Wide Open Channel Research Articles

Origination of Cellular Secondary Cells in Wide Open-Channel Turbulent Unsteady Flows over Spanwise-Oriented Alternate Rough and Smooth Bed Strips

Origination of Cellular Secondary Cells in Wide Open-Channel Turbulent Unsteady Flows over Spanwise-Oriented Alternate Rough and Smooth Bed Strips

Trans-9,10-Dihydro-9,10-ethanoanthracene-11,12-dicarboxylic Acid: Remarkable Host Selectivities in the Presence of Mixtures of Anisole and Bromoanisole Isomers

In this work, trans-9,10-dihydro-9,10-ethanoanthracene-11,12-dicarboxylic acid (DED), a host compound that formed complexes with anisole (ANI), 2-bromoanisole (2-BA), and 4-bromoanisole (4-BA), was revealed to possess a remarkable affinity (93.1–98.6%) for 4-BA when recrystallized from any mixtures containing this guest species, be these binary (ANI/4-BA, 2-BA/4-BA, or 3-BA/4-BA), ternary (ANI/2-BA/4-BA, ANI/3-BA/4-BA, or 2-BA/3-BA/4-BA), or quaternary (ANI/2-BA/3-BA/4-BA) in nature. Furthermore, this striking host selectivity persisted even in binary solutions containing low concentrations of 4-BA (18–25%); near-complete selectivities in favor of 4-BA were observed here too. Single crystal X-ray diffraction analyses showed that the favored guest species, 4-BA, was the only one to experience short intermolecular contacts, through its phenyl ring, with the host molecule’s carbonyl oxygen atom and the host hydrogen atom on the carbon bearing the second carboxylic acid group. Furthermore, ANI and 2-BA were accommodated in wide open channels, while the guest accommodation type for 4-BA was significantly more constricted. Thermal analyses confirmed that 2(DED)·3(ANI) and DED·2-BA, having the disfavored guests in open channels, were unstable at room temperature, while the preferred guest species, 4-BA, in its decidedly constricted channels, was only released at significantly higher temperatures (116.8 °C). Therefore, by employing host–guest chemistry strategies, DED has the ability to purify and separate any of these 4-BA-containing mixtures, a distinct advantage over fractional distillations which are cumbersome owing to physical property similarities of 2-, 3-, and 4-BA.

Selected Tricyclic Fused Systems: Host Behavior in the Presence of Mixed Xylenes and Ethylbenzene

Two tricyclic fused host systems, namely, N,N′-bis(5-phenyl-5-dibenzo[a,d]cycloheptenyl)ethylenediamine H1 and N,N′-bis(5-phenyl-10,11-dihydro-5-dibenzo[a,d]cycloheptenyl)ethylenediamine H2, were investigated for their selectivity behavior in xylene isomer (o-Xy, m-Xy, p-Xy) and ethylbenzene (EB) mixtures. H1 only formed a complex with p-Xy, and H2 only with o-Xy, in single solvent recrystallization experiments. When H1 and H2 were recrystallized from equimolar mixtures of every combination of the four organic solvents, contrasting host selectivities were observed: H1 favored p-Xy and H2 favored o-Xy. High selectivities for p-Xy (87.2–88.7%) were noted in equimolar binary experiments with H1, while H2 furnished crystals with 91.3% o-Xy from an o-Xy/p-Xy mixture. Remarkably, only the apohost was recovered when p-Xy or o-Xy (for H1 or H2) was absent from the mixtures. Constructed selectivity profiles affirmed observations from these equimolar experiments. Thermal analyses demonstrated that H1·p-Xy was significantly less stable than H2·o-Xy, and SCXRD analyses explained these observations: p-Xy, in its complex with H1, was accommodated in wide open channels and did not experience any noncovalent interactions with the host molecule; on the other hand, o-Xy in H2 was bound tightly in near-discrete cages and, furthermore, interacted with H2. Both host compounds demonstrated the ability to separate various of the mixtures reported in this work.

Analysis and validation of mathematical models of secondary velocities along vertical and transverse directions in wide open-channel turbulent flows

Cellular secondary flows are inevitably present in turbulent flows through ducts, natural or artificial channels, and compound channels. Secondary currents significantly modify the characteristics of turbulent quantities, the pattern of primary flow velocity by causing dip-phenomenon. To understand the detailed mechanism and hidden cause, modelling of secondary flow velocities is crucial. In this study, proper mathematical models of secondary flow velocities along vertical and transverse directions are proposed for steady and uniform turbulent flow through wide open channels with equal smooth and rough bed strips. Starting from the continuity and the Reynolds averaged Navier–Stokes equations, governing equation for secondary velocity is derived first and then using appropriate boundary conditions (no-slip boundary conditions at channel bottom and free surface, and maximum vertical velocity in magnitude at the interface of two cellular secondary cells and at mid-depth of the channel. All these conditions are consistent with several experimental observations). A new model of the streamwise Reynolds shear stress is proposed for the entire cross-sectional plane and using it, the analytical solutions are obtained. Proposed models include the effects of viscosity of the fluid and the eddy viscosity model of turbulence. All suggested models are validated with existing experimental data in rectangular open-channel flows, compound open channel flows, and duct flows, and satisfactory results are obtained. Furthermore, models are also compared with existing empirical models from literature to show the effectiveness and superiority of proposed models. Apart from these, the obtained results from this study are used to investigate the effects of vertical and transverse secondary flow velocities on the settling velocity vector in a cross-sectional plane. Effective alternative models for the settling velocity vector are suggested. The model of settling velocity vector is also compared with the existing model. Finally, all results are justified from physical viewpoints.

Mathematical modelling of streamwise velocity profile in open channels using Tsallis entropy

This study derived the vertical distribution of streamwise velocity in wide-open channels by maximizing Tsallis entropy, in accordance with the maximum entropy principle, subject to the total probability rule, and the constraints based on the conservation of mass, momentum, and energy. Entropy maximizing leads to a highly nonlinear differential equation for velocity, which was transformed into a relatively weaker nonlinear equation and then solved analytically using a non-perturbation approach that yielded a series solution. The convergence of the series solution was proved using both theoretical and numerical procedures. For the velocity profile assessment, we calculated the Lagrange multipliers and the entropy index by solving a system of nonlinear equations using the Gauss-Newton method after approximating the constraint integrals using Gauss-Legendre quadrature rule. The derived velocity profile was validated for some selected sets of laboratory and field data and was compared with the existing velocity profiles based on Rényi, Tsallis, and Shannon (with additional constraints) entropies. We found that the incorporation of additional constraints and the effect of the entropy index improved the velocity profile compared to the existing Tsallis and Rényi entropy-based velocity equations. Further, it was observed that the proposed model and the Shannon entropy-based model with additional constraints behaved the same for most of the data sets considered, as the corresponding values of the entropy index were close to 1, which is in agreement with the theoretical consideration. The methodology reported in this study can also be employed for addressing other open channel flow problems, such as sediment concentration and shear stress distribution.

Evaluating Different Methods for Determining the Velocity-Dip Position over the Entire Cross Section and at the Centerline of a Rectangular Open Channel.

The velocity profile of an open channel is an important research topic in the context of open channel hydraulics; in particular, the velocity-dip position has drawn the attention of hydraulic scientists. In this study, analytical expressions for the velocity-dip position over the entire cross section and at the centerline of a rectangular open channel are derived by adopting probability methods based on the Tsallis and general index entropy theories. Two kinds of derived entropy-based expressions have the same mathematical form as a function of the lateral distance from the sidewall of the channel or of the aspect ratio of the channel. Furthermore, for the velocity-dip position over the entire cross section of the rectangular open channel, the derived expressions are compared with each other, as well as with two existing deterministic models and the existing Shannon entropy-based expression, using fifteen experimental datasets from the literature. An error analysis shows that the model of Yang et al. and the Tsallis entropy-based expression predict the lateral distribution of the velocity-dip position better than the other proposed models. For the velocity-dip position at the centerline of the rectangular open channel, six existing conventional models, the derived Tsallis and general index entropy-based expressions, and the existing Shannon entropy-based models are tested against twenty-one experimental datasets from the literature. The results show that the model of Kundu and the Shannon entropy-based expression have superior prediction accuracy with respect to experimental data compared with other models. With the exception of these models, the Tsallis entropy-based expression has the highest correlation coefficient value and the lowest root mean square error value for experimental data among the other models. This study indicates that the Tsallis entropy could be a good addition to existing deterministic models for predicting the lateral distribution of the velocity-dip position of rectangular open channel flow. This work also shows the potential of entropy-based expressions, the Shannon entropy and the Tsallis entropy in particular, to predict the velocity-dip position at the centerline of both narrow and wide rectangular open channels.

Re-examining log law velocity profile in smooth open channel flows

A comprehensive investigation of velocity distribution is presented, and the log law is re-examined using experimental data from a smooth uniform open channel flow. It is widely reported that the coefficients of the log law in channel flows deviate from those obtained from circular pipe flows by Nikuradse (Laws of flow in rough pipes, 1933), but the mechanism is not clear and no theoretical formulae are available to express these deviations. A Laser Doppler Velocimetry system was used to measure velocity profiles at the centre of the channel. The data obtained support previous conclusions that the additive constant B of the log law in channel flows can no longer be considered to be 5.5. Interestingly, the experimental data also support Tracy and Lester’s discovery that the shear velocities on both sides of the log law are different. Better agreement can be achieved if the global shear velocity (U*1) is used to normalize the measured velocity, and the local shear velocity (U*2) is used to normalize the distance from the wall. Other researchers’ data in the literature also validates this new relationship. Based on this new relationship, a theoretical value of B is obtained, which agrees well with the observed B, thus a new form of log law for channel flow is suggested. The new relationship developed was verified with present experimental and past literature data suggesting its universality irrespective of wide or narrow open channels, or subcritical or super critical flow conditions. By using the developed relationship the large scatter associated with the additive constant B associated with the log law has been explained. It is found that the additive constant B in the log law is a function of channel aspect ratio. The developed relationship for B is validated from a wide range of data from the literature to confirm its universality.

Information Entropy Theory Applied to the Dip-Phenomenon Analysis in Open Channel Flows.

The knowledge of the fluid discharge in free surface flows requires a great number of velocity measurements along the whole cross-section, taking up a large amount of time, using expensive equipment, and employing specialized labor. To overcome these obstacles, various models have been developed thus far that show how to estimate the mean velocity through the maximum velocity. In three-dimensional open channels, the maximum velocity can be located below the free surface because of the presence of secondary flows mainly originating by the sidewalls, an occurrence known as dip-phenomenon. In this condition, predicting the maximum velocity position is quite difficult and has always represented a challenge to most hydraulic engineers and researchers. In the present study, a mathematical model derived from the information entropy theory is proposed to evaluate the velocity-dip-position over the entire cross-section of both wide and narrow open channels, thus overcoming the limitations of the existing methods. Large literature measurement sets, collected in uniform and non-uniform flows, were used to test the validity of the model, showing good agreement with the experimental data and providing an accurate estimation of the dip-position.

Application of relative entropy theory to streamwise velocity profile in open-channel flow: effect of prior probability distributions

Applying the concept of relative or cross-entropy and the principle of minimum cross-entropy, this study derives the velocity distribution in a wide open channel. Previous studies have employed Shannon entropy and the principle of maximum entropy to derive distributions of various flow variables, including velocity. Relative entropy is a generalized form of entropy and can specialize into Shannon entropy if the prior probability distribution is taken to be a uniform distribution. The prior distribution is often formulated, based on intuition, experience, or thought experiment. When deriving the velocity distribution in wide open channels, this study assumes four prior probability distributions and analyzes the effect of these assumed priors. It is found that a normal-type and a gamma-type prior can significantly influence the velocity profile, especially near the channel bed, and their prediction accuracies are superior to the previously obtained velocity distribution based on Shannon entropy. Furthermore, closed-form explicit analytical solutions are obtained for the nonlinear differential equations that arise while incorporating these priors. Experimental and field data are used to verify the derived velocity distributions, and an error analysis is carried out to evaluate their performance.

Effect of aspect ratio on developing and developed narrow open channel flow with rough bed

This study attempts to unravel the effect of aspect ratio on the turbulence characteristics in developing and fully developed narrow open channel flows. In this regard, experiments were conducted in a rough bed open channel flow. Instantaneous 3D velocities were acquired using an acoustic Doppler velocimeter at various locations along the centerline of the flume. The variables of interest include the mean components of the flow velocity, turbulence intensity, wall normal Reynolds shear stress, correlation coefficient, turbulence kinetic energy, and anisotropy. A new correlation between the equivalent roughness and velocity shift from the smooth wall logarithmic equation as a function of aspect ratio is proposed. Aspect ratio was found to influence the velocity characteristics throughout the depth in the developing flow region, while the effects are confined to the outer layer for fully developed flows. New equations to describe the variation of turbulence intensities and turbulent kinetic energy are proposed for narrow open channel flows. Reynolds stress anisotropy analysis reveals that level of anisotropy in narrow open channel flow is less than in wide open channel flows. Finally, a linear regression model is proposed to predict flow development length in narrow open channel flows with a rough bed.

Theoretical analysis of pollutant mixing zone considering lateral distribution of flow velocity and diffusion coefficient.

Theoretical formulae have shown significant advantages in describing the characteristic geometric scales of the pollutant mixing zone (PMZ) formed by offshore pollutant discharged by a single general form. They, however, fail to predict the influence of the lateral inhomogeneity of the river flow because constant flow velocity and the lateral diffusion coefficient are assumed during the derivation. The realistic flow velocity in a river is fitted by an exponential law in this study and the lateral diffusion coefficient is proposed to have the same form. Similar idea has been used in previous studies on the vertical dispersion of scalar in the lower atmosphere. Pollutant discharged from a steady onshore point source into a wide straight open channel is examined to characterize the concentration taking into consideration of these lateral variations. Theoretical formulae describing the maximum length, maximum width and its corresponding longitudinal position, as well as the area of the PMZ are derived. A non-dimensional standard curve equation for the isoconcentration boundary of PMZ is also obtained. The results show that the shape of the dimensionless standard curve of PMZ depends only on the exponential constants in the exponential laws. The exponential profiles that fit the near-shore velocity give good prediction, while the ones that match the entire lateral range up to the center of the river underpredict the PMZ significantly. These findings are of great importance for practitioners to characterize the geometry of the PMZ in rivers and for water quality modeling.

Comparative study of 1D entropy-based and conventional deterministic velocity distribution equations for open channel flows

Velocity distributions for open channel flows have been investigated using deterministic and probabilistic approaches. It is well known that the vertical velocity profiles in wide open channels (i.e. aspect ratio width/depth > 5) can be approximated by logarithmic velocity laws and power laws. Recently the entropy concept in the forms of Shannon entropy and Tsallis entropy has been employed to estimate velocity distributions in open channels with different aspect ratios. The accuracy of conventional velocity equations is highly dependent on their parameters that can only be estimated by empirical or semi-empirical analytical relations which requires either a good knowledge of velocity field and/or physical properties of the channel, such as topographic conditions, sedimentation conditions and boundary roughness. In contrast, the entropy based velocity distributions derived based on the least-biased probability density function (PDF) by treating time-averaged velocities as random variables are resilient regardless of the flow and channel conditions. However, a comparison of the velocity profiles computed using deterministic approaches and probabilistic approaches has not been rigorously conducted. Furthermore, the accuracy and reliability of associated velocity distribution equations have not been tested thoroughly using data sets collected using advanced techniques. This paper presents a comprehensive and comparative study to analyze the distinctions and linkages between four commonly used velocity laws and two entropy-based velocity distributions theoretically and quantitatively using selective laboratory and field measurements available in the literature, considering typical sedimentation and channel hydraulic conditions. Amongst all, Tsallis entropy based velocity distribution developed from a generalized form of informational entropy exhibits universal validity to sediment-laden flows in wide alluvial open channels, and is found to be superior to others to predict velocity profiles in large waterways with unmanageable rough beds.

An Entropic Model for the Assessment of Streamwise Velocity Dip in Wide Open Channels.

The three-dimensional structure of river flow and the presence of secondary currents, mainly near walls, often cause the maximum cross-sectional velocity to occur below the free surface, which is known as the “dip” phenomenon. The present study proposes a theoretical model derived from the entropy theory to predict the velocity dip position along with the corresponding velocity value. Field data, collected at three ungauged sections located along the Alzette river in the Grand Duchy of Luxembourg and at three gauged sections located along three large rivers in Basilicata (southern Italy), were used to test its validity. The results show that the model is in good agreement with the experimental measurements and, when compared with other models documented in the literature, yields the least percentage error.

Effect of aspect ratio on higher order moments of velocity fluctuations in hydraulically rough open channel flow

The effect of aspect ratio on the higher order statistics of velocity fluctuations in a hydraulically rough turbulent open channel flow is investigated. In this regard, an experiment was conducted in a rough bed narrow open channel flow of aspect ratio equal to three and the instantaneous flow velocities were measured using a Nortek Vectrino+ Acoustic Doppler Velocimeter. To understand the effect of aspect ratio, the results obtained from the present study are compared with the literature data of approximately same Reynolds number and bed roughness in a wide open channel flow for turbulence intensities and higher order statistics of velocity fluctuations. Comparison of turbulence intensities between Narrow OCF and Wide OCF shows occurrence of higher streamwise and vertical turbulence intensities in the outer region of Narrow OCF. The results of third order moments of velocity fluctuations are sensitive to aspect ratio in the outer region.

Reinvestigation on mixing length in an open channel turbulent flow

The present study proposes a model on vertical distribution of streamwise velocity in an open channel turbulent flow through a newly proposed mixing length, which is derived for both clear water and sediment-laden turbulent flows. The analysis is based on a theoretical consideration which explores the effect of density stratification on the streamwise velocity profile. The derivation of mixing length makes use of the diffusion equation where both the sediment diffusivity and momentum diffusivity are taken as a function of height from the channel bed. The damping factor present in the mixing length of sediment-fluid mixture contains velocity and concentration gradients. This factor is capable of describing the dip-phenomenon of velocity distribution. From the existing experimental data of velocity, the mixing length data are calculated. The pattern shows that mixing length increases from bed to the dip-position, having a larger value at dip-position and then decreases up to the water surface with a zero value thereat. The present model agrees well with these data sets and this behavior cannot be described by any other existing model. Finally, the proposed mixing length model is applied to find the velocity distribution in wide and narrow open channels. The derived velocity distribution is compared with laboratory channel data of velocity, and the comparison shows good agreement.

Prediction of velocity-dip-position over entire cross section of open channel flows using entropy theory

An analytical model to predict the velocity-dip-position is presented in this study. Unlike the previous studies where empirical or semiempirical models were suggested, in this study the model is derived by using entropy theory. Using the principle of maximum entropy, the model for dip-position is derived by maximizing the Shannon entropy function after assuming dimensionless dip-position as a random variable. No estimation of empirical parameter is required for calculating dip-position from the proposed model. The model is capable of predicting the velocity-dip-position at any section from sidewall region (along lateral direction) of an open channel with any aspect ratio. The ratio of mean to maximum value of dip-position is analyzed from data, and it is found that ratio is almost constant for narrow open channels and it increases with aspect ratio for wide open channels. A relation is also proposed to predict this ratio in case of wide open channels. The developed model of velocity-dip-position is tested with existing experimental data for a wide range of aspect ratio and is also compared with other empirical models. The present model shows good agreement with the observed data and is comparable with the existing models.

Estimating bed shear stress from remotely measured surface turbulent dissipation fields in open channel flows

AbstractSynoptic information on bed shear stress is necessary in predicting the transport of sediments and environmental contaminants in rivers and open channels. Existing methods of estimating bed shear stress typically involve measuring vertical profiles of streamwise velocity or Reynolds stress and taking advantage of the logarithmic or the constant stress region, respectively, to determine friction velocity and subsequently, bed shear stress. While effective, these methods yield local measurements of bed shear stress only. Direct measurements of bed shear stress can also be obtained through measurements with a drag plate. However, this method yields average shear stress information over the area of the plate and is impractical for large‐scale implementation in the field.Here we present a method capable of providing continuous synoptic measurements of bed shear stress over a large field‐of‐view. A series of Large‐Scale Particle Image Velocimetry (LSPIV) and Acoustic Doppler Velocimetry (ADV) measurements were made in a variety of flows generated in a wide‐open channel facility. Turbulent dissipation is calculated on the free surface from the 2‐D LSPIV results and is correlated with near‐surface ADV measurements of turbulent dissipation in the water column. The ADV results are consistent with the Nezu (1977) established relationship for the vertical variation of turbulent dissipation in the water column. Knowledge of the correlation between free‐surface and near‐surface dissipation values coupled with Nezu's (1977) relationship allow a robust and accurate estimate of friction velocity to be made and subsequently, shear stress at the bed can be estimated.

Synthesis and Electrochemical Properties of Nanocuboid and Nanofiber WO3

The present research work focused on the one step hydrothermal synthesis of monoclinic nanocuboid and hexagonal nanofiber WO3 from common precursors and different structure directing reagents for high performance electrochromism. The characteristic phase and bonding behaviour depicts the probable intercalation space difference between these nanostructured particles. Electron – proton interaction mechanism of both WO3 coated transparent indium doped tin oxide (ITO) electrode depends on the concentration of acid electrolyte solution and electrochemical scan rate. A correlation between the crystal structures and morphology is evaluated for the fabricated films that exhibits enhanced electrochromic performance. Nanofiber WO3 exhibits high electrochromic reversibility of 78.94%, fast switching speeds of 6.02 s and 5.5 s for coloration and bleaching, respectively and high coloration efficiency of 54.09 cm2 C−1 in comparison to the monoclinic WO3 nanocuboid because of specious intercalation zone and wide open channels in hexagonal crystal structure, and interactive fiber morphology.

On the stability of gradually varying mud-flows in open channels

In the present paper the stability of gradually varying mud-flows in wide open channel is analyzed. The governing equations derive from a Saint–Venant-like approach with a Herschel & Bulkley model for the fluid rheology. In order to define the stability of a given initial accelerating or decelerating flow depth profile, the spatial evolution of a wavefront is analyzed. A stability criterion based on the linearized flow model is introduced, showing that the streamwise non-uniformity of the flow substantially influences the stability limits. A positive flow depth slope induces a stabilization while a negative one determines a destabilizing effect. The dependency of the linear stability on the values of rheological parameters is deeply discussed. A non-linear analysis of the wavefront propagation, along with the full non-linear solution of the problem, has been carried out to confirm the role played by the initial profile on the flow stability. The results of the presented study may be useful for engineering applications in order to prevent or to inhibit roll-waves formation in subcritical flows of a Herschel & Bulkley fluid.

Turbulence Characteristics in Supercritical Open Channel Flows: Effects of Froude Number and Aspect Ratio

AbstractAn experimental investigation of supercritical uniform and gradually varied open channel flows is presented for a wide range of Froude numbers and flume width-to-flow depth aspect ratios. The instantaneous streamwise and vertical flow velocities were measured in a laboratory flume over the entire width using a two dimensional–laser Doppler anemometry (2D-LDA) system to determine turbulence intensities, and bed and Reynolds shear stresses. The mean velocity patterns show undulation across the flume, indicating the presence of counterrotating secondary current cells. These currents redistribute turbulence intensities and bed and Reynolds shear stresses across the flume. For aspect ratios ≤ 4−5, i.e., narrow open channel flow, the velocity-dip phenomenon is identified both in the streamwise velocity and the Reynolds shear stress distributions. For high aspect ratios, i.e., wide open channel flow, the strength of secondary currents diminish toward the flume center, resulting in a 2D flow farther away ...