## Abstract

This paper investigates the velocity and the trajectory of water particles under surface waves, which propagate at a constant water depth, using particle image velocimetry (PIV). The vector fields and vertical distributions of velocities are presented at several phases in one wave cycle. The third-order Stokes wave theory was employed to express the physical quantities. The PIV technique's ability to measure both temporal and spatial variations of the velocity was proved after a series of attempts. This technique was applied to the prediction of particle trajectory in an Eulerian scheme. Furthermore, the measured particle path was compared with the positions found theoretically by integrating the Eulerian velocity to the higher order of a Taylor series expansion. The profile of average travelling distance is also presented with a solution of zero net mass flux in a closed wave flume.

## 1. Introduction

Nonlinear wave problems have piqued the interest of a considerable number of mathematicians over the years, dating back to Gerstner [1], who provided a solution for periodic and permanent waves in canals of infinite depth. Gerstner's theory, called trochoidal wave theory, was the first to be developed for nonlinear waves. The theory consists of the exact solutions required to express a possible form of gravity wave motion in the Lagrangian system, despite the fact that the motion is rotational. Modern approaches towards Gerstner's wave with a particular distribution of vorticity can be found in Constantin [2] and Henry [3]. In the theory, water particles are described as circular orbits whose radii diminish with water depth. Froude [4] and Rankine [5] developed the same theory in a different manner, starting from the assumption of a trochoidal waveform. By analysing the kinematics of the motion of a fluid, and checking the distribution of pressure in a flowing fluid, they arrived at the conditions for the distribution of fluid velocity, which satisfied Euler's equations in Lagrangian form. The trochoidal wave theory is more of a curiosity than a cornerstone of nonlinear surface-wave theory. However, extending Gerstner's theory, the edge wave motion was investigated by Constantin [6] for mono-stratified water and by Stuhlmeier [7] for stratified water.

In the nineteenth century, several applied mathematicians and physicists attempted to obtain satisfactory solutions in the Eulerian system. Airy [8], for instance, approached the case of irrotational waves by applying Gerstner's results and limiting the analysis to first-order approximations. In the theory of a small-amplitude wave (Airy wave) for a horizontal bottom in water of arbitrary depth, the surface profile and velocity motion are sinusoidal and the trajectory is circular in deep water and elliptical in intermediate water. Stokes [9] first used a systematic perturbation technique while assuming that an infinite Fourier series may represent the unknown functions such as velocity potential, surface displacement and wave speed. Stokes arrived at a second-order solution for waves in water of finite depth and a third-order solution for waves in water of infinite depth. However, the particle trajectory becomes an incomplete closure due to nonlinearity. Later, Stokes [10] investigated the motion to the fifth-order approximation for waves in water of infinite depth and the third-order approximation for waves in water of finite depth. Rayleigh [11] further examined the higher approximations for waves in finite deep water, and a systematic treatment of the theory of canal waves was developed by Levi-Civita [12]. Struik [13] and Bowden [14] were influenced by Levi-Civita's work, and extended his procedure to more general cases for waves of small amplitude.

Significant attention has been paid to the problem of higher-order wave profiles in the Eulerian system. Nearly 80 years after Stokes' work, the method for finite water depth was extended to the third order by Borgman & Chappelear [15] and to the fifth order by Skjelbreia & Hendrickson [16] and Fenton [17]. In the last two decades, the power series approach has been replaced by a global bifurcation approach in the analysis of waves of large amplitude [18]. For practical purposes, Sainflou [19] derived a nonlinear standing-wave equation based on the trochoidal wave theory, and later Miche [20] obtained the second-order equation for zero mass transport. The boundary value problem solved by Tadjbakhsh & Keller [21] gave a third-order solution that was the most common method for analysing standing waves in water of finite depth.

The subject of water particle velocity and trajectory may be of considerable importance for both progressive and standing-wave problems; however, the number of such investigations is still limited compared with those concerning nonlinear wave shapes. This is due to the inconvenience of describing the particle velocity and path experimentally under a wave motion in the Lagrangian system. Thus, the Eulerian approach is more frequently used for solving fluid dynamic problems. Recently, Constantin [22] and Constantin & Strauss [23] solved a set of differential equations to investigate the particle path in a fixed Eulerian system. Generally, to obtain the particle trajectory from a computational point of view, the particle velocity from the Eulerian solutions is expanded to higher order and integrated about its mean position.

Flow visualization in the early stages was performed by Wiegel & Johnson [24] and Wallet & Ruellan [25]. Wiegel & Johnson [24] showed that water particle velocity is greater in its forward movement under the wave crest than in its backward movement under the wave trough. This results in mass transport in the direction of wave propagation. The orbital motion near the surface becomes nearly circular in shape, and the orbital paths become flatter as distance from the surface increases. The vertical motion is zero at the bottom where the particle moves back and forth in a purely horizontal motion. Wallet & Ruellan [25] took a series of photographs that show the trajectory motion in a wave flume. Two wave trains of the same frequency travelling in opposite directions were produced by an artificial progressive wave that was reflected by a barrier set at the end of the flume. The corresponding reflection coefficient varied from 0 (no reflection) to 1 (perfect reflection). White seeding particles suspended in the water were photographed during one wave period. In the case of pure progressive waves, most trajectories were ellipses, but they were circular near the free surface and flattened towards the bottom. In addition, there were some open loops indicating a slow drift to the wave direction near the surface and in the opposite direction near the bottom. As the reflection increased, the orbits became increasingly flattened and inclined, and, finally, the trajectories became streamlines in the case of perfect standing waves.

Visualization techniques have played a more important role, as these yield both qualitative and quantitative insights in fluid flows. Significant developments in particle image velocimetry (PIV) and particle tracking velocimetry (PTV) have enabled the visualization of velocity fields and water particle paths. These modern flow visualization techniques can be seen in Adrian [26] and Willert & Gharib [27]. PIV constitutes a powerful technique to perform two-dimensional quantitative measurements for a large variety of flows [28]. Many investigations have been conducted to improve the performance of PIV through progress in image processing techniques. For instance, the Coastal and Ocean Engineering group of Tokyo Metropolitan University has investigated the mechanism of internal waves in a two-layer system comprising homogeneous fluids of slightly different densities using PIV. Shimizu *et al*. [29] performed laboratory tests, attempting a PIV system with a double-pulse Nd:YAG laser and a charge-coupled device camera. Later, Umeyama [30] and Umeyama & Shinomiya [31] set up an apparatus to measure the instantaneous velocity of internal waves using a halogen lamp as a light source and some high-definition digital video cameras. Umeyama & Matsuki [32] recently measured similar physical quantities with two frequency-doubled Nd:YAG lasers of 50 mW energy at 532 nm as the illumination source.

From an Eulerian prospective, the motion of an incompressible fluid is distinctive if the velocity vectors occupy an instantaneous velocity field. As PIV is used to represent a regular array of velocity vectors, it is convenient to define the Eulerian velocity based on the average particle motion in the possible space. In contrast, PTV traces the individual particle path from a sequence of images in a system. From a Lagrangian viewpoint, PTV is better suited than PIV for handling unsteady flow. Quantitative results for Lagrangian fluid motion can be obtained through computerized analyses of the particle images in the modern PTV technique. Umeyama [33] employed a PTV system with single-exposure images to track particle displacements for surface waves with or without a steady current. In addition to the basic use of PTV, an alternative measurement technique was proposed to describe particle trajectories in an Eulerian scheme through PIV analysis. Umeyama *et al.* [34] measured and analysed the water particle velocity and trajectory using a PIV system with two Nd:YAG lasers. Tracers used for this experiment were Diaion and micro bubbles.

In this study, we focus on surface waves propagating in water of finite depth. A third-order perturbation solution is derived to express the nonlinearity of water particle velocity. In addition, the particle displacement is obtained by integrating the second-order horizontal and vertical velocities over time. The water particle velocity and trajectory are measured using a new PIV system with an 8 W Nd:YAG laser. The validation results are presented by comparing the measurements of the PIV system to those of the electromagnetic current (EC) meter. An alternative measurement technique is proposed to describe the instantaneous velocity fields, horizontal and vertical velocity distributions and water particle trajectories of the Stokes waves.

## 2. Theory

### (a) Governing equations

We begin by reviewing some theoretical aspects of two-dimensional irrotational wave motion in incompressible and inviscid fluids. Consider a cross section of a wave field that is perpendicular to the crest line with Cartesian coordinates (*x*,*z*), with the *x*-axis pointing in the direction of wave propagation and the *z*-axis pointing vertically upwards, while the origin is located at the mean water level. Let *u*(*x*,*z*,*t*) and *w*(*x*,*z*,*t*) be the horizontal and vertical components of the velocity at time *t*, respectively. In terms of a velocity potential *ϕ*(*x*,*z*,*t*), the velocity components can be expressed as
2.1
2.2
The equation of mass conservation satisfies the Laplace equation:
2.3
The boundary conditions for surface waves are as follows:
2.4
2.5
2.6
where *g* is acceleration due to gravity, *η* is water surface elevation and *h* is water depth.

### (b) Nonlinear water waves with a uniform current

In the finite-amplitude wave theory, the perturbation technique that yields uniformly valid expansions is chosen to solve equations (2.3)–(2.6). Assuming the wave train is weakly nonlinear, its potential, elevation and angular frequency are perturbed in the order of *a*/*k*≪1, where *a* is wave amplitude and *k* is wavenumber. Thus, the perturbation variables are expressed by a series in which each term is a power of the *n*th order of the dimensionless perturbation parameter:
2.7
2.8
2.9
Here, *U* is steady current, *ε* (=*a*/*k*) is perturbation parameter and *σ* is angular frequency. The quantities *ϕ*^{(n)}, *η*^{(n)} and *σ*^{(n)} are of the order *O*(*ε*^{n}).

We replace the free-surface boundary conditions in equations (2.4) and (2.5) with the conditions to be satisfied for *z*=0 instead of *z*=*η*. The boundary conditions for the first, second and third powers can be obtained by expanding equations (2.4) and (2.5) into a Taylor series in *z*, and substituting equations (2.7)–(2.9) into them and equation (2.7) into equation (2.6). According to Umeyama [35] the results of a third-order solution may be obtained as follows:

The first-order solution is 2.10 2.11 2.12 The second-order solution is 2.13 2.14 2.15 in which The third-order solution is 2.16 2.17 2.18 in which

### (c) Particle trajectories

If (*x*(*t*),*z*(*t*)) is the instantaneous water particle position below the wave at time *t*, then the horizontal and vertical velocities become
2.19
When the mean position of a water particle is given at , the path of the particle is denoted as
2.20
where *ς* is horizontal displacement and *ξ* is vertical displacement.

Expansion of equations (2.19) in Taylor series yields
2.21
2.22
For a first approximation, we assume that *ς* and *ξ* are small quantities, and therefore and can be replaced with and . Integrating equations (2.21) and (2.22) into the first term with respect to *t* yields the first-order displacements. Again, submitting these results into equations (2.21) and (2.22), the solution for the instantaneous water particle position becomes
2.23
2.24
where *A*^{(n)} and *B*^{(n)} are constants to the *n*th order. For a discussion of the convergence of the series, refer to Toland [36] or Constantin & Escher [37].

### (d) Mass-transport velocity

Longuet-Higgins [38] described a general method for finding the mass-transport velocity in an oscillatory motion of small amplitude, and applied it to the case of waves in water of uniform depth. The motion in the interior of the fluid depends on the ratio of the wave amplitude to the thickness of the boundary layer. When the ratio is small, the drift velocity is given by the conduction solution. However, when the ratio is large, the velocity profile is not predicted in the interior. The analytical solution of the mass-transport velocity is given by 2.25 where is the mass-transport velocity. The solution satisfies the condition of zero net mass flux integrated over the depth, i.e. a net flow in the direction of wave advance near the surface and the bed, balanced by a net flow in the opposite direction at mid-depth.

## 3. Experiments

Umeyama presented a detailed sketch of the experimental facility [39–41], in which the wave flume was 25 m long, 0.7 m wide and 1 m deep. A piston-type wave generator was placed at one end of the tank, and a wave absorber was installed at the other end. The wave absorber consisted of a vinylidene chloride mat and limited the reflection to 5 per cent over a wide range of water depth, wave period and wave height. The test section was located in the area from 14 to 15 m downstream from the wave paddle. The water depth was 30–40 cm. All tests had a fixed wave period of 1 s, but the wave height varied from 1.6 to 3.4 cm. Three different techniques were employed for laboratory measurements: PIV, EC meter and capacitance wave gauge.

The PIV technique may be better suited for a study of wave motion in an Eulerian system. The representation of the velocity vector field is a typical example of the PIV. However, the PIV result may be applied to particle tracking and mass-transport processes [33]. The water particle velocity was measured using a PIV system of a single-exposure image. The PIV system consisted of frequency-doubled dual Nd:YAG lasers having 8 W energy at 532 nm. A 2 mm light sheet of uniform intensity was emitted from the upper side, which covered one glass panel area between two steel flanges of the flume. The system used a high-definition digital video camera (SONY HXR-NX5J) with a maximum resolution of 1920×1080 pixels. The video camera was arranged linearly 2 m from the sidewall of the flume. The video camera had a field of view that covered an area up to 88.9×50 cm (frame rate 16:9). The water was seeded with Diaion (DK-Fine HP21) with a grain size and specific gravity of 0.11 mm and 1.01, respectively. A standard cross-correlation interrogation was performed with an interrogation window size of 21×21 pixels and a candidate window size of 42×42 pixels. Vector fields were obtained with the PIV analysis by processing a pair of image frames taken with Δ*t*=0.1 s.

After the PIV measurement provided the velocity available at spatially discrete nodal locations in an Eulerian scheme, we estimated the particle velocity and location. Figure 1 depicts the motion of a particle within a tracking time step Δ*t* along an arbitrary trajectory of a particle across a general mesh of quadrilateral cells. First, the velocity value of a Lagrangian point (A) at time *t* is obtained by interpolating the neighbouring velocity values (*u*_{1} at P_{1}, *u*_{2} at P_{2}, *u*_{3} at P_{3} and *u*_{4} at P_{4}). Second, the particle associated with the Lagrangian point at *t* is traced to a hypothetical location (A^{′}) at *t*+Δ*t*. Thus, these Lagrangian velocities become
and
In addition to the PIV measurement, the horizontal and vertical velocity components were measured using an EC meter placed 10.5 m from the wave-generating board. The data were sampled at approximately 14 points for *h*=30 cm and 19 points for *h*=40 cm over the entire water depth. The surface displacement was measured using a resistance-type probe located near the EC meter. Under each set of experimental conditions, wave data were collected continuously at a frequency of 47 Hz.

## 4. Results

Figure 2 shows the temporal surface displacements for two different experimental cases: (i) *H*=1.9 cm for *h*=30 cm, and (ii) *H*=3.4 cm for *h*=40 cm. The wave steepness is *H*/*L*=0.0138 and 0.0232, respectively. The abscissa is the dimensionless value in terms of one wave cycle, and the ordinate is the surface displacement in centimetres. The circles show the measured instantaneous positions of the free surface, while the solid curve represents the corresponding profile obtained by the third-order Stokes wave theory. Generally, the nonlinear wave profile exhibits a deeper depression near the trough and a lower rise near the crest. This tendency is not very noticeable in both test conditions, but it is relatively prominent for larger wave steepness. These comparisons reveal that the third-order theory predicts the measured interfacial displacement with a reasonable degree of accuracy. The nonlinear nature of the governing equations and boundary conditions is the reason why the wave profile exhibits a form that is not sinusoidal.

Figure 3 shows the instantaneous velocity vector maps at an interval of 0.25 s in PIV measurements. These pictures of the field indicate the typical Eulerian view. At each phase, a water particle under a progressive wave rushes from the area near a node positioned not far behind the crest and gathers around another node to maintain a surface profile. The extreme value of the horizontal velocity appears at the crest or trough positions where the water surface is at a maximum or minimum. An array of clockwise and anticlockwise circulations exists in the field of view and moves in the wave direction. Each pair of counter-rotating circulations appears to be nearly symmetric for *h*=30 cm. The experimental result illustrates that the velocities increase exponentially with elevation above the bottom. We cannot define whether a particle beneath a crest moves faster in the forward direction than beneath a trough in the backward direction. A pair of counter-rotating vortices becomes asymmetric for *h*=40 cm. The nonlinear effect appears with increasing wave steepness.

Figure 4 compares the experimental velocity at a cross section with the theoretical velocity. The measured horizontal and vertical velocity components at four different phase values over a wave cycle are plotted with those calculated by the third-order Stokes approximation. The experimental data were sampled near the centre of the video frame. As seen in the horizontal-velocity profiles, the PIV result fits the theoretical prediction for most phases other than a couple of points near the bottom at the moment when the flow reverses. The EC plot of horizontal velocity occasionally shows significant differences in comparison with the theoretical curve in the total depth. The theoretical vertical velocity agrees with the PIV and EC data in both experimental cases.

The trajectory of a particle during three wave periods is plotted for every time step (Δ*t*) in figure 5, which displays orbital geometries at several depths. The small circles indicate the instantaneous positions of a water particle based on the PIV measurement, while the solid line denotes the path computed by equations (2.23) and (2.24). In the case of *H*=30 cm, the measured trajectories seem to be closed in the layer from the surface to the mid-depth, but the particles in the layer from the mid-depth to the bottom move slightly backwards from the wave direction in order to maintain a balance between the following and opposing fluxes. In contrast, the theoretical solution shows that the particle marches forwards in a non-closed loop in the total depth, implying that each particle has a periodic motion per period but yields an overall forward drift. The theoretical trajectory leads to an asymmetry of particle orbit, and this asymmetry decreases with an increase in depth. This measured result is in agreement with the theoretical considerations by Constantin [22] and Constantin & Strauss [42]. Note that ideal trajectory data cannot be obtained without removing extraneous effects such as reflection from the wall and higher harmonics generated by the wave maker, among others. The trajectory is thinner and flatter owing to the larger horizontal excursion and smaller vertical excursion near the bottom. The experimental result in the case of *H*=40 cm proves that there are no closed particle orbits for Stokes waves of larger wave steepness. The measured orbital size is slightly larger than the predicted one, but the difference between both forward drifts caused by the mass-transport velocity depends on elevation from the bottom: it is conspicuous in the near-surface layer and is rather inconspicuous in the intermediate layer.

Figure 6 shows the vertical distribution of the average travelling distance caused by mass transport under progressive waves in two different test cases. The position of travelling distance at an interval of 2 cm between the surface and the bottom is indicated by several kinds of symbols: each symbol represents a position at a time from the start of measurement. A test of the present dataset shows a tendency for the data to deviate from the theoretical profile in the total depth. It is uncertain whether the deviation is caused by the inadequacy of the theory or an inaccuracy in the measurement. However, the trend of the measured vertical distribution is in agreement with the theoretical prediction by Longuet-Higgins [38].

## 5. Conclusions

The physical characteristics of nonlinear surface waves were theoretically and experimentally examined in this study, which derived the third-order perturbation solution for computing the surface displacement and water particle velocity of Stokes waves in the Eulerian scheme. The temporal surface variation was measured using a wave gauge, and the two-dimensional cross-sectional velocity was accurately observed with a PIV system. An EC meter was also provided to measure the horizontal and vertical velocity components at a vertical section, and the distributions of horizontal and vertical velocity components at an arbitrary phase were compared with the corresponding distributions by the PIV and the Stokes third-order solution. The satisfactory agreement between these experimental and theoretical results confirms the accuracy of the PIV measuring technique. The particle trajectories of a flow were simulated using a solution based on the definition of the Lagrangian approach to the second order of a Taylor series expansion. The PIV measurement with Diaion was good because the seeding particles were not affected by gravity and buoyancy. We demonstrated how the PIV algorithm should be employed to compute the Lagrangian velocity and track water particle paths in Eulerian grids. Reasonable agreement between the computed and measured results with theoretical findings showed that the high accuracy of the proposed approach could be applied to the Lagrangian description of the trajectory of a water particle. Finally, the measured Stokes drift using the Lagrangian residual velocity was evaluated using the mass-transport velocity in a closed wave flume.

## Acknowledgements

The author wishes to thank Shinya Watanabe and Yasuhiro Takei for preparing all experimental data. In addition, the author greatly appreciates support for this publication from the Erwin Schrödinger International Institute for Mathematical Physics.

## Footnotes

One contribution of 13 to a Theme Issue ‘Nonlinear water waves’.

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