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 Environmental Fluid Mechanics    [4 followers]  Follow        Hybrid journal (It can contain Open Access articles)      ISSN (Print) 1573-1510 - ISSN (Online) 1567-7419      Published by Springer-Verlag  [2210 journals]   [SJR: 0.732]   [H-I: 23]
• Preface
• PubDate: 2014-10-01

• LE of shallow mixing interfaces: A review
• Abstract: Abstract Eddy-resolving techniques have become a powerful tool to investigate shallow flows at both laboratory and field scale. In this paper several examples are given where high-resolution 3D numerical simulation are used to investigate the spatial development of mixing interfaces (MIs) forming in shallow environments like open channels with idealized and natural bathymetry where the bed friction plays a major role in the spatial development of the MI and associated large-scale turbulence. The focus is on the coherent structures forming within the MI and in its vicinity that control the momentum and mass exchange and heat transfer between the two sides of the MI. Examples include: (1) a MI developing in a flat-bed open channel downstream of a splitter wall separating two parallel fully-turbulent streams of different velocities, (2) a MI developing in a flat-bed open channel downstream of a 60 $$^{\circ }$$ wedge separating two non-parallel fully turbulent streams of different velocities, (3) a MI developing downstream of a river confluence for cases with a large and, respectively, a small difference between the mean velocities of the two streams. Stratification effects due to unequal densities of the two incoming streams are also discussed, (4) a MI developing between a main rectangular straight channel and a series of shallow embayments present at one of the channel banks. Besides using available experimental data to demonstrate that eddy resolving techniques can accurately predict the structure of the MI and its development, the paper discusses new insights into the physics of these flows obtained based on the simulations. The paper also provides an overview of the main numerical approaches that can be used to simulate the unsteady dynamics of the large scale turbulence in flows containing shallow MIs.
PubDate: 2014-10-01

• Fourier analysis of the roll-up and merging of coherent structures in
shallow mixing layers
• Abstract: Abstract The current study investigates the role of nonlinearity in the development of two-dimensional coherent structures (2DCS) in shallow mixing layers. A nonlinear numerical model based on the depth-averaged shallow water equations is used to investigate temporal shallow mixing layers, where the mapping from temporal to spatial results is made using the velocity at the center of the mixing layer. The flow is periodic in the stream-wise direction and the transmissive boundary conditions are used in the cross-stream boundaries to prevent reflections. The numerical results are examined with the aid of Fourier decomposition. Results show that the previous success in applying local linear theory to shallow mixing layers does not imply that the flow is truly linear. Linear stability theory is confirmed to be only valid within a short distance from the inflow boundary. Downstream of this linear region, nonlinearity becomes important for the roll-up and merging of 2DCS. While the energy required for the merging of 2DCS is still largely provided by the velocity shear, the merging mechanism is one where nonlinear mode interaction changes the velocity field of the subharmonic mode and the gradient of the along-stream velocity profile which, in turn, changes the magnitude of the energy production of the subharmonic mode by the velocity shear implicitly. The nonlinear mode interaction is associated with energy up-scaling and is consistent with the inverse energy cascade which is expected to occur in shallow shear flows. Current results also show that such implicit nonlinear interaction is sensitive to the phase angle difference between the most unstable mode and its subharmonic. The bed friction effect on the 2DCS is relatively small initially and grows in tandem with the size of the 2DCS. The bed friction also causes a decrease in the velocity gradient as the flow develops downstream. The transition from unstable to stable flow occurs when the bed friction balances the energy production. Beyond this point, the bed friction is more dominant and the 2DCS are progressively damped and eventually get annihilated. The energy production by the velocity shear plays an important role from the upstream end all the way to the point of transition to stable flow. The fact that linear stability theory is valid only for a short distance from the inflow boundary suggests that some elements of nonlinearity is incorporated in the mean velocity profile in experiments by the averaging process. The implicit nature of nonlinear interaction in shallow mixing layers and the sensitivity of the nonlinear interaction to phase angle difference between the most unstable mode and its subharmonic allows local linear theory to be successful in reproducing features of the instability such as the dominant mode of the 2DCS and its amplitude.
PubDate: 2014-10-01

• Mixing layer in open-channel junction flows
• Abstract: Abstract When two open-channel flows merge in a three-branch subcritical junction, a mixing layer appears at the interface between the two inflows. If the width of the downstream channel is equal to the width of each inlet channel, this mixing layer is accelerated and is curved due to the junction geometry. The present work is dedicated to simplified geometries, considering a flat bed and a $$90^{\circ }$$ angle where two configurations with different momentum ratios are tested. Due to the complex flow pattern in the junction, the so-called Serret–Frenet frame-axis based on the local direction of the velocity must be employed to characterize the flow pattern and the mixing layer as Cartesian and cylindrical frame-axes are not adapted. The analysis reveals that the centerline of the mixing layer, defined as the location of maximum Reynolds stress and velocity gradient, fairly fits the streamline separating at the upstream corner, even though a slight shift of the mixing layer towards the center of curvature is observed. The shape of the mixing layer appears to be strongly affected by the streamwise acceleration and the complex lateral confinement due to the side walls and the corners of the junction, leading to a streamwise increase of the mean velocity along the centerline and a decrease of the velocity difference. This results in a specific streamwise evolution of the mixing layer width, which reaches a plateau in the downstream region of the junction. Finally, the evaluation of the terms in the Reynolds-Averaged-Navier–Stokes equations reveals that the streamwise and normal acceleration and the pressure gradient remain dominant, which is typical of accelerated and rotational flows.
PubDate: 2014-10-01

• A three-dimensional experimental investigation of the structure of the
spanwise vortex generated by a shallow vortex dipole
• Abstract: Abstract The three-dimensional dynamics of shallow vortex dipoles is investigated by means of an innovative three-dimensional, three-component (3D-3C) scanning PIV technique. In particular, the three-dimensional structure of a frontal spanwise vortex is characterized. The technique allows the computation of the three-dimensional pressure field and the planar (x, y) distribution of the wall shear stress, which are not available using standard 2D PIV measurements. The influence of such a complex vortex structure on mass transport is discussed in the context of the available pressure and wall shear stress fields.
PubDate: 2014-10-01

• Horizontal transport, mixing and retention in a large, shallow estuary:
Río de la Plata
• Abstract: Abstract We use field data and a high-resolution three-dimensional (3D) hydrodynamic numerical model to investigate the horizontal transport and dispersion characteristics in the upper reaches of the shallow Río de la Plata estuary, located between the Argentinean and Uruguayan coasts, with the objective of relating the mixing characteristics to the likelihood of algal bloom formation. The 3D hydrodynamic model was validated with an extensive field experiment including both, synoptic profiling and in situ data, and then used to quantify the geographic variability of the local residence time and rate of dispersion. We show that during a high inflow regime, the aquatic environment near the Uruguayan coast, stretching almost to the middle of the estuary, had short residence time and horizontal dispersion coefficient of around 77 $$\mathrm {m}^{2}\,\mathrm {s}^{-1}$$ , compared to the conditions along the Argentinean coastal regime where the residence time was much longer and the dispersion coefficient (40 $$\mathrm {m}^{2}\,\mathrm {s}^{-1}$$ ) much smaller, making the Argentinian coastal margin more susceptible for algae blooms.
PubDate: 2014-10-01

• Spatial–temporal structure of mixing interface turbulence at two
large river confluences
• Abstract: Abstract Converging flows at stream confluences often produce highly turbulent conditions. The shear layer/mixing interface that develops within the confluence hydrodynamic zone (CHZ) is characterized by complex patterns of three-dimensional flow that vary both spatially and temporally. Previous research has examined in detail characteristics of mean flow and turbulence along mixing interfaces at small stream confluences and laboratory junctions; however few, if any, studies have examined these characteristics within mixing interfaces at large river confluences. This study investigates the structure of mean velocity profiles as well as spatial and temporal variations in velocity, backscatter intensity, and temperature within the mixing interfaces of two large river confluences. Velocity, temperature, and backscatter intensity data were obtained at stationary locations within the mixing interfaces and at several cross sections within the CHZ using acoustic Doppler current profilers. Results show that mean flow within the mixing interfaces accelerates over distance from the junction apex. Turbulent kinetic energy initially increases rapidly over distance, but the rate of increase diminishes downstream. Hilbert–Huang transform analysis of time series data at the stationary locations shows that multiple dominant modes of fluctuations exist within the original signals of velocity, backscatter intensity, and temperature. Frequencies of the largest dominant modes correspond well with predicted frequencies for shallow wake flows, suggesting that mixing-interface dynamics include wake vortex shedding—a finding consistent with spatial patterns of depth-averaged velocities at measured cross sections. Spatial patterns of temperature and backscatter intensity show that the converging flows at both confluences do not mix substantially, indicating that turbulent structures within the mixing interfaces are relatively ineffective at producing mixing of the flows in the CHZ.
PubDate: 2014-10-01

• Hele-Shaw beach creation by breaking waves: a mathematics-inspired
experiment
• Abstract: Abstract Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.
PubDate: 2014-10-01

• Surface signatures of submerged heated jet
• Abstract: Abstract The flow induced at the surface of a water body by a submerged heated horizontal turbulent jet was investigated experimentally with the aim of developing parameterizations for surface mean temperature/velocity fields. The jet nozzle diameter was fixed, the depth of the jet beneath the free surface was varied, and two jet Reynolds numbers (5020, 11300) were considered. The surface temperature was measured using a highly sensitive infrared camera, and the near-surface horizontal velocity field was measured using particle image velocimetry. The experimental results were explained using a model based on similarity solutions with variable turbulent viscosity. While classical Schlichting’s solution with constant turbulent viscosity predicts complete similarity for transverse velocity/temperature distributions only in a plane that coincides with the flow axis, the present solution predicts similarity in an arbitrary plane parallel to the flow axis, which was confirmed using data collected at the surface. Comparisons of present data with available previous results also showed general agreement.
PubDate: 2014-10-01

• Using surface drifter observations to measure tidal vortices and relative
diffusion at Aransas Pass, Texas
• Abstract: Abstract 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.
PubDate: 2014-10-01

• Physical and biological controls of algal blooms in the Río de la
Plata
• Abstract: Abstract 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 $$R= 0.4$$ .
PubDate: 2014-10-01

• Shallow wake behind exposed wood-induced bar in a gravel-bed river
• Abstract: Abstract Recent theoretical research indicates that dynamics of shallow flows can be strongly affected by waves developing on the free surface. In this study a shallow wake with an oblique pressure wave behind a model of a tree-centered emergent bar is investigated in a gravel-bed river. A bar was constructed in a shallow river reach with nearly uniform flow. The structure of flow was assessed with an array of velocimeters. Flow visualization with a solute of fluorescent dye complemented the measurements and provided qualitative information on the wake behavior. This study indicates that quantitative criteria for shallow wakes classification developed in laboratory setups are not straightforward in explaining the field results. According to the wake stability criteria, the expected dynamics for examined wake flow is a vortex street (VS) type. Contrary to this expectation, measurements and visualizations in this study show that mean momentum differential and turbulent vortices in the wake decrease stronger than expected in VS type and therefore the wake should be classified as unsteady bubble type with a weak downstream instability. Analysis of velocity differential dynamics in the examined shallow wake suggests that the bed friction alone is insufficient to explain the inconsistency of VS criterion whereas accounting for advective fluxes driven by inhomogeneous pressure field leads to a correct prediction of the wake behavior.
PubDate: 2014-10-01

• Shallow flows: 2D or not 2D?
• Abstract: Abstract It is commonly assumed that shallow flows are in good approximation two-dimensional (2D) or quasi-2D. We will provide evidence that this is not always the case, and that the simple scaling argument based on the continuity equation does not always apply. Laboratory experiments on vortex flows in shallow fluid layers have revealed that locally significant three-dimensional (3D) effects and substantial vertical motions may occur, clearly destroying the assumed 2D character of the flow. For example, in the case of a dipolar vortex structure, an oscillatory vertical motion is observed in the vortex cores, while a spanwise circulation roll is present in front of the travelling dipole. These laboratory observations are confirmed by 3D numerical flow simulations. Attention will be given to a correct scaling analysis, in which both the aspect ratio of the fluid depth and a typical horizontal scale and the Reynolds number play a role.
PubDate: 2014-10-01

• Gravity currents in rotating, wedge-shaped, adverse channels
• Abstract: Abstract Results are presented from a series of parametric experimental and analytical studies of the behaviour of dense gravity currents along rotating, up-sloping, wedge-shaped channels. High resolution density profile measurements at fixed cross- and along-channel locations reveal the outflowing bottom gravity currents to adjust to quasi-steady, geostrophically-balanced conditions along the channels, with the outflow layer thickness and cross-channel interface slope shown to scale with the inlet Burger number for all experimental conditions tested. A general analytical solution to the classic rotating hydraulics problem has been developed under the assumption of inviscid, zero-potential-vorticity conditions to model dense water flow through a triangular constriction and thus simulate the vee-channel configurations under consideration. Predictions from this zero-PV model are shown to provide good overall quantitative agreement with experimental measurements obtained both under hydraulically-controlled conditions at the channel exit and for subcritical conditions generated along the channel length. Quantitative discrepancies between measurements and analytical predictions are attributed primarily to assumptions and limitations associated with the zero-PV modelling approach adopted, as well as the to the rapid adjustment in outflow characteristics as the channel exit is approached, as characterised by the along-channel variation in densimetric Froude number for the outflows.
PubDate: 2014-10-01

• Instabilities in non-rotating and rotating shallow shear flows
• Abstract: Abstract Numerical simulations for the wave radiation effect on the linear and nonlinear instabilities of rotating and non-rotating shallow flows are conducted using shallow-water equations. At a low convective Froude number, the results of the instabilities is a string of eddies. The coalescence between the neighbouring eddies decides the transverse mixing of the shallow shear flow. At a higher convective Froude number, the development of the shear flow is characterized by wave radiation and the production of shocklets. The radiation of waves in the non-rotating shallow flow is a phenomenon analogous to the radiation of sound in gas dynamics. In the rotating flow on the other hand, the shallow-flow instabilities are intensified due to rotational interference within a window of instability over a narrow range of Rossby numbers.
PubDate: 2014-10-01

• Spatial structure of internal Poincaré waves in Lake Michigan
• Abstract: Abstract In this paper we examine the characteristics of near-inertial internal Poincaré waves in Lake Michigan (USA) as discerned from field experiments and hydrodynamic simulations. The focus is on the determination of the lateral and vertical structure of the waves. Observations of near-inertial internal wave properties are presented from two field experiments in southern Lake Michigan conducted during the years 2009 and 2010 at Michigan City (IN, USA) and Muskegon (MI, USA), respectively. Spectra of thermocline displacements and baroclinic velocities show that kinetic and potential baroclinic energy is dominated by near-inertial internal Poincaré waves. Vertical structure discerned from empirical orthogonal function analysis shows that this energy is predominantly vertical mode 1. Idealized hydrodynamic simulations using stratifications from early summer (June), mid-summer (July) and fall (September) identify the basin-scale internal Poincaré wave structure as a combination of single- and two-basin cells, similar to those identified in Lake Erie by Schwab, with near-surface velocities largest in the center of the northern and southern basins. Near-inertial bottom kinetic energy is seen to have roughly constant magnitude over large swathes across the basin, with higher magnitude in the shallower areas like the Mid-lake Plateau, as compared with the deep northern and southern basins. The near-bottom near-inertial kinetic energy when mapped appears similar to the bottom topography map. The wave-induced vertical shear across thermocline is concentrated along the longitudinal axis of the lake basin, and both near-bottom velocities and thermocline shear are reasonably explained by a simple conceptual model of the expected transverse variability.
PubDate: 2014-10-01

• Unsteady turbulence in expansion waves in rivers and estuaries: an
experimental study
• Abstract: Abstract A sudden decrease in water depth, called a negative surge or expansion wave, is characterised by a gentle change in free-surface elevation. Some geophysical applications include the ebb tide flow in macro-tidal estuaries, the rundown of swash waters and the retreating waters after maximum tsunami runup in a river channel. The upstream propagation of expansion waves against an initially steady flow was investigated in laboratory under controlled flow conditions including detailed free-surface velocity and Reynolds stress measurements. Both non-intrusive free-surface measurements and intrusive velocity measurements were conducted for relatively large Reynolds numbers with two types of bed roughness. The data showed that the propagation of expansion waves appeared to be a relatively smooth lowering to the water surface. The wave leading edge celerity data showed a characteristic trend, with a rapid acceleration immediately following the surge generation, followed by a deceleration of the leading edge surge towards an asymptotical value: $$(\mathrm{U}+\mathrm{V}_\mathrm{o})/(\mathrm{g}\times \mathrm{d}_\mathrm{o})^{1/2}=1$$ for both smooth and rough bed experiments. The results indicated that the bed roughness had little to no effect, within the experimental flow conditions. Relatively large fluctuations in free-surface elevation, velocity and turbulent shear stress were recorded beneath the leading edge of the negative surge for all flow conditions. The instantaneous turbulent shear stress levels were significantly larger than the critical shear stress for sediment erosion. The present results implied a substantial bed erosion during an expansion wave motion.
PubDate: 2014-09-19

• Effects of a sharp change of emergent vegetation distributions on
thermally driven flow over a slope
• Abstract: Abstract In this study, thermally driven flow within a sharp change of rooted emergent vegetation distributions is discussed. A conceptual and linear model is developed to study the interferences and competitions between two forcing mechanisms (vegetation shading and sloping bottom effects) on circulation patterns. The two forcing mechanisms can counteract or reinforce each other according to the presence of emergent vegetation in shallow or deep regions. For tall vegetation in shallows and open water in deep regions, vegetation shading leads to opposite temperature gradients against those from a slope. A critical vegetation blockage $$B_{\textit{critical}}$$ is found to cause a minimum horizontal exchange flowrate and divide the flow regime as topographic or vegetation shading dominated. In clear water, opposite circulation from the bottom is produced and gradually spreads to the whole water column. On the contrary, in turbid water, the opposite circulation emerges from the water surface and gradually propagates to the sloping bottom. The circulation in turbid water is more easily affected by a sharp change of vegetation distributions rather than clear water. For open in shallows and tall vegetation in deep water, pressure gradients from vegetation shading can enhance buoyancy from the sloping bottom and lead to greater horizontal exchange flowrates. Due to a sharp change of emergent vegetation distributions, horizontal exchange flowrates are possibly increased and greater than those without vegetation. In short, circulation patterns and strength can be significantly affected by a sharp change of emergent vegetation distributions.
PubDate: 2014-08-29

• Comment on: “Effect of flow confinement on the hydrodynamics of
circular impinging jets: implications for erosion assessment”; by
Environ Fluid Mech: DOI 10.1007/s10652-014-9354-3
• Abstract: Abstract This brief comment clarifies certain aspects concerning the distribution of the radial wall shear stress that is generated by jets impinging normally on flat plates.
PubDate: 2014-08-28

• Shallow-water solutions for gravity currents in non-rectangular cross-area
channels with stratified ambient
• Abstract: Abstract We consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $$x$$ . The current is of constant density, $$\rho _c$$ , and the ambient has a linear stable stratification, from $$\rho _{b}$$ at the bottom $$z=0$$ to $$\rho _{o}$$ at the top $$z=H$$ . The cross-section of the channel is given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . We develop a one-layer shallow-water (SW) formulation, and we implement it for the solution of a gravity current of fixed volume released from a lock (we focus on a “heavy” bottom current with $$\rho _{c}\ge \rho _{b}$$ ). The dependent variables are the position of the interface, $$h(x,t)$$ , and the speed (averaged over the area of the current), $$u(x,t)$$ , where $$t$$ is the time. The non-rectangular cross-section geometry enters the formulation via $$f(h)$$ and integrals of $$f(z), z f(z)$$ and $$z^2 f(z)$$ , where $$f(z) = f_1(z) + f_2(z)$$ is the width of the channel. For a given geometry $$f(z)$$ , the free input parameters are (1) the height ratio $$H/h_0$$ of ambient to lock; and (2) the stratification parameter $$S = (\rho _{b}- \rho _{o})/(\rho _{c}- \rho _{o})$$ . In general, the SW equations of motion are a hyperbolic PDE system which we solve by a finite-difference method. However, we show that the initial motion displays a “slumping” stage with constant speed of propagation; this can be calculated exactly by the method of characteristics. An analytical solution for the long-time self-similar propagation is also presented for $$S=1$$ and the power-law $$f(z) = bz^\alpha$$ cross section profile. Solutions are presented for various stratification and typical geometries: power-law ( $$f(z) = b z^\alpha$$ , where $$b, \alpha$$ are positive constants), power-law B ( $$f(z) = b( H-z)^\alpha$$ ), trapezoidal, and circle. In general, the increase of the stratification parameter $$S$$ causes a reduction of the speed of propagation, but the details depend on the geometry of the cross section. We also show that, upon a simple transformation, the solutions for the bottom (“heavy”) current can be used for the prediction of top (“light”) currents The present solution is a significant generalization of the classical a...
PubDate: 2014-08-22

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