Experimental assessment of characteristic turbulent scales in two-phase flow of hydraulic jump: from bottom to free surface

A hydraulic jump is a turbulent shear flow with a free-surface roller. The turbulent flow pattern is characterised by the development of instantaneous three-dimensional turbulent structures throughout the air–water column up to the free surface. The length and time scales of the turbulent structures are key information to describe the turbulent processes, which is of significant importance for the improvement of numerical models and physical measurement techniques. However, few physical data are available so far due to the complexity of the measurement. This paper presents an investigation of a series of characteristic turbulent scales for hydraulic jumps, covering the length and time scales of turbulent flow structures in bubbly flow, on free surface and at the impingement point. The bubbly-flow turbulent scales are obtained for Fr = 7.5 with 3.4 × 10⁴ < Re < 1.4 × 10⁵ in both longitudinal and transverse directions, and are compared with the free-surface scales. The results highlight three-dimensional flow patterns with anisotropic turbulence field. The turbulent structures are observed with different length and time scales respectively in the shear flow region and free-surface recirculation region. The bubbly structures next to the roller surface and the free-surface fluctuation structures show comparable length and time scales, both larger than the scales of vortical structures in the shear flow and smaller than the scales of impingement perimeter at the jump toe. A decomposition of physical signals indicates that the large turbulent scales are related to the unsteady motion of the flow in the upper part of the roller, while the high-frequency velocity turbulence dominates in the lower part of the roller. Scale effects cannot be ignored for Reynolds number smaller than 4 × 10⁴, mainly linked to the formation of large eddies in the shear layer. The present study provides a comprehensive assessment of turbulent scales in hydraulic jump, including the analyses of first data set of longitudinal bubbly-flow integral scales and transverse jump toe perimeter integral scales.     

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.     

Stability and Mixing of a Vertical Axisymmetric Buoyant Jet in Shallow Water

The stability, mixing and effect of downstream control on axisymmetric turbulent buoyant jets discharging vertically into shallow stagnant water is studied using 3D Reynolds-averaged Navier–Stokes equations (RANS) combined with a buoyancy-extended k –ε model. The steady axisymmetric turbulent flow, temperature (or tracer concentration) and turbulence fields are computed using the finite volume method on a high resolution grid. The numerical predictions demonstrate two generic flow patterns for different turbulent heated jet discharges and environmental parameters (i) a stable buoyant discharge with the mixed fluid leaving the vertical jet region in a surface warm water layer; and (ii) an unstable buoyant discharge with flow recirculation and re-entrainment of heated water. A stratified counterflow region always appears in the far-field for both stable and unstable buoyant discharges. Provided that the domain radius L exceeds about 6H, the near field interaction and hence discharge stability is governed chiefly by the jet momentum length scale to depth ratio l_M/H, regardless of downstream control. The near field jet stability criterion is determined to be l_M/H = 3.5. A radial internal hydraulic jump always exists beyond the surface impingement region, with a 3- to 6-fold increase in dilution across the jump compared with vertical buoyant jet mixing. The predicted stability category, velocity and temperature/concentration fields are well-supported by experiments of all previous investigators.     

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.     

Coupling between free-surface fluctuations,velocity fluctuations and turbulent Reynolds stresses during the upstream propagation of positive surges,bores and compression waves

In open channel, canals and rivers, a rapid increase in flow depth will induce a positive surge, also called bore or compression wave. The positive surge is a translating hydraulic jump. Herein new experiments were conducted in a large-size rectangular channel to characterise the unsteady turbulent properties, including the coupling between free-surface and velocity fluctuations. Experiments were repeated 25 times and the data analyses yielded the instantaneous median and instantaneous fluctuations of free-surface elevation, velocities and turbulent Reynolds stresses. The passage of the surge front was associated with large free-surface fluctuations, comparable to those observed in stationary hydraulic jumps, coupled with large instantaneous velocity fluctuations. The bore propagation was associated with large turbulent Reynolds stresses and instantaneous shear stress fluctuations, during the passage of the surge. A broad range of shear stress levels was observed underneath the bore front, with the probability density of the tangential stresses distributed normally and the normal stresses distributed in a skewed single-mode fashion. Maxima in normal and tangential stresses were observed shortly after the passage of a breaking bore roller toe. The maximum Reynolds stresses occurred after the occurrence of the maximum free-surface fluctuations, and this time lag implied some interaction between the free-surface fluctuations and shear stress fluctuations beneath the surge front, and possibly some causal effect.     

Analysis of the velocity field in a large rectangular channel with lateral shockwave

In this work the authors describe the main characteristics of the velocity field of hydraulic jumps in a very large channel where lateral shockwaves occur. Experiments were carried out at the Coastal Engineering Laboratory of the Water Engineering and Chemistry Department of the Technical University of Bari (Italy). Extensive flow velocity measurements were investigated in order to have a clearer understanding of both hydraulic jump development and lateral shockwave formation in a very large channel. Eight experiments were performed in a 4m wide rectangular channel; the experiments differed in the inlet Froude number F ₀ and the jump type. Seven tests were carried out with undular jumps and one with a roller jump. The flow velocity and the flow free surface measurements were taken using a two-dimensional Acoustic Doppler Velocimeter (ADV) and an ultrasonic profiler, respectively. The experimental results can be summarized as follow: (i) the formation of well developed lateral shockwaves similar to those of oblique jumps were observed; (ii) the comparison of the experimental and theoretical data shows that the classic shockwave theory is sufficiently confirmed in the analyzed range of Reynolds number, taking into account the experimental errors and the difference between the theoretical and experimental assumptions; (iii) the transversal flow velocity profiles in the recirculating zone show a good agreement with the numerical simulations presented in literature in the case of a separated turbulent boundary layer over a flat plate. This conclusion enables us to confirm the hypothesis that the lateral shockwaves in the channel are the result of a boundary layer which, as observed, forms on the channel sidewalls.     

A closure approach with multi-scale mixing length,and its application to wind and shear stress profiles within the atmospheric boundary layer

A method to determine flow specific first-order closure for the turbulent flux of momentum in the atmospheric boundary layer (ABL) is presented. This is based on the premise that eddy viscosity is a flow rather than a fluid property, and the physically more realistic assumption that the transfer of momentum and other scalar quantities in a turbulent flow takes place by a large, but finite number of length scales, than the often used single length scale, the ‘mixing length’. The resulting eddy viscosity is flow specific and when applied to the study of the ABL, yields the vertical profiles of shear stress and mean wind velocity in good agreement with observations. The method may be extended to other types of turbulent flows, however it should be recognized that each type of flow may yield a different eddy viscosity profile. Using the derived eddy viscosity the paper presents simple analytical solutions of the ABL equations to determine observationally consistent wind speed and shear stress profiles in the ABL for a variety of practical applications including air pollution modelling.     

Experimental analysis of Froude number effect on air entrainment in the hydraulic jump

A hydraulic jump is characterized by strong energy dissipation and mixing, large-scale turbulence, air entrainment, waves, and spray. Despite recent pertinent studies, the interaction between air bubbles diffusion and momentum transfer is not completely understood. The objective of this paper is to present experimental results from new measurements performed in a rectangular horizontal flume with partially developed inflow conditions. The vertical distributions of the void fraction and the air bubbles count rate were recorded for inflow Froude number Fr ₁ in the range from 5.2 to 14.3. Rapid detrainment process was observed near the jump toe, whereas the structure of the air diffusion layer was clearly observed over longer distances. These new data were compared with previous data generally collected at lower Froude numbers. The comparison demonstrated that, at a fixed distance from the jump toe, the maximum void fraction C _max increases with the increasing Fr ₁. The vertical locations of the maximum void fraction and bubble count rate were consistent with previous studies. Finally, an empirical correlation between the upper boundary of the air diffusion layer and the distance from the impingement point was derived.     

A stochastic Lagrangian micromixing model for the dispersion of reactive scalars in turbulent flows: role of concentration fluctuations and improvements to the conserved scalar theory under non-homogeneous conditions

Several reaction schemes, based on the conserved scalar theory, are implemented within a stochastic Lagrangian micromixing model to simulate the dispersion of reactive scalars in turbulent flows. In particular, the formulation of the reaction-dominated limit (RDL) reaction scheme is here extended to improve the model performance under non-homogeneous conditions (NHRDL scheme). The validation of the stochastic model is obtained by comparison with the available measurements of reactive pollutant concentrations in a grid-generated turbulent flow. This test case describes the dispersion of two atmospheric reactant species (NO and O₃) and their reaction product (NO₂) in an unbounded turbulent flow. Model inter-comparisons are also assessed, by considering the results of state-of-the-art models for pollutant dispersion. The present validation shows that RDL reaction scheme provides a systematic overestimation (relative error of ca. 85% around the centreline) in computing the local reactant consumption/production rate, whereas the NHRDL scheme drastically reduces this gap (relative error lower than 5% around the centreline). In terms of NO₂ production (or reactant consumption), neglecting concentration fluctuations determines overestimations of the product mean of around 100% and a NO₂ local production of one order of magnitude higher than the reference simulation. In terms of standard deviations, the concentration fluctuations of both the passive and reactive scalars are generally of the same order of magnitude or up to 1 or 2 orders of magnitudes higher than the corresponding ensemble mean values, except for the background reactant close to the plume edges. The study highlights the importance of modelling pollutant reactions depending on the instantaneous instead of the mean concentrations of the reactants, thus quantifying the role of the turbulent fluctuations of concentration, in terms of scalar statistics (mean, standard deviation, intensity of fluctuations, skewness and kurtosis of concentration, segregation coefficient, simulated reaction rate). This stochastic particle method represents an efficient numerical technique to solve the convection–diffusion equation for reactive scalars and involves several application fields: micro-scale air quality (urban and street-canyon scales), accidental releases, impact of odours, water quality and fluid flow industrial processes (e.g. combustion).     

Mass Transport in Vegetated Shear Flows

Submerged aquatic vegetation has the potential to greatly improve water quality through the removal of nutrients, particulates and trace metals. The efficiency of this removal depends heavily upon the rate of vertical mixing, which dictates the timescale over which these constituents remain in the canopy. Continuous dye injection experiments were conducted in a flume with model vegetation to characterize vertical mass transport in vegetated shear flows. Through the absorbance–concentration relationship of the Beer–Lambert Law, digital imaging was used to provide high-resolution concentration profiles of the dye plumes. Vertical mass transport is dominated by the coherent vortices of the vegetated shear layers. This is highlighted by the strong periodicity of the transport and its simple characterization based on properties of the shear layer. For example, the vertical turbulent diffusivity is directly proportional to the shear and thickness of the layer. The turbulent diffusivity depends upon the size of the plume, such that the rate of plume growth is lower near the source. In the far-field, mass is mixed more than twice as rapidly as momentum. Finally, plume size is dictated predominantly by X, a dimensionless distance that scales upon the number of vortex rotations experienced by the plume.     

Estimation of far-field horizontal and vertical turbulent diffusion coefficients from the concentration field of a wastewater plume near the Akashi Strait

We carried out a field study of the plume discharged by a near-shore wastewater outfall near the Akashi Strait, Japan. Using an Acoustic Doppler Current Profiler and a tow-body CTD, we measured the near-surface salinity and temperature fields in the region throughout an M₂ tidal cycle. We filtered the data in T–S space to remove water masses other than the wastewater, and then used the adiabatic mixing assumption to calculate the concentration of wastewater in the far field of this plume. Averaging the T–S fields of repeated surveys over a time period during which the tidal regime did not change substantially, allowed comparison of the time-averaged plume with the analytical solution for a plume diffusing in both the horizontal and vertical dimensions. The resulting vertical turbulent diffusion coefficients agreed well with those resulting from Thorpe scales determined via a vertically-profiling CTD, as well as with the canonical value for open channel flow of D _z = 0.067hu _*. The corresponding horizontal turbulent diffusion coefficients, however, were two orders of magnitude larger than those typically observed in straight channels, and an order of magnitude larger than those observed in meandering rivers. This is likely a result of enhanced horizontal mixing due to barotropic eddies generated by the interaction of strong tidal flow with headlands and levees, as well as due to the time-varying nature of tidal flow, and baroclinic spreading of the buoyant wastewater plume.     

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.     

Turbulent velocity profile in fully-developed open channel flows

The determination of velocity profile in turbulent narrow open channels is a difficult task due to the significant effects of the anisotropic turbulence that involve the Prandtl’s second type of secondary flow occurring in the cross section. With these currents the maximum velocity appears below the free surface that is called dip phenomenon. The well-known logarithmic law describes the velocity distribution in the inner region of the turbulent boundary layer but it is not adapted to define the velocity profile in the outer region of narrow channels. This paper relies on an analysis of the Navier–Stokes equations and yields a new formulation of the vertical velocity profile in the center region of steady, fully developed turbulent flows in open channels. This formulation is able to predict time averaged primary velocity in the outer region of the turbulent boundary layer for both narrow and wide open channels. The proposed law is based on the knowledge of the aspect ratio and involves a parameter C_Ar depending on the position of the maximum velocity (ξ_dip). ξ_dip may be derived, either from measurements or from an empirical equation given in this paper. A wide range of longitudinal velocity profile data for narrow open channels has been used for validating the model. The agreement between the measured and the computed velocities is rather good, despite the simplification used.     

Characteristics of clustered particles in skimming flows on a stepped spillway

Air–water flows at hydraulic structures are commonly observed and called white waters. The free-surface aeration is characterised by some intense exchanges of air and water leading to complex air–water structures including some clustering. The number and properties of clusters may provide some measure of the level of particle-turbulence and particle–particle interactions in the high-velocity air–water flows. Herein a re-analysis of air–water clusters was applied to a highly aerated free-surface flow data set (Chanson and Carosi, Exp Fluids 42:385–401, 2007). A two-dimensional cluster analysis was introduced combining a longitudinal clustering criterion based on near-wake effect and a side-by-side particle detection method. The results highlighted a significant number of clustered particles in the high-velocity free-surface flows. The number of bubble/droplet clusters per second and the percentage of clustered particles were significantly larger using the two-dimensional cluster analysis than those derived from earlier longitudinal detection techniques only. A number of large cluster structures were further detected. The results illustrated the complex interactions between entrained air and turbulent structures in skimming flow on a stepped spillway, and the cluster detection method may apply to other highly aerated free-surface flows.     

Blockage of saline intrusions in restricted,two-layer exchange flows across a submerged sill obstruction

Results are presented from a series of large-scale experiments investigating the internal and near-bed dynamics of bi-directional stratified flows with a net-barotropic component across a submerged, trapezoidal, sill obstruction. High-resolution velocity and density profiles are obtained in the vicinity of the obstruction to observe internal-flow dynamics under a range of parametric forcing conditions (i.e. variable saline and fresh water volume fluxes; density differences; sill obstruction submergence depths). Detailed synoptic velocity fields are measured across the sill crest using 2D particle image velocimetry, while the density structure of the two-layer exchange flows is measured using micro-conductivity probes at several sill locations. These measurements are designed to aid qualitative and quantitative interpretation of the internal-flow processes associated with the lower saline intrusion layer blockage conditions, and indicate that the primary mechanism for this blockage is mass exchange from the saline intrusion layer due to significant interfacial mixing and entrainment under dominant, net-barotropic, flow conditions in the upper freshwater layer. This interfacial mixing is quantified by considering both the isopycnal separation of vertically-sorted density profiles across the sill, as well as calculation of corresponding Thorpe overturning length scales. Analysis of the synoptic velocity fields and density profiles also indicates that the net exchange flow conditions remain subcritical (G < 1) across the sill for all parametric conditions tested. An analytical two-layer exchange flow model is then developed to include frictional and entrainment effects, both of which are needed to account for turbulent stresses and saline entrainment into the upper freshwater layer. The experimental results are used to validate two key model parameters: (1) the internal-flow head loss associated with boundary friction and interfacial shear; and (2) the mass exchange from the lower saline layer into the upper fresh layer due to entrainment.     

Experimental study of recirculating flows generated by lateral shock waves in very large channels

Due to the lack of data on hydraulic-jump dynamics in very large channels, the present paper describes the main characteristics of the velocity field and turbulence in a large rectangular channel with a width of 4 m. Although a hydraulic jump is always treated as a wave that is transversal to the channel wall, in the case of this study it has a trapezoidal front shape, first starting from a point at the sidewalls and then developing downstream in an oblique manner, finally giving rise to a trapezoidal shape. The oblique wave front may be regarded as a lateral shockwave that arises from a perturbation at a certain point of the lateral wall and travels obliquely toward the centreline of the channel. The experimental work was carried out at the Coastal Engineering Laboratory of the Water Engineering and Chemistry Department of the Technical University of Bari (Italy). In addition to the hydraulic jump formation, a large recirculating flow zone starts to develop from the separating point of the lateral shock wave and a separate boundary layer occurs. Intensive measurements of the streamwise and spanwise flow velocity components along one-half width of the channel were taken using a bidimensional Acoustic Doppler Velocimeter (ADV). The water surface elevation was obtained by means of an ultrasonic profiler. Velocity vectors, transversal velocity profiles, turbulence intensities and Reynolds shear stresses were all investigated. The experimental results of the separated boundary layer were compared with numerical predictions and related work presented in literature and showed good agreement. The transversal velocity profiles indicated the presence of adverse pressure gradient zones and the law of the wall appears to govern the region around the separated boundary layer.     

Spatial evolution of coherent motions in finite-length vegetation patch flow

A number of experimental studies on submerged canopy flows have focused on fully-developed flow and turbulent characteristics. In many natural rivers, however, aquatic vegetation occurs in patches of finite length. In such vegetated flows, the shear layer is not formed at the upstream edge of the vegetation patch and coherent motions develop downstream. Therefore, more work is neededz to reveal the development process for large-scale coherent structures within vegetation patches. For this work, we considered the effect of a limited length vegetation patch. Turbulence measurements were intensively conducted in open-channel flows with submerged vegetation using Particle Image Velocimetry (PIV). To examine the transition from boundary-layer flow upstream of the vegetation patch to a mixing-layer-type flow within the patch, velocity profiles were measured at 33 positions in a longitudinal direction. A phenomenological model for the development process in the vegetation flow was developed. The model decomposed the entire flow region into four zones. The four zones are the following: (i) the smooth bed zone, (ii) the diverging flow zone, (iii) the developing zone and (iv) the fully-developed zone. The PIV data also confirmed the efficiency of the mixing-layer analogy and provided insight into the spatial evolution of coherent motions.     

Double-average mean flow and local turbulence intensity profiles from PIV measurements for an open channel flow with rigid vegetation

This contribution presents particle image velocimetry measurements for an open channel stationary uniform and fully developed flow of water over a horizontal flat bed of uniform glass beads in presence of a staggered array of vertical cylindrical stems. The main objective was to explore and quantify the influence of the stems-to-flow relative submergence, h _v /h, over the mean flow and local turbulence intensities. A comparison with measurements for the non-vegetated flow over the same granular bed is presented. Results indicate a remarkable influence of h _v /h over the whole flow field. The time-average mean flow presents a strong spatial variation in the layer of the flow occupied by the stems. The local velocity fluctuations are strongly affected by the presence of the stems, with regions in between the stems where they reach peaks that are several times larger than those encountered in the flow in absence of vegetation. The turbulence intensity profiles are noticeably different when compared to those measured in the non-vegetated flow conditions. From previous works it was possible to derive an equation for the mean velocity, U _v , of the flow through the vegetated layer of height h _v . The prediction of this equation is in good agreement with the uniform value for the double-average longitudinal velocity profile in this layer. A final brief discussion about the possible impact of these vegetated-flow features on the sediment transport is presented.     

Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows. Part I: Single Round Jet

The mechanics of buoyant jet flows issuing with a general three-dimensional geometry into an unbounded ambient environment with uniform density or stable density stratification and under stagnant or steady sheared current conditions is investigated. An integral model is formulated for the conservation of mass, momentum, buoyancy and scalar quantities in the turbulent jet flow. The model employs an entrainment closure approach that distinguishes between the separate contributions of transverse shear (leading to jet, plume, or wake internal flow dynamics) and of azimuthal shear mechanisms (leading to advected momentum puff or thermal flow dynamics), respectively. Furthermore, it contains a quadratic law turbulent drag force mechanism as suggested by a number of recent detailed experimental investigations on the dynamics of transverse jets into crossflow. The model is validated in several stages: First, comparison with basic experimental data for the five asymptotic, self-similar stages of buoyant jet flows, i.e., the pure jet, the pure plume, the pure wake, the advected line puff, and the advected line thermal, support the choice and magnitude of the turbulent closure coefficients contained in the entrainment formulation. Second, comparison with many types of non-equilibrium flows support the proposed transition function within the entrainment relationship, and also the role of the drag force in the jet deflection dynamics. Third, a number of spatial limits of applicability have been proposed beyond which the integral model necessarily becomes invalid due to its parabolic formulation. These conditions, often related to the breakdown of the boundary layer nature of the flow, describe features such as terminal layer formation in stratification, upstream penetration in jets opposing a current, or transition to passive diffusion in a turbulent ambient shear flow. Based on all these comparisons, that include parameters such as trajectories, centerline velocities, concentrations and dilutions, the model appears to provide an accurate and reliable representation of buoyant jet physics under highly general flow conditions.     

Integral Model for Turbulent Buoyant Jets in Unbounded Stratified Flows Part 2: Plane Jet Dynamics Resulting from Multiport Diffuser Jets

An integral model for the plane buoyant jet dynamics resulting from the interaction of multiple buoyant jet effluxes spaced along a diffuser line is considered as an extension of the round jet formulation that was proposed in Part I. The receiving fluid is given by an unbounded ambient environment with uniform density or stable density stratification and under stagnant or steady sheared current conditions. Applications for this situation are primarily for submerged multiport diffusers for discharges of liquid effluents into ambient water bodies, but also for multiple cooling tower plumes and building air-conditioning. The CorJet model formulation describes the conservation of mass, momentum, buoyancy and scalar quantities in the turbulent jet flow in the plane jet geometry. It employs an entrainment closure approach that distinguishes between the separate contributions of transverse shear and of internal instability mechanisms, and contains a quadratic law turbulent pressure force mechanism. But the model formulation also includes several significant three-dimensional effects that distinguish actual diffuser installations in the water environment. These relate to local merging processes from the individual multiple jets, to overall finite length effects affecting the plume geometry, and to bottom proximity effects given by a “leakage factor” that measures the combined affect of port height and spacing in allowing the ambient flow to pass through the diffuser line in order to provide sufficient entrainment flow for the mixing downstream from the diffuser. The model is validated in several stages: First, comparison with experimental data for the asymptotic, self-similar stages of plane buoyant jet flows, i.e. the plane pure jet, the pure plume, the pure wake, the advected line puff, and the advected line thermal, support the choice of the turbulent closure coefficients contained in the entrainment formulation. Second, comparison with data for many types of non-equilibrium flows with a plane geometry support the proposed functional form of the entrainment relationship, and also the role of the pressure force in the jet deflection dynamics. Third, the observed behavior of the merging process from different types of multiport diffuser discharges in both stagnant and flowing ambient conditions and with stratification appears well predicted with the CorJet formulation. Fourth, a number of spatial limits of applicability, relating to terminal layer formation in stratification or transition to passive diffusion in a turbulent ambient shear flow, have been proposed. In sum, the CorJet integral model appears to provide a mechanistically sound, accurate and reliable representation of complex buoyant jet mixing processes, provided the condition of an unbounded receiving fluid is satisfied.