Comments on “Steady open channel flows with curved streamlines: The Fawer approach revised”

Steady open channel flow with curved streamlines cannot be treated with a hydraulic approach unless an extended Boussinesq-type energy equation is considered to account for the non-hydrostatic pressure distribution. The paper reviews approximate modeling of these flows by the notable theory of Carlos Fawer. This discussion intends to add to Fawer’s model by comparing it with the Dressler curved-flow equations, not quoted in the original paper. Based on a comparative analysis of the present comments Fawer’s theory is demonstrated to be relevant as compared to other formulations, supporting its importance, as highlighted in the original paper.     

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

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

Reply to comment on “steady open channel flows with curved streamlines: the Fawer approach revised”

In this reply to a comment to (Environ Fluid Mech 10(3):297–310 [1]), the comparison between the Dressler curved streamline equations and Fawer’s theory is further expanded by momentum considerations. The importance of Fawer’s theory for steady curved open channel flow is thereby highlighted, both from energy and momentum considerations.     

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

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

One-dimensional modelling of sediment deposition in reservoirs with sinking streams

It is common in karst regions that rivers are occasionally cut by mountains, resulting in the alternate appearances of open channel flow and pressurized flow. With more and more reservoirs being built in this region, the complicated sediment transport processes of such mixture flow are urgently needed to study. In this paper a one-dimensional numerical model with free surface and pressurized flows coupled together is presented. The simulated results are analyzed to explore deposition process in reservoirs with sinking streams; impacts of different hydraulic conditions on the sedimentation are also studied. To verify the computed results, a flume experiment is also conducted. The results show that deposition of sediments mainly occurs in open channel reaches where the longitudinal profile is similar to that of conventional reservoirs, i.e. typical delta has formed, demonstrating characteristics of deltaic deposition morphology in that the crest of delta moves to the downstream direction over time. The model provided by this paper is not only proved to simulate the characteristics of deposition in karst reservoirs successfully, but also reveal the impacts of hydraulic conditions in such circumstances.     

Hydrodynamics and turbulence in emergent and sparsely vegetated open channel flow

This present study reports the results of an experimental study characterizing thorough variation of turbulent hydrodynamics and flow distribution in emergent and sparsely vegetated open channel flow. An emergent and rigid sparse vegetation patch with regular spacing between stems along the flow and transverse directions was fixed in the central region of the cross-section of open channel. Experiments were conducted in subcritical flow conditions and velocity measurements were obtained with an acoustic Doppler Velocimetry system. Large variations of the turbulence intensities, Reynolds shear stress, turbulent kinetic energy and vortical motions are found in and around the vegetation patch. At any cross-section through the interior of the vegetation patch, streamwise velocity decreases with increase in streamwise length and the velocity profiles converge from the log-law to a linear profile with increasing slope. Time-averaged lateral and vertical velocities inside the vegetation patch increase with increasing streamwise distance and converge from negative values to positive values. Turbulence intensities interior of the sparse vegetation patch are more than those of without the vegetation patch. Similar to the trend of streamwise velocity profiles inside the vegetation, turbulence intensities and longitudinal-normal Reynolds shear stress profile decreases with streamwise direction. In the interior of the vegetation patch and downstream of the trailing edge, turbulent kinetic energy profiles are exhibiting irregular fluctuations and the maximum values are occurring in the outer layer. Analysis of flow distribution confirms sparse vegetation patch is inducing a serpentine flow pattern in its vicinity. At the leading edge, flow is rushing towards the right hand sidewall, and at the trailing edge, flow is turning to the left hand sidewall. In between the leading and trailing edges, the streamlines are following a zig-zag fashion at varied degree along the streamwise and lateral directions. Immediate upstream of the leading edge and in the interior of the vegetation patch, vortex motion is clearly visible and the vortices are stretched along the width of the channel with streamwise direction.     

An ecological risk assessment model for a pulsed contaminant emission into a wetland channel flow

As a continuation of the modelling on ecological degradation and hydraulic dispersion of pollutant emission into an idealized two-dimensional free-surface wetland flow (Zeng, L., Chen, G.Q., 2009b. Ecological degradation and hydraulic dispersion of contaminant in wetland. Ecol. Model., doi:10.1016/j.ecolmodel.2009.10.024), an ecological risk assessment model for the typical case of a pulsed contaminant emission into a realistic three-dimensional wetland channel flow is presented in this paper for the fate of cross-sectional mean concentration under environmental dispersion. An environmental dispersion model for the mean concentration is devised as an extension of Taylor’s classical analysis on dispersion in fluid flows. The velocity distribution and the environmental dispersivity in the fully developed steady flow through the wetland is found and illustrated with limiting cases covering various known solutions for the porous media flow between parallel plates, flow in a shallow wetland, sweeping flow in a densely vegetated wetland, and single phase flow in a channel. Obtained by Aris’s method of moments, the environmental dispersivity is shown characterized with multi-scale asymptotic time variations with stem dominated stage, transitional stage, and width-depth-stem dominated stage. Based on the solution for the evolution of contaminant cloud in the wetland channel flow, critical length and duration of the contaminant cloud with concentration beyond given environmental standard level are concretely illustrated for typical pollutant constituents in wastewater emission. Under the same emission intensity and environmental standard, the duration of contaminant cloud in the wetland channel is revealed shorter than that in a free surface wetland, due to the lateral effect.     

A new model for soil–water drainage problems

Seepage flow is an agent related to the transport and dispersion of contamination in groundwater. Steady two-dimensional seepage flow is governed by Laplace’s equation, for which several solution techniques are available. Because computations are complex from a practical point of view, simplified models encompass the Dupuit-Forchheimer approach assuming a horizontal flow. However this approach is inaccurate in seepage problems involving steep drawdowns. In this research, a new theoretical model for 2D seepage flow is proposed based on Fawer’s theory for curved flows Castro-Orgaz (Environ Fluid Mech 10(3):2971–2310, 2010), from which a second-order equation results describing the seepage surface. From this development, a numerical solution for the rectangular dam problem based on the second-order model is presented, whereas a simple first-order equation is found to describe flow to drains under a uniform rainfall. The results of this new model are compared with the full 2D solution of Laplace’s equation for typical test cases, resulting in an excellent agreement.     

Dressler’s theory for curved topography flows: iterative derivation,transcritical flow solutions and higher-order wave-type equations

The Dressler equations are a system of two non-linear partial differential equations for shallow fluid flows over curved topography. The theory originated from an asymptotic stretching method formulating the equations of motion in terrain-fitted curvilinear coordinates. Apparently, these equations failed to produce a transcritical flow profile changing from sub- to supercritical flow conditions. Further, wave-like motions over a flat bottom are excluded because the bed-normal velocity component is not accounted for. However, the theory was found relevant for several environmental flow problems including density currents over mountains and valleys, seepage flow in hillslope hydrology, the development of antidunes, the formation of geological deposits from hyper-concentrated flows, and shallow-water flow modeling in hydraulics. In this work, Dressler’s theory is developed in an alternative way by a systematic iteration of the stream and potential functions in terrain-fitted coordinates. The first iteration was found to be the former Dressler’s theory, whereas a second iteration of the governing equations results in velocity components generalizing Dressler’s theory to wave-like motion. Dressler’s first-order theory produces a transcritical flow solution over topography only if the total head is fixed by a minimum value of the specific energy at the transition point. However, the theory deviates from measurements under subcritical flow conditions, given that the bed-normal velocity component is significant. A second iteration to the velocity field was used to produce a second-order differential equation that resembles the cnoidal-wave theory. It accurately produces flow over an obstacle including the critical point and the minimum specific energy as part of the numerical solution. The new cnoidal-wave model compares well with the theory of a Cosserat surface for directed fluid sheets, whereas the Saint-Venant theory appears to be poor.     

A three-dimensional numerical study of flow characteristics in strongly curved channel bends with different side slopes

Environmental Fluid Mechanics - The influence of channel side slope on flow in strongly curved channel bends is studied numerically. The performances of five different turbulence models are...     

Simulation of dam/levee-break hydrodynamics with a three-dimensional implicit unstructured-mesh finite element model

Dam failures usually cause huge economic and life losses , especially in urban areas where there is a high concentration of inhabitants and economic actors. In order to understand the physical mechanisms of the formation and development of dam-break flooding, lots of efforts have been put into different types of modelling techniques. However, most of existing models are 1D (one-dimensional) or 2D models based on the shallow water equations. In this paper, we present a 3D numerical modelling investigation of dam-break flow hydrodynamics in an open L-shape channel. A newly developed 3D unstructured mesh finite element model is used here. An absorption-like term is introduced to the Navier–Stokes equations in order to control the conditioning of the matrix equation in the numerical solution process and thus improve the stability. A wetting and drying algorithm is used here to allow the free surface height to be treated with a high level of implicitness and stability. The 3D model has been validated by comparing the results with the published experimental data. Good agreement has been achieved at six selected locations. This study shows that the 3D unstructured mesh model is capable of capturing the 3D hydraulic aspects and complicated local flows around structures in simulation of dam-break flows.     

Mixing layer in open-channel junction flows

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.     

Dynamic environ analysis of compartmental systems: A computational approach

Ecosystems are often modeled as stocks of matter or energy connected by flows. Network environ analysis (NEA) is a set of mathematical methods for using powers of matrices to trace energy and material flows through such models. NEA has revealed several interesting properties of flow–storage networks, including dominance of indirect effects and the tendency for networks to create mutually positive interactions between species. However, the applicability of NEA is greatly limited by the fact that it can only be applied to models at constant steady states. In this paper, we present a new, computationally oriented approach to environ analysis called dynamic environ approximation (DEA). As a test of DEA, we use it to compute compartment throughflow in two implementations of a model of energy flow through an oyster reef ecosystem. We use a newly derived equation to compute model throughflow and compare its output to that of DEA. We find that DEA approximates the exact results given by this equation quite closely – in this particular case, with a mean Euclidean error ranging between 0.0008 and 0.21 – which gives a sense of how closely it reproduces other NEA-related quantities that cannot be exactly computed and discuss how to reduce this error. An application to calculating indirect flows in ecosystems is also discussed and dominance of indirect effects in a nonlinear model is demonstrated.     

3D numerical modelling of turbidity currents

During floods, the density of river water usually increases due to a subsequent increase in the concentration of the suspended sediment that the river carries, causing the river to plunge underneath the free surface of a receiving water basin and form a turbidity current that continues to flow along the bottom. The study and understanding of such complex phenomena is of great importance, as they constitute one of the major mechanisms for suspended sediment transport from rivers into oceans, lakes or reservoirs. Unlike most of the previous numerical investigations on turbidity currents, in this paper, a 3D numerical model that simulates the dynamics and flow structure of turbidity currents, through a multiphase flow approach is proposed, using the commercial CFD code FLUENT. A series of numerical simulations that reproduce particular published laboratory flows are presented. The detailed qualitative and quantitative comparison of numerical with laboratory results indicates that apart from the global flow structure, the proposed numerical approach efficiently predicts various important aspects of turbidity current flows, such as the effect of suspended sediment mixture composition in the temporal and spatial evolution of the simulated currents, the interaction of turbidity currents with loose sediment bottom layers and the formation of internal hydraulic jumps. Furthermore, various extreme cases among the numerical runs considered are further analyzed, in order to identify the importance of various controlling flow parameters.     

Are breaking waves,bores, surges and jumps the same flow?

The flow structure in the aerated region of the roller generated by breaking waves remains a great challenge to study, with large quantities of entrained air and turbulence interactions making it very difficult to investigate in details. A number of analogies were proposed between breaking waves in deep or shallow water, tidal bores and hydraulic jumps. Many numerical models used to simulate waves in the surf zone do not implicitly simulate the breaking process of the waves, but are required to parameterise the wave-breaking effects, thus relying on experimental data. Analogies are also assumed to quantify the roller dynamics and the energy dissipation. The scope of this paper is to review the different analogies proposed in the literature and to discuss current practices. A thorough survey is offered and a discussion is developed an aimed at improving the use of possible breaking proxies. The most recent data are revisited and scrutinised for the use of most advanced numerical models to educe the surf zone hydrodynamics. In particular, the roller dynamics and geometrical characteristics are discussed. An open discussion is proposed to explore the actual practices and propose perspectives based on the most appropriate analogy, namely the tidal bore.     

The modified niche model: Including detritus in simple structural food web models

Food webs are constructed as structural directed graphs that describe “who eats whom,” but it is common to interpret them as energy flow diagrams where predation represents an energy transfer from the prey to the predator. It is the aim of this work to demonstrate that food webs are incomplete as energy flow diagrams if they ignore passive flows to detritus (dead organic material). While many ecologists do include detritus in conceptual and mathematical models, the detrital omission is still commonly found. Often detritus is either ignored or treated as an unlimited energy source, yet all organisms contribute to the detritus pool, which can be an energy source for other species in the system. This feedback loop is of high importance, since it increases the number of pathways available for energy flows, revealing the significance of indirect effects, and making the functional role of the top predators less clear. In this work we propose the modified niche model by adding a detritus compartment to the niche model. We demonstrate the effect of structural loops that result from feeding on detritus, by comparing empirical data sets to five different assembly models: (1) cascade, (2) constant connectance, (3) niche, (4) modified niche (original in this work), and (5) cyber-ecosystem. Of these models, only the last two explicitly include detritus. We show that when passive flows to detritus are included in the food web structure, the structure becomes more robust to the removal of individual nodes or connections. In addition, we show that food web models that include the detritus feedback loop perform better with respect to several structural network metrics.     

Experimental investigation of bubbly flow and turbulence in hydraulic jumps

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

Modelling hydraulic jump using the bubbly two-phase flow method

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

On the consequences of aggregation and balancing of networks on system properties derived from ecological network analysis

In the ecological network analysis (ENA) of complex flow food webs the assumption is often made that the models characterizing the flows and stocks of ecosystems occur in a steady state where inflows equals outflows. An assessment of the system indices derived from ENA of six balanced and unbalanced system models, respectively, indicate to differences between indices. The aggregation of highly articulated flow models into models with fewer compartments also has drastic effects on the system metrics, particularly on the information indices.