Journal of Hydraulic Engineering

Forum papers are thought-provoking opinion pieces or essays founded in fact, sometimes containing speculation, on a civil engineering topic of general interest and relevance to the readership of the journal. The views expressed in this Forum article do not necessarily reflect the views of ASCE or the Editorial Board of the journal.

This paper, written to mark the 60th anniversary of the Journal of Hydraulic Engineering, focuses on the nature of meandering flow and its coupling to bed and bank deformation. An outline of the present understanding of the kinematics of meandering flow and how the flow shapes the bed, with a view towards the conditions in real alluvial meandering rivers, is presented. The flow and its interaction with the bed are analyzed by treating separately the effects on the flow of channel curvature and streamwise variation of channel curvature and by considering the results of numerous laboratory and numerical experiments carried out to date. The approach is used to explain essential differences in meandering bed topography exhibited by streams with varying values of sinuosity and width-to-depth ratio. The paper is also used as an opportunity to address the question of why, in the absence of geological constraints, some streams tend to remain regular in plan shape (i.e., symmetric in plan view with regard to the axis of bend) even when their loops actively expand laterally, whereas others acquire irregular plan shapes. This question is considered in view of the intrinsically different mechanics of bed deformation and bank erosion and a new experiment on bank erosion. The paper suggests that differences in the erodibility of the bed and banks may be a significant contributing factor to the planimetric fate of the stream.

This paper, written to celebrate the 60th anniversary of the Journal of Hydraulic Engineering, presents a structured, contemporary approach to scour-depth estimation that matches design method practicality to pier flow-field complexity. The approach involves a mix of semiempirical formulation, advanced experimentation aided by new instruments, and computational fluid dynamics (CFD). Highly useful for understanding complex pier flow fields, CFD holds patent promise for substantial use in design estimation of scour depth. Presently, however, CFD’s limited ability to simulate erosion and scour at a pier foundation hampers CFD’s practicality for design estimation of pier-scour depth. The writers reflect back 60 years when Laursen and Toch’s milestone publication provided major insights into pier scour, and when the hot-film anemometer first became available for investigating complex flow fields. Sixty years ago, pier flow fields were thought too complex to measure, or even visualize. The writers indicate where, today, further research into pier flow fields would benefit design estimation of scour depth.

Constructed wetlands, referred to as stormwater treatment areas (STAs), are important components of the water management infrastructure in south Florida. These systems have complex internal flow patterns due to uneven growth of vegetation, short-circuiting flows, and variable microtopographic features. Artificially generated water waves were used in an experiment to determine the combined effects of these influences on overall flow rates and water distribution in the treatment areas. Wave speeds and wave decay characteristics measured at points in a triangulated network were used to calculate resistance to flow, which depends on energy gradient and depth. The results were used to produce two-dimensional maps of hydraulic transmissivities, stem-drag resistance coefficients, and parameters of power law equations describing resistance. The results also included bulk parameters or single parameters defined for the entire wetland. They show that the patterns of many resistance parameters plotted in two-dimensions (2D) are similar to the general patterns of the distribution of vegetation in areal maps. Moreover, they indicate that power law equations can represent not just the vegetation resistance but also the effects of topography, resistance due to blockages, and the effects of short-circuiting and turbulent behaviors. The dimensionless parameters provided can be used to determine if the governing equation is primarily hyperbolic or parabolic, and to what extent the flow behaves like porous-media flow with diffusive wave behavior or short-circuiting stream flow with kinematic wave behavior.

The three-dimensional unsteady vortex flow in a sump pump is investigated by the computational fluid dynamics (CFD) method with the shear-stress transport (SST) $k-\omega$ model and the volume of fluid (VOF) multiphase model. The helicity density, involving both velocity and vortex, is especially used to discuss the mechanism of the process for air entrainment. The calculated results are qualitatively validated compared with the vortex observed in a previous experiment. The results investigated show that the helicity density takes a very important role in forming the process of the air entrainment. When the dimple appears, the helicity density near the intake pipe is larger than that close to the surface. When the air begins to flow into the pipe, the helicity density tube close to the surface reaches a certain level to connect the helicity density tube from the intake pipe. Moreover, the air entrainment can be suppressed if the helicity density tube is broken by placing a circular plate in a certain position.

Performance of an early warning system composed of online monitoring sensors for protecting municipal water supply is dependent on the number of sensors deployed. The inherent trade-off of performance versus scale of the system implemented is explored in this paper through multiobjective optimization using an augmented messy genetic algorithm (mGA). The augmented messy GA facilitated the comparison of solutions with variability in the number of sensors deployed. In this paper an early warning system is represented by a system of fixed sensors placed at network junctions, inline mobile sensors deployed from network junctions carried by flow within network pipes, and surface transceivers to communicate wirelessly with mobile sensors for data transmission and analysis. Performance of the implemented early warning system was measured as the time required for contamination detection, the detection likelihood, the population affected prior to event detection, and the total system cost for a small-, medium-, and large-scale municipal network. Results show well-defined Pareto fronts for each objective versus the cost of each solution, providing a tool for designers to optimize budget decisions.

A physical model study was conducted to examine the development of submerged sand erosion through a slot on a defective sewer pipe. Various parameters that affect the erosion process were studied, including particle size, defect size, position on the pipe, and water depth. This experimental study used particle image velocimetry analysis to provide a deeper understanding of the mechanism of sand erosion due to defective sewer pipes. The sand velocities were determined, and the sand and water flow rates were measured. After the erosion process was initiated, the sand and water outflow rates quickly reached the steady state until the eroded zone reached the slot opening. During the steady erosion process, the region of sand motion was restricted to a narrow zone above the opening, with sand particles sliding into the newly formed cavity at the angle of repose. An increase in slot width significantly increased the sand and water flow rates. A decrease in sand particle size from medium to fine resulted in a decreased sand flow rate mainly due to decreased water flow. An increase in sand particle size from medium to coarse resulted in a decreased sand flow rate due to the arching resistance close to the slot for larger sand particles. In this study, the sand flow ceased when the ratio between the slot width and particle size was less than 3.1. The ratio between the sand and water flow rates was found to be a function of the ratio of slot width to particle size. The water level above the pipe and the slot position affected the sand and water flow rates. A simple analytical model was developed to estimate the sand and water flow rates in the erosion process.

Knowledge of turbulent jets in shallow water depths is of utmost importance to adequately characterize municipal and industrial discharges. Propagation and dilution of such discharges are influenced by the presence of channel bed and water surface at close proximity. This study numerically investigates the bed/water surface confinement effects on circular turbulent jets by analyzing key parameters such as decay of maximum velocity, velocity profiles, growth of jet, locus of maximum velocity, and turbulence properties. Results reveal that the confinement has a profound impact on the entrainment and mixing characteristics of a jet. Entrainment is suppressed at the confining surface, and as a consequence velocity decays at a lower rate. The locus of maximum velocity shifts toward the closest confining surface for asymmetrical cases, whereas it shifts toward the bed for symmetrical cases. Mixing of a jet both in the horizontal and the vertical planes is affected significantly by the confinement. Turbulence characteristics differ significantly in the vertical plane only. Findings from this study will be useful for characterizing properties of circular jets discharging in a confined ambient conditions.

The current study aims at examining the internal flow field and the corresponding phenomena in the developing stage of forced hydraulic jumps. Experiments with a range of Froude numbers were performed by adjusting the inclination angle of the channel. A nonintrusive measuring technique, particle image velocimetry (PIV), was used in the experiments. Numerically, the weakly compressible moving particle semi-implicit (WC-MPS) method coupled with the subparticle-scale turbulence model was applied to simulate the flow phenomena. The simulation results for both flow surface profiles and velocity distributions at different sections were compared to the experimental data from the published literature and experimental data conducted in the current study. Flow phenomena showed good agreement with the numerical simulation results. Finally, the results from the experiments and simulation provided a detailed examination of the flow field during the developmental stages of forced hydraulic jumps.

Although derived from the principles of conservation of mass and momentum, the water-hammer equations integrating dynamic friction are almost never expressed in conservative form. This is because the pressure and volume discharge are used as variables but these are not conserved quantities, especially when the one-dimensional velocity profile is distorted from its assumed steady state shape due to the large accelerations imposed on the fluid particles across the cross section. This paper presents the derivation of the water-hammer equations in conservation form incorporating dynamic friction. With the dynamic friction taken into account, a source term appears in the basic partial differential equations as presented by Guinot. The numerical algorithm implements the Godunov approach to one-dimensional hyperbolic systems of conservation laws on a finite-volume stencil. Two case studies are used to illustrate the influence of the various formulations. A comparative study between the analytical solution, the numerical solution with quasi-steady friction only, the numerical solution with dynamic friction, and the measurements has been presented. The results indicate that the dynamic friction formulation reduces the peak water hammer pressures when compared with a quasi-steady representation.

The hydraulics of leakage and intrusion flows through leak openings in pipes is complicated by variations in the leak areas owing to changes in pressure. This paper argues that the pressure–area relationship can reasonably be assumed to be a linear function, and a modified orifice equation is proposed for more realistic modeling of leakage and intrusion flows. The properties of the modified orifice equation are explored for different classes of leak openings. The implications for the current practice of using a power equation to model leakage and intrusion flows are then investigated. A mathematical proof is proposed for an equation linking the parameters of the modified orifice and power equations using the concept of a dimensionless leakage number. The leakage exponent of a given leak opening is shown to generally not be constant with variations in pressure and to approach infinity when the leakage number approaches a value of minus one. Significant modeling errors may result if the power equation is extrapolated beyond its calibration pressure range or at high exponent values. It is concluded that the modified orifice equation and leakage number provide a more realistic description of leakage and intrusion flows, and it is recommended that this approach be adopted in modeling studies.

Large-scale roughness renders the flow resistance underestimated with the Manning-Strickler (MS) formula. This can be corrected simply by considering the roughness-induced flow constriction. Two modified MS formulas have been therefore derived for pipe and open-channel flows, respectively. They are applicable for small- to large-scale roughness. Without making any adjustment of the constants involved, both formulas are successfully validated by experimental data, particularly for flows significantly affected by the large-scale roughness.

The unsteady three-dimensional (3D), nonaerated, skimming flow over a stepped spillway is investigated by means of a detached eddy simulation (DES). Special emphasis is placed on the description of the instantaneous velocity and vorticity fields on the step cavities, as well as on the 3D, long-term, average flow structure at a Reynolds number of approximately ${10}^5$. Mean-flow velocities obtained from simulations on both relatively-coarse and fine meshes at different locations in the spillway are in very good agreement with particle image velocimetry (PIV) data published elsewhere. Nondimensional profiles of turbulence intensity and Reynolds stresses are also close to data, although numerical values are larger than the experimental counterparts. Snapshots of velocity vectors reveal that the flow near the pseudo-bottom is continuously flapping, and that it intermittently interacts with ejections of positive vorticity emanating from the cavities. As a result of the interaction between the shear flow and the 3D cavity flow, patches of positive and negative vorticity are carried away downstream, eventually interacting with the free surface. It is also found that the ratios of redistribution among components (terms) of the turbulent kinetic energy (TKE) at the step edges are very similar to those universal values for the intermediate region of open-channel flows. It is additionally shown that large-scale structures extracted from isosurfaces of the $Q$-criterion correspond primarily to well-organized, randomly-distributed thin tubes of vorticity. The instantaneous flow field revealed in this paper is considered crucial for modeling the even more complex aerated region.

Diffuse optical sensors estimate attenuation of natural light (solar irradiance) by measuring its vertical distribution in the water column, and the sensors are useful for estimating both suspended sediment concentration and light availability for organisms in rivers, lakes, and estuaries. However, currently there is a lack of reliable light attenuation models that account for particle size and density, have widely applicable dimensionless form, and show usefulness for coupling with sensors. To overcome these deficiencies, the authors revisit earlier dimensional analysis and relax a number of assumptions to provide a more robust and applicable dimensionless number for light attenuation by sediments. Second, the authors formulate an analytical light attenuation model that is found to be in general agreement with the dimensionless model and provides a theoretical description of the controlling dimensionless number. Third, the authors perform a meta-analysis for a wide range of data sets in diverse waters to test the model across a range of particle sizes and sediment concentrations from 9 to 90 μm and $0\mathrm{t}\mathrm{o}670\mathrm{mg}{\mathrm{L}}^{-1}$ respectively. The dimensionless light attenuation number clearly divides the light attenuation data into two categories: one linear and one with nonlinear dependence on suspended sediment concentration, both of which collapse the respective data sets. About 90% of the data displays linear properties, and the linear empirical coefficient found for the dimensionless model agrees almost exactly with the equation derived with the analytical model. Results that exhibit nonlinearity when scaled with the dimensionless number suggest the influence of secondary processes upon attenuation that most likely arise from the measurement methods for sediment and light properties. The authors recommend ways to avoid nonlinearity, when it can be avoided, that may be followed in future research to enhance the usefulness of sensors and sensor networks.

Numerical modeling of bed-load transport in shallow flows, particularly oriented toward environmental flows, is an active field of research. Nevertheless, other possible applications exist. In particular, bed-load transport phenomena are relevant in urban drainage systems, including sewers. However, few applications of coupled two-dimensional (2D) shallow-water and bed-load transport models can be found, and their transfer from environmental applications—usually river and floodplain—into sewer applications requires some adaptation. Unlike to river systems, where there is a thick layer of sediment that constitutes a movable riverbed, sewer systems have thin layers of sediment that need to be removed, thus exposing a rigid, nonerodible surface. This problem requires careful numerical treatment to avoid generating errors and instability in the simulation. This paper deals with a numerical approach to tackle this issue in an efficient way that allows large-scale studies to be performed and provides empirical evidence that the proposed approach is accurate and applicable for sewage and channel-flushing problems.

A constant flow density current flowing on an inverse slope incline in the range of 1–6% slope is investigated using two-dimensional (2D) large-eddy simulations (LES). The dynamic Smagorinsky subgrid-scale model is used to improve the mixing accuracy of the flow field. The results of the simulation show that, as the inverse slope of the incline surface increases, the mixing in the body of the dense fluid with the ambient fluid increases at the beginning of the slope because of vortices generated by the flow reversal, whereas the mixing in the head of the density current generally decreases. As a result, the entrainment velocity along the free surface decreases by an increasing slope. As the inverse slope of the incline surface increases, the head thickness of the density current is depressed and departed from the body of the density current flow for the higher slopes (4 and 6% slopes). The entrainment result of a 2D large-eddy simulation is compared with 3D simulation to verify the suitability of 2D large-eddy simulation method, and it is shown that the 2D simulation can model the flow field of upslope flow with reasonable accuracy. The numerical simulations are further validated by comparing these numerical results with experimental data for the downslope density current flow and entrainment results of lock-exchange dense current propagating upslope.

A key feature of a schematized open-channel confluence is the separation zone that is present when the tributary flow detaches from the downstream corner of the confluence. This zone of recirculating fluid influences head losses, as well as sediment and solute balances. Most laboratory studies on the separation zone were, however, made in flumes having a rectangular cross-sectional shape. The objective of this research is therefore to elaborate on the influence of the cross-sectional shape on the flow patterns in a 90° confluence. A validated large-eddy simulation (LES) model is used to study the flow in confluences with four different cross-sectional shapes. The results indicate that the dimensions of the separation zone are significantly different for nonrectangular shapes due to lateral currents that might reduce the local momentum deficit. This study contributes to finding the optimal shape of artificial confluences with respect to head losses, as well as to understanding the flow processes that are present in a confluence.

Bendway weirs are hydraulic control structures found in watercourses that range in scale from small streams to large rivers. They are designed to be submerged, broad-crested weirs, typically placed in groups on the outer bank of river bends. They are angled upstream, in order to redirect the flow that passes over them away from the bankline and toward the center of the channel. Although there exists a wealth of valuable guidance concerning best practices for bendway weir design, this guidance is derived mostly from engineering experience, and from physical and numerical model studies. To date, there exists no fundamental theoretical development of the mechanism by which bendway weirs function. Therefore, in order to provide this theoretical foundation, this paper presents a formal mass, energy, and momentum flux analysis of a bendway weir. The analysis yields a detailed description of the basic functionality of a bendway weir and suggests a means of reconciling the results from several extant modeling studies that appear to conflict. In addition, the paper provides design recommendations resulting from the analysis for both individual bendway weirs and for bendway weir groups.

Theoretical analysis and experimental testing of a debris-flow drainage channel with a step-pool configuration are conducted in this paper. The design conditions of the drainage channel with step-pool configuration were obtained by theoretical analysis, including the design shape of the step-pool configuration, design slope of the step section, and total length of a single step-pool configuration. In addition, the correction coefficients of the debris-flow jet length were obtained with different debris-flow densities by comparing the experimental and calculation results. Moreover, the debris flow repeats an acceleration-deceleration motion in the drainage channel to achieve a balanced discharge without scouring or deposition. At the downstream of the step-pool configuration, the average debris-flow velocity decreased by approximately 28 and 40% with debris-flow densities of $\rho =\mathrm{1,700}$ and $\mathrm{2,000}\mathrm{kg}/{\mathrm{m}}^3$. Furthermore, the energy-dissipation ratio varied between debris-flow densities of $\rho =\mathrm{1,700}$ and $\mathrm{2,000}\mathrm{kg}/{\mathrm{m}}^3$. Finally, the authors discuss different types of drainage channels in practical debris-flow hazard-mitigation engineering and parameter variations of these drainage channels.

Geysers are explosive eruptions of air-water mixture from manholes in drainage systems. When the design capacity of storm water drainage systems is exceeded during heavy rainfall, rapid inflows can lead to air-water interactions that give rise to geysers, causing damages and threatening human lives. Over the past two decades, the dynamics of air pockets in vertical shafts and horizontal pipes has been extensively studied. Although the role of entrapped air in causing large pressure transients is revealed, the mechanism of geyser formation remains elusive primarily because of the lack of detailed observations. A comprehensive series of experiments has been performed on a physical model of a simplified drainage system, which consists of a vertical riser and a horizontal pipe with diameter $D$ connected to a constant head tank. The system is filled with water; an air pocket is then released into the horizontal pipe and the trajectories of the air pockets in horizontal pipe and vertical riser are measured by videos and a high speed camera. Pressures are measured using pressure transducers near the pipe end and at the bottom of the riser. Parameters considered include riser diameter ($D_r$), upstream head ($H_0$), and initial air pocket volume ($V_{\mathrm{air}}$). Without an external pressure head, the air pocket migration in a water-filled vertical riser is found to be similar to a slug flow. The rising velocity of the air pocket relative to the free surface ($v_{\mathrm{net}}$) is nearly constant and close to the speed of a Taylor bubble; no geyser is observed. When an external pressure head is applied to the horizontal pipe, two types of flow are observed. For large riser diameter and small air volume, the rise of the air pocket in the vertical riser resembles a Taylor bubble, with an air pressure equal to the hydrostatic pressure of the water column above. The air breaks within the riser and no geyser is observed. For small riser diameter and large air volume, the horizontal air pocket shoots past the riser junction and only partially enters the shaft; the vertical air pocket migration is very different from that of a Taylor bubble. The air pocket pressure is found to be significantly higher than the hydrostatic value, resulting in rapid acceleration of air and water, and jetting out of air-water mixture from the riser top. Experimental results show that geysers are more likely to occur for small risers ($D_r/D\le 0.62$) and large air volumes $\{V_{\mathrm{air}}/[(\pi D_r^2/4)H_0]\ge 3.42\}$.

Near-field drastic vibrations caused by flood discharges have a serious impact on nearby residents and can damage buildings. A novel energy dissipator was invented to solve the technical challenge of energy dissipation in the Xiangjiaba Dam project, a giant hydraulic project in China. However, since this project is located near civil buildings, the obvious near-field vibrations at the wake of first-time flood release have become a serious threat for the surrounding residents and need to be solved urgently. Focusing on hydraulic experiments combined with prototype observations, an effective method was proposed to investigate the responsive relationship between discharge floods and vibrations. In the model experiments, 138 cases were carried out under different combinations of discharge and gate-opening modes. During the experiments, fluctuating pressures acting on the structures were measured; 226 cases were tested in the prototype and the corresponding ground acceleration was measured; the relationship between the root-mean square (RMS) of fluctuating pressure and discharge was obtained. Furthermore, based on the prototype observations, the correlation formula between the acceleration of near-field vibrations and discharge was also proposed. Consequently, the study shows that the prediction of acceleration in near-field vibrations is acceptable and the proposed method can be used to study the near-field vibrations induced by flood releases.

A model to compute reservoir sedimentation rates at daily timescales is presented. The model uses streamflow and sediment load data from nearby stream gauges to obtain an initial estimate of sediment yield for the reservoir’s watershed; it is then calibrated to the total deposition calculated from repeat bathymetric surveys. Long-term changes to reservoir trapping efficiency are also taken into account. The model was applied to the Fena Valley Reservoir, a water supply reservoir on the island of Guam. This reservoir became operational in 1951 and was recently surveyed in 2014. The model results show that the highest rate of deposition occurred during two typhoons (Typhoon Alice in 1953 and Typhoon Tingting in 2004); each storm decreased reservoir capacity by approximately 2–3% in only a few days. The presented model can be used to evaluate the impact of an extreme event, or it can be coupled with a watershed runoff model to evaluate potential impacts to storage capacity as a result of climate change or other hydrologic modifications.

An experimental study of the flow pattern in a V-shaped stepped spillway reveals that the flow structure is quite different from that of a traditional stepped spillway. This paper presents the flow structure, which comprises a downstream zone, collision zone, and spiral zone. Notably, the spiral zone includes a spiral core zone and a spiral convolution zone. Within the spiral core zone, a nested zone of spiral flow exists with opposite rotational directions in the inner and outer layers. The flow in a V-shaped stepped spillway is three-dimensional, unlike that in a traditional stepped spillway. Additionally, the $k\text{-}ϵ$ turbulence model was used to study the flow structure of a V-shaped stepped spillway.

Debris flows represent some of the most relevant phenomena in geomorphological events. Because of the potential destructiveness of such flows, they are the subject of a vast amount of research. This paper addresses the need for a numerical model applicable to granular-fluid mixtures featuring high spatial and temporal resolution, and thus capable of resolving the motion of individual particles, including all interparticle contacts. The DualSPHysics meshless numerical implementation based on smoothed particle hydrodynamics (SPH) is expanded with a distributed-contact discrete-element method (DCDEM) in order to explicitly solve the fluid and solid phases. The specific objective is to test the SPH-DCDEM approach by comparing its results with experimental data. An experimental setup for stony debris flows in a slit check dam is reproduced numerically, where solid material is introduced through a hopper, assuring a constant solid discharge for the considered time interval. With each sediment particle possibly undergoing several simultaneous contacts, thousands of time-evolving interactions are efficiently treated because of the model’s algorithmic structure and the HPC implementation of DualSPHysics. The results, composed of mainly retention curves, are in good agreement with the measurements, correctly reproducing the changes in efficiency with slit spacing and density.

Traditional applications of chemicals to flows in water and wastewater plants are aimed at the flow itself, across its full cross section. However, at seawater intakes the need is to apply antifoulant chemicals not to the whole flow, but just around the inside wall of the intake pipe, as close to its entrance as possible. This desktop study combines the distribution hydraulics of multiport diffusers with an established design for a streamlined pipe entrance. The resulting design is of a ring diffuser incorporated within a streamlined entrance to a seawater intake pipe to provide uniform and well-positioned dosage of an antifoulant chemical stream all around the interior wall of the pipe, for effective and economical dosant application. Streamlining also minimizes head loss at the intake.

The occurrence of hysteresis in a supercritical, open-channel flow approaching an obstacle has been recognized and investigated both experimentally and theoretically over the last few decades. However, the available theory and experimental investigations in the literature do not include the case when subcritical flow, controlled from downstream, can establish across the obstacle. The present work fills this gap by proposing a new theory that includes this occurrence and shows that two different steady flow states can establish for the same obstacle geometry and flow conditions—one with supercritical to subcritical transition far downstream from the obstacle, and the other with supercritical to subcritical transition far upstream from the obstacle. The proposed, more general theory includes the existing theory as a special case. Finally, two specific examples are illustrated and discussed, i.e., the case of flow over a raised bed hump, and the case of flow through a channel contraction.

This paper simplifies the partial differential equation (PDE) of total mean energy in the plumelike region of turbulent buoyant jets under the well-known distributions in the longitudinal and transverse direction for the mean values of velocities and concentrations, in conjunction with experimental evidence for turbulence contributions. The energy fluxes are found to be self-similar. The PDE is separated into two linear ordinary differential equations (ODEs) designating particular solutions for the axial and transverse direction. The linearity of the ODEs implies uniqueness of the superposition method (SM) for the acquisition of the general solution of the mean flow field arising from the merger of shortly-spaced individual plumes. The SM is validated by the successful predictions of the well-known centerline velocity variation of an actual line, plane, or round plume composed of infinitesimal sources. Moreover, a successful application of SM highlights its ability to take into account the reciprocal displacement of interacting plumes in a simple manner. The SM predictions have errors of less than 10% simulating both zones, i.e. the zone of flow establishment (core region) and the zone of established flow. The SM may be applied to plumes formed by ocean outfall diffusers, cooling towers, chimneys, forest fires, and volcanoes.