• Adrian, R. J., 2005: Twenty years of particle image velocimetry. Exp. Fluids, 39, 159169, https://doi.org/10.1007/s00348-005-0991-7.

  • Adrian, R. J., and J. Westerweel, 2011: Particle Image Velocimetry. Cambridge University Press, 558 pp.

  • Bakhoday-Paskyabi, M., and I. Fer, 2014: Turbulence structure in the upper ocean: A comparative study of observations and modeling. Ocean Dyn., 64, 611631, https://doi.org/10.1007/s10236-014-0697-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bever, A. J., J. E. McNinch, and C. K. Harris, 2011: Hydrodynamics and sediment-transport in the nearshore of Poverty Bay, New Zealand: Observations of nearshore sediment segregation and oceanic storms. Cont. Shelf Res., 31, 507526, https://doi.org/10.1016/j.csr.2010.12.007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bühler, O., 2014: Waves and Mean Flows. Cambridge University Press, 378 pp.

  • Calvert, R., C. Whittaker, A. Raby, P. H. Taylor, A. G. L. Borthwick, and T. S. van den Bremer, 2019: Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth. Phys. Rev. Fluids, 4, 114801, https://doi.org/10.1103/PhysRevFluids.4.114801.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Craik, A. D. D., and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, 401426, https://doi.org/10.1017/S0022112076001420.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dalzell, J. F., 1999: A note on finite depth second-order wave-wave interactions. Appl. Ocean Res., 21, 105111, https://doi.org/10.1016/S0141-1187(99)00008-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Drivdal, M., G. Broström, and K. H. Christensen, 2014: Wave-induced mixing and transport of buoyant particles: Application to the Statfjord A oil spill. Ocean Sci., 10, 977991, https://doi.org/10.5194/os-10-977-2014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fenton, J. D., 1988: The numerical solution of steady water wave problems. Comput. Geosci., 14, 357368, https://doi.org/10.1016/0098-3004(88)90066-0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerstner, F., 1809: Theorie der wellen. Ann. Phys., 32, 412445, https://doi.org/10.1002/andp.18090320808.

  • Grue, J., and J. Kolaas, 2017: Experimental particle paths and drift velocity in steep waves at finite water depth. J. Fluid Mech., 810, R1, https://doi.org/10.1017/jfm.2016.726.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henderson, S. M., J. S. Allen, and P. A. Newberger, 2004: Nearshore sandbar migration predicted by an eddy-diffusive boundary layer model. J. Geophys. Res., 109, C06024, https://doi.org/10.1029/2003JC002137.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holmedal, L. E., and D. Myrhaug, 2009: Wave-induced steady streaming, mass transport and net sediment transport in rough turbulent ocean bottom boundary layers. Cont. Shelf Res., 29, 911926, https://doi.org/10.1016/j.csr.2009.01.012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, Z., 2007: An experimental study of the surface drift currents in a wave flume. Ocean Eng., 34, 343352, https://doi.org/10.1016/j.oceaneng.2006.01.005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lamb, H., 1932: Hydrodynamics. Cambridge University Press, 738 pp.

  • Larsen, J., and H. Dancy, 1983: Open boundaries in short wave simulations - a new approach. Coastal Eng., 7, 285297, https://doi.org/10.1016/0378-3839(83)90022-4.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Le Méhauté, B., 1976: Introduction to Hydrodynamics and Water Waves. Springer-Verlag, 323 pp.

  • Massel, S. R., 1996: On the largest wave height in water of constant depth. Ocean Eng., 23, 553573, https://doi.org/10.1016/0029-8018(95)00049-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mei, C. C., P. L.-F. Liu, and T. G. Carter, 1972: Mass transport in water waves. Massachusetts Institute of Technology Tech. Rep. 146, 278 pp.

  • Monismith, S., 2020: Stokes drift: Theory and experiments. J. Fluid Mech., 884, F1, https://doi.org/10.1017/jfm.2019.891.

  • Myrhaug, D., H. Wang, and L. E. Holmedal, 2014: Stokes drift estimation for deep water waves based on short-term variation of wave conditions. Coast. Eng., 88, 2732, https://doi.org/10.1016/j.coastaleng.2014.01.014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Myrhaug, D., H. Wang, and L. E. Holmedal, 2019: Stokes transport in layers in the water column based on long-term wind statistics. Oceanologia, 61, 522526, https://doi.org/10.1016/j.oceano.2019.03.003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Onoszko, J., 1968: Mass transport in forced wave motion in laboratory investigation. Ann. Soc. Geol. Pol., 38, 110.

  • Paprota, M., 2019: Particle image velocimetry measurements of standing wave kinematics in vicinity of a rigid vertical wall. Instrum. Exp. Tech., 62, 277282, https://doi.org/10.1134/S0020441219020234.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Paprota, M., and W. Sulisz, 2018: Particle trajectories and mass transport under mechanically generated nonlinear water waves. Phys. Fluids, 30, 102101, https://doi.org/10.1063/1.5042715.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Paprota, M., W. Sulisz, and A. Reda, 2016: Experimental study of wave-induced mass transport. J. Hydraul. Res., 54, 423434, https://doi.org/10.1080/00221686.2016.1168490.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Phillips, O., 1977: The Dynamics of the Upper Ocean. Cambridge University Press, 336 pp.

  • Qiao, F., Y. Yuan, T. Ezer, C. Xia, Y. Yang, X. , and Z. Song, 2010: A three-dimensional surface wave-ocean circulation coupled model and its initial testing. Ocean Dyn., 60, 13391355, https://doi.org/10.1007/s10236-010-0326-y.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Russel, R. C. H., and J. D. C. Osorio, 1957: An experimental investigation of drift profile in a closed channel. Proc. Sixth Int. Conf. on Coastal Engineering, New York, New York, ASCE, 293–305.

  • Scandura, P., and E. Foti, 2011: Measurements of wave-induced steady currents outside the surf zone. J. Hydraul. Res., 49, 6471, https://doi.org/10.1080/00221686.2011.591046.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schäffer, H. A., 1996: Second-order wavemaker theory for irregular waves. Ocean Eng., 23, 4788, https://doi.org/10.1016/0029-8018(95)00013-B.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stokes, G. G., 1847: On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8, 441455.

  • Sulisz, W., and M. Paprota, 2015: Theoretical and experimental investigations of wave-induced vertical mixing. Math. Probl. Eng., 2015, 950849, https://doi.org/10.1155/2015/950849.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sulisz, W., and M. Paprota, 2019: On modeling of wave-induced vertical mixing. Ocean Eng., 194, 106622, https://doi.org/10.1016/j.oceaneng.2019.106622.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Suzuki, N., and B. Fox-Kemper, 2016: Understanding Stokes forces in the wave-averaged equations. J. Geophys. Res. Oceans, 121, 35793596, https://doi.org/10.1002/2015JC011566.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Swan, C., 1990: Convection within an experimental wave flume. J. Hydraul. Res., 28, 273282, https://doi.org/10.1080/00221689009499069.

  • Tsuchiya, Y., and T. Y. T. Yamaschita, 1980: Mass transport in progressive waves of permanent type. Proc. 17th Int. Conf. on Coastal Engineering, Sydney, Australia, ASCE, 70–81.

    • Crossref
    • Export Citation
  • Umeyama, M., 2012: Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philos. Trans. Roy. Soc. London, 370A, 16871702, https://doi.org/10.1098/rsta.2011.0450.

    • Search Google Scholar
    • Export Citation
  • van den Bremer, T. S., and Ø. Breivik, 2018: Stokes drift. Philos. Trans. Roy. Soc. London, 376A, 20170104, https://doi.org/10.1098/rsta.2017.0104.

    • Search Google Scholar
    • Export Citation
  • van den Bremer, T. S., C. Whittaker, R. Calvert, A. Raby, and P. H. Taylor, 2019: Experimental study of particle trajectories below deep-water surface gravity wave groups. J. Fluid Mech., 879, 168186, https://doi.org/10.1017/jfm.2019.584.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yu, Q., Y. P. Wang, B. Flemming, and S. Gao, 2012: Tide-induced suspended sediment transport: Depth-averaged concentrations and horizontal residual fluxes. Cont. Shelf Res., 34, 5363, https://doi.org/10.1016/j.csr.2011.11.015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    A schematic presentation of a data acquisition system in the wave flume: (a) side view and (b) top view.

  • View in gallery

    A schematic presentation of the PIV system arrangement in the wave flume and the location of the coordinate system: (a) side view and (b) front view.

  • View in gallery

    Experimental wave conditions according to Le Méhauté (1976).

  • View in gallery

    Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 1.517 s, L = 2.00 m, H = 0.007 m (W1); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

  • View in gallery

    Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 1.517 s, L = 2.04 m, H = 0.04 m (W2); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

  • View in gallery

    Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 3.607 s, L = 5.01 m, H = 0.006 m (W3); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

  • View in gallery

    Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 3.607 s, L = 5.32 m, H = 0.045 m (W4); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

  • View in gallery

    Particle trajectories at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 1.517 s, L = 2.00 m, H = 0.007 m (W1) for the smaller embedded graph; T = 1.517 s, L = 2.04 m, H = 0.04 m (W2) for the larger main graph. Symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

  • View in gallery

    Particle trajectories at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 3.607 s, L = 5.01 m, H = 0.006 m (W3) for the smaller embedded graph; T = 3.607 s, L = 5.32 m, H = 0.045 m (W4) for the larger main graph. Symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

  • View in gallery

    Lagrangian mass transport velocity vertical profiles at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 1.517 s, L = 2.04 m, H = 0.04 m (W2); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

  • View in gallery

    Lagrangian mass transport velocity vertical profiles at a distance of 2.5 m (black), 10 m (blue), and 20 m (green) from the wavemaker for regular water waves: T = 3.607 s, L = 5.32 m, H = 0.045 m (W4); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

All Time Past Year Past 30 Days
Abstract Views 363 41 0
Full Text Views 159 98 5
PDF Downloads 193 106 1

Experimental Study on Spatial Variation of Mass Transport Induced by Surface Waves Generated in a Finite-Depth Laboratory Flume

View More View Less
  • 1 Institute of Hydro-Engineering, Polish Academy of Sciences, Gdańsk, Poland
Free access

Abstract

The aim of laboratory experiments conducted in a wave flume is to confirm that wave-induced mass transport varies along the way of propagation of regular mechanically generated waves in water of finite depth. This effect is attributed to the interactions between primary waves generated in the flume and associated free waves. Measurements of wave kinematics in several locations along the flume are performed to evaluate mass transport. Wave gauges and the particle image velocimetry technique are employed, respectively, to provide information on the evolution of the oscillating water surface and the wave velocity field under regular waves. Laboratory data are compared with numerical predictions of the derived nonlinear wavemaker model based on a pseudospectral approach. The experimental data exhibit good agreement with the numerical results with respect to particle trajectories and mass transport for nonlinear transitional and shallow water waves. Numerical and experimental investigations confirm that free-surface oscillatory motion and particle kinematics are affected by second-order free waves, which modify the characteristics of laboratory waves, especially in a strongly nonlinear shallow water regime. Moreover, the present study gives valuable insight into the effects of the interaction between two independent waves on wave-induced mass transport, which highlights the need of including nonlinear energy transfers between higher harmonics for the reliable estimation of wave processes in ocean circulation models.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Maciej Paprota, mapap@ibwpan.gda.pl

Abstract

The aim of laboratory experiments conducted in a wave flume is to confirm that wave-induced mass transport varies along the way of propagation of regular mechanically generated waves in water of finite depth. This effect is attributed to the interactions between primary waves generated in the flume and associated free waves. Measurements of wave kinematics in several locations along the flume are performed to evaluate mass transport. Wave gauges and the particle image velocimetry technique are employed, respectively, to provide information on the evolution of the oscillating water surface and the wave velocity field under regular waves. Laboratory data are compared with numerical predictions of the derived nonlinear wavemaker model based on a pseudospectral approach. The experimental data exhibit good agreement with the numerical results with respect to particle trajectories and mass transport for nonlinear transitional and shallow water waves. Numerical and experimental investigations confirm that free-surface oscillatory motion and particle kinematics are affected by second-order free waves, which modify the characteristics of laboratory waves, especially in a strongly nonlinear shallow water regime. Moreover, the present study gives valuable insight into the effects of the interaction between two independent waves on wave-induced mass transport, which highlights the need of including nonlinear energy transfers between higher harmonics for the reliable estimation of wave processes in ocean circulation models.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Maciej Paprota, mapap@ibwpan.gda.pl

1. Introduction

Due to the fact that orbits of water waves are not fully closed, a net movement of water mass accompanies the oscillatory motion of the ocean surface. The forward drift of water may be defined as the difference between Lagrangian and Eulerian averages of a flow field (Bühler 2014; van den Bremer and Breivik 2018) and is named after Sir G. G. Stokes, who first proposed a theoretical basis for wave-induced mass transport in his pioneering work on irrotational nonlinear wave motion (Stokes 1847). Stokes waves and Stokes drift stand in opposition to Gerstner waves, which are rotational and follow closed circular loops in deep water (Gerstner 1809). The rotational nature of Gerstner waves, however, precludes them from being initiated by conservative forces (Lamb 1932). Nevertheless, Gerstner waves are still considered an intriguing subject, which deserves scientific attention (Monismith 2020).

The knowledge on mass transport processes driven by surface waves is indispensable for reliable estimates of the transport of suspended sediments and marine litter in the near-surface layer of the ocean (see, e.g., Phillips 1977; Henderson et al. 2004; Bever et al. 2011; Myrhaug et al. 2014, 2019). Stokes drift plays an important role in the advection of solute plumes, floating pollutants and submerged organic objects, as well as in wave-induced vertical mixing (see, e.g., Holmedal and Myrhaug 2009; Yu et al. 2012; Drivdal et al. 2014; Sulisz and Paprota 2015, 2019). Moreover, the accuracy of prediction in ocean circulation models depends on a reliable assessment of mass transport processes induced by waves (Qiao et al. 2010). Stokes drift is also an essential process in predicting and understanding Langmuir circulations (Craik and Leibovich 1976; Bakhoday-Paskyabi and Fer 2014; Suzuki and Fox-Kemper 2016).

The number of studies involving Lagrangian, Eulerian, and Stokes mean flows and their effect on ocean dynamics is constantly increasing. For a comprehensive review on the subject, the reader is referred to the article by van den Bremer and Breivik (2018). This work provides valuable information on the advances in the knowledge on wave-induced transport processes and related problems, including a theoretical background to Lagrangian and Eulerian mean currents and Stokes drift.

The interest in wave kinematics and time-averaged flows induced by waves is equally important in experimental ocean research and engineering. Intrinsic time-averaged features of a mechanically generated oscillatory motion and the effect of laboratory wave generation on mass redistribution in a closed domain are of practical importance for both ocean and coastal studies. First experimental attempts at describing particle kinematics and mean-flow velocities in wave flumes were made using particle tracking methods (Russel and Osorio 1957; Onoszko 1968; Mei et al. 1972; Tsuchiya and Yamaschita 1980). More sophisticated approaches involving laser Doppler anemometry (Swan 1990) and acoustic Doppler velocimetry (Scandura and Foti 2011) were adopted to measure Eulerian velocity. To determine the characteristics of surface drift in a wave flume, Huang (2007) employed imaging techniques based on the motion of floating tracers illuminated by lamps and recorded by camera.

More accurate measurements of wave-induced mass transport became possible with the invention and development of particle image velocimetry (PIV) methods (Adrian 2005). The following scientific works involving PIV experimental techniques are a significant step toward a better understanding of mass transport processes associated with water waves. First, Umeyama (2012) presented reasonable agreement between experimental and theoretical particle trajectories under regular surface waves. Next, a successful validation of weakly nonlinear theory with respect to wave-induced flows in a closed flume was reported by Paprota et al. (2016) regarding transitional water waves of relatively low steepness. PIV techniques were also employed to investigate the kinematics of low-amplitude standing waves (Paprota 2019).

The particle paths and the drift velocity distribution of higher waves were studied using PIV-based particle tracking velocimetry (PTV) techniques by Grue and Kolaas (2017). The research concluded that velocities of steep waves in a region confined to bottom and surface boundary layers contributed to a strongly enhanced forward drift velocity (Grue and Kolaas 2017). Consequently, a series of PTV experiments was performed by van den Bremer et al. (2019) to determine particle kinematics, Stokes drift and return current velocity under surface gravity wave groups in a deep-water flume, and they were in good agreement with the presented irrotational flow theory. Finally, admitting the effect of a setdown, PTV measurements were made to validate theoretical particle trajectories and the mean flow under wave packets in finite-depth water (Calvert et al. 2019).

In basic experimental research on water surface hydrodynamics in wave flumes, it is customary to use monochromatic wavemaker signals to generate regular waves with a fixed period and height. However, the first-order oscillatory motion of the wave paddle induces additional free waves at double the frequency of the primary wave (Schäffer 1996). These free oscillations interact with the primary wave and violate its permanent form along the flume. In this way, an experimental trial investigating nonlinear regular waves composed of a sum of fundamental and higher-order bound wave components becomes, in fact, an analysis of the interaction between two independent waves.

The resulting free-surface oscillations η of unidirectional mechanically generated waves may be expressed in a simplified form, for a given time t and longitudinal position x as a linear superposition of primary and free waves (Massel 1996)
η(x,t)=a1Pcos(kPxωt)+a2Pcos(2kPx2ωt)+a2Fcos(kFx2ωt+π),
where a1P and a2P are the amplitudes of first- and second-order bound wave components of the generated primary waves, respectively, while a2F denotes the amplitude of extraneous free waves. The fundamental angular frequency ω and wavenumbers kP and kF satisfy the following dispersion relations ω2 = gkPtanh(kPh) and 4ω2 = gkFtanh(kFh), where g is the acceleration due to gravity and h is water depth. This immediately implies that primary and free waves travel at different phase velocities expressed as cP = ω/kP and cF = 2ω/kF, respectively. Due to the fact that free waves are shorter, they propagate slower with an increasing difference in celerities toward deeper waters and create an interference pattern, as they overlap primary waves.

The problem of the superposition of primary and free waves generated in a closed flume expressed in a simplified way in Eq. (1) becomes more complicated upon considering nonlinear energy transfers between two individual waves. According to the second-order solution provided by Dalzell (1999), additional products of interactions of a pair of waves with fixed frequencies ω1 and ω2 appear at frequencies ω1ω2 and ω1 + ω2. In the case of primary free wave interaction, where ω1 = ω and ω2 = 2ω, the transfer of wave energy between the harmonic components of a generated bimodal wave system leads to an oscillation of the primary wave amplitude (interference at 2ωω = ω) and generation of a superharmonic wave component with 3 times the fundamental wave frequency (2ω + ω = 3ω). These second-order wave interactions affect not only the profile of laboratory waves as they propagate but also modify the magnitude of wave-induced mass transport along the flume.

It is now relatively clear that from the point of view of a scientist willing to recognize and quantify the hydrodynamics of regular waves of permanent form, additional free oscillations are considered spurious effects of mechanical wave generation. On the other hand, the bichromatic system composed of primary and second-order free waves may be regarded as a unidirectional bimodal sea state (e.g., comprising wind waves and swell), which occurs frequently at the ocean surface. In this context, inevitable side effects of mechanical generation of laboratory waves in the form of additional free oscillations may be used to achieve an important scientific outcome and provide insight into the effect of wave–wave interaction on mass transport processes.

Recently, Paprota and Sulisz (2018) provided theoretical evidence that particle trajectories and mass transport under relatively long surface water waves vary substantially depending on the distance from the wave generating boundary of a closed hydraulic flume. In their nonlinear solution to the wavemaker problem, spatial fluctuations of mass transport velocity as well as apparent changes in wave orbital movement were attributed to the evanescent modes effect (which emerged due to the presence of a sinusoidally moving wavemaker paddle) and an interplay between primary waves and spurious free waves along the flume. The latter is a result of the wavemaker paddle leaving its mean position when operating according to the first-order control signal [see, e.g., Schäffer (1996) for more details]. In fact, the influence of free waves is considerably more significant with respect to mass transport variation and particle trajectories deformation along the flume than the evanescent modes effect, which is limited only to a small region of direct wavemaker action (Paprota and Sulisz 2018).

The primary objective of the present study is to confirm theoretical results provided by Paprota and Sulisz (2018), which support the conclusion that mass transport induced by nonlinear laboratory waves varies along the flume due to the interaction between primary waves and free waves propagating in water of finite depth. In the course of the laboratory experiments, the registered free-surface oscillations along with the recorded velocity fields under an undulating air–water interface are used to analyze the effect of mechanically generated waves on time-averaged characteristics of wave-induced flow kinematics in a number of locations along the wave flume. Measurements are carried out by employing a set of resistant-type wave gauges and a state-of-the-art PIV system. Experimental values are compared with nonlinear wavemaker model predictions (Paprota and Sulisz 2018), and they are also used to estimate the amplitude of free wave oscillations. The successfully verified numerical model may be used in a more detailed follow-up research on mass transport characteristics by extending the range of wave conditions. The paper is organized as follows. In section 2, an outline of the nonlinear wavemaker model is presented. A description of the experimental procedure is provided in section 3. A comprehensive overview of main results and their discussion is covered in section 4. Finally, the main conclusions are presented in section 5.

2. Theoretical approach

a. Wavemaker problem

A two-dimensional mechanically induced irrotational motion of an inviscid and incompressible fluid is considered. Within the framework of the Eulerian description, a fluid element is defined by a Cartesian coordinate system with the horizontal x axis and the upward-pointing vertical z axis coinciding with the undisturbed water surface and the wavemaker zero position, respectively. The computational domain is enclosed by the water surface with its position defined by the function of free-surface elevation η(x, t), a horizontal bottom, a piston-type wavemaker paddle that moves according to the displacement function χ(t), and a vertical wall. Under the potential flow assumptions, the velocity vector v(x, z, t) may be expressed in terms of the function ϕ(x, z, t) such that v = ∇ϕ. The following boundary-value problem, comprising the Laplace equation, the kinematic and dynamic free-surface boundary conditions, as well as the requirement that the normal velocity is zero at impermeable boundaries, i.e., the bottom, the wavemaker paddle, and the vertical wall, is formulated, respectively, as
2ϕ=0,hzη,
ηt+ϕxηxϕz=0,z=η,
ϕt+12(ϕx2+ϕz2)+gη=0,z=η,
ϕz=0,z=h.
χtϕx=0,x=χ,
ϕx=0,x=b,
where b is the length of the flume.

b. Spectral solution

A brief outline of the solution to the formulated boundary-value problem is provided here according to the method proposed by Paprota and Sulisz (2018). First, due to the fact that Eqs. (3) and (4) are applied at the position of moving boundaries of the wavemaker paddle and the surface defined by χ and η, respectively, Taylor series expansions are used to preserve the rectangular domain, i.e.,
m=0ηmm!mzm(ηt+ϕxηxϕz)=0,z=0,
m=0ηmm!mzm[ϕt+12(ϕx2+ϕz2)+gη]=0,z=0,
m=0χmm!mxm(χtϕx)=0,x=0.
The solution is then sought in the following form
η(x,t)=n=0ancos(λnx),
ϕ(x,z,t)=n=0Ancosh[λn(z+h)]cosh(λnh)cos(λnx)+B0[(xb)2(z+h)2]+j=1Bjcosh[μj(xb)]cosh(μjl)cos[μj(z+h)],
where λn = /b and μj = /h are eigenvalues of the expansions, an and An are the unknown solution coefficients which are to be determined using the Fourier transform technique, while Bj are the unknown coefficients calculated from the following formulas
B0=χth+m=1χmm!sinmπ2n=0λnmAntanh(λnh)2h(χb),
and for (j > 0)
Bj=2cos(μjh)m=1χmm!sin(mπ2)n=0λnm+2Antanh(λnh)λn2+μj2hm=0μjm+1χmm!cos(mπ)tanh(μjb)|cosmπ2|.
Starting from the initial conditions ϕ(x, 0, 0) = 0 and η(x, 0) = 0 at t = 0, an iterative procedure based on fast Fourier transforms of η and ϕ and the Adams–Bashforth–Moulton time-stepping technique are applied to determine unknown coefficients and advance the solution in time. Then, spatial derivatives of velocity potential functions ϕx and ϕz are used to calculate the horizontal and vertical components of the velocity vector field, respectively. Finally, a numerical integration of dx/dt = ϕx(x, z, t) and dz/dt = ϕz(x, z, t) for the initial particle location (x0, z0) at time t = t0 allows the particle paths to be determined.

3. Laboratory investigations

a. Experimental setup

The experimental activities are carried out in a wave flume filled with water of uniform depth h = 0.2 m. The flume is 64 m long and 0.6 m wide. A piston-type wavemaker paddle driven by an electric motor is capable of generating unidirectional waves with prescribed characteristics including regular waves, wave groups, solitons, and random wave fields. A porous wave absorber is installed to prevent incident waves from being reflected from the wall at the opposite end of the flume (relative to the wavemaker). Free-surface oscillations are registered using resistant-type gauges with ±1-mm accuracy. In the experiments, a system of three wave gauges located 5, 10, and 20 m away from the mean position of the wavemaker paddle allows free-surface oscillations of water to be simultaneously registered at a 200-Hz sampling rate. A schematic view of the experimental setup is presented in Figs. 1 and 2 .

Fig. 1.
Fig. 1.

A schematic presentation of a data acquisition system in the wave flume: (a) side view and (b) top view.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 2.
Fig. 2.

A schematic presentation of the PIV system arrangement in the wave flume and the location of the coordinate system: (a) side view and (b) front view.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Velocity measurements are performed using the PIV methodology. The imaging acquisition system, which operates at a 50-Hz repetition rate, comprises a high-speed camera with a 1280 × 1024 pixel resolution and a dual laser-head system. The principle of a velocity field measurement procedure is as follows (see also Fig. 2). The pulse of a laser light illuminates tracer particles which are suspended in the water. The camera records an image of illuminated seeding particles, which form a plane perpendicular to the wave flume walls, and provides information on the distribution of tracers in a rectangular field of view covering an area of approximately 0.3 m × 0.25 m (presented in red color in Fig. 2). The rough estimate of the displacement vector uncertainty corresponds to ~0.1 pixel (~0.02 mm for a given region of interest) (Adrian and Westerweel 2011). The origin of a local coordinate system is positioned at the surface in the middle of the field of view with the x axis coinciding with the direction of wave propagation and the z axis pointing vertically upward. PIV measurements are carried out in three positions along the flume (Fig. 1), i.e., 2.5, 10, and 20 m away from the mean position of the wavemaker paddle to capture different stages of regular wave transformation as it propagates in the flume.

Based on the time delay between consecutive images and the average displacement of seeding material calculated by the cross correlation between two images, a velocity vector is determined in selected subareas called interrogation windows (Adrian 2005). Here, a multipass method with a decreasing size of interrogation windows is used. The initial and the resulting size of windows is 256 × 256 pixels and 64 × 64 pixels, respectively. The idea behind this data processing technique is to use the information on velocity from a larger area to improve the accuracy of the velocity calculation in smaller windows, which reduces the error of measurements for the current PIV system setup to approximately 0.1% of the mean registered velocity. The velocity vector processing is applied only to that part of the field of view plane which is occupied by water. The remaining part above the oscillating free surface is eliminated from the analysis using a geometric mask. This improves the velocity calculation accuracy in a subsurface water layer. A detailed description of the PIV system setup and methods used in the study is provided in Table 1.

Table 1.

A summary of PIV equipment and methods.

Table 1.

b. Wave tests

Regular progressive waves are generated by a monochromatic motion of the wavemaker paddle in water of uniform depth (h = 0.2 m). The waves are characterized by a fixed wave period T and wave height H. The corresponding wavelengths L are calculated by applying a nonlinear dispersion relation (Fenton 1988). The list of laboratory tests with basic wave characteristics is provided in Table 2. A standard ramp function is applied to the first few periods of the wavemaker paddle oscillatory displacement in order to achieve a smooth transition between hydrostatic conditions and the fully developed, regular wave motion in the flume. The analysis of regular wave parameters is based on a time window, which starts after the transient effects of the head of the wave train become negligible. A limited length of the window corresponding to three wave periods guarantees that the analysis is free from partial reflections from the rear end wall of the flume.

Table 2.

Basic parameters of regular waves generated in the flume.

Table 2.

In the study, transitional (W1 and W2) and shallow water (W3 and W4) wave cases are analyzed with the relative depth parameter L/h of approximately 10 and 25, respectively. In each depth regime, the low (W1 and W3) and moderate (W2 and W4) amplitude waves are generated in order to show the effect of nonlinearity on wave characteristics. In Fig. 3, wave conditions for the considered test cases are presented in a classic diagram form showing ranges of suitability of recognized wave theories (Le Méhauté 1976). It may be seen that generated waves are nonbreaking and may be classified as relatively long, because their steepness and depth-relative wavelength parameters fix them below the wave breaking line and close to the shallow water limit (Fig. 3). Waves of lower amplitude (W1 and W3) may be considered linear or weakly nonlinear. In the case of higher waves (W2 and W4), strongly nonlinear profiles are expected to emerge and eventually to take cnoidal forms in a shallower regime (W4). The free wave relative amplitude for the presented experimental tests is estimated to be as high as ~2%, ~20%, ~15%, and ~50% for the W1, W2, W3, and W4 cases, respectively. The assessment is based on long-term predictions provided by the nonlinear wavemaker model for large longitudinal dimensions of the numerical flume. This procedure allows for the identification of the spatial variation of the amplitude of the second harmonic. In this way, bound and free parts of the secondary wave component may be separated and the free wave amplitude is roughly approximated based on a weakly nonlinear solution to the interaction between two waves derived by Dalzell (1999).

Fig. 3.
Fig. 3.

Experimental wave conditions according to Le Méhauté (1976).

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

4. Results and discussion

a. Free-surface kinematics

Mechanically generated regular waves are first analyzed with respect to free-surface profiles recorded along the flume. In Figs. 47, free-surface oscillations are presented for three positions along the flume. The experimental data correspond to the mean surface profile recorded by wave gauges for three repetitions and three subsequent wave periods. The repeatability of experiments is excellent due to the fact that the maximum random error represented by the standard deviation of the mean of the surface elevation data for all wave cases is ~0.2 mm and the highest standard error of the calculated Fourier amplitudes is ~0.03 mm, which is much less than the 1-mm accuracy of the gauges.

Fig. 4.
Fig. 4.

Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 1.517 s, L = 2.00 m, H = 0.007 m (W1); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 5.
Fig. 5.

Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 1.517 s, L = 2.04 m, H = 0.04 m (W2); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 6.
Fig. 6.

Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 3.607 s, L = 5.01 m, H = 0.006 m (W3); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 7.
Fig. 7.

Free-surface oscillations and corresponding Fourier amplitudes at a distance of 5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular waves: T = 3.607 s, L = 5.32 m, H = 0.045 m (W4); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars, which form envelopes in the case of time series data.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

It may be noted that lower amplitude waves (W1 and W3) generally follow a sinusoidal pattern as predicted by the linear Airy theory (Figs. 4 and 6). However, in the case of longer waves (W3), a noticeable departure from the sinusoidal profile may be observed, especially at the gauge positioned 20 m away from the wavemaker. Upon increasing the wavemaker stroke (W2 and W4), the wave profile becomes more nonlinear (Figs. 5 and 7). In the case of waves traveling in transitional waters (W2), where the effect of spurious free waves is weaker, the free surface may be described according to a weakly nonlinear second-order solution (Stokes 1847). The free wave influence manifests itself in relatively small modifications of wave troughs (Fig. 5). In the shallow water limit, a cnoidal-like wave profile (W4) with long and flat troughs and short and steep crests is significantly affected by free waves which results in a clearly visible additional bump of water. This bump overlaps the primary wave profile at subsequent phase instants as the wave propagates along the flume (Fig. 7).

The free-surface elevation graphs are complemented by the results of Fourier analysis, which provides important information on nonlinear energy transfers between primary wave components and higher harmonics (Figs. 47). The Fourier amplitude spectrum determined for transitional water waves of low steepness (W1) is dominated by a primary wave component (Fig. 4), which exhibits a slight decrease in subsequent wave gauge positions due to small wave energy losses along the flume of approximately 5% per 10 m of the wave flume length. As a result, the sinusoidal profile of free-surface oscillations remains relatively unchanged with a slight reduction of wave amplitude. In the previous paragraph it is mentioned that transitional waves of higher steepness experience small modifications of the wave profile along the flume due to the free wave effect. In Fig. 5, it may be seen that the free surface motion becomes nonlinear, which is confirmed by the presence of higher harmonics in the wave amplitude spectrum. The reduction of primary wave amplitude is triggered not only by small energy dissipation (~6% per 10 m) but also by the transfer of wave energy from the primary wave component to higher harmonics. Spectral evolution is driven by interactions between primary and free waves.

The situation becomes more interesting in the longer wave cases (W3 and W4). These waves are characterized by higher values of the Ursell number, which is a measure of wave nonlinearity. It may be seen in Fig. 6 that even the lower-amplitude wave (W3) is accompanied by a free wave, which is capable of increasing the second harmonic component by a factor of 3 when comparing wave gauge registrations 5 and 20 m away from the wavemaker. As a result, the wave profile becomes vertically asymmetric due to a small lift of the front slope of the crest (cf. the wave profiles registered 5 and 20 m away from the wavemaker and presented in Fig. 6). The longest and highest analyzed wave (W4) is characterized by the Ursell number of 159.2 indicating that the wave is highly nonlinear. This is also confirmed by the spectral structure of the wave profile presented in Fig. 7. The Fourier analysis of free-surface registration results in six distinguishable harmonics, which substantially change along the flume due to strong interactions between free and primary waves. The energy transfer from the primary wave component to higher harmonics reduces the first harmonic wave component amplitude by half over a distance of 15 m in the direction of propagation. At a distance of 20 m away from the wavemaker paddle, the second harmonic wave component amplitude is almost twice as large as the first one. The nonlinear energy transfers between component waves result in a considerable deformation of the wave profile. The wave energy dissipation rate for both shallow water wave cases ranges from 4% to 5% per 10 m of wave flume length.

In Figs. 47, a comparison between numerical results and experimental data is presented. Because some energy is lost along the experimental flume, laminar damping is introduced to the nonlinear wavemaker model according to Larsen and Dancy (1983) to take into account wave amplitude reduction caused by wave energy dissipation driven by the presence of wave flume walls and the bottom. It may be seen that good agreement is achieved between mathematical model predictions and measurements with respect to the free-surface elevation registered at three gauge positions (Figs. 47). Some discrepancies are only visible in the Fourier analysis graph in the case of the highest and the longest of the analyzed waves (Fig. 7). The greatest differences between component amplitudes in relation to the first-order harmonic for W4 case reach approximately 3% for the first harmonic (wave gauge at a distance of 5 m from the wavemaker), 22% for the second-order harmonic (20 m), 19% for the third-order (20 m), and 10% for the fourth-order (10 m) components. In all remaining cases the differences are less than 2%, and only occasionally slightly exceed 4%.

b. Particle trajectories

The velocity fields calculated using the PIV methodology provide valuable information on wave kinematics under an oscillating water surface. The postprocessed images of the illuminated seeding motion recorded by the PIV camera in the form of evolving vector fields are numerically integrated to map water particle trajectories. The method reported by Paprota et al. (2016) is used to retrieve the shape of orbits of mechanically generated regular waves in a closed flume. Based on discrete velocity values, the nearest neighbor algorithm is applied to determine functions which interpolate horizontal and vertical components of the ith velocity vector map ui(x, z) and wi(x, z), respectively, at a given time ti. As previously stated, the trajectory of a particle is calculated by numerically solving the system of equations dx/dt = ui(x, z, ti) and dz/dt = wi(x, z, ti). Starting from an initial particle location (x0, z0) at time t0, the numerical solution is time stepped using the four-step Adams–Bashforth scheme. Because higher-order multistep methods require values from preceding time instants, first-, second-, and third-order Adams–Bashforth formulas are consequently used for the first three steps. The initial particle location of coordinates x0 and z0 coincides with a zero down-crossing point of the registered profile of the considered wave. In Figs. 8 and 9, particle trajectories corresponding to one wave period are presented for transitional (W1 and W2) and shallow water (W3 and W4) waves, respectively. In the main larger graphs, regular waves of higher steepness are shown. The waves of lower amplitudes are depicted in the embedded smaller charts attached in the lower right corners of the main graphs. The same scale is preserved for vertical and horizontal axes as well as the main and embedded graphs for the convenience of comparison. The error bars presented in Figs. 8 and 9 correspond to random error estimates based on three subsequent waves and three repetitions, and they are calculated as the standard deviation of the mean of the resulting nine individual recordings for each analyzed wave case. The maximum values of standard error remain below 0.3 and 0.8 mm for lower (W1 and W3) and higher (W2 and W4) wave amplitudes, respectively, and confirm good repeatability of the experiments.

Fig. 8.
Fig. 8.

Particle trajectories at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 1.517 s, L = 2.00 m, H = 0.007 m (W1) for the smaller embedded graph; T = 1.517 s, L = 2.04 m, H = 0.04 m (W2) for the larger main graph. Symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 9.
Fig. 9.

Particle trajectories at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 3.607 s, L = 5.01 m, H = 0.006 m (W3) for the smaller embedded graph; T = 3.607 s, L = 5.32 m, H = 0.045 m (W4) for the larger main graph. Symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

The particle trajectories of transitional water waves of higher amplitude (W2) form ellipses with a flattened bottom (Fig. 8). On the way of wave propagation, the path of a water particle corresponding to a trough phase is modified due to primary and free wave interaction. The considered movement of the particle is in line with the motion of the free surface recorded on consecutive gauges as presented in Fig. 5. In the case of the shallow water wave of higher steepness (W4), the water particle follows an elliptical path close to the wavemaker (Fig. 9). When moving farther, the contribution of the free wave becomes apparent. The interplay between the primary and the spurious free wave results in an additional loop, which is formed by a particle in different phase instants of a regular long wave depending on the distance from the wavemaker. By comparing the wave profile evolution presented in Fig. 7 and the corresponding particle trajectory shown in Fig. 9, a link between the bump in the trough phase of the wave profile and the secondary loop in the particle trajectory may be noticed. Actually, the presence of the bump requires an additional upward and downward movement of the particle, which results in an extra loop. This phenomenon was first noticed by Paprota and Sulisz (2018), who investigated water particle kinematics theoretically using an arbitrary-order potential flow nonlinear wavemaker model. The experimental particle trajectories presented in the study are consistent with the numerical trajectories. Good agreement is observed for the W2 case (Fig. 8), while the differences between experimental and numerical results relative to horizontal and vertical diameters of wave orbits are less than 4%. In the longer wave case (W4), the relative differences are greater reaching approximately 9% and 15% for horizontal and vertical displacements, respectively.

c. Mass transport

The analysis of mechanically generated waves reveals that water particle trajectories form ellipses which are not fully closed and deformed to a certain degree. This property of nonlinear wave motion is more pronounced in the case of waves of higher steepness. In consequence, near the surface, water particles drift in the same direction as the wave propagates (see Figs. 8 and 9). By calculating particle trajectories for the entire water depth using the methodology from the previous section, time-averaged Lagrangian mass transport velocity uL may be determined as the horizontal displacement of particles between two successive in-phase positions divided by the time interval between their occurrence, which is considered one Lagrangian wave period (Paprota et al. 2016). The starting positions of a set of particles uniformly distributed over the depth are selected at the zero down-crossing phase of the registered profile of a particular wave.

The calculated vertical profiles of Lagrangian mass transport velocity are presented for wave cases W2 and W4 at three locations along the flume in Figs. 10 and 11 . The same procedure is applied to obtain error values presented in Figs. 10 and 11 as well as in the case of the particle trajectory calculations. The spatial variation of mass transport intensity is clearly seen in both presented graphs. The characteristic mean velocity distribution confirms that, in a closed wave flume and irrotational flow, water mass follows the waves in the subsurface region. On the other hand, the backward drift moves the water near the bottom toward the wavemaker paddle. The experimental results indicate that, in the case of transitional water depths (W2), the absolute drift velocity at the surface and near the bottom decreases by ~15% and ~30%, respectively, for profiles located 2.5 and 20 m away from the wavemaker (Fig. 10). In shallow water (W4), the lowest transport velocities are estimated for the position closest to the wavemaker. The Lagrangian time-averaged velocity increases with the distance from the wavemaker (Fig. 11).

Fig. 10.
Fig. 10.

Lagrangian mass transport velocity vertical profiles at a distance of 2.5 (black), 10 (blue), and 20 m (green) from the wavemaker for regular water waves: T = 1.517 s, L = 2.04 m, H = 0.04 m (W2); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

Fig. 11.
Fig. 11.

Lagrangian mass transport velocity vertical profiles at a distance of 2.5 m (black), 10 m (blue), and 20 m (green) from the wavemaker for regular water waves: T = 3.607 s, L = 5.32 m, H = 0.045 m (W4); symbols and lines correspond to experimental data and numerical predictions, respectively. The inaccuracy of measurements is represented with error bars.

Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-20-0092.1

The experimental results of mass transport variation along the flume should be, however, treated with caution. The magnitude of mass transport is small in relation to orbital velocities of generated waves (Paprota et al. 2016). That is why even small errors in the experimental wave velocity field may hinder the correct estimation of the vertical distribution of mass transport velocity along the flume. As it is clearly seen in Figs. 8 and 9, reasonable agreement between experimental and numerical results is achieved, but the applied PIV system is not capable of capturing a small drift of particles, represented by a horizontal shift of wave orbits, with sufficient accuracy in all cases. A satisfactory estimate of mass transport velocity may be seen for the W2 case in Fig. 10 with the repeatability corresponding to a maximum random error of 0.3 mm s−1. In the long wave limit (W4), the time-averaged velocities are of the order of a millimeter per second, and only a quantitative conclusion of some variation of mass transport velocity along the flume may be drawn from the experimental data. The repeatability is also less satisfactory, as it corresponds to maximum random errors of the order of half a millimeter. However, it is to say that the interaction of primary waves and spurious free waves leads to a significant modification of time-mean Lagrangian particle kinematics of mechanically generated water waves, which may be confirmed by the presented experimental data supported by the reliable numerical wavemaker model results.

5. Conclusions

In a series of experiments carried out in a wave flume, the theoretical result stating that the mass transport velocity varies along the flume in shallower waters due to nonlinear interaction of mechanically generated waves and second-order free waves is confirmed. The analysis of the oscillatory motion induced by regular laboratory waves propagating at a uniform depth supported by numerical predictions indicates that for transitional and shallow water conditions, the effect of free waves qualitatively and quantitatively modifies the kinematics of water particles under an undulating free surface.

A significant contribution of additional oscillations is clearly seen in the trough phase of the wave profile. In the case of transitional water waves, the trough deformation is relatively weak. This effect becomes apparent for longer waves. A bump of water appears and moves with respect to the shallow water wave profile along the flume. Due to the additional free-surface oscillation, a moving water particle forms an extra loop as it follows its substantially deformed elliptical orbit.

Although the current research is focused on intrinsic features of mechanical generation of regular waves in closed flumes, and it explains spurious laboratory wave processes associated with the generation of second-order free waves, the present study gives valuable insight into the effects of interaction between two independent waves on wave-induced mass transport. In this context, the obtained results confirm the need of including nonlinear energy transfers between higher harmonics of a hydrodynamic system formed by two or more independent wave components for a correct and reliable estimation of wave-induced mass transport processes.

Data availability statement

Data will be made available on request.

REFERENCES

  • Adrian, R. J., 2005: Twenty years of particle image velocimetry. Exp. Fluids, 39, 159169, https://doi.org/10.1007/s00348-005-0991-7.

  • Adrian, R. J., and J. Westerweel, 2011: Particle Image Velocimetry. Cambridge University Press, 558 pp.

  • Bakhoday-Paskyabi, M., and I. Fer, 2014: Turbulence structure in the upper ocean: A comparative study of observations and modeling. Ocean Dyn., 64, 611631, https://doi.org/10.1007/s10236-014-0697-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bever, A. J., J. E. McNinch, and C. K. Harris, 2011: Hydrodynamics and sediment-transport in the nearshore of Poverty Bay, New Zealand: Observations of nearshore sediment segregation and oceanic storms. Cont. Shelf Res., 31, 507526, https://doi.org/10.1016/j.csr.2010.12.007.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bühler, O., 2014: Waves and Mean Flows. Cambridge University Press, 378 pp.

  • Calvert, R., C. Whittaker, A. Raby, P. H. Taylor, A. G. L. Borthwick, and T. S. van den Bremer, 2019: Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth. Phys. Rev. Fluids, 4, 114801, https://doi.org/10.1103/PhysRevFluids.4.114801.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Craik, A. D. D., and S. Leibovich, 1976: A rational model for Langmuir circulations. J. Fluid Mech., 73, 401426, https://doi.org/10.1017/S0022112076001420.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dalzell, J. F., 1999: A note on finite depth second-order wave-wave interactions. Appl. Ocean Res., 21, 105111, https://doi.org/10.1016/S0141-1187(99)00008-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Drivdal, M., G. Broström, and K. H. Christensen, 2014: Wave-induced mixing and transport of buoyant particles: Application to the Statfjord A oil spill. Ocean Sci., 10, 977991, https://doi.org/10.5194/os-10-977-2014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fenton, J. D., 1988: The numerical solution of steady water wave problems. Comput. Geosci., 14, 357368, https://doi.org/10.1016/0098-3004(88)90066-0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gerstner, F., 1809: Theorie der wellen. Ann. Phys., 32, 412445, https://doi.org/10.1002/andp.18090320808.

  • Grue, J., and J. Kolaas, 2017: Experimental particle paths and drift velocity in steep waves at finite water depth. J. Fluid Mech., 810, R1, https://doi.org/10.1017/jfm.2016.726.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Henderson, S. M., J. S. Allen, and P. A. Newberger, 2004: Nearshore sandbar migration predicted by an eddy-diffusive boundary layer model. J. Geophys. Res., 109, C06024, https://doi.org/10.1029/2003JC002137.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holmedal, L. E., and D. Myrhaug, 2009: Wave-induced steady streaming, mass transport and net sediment transport in rough turbulent ocean bottom boundary layers. Cont. Shelf Res., 29, 911926, https://doi.org/10.1016/j.csr.2009.01.012.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Huang, Z., 2007: An experimental study of the surface drift currents in a wave flume. Ocean Eng., 34, 343352, https://doi.org/10.1016/j.oceaneng.2006.01.005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lamb, H., 1932: Hydrodynamics. Cambridge University Press, 738 pp.

  • Larsen, J., and H. Dancy, 1983: Open boundaries in short wave simulations - a new approach. Coastal Eng., 7, 285297, https://doi.org/10.1016/0378-3839(83)90022-4.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Le Méhauté, B., 1976: Introduction to Hydrodynamics and Water Waves. Springer-Verlag, 323 pp.

  • Massel, S. R., 1996: On the largest wave height in water of constant depth. Ocean Eng., 23, 553573, https://doi.org/10.1016/0029-8018(95)00049-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mei, C. C., P. L.-F. Liu, and T. G. Carter, 1972: Mass transport in water waves. Massachusetts Institute of Technology Tech. Rep. 146, 278 pp.

  • Monismith, S., 2020: Stokes drift: Theory and experiments. J. Fluid Mech., 884, F1, https://doi.org/10.1017/jfm.2019.891.

  • Myrhaug, D., H. Wang, and L. E. Holmedal, 2014: Stokes drift estimation for deep water waves based on short-term variation of wave conditions. Coast. Eng., 88, 2732, https://doi.org/10.1016/j.coastaleng.2014.01.014.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Myrhaug, D., H. Wang, and L. E. Holmedal, 2019: Stokes transport in layers in the water column based on long-term wind statistics. Oceanologia, 61, 522526, https://doi.org/10.1016/j.oceano.2019.03.003.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Onoszko, J., 1968: Mass transport in forced wave motion in laboratory investigation. Ann. Soc. Geol. Pol., 38, 110.

  • Paprota, M., 2019: Particle image velocimetry measurements of standing wave kinematics in vicinity of a rigid vertical wall. Instrum. Exp. Tech., 62, 277282, https://doi.org/10.1134/S0020441219020234.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Paprota, M., and W. Sulisz, 2018: Particle trajectories and mass transport under mechanically generated nonlinear water waves. Phys. Fluids, 30, 102101, https://doi.org/10.1063/1.5042715.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Paprota, M., W. Sulisz, and A. Reda, 2016: Experimental study of wave-induced mass transport. J. Hydraul. Res., 54, 423434, https://doi.org/10.1080/00221686.2016.1168490.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Phillips, O., 1977: The Dynamics of the Upper Ocean. Cambridge University Press, 336 pp.

  • Qiao, F., Y. Yuan, T. Ezer, C. Xia, Y. Yang, X. , and Z. Song, 2010: A three-dimensional surface wave-ocean circulation coupled model and its initial testing. Ocean Dyn., 60, 13391355, https://doi.org/10.1007/s10236-010-0326-y.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Russel, R. C. H., and J. D. C. Osorio, 1957: An experimental investigation of drift profile in a closed channel. Proc. Sixth Int. Conf. on Coastal Engineering, New York, New York, ASCE, 293–305.

  • Scandura, P., and E. Foti, 2011: Measurements of wave-induced steady currents outside the surf zone. J. Hydraul. Res., 49, 6471, https://doi.org/10.1080/00221686.2011.591046.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schäffer, H. A., 1996: Second-order wavemaker theory for irregular waves. Ocean Eng., 23, 4788, https://doi.org/10.1016/0029-8018(95)00013-B.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stokes, G. G., 1847: On the theory of oscillatory waves. Trans. Cambridge Philos. Soc., 8, 441455.

  • Sulisz, W., and M. Paprota, 2015: Theoretical and experimental investigations of wave-induced vertical mixing. Math. Probl. Eng., 2015, 950849, https://doi.org/10.1155/2015/950849.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sulisz, W., and M. Paprota, 2019: On modeling of wave-induced vertical mixing. Ocean Eng., 194, 106622, https://doi.org/10.1016/j.oceaneng.2019.106622.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Suzuki, N., and B. Fox-Kemper, 2016: Understanding Stokes forces in the wave-averaged equations. J. Geophys. Res. Oceans, 121, 35793596, https://doi.org/10.1002/2015JC011566.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Swan, C., 1990: Convection within an experimental wave flume. J. Hydraul. Res., 28, 273282, https://doi.org/10.1080/00221689009499069.

  • Tsuchiya, Y., and T. Y. T. Yamaschita, 1980: Mass transport in progressive waves of permanent type. Proc. 17th Int. Conf. on Coastal Engineering, Sydney, Australia, ASCE, 70–81.

    • Crossref
    • Export Citation
  • Umeyama, M., 2012: Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Philos. Trans. Roy. Soc. London, 370A, 16871702, https://doi.org/10.1098/rsta.2011.0450.

    • Search Google Scholar
    • Export Citation
  • van den Bremer, T. S., and Ø. Breivik, 2018: Stokes drift. Philos. Trans. Roy. Soc. London, 376A, 20170104, https://doi.org/10.1098/rsta.2017.0104.

    • Search Google Scholar
    • Export Citation
  • van den Bremer, T. S., C. Whittaker, R. Calvert, A. Raby, and P. H. Taylor, 2019: Experimental study of particle trajectories below deep-water surface gravity wave groups. J. Fluid Mech., 879, 168186, https://doi.org/10.1017/jfm.2019.584.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yu, Q., Y. P. Wang, B. Flemming, and S. Gao, 2012: Tide-induced suspended sediment transport: Depth-averaged concentrations and horizontal residual fluxes. Cont. Shelf Res., 34, 5363, https://doi.org/10.1016/j.csr.2011.11.015.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save