Mass Transport In Water Waves Research Articles

Mass transport in water waves over a thin layer of soft viscoelastic mud

A theory is presented for the mass transport induced by a small-amplitude progressive wave propagating in water over a thin layer of viscoelastic mud modelled as a Voigt medium. Based on a sharp contrast in length scales near the bed, the boundary-layer approximation is applied to the Navier–Stokes equations in Lagrangian form, which are then solved for the first-order oscillatory motions in the mud and the near-bed water layers. On extending the analysis to second order for the mass transport, it is pointed out that it is inappropriate, as was done in previous studies, to apply the complex viscoelastic parameter to a higher-order analysis, and also to suppose that a Voigt body can undergo continuous steady motion. In fact, the time-mean motion of a Voigt body is only transient, and will stop after a time scale given by the ratio of the viscosity to the shear modulus. Once the mud has attained its steady deformation, the mass transport in the overlying water column can be found as if it were a single-layer system. It is found that the near-bed mass transport has non-trivial dependence on the mud depth and elasticity, which control the occurrence of resonance. Even when the resonance is considerably damped by viscosity, the mass transport in water over a viscoelastic layer can be dramatically different, in terms of magnitude and direction, from that over a rigid bed.

Mass transport in water waves over an elastic bed

The mass transport velocity in water waves propagating over an elastic bed is investigated. Water is assumed to be incompressible and slightly viscous. The elastic bed is also incompressible and satisfies the Hooke’s law. For a small amplitude progressive wave perturbation solutions via a boundary-layer approach are obtained. Because the wave amplitude is usually larger than the viscous boundary layer thickness and because the free surface and the interface between water and the elastic bed are moving, an orthogonal curvilinear coordinate system (Longuet-Higgins 1953) is used in the analysis of free surface and interfacial boundary layers so that boundary conditions can be applied on the actual moving surfaces. Analytical solutions for the mass transport velocity inside the boundary layer adjacent to the elastic seabed and in the core region of the water column are obtained. The mass transport velocity above a soft elastic bed could be twice of that over a rigid bed in the shallow water.

MASS TRANSPORT IN PROGRESSIVE WAVES OF PERMANENT TYPE

Mass transport phenomenon was first recognized by Stokes in 1847 using a Lagrangian description. Later, a basic theory for the mass transport in water waves in viscous fluid and of finite depth was derived by Longuet-Higgins in 1953. Theoretical solutions of mass transport in progressive waves of permanent type are subjected to the definitions of wave celerity in deriving the various finite amplitude wave theories. As it has been generally acknowledged that the Stokes wave theory can not yield a correct prediction of mass transport in the shallow depths, some new theories have been developed. Recently the authors(1974 § 1977) have derived a new finite amplitude wave theory in shallow water for quasi- Stokes and cnoidal waves by the so-called reductive perturbation method, in which the mass transport is formulated both in Lagrangian and Eulerian descriptions. On the experimental verification, Russell and 0sorio(1957) investigated and compared Longuet-Higgins' solution with experimental data of Lagrangian mass transport velocity obtained in a normal closed wave tank of finite length. Since then, many investigations, and nearly all of them, have employed the finite length of wave tank in carrying out their experiments. However, no experiment has yet been attempted at verifying the Stokes drift in progressive waves of permanent type in a wave tank of infinite length. It is not realistic nor economical in constructing such an infinitely long flume to investigate experimentally the mass transport velocity in progressive waves. Instead of using such an ideal wave tank, a new one incorporated with natural water re-circulation was equipped to carry out experiments by the authors(1978). It was confirmed from these experiments that mass transport in progressive waves of permanent type exists in the Same direction of wave propagation throughout the depth, and agrees with both the Stokes drift and the authors' new formulations, within the test range of experiments.

Mass transport in water waves propagated over a permeable bed (Liu, 1977)

Mass transport in water waves propagated over a permeable bed (Liu, 1977)

Mass transport in water waves propagated over a permeable bed

Mass transport in water waves propagated over a permeable bottom is investigated. Boundary-layer approximations are incorporated into the Lagrangian equations of motion, and the mass transport velocity is obtained for both monochromatic and random waves. It is found that inside the permeable bed, although the Eulerian steady streaming vanishes, the mass transport always exists in the direction of wave propagation. The effects of porosity and permeability of the porous bed on the mass transport are discussed.

Wave-Induced Mass Transport in Water Waves

The Eulerian mass transport under water waves is studied for an inviscid fluid using finite amplitude wave theories. The distribution of the transport over the water depth for example waves is shown to be confined to a region above the wave trough and is of lesser magnitude than predicted by Airy wave theory. A measured mean velocity is defined to examine the effect of time—averaging over the duration of immersion, and its variation over the water column is also shown. Finally, the presence of a linear shear current in the direction of the wave is shown to alter the total mass transport principally by superposition.

Correction [to “Mass transport in water waves’ by Ünluata, Ü, and C. C. Mei

Journal of Geophysical Research (1896-1977)Volume 76, Issue 12 p. 2903-2903 Correction [to “Mass transport in water waves’ by Ünluata, Ü, and C. C. Mei] Ü. Ünlüata, Ü. ÜnlüataSearch for more papers by this authorC. C. Mei, C. C. MeiSearch for more papers by this author Ü. Ünlüata, Ü. ÜnlüataSearch for more papers by this authorC. C. Mei, C. C. MeiSearch for more papers by this author First published: 20 April 1971 https://doi.org/10.1029/JC076i012p02903AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL No abstract is available for this article. Volume76, Issue12Oceans and Atmospheres20 April 1971Pages 2903-2903 RelatedInformation

Mass transport in water waves

The theory of mass transport in water waves is re-examined. Boundary-layer arguments are incorporated into the Lagrangian equations of motion, and the mass transport velocity is conveniently derived for both monochromatic and random waves. For finite depth, Longuet-Higgins' result is confirmed. Conditions at the far end of a wave tank are found to be related to a mean pressure gradient, and results are also obtained for a truly infinite channel. The nonuniformity of the solution for the deep-water limit is discussed.

Mass transport in water waves

It was shown by Stokes that in a water wave the particles of fluid possess, apart from their orbital motion, a steady second-order drift velocity (usually called the mass-transport velocity). Recent experiments, however, have indicated that the mass-transport velocity can be very different from that predicted by Stokes on the assumption of a perfect, non-viscous fluid. In this paper a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation. Part I deals especially with the interior of the fluid. It is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness d of the boundary layer: when a 2 / d 2 is small the diffusion of vorticity takes place by viscous ‘conduction’; when a 2 / d 2 is large, by convection with the mass-transport velocity. Appropriate field equations for the stream function of the mass transport are derived. The boundary layers, however, require separate consideration. In part II special attention is given to the boundary layers, and a general theory is developed for two types of oscillating boundary: when the velocities are prescribed at the boundary, and when the stresses are prescribed. Whenever the motion is simple-harmonic the equations of motion can be integrated exactly. A general method is described for determining the mass transport throughout the fluid in the presence of an oscillating body, or with an oscillating stress at the boundary. In part III, the general method of solution described in parts I and II is applied to the cases of a progressive and a standing wave in water of uniform depth. The solutions are markedly different from the perfect-fluid solutions with irrotational motion. The chief characteristic of the progressivewave solution is a strong forward velocity near the bottom. The predicted maximum velocity near the bottom agrees well with that observed by Bagnold.