Fourier analysis of the roll-up and merging of coherent structures in shallow mixing layers

The current study investigates the role of nonlinearity in the development of two-dimensional coherent structures (2DCS) in shallow mixing layers. A nonlinear numerical model based on the depth-averaged shallow water equations is used to investigate temporal shallow mixing layers, where the mapping from temporal to spatial results is made using the velocity at the center of the mixing layer. The flow is periodic in the stream-wise direction and the transmissive boundary conditions are used in the cross-stream boundaries to prevent reflections. The numerical results are examined with the aid of Fourier decomposition. Results show that the previous success in applying local linear theory to shallow mixing layers does not imply that the flow is truly linear. Linear stability theory is confirmed to be only valid within a short distance from the inflow boundary. Downstream of this linear region, nonlinearity becomes important for the roll-up and merging of 2DCS. While the energy required for the merging of 2DCS is still largely provided by the velocity shear, the merging mechanism is one where nonlinear mode interaction changes the velocity field of the subharmonic mode and the gradient of the along-stream velocity profile which, in turn, changes the magnitude of the energy production of the subharmonic mode by the velocity shear implicitly. The nonlinear mode interaction is associated with energy up-scaling and is consistent with the inverse energy cascade which is expected to occur in shallow shear flows. Current results also show that such implicit nonlinear interaction is sensitive to the phase angle difference between the most unstable mode and its subharmonic. The bed friction effect on the 2DCS is relatively small initially and grows in tandem with the size of the 2DCS. The bed friction also causes a decrease in the velocity gradient as the flow develops downstream. The transition from unstable to stable flow occurs when the bed friction balances the energy production. Beyond this point, the bed friction is more dominant and the 2DCS are progressively damped and eventually get annihilated. The energy production by the velocity shear plays an important role from the upstream end all the way to the point of transition to stable flow. The fact that linear stability theory is valid only for a short distance from the inflow boundary suggests that some elements of nonlinearity is incorporated in the mean velocity profile in experiments by the averaging process. The implicit nature of nonlinear interaction in shallow mixing layers and the sensitivity of the nonlinear interaction to phase angle difference between the most unstable mode and its subharmonic allows local linear theory to be successful in reproducing features of the instability such as the dominant mode of the 2DCS and its amplitude.     

Flow structure and turbulence characteristics downstream of a spanwise suspended linear array

Laboratory experiments are conducted to quantify the mean flow structure and turbulence properties downstream of a spanwise suspended linear array in a uniform ambient water flow using Particle Tracking Velocimetry. Eighteen experimental scenarios, with four depth ratios (array depth to water column depth) of 0.35, 0.52, 0.78, and 0.95 and bulk Reynolds number (length scale is the array depth) from 11,600 to 68,170, are investigated. Three sub-layers form downstream of the array: (1) an internal wake zone, where the time-averaged velocity decreases with increasing distance downstream, (2) a shear layer which increases in vertical extent with increasing distance downstream of the array, and the rate of the increase is independent of the bulk Reynolds number or the depth ratio, and (3) an external wake layer with enhanced velocity under the array. The location of the shear layer is dependent on the depth ratio. The spatially averaged and normalized TKE of the wake has a short production region, followed by a decay region which is comparable to grid turbulence decay and is dependent on the depth ratio. The results suggest that the shear layer increases the transfer of horizontal momentum into the internal wake zone from the fluid outside of the array and that the turbulence in the internal wake zone can be modeled similarly to that of grid turbulence.     

Laboratory study investigating the regeneration potential of iron particles by and the hydrodynamics of a dam-break generated flow from an infinite reservoir into a channel with an adverse slope

To determine the feasibility of using a dam-break generated flow from the sea into a storm-drain to aid in the regeneration of iron particles that control the production of H₂S in the storm-drain, a laboratory experimental investigation is carried out to measure the regeneration potential and the detailed hydrodynamics of the dam-break generated flow that causes the regeneration. The experiments are carried out using a reservoir of essentially infinite size, the sea, and a channel of limited width and adverse slope 1:20, the storm-drain. The regeneration experiments confirmed the ability of the dam-break generated flow to aid in the regeneration of the iron particles, however the regeneration potential varies from good to poor with distance away from the gate into the channel. The detailed measurements of the hydrodynamics highlighted that the dam-break generated flow from an infinite reservoir diverges little during the first uprush, has much smaller velocities during the first backwash and includes significant free surface waves. An initially wet channel bed reduces the flux into the channel. Close to the gate the flow depth increases more quickly but the velocity, and therefore the regeneration potential, is smaller.     

Effect of initial excess density and discharge on constant flux gravity currents propagating on a slope

The effect of the upstream conditions on propagation of gravity current over a slope is investigated using three-dimensional numerical simulations. The current produced by constant buoyancy flux, is simulated using a large eddy simulation solver. The dense saline solution used at the inlet is the driving force of the flow. Higher replenishment of the current is possible either by a high inflow discharge or high initial fractional density excess. In the simulations, it is observed that these two parameters affect the flow in different ways. Results show that the front speed of the descending current is proportional to the cube root of buoyancy flux, $(g_o^{\prime } Q)^{1/3}$ , which agrees with the previous experimental and numerical observations. The height of the tail of the current grows linearly in the streamwise direction. Formation of a strong shear layer at the boundary of mixed upper layer and dense lower layer is observed within the body and the tail of the current. Over the tail of the current far enough from the inlet, the vertical velocity and density profiles are compared to the ones from an experimental study. Distance from the bed to the point of maximum velocity increases with an increase in inflow discharge, while it remains practically unchanged with increasing initial fractional excess density in the simulations. Even though the velocity profiles are in good agreement, some discrepancies are observed in fractional excess density profiles among experimental and numerical results. Possible reasons for these discrepancies are discussed. Generally, gravity current type of flows could be expressed in layer-integrated formulation of governing equations. However, layer integration introduces several constants, commonly known as shape factors, to the equations of motion. The values of these shape factors are calculated based on simulation results and compared to the values from experiments and to the favorably used ‘top hat’ assumption.     

Modelling hydraulic jump using the bubbly two-phase flow method

Hydraulic jumps have complex flow structures, characterised by strong turbulence and large air contents. It is difficult to numerically predict the flows. It is necessary to bolster the existing computer models to emphasise the gas phase in hydraulic jumps, and avoid the pitfall of treating the phenomenon as a single-phase water flow. This paper aims to improve predictions of hydraulic jumps as bubbly two-phase flow. We allow for airflow above the free surface and air mass entrained across it. We use the Reynolds-averaged Navier–Stokes equations to describe fluid motion, the volume of fluid method to track the interface, and the k–ε model for turbulence closure. A shear layer is shown to form between the bottom jet flow and the upper recirculation flow. The key to success in predicting the jet flow lies in formulating appropriate bottom boundary conditions. The majority of entrained air bubbles are advected downstream through the shear layer. Predictions of the recirculation region’s length and air volume fraction within the layer are validated by available measurements. The predictions show a linear growth of the shear layer. There is strong turbulence at the impingement, and the bulk of the turbulence kinetic energy is advected to the recirculation region via the shear layer. The predicted bottom-shear-stress distribution, with a peak value upstream of the toe of the jump and a decaying trend downstream, is realistic. This paper reveals a significant transient bottom shear stress associated with temporal fluctuations of mainly flow velocity in the jump. The prediction method discussed is useful for modelling hydraulic jumps and advancing the understanding of the complex flow phenomenon.     

Measurements and modeling of open-channel flows with finite semi-rigid vegetation patches

The hydrodynamics of flows through a finite length semi-rigid vegetation patch (VP) were investigated experimentally and numerically. Detailed measurements have been carried out to determine the spatial variation of velocity and turbulence profiles within the VP. The measurement results show that an intrusion region exists in which the peak Reynolds stress remains near the bed. The velocity profile is invariant within the downstream part of the VP while the Reynolds stress profile requires a longer distance to attain the spatially invariant state. Higher vegetation density leads to a shorter adjustment length of the transition region, and a higher turbulence level within the VP. The vegetation density used in the present study permits the passing through of water and causes the peak Reynolds stress and turbulence kinetic energy each the maximum at the downstream end of the patch. A 3D Reynolds-averaged Navier–Stokes model incorporating the Spalart–Allmaras turbulence closure was employed subsequently to replicate the flow development within the VP. The model reproduced transitional flow characteristics well and the results are in good agreement with the experimental data. Additional numerical experiments show that the adjustment length can be scaled by the water depth, mean velocity and maximum shear stress. Empirical equations of the adjustment lengths for mean velocity and Reynolds stress were derived with coefficients quantified from the numerical simulation results.     

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.     

Spatial evolution of coherent motions in finite-length vegetation patch flow

A number of experimental studies on submerged canopy flows have focused on fully-developed flow and turbulent characteristics. In many natural rivers, however, aquatic vegetation occurs in patches of finite length. In such vegetated flows, the shear layer is not formed at the upstream edge of the vegetation patch and coherent motions develop downstream. Therefore, more work is neededz to reveal the development process for large-scale coherent structures within vegetation patches. For this work, we considered the effect of a limited length vegetation patch. Turbulence measurements were intensively conducted in open-channel flows with submerged vegetation using Particle Image Velocimetry (PIV). To examine the transition from boundary-layer flow upstream of the vegetation patch to a mixing-layer-type flow within the patch, velocity profiles were measured at 33 positions in a longitudinal direction. A phenomenological model for the development process in the vegetation flow was developed. The model decomposed the entire flow region into four zones. The four zones are the following: (i) the smooth bed zone, (ii) the diverging flow zone, (iii) the developing zone and (iv) the fully-developed zone. The PIV data also confirmed the efficiency of the mixing-layer analogy and provided insight into the spatial evolution of coherent motions.     

Hydrodynamic characteristics and related mass-transfer properties in open-channel flows with rectangular embayment zone

Consecutive groynes and embayments form dead water zones, where sedimentation and high concentrations of pollutants are often observed. It is thus very important to understand the mass and momentum exchange between the main channel and side cavities in rivers and hydraulic engineering structures. The spanwise gradient of the streamwise velocity near the junction produces small-scale turbulent vortices because of shear instability. Furthermore, large-scale horizontal circulation is also generated in the cavity zone. These coherent turbulent structures play a significant role in mass and sediment transfer at the boundary between the mainstream and embayment. However, the relation between turbulence and mass transfer is poorly understood. In this study, we performed particle image velocity and laser-induced fluorescence experiments using a laboratory flume, laser light sheets and a high-speed CMOS camera. We examined the exchange properties of a dye as a function of bed configuration and sedimentation effect. Both primary and secondary gyres were observed in the flat bed and downward-sloping bed, whereas the primary gyre was prevalent in the upward-sloping bed. Moreover, the horizontal circulation strongly affected the mass-transfer properties between the mainstream and side cavity.     

Experimental study of recirculating flows generated by lateral shock waves in very large channels

Due to the lack of data on hydraulic-jump dynamics in very large channels, the present paper describes the main characteristics of the velocity field and turbulence in a large rectangular channel with a width of 4 m. Although a hydraulic jump is always treated as a wave that is transversal to the channel wall, in the case of this study it has a trapezoidal front shape, first starting from a point at the sidewalls and then developing downstream in an oblique manner, finally giving rise to a trapezoidal shape. The oblique wave front may be regarded as a lateral shockwave that arises from a perturbation at a certain point of the lateral wall and travels obliquely toward the centreline of the channel. The experimental work was carried out at the Coastal Engineering Laboratory of the Water Engineering and Chemistry Department of the Technical University of Bari (Italy). In addition to the hydraulic jump formation, a large recirculating flow zone starts to develop from the separating point of the lateral shock wave and a separate boundary layer occurs. Intensive measurements of the streamwise and spanwise flow velocity components along one-half width of the channel were taken using a bidimensional Acoustic Doppler Velocimeter (ADV). The water surface elevation was obtained by means of an ultrasonic profiler. Velocity vectors, transversal velocity profiles, turbulence intensities and Reynolds shear stresses were all investigated. The experimental results of the separated boundary layer were compared with numerical predictions and related work presented in literature and showed good agreement. The transversal velocity profiles indicated the presence of adverse pressure gradient zones and the law of the wall appears to govern the region around the separated boundary layer.     

Experimental surface flow visualization and numerical investigation of flow structure around an oblong stockpile

Emission factors are largely used to quantify particle emissions from industrial open storage piles. These factors are based on the knowledge of velocity distribution and flow patterns over the stockpile surface which still requires further research. The aim of the present work is to investigate the airflow characteristics over a single typical oblong pile and in its near-ground surroundings for various wind flow directions. Wind tunnel experiments using an oil-film surface coating technique were carried out for near-wall flow visualization. Numerical simulation results, favorably compared to PIV measurements, were used to allow comparison analysis of flow features. For the stockpile oriented 90° to the wind main direction, typical topology of flow around wall-mounted obstacles were observed, notably a wake zone downstream the pile including two main counter-rotating vortices. Further analysis of numerical wall shear stress distribution and streamlines indicates that two complex three-dimensional vortical flow structures develop downstream the pile. For other incoming wind flow directions (30 and 60°), the flow characteristics over the storage pile greatly differ as a single helical main vortex develops from the pile’s crest. Corresponding high values of wall shear stress are noticed downstream the storage pile. For each configuration studied, downwash and upwash zones are induced by the vortical structures developed. This near-wall flow topology combined with areas of high friction levels may be linked to potential dust emission from the ground surface surrounding industrial stockpiles.     

Mass-based depth and velocity scales for gravity currents and related flows

Gravity driven flows on inclines can be caused by cold, saline or turbid inflows into water bodies. Another example are cold downslope winds, which are caused by cooling of the atmosphere at the lower boundary. In a well-known contribution, Ellison and Turner (ET) investigated such flows by making use of earlier work on free shear flows by Morton, Taylor and Turner (MTT). Their entrainment relation is compared here with a spread relation based on a diffusion model for jets by Prandtl. This diffusion approach is suitable for forced plumes on an incline, but only when the channel topography is uniform, and the flow remains supercritical. A second aspect considered here is that the structure of ET’s entrainment relation, and their shallow water equations, agrees with the one for open channel flows, but their depth and velocity scales are those for free shear flows, and derived from the velocity field. Conversely, the depth of an open channel flow is the vertical extent of the excess mass of the liquid phase, and the average velocity is the (known) discharge divided by the depth. As an alternative to ET’s parameterization, two sets of flow scales similar to those of open channel flows are outlined for gravity currents in unstratified environments. The common feature of the two sets is that the velocity scale is derived by dividing the buoyancy flux by the excess pressure at the bottom. The difference between them is the way the volume flux is accounted for, which—unlike in open channel flows—generally increases in the streamwise direction. The relations between the three sets of scales are established here for gravity currents by allowing for a constant co-flow in the upper layer. The actual ratios of the three width, velocity, and buoyancy scales are evaluated from available experimental data on gravity currents, and from field data on katabatic winds. A corresponding study for free shear flows is referred to. Finally, a comparison of mass-based scales with a number of other flow scales is carried out for available data on a two-layer flow over an obstacle. Mass-based flow scales can also be used for other types of flows, such as self-aerated flows on spillways, water jets in air, or bubble plumes.     

Steady Shallow-Water Current and Solute Transport around a Semi-Conical Headland

Laboratory experiments have been carried out to investigate the effects of a sloping wall headland on the flow characteristics and the associated concentration distributions from a point source around the headland. A semi-conical headland with a slope of 1:2 was set up in a flow basin, 4.8 m long and 3.8 m wide. In this paper, the experimental results of a steady shallow-water current are reported. Three dimensional flow velocities in the basin were measured using Sontek-ADV instrument. The dye concentration levels in the basin were measured by two fluorometers. The experimental results showed a large-scale re-circulation region behind the semi-conical headland. The peak turbulence energy, at about 53% of the local kinetic flow energy, coincides with the region of high velocity gradient. Significant vertical flows were observed around the area near the downhill slope of the headland, with a maximum ratio of vertical to horizontal velocities being about 22%. Such relatively significant vertical scouring velocities, coupled with strong turbulence energy and high horizontal velocity gradients in the same region, could cause severe bed erosion. The experimental results have also been compared with the predicted results of a depth-averaged numerical model. The predicted eddy structure and the concentration distribution in the re-circulation area were found to compare favourably with the experimental results. However, the discrepancies in the flow velocities and the concentration levels near the headland were apparent. It was observed that the dye concentration continued to spread in the cross-stream direction after passing the headland, whereas only a limited extent of the lateral spreading was predicted by the numerical model further downstream of the headland.     

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.     

Gravity currents with tailwaters in Boussinesq and non-Boussinesq systems: two-layer shallow-water dam-break solutions and Navier–Stokes simulations

We consider the dam-break initial stage of propagation of a gravity current of density $\rho _{c}$ released from a lock (reservoir) of height $h_0$ in a channel of height $H$ . The channel contains two-layer stratified fluid. One layer, called the “tailwater,” is of the same density as the current and is of thickness $h_T (< h_0)$ , and the other layer, called the “ambient,” is of different density $\rho _{a}$ . Both Boussinesq ( $\rho _{c}/\rho _{a}\approx 1$ ) and non-Boussinesq systems are investigated. By assuming a large Reynolds number, we can model the flow with the two-layer shallow-water approximation. Due to the presence of the tailwater, the “jump conditions” at the front of the current are different from the classical Benjamin formula, and in some circumstances (clarified in the paper) the interface of the current matches smoothly with the horizontal interface of the tailwater. Using the method of characteristics, analytical solutions are derived for various combinations of the governing parameters. To corroborate the results, two-dimensional direct numerical Navier–Stokes simulations are used, and comparisons for about 80 combinations of parameters in the Boussinesq and non-Boussinesq domains are performed. The agreement of speed and height of the current is very close. We conclude that the model yields self-contained and fairly accurate analytical solutions for the dam-break problem under consideration. The results provide reliable insights into the influence of the tailwater on the propagation of the gravity current, for both heavy-into-light and light-into-heavy motions. This is a significant extension of the classical gravity-current theory from the particular $h_T=0$ point to the $h_T > 0$ domain.     

Hydrodynamics and bed stability of open channel flows with submerged foliaged plants

In this work we investigate experimentally and numerically the flow structure around foliaged plants deployed in a channel with gravels on the bed under submerged and barely submerged conditions. Velocity and Reynolds stress were measured by using a NORTEK Vectrino profiler. Visual observation shows that the initial motion of gravels is easier to be triggered under the condition of flow with barely submerged vegetation. This is confirmed by the measured velocity, Reynolds stress and total kinetic energy (TKE) profiles. The velocity exhibits a speed up in the near-bed region, and the associated Reynolds stress and TKE increase there. A 3D numerical model is then verified against the experiments and used to investigate systematically the effect of degree of submergence of foliaged plants on the channel bed shear stress. The results show that the maximum bed shear stress occurs when the foliage is situated slightly below the water surface, which can enhance channel bed instability.     

Gravity currents in two-layer stratified media

An analytical, experimental and numerical study of boundary gravity currents propagating through a two-layer stratified ambient of finite vertical extent is presented. Gravity currents are supposed to originate from a lock-release apparatus; the (heavy) gravity current fluid is assumed to span the entire channel depth, H, at the initial instant. Our theoretical discussion considers slumping, supercritical gravity currents, i.e. those that generate an interfacial disturbance whose speed of propagation matches the front speed, and follows from the classical analysis of Benjamin (J Fluid Mech 31:209?C248, 1968). In contrast to previous investigations, we argue that the interfacial disturbance must be parameterized so that its amplitude can be straightforwardly determined from the ambient layer depths. Our parameterization is based on sensible physical arguments; its accuracy is confirmed by comparison against experimental and numerical data. More generally, measured front speeds show positive agreement with analogue model predictions, which remain strictly single-valued. From experimental and numerical observations of supercritical gravity currents, it is noted that this front speed is essentially independent of the interfacial thickness, ??, even in the limiting case where ?? = H so that the environment is comprised of a uniformly stratified ambient with no readily discernible upper or lower ambient layer. Conversely, when the gravity current is subcritical, there is a mild increase of front speed with ??. Our experiments also consider the horizontal distance, X, at which the front begins to decelerate. The variation of X with the interface thickness and the depths and densities of the ambient layers is discussed. For subcritical gravity currents, X may be as small as three lock lengths whereas with supercritical gravity currents, the gravity current may travel long distances at constant speed, particularly as the lower layer depth diminishes.