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The study on solitary waves generated by a piston-type wave maker
Ocean Engineering 117 (2016) 114–129
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The study on solitary waves generated by a piston-type wave maker Nan-Jing Wu a,n, Shih-Chun Hsiao b, Hsin-Hung Chen c, Ray-Yeng Yang c,d a
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi City 600, Taiwan Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan City 701, Taiwan Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan City 709, Taiwan d International Wave Dynamics Research Center, National Cheng Kung University, Tainan City 709, Taiwan b c
art ic l e i nf o
a b s t r a c t
Article history: Received 2 April 2015 Received in revised form 21 January 2016 Accepted 16 March 2016
The focus of present study is on how to generate solitary waves in a wave flume using a piston type wave maker. Experimental observations are implemented to evaluate the stability of the generated solitary waves. A better “stability” implies that the generated solitary wave can travel a longer distance without an obvious decay. Another discovery of this study is the imperfect fitness of wave paddle to the flume could degenerate the solitary wave heights in a great amount. Numerical simulations are carried out to verify this. This study concludes that the method proposed by Wu et al. (2014) is effective for generating solitary waves, even if the wave paddle fits the wave flume imperfectly. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Solitary wave Wave generation Piston-type wave maker Numerical simulation Experiment
1. Introduction It has been more than one century since the remarkable discovery of solitary waves was made. Till now, many studies on the characters and behaviors of solitary waves have been done via experimental approaches. An essence of these studies is how to generate solitary waves in a wave flume. Methods for laboratory generation of solitary waves include dropping weights (Hammack and Segur, 1974; Wiegel, 1955), displacing a given mass of water by a rising bottom (Daily and Stephan, 1952), releasing a prescribed amount of water behind a barrier (Kishi and Saeki, 1966), and horizontal movement of a vertical paddle by a piston-type wave maker (Camfield and Street, 1969; Goring, 1978). Among these methods, Goring's method which prescribes the velocity of the wave paddle of a piston-type wave maker has for several decades been the most commonly employed. Nevertheless, one could find in some studies that Goring's method only works for small solitary waves. When a higher solitary wave is generated, a slight depression of the free surface could be observed behind the main impulse accompanying with the wave height decrease as the wave propagates. This could be found in some papers (Grilli and Svendsen, 1991; Grilli et al., n
Corresponding author. E-mail address: [email protected] (N.-J. Wu).
http://dx.doi.org/10.1016/j.oceaneng.2016.03.020 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
1994; Hsiao and Lin, 2010) though the focuses of these studies were neither on the generation nor the propagation of solitary waves. Alternative methods were proposed for improving the “stability” of the generated solitary waves (Guizien and Barthélemy, 2002; Malek-Mohammadi and Testik, 2010; Wu et al., 2014). A better “stability” implies that the generated solitary wave can travel a longer distance without an obvious decay. The method proposed in the work of Wu et al. (2014) which just slightly modifies Goring's method seems to have the most effective improvement. However, in that paper the verification was done by using just a potential flow numerical model. Further verification on the method of Wu et al. (2014) was carried out by Farhadi et al. (2016) in which numerical simulations with the incompressible smooth particle hydrodynamics model (ISPH) was employed and the viscosity of the water was considered. For showing how the method of Wu et al. (2014) works in a real wave flume, we conduct several physical tests in the wave flume at Tainan Hydraulics Laboratory, National Chung Kung University, Taiwan (THL, NCKU). The observed wave heights are systematically smaller than expected. During the experiment, we found that the wave paddle fits the flume imperfectly and water leaks from the gaps during motion of the wave paddle. This is presumed as the reason why the observed wave heights are systematically smaller than the expected and then this is verified by using FLOW 3Ds, a commercial software for free surface viscous and turbulent flows, to simulate the process of
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Table 1 Formulae and coefficients of the Fenton's ninth order solitary wave solution. Order i
Ki
ηi
Ci
1
S2
1.000000
1.000000
2
−0.75S2 + 0.75S 4
0.625000
0.500000
0.554688
0.042857
0.561535
0.034286
3
0.625S − 1.8875S + 1.2625S
4
−1.36817S2 + 3.88033S 4 − 4.68304S 6 + 2.17088S 8
5
1.86057S2 − 7.45136S 4 + 12.7637S 6 − 11.4199S 8 + 4.24687S10
0.567095
0.031520
6
−2.57413S2 + 13.2856S 4 − 31.1191S 6 + 40.1068S 8 − 28.4272S10 + 8.728S12
0.602969
0.029278
7
3.4572S2 − 22.782S 4 + 68.258S 6 − 116.794S 8 + 120.49S10 − 71.057S12 + 18.608S14
0.624914
0.026845
8
−4.6849S2 + 37.67S 4 − 139.28S 6 + 301.332S 8S12 − 411.416S10 + 355.069
0.670850
0.030263
9
6.191S2 − 60.67S 4 + 269.84S 6 − 712.125S 8 + 1217.98S10 − 1384.37S12 + 1023.07S14
0.700371
0.021935
2
4
6
− 180.212S14 + 41.412S16 − 450.29S16 + 90.279S18
Cη x = ξ dξ = u¯ (ξ , t ) = dt h + η x=ξ
(1)
where ξ(t ) is the position of the wave paddle at time t , while t is elapsed time since the start of the wave paddle motion, C is the wave celerity or say wave speed, η is the free surface displacement, and h is the still water depth. In the dissertation of Goring (1978) the solitary wave solution of Boussinesq (1871) was used to determine the free surface displacement η and the wave celerity C for Eq. (1). That is
Fig. 1. Experiment setup.
η = H S2 Table 2 The list of experimental conditions.
(2)
where H is the wave height, and
Case Wave generation method
Still water depth (cm)
Target wave height (cm)
Stroke of the wave paddle (cm)
Re (water at Maximum speed of the 24 °C) paddle (m/ s)
in which K is the boundary outskirt decay coefficient, and
1
Goring
40
20
65.32
0.8086
2.886 × 105
X = ξ − Ct − x 0
2
Goring
40
14
54.65
0.5966
5
1.782 × 10
3
Goring
40
8
41.31
0.3615
8.161 × 104
4
Modified
40
20
71.62
0.8023
3.140 × 105
5
Modified
40
14
59.62
0.5947
1.937 × 105
6
Modified
40
8
43.91
0.3613
8.668 × 104
solitary wave generation under this condition.
S = sech⎡⎣ KX ⎤⎦
(3)
(4)
where x0 is the initial position of the wave crest. Because the solitary wave is generated in a quiescent water flume, x0 must be chosen as a negative value, which means the wave crest is initially out of the domain. Its value is chosen by considering the length of the wave. The boundary outskirt decay coefficient K is determined as
K=
3H 4h3
(5)
whereas the wave speed is determined as 2. Goring's solitary wave generation method and its modification By assuming the average horizontal water particle velocity adjacent to the wave paddle, u¯ , equals the wave paddle velocity, Goring (1978) derived a formula to determine the wave paddle trajectory during the solitary wave generating procedure
C=
g (h + H )
(6)
This is a well-known solitary wave generation method proposed by Goring. By Malek-Mohammadi and Testik (2010) Eq. (1) was challenged while in other papers (Guizien and Barthélemy, 2002; Wu et al., 2014) Eq. (1) was considered acceptable but the way of determining η and C were suggested to be altered. The Fenton's (1972) ninth order solitary wave solution was
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Fig. 2. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 20 cm while the still water depth is 40 cm. (i.e. case 1 vs case 4).
preferred for determining η and C in the paper of Wu et al. (2014). The free surface displacement η of Fenton's ninth order solution reads
find them in the paper of Fenton (1972). In a similar way, the boundary outskirt decay coefficient and the wave celerity of Fenton's solitary wave solution are determined as
9 ⎛ H ⎞i η = h ∑ ηi⎜ ⎟ ⎝ h⎠
K=
i=1
(7)
in which the shape of the wave is related to the S in Eq. (2), but in a much more complicated way. The formulae to determine the values of η1 to η9 are listed in Table 1 while one could also
and
3H 3
4h
9
⎛ H ⎞i ⎟ h⎠
∑ Ki⎜⎝ i=1
(8)
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Fig. 3. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 14 cm while the still water depth is 40 cm. (i.e. case 2 vs case 5).
⎛ C2 = gh⎜⎜ 1 + ⎝
9
⎛ H ⎞i ⎞ ⎟⎟ h ⎠ ⎟⎠
∑ Ci⎜⎝ i=1
(9)
where coefficients K1, ⋯ , K9 , and coefficients C1, ⋯ , C9 are also listed in Table 1 and one could find them in the paper of Fenton (1972) too. It should be kept in mind that when determining η by Eq. (7), K in Eq. (3) and C in Eq. (4) for the ninth order solution are obtained by using Eqs. (8) and (9). In this paper we denominate the method applying Eqs. (7) to (9) into Eq. (1) as the modified method. It is worthily noted that one could obtain the third order solitary wave profile of Grimshaw (1971) by omitting terms higher than 3. An analogous method
employing the third order solitary wave profile to Eq. (1) could be found in Liu et al. (2006), which focused on the viscous effects of solitary wave propagation in a wave tank experimentally and numerically. It is necessary to explain how to implement the solitary wave generating process here so readers can generate solitary waves in their own flumes following this paper. In many machines the Goring's method is built-in, but (ξ v.s. t ) or (u¯ v.s. t ) format are also acceptable for prescribing the motion of the wave paddle. The (ξ v.s. t ) and (u¯ v.s. t ) data for the input can be obtained by numerical integration of Eq. (1). Before starting the numerical integration, the initial position of the wave crest, x0 in Eq. (4), has to be chosen. According to our experiences, it is
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Fig. 4. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 8 cm while the still water depth is 40 cm. (i.e. case 3 vs case 6).
Table 3 wave heights observed at wave gauges. case wave generation method
1 4 2 5 3 6
Goring Modified Goring Modified Goring Modified
Table 4 Times that wave passes the wave gauges.
target wave height (cm)
wave heights observed at gauges (cm)
20 20 14 14 8 8
18.85 18.25 12.99 12.84 7.20 7.19
A
B
18.16 18.12 12.78 12.74 7.36 7.21
C
17.58 17.99 12.45 12.66 7.22 7.18
case wave generation method
target wave height (cm)
times that wave peak pass the gauges (s)
D
17.08 17.79 12.16 12.56 6.97 7.14
1 4 2 5 3 6
Goring Modified Goring Modified Goring Modified
20 20 14 14 8 8
A
B
C
D
3.24 3.24 3.42 3.41 3.61 3.61
4.53 4.52 4.77 4.74 5.04 5.03
5.81 5.79 6.11 6.08 6.46 6.45
8.38 8.34 8.80 8.75 9.29 9.26
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Fig. 5. Comparisons of observed wave heights with analytical solutions. The initial wave heights are presumed as 18.28 cm, 12.86 cm and 7.26 cm respectively. Because the initial position of the wave paddle is at x = − 0.35 m , the traveling distance at the gauges are 1.85 m, 4.85 m, 7.85 m and 13.85 m respectively.
Fig. 6. The grid map for the numerical simulation.
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Fig. 7. Results of model calibration. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 1).
adequate to set x0 farther than 15 times of the still water depth away from the front face of the wave paddle. The procedures for the numerical integration are listed as follows. Step 1. Set the initial position of the wave paddle as ξ0 . It can be expressed as ξi at i = 0. The elapsed time t at this instant is
t = i Δt = 0 where Δt is the time increment. Step 2. Calculate η x = ξ by using Eq. (7). i Step 3. Calculate u¯ x = ξi by employing the result obtained in the previous step to Eq. (1). It should be noted that C and K should be obtained by using Eqs. (8) and (9). ⎧ ¯ x = ξ Δt , if i = 0 ⎪ ξi + u i Step 4. Predict ξi + 1 by using ξi + 1 = ⎨ and ⎪ ¯ ⎩ ξi − 1 + 2u x = ξi Δt , if i > 0
then obtained the predicted values of η x = ξ + 1 and u¯ x = ξi + 1 with Eq. i
(7) and Eq. (1). Step 5. Modify ξi + 1 by using ξi + 1 = ξi + (u¯ x = ξi + u¯ x = ξi + 1)Δt/2. Step 6. If the number of inquired time steps is achieved, stop. Otherwise, set i = i + 1 and repeat steps 2–5. Once the (ξ v.s. t ) or (u¯ v.s. t ) format data are obtained, one could use them as the input to their wave maker and generate a solitary wave with a better stability in their own flume.
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Fig. 8. Snapshots of water leakage during the wave generating process. Water leaks from the gaps at both sides of the wave paddle. (a) Numerical simulation (b) photo taken in the experiment.
3. Experimental setup The experiment was carried out in the wave flume at Tainan Hydraulics Laboratory, National Chung Kung University, Taiwan (THL, NCKU). The total length of the wave flume is 22 m. The width of the wave flume is 0.5 m. The limitation of the piston stroke is 0.75 m. The central position of the wave paddle is 0.5 m away from the vertical wall at one end while the wave absorber is installed at the other end. Four wave gauges are placed in the wave flume during the experiment. Setting the central position of the wave paddle as x = 0, the positions of the four wave gauges are x = 1.5 m , x = 4.5 m , x = 7.5 m , and x = 13.5 m , respectively. The still water depth is 0.4 m. In each run, the wave paddle rests at x = − 0.35 m initially, i.e. ξ0 = − 0.35 m . A sketch of the experimental setup is shown in Fig. 1. Three target wave heights are considered. They are H = 0.2 m , H = 0.14 m , and H = 0.08 m , respectively. They correspond to relative wave heights 0.5, 0.35, and 0.2. Waves generated by both Goring's method and the modified method are tested. Therefore, there
are totally 6 cases. Each case is repeated twice, so there are 18 runs in total. The experimental conditions of the cases are listed in Table 2. Following Sumer et al. (2010), Reynolds numbers in this table are defined as
Re =
a u¯ max ν
(10)
in which ν is the kinematic viscosity of the fluid, a is the half stroke of the wave paddle while u¯ max is the maximum speed of the wave paddle. The temperature in the laboratory was 24 °C when the air conditioner was on, so ν = 9.15 × 10−7 Pa s.
4. Experimental results The comparisons of the observed solitary waves generated by using Goring's method with those by using the modified method are shown in Figs. 2–4 and Tables 3 and 4. In these figures and
Fig. 9. Results of grid convergence test. Free surface profiles of three different grid resolutions at t = 10 s are plotted.
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Fig. 10. Results of model verification. The target wave height is 14 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 2).
tables, one can find that the modified method performs better than Goring's method. Each solitary wave generated using Goring's method decays more obviously than its counterpart that is generated using the modified method. Such a diminishing wave height in Goring's method is due to the asymmetric solitary wave shape which leads to wave dispersion. Additionally, because of wave height decay, the traveling speeds of these waves generated by using Goring's method are smaller. Therefore, discrepancy between the waves generated by two different methods is more perspicuous at gauges farther from the wave maker. This situation is more apparent in case the target wave height is larger.
It is worthy to note that the height of each generated wave is systematically smaller than the target about 10%. In the experiment, we found during the wave generating process water leaks from the small gaps between the wave paddle and the side walls. It is presumed that water might also leak from the small gap between the wave paddle and the bottom. We postulate this as the reason why the wave height discount occurs. Further discussion and numerical simulations for verifying this assumption are in the following sub-sections.
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Fig. 11. Results of model verification. The target wave height is 8 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 3).
5. Discussion on the viscous and turbulent effects In Liu et al. (2006) it was pointed that the measured wave height decay was close but slightly smaller than the laminar-based analytical expression for damping (Mei and Liu, 1973). It should be noted that the first wave gauge was 43.33 times of the constant depth away from the wave maker. Therefore the solitary wave propagation could be deemed a phenomenon close to laminar flows and the viscous effect might be even smaller. When the wave paddle is rapidly moving during the solitary wave generation process, there could be large velocity gradient near the wave paddle, especially when the wave paddle does not fit the wave flume tightly. Therefore, it is reasonable to presume turbulent effect might be dominant in this case. One could also
find descriptions on the complicated turbulence flows caused by the generation and propagation of solitary waves (Winarta et al., 2012) Due to the water leakage during the solitary wave generation, it is really difficult to predict what the generated wave height really is. Besides, it is a big problem to determine within the region how far from the wave paddle should the turbulent effect be taken into consideration. As a compromise, we try to estimate for each case an equivalent initial wave height whose decay rate is close to the observed data. And then we compare the estimated wave decay with the observed data. According to Keulegan (1948), the damping rate of the solitary wave is
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Fig. 12. Results of model verification. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 4).
⎛ H ⎞−1/4 ⎛ H ⎞−1/4 xl 2h 0 ⎞ ν 1⎛ = ⎜ i⎟ + ⎜ ⎟ ⎜1 + ⎟ ⎝ h h 12 B ⎠ g1/2h03/2 h0 ⎝ 0⎠ ⎝ 0⎠
(11)
where Hi is the initial wave height, xl is the distance traveled by the solitary wave, ν is the kinematic viscosity of the fluid and B is the width of the channel. What we do here is to find an Hi whose total square error between the wave heights calculated by using Eq. (11) and the observed ones is the smallest. Only the solitary waves generated by using the modified method are considered. This is for the sake that those waves generated by using Goring's method decay more rapidly due to the asymmetric shape which leads to wave dispersion. The comparisons are shown in Fig. 5. The comparisons give us three hints. First, for case 6, gauge A might be
too close to the wave paddle so the undeveloped solitary wave is measured. The target wave height of this case is just 8 cm, which implies the characteristic wave length is larger. Second, between gauges A and C, the turbulent effect might be important, for the decay rate of the wave height is greater than the analytical solution of laminar solitary wave propagation. Third, between gauge C and gauge D, the observed data coincide with what was pointed out in Liu et al. (2006).
6. Numerical simulation on the condition that the wave
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Fig. 13. Results of model verification. The target wave height is 14 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 5).
paddle fits the wave flume imperfectly In this study FLOW 3Ds is employed to simulate the solitary wave generating process under the condition that the wave paddle fits the wave flume imperfectly. All the numerical setup follows the experiment. The thickness of the wave paddle is set as 0.03 m. The grid resolution for the numerical simulation is generally 0.01 m and toward the region around the side walls and the bottom the grid spacing is gradually reduced to 0.002 m. The grid map is shown in Fig. 6. All the boundary conditions used for the bottom and side walls are no-slip conditions. Firstly, we chose the case that the target wave height is 0.2 m and the wave generation method is Goring's
(i.e. case 1) for the model calibration. The gap width between the wave paddle and the side walls is considered the same as the gap width between the wave paddle and the bottom. Gap widths from 1.0 mm to 1.5 mm are tested. Because of using the FAVOR (Fractional Area–Volume Obstacle Representation) technique, FLOW 3Ds can do the simulation that distances between obstacles to obstacles/boundaries are smaller than the grid size. As discussed in the previous sub-section, turbulence is taken into consideration. Therefore, several turbulence models, including LES, standard k − ε , and RNG k − ε , are tested as well as the laminar model. Among these tests, the result of setting the gap width as 1.2 mm and using the LES model is closest to the observed data. Fig. 7 shows the result of calibration. Fig. 8 shows the
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Fig. 14. Results of model verification. The target wave height is 8 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 6).
snapshots of water leakage during the wave generating process. It shows that even the gap width is just 0.24 % of the width of the wave flume, the leakage is still quite apparent. Before applying the result of model calibration to other cases, we test the convergence of the grid resolution. The grid spacing is magnified or reduced by 20% for the two additional numerical runs. For the case that grid spacing is magnified by 20%, the grid resolution is generally 0.012 m and the grid spacing is gradually reduced to 0.0024 m toward the region around the side walls and the bottom. For the case that that grid spacing is reduced by 20%, the grid resolution is generally 0.008 m and the grid spacing is
gradually reduced to 0.0016 m toward the region around the side walls and the bottom. The free surface profiles of three different grid resolutions at t = 10 s are plotted in Fig. 9. This figure shows that the grid resolution used in the numerical simulation is fine enough. Figs. 10–14 are the results of model verification, which are the comparisons of the numerical results with experimental observations. Good agreement could be found in these figures. This confirms the postulation of how well the wave paddle fits the wave flume is an important issue in solitary wave generation and explains why the heights of generated waves in our
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Fig. 15. Snapshots of the free surface profile. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 1).
experiment are systematically 10% lower than the target. It might also be the reason why in Guizien and Barthélemy (2002) a solitary wave solution with more water volume was suggested for determining the η in Eq. (1) and why what suggested in Malek-Mohammadi and Testik (2010) was an alternative of Eq. (1) which makes more water volume pushed to form the wave. Snapshots of the free surface profiles in case 1 and case 4 are shown in Figs. 15 and 16. In these two figures one could find that near the wave paddle the free surface goes to a status very close to the target wave height when Goring's method is employed. A free surface depression behind the main pulse is very perspicuous. The wave generated by the modified method also decays as it travels, but it is not so apparent. Even though between
gauges C and D the flow of wave propagation might be laminar, the numerical result using the LES model is still quite close to the experimental data. This could be due to the distance between these two gauges is just 15 times of the water depth so the difference is barely distinguished.
7. Conclusions In this study, we carried out experimental observations to evaluate the stability of solitary waves generated by two methods. One of them is the method proposed by Wu et al. (2014) which is a modification of Goring's method (1978) while
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Fig. 16. Snapshots of the free surface profile. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 4).
the other one is the famous Goring's method. The procedure of implementing the modified solitary wave generating method is expounded. In the experiment we found that the generated wave heights in our experiment are roundly 10% off the targets. The imperfectly fitness of the wave paddle to the wave flume was presumed as the reason and then it is verified by numerical simulations. The results of numerical simulations show that even if the gap width is just 0.24% of the wave flume width, the effect of water leakage is still quite apparent. The viscous and turbulence effects are just discussed a little in this paper. But since the main focus of present study is to show how the modified solitary wave generating method works in a real wave flume, we spare further discussion on effects of viscous/turbulence or the water leakage through the gaps as our future research topics. This study concludes that the method proposed by Wu et al. (2014) works better than the Goring's method
(1978) in a real wave flume even if the wave paddle fits the flume imperfectly.
Acknowledgment The authors would like to thank the financial support from the Ministry of Science and Technology, Taiwan (Grant nos.: MOST 104-2911-I-006-301- and MOST 104-2221-E-415 -020 -).
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Hsiao, S.-C., Lin, T.-C., 2010. Tsunami-like solitary waves impinging and overtopping an impermeable seawall: experiment and RANS modeling. Coast. Eng. 57 (1), 1–18. Kishi, T., Saeki, H., 1966. The shoaling, breaking, and runup of the solitary wave on impermeable rough slopes, In: Proceedings of the 10th International Conference on Coastal Engineering, pp. 322–348. Keulegan, G.H., 1948. Gradual damping of solitary waves. J. Res. Natl. Bur. Stand. 40, 487–498, U.S. Department of Commerce, RP1895. Liu, P.L. -F., Simarro, G., Vandever, J., Orfila, A., 2006. Experimental and numerical investigation of viscous effects on solitary wave propagation in a wave tank. Coast. Eng. 53 (2–3), 181–190. Malek-Mohammadi, S., Testik, F.Y., 2010. New methodology for laboratory generation of solitary waves. ASCE J. Waterw. Port Coast. Ocean Eng. 136 (5), 286–294. Mei, C.C., Liu, P.L. -F., 1973. The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59 (2), 239–256. Sumer, B.M., Jensen, P.M., Sørensen, L.B., Fredsøe, J., Liu, P.L. -F., Carstensen, S., 2010. Coherent structures in wave boundary layers. Part 2. Solitary motion. J. Fluid Mech. 646, 2079–2231. Wiegel, R.L., 1955. Laboratory studies of gravity waves generated by the movement of a submarine body. Trans. Am. Geophys. Union 36 (5), 759–774. Winarta, B., Tanaka, H., Yamagi, H. 2012. Characterization of transition of turbulence in solitary wave boundary layer. In: Proceedings of the 33rd International Conference on Coastal Engineering. Wu, N. -J., Tsay, T. -K., Chen, Y. -Y., 2014. Generation of stable solitary waves by a piston-type wave maker. Wave Motion 51 (2), 240–255.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
The study on solitary waves generated by a piston-type wave maker Nan-Jing Wu a,n, Shih-Chun Hsiao b, Hsin-Hung Chen c, Ray-Yeng Yang c,d a
Department of Civil and Water Resources Engineering, National Chiayi University, Chiayi City 600, Taiwan Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan City 701, Taiwan Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan City 709, Taiwan d International Wave Dynamics Research Center, National Cheng Kung University, Tainan City 709, Taiwan b c
art ic l e i nf o
a b s t r a c t
Article history: Received 2 April 2015 Received in revised form 21 January 2016 Accepted 16 March 2016
The focus of present study is on how to generate solitary waves in a wave flume using a piston type wave maker. Experimental observations are implemented to evaluate the stability of the generated solitary waves. A better “stability” implies that the generated solitary wave can travel a longer distance without an obvious decay. Another discovery of this study is the imperfect fitness of wave paddle to the flume could degenerate the solitary wave heights in a great amount. Numerical simulations are carried out to verify this. This study concludes that the method proposed by Wu et al. (2014) is effective for generating solitary waves, even if the wave paddle fits the wave flume imperfectly. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Solitary wave Wave generation Piston-type wave maker Numerical simulation Experiment
1. Introduction It has been more than one century since the remarkable discovery of solitary waves was made. Till now, many studies on the characters and behaviors of solitary waves have been done via experimental approaches. An essence of these studies is how to generate solitary waves in a wave flume. Methods for laboratory generation of solitary waves include dropping weights (Hammack and Segur, 1974; Wiegel, 1955), displacing a given mass of water by a rising bottom (Daily and Stephan, 1952), releasing a prescribed amount of water behind a barrier (Kishi and Saeki, 1966), and horizontal movement of a vertical paddle by a piston-type wave maker (Camfield and Street, 1969; Goring, 1978). Among these methods, Goring's method which prescribes the velocity of the wave paddle of a piston-type wave maker has for several decades been the most commonly employed. Nevertheless, one could find in some studies that Goring's method only works for small solitary waves. When a higher solitary wave is generated, a slight depression of the free surface could be observed behind the main impulse accompanying with the wave height decrease as the wave propagates. This could be found in some papers (Grilli and Svendsen, 1991; Grilli et al., n
Corresponding author. E-mail address: [email protected] (N.-J. Wu).
http://dx.doi.org/10.1016/j.oceaneng.2016.03.020 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
1994; Hsiao and Lin, 2010) though the focuses of these studies were neither on the generation nor the propagation of solitary waves. Alternative methods were proposed for improving the “stability” of the generated solitary waves (Guizien and Barthélemy, 2002; Malek-Mohammadi and Testik, 2010; Wu et al., 2014). A better “stability” implies that the generated solitary wave can travel a longer distance without an obvious decay. The method proposed in the work of Wu et al. (2014) which just slightly modifies Goring's method seems to have the most effective improvement. However, in that paper the verification was done by using just a potential flow numerical model. Further verification on the method of Wu et al. (2014) was carried out by Farhadi et al. (2016) in which numerical simulations with the incompressible smooth particle hydrodynamics model (ISPH) was employed and the viscosity of the water was considered. For showing how the method of Wu et al. (2014) works in a real wave flume, we conduct several physical tests in the wave flume at Tainan Hydraulics Laboratory, National Chung Kung University, Taiwan (THL, NCKU). The observed wave heights are systematically smaller than expected. During the experiment, we found that the wave paddle fits the flume imperfectly and water leaks from the gaps during motion of the wave paddle. This is presumed as the reason why the observed wave heights are systematically smaller than the expected and then this is verified by using FLOW 3Ds, a commercial software for free surface viscous and turbulent flows, to simulate the process of
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Table 1 Formulae and coefficients of the Fenton's ninth order solitary wave solution. Order i
Ki
ηi
Ci
1
S2
1.000000
1.000000
2
−0.75S2 + 0.75S 4
0.625000
0.500000
0.554688
0.042857
0.561535
0.034286
3
0.625S − 1.8875S + 1.2625S
4
−1.36817S2 + 3.88033S 4 − 4.68304S 6 + 2.17088S 8
5
1.86057S2 − 7.45136S 4 + 12.7637S 6 − 11.4199S 8 + 4.24687S10
0.567095
0.031520
6
−2.57413S2 + 13.2856S 4 − 31.1191S 6 + 40.1068S 8 − 28.4272S10 + 8.728S12
0.602969
0.029278
7
3.4572S2 − 22.782S 4 + 68.258S 6 − 116.794S 8 + 120.49S10 − 71.057S12 + 18.608S14
0.624914
0.026845
8
−4.6849S2 + 37.67S 4 − 139.28S 6 + 301.332S 8S12 − 411.416S10 + 355.069
0.670850
0.030263
9
6.191S2 − 60.67S 4 + 269.84S 6 − 712.125S 8 + 1217.98S10 − 1384.37S12 + 1023.07S14
0.700371
0.021935
2
4
6
− 180.212S14 + 41.412S16 − 450.29S16 + 90.279S18
Cη x = ξ dξ = u¯ (ξ , t ) = dt h + η x=ξ
(1)
where ξ(t ) is the position of the wave paddle at time t , while t is elapsed time since the start of the wave paddle motion, C is the wave celerity or say wave speed, η is the free surface displacement, and h is the still water depth. In the dissertation of Goring (1978) the solitary wave solution of Boussinesq (1871) was used to determine the free surface displacement η and the wave celerity C for Eq. (1). That is
Fig. 1. Experiment setup.
η = H S2 Table 2 The list of experimental conditions.
(2)
where H is the wave height, and
Case Wave generation method
Still water depth (cm)
Target wave height (cm)
Stroke of the wave paddle (cm)
Re (water at Maximum speed of the 24 °C) paddle (m/ s)
in which K is the boundary outskirt decay coefficient, and
1
Goring
40
20
65.32
0.8086
2.886 × 105
X = ξ − Ct − x 0
2
Goring
40
14
54.65
0.5966
5
1.782 × 10
3
Goring
40
8
41.31
0.3615
8.161 × 104
4
Modified
40
20
71.62
0.8023
3.140 × 105
5
Modified
40
14
59.62
0.5947
1.937 × 105
6
Modified
40
8
43.91
0.3613
8.668 × 104
solitary wave generation under this condition.
S = sech⎡⎣ KX ⎤⎦
(3)
(4)
where x0 is the initial position of the wave crest. Because the solitary wave is generated in a quiescent water flume, x0 must be chosen as a negative value, which means the wave crest is initially out of the domain. Its value is chosen by considering the length of the wave. The boundary outskirt decay coefficient K is determined as
K=
3H 4h3
(5)
whereas the wave speed is determined as 2. Goring's solitary wave generation method and its modification By assuming the average horizontal water particle velocity adjacent to the wave paddle, u¯ , equals the wave paddle velocity, Goring (1978) derived a formula to determine the wave paddle trajectory during the solitary wave generating procedure
C=
g (h + H )
(6)
This is a well-known solitary wave generation method proposed by Goring. By Malek-Mohammadi and Testik (2010) Eq. (1) was challenged while in other papers (Guizien and Barthélemy, 2002; Wu et al., 2014) Eq. (1) was considered acceptable but the way of determining η and C were suggested to be altered. The Fenton's (1972) ninth order solitary wave solution was
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Fig. 2. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 20 cm while the still water depth is 40 cm. (i.e. case 1 vs case 4).
preferred for determining η and C in the paper of Wu et al. (2014). The free surface displacement η of Fenton's ninth order solution reads
find them in the paper of Fenton (1972). In a similar way, the boundary outskirt decay coefficient and the wave celerity of Fenton's solitary wave solution are determined as
9 ⎛ H ⎞i η = h ∑ ηi⎜ ⎟ ⎝ h⎠
K=
i=1
(7)
in which the shape of the wave is related to the S in Eq. (2), but in a much more complicated way. The formulae to determine the values of η1 to η9 are listed in Table 1 while one could also
and
3H 3
4h
9
⎛ H ⎞i ⎟ h⎠
∑ Ki⎜⎝ i=1
(8)
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Fig. 3. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 14 cm while the still water depth is 40 cm. (i.e. case 2 vs case 5).
⎛ C2 = gh⎜⎜ 1 + ⎝
9
⎛ H ⎞i ⎞ ⎟⎟ h ⎠ ⎟⎠
∑ Ci⎜⎝ i=1
(9)
where coefficients K1, ⋯ , K9 , and coefficients C1, ⋯ , C9 are also listed in Table 1 and one could find them in the paper of Fenton (1972) too. It should be kept in mind that when determining η by Eq. (7), K in Eq. (3) and C in Eq. (4) for the ninth order solution are obtained by using Eqs. (8) and (9). In this paper we denominate the method applying Eqs. (7) to (9) into Eq. (1) as the modified method. It is worthily noted that one could obtain the third order solitary wave profile of Grimshaw (1971) by omitting terms higher than 3. An analogous method
employing the third order solitary wave profile to Eq. (1) could be found in Liu et al. (2006), which focused on the viscous effects of solitary wave propagation in a wave tank experimentally and numerically. It is necessary to explain how to implement the solitary wave generating process here so readers can generate solitary waves in their own flumes following this paper. In many machines the Goring's method is built-in, but (ξ v.s. t ) or (u¯ v.s. t ) format are also acceptable for prescribing the motion of the wave paddle. The (ξ v.s. t ) and (u¯ v.s. t ) data for the input can be obtained by numerical integration of Eq. (1). Before starting the numerical integration, the initial position of the wave crest, x0 in Eq. (4), has to be chosen. According to our experiences, it is
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Fig. 4. Comparison of the experimental observation by two different solitary wave generation methods. The target wave height is 8 cm while the still water depth is 40 cm. (i.e. case 3 vs case 6).
Table 3 wave heights observed at wave gauges. case wave generation method
1 4 2 5 3 6
Goring Modified Goring Modified Goring Modified
Table 4 Times that wave passes the wave gauges.
target wave height (cm)
wave heights observed at gauges (cm)
20 20 14 14 8 8
18.85 18.25 12.99 12.84 7.20 7.19
A
B
18.16 18.12 12.78 12.74 7.36 7.21
C
17.58 17.99 12.45 12.66 7.22 7.18
case wave generation method
target wave height (cm)
times that wave peak pass the gauges (s)
D
17.08 17.79 12.16 12.56 6.97 7.14
1 4 2 5 3 6
Goring Modified Goring Modified Goring Modified
20 20 14 14 8 8
A
B
C
D
3.24 3.24 3.42 3.41 3.61 3.61
4.53 4.52 4.77 4.74 5.04 5.03
5.81 5.79 6.11 6.08 6.46 6.45
8.38 8.34 8.80 8.75 9.29 9.26
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Fig. 5. Comparisons of observed wave heights with analytical solutions. The initial wave heights are presumed as 18.28 cm, 12.86 cm and 7.26 cm respectively. Because the initial position of the wave paddle is at x = − 0.35 m , the traveling distance at the gauges are 1.85 m, 4.85 m, 7.85 m and 13.85 m respectively.
Fig. 6. The grid map for the numerical simulation.
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Fig. 7. Results of model calibration. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 1).
adequate to set x0 farther than 15 times of the still water depth away from the front face of the wave paddle. The procedures for the numerical integration are listed as follows. Step 1. Set the initial position of the wave paddle as ξ0 . It can be expressed as ξi at i = 0. The elapsed time t at this instant is
t = i Δt = 0 where Δt is the time increment. Step 2. Calculate η x = ξ by using Eq. (7). i Step 3. Calculate u¯ x = ξi by employing the result obtained in the previous step to Eq. (1). It should be noted that C and K should be obtained by using Eqs. (8) and (9). ⎧ ¯ x = ξ Δt , if i = 0 ⎪ ξi + u i Step 4. Predict ξi + 1 by using ξi + 1 = ⎨ and ⎪ ¯ ⎩ ξi − 1 + 2u x = ξi Δt , if i > 0
then obtained the predicted values of η x = ξ + 1 and u¯ x = ξi + 1 with Eq. i
(7) and Eq. (1). Step 5. Modify ξi + 1 by using ξi + 1 = ξi + (u¯ x = ξi + u¯ x = ξi + 1)Δt/2. Step 6. If the number of inquired time steps is achieved, stop. Otherwise, set i = i + 1 and repeat steps 2–5. Once the (ξ v.s. t ) or (u¯ v.s. t ) format data are obtained, one could use them as the input to their wave maker and generate a solitary wave with a better stability in their own flume.
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Fig. 8. Snapshots of water leakage during the wave generating process. Water leaks from the gaps at both sides of the wave paddle. (a) Numerical simulation (b) photo taken in the experiment.
3. Experimental setup The experiment was carried out in the wave flume at Tainan Hydraulics Laboratory, National Chung Kung University, Taiwan (THL, NCKU). The total length of the wave flume is 22 m. The width of the wave flume is 0.5 m. The limitation of the piston stroke is 0.75 m. The central position of the wave paddle is 0.5 m away from the vertical wall at one end while the wave absorber is installed at the other end. Four wave gauges are placed in the wave flume during the experiment. Setting the central position of the wave paddle as x = 0, the positions of the four wave gauges are x = 1.5 m , x = 4.5 m , x = 7.5 m , and x = 13.5 m , respectively. The still water depth is 0.4 m. In each run, the wave paddle rests at x = − 0.35 m initially, i.e. ξ0 = − 0.35 m . A sketch of the experimental setup is shown in Fig. 1. Three target wave heights are considered. They are H = 0.2 m , H = 0.14 m , and H = 0.08 m , respectively. They correspond to relative wave heights 0.5, 0.35, and 0.2. Waves generated by both Goring's method and the modified method are tested. Therefore, there
are totally 6 cases. Each case is repeated twice, so there are 18 runs in total. The experimental conditions of the cases are listed in Table 2. Following Sumer et al. (2010), Reynolds numbers in this table are defined as
Re =
a u¯ max ν
(10)
in which ν is the kinematic viscosity of the fluid, a is the half stroke of the wave paddle while u¯ max is the maximum speed of the wave paddle. The temperature in the laboratory was 24 °C when the air conditioner was on, so ν = 9.15 × 10−7 Pa s.
4. Experimental results The comparisons of the observed solitary waves generated by using Goring's method with those by using the modified method are shown in Figs. 2–4 and Tables 3 and 4. In these figures and
Fig. 9. Results of grid convergence test. Free surface profiles of three different grid resolutions at t = 10 s are plotted.
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Fig. 10. Results of model verification. The target wave height is 14 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 2).
tables, one can find that the modified method performs better than Goring's method. Each solitary wave generated using Goring's method decays more obviously than its counterpart that is generated using the modified method. Such a diminishing wave height in Goring's method is due to the asymmetric solitary wave shape which leads to wave dispersion. Additionally, because of wave height decay, the traveling speeds of these waves generated by using Goring's method are smaller. Therefore, discrepancy between the waves generated by two different methods is more perspicuous at gauges farther from the wave maker. This situation is more apparent in case the target wave height is larger.
It is worthy to note that the height of each generated wave is systematically smaller than the target about 10%. In the experiment, we found during the wave generating process water leaks from the small gaps between the wave paddle and the side walls. It is presumed that water might also leak from the small gap between the wave paddle and the bottom. We postulate this as the reason why the wave height discount occurs. Further discussion and numerical simulations for verifying this assumption are in the following sub-sections.
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Fig. 11. Results of model verification. The target wave height is 8 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 3).
5. Discussion on the viscous and turbulent effects In Liu et al. (2006) it was pointed that the measured wave height decay was close but slightly smaller than the laminar-based analytical expression for damping (Mei and Liu, 1973). It should be noted that the first wave gauge was 43.33 times of the constant depth away from the wave maker. Therefore the solitary wave propagation could be deemed a phenomenon close to laminar flows and the viscous effect might be even smaller. When the wave paddle is rapidly moving during the solitary wave generation process, there could be large velocity gradient near the wave paddle, especially when the wave paddle does not fit the wave flume tightly. Therefore, it is reasonable to presume turbulent effect might be dominant in this case. One could also
find descriptions on the complicated turbulence flows caused by the generation and propagation of solitary waves (Winarta et al., 2012) Due to the water leakage during the solitary wave generation, it is really difficult to predict what the generated wave height really is. Besides, it is a big problem to determine within the region how far from the wave paddle should the turbulent effect be taken into consideration. As a compromise, we try to estimate for each case an equivalent initial wave height whose decay rate is close to the observed data. And then we compare the estimated wave decay with the observed data. According to Keulegan (1948), the damping rate of the solitary wave is
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Fig. 12. Results of model verification. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 4).
⎛ H ⎞−1/4 ⎛ H ⎞−1/4 xl 2h 0 ⎞ ν 1⎛ = ⎜ i⎟ + ⎜ ⎟ ⎜1 + ⎟ ⎝ h h 12 B ⎠ g1/2h03/2 h0 ⎝ 0⎠ ⎝ 0⎠
(11)
where Hi is the initial wave height, xl is the distance traveled by the solitary wave, ν is the kinematic viscosity of the fluid and B is the width of the channel. What we do here is to find an Hi whose total square error between the wave heights calculated by using Eq. (11) and the observed ones is the smallest. Only the solitary waves generated by using the modified method are considered. This is for the sake that those waves generated by using Goring's method decay more rapidly due to the asymmetric shape which leads to wave dispersion. The comparisons are shown in Fig. 5. The comparisons give us three hints. First, for case 6, gauge A might be
too close to the wave paddle so the undeveloped solitary wave is measured. The target wave height of this case is just 8 cm, which implies the characteristic wave length is larger. Second, between gauges A and C, the turbulent effect might be important, for the decay rate of the wave height is greater than the analytical solution of laminar solitary wave propagation. Third, between gauge C and gauge D, the observed data coincide with what was pointed out in Liu et al. (2006).
6. Numerical simulation on the condition that the wave
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Fig. 13. Results of model verification. The target wave height is 14 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 5).
paddle fits the wave flume imperfectly In this study FLOW 3Ds is employed to simulate the solitary wave generating process under the condition that the wave paddle fits the wave flume imperfectly. All the numerical setup follows the experiment. The thickness of the wave paddle is set as 0.03 m. The grid resolution for the numerical simulation is generally 0.01 m and toward the region around the side walls and the bottom the grid spacing is gradually reduced to 0.002 m. The grid map is shown in Fig. 6. All the boundary conditions used for the bottom and side walls are no-slip conditions. Firstly, we chose the case that the target wave height is 0.2 m and the wave generation method is Goring's
(i.e. case 1) for the model calibration. The gap width between the wave paddle and the side walls is considered the same as the gap width between the wave paddle and the bottom. Gap widths from 1.0 mm to 1.5 mm are tested. Because of using the FAVOR (Fractional Area–Volume Obstacle Representation) technique, FLOW 3Ds can do the simulation that distances between obstacles to obstacles/boundaries are smaller than the grid size. As discussed in the previous sub-section, turbulence is taken into consideration. Therefore, several turbulence models, including LES, standard k − ε , and RNG k − ε , are tested as well as the laminar model. Among these tests, the result of setting the gap width as 1.2 mm and using the LES model is closest to the observed data. Fig. 7 shows the result of calibration. Fig. 8 shows the
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Fig. 14. Results of model verification. The target wave height is 8 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 6).
snapshots of water leakage during the wave generating process. It shows that even the gap width is just 0.24 % of the width of the wave flume, the leakage is still quite apparent. Before applying the result of model calibration to other cases, we test the convergence of the grid resolution. The grid spacing is magnified or reduced by 20% for the two additional numerical runs. For the case that grid spacing is magnified by 20%, the grid resolution is generally 0.012 m and the grid spacing is gradually reduced to 0.0024 m toward the region around the side walls and the bottom. For the case that that grid spacing is reduced by 20%, the grid resolution is generally 0.008 m and the grid spacing is
gradually reduced to 0.0016 m toward the region around the side walls and the bottom. The free surface profiles of three different grid resolutions at t = 10 s are plotted in Fig. 9. This figure shows that the grid resolution used in the numerical simulation is fine enough. Figs. 10–14 are the results of model verification, which are the comparisons of the numerical results with experimental observations. Good agreement could be found in these figures. This confirms the postulation of how well the wave paddle fits the wave flume is an important issue in solitary wave generation and explains why the heights of generated waves in our
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Fig. 15. Snapshots of the free surface profile. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is Goring's method. (i.e. case 1).
experiment are systematically 10% lower than the target. It might also be the reason why in Guizien and Barthélemy (2002) a solitary wave solution with more water volume was suggested for determining the η in Eq. (1) and why what suggested in Malek-Mohammadi and Testik (2010) was an alternative of Eq. (1) which makes more water volume pushed to form the wave. Snapshots of the free surface profiles in case 1 and case 4 are shown in Figs. 15 and 16. In these two figures one could find that near the wave paddle the free surface goes to a status very close to the target wave height when Goring's method is employed. A free surface depression behind the main pulse is very perspicuous. The wave generated by the modified method also decays as it travels, but it is not so apparent. Even though between
gauges C and D the flow of wave propagation might be laminar, the numerical result using the LES model is still quite close to the experimental data. This could be due to the distance between these two gauges is just 15 times of the water depth so the difference is barely distinguished.
7. Conclusions In this study, we carried out experimental observations to evaluate the stability of solitary waves generated by two methods. One of them is the method proposed by Wu et al. (2014) which is a modification of Goring's method (1978) while
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Fig. 16. Snapshots of the free surface profile. The target wave height is 20 cm while the still water depth is 40 cm. The method for generating the solitary wave is the modified method. (i.e. case 4).
the other one is the famous Goring's method. The procedure of implementing the modified solitary wave generating method is expounded. In the experiment we found that the generated wave heights in our experiment are roundly 10% off the targets. The imperfectly fitness of the wave paddle to the wave flume was presumed as the reason and then it is verified by numerical simulations. The results of numerical simulations show that even if the gap width is just 0.24% of the wave flume width, the effect of water leakage is still quite apparent. The viscous and turbulence effects are just discussed a little in this paper. But since the main focus of present study is to show how the modified solitary wave generating method works in a real wave flume, we spare further discussion on effects of viscous/turbulence or the water leakage through the gaps as our future research topics. This study concludes that the method proposed by Wu et al. (2014) works better than the Goring's method
(1978) in a real wave flume even if the wave paddle fits the flume imperfectly.
Acknowledgment The authors would like to thank the financial support from the Ministry of Science and Technology, Taiwan (Grant nos.: MOST 104-2911-I-006-301- and MOST 104-2221-E-415 -020 -).
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