Martin C. Miller

The Threshold of Sediment Movement Under Oscillatory Water Waves

ABSTRACT As the velocity of the to-and-fro water motion near the bottom under oscillatory waves is increased, there comes a stage when the water exerts a stress on the particles sufficient to cause them to move. This study reviews the analyses and available data on this threshold of sediment motion under wave action. For grain diameters less than about 0.05 cm (medium sands and finer) the threshold is reached while the flow in the boundary layer is still laminar and the threshold is best related by the equation um2/(s-) gD = 0.30 (do/D), where um and do are the near-bottom velocity and orbital diameter of the wave motion, is the density of water, and s and D are respectively the density and diameter of the sediment grains. This relationship is modified after an empirical equation deduced by Bagnold but has a theoretical basis. For grain diameters greater than 0.05 cm (coarse sands and coarser) the threshold occurs after the boundary layer has become turbulent and is best predicted with an empirical curve relating do/D to um/(s - ) g T where T is the wave period. This latter dimensionless number represents the ratio of the acceleration forces to the effective gravity force acting on the grains.

SEDIMENT THRESHOLD UNDER OSCILLATORY WAVES

The steady state profile of the longshore current induced by regular, obliquely incident, breaking waves, over a bottom with arbitrary parallel bottom contours, is predicted. A momentum approach is adopted. The wave parameters must be given at a depth outside the surf zone, where the current velocity is very small. The variation of the bottom roughness along the given bottom profile must be prescribed in advance. Depth refraction is included also in the calculation of wave set-down and set-up. Current refraction and rip-currents are excluded. The model includes two new expressions, one for the calculation of the turbulent lateral mixing, and one for the turbulent bottom friction. The term for the bottom friction is non-linear. Rapid convergent numerical algorithms are described for the solution of the governing equations. The predicted current profiles are compared with laboratory experiments and field measurements. For a plane sloping bottom, the influence of different eddy viscosities and constant values of bottom roughness is examined.The calculation of turbulent flow using Naviers equations assumes the introduction of a turbulent viscosity coefficient the value of which is normally constant, conforming with Boussinesqs hypothesis. It was shown that setting aside this hypothesis, a velocity profile quite different to that resulting from the classic theory is obtained in the case of flow induced by wind. This result appears to be confirmed by the tests carried out in the Mediterranean. The advantage of this method is that it gives the vertical turbulent diffusion which is of particular interest to pollution studies.In the numerical method of prediction of wind waves in deep water, Hasselmanns nonlinear interaction theory is applied. This method assumes the energy balance of individual component waves. However, the total energy balance must exist in the transformation of irregular waves in shoaling water. In this investigation, experiments were carried out on the transformations in shoaling water of composite waves having two components and random waves having one or two main peaks. It was found that the elementary component wave height of the composite waves and the elementary peak power of the random waves decrease with decrease in the water depth. This reason can be explained qualitatively by the theory of the elementary component wave height change of finite amplitude waves in shoaling water. The secondary component wave height of the composite waves and the secondary peak power of the random waves increase with decrease in the water depth. This can be explained qualitatively by Hamadas theory of nonlinear interaction in uniform depth.Experiments have been carried out by using non-breaking waves and breaking waves to investigate the wave forces on a vertical circular cell located in the shallow water. Based on the experimental data, the drag coefficient and the inertia coefficient of a circular cylinder and the curling factor of breaking waves are estimated, and the computation methods of wave forces are examined. As a result, it is shown that the phase lag of inertia forces behind the accelerations of water particles should be considered for the estimation of the drag coefficient as well as the inertia coefficient. In addition the previous formula of the maximum breaking wave forces acting on a cell or a pile is revised by introducing the effects of the above-mentioned phase lag and another phase difference, both of which are functions of the ratio of the cell diameter to the wave length. • It is confirmed that the proposed formula is applicable even to the large cell with the diameter comparable to the wave length. INTRODUCTION Many studies have been done on the impulsive pressures acting on a vertical wall, but there has been very little investigation of breaking wave forces on a cell-type structure. The breaking wave forces should be taken into consideration all the same in the design of pile-type or cell-type structures in nearshore area, because breaking waves cause extreme shock pressures on a cell structure asThe air bubble plume induced by the steady release of air into water has been analyzed with an integral technique based on the equations for conservation of mass, momentum and buoyancy. This approach has been widely used to study the behavior of submerged turbulent jets and plumes. The case of air-bubble induced flow, however, includes additional features. In this study the compressibility of the air and the differential velocity between the rising air bubbles ,and the water are introduced as basic propertie s of the air bubble plume in addition to a fundamental coefficient of entrainment and a turbulent Schmidt number characterizing the lateral spreading of the air bubbles. Theoretical solutions for twoand three-dimensional air-bubble systems in homogeneous, stagnant water are presented in both dimensional and normalized form and compared to existing experimental data. The further complication of a stratified environment is briefly discussed since this case is of great practical interest. This paper is to be considered as a progress report, as future experimental verification of various hypotheses is needed.

Oscillation Sand Ripples Generated by Laboratory Apparatus

ABSTRACT Laboratory data on the geometry of oscillation ripples, collected by a number of investigators using a variety of experimental devices, have been compared to determine differences among oscillating bed, wave flume and water tunnel devices. Additional data have been obtained employing a wave flume that generates large waves. It was found that ripples formed in oscillating-bed devices do not correspond well with those developed under progressive waves in flumes and in water tunnels. The remaining vortex ripple data, the class of ripples primarily formed under ocean waves, are found to correspond to the linear relationship, = 0.65do, where is the ripple spacing and do is the near-bottom orbital diameter. The data depart from this relationship as the orbital diameter increases, the departure occurring at higher orbital diameters as the grain size increases. A tentative diagram is developed relating the maximum possible tipple spacing to the median sand diameter.

The Initiation of Oscillatory Ripple Marks and the Development of Plane-bed at High Shear Stresses Under Waves

ABSTRACT Data on the development of oscillatory ripple marks under waves are utilized as a further check on the equations that have been proposed for the threshold of sediment motion. If the threshold curve is correct, the data on ripple occurrence should lie in a stress field above the curve for threshold. Analysis of the data shows this mainly to be true but suggests that the curve for the threshold should be lowered slightly to smaller stress values as a number of ripple occurrences would otherwise fall below the threshold curve. As the stress of the wave orbital motion increases, the ripple heights progressively decrease and ultimately disappear at a critical stress value. Data on this disappearance and plane-bed sheet sand transport development are examined and compared with the data on the high-stress presence of ripples. There is good agreement of the two data sets and also confirmation of the theoretical criterion of Bagnold for the disappearance of ripples.

Measurements of Sand Spreading Rates under Near-Bottom Wave Orbital Motions

Laboratory and field sand tracer experiments have been conducted to determine the longitudinal (stream-wise) mass spreading coefficient of sand under the oscillatory motions of progressive surface water waves. The laboratory data were obtained in a wave channel sufficiently large to generate monochromatic waves with periods up to 5 sec and heights of about 38 cm. The field experiments were conducted off the Oregon coast, outside the breaker zone in water depths of 16-18 m. Mass spreading coefficients, determined from the time-rate of change of variance of the tracer distribution, were found to be proportional to the entrainment rate of tracer at the injection site. A direct relationship based upon random-walk considerations was also found between the sand spreading rate and the wave orbital parameters. A spreading model based upon the continuous release of tracer from a line source was compared with favorable results to the observed tracer distribution patterns.

The Spreading of Sand Tracers Under Water Waves: Laboratory and Field Investigations

A series of Eularian sand tracer experiments using fluorescent dyed sand were conducted in a large laboratory wave flume and in the high energy environment off the Oregon coast in order to determine the longitudinal (streamwise) mass spreading (dispersion) coefficient of sand under the oscillatory motion of progressive water waves. Laboratory wave periods ranged from three to five seconds with wave heights of 38 cm. Wave periods during the field experiment were ten seconds with wave heights of about one metre. Water depths were 3.17 and 16 to 18 m in the laboratory and field, respectively. Mass spreading coefficients, determined from the time-rate of change of variance of the tracer distribution, were found to be proportional to the entrainment rate of tracer at the injection site. A direct relationship based upon random-walk considerations was also found between the sand spreading rate and the wave orbital parameters. A spreading model based upon the continuous release of tracer from a line source was compared with favorable results to the observed tracer distribution patterns.