Experiments on gravity currents propagating on unbounded uniform slopes

Gravity currents propagating on \(12^\circ \), \(9^\circ \), \(6^\circ \), \(3^\circ \) unbounded uniform slopes and on an unbounded horizontal boundary are reported. Results show that there are two stages of the deceleration phase. In the early stage of the deceleration phase, the front location history follows \({(x_f+x_0)}^2 = {(K_I B)}^{1/2} (t+t_{I})\), where \((x_f+x_0)\) is the front location measured from the virtual origin, \(K_I\) an experimental constant, B the total buoyancy, t time and \(t_I\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, the front location history follows \({(x_f+x_0)}^{8/3} = K_{VS} {{B}^{2/3} V^{2/9}_0 }{\nu }^{-1/3} ({t+t_{VS}})\), where \(K_{VS}\) is an experimental constant, \(V_0\) the initial volume of heavy fluid, \(\nu \) the kinematic viscosity and \(t_{VS}\) the t-intercept. In the late stage of the deceleration phase for the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the front location history follows \({(x_f+x_0)}^{4} = K_{VM} {{B}^{2/3} V^{2/3}_0 }{\nu }^{-1/3} ({t+t_{VM}})\), where \(K_{VM}\) is an experimental constant and \(t_{VM}\) the t-intercept. Two qualitatively different flow morphologies are identified in the late stage of the deceleration phase. For the gravity currents on \(12^\circ \), \(9^\circ \), \(6^\circ \) unbounded uniform slopes, an ‘active’ head separates from the body of the current. For the gravity currents on a \(3^\circ \) unbounded uniform slope and on an unbounded horizontal boundary, the gravity currents maintain an integrated shape throughout the motion. Results indicate two possible routes to the final stage of the gravity currents on unbounded uniform slopes.

     

Penetrative convection in slender containers

An experimental study was conducted to investigate the penetration of a convective mixed layer into an overlying stably (solutally) stratified layer contained in a narrow, tall vessel when the fluid is subjected to a destabilizing heat flux from below. The interest was the evolution of the bottom mixed-layer height (\(h\)) with time (\(t\)) in the presence of side-wall effects, but without the formation of conventional double-diffusive layers. The side-wall effects are expected at small mixed-layer aspect ratios, \(\varGamma_{h} = (W/h)\), where \(W\) is the container width. This case has not been studied hitherto, although there are important environmental and industrial applications. The mixed-layer growth laws for low aspect ratio convection were formulated by assuming a balance between the vertical kinetic energy flux at the interface and the rate of change of potential energy of the fluid system due to turbulent entrainment. The effects of sidewalls were considered using similarity arguments, by taking characteristic rms velocities to be a function of \(\varGamma_{h}\), in addition to buoyancy flux (\(q_{0}\)) and \(h\). In all stages of evolution, the similarity variables \(\xi = h/W\) and \(t^{\prime } = Nt/A\), where \(A = N^{3} W^{2} /4q_{0}\) and \(N\) is the buoyancy frequency, scaled the mixed-layer evolution data remarkably well. Significant wall effects were noted when \(\varGamma_{h} < 1\), and for this case the interfacial vertical turbulent velocity and length scales were identified via scaling arguments and experimental data.     

Warm discharges in cold fresh water: 2. Numerical simulation of laminar line plumes

The behaviour of a discharge of warm water upwards into a homogeneous body of cold fresh water was investigated by means of a numerical model. The discharge has a parabolic velocity profile, with Reynolds number \(Re=50\), Prandtl number \(Pr=7\) and Froude number varied over the range \(0.2 \le {\rm Fr} \le 2.5\). Water density is taken to be a quadratic function of temperature, so that an initially positively buoyant discharge will experience buoyancy reversal as it mixes with an ambient below the temperature of maximum density. The resulting plume has some similarities to a fountain resulting from injection of negatively buoyant fluid upward into a less dense ambient. The plume is initially symmetric, but then its head detaches as it approaches its maximum height. The detached head is denser than the fluid in the plume below it, and the interaction between the sinking head and the rising plume causes a sideways deflection; as this cycle is repeated, the plume displays side-to-side flapping motion and vertical bobbing. As Froude number is increased (i.e. buoyancy reduced) the growth of the plume becomes slower, but the plume eventually reaches a greater height. We obtain empirical power-law scalings for maximum height and time taken to reach that height as functions of Froude number; these scalings are simlar to those for fountains with a linear dependence of density on temperature in the very weak regime.     

Mixing at a sediment concentration interface in turbulent open channel flow

In this work we address the role of turbulence on mixing of clear layer of fluid with sediment-laden layer of fluid at a sediment concentration interface. This process can be conceived as the entrainment of sediment-free fluid into the sediment-laden layer, or alternatively, as the transport of sediment into the top sediment-free flow. This process is governed by four parameters—Reynolds number of the flow \(Re_\tau\), non-dimensional settling velocity of the sediment (proxy for sediment size) \(\tilde{V}\), Richardson number \(Ri_\tau\) and Schmidt number Sc. For this work we have performed direct numerical simulations for fixed Reynolds and Schmidt numbers while varying the values of Richardson number and particle settling velocity. In the simple model considered here, the flow’s momentum and turbulence pre-exists over the entire layer of fluid, while the sediment is initially confined to a layer close to the bed. Mixing of sediment-free fluid with the sediment-laden layer is associated primarily with upward transport of sediment and buoyancy. There is no simultaneous upward transport of fluid momentum and turbulence into the sediment-free fluid layer, which is already in motion and turbulent. The analysis performed shows that the ability of the flow to transport a given sediment size decreases with the distance from the bottom, and thus only fine enough sediment particles are transported across the sediment concentration interface. For these cases, the concentration profiles evolve to a final steady state in good agreement with the well-known Rouse profile. The approach towards the Rouse profile happens through a transient self-similar state. This behavior of the flow is not seen for larger particles. Detailed analysis of the three dimensional structure of the sediment concentration interface shows the mechanisms by which sediment particles are lifted up by tongues of sediment-laden fluid with positive correlation between vertical velocity and sediment concentration. Finally, the mixing ability of the flow is addressed by monitoring the time evolution of the center of mass of the sediment-laden layer and the vertical location of the sediment-free/sediment-laden interface.     

Integral models for bubble,droplet, and multiphase plume dynamics in stratification and crossflow

We present the development and validation of a numerical modeling suite for bubble and droplet dynamics of multiphase plumes in the environment. This modeling suite includes real-fluid equations of state, Lagrangian particle tracking, and two different integral plume models: an Eulerian model for a double-plume integral model in quiescent stratification and a Lagrangian integral model for multiphase plumes in stratified crossflows. Here, we report a particle tracking algorithm for dispersed-phase particles within the Lagrangian integral plume model and a comprehensive validation of the Lagrangian plume model for single- and multiphase buoyant jets. The model utilizes literature values for all entrainment and spreading coefficients and has one remaining calibration parameter \( \kappa \), which reduces the buoyant force of dispersed phase particles as they approach the edge of a Lagrangian plume element, eventually separating from the plume as it bends over in a crossflow. We report the calibrated form \( \kappa = [(b - r) / b]^4 \), where b is the plume half-width, and r is the distance of a particle from the plume centerline. We apply the validated modeling suite to simulate two test cases of a subsea oil well blowout in a stratification-dominated crossflow. These tests confirm that errors from overlapping plume elements in the Lagrangian integral model during intrusion formation for a weak crossflow are negligible for predicting intrusion depth and the fate of oil droplets in the plume. The Lagrangian integral model has the added advantages of being able to account for entrainment from an arbitrary crossflow, predict the intrusion of small gas bubbles and oil droplets when appropriate, and track the pathways of individual bubbles and droplets after they separate from the main plume or intrusion layer.     

On the front conditions for gravity currents in channels of general cross-section

We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel with general cross-section whose width is \(f(z), 0 \le z\le H\), and the gravity acceleration g acts in \(-z\) direction. (The classical rectangular cross-section is covered by the particular case \(f(z) =\) const.) We assume a two-layer system of homogeneous fluids of constant densities \(\rho _{c}\) (current, of height \(h < H \)) and smaller \(\rho _{a}\) (ambient, filling the remaining part of the channel). We focus attention on the calculation and assessment of the nose Froude-number condition \(Fr = U/(g' h)^{1/2}\); here U is the speed of propagation of the current and \(g' = (\rho _{c}/\rho _{a}-1) g\) is the reduced gravity. We first revisit the steady-state current, and derive compact insightful expressions of Fr and energy dissipation as a function of \(\varphi \) (\(=\) area fraction occupied by the current in the cross-section). We show that the head loss \(\delta _0\) on the stagnation line is formally a degree of freedom in the determination of \(Fr(\varphi )\), and we clarify the strong connections with the head loss \(\delta \) in the ambient fluid, and with the overall rate of dissipation \(\dot{{\mathcal{D}}}\). We demonstrate that the closure \(\delta _0 = 0\) [suggested by Benjamin (J Fluid Mech 31, 209–248, 1968) for the rectangular cross-section] produces in general the smallest Fr for a given \(\varphi \); the results are valid for a significant range \([0, \varphi _{\max }]\), in which the current is dissipative, except for the point \(\varphi _{\max }\) where \(\delta = \dot{{\mathcal{D}}} = 0\). We show that imposing the closure \(\delta = \dot{{\mathcal{D}}} = 0\), which corresponds to an energy-conserving or non-dissipative current, produces in general unacceptable restrictions of the range of validity, and large values of Fr; in particular, deep currents (\(\varphi < 0.3\) say) must be excluded because they are inherently dissipative. On the other hand, the compromise closure \(\delta (\varphi ) =\delta _0(\varphi )\) produces the simple \(Fr(\varphi ) = \sqrt{2}(1 - \varphi )\) formula whose values and dissipation properties are very close, and the range of validity is identical, to these obtained with Benjamin’s closure (moreover, we show that this corresponds to circulation-conservation solutions). The results are illustrated for practical cross-section geometries (rectangle, \(\Delta \) and \(\nabla \) triangle, circle, and the general power-law \(f(z) = b z ^\alpha \) (\(b>0, \alpha \ge 0, 0< z \le H\)). Next, we investigate the connection of the steady-state results with the time-dependent current, and show that in a lock-released current the rate of dissipation of the system is equal to, or larger than, that obtained for Fr corresponding to the conditions at the nose of the current. The results and insights of this study cover a wide range of cross-section geometry and apply to both Boussinesq and non-Boussinesq systems; they reveal a remarkable robustness of Fr as a function of \(\varphi \).     

On the periodicity of atmospheric von Kármán vortex streets

For over 100 years, laboratory-scale von Kármán vortex streets (VKVSs) have been one of the most studied phenomena within the field of fluid dynamics. During this period, countless publications have highlighted a number of interesting underpinnings of VKVSs; nevertheless, a universal equation for the vortex shedding frequency ( \(N\) ) has yet to be identified. In this study, we have investigated \(N\) for mesoscale atmospheric VKVSs and some of its dependencies through the use of realistic numerical simulations. We find that vortex shedding frequency associated with mountainous islands, generally demonstrates an inverse relationship to cross-stream obstacle length ( \(L\) ) at the thermal inversion height of the atmospheric boundary layer. As a secondary motive, we attempt to quantify the relationship between \(N\) and \(L\) for atmospheric VKVSs in the context of the popular Strouhal number ( \(Sr\) )–Reynolds number ( \(Re\) ) similarity theory developed through laboratory experimentation. By employing numerical simulation to document the \(Sr{-}Re\) relationship of mesoscale atmospheric VKVSs (i.e., in the extremely high \(Re\) regime) we present insight into an extended regime of the similarity theory which has been neglected in the past. In essence, we observe mesoscale VKVSs demonstrating a consistent \(Sr\) range of 0.15–0.22 while varying \(L\) (i.e, effectively varying \(Re\) ).     

Effect of atmospheric boundary layer stability on the inclination angle of turbulence coherent structures

Coherent structures in the atmospheric boundary layer are fundamental to the transport of momentum and heat as well as to the production of turbulence. The present work attempts to investigate the behavior of the inclination angle of the vortex packet structures (\(\gamma\)) under different stability conditions. The data were collected from the Marine Ecosystem Research Centre (EKOMAR) site at the east coast of Peninsular Malaysia. The main measurements were conducted by placing two hotwires 3 and 12 m above ground. The two-point correlation method was used to calculate the vortex packet structure inclination angle, while the one-point correlation method was employed to calculate its length-scale. The inclination angle was found to increase under both stable and unstable conditions. As the Obukhov stability parameter (\(\zeta\)) approaches 0, the inclination angle ranged between \(\gamma = 15^\circ\) to \(\gamma = 18^\circ\) for the stable and unstable conditions, respectively, which agrees with the findings of previous research. The vertical gradient of velocity is the dominant parameter affecting the inclination angle under different stability conditions.     

Thin-layer models for gravity currents in channels of general cross-section area,a review

We present a brief review of the recent investigations on gravity currents in horizontal channels with non-rectangular cross-section area (such as triangle, \(\bigvee \)-valley, circle/semi-circle, trapezoid) which occur in nature (e.g., rivers) and constructed environment (tunnels, reservoirs, canals). To be specific, we discuss the propagation of a gravity current (GC) in a horizontal channel along the horizontal coordinate x, with gravity g acting in the \(-z\) direction, and y the horizontal–lateral coordinate. The bottom and top of the channel are at \(z=0,H\). The “standard” problem is concerned with 2D flow in a channel with rectangular (or laterally unbounded) cross-section area (CSA). Recent investigations have successfully extended the standard knowledge to the channels of CSA given by the quite general \(-f_1(z)\le y \le f_2(z)\) for \(0 \le z \le H\). This includes the practical \(\bigvee \)-valley, triangle, circle/semi-circle and trapezoid; these geometries may be in “up” or “down” setting with respect to gravity, e.g., \(\bigtriangleup \) and \(\bigtriangledown \). The major objective of the extended theory is to predict the height of the interface \(z=h(x,t)\) and the velocity (averaged over the CSA) u(x, t), where t is time; the prediction includes the speed and position of the nose \(u_N(t), x_N(t)\). We show that the motion is governed by a set of simplified equations, called “model,” that provides versatile and insightful solutions and trends. The emphasis in on a high-Reynolds-number current whose motion is dominated by buoyancy–inertia balance; in particular a GC released from a lock, which also contains general effects such as front and internal jumps (shocks), and reflected bore. We discuss two-layer, one-layer, and box models; Boussinesq and non-Boussinesq systems; compositional and particle-driven cases; and the effect of stratification of the ambient fluid. The models are self-contained, and admit realistic initial and boundary conditions. The governing equations are amenable to analytical solutions in some special circumstances. Some salient features of the buoyancy-viscous regime, and the estimate for the length at which transition to this regime takes place, are also presented. Some experimental support to the theory, and open questions for further investigations, are also mentioned. The major conclusions are (1) The CSA geometry has significant influence on the motion of the GC; and (2) The new theory is a useful, very significant, extension of the standard two-dimensional GC problem. The standard current is just a particular case, \(f_{1,2} =\) constants, among many other covered by the new theory.     

Statistical properties of concentration fluctuations in two merging plumes

The statistics of the fluctuating concentration field within a plume is important in the analysis of atmospheric dispersion of toxic, inflammable and odorous gases. Previous work has tended to focus on concentration fluctuations in single plumes released in the surface layer or at ground level and there is a general lack of information about the mixing of two adjacent plumes and how the statistical properties of the concentration fluctuations are modified in these circumstances. In this work, data from wind tunnel experiments are used to analyse the variance, skewness, kurtosis, intermittency, probability density function and power spectrum of the concentration field during the mixing of two identical plumes and results are compared with those obtained for an equivalent single plume. The normalised variance, skewness and kurtosis on the centre-lines of the combined plume increase with distance downwind of the stack and, in the two-source configuration, takes lower values than those found in the single plumes. The results reflect the merging process at short range, which is least protracted for cases in which the sources are in-line or up to 30 \(^{\circ }\) off-line. At angles of 45 \(^{\circ }\) and more, the plumes are effectively side-by-side during the merging process and the interaction between the vortex pairs in each plume is strong. Vertical asymmetry is observed between the upper and the lower parts of the plumes, with the upper part having greater intermittency (i.e. the probability that no plume material is present) and a more pronounced tail to the concentration probability distribution. This asymmetry tends to diminish at greater distances from the source but occurs in both buoyant and neutral plumes and is believed to be associated with the ‘bending-over’ of the emission in the cross-flow and the vortex pair that this generates. The results allowed us to identify three phases in plume development. The first, very near the stack, is dominated by turbulence generated within the plume and characterised by concentration spectra with distinct peaks corresponding to scales comparable with those of the counter-rotating vortex pair. A second phase follows at somewhat greater distances downwind, in which there are significant contributions to the concentration fluctuations from both the turbulence internal to the plume and the external turbulence. The third phase is one in which the concentration fluctuations appear to be controlled by the external turbulence present in the ambient flow.     

Evolution and decay of gravity wavefields in weak-rotating environments: a laboratory study

Gravity waves are prominent physical features that play a fundamental role in transport processes of stratified aquatic ecosystems. In a two-layer stratified basin, the equations of motion for the first vertical mode are equivalent to the linearised shallow water equations for a homogeneous fluid. We adopted this framework to examine the spatiotemporal structure of gravity wavefields weakly affected by the background rotation of a single-layer system of equivalent thickness \(h_{\ell }\), via laboratory experiments performed in a cylindrical basin mounted on a turntable. The wavefield was generated by the release of a diametral linear tilt of the air–water interface, \(\eta _{\ell }\), inducing a basin-scale perturbation that evolved in response to the horizontal pressure gradient and the rotation-induced acceleration. The basin-scale wave response was controlled by an initial perturbation parameter, \({\mathcal{A}}_{*} = \eta _{0}/h_{\ell }\), where \(\eta _{0}\) was the initial displacement of the air–water interface, and by the strength of the background rotation controlled by the Burger number, \({\mathcal{S}}\). We set the experiments to explore a transitional regime from moderate- to weak-rotational environments, \(0.65\le {\mathcal{S}} \le 2\), for a wide range of initial perturbations, \(0.05\le {\mathcal{A}}_{*}\le 1.0\). The evolution of \(\eta _{\ell }\) was registered over a diametral plane by recording a laser-induced optical fluorescence sheet and using a capacitive sensor located near the lateral boundary. The evolution of the gravity wavefields showed substantial variability as a function of the rotational regimes and the radial position. The results demonstrate that the strength of rotation and nonlinearities control the bulk decay rate of the basin-scale gravity waves. The ratio between the experimentally estimated damping timescale, \(T_{d}\), and the seiche period of the basin, \(T_{g}\), has a median value of \(T_{d}/T_{g}\approx 11\), a maximum value of \(T_{d}/T_{g}\approx 10^{3}\) and a minimum value of \(T_{d}/T_{g}\approx 5\). The results of this study are significant for the understanding the dynamics of gravity waves in waterbodies weakly affected by Coriolis acceleration, such as mid- to small-size lakes.     

Physical and biological controls of algal blooms in the Río de la Plata

Coupled three-dimensional hydrodynamic and ecological numerical simulations were used to investigate the role of transport, stagnation zones and dispersion on inter-annual blooms of the diatom Aulacoseira sp. in the vicinity of the drinking water intakes of the Buenos Aires city (Argentina) in the upper Río de la Plata. Three different summer events were analyzed. First, a mild biomass bloom year (2006–2007), second, a high biomass bloom year (2007–2008) and third, a “normal” no bloom year (2009–2010). Simulated water height, water temperature, suspended solids and chlorophyll \(a\) concentrations patterns compared well with field data. Results revealed that the advection of phytoplankton cells via inflows to the Río de la Plata triggered Aulacoseira sp. blooms in the domain. In addition, excessive growth observed near the drinking water intakes, along the Argentinean margin, were associated with long retention times (stagnant region) and weak horizontal dispersion. Increased concentrations of suspended solids in the water column, in response to re-suspension events, did not prevent the blooms, however, were found to also play a key role in controlling the rate of phytoplankton growth. Finally, a non-dimensional parameter, R, that considers phytoplankton patch size, e-folding growth and dispersion time scales is shown to determine the potential bloom occurrences, as well as bloom intensity; R values higher than 5.7 suggest intense phytoplankton growth. For the mild biomass bloom year, \(R = 7.5\) , for the high biomass bloom year, \(R = 11\) and for the “normal” no bloom year \(R= 0.4\) .     

Partitioning of \alpha and \beta diversity using hierarchical Bayesian modeling of species distribution and abundance

Diversity partitioning is becoming widely used to decompose the total number of species recorded in an area or region \((\gamma )\) into the average number of species within samples \((\alpha )\) and the average difference in species composition \((\beta )\) among samples. Single-value metrics of \(\alpha \) and \(\beta \) diversity are popular because they may be applied at multiple scales and because of their ease in computation and interpretation. Studies thus far, however, have emphasized observed diversity components or comparisons to randomized, null distributions. In addition, prediction of \(\alpha \) and \(\beta \) components using environmental or spatial variables has been limited to more extensive data sets because multiple samples are required to estimate single \(\alpha \) and \(\beta \) components. Lastly, observed diversity components do not incorporate variation in detection probabilities among species or samples. In this study, we used hierarchical Bayesian models of species abundances to provide predictions of \(\alpha \) and \(\beta \) components in species richness and composition using environmental and spatial variables. We illustrate our approach using butterfly data collected from 26 grassland remnants to predict spatially nested patterns of \(\alpha \) and \(\beta \) based on the predicted counts of butterflies. Diversity partitioning using a Bayesian hierarchical model incorporated variation in detection probabilities by butterfly species and habitat patches, and provided prediction intervals for \(\alpha \) and \(\beta \) components using environmental and spatial variables.     

Numerical simulations of intrusive gravity currents interacting with a bottom-mounted obstacle in a continuously stratified ambient

In this study, the flow dynamics of intrusive gravity currents past a bottom-mounted obstacle were investigated using highly resolved numerical simulations. The propagation dynamics of a classic intrusive gravity current was first simulated in order to validate the numerical model with previous laboratory experiments. A bottom-mounted obstacle with a varying non-dimensional height of \(\tilde{D}=D/H\), where D is the obstacle height and H is the total flow depth, was then added to the problem in order to study the downstream flow pattern of the intrusive gravity current. For short obstacles, the intrusion re-established itself downstream without much distortion. However, for tall obstacles, the downstream flow was found to be a joint effect of horizontal advection, overshoot-springback phenomenon, and associated Kelvin-Helmholtz instabilities. Analysis of the numerical results show that the relationship between the downstream propagation speed and the obstacle height can be subdivided into three regimes: (1) a retarding regime (\(\tilde{D}\) \(\approx \) 0–0.3) where a 30 % increase in obstacle height leads to a 20 % reduction in propagation speed, simply due to the obstacle’s retarding effect; (2) an impounding regime (\(\tilde{D}\) \(\approx \) 0.3–0.6) where the additional 30 % increase in obstacle height only leads to a further (negligible) 5 % reduction in propagation speed, due to the accelerating effect of upstream impoundment and downstream enhanced mixing; and (3) a choking regime (\(\tilde{D}\) \(\approx \) 0.6–1.0) where the propagation speed is dramatically reduced due to the dominance of the obstacle’s blocking effect. The obstacle thickness was found to be irrelevant in determining the downstream propagation speed at least for the parameter range explored in this study. The present work highlights the significance of topographic effects in stratified flows with horizontal pressure forcing.