of submerged breakwaters. Comparison of measured and calculated Kt using Eq. (17) for different series of experiments is shown in Fig. 14. As implied from the figure, the proposed equation provides reasonable approximations of transmission coefficients through flushing culverts for a range of wave conditions and different
Kt predicted
0.0
0.3
0.2
0.1
0.0 0.0
0.1
0.2 0.3 Kt measured
0.4
0.5
Fig. 15. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (9) (Model 7) and Eq. (10) (Models 8 and 9).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
equation and the optimisation of the coefficients derived from the regression analysis (Figs. 15 and 16).
0.5
Kt predicted
0.4
0.3
I2 I3 I4 I5 K2 K3 K4 K6 K7 K8
References
0.2
0.1
0.0 0.0
0.1
445
0.2 0.3 Kt measured
0.4
0.5
Fig. 16. Comparison of measured and calculated transmission coefficients using the final proposed model (Eq. (12)).
conducted to examine the effect of different parameters on wave transmission through flushing culverts. The transmission coefficient Kt was found to increase with increased submergence depth, width of the flushing culvert, and wave period, whereas flushing culvert length and wave height had inverse effects on Kt. It was also observed that Kt was higher for waves with perpendicular angles of incidence. Analysis of the effect of dimensionless parameters on the transmission coefficient showed that the inverse wave height to flushing culvert width ratio b/Hi, the submergence ratio hs/Hi, the inverse wave steepness Hi, L0 and the ratio of wave height to flushing culvert length Hi/l, influence the transmission coefficient; the relative magnitude of the effect of each parameter is in the order given above. Considering the effect of dimensionless parameters four different forms of empirical equations were evaluated via nonlinear regression analysis in order to best correlate the wave transmission coefficient with the characteristics of the waves and the geometrical characteristics of the flushing culvert. The proposed equation seems to be a satisfactory preliminary approximation and could be used for the estimation of the transmission coefficient. According to the scales of the tested physical models, a quite well prediction is expected for waves with heights between 0.4 and 2.5 m and periods 4.0–7.8 s, for flushing culverts with the mean axis placed at the sea water level. However, as this equation derived from laboratory experiments possible scale effects are fomented in the results. To overcome this uncertainty, further experiments should be undertaken in different scales – mainly greater than the selected in the present study – for both regular and irregular waves. An active wave absorber system is also recommended for the scale comparison of results from different faculties. Finally measurements in nature are needed to be conducted for the validation of the proposed
Bersford, P.J., 2007. HR Wave Maker Wave Generation control program, Software manual, Report IT 453 Issue 5, H.R. Wallingford. Bersford, P.J., Spencer J.M.A, Clarke, J., 2005. HR Wave Data—Data Acquisition and analysis software program User manual, Report IT 493, H.R. Wallingford. Cardwell, R.D., Koons, R.R., 1981. Biological considerations for the siting and design of marina and affiliated structures in puget sound. Technical Report no. 60. State of Washington, Department of Fisheries. d’ Angremond, K., Van der Meer, J.W., De Jong, R.J., 1996. Wave transmission at low crested structures. In: Proceedings of the 25th International Conference on Coastal Engineering. ASCE, New York, pp. 2418–2427. Draper, N.R., 1984. The box-wetz criterion versus R2. Journal of Statistical Society 147, 100–103. Driscoll, A.M., Darlymple, R.A., Grilli, S.T., 1992. Harmonic generation and transmission past a submerged rectangular obstacle. In: Proceedings of the 23th International Conference on Coastal Engineering. ASCE, New York, pp. 1142–1152. Fountoulis, G., Memos, C., 2005. Optimization of openings for water renewal in a harbor basin. Journal of Marine Environmental Engineering 7 (4), 297–306. Goda, Y., Okazaki, K., Kagawa, M., 1999. Generation and evolution of harmonic wave components by abrupt depth changes. In: Proceedings of the Coastal Structures ’99. ASCE, New York, pp. 649–658. Haller, M.C., Darlymple, R.A., Svendsen, I.A., 2002. Experimental study of nearshore dynamics on a barred beach with rip channels. Journal of Geophysical Research 107 (C6), 14-1–14-21. Hughes, S.A., 1993. Physical Model and Laboratory Techniques in Coastal Engineering, Advanced Series on Ocean Engineering, Vol. 7. World Scientific Publishing, Ltd., Singapore, p. 568. Longuet-Higgins, M.S., 1977. The mean forces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines. Proceedings of the Royal Society of London A352, 463–489. Losada, I.J., Patterson, M.D., Losada, M.A., 1997. Harmonic generation past a submerged porous step. Coastal Engineering 31, 281–304. Massel, S.R., 1983. Harmonic generation by waves propagating over a submerged step. Journal of Coastal Engineering 7, 357–380. Mei, C.C., Black, J.L., 1969. Scattering of surface waves by rectangular obstacles in water of finite depth. Journal of Fluid Mechanics 38 (3), 499–511. Papaioannou, A.I., Azorakos, S., Moutzouris, C.I., 1999. Experimental study of wave energy penetration inside a harbor basin through a gap in the external breakwater. In: Moutzouris, C.I. (Ed.), Second Conference on Coastal Zone Management and Improvement. Athens, Greece, pp. 211–218. Schwartz, R.A., 1989. The flushing characteristics of Hillarys boat harbor. M. Eng. Sc. Thesis, University of Western Australia, Centre for Water Research Reference ED, 259 RS. Schwartz, R.A., Imberger, J., 1988. Flushing behavior of a coastal marina. In: Proceedings of the 21st International Coastal Engineering Conference, Spain, ED 439. SPSS Inc, 2004. SYSTATs v.13.0, Technical Software. Stamou, A.I., Katsiris, I.K., Moutzouris, C.I., Tsoukala, V.K., 2004. Improvement of marina design technology using hydrodynamic models. Global Nest 6 (1), 63–72. Tsoukala, V.K., Moutzouris, C.I., 2009. Experimental investigation of the beneficial role of wave transmission in harbors through flushing culverts. In: Proceedings of the 31st International Coastal Engineering Conference, Germany, in press. Tsoukala, V.K., Michalopoulou, P.N., Stamou, A.I., Moutzouris, C.I., 2009. A two dimensional experimental and numerical study on wave transmission in harbor basins through flushing culverts. Coastal Engineering Journal, submitted for publication. US Army Corps of Engineers, 2002. Coastal Engineering Manual (CEM), EM 1110-21100. US Environmental Protection Agency Staff, 1993. Environmental Engineering for Small Boat Basins, Engineering Manual, EMP 10-2-1206. US Environmental Protection Agency Staff, 2001. National management measures guidance to control non-pointsource pollution from marinas and recreational boating, EPA841-B-01-005. van der Meer, J.W., Briganti, R., Wang, B., Zanuttigh, B., 2004. Wave transmission at low-crested structures, including oblique wave attack. In: Proceedings of the 29th International Conference on Coastal Engineering. ASCE, New York, pp. 4152–4164. Wilmott, C.J., 1981. On the validiation of models. Progress in Physical Geography 2, 184–194.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Wave transmission in harbors through flushing culverts V.K. Tsoukala , C.I. Moutzouris National Technical University of Athens, School of Civil Engineering, Laboratory of Harbor Works, 5 Iroon Polytechniou, 15780 Zografou, Athens, Greece
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 August 2008 Accepted 8 January 2009 Available online 20 January 2009
An important aspect that must be considered in harbor design and construction is the water quality within the basin, which depends on the water exchange between the harbor and the surrounding water body. The most internationally distinguished methods to counteract diminished flushing and insufficient renewal occurring leeward of coastal structures include mechanical shakers, overflows, pumps, permeable breakwaters, and flushing culverts. Among these methods, the construction of flushing culverts is favorable due to the low costs of construction and operation. In the present paper, wave transmission through flushing culverts was investigated experimentally in two physical models. Incident wave and transmitted wave heights were determined using wave gauges for various combinations of wave characteristics and geometric characteristics of the flushing culverts. Wave height transformation through the flushing culvert was processed and analyzed for all experimental conditions. The sensitivity of the wave transmission coefficient with respect to other dimensional and non-dimensional parameters was comprehensively investigated in order to define which parameters could most effectively predict the wave transmission coefficient. The wave transmission coefficient increased when the incident wave period, the width, and the height of the flushing culvert increased, the incident wave became steeper, the length of the flushing culvert decreased, and the incident wave angle approached 90o. An empirical equation that correlates the wave transmission coefficient with the wave characteristics and the geometrical characteristics of the flushing culvert was derived using nonlinear regression analysis. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Breakwater Harbor Physical model Wave transmission Water renewal
1. Introduction An important aspect that must be considered in harbor design and construction is the water quality within the basin, which depends on the water exchange between the harbor and the surrounding water body (US Army Corps of Engineers, 2002). Water exchange produces a flushing action in the harbor, which is important, because it reduces the level of pollution by chemical, biological, and floating solids (US EPA, 1993). Studies have shown that adequate flushing improves water quality, reduces or eliminates water stagnation, and helps maintain biological productivity and aesthetic appeal (Schwartz and Imberger, 1988; Schwartz, 1989). Flushing may reduce pollutant concentrations in a harbor basin by 70–90% over a 24-h period (Cardwell and Koons, 1981). Generally, the small dimension and the location of the entrance of the harbor and specially the marinas do not permit the inflow or outflow of significant flow rates, thus resulting in high values of flushing time. Several methods have been developed to counteract diminished flushing and insufficient
Corresponding author. Fax: +30 210 772 2368.
E-mail address: [email protected] (V.K. Tsoukala). 0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.01.005
renewal occurring leeward of coastal structures (US EPA, 2001). The most internationally distinguished methods include mechanical shakers, overflows, pumps, permeable breakwaters, and flushing culverts. Among these methods, the construction of flushing culverts is favorable due to the low costs of construction and operation. The use of flushing culvert enhance wave transmission which results to significant increase of the flow rate passing through the harbor and to increase of the magnitude of water particles velocity. Flushing culverts therefore enhance water circulation through the amplification of the velocity field inside the harbor basin and consequently contribute to the reduction of the flushing times. Previous studies (Stamou et al., 2004) in specific harbor have already shown that the flushing time decreases by 10–32%, depending on the wave characteristics, with the construction of flushing culvert. Underwater placement of culverts is internationally standardized; in this case, tidal hydrodynamics is the main mechanism of enhanced flushing. In Greece, where there are more than 600 harbors of different sizes and operations, construction of flushing culverts for the improvement of water quality is common. Flushing culverts are mainly constructed in protective structures, i.e., breakwaters, to significantly improve water circulation and flushing (Papaioannou et al., 1999). As the ranges of the tides are low, it is preferred that culverts are constructed with their longitudinal axis at sea
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
Nomenclature
L0
b d g h hs Hi Ht Kt l
L R SWL tan j T
width of the flushing culvert water depth gravitational acceleration, g ¼ 9.81 m/s2 height of the flushing culvert submergence depth incident wave height transmitted wave height wave transmission coefficient length of the flushing culvert
water level (Fig. 1). Additionally, as the culvert is placed away from the seabed, sediment transport into the harbor basin is restricted. Although the design of flushing culverts, including determination of their shape, location, and dimensions has been used to improve water circulation and flushing without excessive wave penetration into the harbor basin (Fountoulis and Memos, 2005), the design process has thus far depended on empirical rules. Consequently, the resulting flushing is occasionally inadequate. Additionally, problems can sometimes arise in the functionality of
a x
435
offshore length of the incident wave or deep water wavelength wave length correlation coefficient of the empirical equation surface water level breakwater slope wave period incident wave angle surf similarity parameter or Iribarren number, x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan j= 2pHi =gT 2
the harbor and craft anchorage due to excessive wave penetration into the harbor basin (Tsoukala and Moutzouris, 2009). The effect of the wave penetration is quantified using the wave transmission coefficient, which is defined as the ratio of the transmitted wave height (Ht) to the incident wave height (Hi), i.e., Kt ¼ Ht/Hi. Several efforts have been made to demonstrate the beneficial effect of designing flushing culverts using numerical models (Stamou et al., 2004) or finite difference models of an idealized harbor (Fountoulis and Memos, 2005). However, there is no
Flushing culverts
Marina Mytilini
Flushing culverts Marina Kalamata
Flushing culverts
Marina Mati Fig. 1. Characteristic views of flushing culverts in three marinas in Greece.
ARTICLE IN PRESS 436
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
experimental work in the literature dealing with the influence of the flushing culvert on the characteristics of the transmitted waves or on the flushing time of the harbor. The majority of existing studies have mainly analyzed wave transmission through various types of structures. In all of these studies, Kt was used as the main non-dimensional parameter, and it was correlated with the wave and geometrical characteristics of the structures using dimensional, parametric, and regression analyses. In this paper, wave transmission through flushing culverts was investigated experimentally in two physical models. A constitutive set of experiments was carried out to observe the effect of the geometrical characteristics of flushing culverts on the wave transmission coefficient in order to define which parameters could most effectively predict the wave transmission coefficient. Furthermore, the experimental data were processed using dimensional, parametric, and regression analyses. As a result a preliminary empirical equation that can be used in the design of flushing culverts was derived. The proposed equation correlates the wave transmission coefficient (Kt) to the characteristics of flushing culverts and to the wave characteristics.
intra-gravity waves. In the present study, only regular waves were produced (Fig. 2). 2.2. Test layouts and structures Two physical models of two different harbors were used in order to examine wave transmission through flushing culverts. Model 1 is a physical model of Ikaria Harbor (Fig. 3), and Model 2 is a physical model of Kolybari Harbor (Fig. 4). These models were constructed using Froude similarity at scales of 1:100 and 1:60, respectively. The breakwater of each model was composed of two layers of quarry rocks having densities of 2.65 kg/m3. In Model 1, the breakwater slopes were 1:2.5 and 1:1.5 for the seaward and leeward sides, respectively, while for Model 2, the slopes were 1:1.5 for both sides of the breakwater. Three flushing culverts in Model 1 and one in Model 2 were placed at the windward breakwaters of the harbors, with their axes at the mean water surface level as shown in Fig. 3. The different dimensions of the flushing culverts that were tested for each model are shown in Table 1.
2. Experimental setup and data sampling
2.3. Instrumentation
In order to give a better description of wave transmission through a flushing culvert in the lee of a breakwater, a comprehensive set of laboratory experiments were preformed in one of the three wave basins of the Laboratory of Harbor Works (LHW), National Technical University of Athens (NTUA). The wave basin has the dimensions 26.8 m 24.3 m 1.0 m (length width depth).
2.3.1. Wave gauges Resistance-type wave probes were used (see Fig. 5) to record and determine wave characteristics for each test. A quoted precision of 72% can be achieved with these wave gauges. The resistance from the gauges is converted to a voltage. The voltage outputs of the wave probes are converted to water levels using calibration factors for each wave gauge. The HR Wave Data Calibration Module (Bersford et al., 2005) was used to calibrate the wave gauges; every day prior the experiments, several different water levels (3 or 5 depending on the length of the wave gauge) were measured against the corresponding output voltages from the gauges. The voltage conversion factor was then determined as the slope of the linear curve best fitting the height–voltage diagram.
2.1. Wave generation The hydraulic wave generation system is composed of two paddles side by side, each 6.0 m wide, and can generates waves based on the signal-generator system. Produced waves must be in the range of 0.3–3 Hz and should not be higher than 0.5 m. No active absorption on the wave paddles was used, but passive absorption was placed at the rear end of the basin and at both sides of the basin. Waves are absorbed on sidewalls by rubble mound structures made of stones having a characteristic diameter equal to 0.08 m, and they are also absorbed on wiredrawn cylinders. The input signal to the wave generator is produced by a random wave synthesizer that uses HR Wave-Maker software (Bersford, 2007). HR Wave-Maker is an advanced software package produced by HR Wallingford and it is designed to simulate a variety of sea states for both single and multi-element wavemakers. It can generate regular and irregular waves and can also produce compensation for set down or very long-period
2.3.2. Monitoring All tests were monitored using a video camera, with observations being recorded on video tape and with a digital camera for further analysis.
3. Test conditions and data sampling Preliminary experiments were conducted to understand the flow characteristics better as well as to select the final wave conditions. Waves that generated agitation due to entering the harbors and interdigited flows between the three flushing
Fig. 2. Schematic representation of the flushing culvert placed in the breakwater.
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
437
Fig. 3. Model 1: (a) top view of Ikaria Harbor and (b) physical model of Ikaria Harbor in LHW.
Fig. 4. Model 2: (a) top view of Kolybari Harbor and (b) physical model of Kolybari Harbor in LHW.
culverts in Model 1 were discarded. Based on these tests, it was decided that the results from the middle flushing culvert would be utilized. Based on the preliminary observations and in order to avoid the diffraction observed in Model 2, the seabed on the seaward side of the breakwater was also demolished. A total of 105 tests performed under different combinations for 24 monochromatic wave climates were finally separated for further analysis. As no active absorption on the wave paddles was used, the constructed Models may reflect from 20% to 80% of the incident wave amplitude back towards the wave generator. To overcome this problem all the measurements of incident wave height used in the following analysis were made by repeating the test sequence without the structure in place. The experiments were organized into 13 series: five in Model 1 and eight in Model 2. The ranges of tested variations in water depth (d), wave characteristics including height (Hi), wave period (T), and incident wave angle (a), as well as the geometrical characteristics of the flushing culverts including length (l), height (h), and width (b), are presented in Table 1. The list of wave conditions is presented in Table 2. To avoid surface tension effects and the production of
capillary waves, the wave periods were selected to be higher than 0.35 s (Hughes, 1993). The effect of flushing culvert width in wave transmission coefficient was investigated through different wave conditions for four different flushing culvert widths (see Table 1) through the experiments conducted in Model 1 (series I2–I6). In Model 2 the influence of the length, the height, and the angle of wave attack on transmission coefficient was also investigated. Experimental series I1, K1, and K5 represent the experiments conducted without the harbor Models in the wave basin in order to estimate the incident wave characteristics. Water level time series were measured using resistance-type wave probes. The location of the probes and their distances from each other depended on the length of the flushing culvert employed in the test. Two wave gauges (named 1 and 2 in Fig. 5) were used to monitor generated wave conditions. Generated wave conditions were calibrated at those wave gauges for all experiments. Additional wave gauges were placed at other points inside the wave basin to obtain data for additional numerical analysis; those data will not be reported here. Fig. 5 shows the locations of the probes in each test condition (with and without the harbor model).
ARTICLE IN PRESS 438
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
Table 1 Characteristics of the experimental series. Series
Harbor
I1 I2 I3 I4 I5
Model 1
K1 K2 K3 K4
Model 2
K5 K6 K7 K8
Model 2
Geometrical characteristics of the flushing culverts
Water depth (m)
Incident wave angle (1)
0.23 0.23
901 901
Height (m)
Length (m)
Width (m)
No flushing culvert 0.025
0.375
0.025 0.050 0.075 0.100
No flushing culvert 0.025 0.025 0.030
0.160 0.320 0.160
0.050
0.25 0.25
901 901
No flushing culvert 0.025 0.025 0.030
0.160 0.320 0.160
0.050
0.25 0.25
601 601
A sampling rate for water surface elevation measurement using wave probes was adopted such that a minimum of 15 samples were collected during one wave period. Data were sampled at a rate of 20 Hz. Test models each consisted of 1000 regular waves. HR Wave Data (Bersford et al., 2005) was used for data acquisition, and the output signals of wave probes (in volts) were stored in different data files for each test.
incident wave. All the acquired time series show strongly nonlinear wave forms that begin to split into a series of smaller waves at the lee of the flushing culvert and further down inside the harbor basin. This includes considerable growth of the second and the third harmonics which are then drastically reduced as the wave continues on the leeward side of the structure. 4.3. Incident wave height
4. Experimental data processing 4.1. Harmonic generation in the wave basin without the harbor models Monitoring of water surface elevation in the water basin without the harbor models was carried out as discussed in 2.4. Recorded signals from the probes were spectrally analyzed using fast Fourier transformation (FFT) method in order to investigate the behavior of water surface elevation and waves traveling along the wave basin. The water-level spectra revealed that higher harmonic wave generation took place under certain test conditions. These tests were discarded from further analysis.
Water surface elevation data collected by wave gauges along the wave basin were used to calculate incident, reflected, and transmitted wave heights. The incident wave height was calculated for each test from data obtained without the harbor models.
5. Analysis and discussion of experimental results Wave height transformations occurring through the flushing culvert under all experimental conditions were processed and analyzed. The effectiveness of several parameters (dimensional and dimensionless) was examined to improve the understanding of the hydrodynamic processes.
4.2. Harmonic generation caused by the flushing culvert
5.1. Transmitted wave height
Harmonic generation due to abrupt changes in water depth is well known, and some experimental and numerical approaches have been developed (e.g., Mei and Black, 1969; Massel, 1983; Goda et al., 1999) to investigate this process. When waves approach the breakwater containing the flushing culvert, part of the wave energy is reflected by the barrier, but, at the bottom of the flushing culvert, harmonic generation with energy transfer to higher harmonics is observed. The higher harmonics generated inside the flushing culvert are transmitted to deeper water behind the breakwater as free waves. The generation and growth of harmonics and their progression leeward were obtained by analyzing all the probe records. The shape of the wave spectra offshore from the breakwater, showed similar trends of harmonic generation to those noted in the flume without the breakwater with the flushing culverts. In the probes in front of the flushing culvert, an increase in wave spectra is observed, which is due to the overlapping of the reflected to the
The height of transmitted waves passing through the experimental flushing culverts was measured using two- and threeresistance wave gauges in Models 1 and 2, respectively, located as shown in Figs. 4 and 5. Waves transmitted onshore included higher harmonic-free waves that were generated over the bottom of the flushing culvert and transmitted to the deeper water behind the breakwater as free waves. Hence, the wave height behind the flushing culvert varied spatially and depended on the number of significant harmonics generated and transmitted inside the culvert. This phenomenon was clearly observed throughout testing. The amplitude of the second harmonic wave was unknown until recently, although this subject has widely been examined in the past for the case of wave transmission over submerged breakwaters (Longuet-Higgins, 1977; Massel, 1983; Driscoll et al., 1992; Losada et al., 1997). An interesting discussion about the observed physics for wave transmission through flushing culverts is given in Tsoukala et al. (2009).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
439
Table 2 Characteristics of the waves.
Model 1
Model 2
a/a
Harbor
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model Model
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
T (s)
Hi (m)
L0 ¼ gT2/2p (m)
Hi/L0
0.52 0.52 0.52 0.52 0.75 0.75 0.75 1.00 1.00 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.50 0.50 0.50 0.50 0.50 0.67 0.67 0.67
0.060 0.014 0.018 0.023 0.012 0.017 0.023 0.022 0.030 0.004 0.007 0.010 0.014 0.018 0.023 0.025 0.005 0.008 0.011 0.015 0.020 0.012 0.017 0.025
0.42 0.42 0.42 0.42 0.88 0.88 0.88 1.56 1.56 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.39 0.39 0.39 0.39 0.39 0.70 0.70 0.70
0.014 0.033 0.043 0.054 0.014 0.019 0.026 0.014 0.019 0.014 0.028 0.040 0.056 0.072 0.092 0.100 0.013 0.020 0.028 0.038 0.051 0.017 0.024 0.036
ˆ rms) calculated from the wave probes at squared wave height (H the leeward side of the flushing culvert was calculated with the following three methods:
Average Hˆrms at each probe calculated from the simple statistic method pffiffiffi ^ rms ¼ 2 2 H
Pn
Average Hˆrms
s
i (1) n calculated from the zero moment of each
i¼1
spectrum Pn pffiffiffiffiffiffiffiffi i¼1 m0i n Square root of the mean zero moment of the spectra sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ rms ¼ 2 2 i¼1 m0i H n
pffiffiffi ^ rms ¼ 2 2 H
Model 2
(2)
(3)
where si is the standard deviation of the water surface level enhanced by the ith wave gauge behind the flushing culvert, n is the number of wave gauges, and m0i is the zero moment of the spectrum of the ith wave gauge.
Fig. 5. Wave basin layout and wave gauge positions: (a) Model 1, (b) Model 2 with 16 cm flushing culvert, and (c) Model 2 with 32 cm flushing culvert.
Comparison of the three above-mentioned methods indicated no significant difference for the calculation of transmitted wave ˆ rms obtained from Eq. (1) was adopted as heights. In this paper, H representative of the transmitted wave height. It should be noted that the calculated transmitted wave height is an average and includes the fundamental and higher generated harmonic waves since the total spectrum (wave energy) was taken into account. This is consistent with the total energy of wave approaches to the breakwater that was considered in the calculation of the incident wave height (see Section 5.1). 5.2. Transmission coefficient
In order to obtain more accurate measurements of the height of waves transmitted through the flushing culvert, different approaches were applied and compared. The average root-mean-
The transmission coefficient was calculated as the ratio of the average transmitted wave height obtained from Eq. (1) to the
ARTICLE IN PRESS 440
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
incident wave height
0.4
^ rms H Kt ¼ Hi
(4)
0.3
Hi ~ 0.004m Hi~ = 0.026m
Analysis of the transmission coefficient and the effects of different variables (dimensional and non-dimensional) on Kt were examined both graphically and statistically. The purpose of this analysis was to define which parameters could most effectively predict the transmission coefficient. The variation of the transmission coefficient is plotted in Figs. 6–11 as a function of the incident wave height Hi, wave period T, flushing culvert length l, flushing culvert width b, submergence depth hs (water depth over the bottom of the flushing culvert, hs ¼ h/2 for all the experiments reported in the present study), and incident wave angle a. Graphical representation demonstrated that the transmission coefficient increased with submergence depth, flushing culvert width, and wave period, whereas culvert length and incident wave height had inverse effects on Kt. The above results provide valuable guidance for the geometric design of flushing culverts. However, non-dimensional parameters
Kt
5.3. Effective parameters for prediction of Kt
0.2
0.1
0.0 0.00
0.16
0.08
0.24
0.32
0.40
l (m) Fig. 8. Plot of transmission coefficient against flushing culvert length (T ¼ 0.40 s and hs ¼ 0.013 m).
0.3 Hi ~ 0.014m
0.4
Hi~ = 0.019m
hs = 0.0125m 0.2
hs = 0.015m
Kt
Kt
0.3
0.2
0.1
0.1 0.0 0.01 0.0 0.00
0.01
0.01
0.02 Hi (m)
0.02
0.03
0.03
0.01
0.01
0.01 hs (m)
0.01
0.02
0.02
Fig. 9. Plot of transmission coefficient against flushing culvert submergence depth (T ¼ 0.40 s and l ¼ 0.16 m).
Fig. 6. Plot of transmission coefficient against incident wave height (T ¼ 0.40 s and l ¼ 0.16 m).
0.3
0.3 Hi ~ 0.014m
Hi ~ 0.013m
Hi~ = 0.019m
Hi ~ 0.024m 0.2
Kt
Kt
0.2
0.1
0.1
0.0 0.00
0.20
0.40 T (sec)
0.60
0.80
Fig. 7. Plot of transmission coefficient against wave period (hs ¼ 0.13 s and l ¼ 0.16 m).
0.0 0.000
0.025
0.050
0.075
0.100
0.125
b (m) Fig. 10. Plot of transmission coefficient against flushing culvert width (T ¼ 0.40 s, l ¼ 0.16 m and hs ¼ 0.0013 m).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
can provide a more effective correlation for the above-dependent parameters. Based on dimensional analysis and a review of the single parameters above, the following dimensionless parameters were considered for further analysis: Kt ¼ f
l hs Hi H H ; ; or x; i ; i ; sin a L0 Hi L0 l b
(5)
0.3
Hi~0.007m Hi~0.014m
0.2
441
in which the deep water wave length L0 is employed instead of the wave period T. The dimensionless parameter, Iribarren number x ðx ¼ tan f= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pHi =gT 2 Þ, can be used to represent wave steepness since it is commonly used for breaking wave analysis in the surf zone, where tan j is the breakwater’s slope. Various plots of the variation of the transmission coefficient as a function of the above non-dimensional parameters are presented in Fig. 12. Graphical representation of the dimensionless parameters demonstrates that the transmission coefficient increases as the length of the flushing culvert decreases. In particular, based on the graphical interpretation of Fig. 12, the effect of the dimensionless parameters on the wave transmission coefficient is described as follows:
Kt
Dimensionless flushing culvert length l/L0 has an inverse effect 0.1
0.0
0
20
60
40
80
100
α (°) Fig. 11. Plot of transmission coefficient against incident wave angle (T ¼ 0.40 s, l ¼ 0.16 m, and hs ¼ 0.0013 m).
on Kt. For the tests with shorter culvert lengths (l/L0o1.0), the transmission coefficient is more sensitive to l/L0; thus, the effect is less for longer culverts. It is therefore likely that, for flushing culverts longer than the incident wave length, variation of the crest width will have a small effect on the transmission coefficient. The submergence ratio hs/Hi has a direct influence on the transmission coefficient. Increasing the submergence ratio allows waves to pass through the flushing culvert with less energy losses, such more energy transfers onshore. Variation of the transmission coefficient with respect to the submergence ratio for different flushing culvert lengths (Fig. 12b) revealed
0.4
0.4
l = 0.16m
0.3
l = 0.32m
0.3
Kt
Kt
l = 0.38m
0.2
0.1
0.1 0.0 0.0
0.2
0.5
1.0
0.0 0.0
1.5
0.5
1.0
0.4
0.4
0.3
0.3 Kt
Kt
l/L0
0.2 0.1 0.0 0.00
1.5 2.0 hs/Hi
2.5
3.0
3.5
0.2 0.1
0.05
0.10
0.15
0.0 0.00
0.05
0.10 Hi/l
Hi/L0
0.15
0.20
0.4 b = 0.025m b = 0.050m b = 0.075m b = 0.010m
Kt
0.3 0.2 0.1 0.0 0.0
0.3
0.6
0.9
1.2
1.5
Hi/b Fig. 12. Plots of transmission coefficient against different dimensionless variables. (a) flushing culvert length ratio, (b) submergence ratio, (c) wave steepness, (d) wave height to flushing culvert length ratio, and (e) wave height to flushing culvert width ratio.
ARTICLE IN PRESS 442
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
greater effects for the submergence ratio in tests with longer flushing culverts. Increasing the incident wave steepness (Hi/L0) causes the transmission coefficient to decrease due to stronger wave breaking. The transmission coefficient decrease as the ratio of incident wave height to flushing culvert length (wave height ratio) increases. The wave height to culvert width ratio has an inverse effect on Kt. Fig. 12(e) shows that, in tests with wider flushing culvert widths (Hi/bo0.3), the transmission coefficient is more sensitive to Hi/b; this effect is decreased for narrower flushing culverts.
The Pearson (linear) correlation test is used to investigate the relative dependency of the transmission coefficient on the above-independent variables. Although the linear correlation coefficient does not conclusively indicate a non-linear relationship, it provides a general estimation of the effectiveness of the variables. The correlation coefficients between Kt and other dimensionless variables were calculated for all the conducted experiments, and the values are provided in Table 3. The correlation coefficients indicate that the wave height to flushing culvert width ratio (Hi/b), the submergence ratio (hs/Hi), the wave steepness (Hi/L0), the ratio of flushing culvert length to wave height (Hi/l), and the flushing culvert ratio (l/L0) are the parameters that most affect the transmission coefficient and are used in the next section for the formation of a design equation for the transmission coefficient. 5.4. Proposed empirical model Based on the graphical and statistical analyses, six dimensionless variables were finally adopted for generation of a design
Correlation coefficient
l/L0 hs/Hi Hi/L0 Hi/l Hi/b
0.28 0.58 0.57 0.31 0.62
(6)
Four functions were considered for developing an appropriate design equation for Kt. The first function was a non-linear equation that included the submergence ratio, wave height to breakwater crest ratio, dimensionless crest width, and inverse ratio of wave steepness K t ¼ b1
hs Hi
b2
þ b3
l Hi
b4
þ b5
L0 Hi
b6
þ b7
b8 b10 l b þ b9 L0 Hi (7)
where the coefficients b1–b10 are defined by non-linear regression analysis. The second form of the equation is similar to Eq. (7) except that the inverse ratio of the wave steepness was replaced by the Iribarren number K t ¼ b1
hs Hi
b2
þ b3
l Hi
b4
þ b5 ðxÞb6 þ b7
b8 b10 l b þ b9 L0 Hi
(8)
The third form of the equation is similar in form to that proposed by d’Angremond et al. (1996) and van der Meer et al. (2004) for wave transmission over submerged breakwaters K t ¼ b1
hs Hi
b2
þ b3
l Hi
b5
ð1 expðb6 xÞÞ þ b7
b Hi
b8
(9)
where the coefficients b1–b8 are determined by non-linear regression analysis. Finally, the fourth form of the equation is similar to Eq. (9) except that the impact of the Iribarren parameter is incorporated into the dimensionless ratios of flushing culvert length and width as follows: b2 " b4 b8 # hs l b K t ¼ b1 þ b3 þ b7 (10) ð1 expðb6 xÞÞ Hi Hi Hi An easy way to include the effect of oblique waves is to add a sin a2/3 (van der Meer et al., 2004) as function to Eq. (10). This leads to the final prediction formulas for wave transmission through flushing culverts, including obliquity. Therefore, the influence of incident wave angle a was also investigated in all the above equations by multiplying the equations by sin a2/3. Non-linear regression analysis was carried out using SPSS15 (SPSS Inc, 2004) software. Several scripts were developed to determine parameters based on non-linear least-squares regressions.
Table 3 Correlation coefficient between Kt and other dimensionless variables. Dimensionless parameter
equation for the transmission coefficient Ht hs Hi l b L0 Kt ¼ ; ; ¼f ; ; x or ; sin a Hi Hi l L0 Hi Hi
Table 4 Calibration coefficients calibrated by non-linear regression analysis for best fitting of data and the corresponding statistical parameters for evaluation of the models. Equation number
Model no.
Calibration coefficients b1
b2
b3
b4
b5
R2
b
RMSE
Iw
e
b6
b7
b8
b9
b10
0.001 0.000 0.000 0.001 0.001 0.000
0.20 0.21 11.34 0.19 0.20 0.19
0.45 0.44 0.01 0.46 0.44 0.46
0.71 0.73 0.64 0.71 0.73 0.73
1.04 1.04 1.05 1.04 1.04 1.04
0.30 0.29 0.34 0.30 0.29 0.29
0.93 0.94 0.92 0.93 0.94 0.94
0.20 0.20 0.22 0.20 0.20 0.20
1 2 3 4 5 6
162.05 342.52 164.79 382.35 374.71 611.97
0.00 0.00 0.00 0.00 0.00 0.00
33.11 286.04 158.35 253.42 256.25 612.15
0.00 0.00 0.00 0.00 0.00 0.00
0.20 0.01 4.82 92.04 71.10 0.19
1.08 0.94 0.01 0.00 0.00 0.79
128.39 56.34 0.00 36.77 47.22 0.00
9
7
0.47
0.04
0.39
0.26
8277.30
1.09
0.00
–
–
–
0.65
1.05
0.32
0.92
0.22
10
8 9
0.00 0.10
3.31 0.76
0.16 2.64
0.58 0.03
0.01 3.00
0.88 0.05
0.53 0.21
– –
– –
– –
0.73 0.79
1.06 1.06
0.27 0.27
0.94 0.98
0.20 0.19
8
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
A total of 10 different design equations for Kt were investigated based on Eqs. (7)–(10). The calculated values for calibration coefficients b1–b10 were derived from regression analysis and are provided in Table 4. Models 1–3 in Table 4 correspond to Eq. (7), and Models 4 and 6 correspond to Eq. (8). In Models 2 and 4, the effect on the predicted equations was examined by incorporating the sin a function. The term l/L0 on the right-hand side of Eqs. (7) and (8) was ignored in Models 3 and 6 to assess the sensitivity of Kt, to the culvert length ratio l/L0; coefficients b7 and b8 were considered to be zero in regression
0.4 Model 1
Kt predicted
0.3
Correlation coefficient R2 P ð N ðX m X¯ m Þi ðX c X¯ c Þi Þ2 R2 ¼ PN i¼1 P ¯ 2 N ¯ 2 i¼1 ðX m X m Þi i¼1 ðX c X c Þi
0.1
0.0
analysis of Models 3 and 6. Models 7 and 8 were extracted from the formulae of d’Angremond et al. (1996) and van der Meer et al. (2004) (Eq. (9)), while Model 9 was extracted from Eq. (10). Graphical comparison of the measured transmission coefficient with the calculated values using the proposed equations is displayed in Fig. 13. Comparison of the figures for Models 1 and 3, as well as Models 4 and 5 in Fig. 13, indicates that the flushing culvert length ratio (l/L0) has little influence on the transmission coefficient. The dimensional parameters l and L0 have both been employed to define other non-dimensional parameters, and their combination can be ignored from the corresponding equations. Statistical evaluation of the models was carried out by calculating the following statistical indicators: Bias or distortion (b): PN ðX =X Þ (11) b ¼ i¼1 c m i N Root-mean-squared error (RMSE) "P #1=2 N 2 i¼1 ððX c X m Þ=X m Þi RMSE ¼ N
0.2
0.2 Kt measured
0
0.4
0.4 Model 2
Kt predicted
¼
2 0.1
0.0
0.2 Kt measured
0
0.4
0.4 Model 3
Kt predicted
0.3
0.2
0.1
0.0
0
0.2 Kt measured
0.4
Fig. 13. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (7).
(12)
(13)
The Wilmott number Iw (Wilmott, 1981) and error function e (Haller et al., 2002), were also implemented for additional discussion of the validity of the various models PN 2 i¼1 ðX c X m Þi (14) Iw ¼ 1 P 2 ¯ ¯ i¼1 ðjX c X m j þ jX m X m jÞ
0.3
0.2
443
"P N
2 i¼1 ðX c X m Þi PN 2 i¼1 X m
#1=2 (15)
where Xc are the calculated values, Xm are the measured (available data) values, and the overbars indicate the average value of the parameter. Perfect agreement is indicated when R2, bias b, and Wilmott index Iw are 1.0, and the error function e and RMSE are zero. R2 is considered unsatisfactory for measurement of fit quality in a multivariate regression-based relationship (Draper, 1984); thus, the validity of the model was mostly assessed using other parameters. The values of the five statistical parameters are provided in Table 4. Graphical and statistical comparison of the measured transmission coefficient with the calculated values from the proposed model revealed that Models 1 through 8, with the exception of Model 3, provided reasonable approximations of the transmission coefficient with respect to laboratory measurements. Iw and b were greater than 0.92, and the RMSE and e error values were less than 0.30 and 0.20, respectively. Models 6 and 9 present marginally more accurate results when compared with the other models hs l K t ¼ 374:7 þ 256:3 þ 71:1ðxÞ0:001 Hi Hi 0:001 0:44 # l b þ 0:204 (16) sin a2=3 L0 Hi " " 0:032 0:76 hs l þ 2:64 K t ¼ 0:1 Hi Hi # 0:05 # b 3 ð1 expð0:21 xÞÞ sin a2=3 Hi
(17)
ARTICLE IN PRESS 444
V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
culvert geometrical characteristics. Therefore, Eq. (17) should be adopted for calculating the transmission coefficient through flushing culverts with axes at the mean sea water level.
0.4 Model 4
Kt predicted
0.3 6. Conclusions
0.2
Wave transmission through flushing culverts placed across a breakwater was investigated in the present study for improvement of harbor basin quality. An extended set of experiments was
0.1 0.4 Model 7
0.2 Kt measured
0
0.4
0.4 Model 5 0.3 Kt predicted
0.3 Kt predicted
0.0
0.2
0.1 0.2
0.0
0
0.2 Kt measured
0.1
0.4
0.5 Model 8 0
0.2 Kt measured
0.4 Model 6 0.3 Kt predicted
0.4
0.4
Kt predicted
0.0
0.3
0.2
0.1 0.2 0.0 0.0
0.1
0.1
0.2 0.3 Kt measured
0.4
0.5
0.5 Model 9
0
0.2 Kt measured
0.4
0.4
Fig. 14. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (8).
The structure of Eq. (17) (Model 9) is less complicated than that of Eq. (16) (Model 6). Moreover, it is more common in the literature, as its predictive form was reported by d’Angremond et al. (1996), and it has subsequently been used by other investigators for predicting the transmission coefficient of submerged breakwaters. Comparison of measured and calculated Kt using Eq. (17) for different series of experiments is shown in Fig. 14. As implied from the figure, the proposed equation provides reasonable approximations of transmission coefficients through flushing culverts for a range of wave conditions and different
Kt predicted
0.0
0.3
0.2
0.1
0.0 0.0
0.1
0.2 0.3 Kt measured
0.4
0.5
Fig. 15. Comparison of measured and calculated transmission coefficients using the proposed models derived from Eq. (9) (Model 7) and Eq. (10) (Models 8 and 9).
ARTICLE IN PRESS V.K. Tsoukala, C.I. Moutzouris / Ocean Engineering 36 (2009) 434–445
equation and the optimisation of the coefficients derived from the regression analysis (Figs. 15 and 16).
0.5
Kt predicted
0.4
0.3
I2 I3 I4 I5 K2 K3 K4 K6 K7 K8
References
0.2
0.1
0.0 0.0
0.1
445
0.2 0.3 Kt measured
0.4
0.5
Fig. 16. Comparison of measured and calculated transmission coefficients using the final proposed model (Eq. (12)).
conducted to examine the effect of different parameters on wave transmission through flushing culverts. The transmission coefficient Kt was found to increase with increased submergence depth, width of the flushing culvert, and wave period, whereas flushing culvert length and wave height had inverse effects on Kt. It was also observed that Kt was higher for waves with perpendicular angles of incidence. Analysis of the effect of dimensionless parameters on the transmission coefficient showed that the inverse wave height to flushing culvert width ratio b/Hi, the submergence ratio hs/Hi, the inverse wave steepness Hi, L0 and the ratio of wave height to flushing culvert length Hi/l, influence the transmission coefficient; the relative magnitude of the effect of each parameter is in the order given above. Considering the effect of dimensionless parameters four different forms of empirical equations were evaluated via nonlinear regression analysis in order to best correlate the wave transmission coefficient with the characteristics of the waves and the geometrical characteristics of the flushing culvert. The proposed equation seems to be a satisfactory preliminary approximation and could be used for the estimation of the transmission coefficient. According to the scales of the tested physical models, a quite well prediction is expected for waves with heights between 0.4 and 2.5 m and periods 4.0–7.8 s, for flushing culverts with the mean axis placed at the sea water level. However, as this equation derived from laboratory experiments possible scale effects are fomented in the results. To overcome this uncertainty, further experiments should be undertaken in different scales – mainly greater than the selected in the present study – for both regular and irregular waves. An active wave absorber system is also recommended for the scale comparison of results from different faculties. Finally measurements in nature are needed to be conducted for the validation of the proposed
Bersford, P.J., 2007. HR Wave Maker Wave Generation control program, Software manual, Report IT 453 Issue 5, H.R. Wallingford. Bersford, P.J., Spencer J.M.A, Clarke, J., 2005. HR Wave Data—Data Acquisition and analysis software program User manual, Report IT 493, H.R. Wallingford. Cardwell, R.D., Koons, R.R., 1981. Biological considerations for the siting and design of marina and affiliated structures in puget sound. Technical Report no. 60. State of Washington, Department of Fisheries. d’ Angremond, K., Van der Meer, J.W., De Jong, R.J., 1996. Wave transmission at low crested structures. In: Proceedings of the 25th International Conference on Coastal Engineering. ASCE, New York, pp. 2418–2427. Draper, N.R., 1984. The box-wetz criterion versus R2. Journal of Statistical Society 147, 100–103. Driscoll, A.M., Darlymple, R.A., Grilli, S.T., 1992. Harmonic generation and transmission past a submerged rectangular obstacle. In: Proceedings of the 23th International Conference on Coastal Engineering. ASCE, New York, pp. 1142–1152. Fountoulis, G., Memos, C., 2005. Optimization of openings for water renewal in a harbor basin. Journal of Marine Environmental Engineering 7 (4), 297–306. Goda, Y., Okazaki, K., Kagawa, M., 1999. Generation and evolution of harmonic wave components by abrupt depth changes. In: Proceedings of the Coastal Structures ’99. ASCE, New York, pp. 649–658. Haller, M.C., Darlymple, R.A., Svendsen, I.A., 2002. Experimental study of nearshore dynamics on a barred beach with rip channels. Journal of Geophysical Research 107 (C6), 14-1–14-21. Hughes, S.A., 1993. Physical Model and Laboratory Techniques in Coastal Engineering, Advanced Series on Ocean Engineering, Vol. 7. World Scientific Publishing, Ltd., Singapore, p. 568. Longuet-Higgins, M.S., 1977. The mean forces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines. Proceedings of the Royal Society of London A352, 463–489. Losada, I.J., Patterson, M.D., Losada, M.A., 1997. Harmonic generation past a submerged porous step. Coastal Engineering 31, 281–304. Massel, S.R., 1983. Harmonic generation by waves propagating over a submerged step. Journal of Coastal Engineering 7, 357–380. Mei, C.C., Black, J.L., 1969. Scattering of surface waves by rectangular obstacles in water of finite depth. Journal of Fluid Mechanics 38 (3), 499–511. Papaioannou, A.I., Azorakos, S., Moutzouris, C.I., 1999. Experimental study of wave energy penetration inside a harbor basin through a gap in the external breakwater. In: Moutzouris, C.I. (Ed.), Second Conference on Coastal Zone Management and Improvement. Athens, Greece, pp. 211–218. Schwartz, R.A., 1989. The flushing characteristics of Hillarys boat harbor. M. Eng. Sc. Thesis, University of Western Australia, Centre for Water Research Reference ED, 259 RS. Schwartz, R.A., Imberger, J., 1988. Flushing behavior of a coastal marina. In: Proceedings of the 21st International Coastal Engineering Conference, Spain, ED 439. SPSS Inc, 2004. SYSTATs v.13.0, Technical Software. Stamou, A.I., Katsiris, I.K., Moutzouris, C.I., Tsoukala, V.K., 2004. Improvement of marina design technology using hydrodynamic models. Global Nest 6 (1), 63–72. Tsoukala, V.K., Moutzouris, C.I., 2009. Experimental investigation of the beneficial role of wave transmission in harbors through flushing culverts. In: Proceedings of the 31st International Coastal Engineering Conference, Germany, in press. Tsoukala, V.K., Michalopoulou, P.N., Stamou, A.I., Moutzouris, C.I., 2009. A two dimensional experimental and numerical study on wave transmission in harbor basins through flushing culverts. Coastal Engineering Journal, submitted for publication. US Army Corps of Engineers, 2002. Coastal Engineering Manual (CEM), EM 1110-21100. US Environmental Protection Agency Staff, 1993. Environmental Engineering for Small Boat Basins, Engineering Manual, EMP 10-2-1206. US Environmental Protection Agency Staff, 2001. National management measures guidance to control non-pointsource pollution from marinas and recreational boating, EPA841-B-01-005. van der Meer, J.W., Briganti, R., Wang, B., Zanuttigh, B., 2004. Wave transmission at low-crested structures, including oblique wave attack. In: Proceedings of the 29th International Conference on Coastal Engineering. ASCE, New York, pp. 4152–4164. Wilmott, C.J., 1981. On the validiation of models. Progress in Physical Geography 2, 184–194.