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.     

Visualization of the internal flow properties and the material exchange interface in an entraining viscous Newtonian gravity current

In order to simulate a simple entraining geophysical flow, a viscous Newtonian gravity current is released from a reservoir by a dam-break and flows along a rigid horizontal bed until it meets a layer of entrainable material of finite depth, identical to the current. The goal is to examine the entrainment mechanisms by observing the interaction between the incoming flow and the loose bed. The sole parameter varied is the initial volume of the gravity current, thus altering its height and velocity. The gravity current plunges or spills into the entrainable bed and the velocity of the flow front becomes linear with time. The bed material is directly affected: motion is generated in the fluid far downstream of, and in that lying beneath the encroaching front. Shear bands are identified, separating horizontal flow downstream from flow with a strong vertical component close to the step. Downstream of the step the flow is horizontal and stratified, with no slip on the bottom boundary and very low shear near the surface. Between these two regions may lie transitional zones with linear velocity profiles, separated by horizontal bands of high shear; the number of transitional zones in the cross-section varies with the initial volume of the dam-break.     

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.     

Small-scale entrainment in inclined gravity currents

We investigate the effect of buoyancy on the small-scale aspects of turbulent entrainment by performing direct numerical simulation of a gravity current and a wall jet. In both flows, we detect the turbulent/nonturbulent interface separating turbulent from irrotational ambient flow regions using a range of enstrophy iso-levels spanning many orders of magnitude. Conform to expectation, the relative enstrophy isosurface velocity \(v_n\) in the viscous superlayer scales with the Kolmogorov velocity for both flow cases. We connect the integral entrainment coefficient E to the small-scale entrainment and observe excellent agreement between the two estimates throughout the viscous superlayer. The contribution of baroclinic torque to \(v_n\) is negligible, and we show that the primary reason for reduced entrainment in the gravity current as compared to the wall-jet are 1) the reduction of \(v_n\) relative to the integral velocity scale \(u_T\); and 2) the reduction in the surface area of the isosurfaces.     

Blockage of saline intrusions in restricted,two-layer exchange flows across a submerged sill obstruction

Results are presented from a series of large-scale experiments investigating the internal and near-bed dynamics of bi-directional stratified flows with a net-barotropic component across a submerged, trapezoidal, sill obstruction. High-resolution velocity and density profiles are obtained in the vicinity of the obstruction to observe internal-flow dynamics under a range of parametric forcing conditions (i.e. variable saline and fresh water volume fluxes; density differences; sill obstruction submergence depths). Detailed synoptic velocity fields are measured across the sill crest using 2D particle image velocimetry, while the density structure of the two-layer exchange flows is measured using micro-conductivity probes at several sill locations. These measurements are designed to aid qualitative and quantitative interpretation of the internal-flow processes associated with the lower saline intrusion layer blockage conditions, and indicate that the primary mechanism for this blockage is mass exchange from the saline intrusion layer due to significant interfacial mixing and entrainment under dominant, net-barotropic, flow conditions in the upper freshwater layer. This interfacial mixing is quantified by considering both the isopycnal separation of vertically-sorted density profiles across the sill, as well as calculation of corresponding Thorpe overturning length scales. Analysis of the synoptic velocity fields and density profiles also indicates that the net exchange flow conditions remain subcritical (G < 1) across the sill for all parametric conditions tested. An analytical two-layer exchange flow model is then developed to include frictional and entrainment effects, both of which are needed to account for turbulent stresses and saline entrainment into the upper freshwater layer. The experimental results are used to validate two key model parameters: (1) the internal-flow head loss associated with boundary friction and interfacial shear; and (2) the mass exchange from the lower saline layer into the upper fresh layer due to entrainment.     

Compound channel flow with a longitudinal transition in hydraulic roughness over the floodplains

Flows in a compound open-channel (two-stage geometry with a main channel and adjacent floodplains) with a longitudinal transition in roughness over the floodplains are experimentally investigated in an 18 m long and 3 m wide flume. Transitions from submerged dense vegetation (meadow) to emergent rigid vegetation (wood) and vice versa are modelled using plastic grass and vertical wooden cylinders. For a given roughness transition, the upstream discharge distribution between main channel and floodplain (called subsections) is also varied, keeping the total flow rate constant. The flows with a roughness transition are compared to flows with a uniformly distributed roughness over the whole length of the flume. Besides the influence of the downstream boundary condition, the longitudinal profiles of water depth are controlled by the upstream discharge distribution. The latter also strongly influences the magnitude of the lateral net mass exchanges between subsections, especially upstream from the roughness transition. Irrespective of flow conditions, the inflection point in the mean velocity profile across the mixing layer is always observed at the interface between subsections. The longitudinal velocity at the main channel/floodplain interface, denoted \(U_{int}\), appeared to be a key parameter for characterising the flows. First, the mean velocity profiles across the mixing layer, normalised using \(U_{int}\), are superimposed irrespective of downstream position, flow depth, floodplain roughness type and lateral mass transfers. However, the profiles of turbulence quantities do not coincide, indicating that the flows are not fully self-similar and that the eddy viscosity assumption is not valid in this case. Second, the depth-averaged turbulent intensities and Reynolds stresses, when scaled by the depth-averaged velocity \(U_{d,int}\) exhibit two plateau values, each related to a roughness type, meadow or wood. Lastly, the same results hold when scaling by \(U_{d,int}\) the depth-averaged lateral flux of momentum due to secondary currents. Turbulence production and magnitude of secondary currents are increased by the presence of emergent rigid elements over the floodplains. The autocorrelation functions show that the length of the coherent structures scales with the mixing layer width for all flow cases. It is suggested that coherent structures tend to a state where the magnitude of velocity fluctuations (of both horizontal vortices and secondary currents) and the spatial extension of the structures are in equilibrium.     

Three-dimensional analysis of coherent turbulent flow structure around a single circular bridge pier

The coherent turbulent flow around a single circular bridge pier and its effects on the bed scouring pattern is investigated in this study. The coherent turbulent flow and associated shear stresses play a major role in sediment entrainment from the bed particularly around a bridge pier where complex vortex structures exist. The conventional two-dimensional quadrant analysis of the bursting process is unable to define sediment entrainment, particularly where fully three-dimensional flow structures exist. In this paper, three-dimensional octant analysis was used to improve understanding of the role of bursting events in the process of particle entrainment. In this study, the three-dimensional velocity of flow was measured at 102 points near the bed of an open channel using an Acoustic Doppler Velocity meter (Micro-ADV). The pattern of bed scouring was measured during the experiment. The velocity data were analysed using the Markov process to investigate the sequential occurrence of bursting events and to determine the transition probability of the bursting events. The results showed that external sweep and internal ejection events were an effective mechanism for sediment entrainment around a single circular bridge pier. The results are useful in understanding scour patterns around bridge piers.     

Near-critical free-surface flows: real fluid flow analysis

An open channel flow with a flow depth close to the critical depth is characterised by a curvilinear streamline flow field that results in steady free surface undulations. Near critical flows of practical relevance encompass the undular hydraulic jump when the flow changes from supercritical (F > 1) to subcritical (F < 1), and the undular weir flow over broad-crested weirs where the flow changes from subcritical (F < 1) to supercritical (F > 1). So far these flows were mainly studied based on ideal fluid flow computations, for which the flow is assumed irrotational and, thus, shear forces are absent. While the approach is accurate for critical flow conditions (F = 1) in weir and flumes, near-critical flows involve long distances reaches, and the effect of friction on the flow properties cannot be neglected. In the present study the characteristics of near-critical free-surface flows are reanalysed based on a model accounting for both the streamline curvature and friction effects. Based on the improved model, some better agreement with experimental results is found, thereby highlighting the main frictional features of the flow profiles.     

Structure of a dense release produced by varying initial conditions

Buoyancy driven flows such as gravity currents, present in nature and human made applications, are conveyors of particles or dissolved substances for long distances with clear implications for the environment. This transport depends on the triggering conditions of the current. Gravity currents are experimentally simulated under varying initial conditions by combining three different initial buoyancies and five volumes of dense fluid released. The horizontal and vertical structures of the gravity currents are analysed and it is shown that the variation on the initial configuration is conditioning for these. Vertical transport through the gravity current is influenced at the bottom by the solid wall over which the current flows, and at the upper interface by the contact with the ambient water. The relative contribution of shear stress at the bottom and at the upper interfaces are estimated and analysed in terms of the initial triggering conditions of the current; these two compete with the buoyancy, the driver of the current, determining mixing and entrainment. By using a proper parametrization, which accounts for both initial volume of release and location of the observation position relative to the lock, a relation between the resistance at the bottom and at the upper interfaces with the initial conditions of release (i.e. the lock-length) has been found; this is found to be independent of the initial density in the lock. The present study shows that the variation of the initial conditions have consequences on (1) the configuration of the currents and on (2) the hydrodynamics of the currents, including mass and momentum exchanges, which are in addition mutually dependent.     

Froude scaling limitations in modeling of turbidity currents

The scaling problem associated with the modeling of turbidity currents has been recognized but is yet to be explored systematically. This paper presents an analysis of the dimensionless governing equations of turbidity currents to investigate the scale effect. Three types of flow conditions are considered: (i) conservative density current; (ii) purely depositional turbidity current; and (iii) mixed erosional/depositional turbidity current. Two controlling dimensionless numbers, the Froude number and the Reynolds number, appear in the non-dimensional governing equations. When densimetric Froude similarity is satisfied, the analysis shows that the results would be scale-invariant for conservative density current under the rough turbulent condition. In the case of purely depositional flows, truly scale-invariant results cannot be obtained, as the Reynolds-mediated scale effects appear in the bottom boundary conditions of the flow velocity and sediment fall velocity. However, the scale effect would be relatively modest. The Reynolds effect becomes more significant for erosional or mixed erosional/depositional turbidity currents as Reynolds-mediated scale effects also appear in the sediment entrainment relation. Numerical simulations have been conducted at three different scales by considering densimetric Froude scaling alone as well as combined densimetric Froude and Reynolds similarity. Simulation results confirm that although the scaling of densimetric Froude number alone can produce scale-invariable results for conservative density currents, variations occur in the case of turbidity currents. The results become scale invariant when densimetric Froude and Reynolds similarities are satisfied simultaneously.     

Steady open channel flows with curved streamlines: the Fawer approach revised

Rapidly varied open channel flows are characterized by curvilinear streamlines, thereby resulting in a pressure field different from the hydrostatic approach proposed in the standard gradually varied flow theory. This problem is related to environmental hydraulic problems such as the undular hydraulic jump and flow over round-crested weirs, for which streamline curvature effects are significant. The inclusion of the curvilinear streamline effect in an extended energy equation was firstly by Fawer. Most of the extended energy equations currently employed are therefore modified forms of the original Fawer approach. The aim of the present study is to highlight and remind engineers of the outstanding theory presented by Fawer. Herein, his approach for steady open channel flow with curved streamlines is revised and compared with experimental observations. Computational methods are presented in detail and based on present results, it can be observed that more recent and complex models for these problems are similar to the original proposal of Fawer, and hardly more accurate in some instances. Based on the proposed study an useful framework for theoretical models for steady open channel flows with curved streamlines is proposed.     

Evaluating the dimensionality and significance of “active periods” in turbulent environmental flows defined using Lipshitz/Hölder regularity

Active periods within perturbed boundary-layer flows are considered in terms of the local roughness of measured velocity time series and defined in terms of Hölder/Lipshitz exponents. Such events are associated with the passage of energetic, coherent flow structures and are responsible for exerting high turbulent stresses because of the rapid changes in velocity that occur at such times. A method is proposed for assessing the effective dimensionality of such active periods, as well as their significance to the flow field, for a particular choice of flow metric. The method is applied to the turbulent flow through a confluence flow geometry, with velocity samples acquired close to the bed of the channel in a zone of complex mixing. The dimensionality of the active periods is consistent with the observed patterns of sediment entrainment from the bed, with the significance of the active periods decaying away from the erosional zone.     

Turbulence and aeration in hydraulic jumps: free-surface fluctuation and integral turbulent scale measurements

In an open channel, a change from a supercritical to subcritical flow is a strong dissipative process called a hydraulic jump. Herein some new measurements of free-surface fluctuations of the impingement perimeter and integral turbulent time and length scales in the roller are presented with a focus on turbulence in hydraulic jumps with a marked roller. The observations highlighted the fluctuating nature of the impingement perimeter in terms of both longitudinal and transverse locations. The results showed further the close link between the production and detachment of large eddies in jump shear layer, and the longitudinal fluctuations of the jump toe. They highlighted the importance of the impingement perimeter as the origin of the developing shear layer and a source of vorticity. The air–water flow measurements emphasised the intense flow aeration. The turbulent velocity distributions presented a shape similar to a wall jet solution with a marked shear layer downstream of the impingement point. The integral turbulent length scale distributions exhibited a monotonic increase with increasing vertical elevation within 0.2 < L_z/d₁ < 0.8 in the shear layer, where L_z is the integral turbulent length scale and d₁ the inflow depth, while the integral turbulent time scales were about two orders of magnitude smaller than the period of impingement position longitudinal oscillations.     

Experimental investigation of bubbly flow and turbulence in hydraulic jumps

Many environmental problems are linked to multiphase flows encompassing ecological issues, chemical processes and mixing or diffusion, with applications in different engineering fields. The transition from a supercritical flow to a subcritical motion constitutes a hydraulic jump. This flow regime is characterised by strong interactions between turbulence, free surface and air–water mixing. Although a hydraulic jump contributes to some dissipation of the flow kinetic energy, it is also associated with increases of turbulent shear stresses and the development of turbulent eddies with implications in terms of scour, erosion and sediment transport. Despite a number of experimental, theoretical and numerical studies, there is a lack of knowledge concerning the physical mechanisms involved in the diffusion and air–water mixing processes within hydraulic jumps, as well as on the interaction between the free-surface and turbulence. New experimental investigations were undertaken in hydraulic jumps with Froude numbers up to Fr = 8.3. Two-phase flow measurements were performed with phase-detection conductivity probes. Basic results related to the distributions of void fraction, bubble frequency and mean bubble chord length are presented. New developments are discussed for the interfacial bubble velocities and their fluctuations, characterizing the turbulence level and integral time scales of turbulence representing a “lifetime” of the longitudinal bubbly flow structures. The analyses show good agreement with previous studies in terms of the vertical profiles of void fraction, bubble frequency and mean bubble chord length. The dimensionless distributions of interfacial velocities compared favourably with wall-jet equations. Measurements showed high turbulence levels. Turbulence time scales were found to be dependent on the distance downstream of the toe as well as on the distance to the bottom showing the importance of the lower (channel bed) and upper (free surface) boundary conditions on the turbulence structure.     

Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows. Part I: Single Round Jet

The mechanics of buoyant jet flows issuing with a general three-dimensional geometry into an unbounded ambient environment with uniform density or stable density stratification and under stagnant or steady sheared current conditions is investigated. An integral model is formulated for the conservation of mass, momentum, buoyancy and scalar quantities in the turbulent jet flow. The model employs an entrainment closure approach that distinguishes between the separate contributions of transverse shear (leading to jet, plume, or wake internal flow dynamics) and of azimuthal shear mechanisms (leading to advected momentum puff or thermal flow dynamics), respectively. Furthermore, it contains a quadratic law turbulent drag force mechanism as suggested by a number of recent detailed experimental investigations on the dynamics of transverse jets into crossflow. The model is validated in several stages: First, comparison with basic experimental data for the five asymptotic, self-similar stages of buoyant jet flows, i.e., the pure jet, the pure plume, the pure wake, the advected line puff, and the advected line thermal, support the choice and magnitude of the turbulent closure coefficients contained in the entrainment formulation. Second, comparison with many types of non-equilibrium flows support the proposed transition function within the entrainment relationship, and also the role of the drag force in the jet deflection dynamics. Third, a number of spatial limits of applicability have been proposed beyond which the integral model necessarily becomes invalid due to its parabolic formulation. These conditions, often related to the breakdown of the boundary layer nature of the flow, describe features such as terminal layer formation in stratification, upstream penetration in jets opposing a current, or transition to passive diffusion in a turbulent ambient shear flow. Based on all these comparisons, that include parameters such as trajectories, centerline velocities, concentrations and dilutions, the model appears to provide an accurate and reliable representation of buoyant jet physics under highly general flow conditions.     

A closure approach with multi-scale mixing length,and its application to wind and shear stress profiles within the atmospheric boundary layer

A method to determine flow specific first-order closure for the turbulent flux of momentum in the atmospheric boundary layer (ABL) is presented. This is based on the premise that eddy viscosity is a flow rather than a fluid property, and the physically more realistic assumption that the transfer of momentum and other scalar quantities in a turbulent flow takes place by a large, but finite number of length scales, than the often used single length scale, the ‘mixing length’. The resulting eddy viscosity is flow specific and when applied to the study of the ABL, yields the vertical profiles of shear stress and mean wind velocity in good agreement with observations. The method may be extended to other types of turbulent flows, however it should be recognized that each type of flow may yield a different eddy viscosity profile. Using the derived eddy viscosity the paper presents simple analytical solutions of the ABL equations to determine observationally consistent wind speed and shear stress profiles in the ABL for a variety of practical applications including air pollution modelling.     

A σ-coordinate transport model coupled with rotational boussinesq-type equations

This paper describes a σ-coordinate scalar transport model coupled with a Boussinesq-type hydrodynamic model. The Boussinesq model has the ability to calculate both three-dimensional velocity distributions and the water surface motion. To capture ‘dispersion’ processes in open channel flow, horizontal vorticity effects induced by a bottom shear stress are included in the Boussinesq model. Thus, a reasonable representation of vertical flow structure can be captured in shallow and wavy flow fields. To solve the coupled Boussinesq and scalar transport system, a finite-volume method, based on a Godunov-type scheme with the HLL Riemann solver, is employed. Basic advection and advection–diffusion numerical tests in a non-rectangular domain were carried out and the computed results show good agreement with analytic solutions. With quantitative comparisons of dispersion experiments in an open channel, it is verified that the proposed coupled model is appropriate for both near and far field scalar transport predictions. From numerical simulations in the surf zone, physically reasonable results showing expected vertical variation are obtained.     

A review of gravity currents formed by submerged single-port discharges in inland and coastal waters

In this paper, the authors review the current state of the science on the dynamics of gravity currents generated by positively and negatively buoyant jet discharges from submerged round outfalls (i.e., a point source) in inland and coastal waters. Specifically, this article focuses on describing gravity currents occurring at both the bottom boundary and the free surface of the receiving fluid. The manmade discharge operations generating both types of gravity currents and their significance to sustainability of the surrounding hydro-environment are first described. The authors then summarize the flow regimes characteristics of these discharges before becoming gravity currents and how those flow regimes influence the dynamics of the gravity currents. The gravity current dynamics in the calm receiving waters are then analyzed. This analysis is followed by an analysis of the influence of the hydrodynamic forces (e.g., currents, turbulence, waves) on the dynamics of gravity currents. Finally, the authors review quantitative modeling approaches for different forms of gravity current, and identify the current knowledge gaps and research needs.     

Turbulent velocity profile in fully-developed open channel flows

The determination of velocity profile in turbulent narrow open channels is a difficult task due to the significant effects of the anisotropic turbulence that involve the Prandtl’s second type of secondary flow occurring in the cross section. With these currents the maximum velocity appears below the free surface that is called dip phenomenon. The well-known logarithmic law describes the velocity distribution in the inner region of the turbulent boundary layer but it is not adapted to define the velocity profile in the outer region of narrow channels. This paper relies on an analysis of the Navier–Stokes equations and yields a new formulation of the vertical velocity profile in the center region of steady, fully developed turbulent flows in open channels. This formulation is able to predict time averaged primary velocity in the outer region of the turbulent boundary layer for both narrow and wide open channels. The proposed law is based on the knowledge of the aspect ratio and involves a parameter C_Ar depending on the position of the maximum velocity (ξ_dip). ξ_dip may be derived, either from measurements or from an empirical equation given in this paper. A wide range of longitudinal velocity profile data for narrow open channels has been used for validating the model. The agreement between the measured and the computed velocities is rather good, despite the simplification used.