Aeolian erosion of storage piles yards: contribution of the surrounding areas

Dust emissions from stockpiles surfaces are often estimated applying mathematical models such as the widely used model proposed by the USEPA. It employs specific emission factors, which are based on the fluid flow patterns over the near surface. But, some of the emitted dust particles settle downstream the pile and can usually be re-emitted which creates a secondary source. The emission from the ground surface around a pile is actually not accounted for by the USEPA model but the method, based on the wind exposure and a reconstruction from different sources defined by the same wind exposure, is relevant. This work aims to quantify the contribution of dust re-emission from the areas surrounding the piles in the total emission of an open storage yard. Three angles of incidence of the incoming wind flow are investigated ( $30^{\circ }, 60^{\circ }$ and $90^{\circ }$ ). Results of friction velocity from numerical modelling of fluid dynamics were used in the USEPA model to determine dust emission. It was found that as the wind velocity increases, the contribution of particles re-emission from the ground area around the pile in the total emission also increases. The dust emission from the pile surface is higher for piles oriented $30^{\circ }$ to the wind direction. On the other hand, considering the ground area around the pile, the $60^{\circ }$ configuration is responsible for higher emission rates (up to 67 %). The global emissions assumed a minimum value for the piles oriented perpendicular to the wind direction for all wind velocity investigated.     

Experimental and numerical study on the shear velocity distribution along one or two dunes in tandem

This paper investigates, experimentally and numerically, the shear velocity distribution along a single transverse dune and along two closely spaced dunes, analyzing the flow effects of one dune upon the other. The paper focuses on two-dimensional models simulating transverse sand dunes. The shape of the two pile geometries studied is described by sinusoidal curves, one having a maximum slope of $32^{\circ }$ and the other $27.6^{\circ }$ , with leeward flow separation. The tests were carried out for two undisturbed wind speeds and the experimental data obtained through wind-tunnel modeling encompass flow visualization and shear-velocity results. A generally good agreement is observed between the experimental measurements and computational results. From the inquiry between shear velocity distributions and published eroded contours for the same geometries, it appears the Bagnold’s approach is insufficient in the prediction of threshold conditions in wake flows formed in the dune’s leeward side.     

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.

     

Using surface drifter observations to measure tidal vortices and relative diffusion at Aransas Pass,Texas

Tidal vortices play an important role in the flushing of coastal regions. At the mouth of a tidal inlet, the input of circulation by the ebb tide may force the formation of a starting-jet dipole vortex. The continuous ebb jet current also creates a periodic sequence of secondary vortices shed from the inlet mouth. In each case, these tidal vortices have a shallow aspect ratio, with a lateral extent much greater than the water depth. These shallow vortices affect the transport of passive tracers, such as nutrients and sediment from the estuary to the ocean and vice versa. Field observation of tidal vortices primarily relies on ensemble averaging over several vortex events that are repeatable in space and can be sampled by a fixed Eulerian measurement grid. This paper presents an adaptive approach for locating and measuring within tidal vortices that propagate offshore near inlets and advect along variable trajectories set by the wind-driven currents. A field experiment was conducted at Aransas Pass, Texas to measure these large-scale vortices. Locations of the vortices produced during ebb tide were determined using near real-time updates from surface drifters deployed near or within the inlet during ebb tide, and the paths of towed acoustic Doppler current profiler (ADCP) transects were selected by analysis of the drifter observations. This method allowed ADCP transects to be collected within ebb generated tidal vortices, and the paths of the drifters indicated the presence of both the starting-jet dipole and the secondary vortices of the unstable ebb tidal jet. Drifter trajectories were also used to estimate the size of each observed vortex as well as the statistics of relative diffusion offshore of Aransas Pass. The field data confirmed the starting-jet spin-up time (time until the vortex dipole begins to propagate offshore) measured in the laboratory by Bryant et al. [6] and that the Strouhal condition of \(St=0.2\) predicts the shedding of secondary vortices from the inlet mouth. The size of the rotational core of the vortex is also shown to be approximated physically by the inlet width or by \(0.02UT\) , where U is the maximum velocity through the inlet channel and T is the tidal period, and confirms results found in laboratory experiments by Nicolau del Roure et al. [23]. Additionally, the scale of diffusion was approximately 1–15 km and the apparent diffusivity was between 2–130  \(m^2/s\) following Richardsons law.     

Turbulent flow characteristics and drag over 2-D forward-facing dune shaped structures with two different stoss-side slopes

The present paper explores the characteristics of turbulent flow and drag over two artificial 2-D forward-facing waveform structures with two different stoss side slopes of $50^{\circ }$ and $90^{\circ },$ respectively. Both structures possessed a common slanted lee side slope of $6^{\circ }.$ Flume experiments were conducted at the Fluvial Mechanics Laboratory of Indian Statistical Institute, Kolkata. The velocity data were analyzed to identify the spatial changes in turbulent flow addressing the flow separation region with recirculating eddy, the Reynolds stresses, the turbulent events associated with burst-sweep cycles and the drag over two upstream-facing bedforms for Reynolds number $Re_h=1.44\times 10^5.$ The divergence at the stoss side slope between the two structures revealed significant changes in the mean flow and turbulence. Comparison showed that during the flood-tide condition there was no flow separation region on the gentle lee side of the structure with smaller slope at the stoss side, while for the other structure with vertical stoss side slope a thick flow separation region with recirculating eddy was observed at the gentle lee side just downstream of the crest. The recirculating eddy induced on the lee-side had a strong influence on the resistance that the structure exerts to the flow due to loss of energy through turbulence. In contrast, a great amount of reduction in drag was observed in the case of smaller stoss side sloped structure as there was no flow separation. The quadrant analysis was also used to highlight the turbulent event evolution along the bed form structures under flood-tide conditions.     

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 \(Re\) ).     

Initial mixing of inclined dense jet in perpendicular crossflow

A comprehensive experimental investigation for an inclined ( $60^{\circ }$ to vertical) dense jet in perpendicular crossflow—with a three-dimensional trajectory—is reported. The detailed tracer concentration field in the vertical cross-section of the bent-over jet is measured by the laser-induced fluorescence technique for a wide range of jet densimetric Froude number $Fr$ and ambient to jet velocity ratios $U_r$ . The jet trajectory and dilution determined from a large number of cross-sectional scalar fields are interpreted by the Lagrangian model over the entire range of jet-dominated to crossflow-dominated regimes. The mixing during the ascent phase of the dense jet resembles that of an advected jet or line puff and changes to a negatively buoyant thermal on descent. It is found that the mixing behavior is governed by a crossflow Froude number $\mathbf{F} = U_r Fr$ . For $\mathbf{F} < 0.8$ , the mixing is jet-dominated and governed by shear entrainment; significant detrainment occurs and the maximum height of rise $Z_{max}$ is under-predicted as in the case of a dense jet in stagnant fluid. While the jet trajectory in the horizontal momentum plane is well-predicted, the measurements indicate a greater rise and slower descent. For $\mathbf{F} \ge 0.8$ the dense jet becomes significantly bent-over during its ascent phase; the jet mixing is dominated by vortex entrainment. For $\mathbf{F} \ge 2$ , the detrainment ceases to have any effect on the jet behavior. The jet trajectory in both the horizontal momentum and buoyancy planes are well predicted by the model. Despite the under-prediction of terminal rise, the jet dilution at a large number of cross-sections covering the ascent and descent of the dense jet are well-predicted. Both the terminal rise and the initial dilution for the inclined jet in perpendicular crossflow are smaller than those of a corresponding vertical jet. Both the maximum terminal rise $Z_{max}$ and horizontal lateral penetration $Y_{max}$ follow a $\mathbf{F}^{-1/2}$ dependence in the crossflow-dominated regime. The initial dilution at terminal rise follows a $S \sim \mathbf{F}^{1/3}$ dependence.     

A flexible spatio-temporal model for air pollution with spatial and spatio-temporal covariates

The development of models that provide accurate spatio-temporal predictions of ambient air pollution at small spatial scales is of great importance for the assessment of potential health effects of air pollution. Here we present a spatio-temporal framework that predicts ambient air pollution by combining data from several different monitoring networks and deterministic air pollution model(s) with geographic information system covariates. The model presented in this paper has been implemented in an R package, SpatioTemporal, available on CRAN. The model is used by the EPA funded Multi-Ethnic Study of Atherosclerosis and Air Pollution (MESA Air) to produce estimates of ambient air pollution; MESA Air uses the estimates to investigate the relationship between chronic exposure to air pollution and cardiovascular disease. In this paper we use the model to predict long-term average concentrations of \(\text {NO}_{x}\) in the Los Angeles area during a 10 year period. Predictions are based on measurements from the EPA Air Quality System, MESA Air specific monitoring, and output from a source dispersion model for traffic related air pollution (Caline3QHCR). Accuracy in predicting long-term average concentrations is evaluated using an elaborate cross-validation setup that accounts for a sparse spatio-temporal sampling pattern in the data, and adjusts for temporal effects. The predictive ability of the model is good with cross-validated \(R^2\) of approximately \(0.7\) at subject sites. Replacing four geographic covariate indicators of traffic density with the Caline3QHCR dispersion model output resulted in very similar prediction accuracy from a more parsimonious and more interpretable model. Adding traffic-related geographic covariates to the model that included Caline3QHCR did not further improve the prediction accuracy.     

Quantifying a significance of sediment particle size to hyporheic sedimentary oxygen demand with a permeable stream bed

A mechanistic model of sedimentary oxygen demand (SOD) for hyporheic flow is presented. The permeable sediment bed, e.g. sand or fine gravel, is considered with hydraulic conductivity in the range $0.1 < K < 20$  cm/s. Hyporheic pore water flow is induced by pressure fluctuations at the sediment/water interface due to near-bed turbulent coherent motions. A 2-D advection–diffusion equation is linked to the pore water flow model to simulate the effect of advection–dispersion driven by interstitial flow on oxygen transfer through the permeable sediment. Microbial oxygen uptake in the sediment is expressed as a function of the microbial growth rate, and is related to the sediment properties, i.e. the grain diameter $(d_{s})$ and porosity $(\phi )$ . The model describes the significance of sediment particle size to oxygen transfer through the sediment and microbial oxygen uptake: With increasing grain diameter $(d_{s})$ , the hydraulic conductivity $(K)$ increases so does the oxygen transfer rate, while particle surface area per volume (the available surface area for colonization by biofilms) decreases reducing the microbial oxygen uptake rate. Simulation results show that SOD increases as the hydraulic conductivity $(K)$ increases before a threshold has been reached. After that, SOD diminishes with the increment of the hydraulic conductivity $(K)$ .     

Flow and drag force around a free surface piercing cylinder for environmental applications

This paper investigates flows around a free surface piercing cylinder with Froude number F > 0.5 and Reynolds number around Re = 50,000. The aim of this work is to gain a better understanding of the flow behaviour in environmental systems such as fishways. The advances are based upon experimental and numerical results. Several flow discharges and slopes are tested to obtain both subcritical and supercritical flows. The drag force exerted on the cylinder is measured with the help of a torque gauge while the velocity field is obtained using particle velocimetry. For the numerical part, two URANS turbulence models are tested, the k-\(\omega\) SST and the RNG k-\(\varepsilon\) models using the OpenFOAM software suite for subcritical cases, and then compared with the corresponding experimental results. With fishways applications in mind, the changes in drag coefficient \(C_d\) versus Froude number and water depth are studied and experimental correlations proposed. We conclude that the most suitable URANS turbulence model for reproducing this kind of flow is the k-\(\omega\) SST model.     

Data depth for the uniform distribution

Given a set $X$ of $k$ points and a point $z$ in the $n$ -dimensional euclidean space, the Tukey depth of $z$ with respect to $X$ , is defined as $m/k$ , where $m$ is the minimum integer such that $z$ is not in the convex hull of some set of $k-m$ points of $X$ . If $z$ belongs to the closed region $B$ delimited by an ellipsoid, define the continuous depth of $z$ with respect to $B$ as the quotient $V(z)/\text{ Vol }(B)$ , where $V(z)$ is the minimum volume of the intersection of $B$ with the halfspaces defined by any hyperplane passing through $z$ , and $\text{ Vol }(B)$ is the volume of $B$ . We consider $z$ a random variable and prove that, if $z$ is uniformly distributed in $B$ , the continuous depth of $z$ with respect to $B$ has expected value $1/2^{n+1}$ . This result implies that if $z$ and $X$ are uniformly distributed in $B$ , the expected value of Tukey depth of $z$ with respect to $X$ converges to $1/2^{n+1}$ as the number of points $k$ goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.     

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.     

Flow aeration, cavity processes and energy dissipation on flat and pooled stepped spillways for embankments

The design floods of several reservoirs were recently re-evaluated and the revised spillway outflow could result in dam overtopping with catastrophic consequences for some embankment structures. Herein a physical study was performed on flat and pooled stepped spillways with a slope typical of embankments $(\uptheta = 26.6^{\circ })$ and four stepped configurations were tested: a stepped spillway with flat horizontal steps, a pooled stepped spillway, and two stepped spillways with in-line and staggered configurations of flat and pooled steps. The focus of the study was on the flow aeration, air–water flow properties, cavity flow processes, and energy dissipation performances. The results demonstrated the strong aeration of the flow for all configurations. On the in-line and staggered configurations of flat and pooled steps, the flow was highly three-dimensional. The residual head and energy dissipation rates at the stepped chute downstream end were calculated based upon the detailed air–water flow properties. The results showed that the residual energy was the lowest for the flat stepped weir. The data for the stepped spillway configuration with in-line and staggered configurations of flat and pooled steps showed large differences in terms of residual head in the transverse direction. Altogether the present results showed that, on a $26.6^{\circ }$ slope stepped chute, the designs with in-line and staggered configurations of flat and pooled steps did not provide any advantageous performances in terms of energy dissipation and flow aeration, but they were affected by three-dimensional patterns leading to some flow concentration.     

On the effective hydraulic conductivity and macrodispersivity for density-dependent groundwater flow

In this paper, semi-analytical expressions of the effective hydraulic conductivity ( $K^{E})$ and macrodispersivity ( $\alpha ^{E})$ for 3D steady-state density-dependent groundwater flow are derived using a stationary spectral method. Based on the derived expressions, we present the dependence of $K^{E}$ and $\alpha ^{E}$ on the density of fluid under different dispersivity and spatial correlation scale of hydraulic conductivity. The results show that the horizontal $K^{E}$ and $\alpha ^{E}$ are not affected by density-induced flow. However, due to gravitational instability of the fluid induced by density contrasts, both vertical $K^{E}$ and $\alpha ^{E}$ are found to be reduced slightly when the density factor ( $\gamma $ ) is less than 0.01, whereas significant decreases occur when $\gamma $ exceeds 0.01. Of note, the variation of $K^{E}$ and $\alpha ^{E}$ is more significant when local dispersivity is small and the correlation scale of hydraulic conductivity is large.     

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.     

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.     

Passive scalar roughness lengths for atmospheric boundary layer flow over complex, fractal topographies

When modeling atmospheric boundary layer flow over rough landscapes, surface fluxes of flow quantities (momentum, temperature, etc.) can be described with equilibrium logarithmic law expressions, all of which require specification of a roughness length that is, physically, the elevation at which the flow quantity equals its surface value. In high Reynolds number flows, such as the atmospheric boundary layer, inertial forces associated with turbulent eddy motions are responsible for surface momentum fluxes (form, or pressure drag). Surface scalar fluxes, on the other hand, occur exclusively via diffusion in the immediate vicinity of the topography—the interfacial region—before being advected by turbulent eddy motions into the bulk of the flow. Owing to this difference in surface transfer mechanism, the passive scalar roughness length, $z_{0S}$ , is known to be less than the momentum roughness length, $z_0$ . In this work, classical relations are used to specify $z_{0S}$ during large-eddy simulation of atmospheric boundary layer flow over aerodynamically rough, synthetic, fractal topographies which exhibit power-law height energy spectrum, $E_h (k) \sim k^{\beta _s}$ , where $\beta _s$ is a (predefined) spectral exponent. These topographies are convenient since they resemble natural landscapes and $\beta _s$ can be varied to change the topography’s aerodynamic roughness (the study considers a suite of topographies with $-2.4 \le \beta _s \le -1.2$ , where $-2.4$ and $-1.2$ are the “most smooth” and “most rough” cases, respectively, corresponding with roughness Reynolds number, $Re_0 \approx 10$ and $300$ ). It is often assumed that $z_{0S}/z_{0} \approx 10^{-1}$ for all $Re_0$ . But results from this work show that the roughness length ratio, $z_{0S}/z_{0}$ , depends strongly on $Re_0$ , ranging between $10^{-3}$ and $10^{-1}$ .     

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.     

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.