Oil spreading in instantaneous and continuous spills on rotating earth

The effect of the Coriolis force on the oil spill spreading in the gravity-viscous regime is examined. A new shallow water model for the transport and spreading of oil slick of arbitrary shape is described in which the Coriolis force is included in the momentum equations and the oil–water friction is parameterized in a frame of the boundary layer theory including the Ekman friction. The numerical Lagrangian method based on smoothed particle dynamics is described. New similarity solutions of the model equations are obtained for unidirectional and axisymmetric spreading in gravity-viscous, gravity-turbulent and gravity-viscous-rotational regimes for instantaneous as well as continuous releases. The numerical simulation extends these results for the case of continuous release in the presence of currents. It was shown that Coriolis term in the momentum equation can be omitted if slick thickness is much less of the laminar Ekman layer thickness. However, the Ekman friction should be retained for slicks of any thickness for larger times. The Ekman friction results in the essential slowdown of the spreading as well as in the deflection of the oil spreading velocity at 45° from the direction of velocity in the non-rotation case. Numerical simulations of large-scale spills showed that after the 2?days the slick area with the Coriolis effect was approximately less than half of that without rotation. Therefore, the earth rotation can be also important in the oil weathering.     

Model of turbulence penetration into a suspension layer on a sediment bed and effect on vertical solute transfer

The vertical diffusional mass (solute) transfer through a suspended sediment layer, e.g. at the bottom of a lake, reservoir or estuary, by the propagation of velocity fluctuations from above was investigated. The attenuation of the velocity fluctuations in the suspension layer and the associated effect on solute transfer through the suspension layer was simulated. To represent large eddies traveling downstream in water over a high-concentration suspended sediment layer, a streamwise velocity fluctuation moving in downstream direction was imposed along the upper boundary of the suspension layer. Velocity fluctuations and downstream velocity were normalized by the shearvelocity (U_*) at the top of the suspension layer. Streamwise and vertical velocity components inside the suspension layer, were obtained from the 2-D continuity and the Navier–Stokes equations. The persistence of turbulence with depth—as it penetrates from the overlying water into the suspension layer—was found to depend on its amplitude, its period, and on the apparent viscosity of the suspension. The turbulence was found to propagate efficiently into the suspension layer when its frequency is low, and the apparent viscosity of the suspension is high. Effects on vertical mass transfer were parameterized by penetration depth and effective diffusion coefficient, and related to apparent viscosity of the suspension, Schmidt number and shear velocity on top of the suspension layer. The enhancement of turbulence penetration by viscosity is similar to the flow near an oscillating flat plate (Stokes’ second problem), but is opposite to turbulence penetration into a stationary porous and permeable sediment bed. The information is applicable to water quality modeling mear the sediment/water interface of lakes, river impoundments and estuaries.     

The Residence Time of Settling Particles in the Surface Mixed Layer

The transport from the upper mixed layer into the pycnocline of particles with negative buoyancy is considered. Assuming the hydrodynamic parameters to be time- independent, an adjoint model is resorted to that provides a general expression of the residence time in the mixed layer of the constituent under study. It is seen that the residence time decreases as the settling velocity increases or the diffusivity decreases. Furthermore, it is demonstrated that the residence time must be larger than z/w and smaller than h/w, where z, h and w denote the distance to the pycnocline, the thickness of the mixed layer and the sinking velocity. In the vicinity of the pycnocline, the residence time is not necessarily zero; its behaviour critically depends on the eddy diffusivity profile in this region. Closed-form solutions are obtained for constant and quadratic diffusivity profiles, which allows for an analysis of the sensitivity of the residence time to the Peclet number. Finally, an approximate value is suggested of the depth-averaged value of the residence time.     

A model for the larval migration of the Japanese eel: roles of the trade winds and salinity front

A model that helps explain the mysterious long-distance migration of the Japanese eel (Anguilla japonica) is presented, based on oceanographic observations, satellite buoy drift experiments, and samplings of eel larvae taken in 1991. The trajectory of a 150 m depth buoy relased in the spawning area strongly suggests that A. japonica larvae spawned just south of the salinity front are transported westward by the North Equatorial Current (NEC). The larvae are then thought to be entrained into the Mindanao Current flowing southward along the Philippine Islands where A. japonica juveniles are scarcely distributed. These controversial results lead to the assumption that eel larvae are transferred from the NEC to the northward flowing Kuroshio, which distributes the eel larvae to the growth habitats of eastern Asia. In this eel larvae transfer model, a northward Ekman transport caused by trade winds plays an important role in explaining the wind-induced northward shift of the larvae together with the onset of diel vertical migration. Assuming that leptocephali greater than 20 mm initiate the vertical migration, a westward wind velocity greater than 5 to 10 m s^-1 should be high enough to diminish the southward current velocity. When the physical and geophysical conditions — such as the salinity front for spawning activity, the water tunnel for westward larval transport, the Ekman transport by the trade wind for transfer of the larvae from the NEC to the Kuroshio, and the strong velocity of the Kuroshio for rapid transport to growth habitats — are well matched with the timing of the onset of vertical migration, large-scale eel migration could result.     

PIV experimental study on the flow characteristics upstream of a floating intake in nonlinear stratified ambient conditions

Selective withdrawal is commonly implemented in nonlinearly stratified ambient, which typically has stratified ambient conditions, for purposes of controlling quality. A floating intake is applied as an effective facility of selective withdrawal. However, the outflow dynamics of a floating intake in a nonlinearly stratified ambient have been disregarded, which has a significant effect on the outflow water quality of a reservoir. Experiments were conducted to investigate the effect of thermal stratification on the flow characteristics using particle image velocimetry at three temperature distributions (no stratification, weak stratification and strong stratification). The flow fields upstream of the floating intake showed that the withdrawal layer was formed inhibited by the thermal stratification. And strong stratification produced the thinner withdrawal layer thickness, leading to a larger nonuniform coefficient of the velocity profile. To quantitatively describe the velocity profiles, formulas of dimensionless velocity profiles were proposed. The flow developments were analysed, and the virtual control points located 0.56d above the floating intake (where d is the straight pipe diameter of the floating intake) were obtained. The positions of virtual control points mainly depended on the withdrawal discharge. The decay rate of the velocity along the horizontal line passing through the virtual control point was inversely proportional to the stratification intensity.

     

Stokes drift through corals

Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers (Phillips in Flow and reactions in permeable rocks, Cambridge University Press, Cambridge, 1991) or only considers the drift above the porous layer (Monismith in Ann Rev Fluid Mech 39:37–55, 2007). Overcoming these limitations, we propose a model where flow is described by a velocity potential above the porous layer and by Darcy’s law in the porous bed, with derived matching conditions at the interface between the two layers. Both a horizontal and a novel vertical drift effect arise from the damping of the porous bed, which requires the use of a complex wavenumber k. This is in contrast to the purely horizontal second-order drift first derived by Stokes (Trans Camb Philos Soc 8:441–455, 1847) when working with solely a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al. In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007). We compare our model with field measurements by Koehl and Hadfield (J Mar Syst 49:75–88, 2004) and also explain the vertical drift effects as documented by Koehl et al. (Mar Ecol Prog Ser 335:1–18, 2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.

     

An engineering model for countercurrent flow under wind-induced waves and current

A general model for the phase-averaged velocity field in wind-induced countercurrent flow is proposed. The influence of waves on the time-averaged velocity is accounted for by introducing a skewness factor in a parabolic eddy viscosity model. The skewness factor represents the net effect of the wavy surface in the engineering model for velocity. The coherent velocity components are described separately by an orbital velocity obtained from linear wave theory and are added to the time-averaged components to give a complete model for the phase-averaged velocity field. The proposed model collapses to the standard model for deep-water conditions, but is also shown to yield the correct behavior for intermediate conditions. Moreover, the bed shear stress, derived from the proposed velocity model, is also shown to be in agreement with experiments.     

Kelvin–Helmholtz instability in the presence of variable viscosity for mudflow resuspension in estuaries

The temporal stability of a parallel shear flow of miscible fluid layers of different density and viscosity is investigated through a linear stability analysis and direct numerical simulations. The geometry and rheology of this Newtonian fluid mixing can be viewed as a simplified model of the behavior of mudflow at the bottom of estuaries for suspension studies. In this study, focus is on the stability and transition to turbulence of an initially laminar configuration. A parametric analysis is performed by varying the values of three control parameters, namely the viscosity ratio, the Richardson and Reynolds numbers, in the case of initially identical thickness of the velocity, density and viscosity profiles. The range of parameters has been chosen so as to mimic a wide variety of real configurations. This study shows that the Kelvin–Helmholtz instability is controlled by the local Reynolds and Richardson numbers of the inflection point. In addition, at moderate Reynolds number, viscosity stratification has a strong influence on the onset of instability, the latter being enhanced at high viscosity ratio, while at high Reynolds number, the influence is less pronounced. In all cases, we show that the thickness of the mixing layer (and thus resuspension) is increased by high viscosity stratification, in particular during the non-linear development of the instability and especially pairing processes. This study suggests that mud viscosity has to be taken into account for resuspension parameterizations because of its impact on the inflection point Reynolds number and the viscosity ratio, which are key parameters for shear instabilities.     

Streamwise velocity profiles in coastal currents

The ability to model marine currents can be a powerful device for many planning activities, for which the knowledge of the velocity field is of pivotal importance, such as the evaluation of current-induced loading on maritime structures or the diffusion and dispersion of polluted flow discharges. Observations of time-averaged velocity profiles, taken with a vessel mounted acoustic Doppler current profiler during a monitoring survey program in the seas of Southern Italy, are analysed in this paper. The measurements were taken under non-breaking conditions, offshore the surf zone, with the aim of reproducing the vertical trends of the streamwise velocity by means of standard theoretical laws. To evaluate also the possible influence of stratification on the current velocity profile shape, together with velocity measurements water temperature and salinity were also measured at the same time and locations, by means of a CTD recorder. The examined surveys referred to different time periods and sites, to guarantee a general validity of deductions. On the basis of the experiments, we verified the actual existence of a log-layer and concluded that the upper limit of the region in which the log law is applicable extends well beyond the inner region. Moreover, the deviations of the measured velocity from the logarithmic profiles above the height of the log layer is consistent with the effects of stratification. The parameters of the log law were estimated, depending on both flow dynamics and stratification in the target area. As a second step, in the most superficial and stratified layer, the velocity profiles were modelled by means of a power law, which fitted the measured data well. According to previous studies, the power law parameters result Reynolds number dependent by means of a new proposed formulation. Finally, the bottom stress and the bottom drag coefficient were investigated.     

RANS-based simulation of transverse turbulent mixing in a 2D geometry

Although transverse mixing is a significant process in river engineering when dealing with the discharge of pollutants from point sources or the mixing of tributary inflows, no theoretical basis exists for the prediction of its rate, which is indeed based upon the results of experimental works carried on in laboratory channels or in streams and rivers. The paper presents the preliminary results of a numerical study undertaken to simulate the transverse mixing of a steady-state point source of a tracer in a two-dimensional rectangular geometry, which is expected to reproduce a shallow flow. This geometry is that of Lau and Krishnappan (J Hydraul Div 13(HY10):1173–1189, 1977), who collected turbulent mixing data for a shallow flow. In the numerical study an approach based on the Reynolds Averaged Navier–Stokes (RANS) equations was applied, where the closure problem was solved by using turbulent viscosity concept. Particularly, the classical two-equations k–? model was used. Two methods were applied to the model results to evaluate the turbulent transverse mixing coefficient. The effect on transverse mixing of a grid located upstream the tracer source was also studied. Numerical results were generally higher than the experimental data. This overestimation could be explained considering the hypothesis of isotropic turbulence underlying the k–? model, which can lead to large turbulent viscosities and rate of mixing. However, RANS-based results may still be considered acceptable also providing the large uncertainties associated with literature predictive equations.     

Surface signatures of submerged heated jet

The flow induced at the surface of a water body by a submerged heated horizontal turbulent jet was investigated experimentally with the aim of developing parameterizations for surface mean temperature/velocity fields. The jet nozzle diameter was fixed, the depth of the jet beneath the free surface was varied, and two jet Reynolds numbers (5020, 11300) were considered. The surface temperature was measured using a highly sensitive infrared camera, and the near-surface horizontal velocity field was measured using particle image velocimetry. The experimental results were explained using a model based on similarity solutions with variable turbulent viscosity. While classical Schlichting’s solution with constant turbulent viscosity predicts complete similarity for transverse velocity/temperature distributions only in a plane that coincides with the flow axis, the present solution predicts similarity in an arbitrary plane parallel to the flow axis, which was confirmed using data collected at the surface. Comparisons of present data with available previous results also showed general agreement.     

Beyond eddy diffusivity: an alternative model for turbulent dispersion

Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.     

Analytical solution for katabatic flow induced by an isolated cold strip

An analytical model for katabatic flow induced by cold strip of finite width in the cross-slope direction but of infinite extent in the downslope direction is presented. The fluid is assumed to have a constant (eddy) viscosity, and the Coriolis force is neglected. A numerical simulation has been used to verify the model, which is shown to revert to the classical Prandtl model if the strip width goes to infinity. The effects of the strip width and slope angle on the katabatic flow are studied. The buoyancy and downslope velocity reach maximum values at the centre of the strip, and spread outwards in the cross-slope direction. The downslope wind maximum weakens for narrow strips and shallow slopes. In contrast to the Prandtl solution, which shows a counter flow above the wind maximum, our model predicts the counter flow to occur outside the strip. The cross-slope variation in the surface forcing induces cross-slope and slope-normal velocities, which are manifested in vortices at the strip edges. Below the wind maximum, the fluid above the cooling surface descends and moves toward the strip edge where it is detrained from the strip region. Replenishment of fluid into the strip region takes place above the wind maximum.     

Theoretical modeling of suspended grain-size distribution in fluvial environment by stratification and secondary current approaches

Here we propose a theoretical model to compute the suspended grain-size distribution in fluvial environment. We derive the model based on the Reynolds averaged Navier–Stokes equation and the continuity equation of sediment phase. The model includes the effects of secondary current and stratification which are the cause of complex interaction between turbulence and grain-size distribution in the sediment-laden flow. Due to an immense importance of particle–particle and particle-turbulence interactions near the channel bed, we include their impacts in the boundary condition of the model. The present model has noteworthy contribution to demonstrate the phenomena of suspended grain-size distribution in the real world. Reported experimental data in literature shows well agreement with the numerical solution computed from the suggested model. The better computational accuracy of the present model is ascertained when the upper bound of calculated error between observed experimental data and computed values is found to be lowest for our model in comparison to a large number of existing models developed from different mathematical viewpoints.     

Drift velocity in sediment-laden downward jets

Environmental Fluid Mechanics - The drift velocity, defined as the velocity of individual phase relative to the water–sediment mixture, is a key variable in two-phase mixture model. In this...     

Turbulent entrainment into fluid mud gravity currents

The entrainment of ambient water into non-Newtonian fluid mud gravity currents was investigated in this study. Constant volume release gravity currents were generated in a lock-exchange tank for a wide range of experimental conditions. A technique similar to the so-called light attenuation technique was used to find the boundary of the current, allowing for the calculation of both temporal and bulk entrainment parameters (in terms of the temporal and bulk entrainment velocities, respectively). It was found that the temporal entrainment velocity is dependent on different parameters in the different propagation phases. The slumping phase begins with an adjustment zone (henceforth, non-established zone) in which the temporal entrainment velocity is not a function of the current front velocity, followed by the established zone in which the temporal entrainment velocity is a function of the current front velocity. This dependence of the temporal entrainment velocity on the current front velocity carries through to the inertia-buoyancy phase. As expected, temporal entrainment velocity in the viscous-buoyancy phase was negligible in comparison to average entrainment velocity in the other phases. It is observed that the temporal entrainment characteristics in the non-established zone is governed by the competition between the entrainment-inhibiting density stratification effects and the entrainment-favouring effects of the Kelvin–Helmholtz billows that are quantified by the Richardson number and the Reynolds number of the gravity current, respectively. In the established zone, Reynolds number effects were observed to dominate over Richardson number effects in dictating temporal entrainment characteristics. A parameterization for the temporal entrainment velocity for non-Newtonian fluid mud gravity currents is developed based upon the experimental observations. This study also found that the bulk entrainment characteristics for the non-Newtonian fluid mud gravity currents can be parameterized by the Newtonian bulk entrainment parameterizations that rely solely on a bulk Richardson number. Interestingly, it was found that the non-Newtonian characteristics of the gravity current have little to no effect on the entrainment of the Newtonian ambient fluid.     

A One-Equation Turbulence Model for Geophysical Applications: Comparison with Data and the k−ε Model

A one-equation turbulence model is presented, in which the turbulent kinetic energy k is calculated with a transport equation whereas the turbulent length scale l is calculated with an algebraic expression. The value of l depends on the local stratification and reduces to the classical |z| scaling for unstratified flows near a wall, where |z| is the distance to the wall. The length scale decreases during stable stratification, and increases for unstable stratification compared to the neutral case. In the limit of strong stable stratification, the so-called buoyancy length scale proportional to k ^1/2 N ^–1 is obtained, where N is the buoyancy frequency. The length scale formulation introduces a single model parameter which is calibrated against experimental data. The model is verified extensively against laboratory measurements and oceanic data, and comparisons are made with the two-equation k- model. It is shown that the performance of the proposed k model is almost identical to that of the k- model. In addition, the stability functions of Launder are revisited and adjusted to obtain better agreement with recent data.     

Idealized study on cohesive sediment flux by tidal asymmetry

The flux of cohesive sediment in an estuary is determined by many factors, including tidal asymmetry, wave effect, fluvial influence, phase difference between tidal velocity and tidal level fluctuations, sediment properties, flocculation, bed erodibility, bathymetry effect and other nonlocal effects. Our capability in predicting sediment fluxes in tide-dominant environments is critical to the morphodynamics and water quality of estuaries. Due to the difficulties in carrying out detailed measurement of sediment flux with high spatial and temporal resolutions, an one-dimensional-vertical (1DV) numerical model for cohesive sediment transport, previously verified and calibrated with field measured cohesive sediment concentration data, is utilized here to study some of the aforementioned factors in affecting tidal-driven sediment fluxes in idealized condition. Tidal-averaged sediment flux is shown to be correlated with tidal velocity skewness with a linear relationship. This linear relationship is different from that of non-cohesive sediment and it is demonstrated here to be mainly due to variable critical shear stress implemented for the mud bed in order to parameterize consolidation. The reason that tidal velocity skewness causes tidal-averaged residual sediment transport is shown to be due to nonlinear intra-tidal interactions between flow velocity and sediment concentration. Moreover, the effects of nonlinear intra-tidal interaction between tidal velocity and tidal level fluctuations is shown to mainly cause seaward transport, which is the most significant under progressive wave system (phase difference 90°) and almost negligible for standing wave system (phase difference 0°).