On the decrease of velocity with depth in an irrotational water wave

1953 ◽  
Vol 49 (3) ◽  
pp. 552-560 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Periodic Motion ◽  
Water Wave ◽  
Pressure Fluctuations ◽  
Periodic Wave ◽  
Finite Amplitude ◽  
Transport Velocity ◽  
Mean Pressure ◽  
Mass Transport Velocity ◽  
The Mean ◽  
The Waves

ABSTRACTThe following theorems are proved for irrotational surface waves of finite amplitude in a uniform, incompressible fluid:(a) In any space-periodic motion (progressive or otherwise) in uniform depth, the mean square of the velocity is a decreasing function of the mean depth z below the surface. Hence the fluctuations in the mean pressure increase with z.(b) In any space-periodic motion in infinite depth, the particle motion tends to zero exponentially as z tends to infinity. The pressure fluctuations at great depths are therefore simultaneous, but they do not in general tend to zero.(c) In a progressive periodic wave in uniform depth the mass-transport velocity is a decreasing function of the mean depth of a particle below the free surface, and the tangent to the velocity profile is vertical at the bottom. This result conflicts with observations in wave tanks, and shows that the waves cannot be wholly irrotational.(d) Analogous results are proved for the solitary wave.

scholarly journals THE EFFECT OF ROUGHNESS ON THE MASS-TRANSPORT OF PROGRESSIVE GRAVITY WAVES

1966 ◽  
Vol 1 (10) ◽  
pp. 11 ◽  
Author(s):  
Arthur Brebner ◽  
J.A. Askew ◽  
S.W. Law
Keyword(s):  
Mass Transport ◽  
Gravity Waves ◽  
Orbital Motion ◽  
Wave Length ◽  
Wave Theory ◽  
Finite Amplitude ◽  
Maximum Value ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
The Mean

On the basis of non-viscous small amplitude firstorder theory the maximum value of the horizontal orbital motion at the bed in water of constant depth his given by /U/n yy* »* " r •»** */i where k = /L, H is the wave height crest to trough, T is the period, and L the wave length (L = Sry2jr Arf 2*%/L ). On the basis of finite amplitude wave theory where the particle orbits are not closed ana by the insertion of the viscous laminar boundary layer (the conducti6n solution) the mean drift velocity or mass transport velocity on a perfectly smooth bed is given by Longuet- Higgins (1952) as 7, K H* kcr where

Sediment transport by waves and currents

1983 ◽  
Vol 10 (1) ◽  
pp. 142-149 ◽  
Author(s):  
Michael C. Quick
Keyword(s):  
Sediment Transport ◽  
Mass Transport ◽  
Power Relationship ◽  
Zero Current ◽  
Transport Direction ◽  
Transport Velocity ◽  
Waves And Currents ◽  
Mass Transport Velocity ◽  
Combined Action ◽  
The Waves

Sediment transport is measured under the combined action of waves and currents. Measurements are made with currents in the direction of wave advance and with currents opposing the wave motion. Theoretical relationships are considered that predict the wave velocity field and the mass transport velocity for zero current and for steady currents.Following Bagnold's approach, a transport power relationship is developed to predict the sediment transport rate. Making assumptions for the mass transport velocity, the transport power is shown to agree with the measured sediment transport rates. It is particularly noted that the sediment transport direction is mainly determined by the direction of wave movement, even for adverse currents, until the waves start to break. Keywords: sediment transport, waves and currents, coastal engineering.

Prediction of pressure fluctuations on sloping stilling basins

2006 ◽  
Vol 33 (11) ◽  
pp. 1379-1388 ◽  
Author(s):  
A Güven ◽  
M Günal ◽  
A Çevik
Keyword(s):  
Neural Network ◽  
Pressure Fluctuation ◽  
Hydraulic Jump ◽  
Regression Models ◽  
Pressure Fluctuations ◽  
Network Models ◽  
Neural Network Models ◽  
Mean Pressure ◽  
Stilling Basins ◽  
The Mean

Various types of hydraulic jump occurring on horizontal and sloping channels have been analyzed experimentally, theoretically, and numerically and the results are available in the literature. In this study, artificial neural network models were developed to simulate the mean pressure fluctuations beneath a hydraulic jump occurring on sloping stilling basins. Multilayers feed a forward neural network with a back-propagation learning algorithm to model the pressure fluctuations beneath such a type of hydraulic jump (B-jump). An explicit formula that predicts the mean pressure fluctuation in terms of the characteristics that contribute most to the hydraulic jump occurring on the sloping basins is presented. The proposed neural network models are compared with linear and nonlinear regression models that were developed using considered physical parameters. The results of the neural network modelling are found to be superior to the regression models and are in good agreement with the experimental results due to relatively small values of error (mean absolute percentage error).Key words: neural networks, pressure fluctuation, hydraulic jump, sloping stilling basin, explicit NN formulation, regression analysis.

scholarly journals An exact theory of nonlinear waves on a Lagrangian-mean flow

1978 ◽  
Vol 89 (4) ◽  
pp. 609-646 ◽  
Author(s):  
D. G. Andrews ◽  
M. E. Mcintyre
Keyword(s):  
Water Waves ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Radiation Stress ◽  
Initial State ◽  
Mean Velocity ◽  
Generalized Lagrangian ◽  
Flow Evolution ◽  
The Mean ◽  
The Waves

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

scholarly journals A PRE-DREDGING SAND MOBILITY STUDY USING A RADIOISOTOPE TRACER

1984 ◽  
Vol 1 (19) ◽  
pp. 138
Author(s):  
A. Davison
Keyword(s):  
Sand Transport ◽  
Dispersion Pattern ◽  
Tracer Dispersion ◽  
Navigation Channel ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
The Mean ◽  
Wave Boundary ◽  
Mobility Study ◽  
Radioisotope Tracer

The wave-driven movement of sand across the alignment of a proposed navigation channel was investigated using radioactive chromium-51 labelled tracer sand. The mean particle velocity and thickness of the mobile layer were determined over a two-month period, and an annual infill rate estimated. Wave height and period were measured concurrently. Despite two storms, during which near-bed oscillating velocities of 1.5 m s-1 were calculated the sand transport at 10 m (BMWL) appears to occur within the wave boundary layer. Onshore transport in the direction of wave propagation, due to mass transport velocity and wave asymmetry effects, was easily identified. Tidal currents up to 1.2 m s-1 (at 3 m above bed) had less than the expected effect on the tracer dispersion pattern.

Mass transport of interfacial waves in a two-layer fluid system

1995 ◽  
Vol 297 ◽  
pp. 231-254 ◽  
Author(s):  
Jiangang Wen ◽  
Philip L.-F. Liu
Keyword(s):  
Mass Transport ◽  
Boundary Layers ◽  
Small Amplitude ◽  
Viscous Damping ◽  
Interfacial Wave ◽  
Fluid System ◽  
Steady Streaming ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
The Mean

Effects of viscous damping on mass transport velocity in a two-layer fluid system are studied. A temporally decaying small-amplitude interfacial wave is assumed to propagate in the fluids. The establishment and the decay of mean motions are considered as an initial-boundary-value problem. This transient problem is solved by using a Laplace transform with a numerical inversion. It is found that thin ‘second boundary layers’ are formed adjacent to the interfacial Stokes boundary layers. The thickness of these second boundary layers is of O(ε1/2) in the non-dimensional form, where ε is the dimensionless Stokes boundary layer thickness defined as $\epsilon = \hat{k}\hat{\delta}=\hat{k}(2\hat{v}/\hat{\sigma})^{1/2}$ for an interfacial wave with wave amplitude â, wavenumber $\hat{k}$ and frequency $\hat{\sigma}$ in a fluid with viscosity $\hat{v}$. Inside the second boundary layers there exists a strong steady streaming of O(α2ε−1/2), where $\alpha = \hat{k}\hat{a}$ is the surface wave slope. The mass transport velocity near the interface is much larger than that in a single-layer system, which is O(α2) (e.g. Longuet-Higgins 1953; Craik 1982). In the core regions outside the thin second boundary layers, the mass transport velocity is enhanced by the diffusion of the mean interfacial velocity and vorticity. Because of vertical diffusion and viscous damping of the mean interfacial vorticity, the ‘interfacial second boundary layers’ diminish as time increases. The mean motions eventually die out owing to viscous attenuation. The mass transport velocity profiles are very different from those obtained by Dore (1970, 1973) which ignored viscous attenuation. When a temporally decaying small-amplitude surface progressive wave is propagating in the system, the mean motions are found to be much less significant, O(α2).

Mass transport in a turbulent boundary layer under a progressive water wave

1984 ◽  
Vol 146 ◽  
pp. 303-312 ◽  
Author(s):  
S. J. Jacobs
Keyword(s):  
Boundary Layer ◽  
Mass Transport ◽  
Water Wave ◽  
Approximate Solutions ◽  
Reynolds Numbers ◽  
Bottom Boundary Layer ◽  
Bottom Boundary ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
Theoretical Results

The bottom boundary layer under a progressive water wave is studied using Saffman's turbulence model. Saffman's equations are analysed asymptotically for the case Re [Gt ] 1, where Re is a Reynolds number based on a characteristic magnitude of the orbital velocity and a characteristic orbital displacement. Approximate solutions for the mass-transport velocity at the edge of the boundary layer and for the bottom stress are obtained, and Taylor's formula for the rate of energy dissipation is verified. The theoretical results are found to agree well with observations for sufficiently large Reynolds numbers.

Double boundary layers in standing interfacial waves

1976 ◽  
Vol 76 (4) ◽  
pp. 819-828 ◽  
Author(s):  
B. D. Dore
Keyword(s):  
Boundary Layer ◽  
Mass Transport ◽  
Layer Structure ◽  
Standing Waves ◽  
Boundary Layer Theory ◽  
Two Dimensional ◽  
Transport Velocity ◽  
Oscillatory Velocity ◽  
Mass Transport Velocity ◽  
The Mean

The double-boundary-layer theory of Stuart (1963, 1966) and Riley (1965, 1967) is employed to investigate the mass transport velocity due to two-dimensional standing waves in a system comprising two homogeneous fluids of different densities and viscosities. The most important double-boundary-layer structure occurs in the neighbourhood of the oscillating interface, and the possible existence of jet-like motions is envisaged at nodal positions, owing to the nature of the mean flows in the layers. In practice, the magnitude of the mass transport velocity can be a significant fraction of that of the primary, oscillatory velocity.

scholarly journals REACH OF WAVES TO THE BED OF THE CONTINENTAL SHELF

1970 ◽  
Vol 1 (12) ◽  
pp. 40 ◽  
Author(s):  
Richard Silvester ◽  
Geoffrey R. Mogridge
Keyword(s):  
Boundary Layer ◽  
Continental Shelf ◽  
Surf Zone ◽  
Sediment Concentration ◽  
Geologic Time ◽  
Continental Shelves ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
Wave Boundary ◽  
The Waves

The physiography of Continental Shelves and their major composition of sediment indicate strongly their terrigenous origin and their smoothing by wave action This premise is supported by the geologic time over which waves have existed and the mass-transport velocity in these relatively shallow depths, particularly the net movement within the wave boundary layer at the bed A given wave tram arriving obliquely to the shore can transport material along the coast, both beyond the breaker line and within the surf zone It is shown that for equal over-all discharge in the two zones, the average sediment concentration offshore close to the bed need be reasonably small, indicating that transport near the beach could be a fraction of that from the breakers to the reach of the waves This latter limit is shown to extend at least half way across the Shelf, with possibilities of greater reach when more realistic prototype conditions are introduced into experiments.

Microscale pressure fluctuations near waves being generated by the wind

1972 ◽  
Vol 54 (3) ◽  
pp. 427-448 ◽  
Author(s):  
J. A. Elliott
Keyword(s):  
Static Pressure ◽  
Pressure Field ◽  
Pressure Fluctuations ◽  
Phase Speed ◽  
Sea Waves ◽  
Mean Wind Speed ◽  
The Mean ◽  
Active Wave ◽  
The Waves ◽  
Active Generation

Measurements of static pressure and wave height are used to describe the waveinduced pressure field above generating sea waves. A large hump in the pressure spectra is observed at the wave frequencies. The amplitude of this hump increases and the rate of its vertical decay decreases as the mean wind speed increases. The phase difference between the pressure and the waves during active generation is about 135°, pressure lagging the waves, and does not change vertically for measurements at heights greater than the wave crests. In the present data, active wave generation appears to occur only when the wind at a height of 5 metres is greater than or about equal to twice the phase speed of the waves.

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