EXPERIMENT STUDY OF KINEMATIC AND DYNAMICAL CHARACTERISITCS OF STANDING WAVES

Li Yanbao¹, Zhang Shaosong¹, Andrew Cornett², Arthur Budvietas²

 ¹Dept. of Port &Coastal Engineering, Tianjin University, Tianjin, 300072, China

E-mail: liyanbao@yahoo.com, Tel: 86-022-27406269

 ²Canadian Hydraulics Center, Ottawa, K1A-0R6, Canada, E-mail: andrew.cornett@nrc.ca

Abstract: The velocity and acceleration of water particles near the nodes, and the wave pressure on the breakwater for full and partial standing waves are measured in a wave flume. The results of experiment are analyzed systematically. The measured maximum of velocity, time history of velocity and acceleration are compared with theoretical ones in order to ascertain the range in which each typical theory is available. The effect of partial standing wave on kinematic characteristics of water particles is also discussed. Based on the analysis of the dynamical characteristics of the standing wave, a new method is introduced to analyze the wave pressure with nonlinear properties. The results of this method are compared with those of experiment and Fourier series approximation theory. It is proved to have almost the same precision as Fourier series approximation. But its operation is much simpler than the latter’s.

Keywords: standing wave, partial standing wave, water particle velocity, water particle acceleration, nonlinear wave pressure

1    Introduction

The kinematic character of the standing wave, which directly influences the scour of sand bed in front of a vertical breakwater and the composing of wave pressure, is one of its important hydrodynamic properties. So far, the experiment study on this subject is limited. De Best et al. (1971), Irie and Nadaoka (1984), Hughes and Fowler (1991) and B.M.Sumer et al. (2000) investigated the 2D scour in front of a vertical breakwater. But they did not analyze the water particle velocity systematically. When studying the scour pattern of a sand bed in front of a vertical breakwater, Xie (1981) measured the distribution of the horizontal velocity of water particles in front of the breakwater and compared the experimental results with those of linear and Miche second order standing wave theory. In his experiment, wave steepnesses H/L ranged from 0.0083 to 0.0375, relative water depths d/L ranged from 0.05 to 0.175, here H is the wave heights of incident wave, d is the water depth, L is the wave length. His study indicates that linear theory has a satisfying precision before the shape of sand bed changed. Zhao (1990) introduced electric analog method and applied it to ascertain bottom velocities in front of a vertical breakwater with high rubble bed. He drew the conclusion that the bottom velocity calculated by Miche second order theory is larger than that obtained by electric analog method while the space distributions of them are consistent with each other.

Large numbers of field and model tests indicate that wave pressure on vertical breakwaters shows some nonlinear properties, such as double humps in time-history of wave pressure. As a general rule, perturbation approach and Fourier series approximation are applied to describe these properties. Although Tsai (1992) has proved the high precision of Fourier series approximation in calculating the nonlinear wave pressure, the operation of this method is very complex.

In the base of former research and the systematic experiments, this paper intends to analyze the kinematic and dynamical characteristics of standing waves. Available range of velocity equations of some typical standing wave theories is judged. Maximum velocity of water particles near nodes, time-history of velocity and acceleration obtained by theories are compared with those of experiments. The kinematic character of water particles of partial standing waves is discussed. The relation between the time-history of wave pressure and the kinematic character of water particles is expounded. A simple method to calculate the nonlinear wave pressure with high precision is introduced. Tests were conducted with both 2D regular and irregular waves, however, only the results of 2D regular waves are considered in this paper. 

2    Experiment

Experiment was conducted in a 63m by 1.25 m by 1.25 m wave flume in Canadian Hydraulics Center (CHC). The flume is equipped with an active wave absorption system, which is an essential ingredient for obtaining consistent and accurate results when conducting two-dimensional studies of highly reflective structures. This technology ensures that the incident wave energy arriving at the test model remains constant throughout a test. The breakwater model was made of a 3/4″plastic board, extending from the base of the flume to an elevation of 60 cm. A steel piano hinge was used to connect the base segment to the movable plastic flap allowing changes in the slope of the top of the wall. Flap slope was 30°, 45°, 60°, 90°, 120°, 135°, 150° respectively. The elevation of the top of the model, in all configurations, was 75cm.

For each case of top slope, 3 test water depths were adopted, 10 test wave conditions were used for every water depth. The test wave conditions were listed in Table 1. Wave steepness H/L ranges from 0.021 to 0.068 and relative water depth d/L ranges from 0.139 to 0.353, which exactly extend the range of test conditions of Xie (1981).

Table    1

The research object of this experiment is to investigate the hydraulic performance of the sloping top vertical breakwater. Performance issues include wave reflection, run-up and overtopping rate, wave pressure and globe forces on the wall, and water particle velocity at the node. Only results related to water particle velocity will be analyzed in this paper.

Water particle velocity at 0.25m from the bottom of the flume was measured using a SonTek acoustic Doppler velocimeter. Because of the limitation of the number of velocimeters, only the velocity of water particles near wave node was measured. The probe of velocimeter was positioned at the anticipated node positions for 0.60m water depth with linear wave theory. The distances from it to the face of the wall were respectively 0.460m, 0.675m, 1.025m corresponding to wave periods of 1.1 s, 1.4s, 1.9s. Although velocities along X-axis and Z-axis have been measured at the same time, this paper only consider the horizontal velocity of water particles. Because the vertical velocity near nodes is quite small, its order of magnitude is the same as, even less than that of the error caused by noise.

3    Applied wave theories

Standing waves have been studied for a long time. Classic standing wave theories include linear theory, Sainflou (1928) shallow water standing wave theory, Miche (1944) second order theory and etc. After that, many scholars including Penny (1952), Tadjbaksh-Keller (1960), Goda (1967), Qiu (1985) and so on developed different kinds of high order standing wave theory in farther. Tsai (1989) introduced Fourier series standing wave theory. In this paper linear theory, Qiu’s third order theory, Fourier series approximation theory are considered.

Linear theory is the most effective and simplest theory for standing waves, which makes the hypothesis that standing wave is the superposition of small-amplitude incident wave and reflected wave.

Qiu (1985) developed third order solutions of standing waves in shallow water based on the solutions of Penny and Price. Wavelength of standing wave is deemed to be equal to that of the incident wave when linear and second order theories are adopted to calculate the water particle velocity. But in fact, they are unequal for the effect of nonlinear properties. The differences between them are considered in third order theory of Qiu (1985).

Many of high order theories, which can describe the nonlinear phenomenon of waves well, were deduced by perturbation method. Tsai (1992) developed Fourier series standing wave theory in the base of stream function wave theory. The procedure of Fourier series approximation involves expressing the general solution of Laplase equation in a Fourier series, which satisfy the free surface boundary conditions, keeping the wave surface function as implicit function, discreting wave profile in terms of time and space. Finally, a series of nonlinear equations will be obtained. By solving these equations with numerical method, solution of the wave fields will be got. Fourier series standing wave theory has a high precision when applied to calculate wave pressure on the breakwaters. Tsai has verified that the calculation results convergent to millionth when the order of Fourier series is 8th. So eighth order Fourier series approximation theory is adopted in this paper in order to save the run time of computer.

4    Analysis of experiment results

4.1    Maximum of the horizontal velocity of water particles

The shoreward water particle velocity reaches maximum (crest) when t=0, T, 2T, etc and seaward velocity reaches maximum (trough) when t= T/2, 3T/2, 5T/2, etc, where T is wave period. Considering the periodicity of wave motions, the water particle velocity mentioned following refer to shoreward maximum velocity for the purpose of convenience.

Measured velocity increase with wave height and decrease with the increase of wave period and water depth, which is consistent with the linear standing wave theory.

When no overtopping occurs, the measured velocities are close to the theoretical ones as shown in figure 1and 2. There  and  are defined as the theoretical and measured maximum velocity of water particles. So  is the coincidence indicator of them.  increase with the increase of d/L as shown in figure 2. Results of linear theory is all larger than those of experiment, the range of  is from 1.07 to 1.30, here the extra subscript l represents the linear theory. The range of  and  is from 0.8 to 1.2 when d/L is nearby 0.15 and form 1.0 to 1.3 when d/L>0.20, here the extra subscript 3 and f represent third order and Fourier series approximation theory respectively. Among the three theories, precision of Fourier series approximation is highest, precision of third order theory takes second place, and precision of linear theory is lowest. When d/L increases, their differences become less gradually, especially the difference between the precision of linear theory and third order theory.

Generally speaking, the reliability of linear theory is satisfying, which is coherent with the conclusion drawn by Xie (1981). But his experiment results differ a bit from those of current experiment, in which most of theoretical velocities are larger than measured ones. It may be caused by the differences of the range of H/L and d/L.

4.2    Velocity and acceleration time-history

Time-history of measured velocity sampled at a rate of 20Hz is compared with theoretical one. Although the differences between magnitudes of measured and theoretical velocity vary with d/L and H/L in a whole period, the change trends of them agree well with each other in all cases. Figure 3 shows a case of little difference

Water particle acceleration, which cannot be measured directly by velocimeter, is obtained by applying Shaprio (1970) numerical filtering method to calculate the change rate of velocity. Time histories of acceleration obtained by this ways are also compared with theoretical ones. As shown in Figure 3, relative high error occurs when velocity direction changes for the low sampling frequency of velocity. So the differences of them concentrate on the time near to T/4 and 3T/4, when the precision of Fourier series theory is obviously higher than that of the other two theories.

4.3    Eeffect of partial standing waves on water particle velocity

4.3.1    Eeffect of overtopping

Overtopping leads to partial reflection of incident wave and the forming of partial standing wave. In current experiment, increasing the water depth and wave height engender overtopping while the elevation of the top of the breakwater is fixed. There is no comparison in the same conditions of water depth and wave height. So a non-dimensional velocity is adopted to evaluate the effect of overtopping. The effect of overtopping on water particle velocity is as shown in Figure 4. The velocity in a overtopping conditions is only about 5% smaller than that in a conditions of full reflections(no overtopping). It may be because the overtopping is not so large, the ratio of thickness of overtopping nappe to incident wave height is only from 0.073 to 0.579 in this test.

4.3.2    Effect of top slope

Comparative experiments between top slope and vertical breakwater in the same other conditions were conducted by altering the position of movable plastic flap. To show the effect of tope slope,  is defined as the ratio of water particle velocity in front of a sloping top breakwater to that in front of a vertical breakwater while other conditions are same. Two conditions related to top slope are considered.

(1) the magnitude of the top slope.

(2) the range of reflected wave acting on the slope top, it can be evaluated by a parameter Sc=(d+H-d₁)/d₂, where d₁ and d₂ are respectively heights of vertical part and sloping part of a breakwater.

Analysis indicates that  has little correlation with Sc. And it is also lack of dependence on H/L and d/L. So  with the same top slope can be averaged and compared with each other. As shown in figure 6, effect of top slope on water particle velocity is slight. Except for 30° top slope, <5%.

4.4    Correlation between dynamics and kinematics of standing wave

Large numbers of field and model tests indicate that wave pressure on vertical breakwaters shows some nonlinear properties, such as double humps in time-history of wave pressure, dynamic pressure exhibiting periodicity at the bed, which oscillates at as twice frequency as wave surface does. Perturbation approach and Fourier series approximation are adopted to describe these properties. Present paper will introduce a new method to calculate the nonlinear wave pressure based on the research of A. I. Kouzinezov.

A. I. Kouzinezov^[1] (1954) deduced from LaGrange kinematic equations of ideal liquid to the conclusion that dynamic pressure of ideal liquid is a superposition of three components, i.e. gravity, inertial forces along the tangent and normal direction of the movement of water particle. Gravity component is in direct proportion to net hydraulic head. Components of tangent and normal inertial forces are in direct proportion to the tangent acceleration and the square of water particle velocity respectively. To increase H/L and the depth of the position to be calculated will lead to reducing the curvature radius of water particle trajectory and  increasing the velocity respectively. So they results in normal inertial force increasing and the phenomenon of double humps becoming more obvious.

A.I.Kouzinezov introduced his calculation method according to above theory. But it is rather burdensome. This paper will simplify it with Sainflou theory, which can describe the kinematic characteristics of water particles well. The procedure is stated as follow.

The upright projection of LaGrange kinematics equation is:

                              (1)

where  is water density, g is the acceleration due to gravity, x and z are the water particle coordinates in still water, X axis is located at still water level and Z axis is downward. The coordinate system of equation (1) is compatible with that adopted by Sainflou theory, which is the premise of following deduction.

Substituting  with Sainflou theory:

，     

Integrate both sides of equation (1). Net wave pressure at z will be obtained:

=    

Properties of pressure fields of fluid indicate that it is just the wave pressure on the breakwaters at z. It should be emphasized that converting z from Euler coordinate system to LaGrange coordinate system is necessary when the wave pressure at z is calculated.

Measured time-history of wave pressure is compared with those obtained by above method and Fourier series approximation as shown in figure 6. It can be seen that precision of theoretical time history increases with H/L and z/L. For all H/L in present experiment, precision of both calculating methods is very high at the positions near bottom.

Moreover, comparison in extended conditions indicates that pressure time-history obtained by present method is agreed well with that obtained by Fourier series approximation. But the operation of present method is much simpler than that of the latter, which includes solving high order non-linear equations. So current method is worth recommending for its almost the same precision as Fourier series approximation when applied to calculate non-linear wave pressure.

5    Conclusion

The following conclusions are drawn from present study.

(1) When adopted to describe the kinematic character of water particles in front of a vertical breakwater, the precision of Fourier series approximation theory is highest. With the decrease of nonlinear effect, the differences of precision of different theories become less gradually. 

(2) Compared with those of full standing waves, water particle velocity of partial standing waves only changes a little in the test condition in this paper. As to sloping top vertical breakwaters, the differences between them are in the range of ±5% except for the case of 30° top slope.

(3) Current method based on Sainflou standing wave theory can describe the nonlinear properties of wave pressure well. The operation procedure is much simpler than Fourier series approximation method.

Acknowledgement

This study is supported by CHC R&D Project(Canadian Hydraulics Center), AMC22000 and China Natural Science Fund No. 59779004 and Fund of State Key Lab. of Coastal and Offshore Eng., Dalian University of Technology. The beneficial discussion with Mansard.Etienne, Pratte.Bruce and Zhang Qinhe is acknowledged.

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