Direct integration of gradually varied flow equation in parabolic channels

The direct integration method is used to compute water surface profiles of gradually varied flow (GVF) in a prismatic open channel. No closed-form solution is available for the GVF equation in the case of general parabolic channels. Open channels with parabolic cross-sections are often a quite good approximation of the real geometry of natural rivers. Technology is also available for constructing this shape of channels. In the present study, by applying the Manning formula, a semi-analytical solution to compute the length of the gradually-varied-flow profile for parabolic channels is developed. The proposed solution uses a single step for the computation of water surface profiles and, as such, provides an accurate and yet simple way to compute GVF flow profiles; thus, it should be of interest to practitioners in the hydraulic engineering community.     

Analytical solution of gradually varied flow equation in circular channels using variable Manning coefficient

The direct integration method is used to solve the equation of gradually varied flow (GVF) in open prismatic channels. The GVF is a non-uniform flow with gradual changes in flow depth. In circular channels, Manning roughness coefficient under the partially full flow condition varies with the flow depth, and thus a variable Manning coefficient should be used to calculate the water surface profile. It is accepted that the Manning coefficient varies with flow depth in accordance with Camp’s curve. Given that circular channels have an important application in sewer systems, this research presents a semi-analytical approach for establishing the gradually varied flow profiles in circular channels through application of a variable Manning coefficient. For this, using two approximation expressions, the integrand of the GVF equation is expanded into a finite set of partial fractions and then every term of them is integrated separately. The proposed semi-analytical solution uses a single step for the computation of water surface profiles and provides an accurate and simple way to compute GVF flow profiles.     

Prediction of roughness coefficient of a meandering open channel flow using Neuro-Fuzzy Inference System

Almost all the natural water resource channels meander. Accurate estimation of discharge capacity in a meander open channel is important from river engineering point of view. It helps the practitioners to provide essential information regarding flood mitigation, construction of hydraulic structures and prediction of sediment loads so as to plan for effective preventive measures. Reliable estimation of discharge capacity of a natural channel depends on selection of proper value of roughness in terms of Manning’s n. Evaluation of Manning’s n for a meandering channel is a complex procedure because of its dependence on many geometrical, hydraulic and surface parameters of the channel. Experimental investigation concerning the variation of roughness coefficient of meandering channels with flow depth, aspect ratio, slope and sinuosity are presented in this paper. An effort has been made to predict the roughness co-efficient of a meandering channel based on ANFIS. The results are compared with well established methods available in the literature. Statistical error analysis is also carried out to know the degree of accuracy of the models. Finally the present model is found to give better results as compared to others. It is concluded that, in practice ANFIS model can be used as a suitable and effective method to predict the non-linear relationship between roughness coefficient and the non-dimensional factors affecting it.     

Free overall in circular channels with flat base: a method of open channel flow measurement

A free overfall at the end of an open channel offers a simple means of measuring flow discharge. In this paper, two methods are presented for the computation of end-depth and discharge of a free overfall from smooth circular channels with flat base. Firstly, applying the momentum equation based on the Boussinesq approximation, the flow upstream of a free overfall is theoretically analyzed to calculate the end-depth-ratio (EDR). This approach eliminates the need of an experimentally determined pressure coefficient. In subcritical flows, the EDR is related to the critical-depth that occurs far upstream. In supercritical flows, the Manning equation is used to express the end-depth as a function of streamwise slope of the channel. Methods to estimate discharge from the end-depth in subcritical and supercritical flows are presented. The upstream flow profiles of a free overfall are computed using the streamline curvature at free surface. Secondly, an alternate method for analyzing free overfall from circular channels with flat base is also presented, where a free overfall in a circular channel with flat base is simulated by the flow over a sharp-crested weir to calculate the EDR. The comparisons of the computed results with the experimental data are satisfactory for subcritical flow and acceptable for supercritical flow.     

Sensitivity analysis of the factors affecting the discharge capacity of side weirs in trapezoidal channels using extreme learning machines

Side weirs are installed on the side walls of main channels to control and regulate flow. In this study, sensitivity analysis is planned using Extreme Learning Machines (ELM) to recognize the factors affecting the discharge coefficient in trapezoidal channels. A total of 31 models with 1 to 5 parameters are developed. The input parameters are ratio of side weir length to trapezoidal channel bottom width (L/b), Froude number (Fr), ratio of side weir length to flow depth upstream of the side weir (L/y₁), ratio of flow depth upstream of the side weir to the main channel bottom width (y₁/b) and trapezoid channel side wall slope (m). Among the models with one input parameter, the model including Froude number modeled the discharge coefficient more accurately (MAPE=4.118, R²=0.835). Between models with two input parameters, the model using Fr and L/b produced MAPE and R² values of 2.607 and 0.913 respectively. Moreover, among the models with four input parameters, the model containing Fr, L/b, L/y₁ and y₁/b was the most accurate (MAPE=2.916, R²=0.925).     

Approximate method of determining the optimum cross section of microhannel heat sink

Microchannels are at the forefront of today’s cooling technologies. They are widely being considered for cooling of electronic devices and in micro heat exchanger systems due to their ease of manufacture. One issue which arises in the use of microchannels is related to the small length scale of the channel or channel cross-section. In this work, the maximum heat transfer and the optimum geometry for a given pressure loss have been calculated for forced convective heat transfer in microchannels of various cross-section having finite volume for laminar flow conditions. Solutions are presented for 10 different channel cross sections: parallel plate channel, circular duct, rectangular channel, elliptical duct, polygonal duct, equilateral triangular duct, isosceles triangular duct, right triangular duct, rhombic duct and trapezoidal duct. The model is only a function of the Prandtl number and the geometrical parameters of the cross-section, i.e., area and perimeter. This solution is performed with two exact and approximate methods. Finally, in addition to comparison and discussion of these two methods, validation of the relationship is provided using results from the open literature.     

Comparison of performances of the various shapes of asymmetric fins

A comparison between the heat loss of the asymmetric triangular fin and the asymmetric trapezoidal fins which have various slopes of the fin’s upper lateral side is performed. The relation between the slope factor of the fin and the non-dimensional fin length for equal amount of heat loss is shown. Further, the relation between the Biot number and the non-dimensional fin length for equal amount of heat loss is given. For these analyses, a forced analytic method is used. In particular, the same equation is used for both the asymmetric triangular fin and the asymmetric trapezoidal fins just by adjusting the value of the slope factor. It is shown that this equation can also be applied to a rectangular fin with very good accuracy. The base temperature, thermal conductivity of fin’s material and the heat transfer coefficient are assumed constant.     

Experimental study on rectangular cut-throated flume: Effects of flume walls slopes and channel longitudinal slope

This research conducted to study the flow through a rectangular cut-throated flume (RCTF). The flume is simply formed by placing two vertical triangular prisms (two vertical folded plates) on either side of a rectangular open channel. Both channel and flume cross-sections are rectangular. The investigated flume is inexpensive, easy to install and does not require high maintenance. A wide experimental investigation, carried out under free outflow conditions and under upstream subcritical flow regime to investigate the effects of the channel longitudinal slope, the flume throat width, and slopes of upstream and downstream flume walls on the stage-discharge relationship. The stage-discharge relationships were deduced by applying the dimensional analysis and were calibrated using the data of this study. The proposed stage-discharge equation for horizontal channel has an average absolute relative error of 2.97% with the relative errors restricted in the range of ±10%, and 80% of the data points are in the error range of ±5%. The proposed stage-discharge equation for sloping channel has an average absolute relative error of 3.97% with the relative errors restricted in the range of ±10%, and 66% of the data points are in the error range of ±5%. The measurements indicate that slopes of upstream and downstream walls affect the stage-discharge relationship of the CTFs only in sloping channels and flow discharge is not influenced by the flume walls slopes in a horizontal channel. The proposed unified stage-discharge equation for both horizontal and sloping channels has an average absolute relative error of 3.38% with the relative errors restricted in the range of ±10%, and 74% of the data points are in the error range of ±5%. The proposed stage-discharge model demonstrates favorable accurate and convenient estimation of discharge for flows through the CTFs.     

Flow measurement using free over-fall in generalized trapezoidal channels based on one velocity point method

A free over-fall offers the possibility of being used as a flow measuring device in hydraulic structures with a single depth measurement of the end section. Due to its practical importance, considerable attention has been paid to investigate free over-falls for different channel cross-sections using various approaches. This paper presents a new theoretical approach for computing the end depth ratio (EDR) relationship for the generalized trapezoidal channel cross-sections at free over-falls in sub critical flow regimes from which the end depth discharge (EDD) can be computed. The generalized trapezoidal channel is a geometric shape that is defined mathematically with a single equation where five widely known prismatic channel cross-sectional shapes can be generated (trapezoidal, inverted triangular (Δ), rectangular, parabolic, and triangular). This suggested theoretical approach uses one velocity point at the geometric center of the end section based on the energy and the continuity equations. Relevant experimental and theoretical results were utilized in order to examine the suggested method through the statistical measuring indices (percentage difference and the correlation coefficient (R²)). The computed results show very close agreements with the earlier works. Furthermore, simple equations are also generated using the regression curve fitting technique in order to estimate the direct discharges (Q) using the end depth (y_e) for each of the above mentioned channel cross-sections.     

An improved method for predicting discharge of homogeneous compound channels based on energy concept

Accurate estimation of flow discharge in a compound river channel is increasingly important in river management and hydro-environment design. In this paper, a new model is developed to improve the prediction of flow based on Energy Concept Method (ECM) and Weighted Divided Channel Method (WDCM) along with the apparent shear stress at the interface between main channel and floodplain. The new model is compared with a wide range of our experimental data and the data available in the literature. The 27 datasets used include homogenous symmetric channels (22 datasets) and asymmetric channels (5 datasets) with various aspect ratios [channel total width (B) at bankfull / main channel bottom (b) =1.5–15.8], and bed slopes (S_o = 4.3 × 10⁻⁴–1.3 × 10⁻²). It was found that the new model has significantly improved the accuracy of flow prediction compared with the traditional Divided Channel Method (DCM), and has also considerably better results than the ECM and WDCM methods against all the datasets, particularly for relatively low flow depths of floodplain where the flow discharges are most difficult to predict correctly. The new model predicts the total discharge well for both symmetric and asymmetric channels, within an averaged relative error of about 5%.     

Flow resistance law under suspended sediment laden conditions

The uniform flow resistance equation, in the form due to Manning or Darcy-Weisbach, is widely applied to establish the stage-discharge relationship of a river cross-section. The application of this equation, namely the slope-area method, allows to indirectly measure the corresponding river discharge by measurements of bed slope, water level, cross-section area, wetted perimeter and an estimate of channel roughness. In this paper, a recently deduced flow resistance equation for open channel flow was tested during conditions of suspended sediment-laden flow. First, the flow resistance equation was determined by dimensional analysis and by applying the condition of incomplete self-similarity for the flow velocity profile. Then the analysis was developed by the following steps: (i) for sediment-laden flows characterized by known values of mean diameter and concentration of suspended sediments, a relationship (Eq. (28)) between the Γ function of the velocity profile, the channel slope and the Froude number was calibrated by the available measurements; and (ii) a relationship for estimating the Γ function (Eq. (29)) which also takes into account the mean concentration of suspended particles was also established. The theoretical flow resistance law (Eq. (26)) coupled with the relationship for estimating the Γ function (Eq. (28) or Eq. (29)), which is characterized by the applicability of a wide range of flow conditions, allowed to estimate the Darcy-Weisbach friction factor for flows with suspended-load. The analysis showed that for large-size mixtures the Darcy-Weisbach friction factor can be accurately estimated neglecting the effect of mean concentration of suspended sediments while for small-size mixtures the friction factor decreases when the mean sediment concentration increases.     

Experimental study on free and submerged multi-jets

The flow characteristics of the hydraulic jump due to parallel jets are different from the classical jump emerging from a single gate. Due to the highly complex flow field at the downstream pool, deciding about the tailwater measuring location is a challenging issue affecting the flow measuring accuracy. Experiments are conducted herein, on different parallel jets’ configurations for both free and submerged flow conditions. To quantify the flow uniformity, for any downstream cross section, the associated momentum correction factors, β₂, were estimated for the free-flow condition. It is found that β₂-values depend significantly on the measuring location, and consequently the available conjugated depths relationship results in poor estimation when measuring location moves downstream. Employing Buckingham analysis, a general formula is proposed to calculate the momentum correction factors associated with the free hydraulic jump at different downstream measuring locations. The experimental results of this study indicated that such a formula enhances distinguishing between free and submerged flow conditions of the gates installed in parallel. Finally, a dimensionless stage-discharge formula is presented to predict the submerged flow rate through parallel gates of different gate openings and widths.     

Estimating the uncertainty of discharge coefficient predicted for oblique side weir using Monte Carlo method

Side weir is a hydraulic structure, which is used in irrigation systems to divert some water from main to side channel. It is installed at the entrance of the side channel to control and measure passing water into the side channel. Many studies provided side weir water surface profile and coefficient of discharge to measure water discharge diverted into the side channel. These studies dealt with different side weir shapes (rectangular, trapezoidal, triangular and circular), which were installed perpendicular to the flow direction. Recently, some studies dealt with skew side weir, but these studies still need to more investigation. Here we report to investigate oblique side weir theoretically using statistical method to supported other studies in this case. Measurement uncertainty discharge coefficient C_d was obtained by two methods: analytical according to the ‘Guide to the expression of uncertainty in measurement’ and the Monte Carlo method. The results indicate that all experimental results are consistent with the analytical results. The relative expanded uncertainty of the discharge coefficient C_d does not exceed 2%.     

Flow measurement using a triangular broad crested weir theory and experimental validation

The self-cleaning and semi-modular triangular broad-crested weir without crest height was firstly subjected to a rigorous theory. The main objective was to establish the discharge relationship as well as that of the resulting discharge coefficient. For this, both energy equation and momentum equation applied between two judiciously chosen sections were necessary and proved to be essential. Contrary to previous studies related to flow metering, the relationship governing the flow rate was established by taking into account the approach flow velocity. Secondarily, the device was subjected to an intense experimental program to confirm the validity of the proposed theoretical relationships. It was observed an excellent agreement between the experimental and theoretical values of the flow rate. It has been found that the experimental and theoretical flow rates are related by a linear relationship such that $Q_{Exp}=1.0057Q_{Th}$. The constant clearly indicates that the flow rate theoretical formula only needs a slight correction. The theoretical stage-discharge formula was then very accurate even no calibration parameter was employed. The theoretical development has shown that the discharge coefficient C_d only depends on the dimensionless parameter M₁ that reflects the effect of the contraction of the cross-section of the approach channel. The variation curve of C_d(M₁) showed that C_d increases in the range [0.233; 0.277] with the increase in M₁.     

Algorithm with improved accuracy for real-time measurement of flow rate in open channel systems

Most methods for flow rate measurement in open channels usually have low accuracy over a range of flow rates due to varying fluid properties, flow conditions and channel length. This paper suggests an algorithm to improve on the accuracy of flow rates computed based on hydraulic structure and slope-hydraulic radius methods. A model for determining flow rates in accelerating flows is also developed. In the proposed algorithm, the parameter used for adapting the flow rate models is obtained by comparing the measured fluid depth with the depth simulated based on the one-dimensional Saint Venant equations. The results show that an improvement from ± 2.3% to ± 0.8% accuracy in the flow rate measurement using the Venturi flume method could be achieved. In unsteady state flow in a straight-run channel, the results based on flow simulation also show possibility of achieving accurate computation over a wide range of flow rates.     

Flow resistance due to shrubs and woody vegetation

In this paper, a theoretical open channel flow resistance equation was verified using flow depth and discharge measurements carried out by Freeman et al. in a large channel, 2.44 m wide, for ten different types of uniform-sized plants (shrubs and woody vegetation). The plants, which are broadleaf deciduous vegetation commonly found in floodplains and riparian zones, were placed in staggered rows inside the channel whose bed was constructed to accept plants with their root systems. For each species, the available measurements were carried out by Freeman et al. with plants having different values of plant density, height, and bending stiffness. The available literature database (87 measurements) was divided into two groups which were separately used to calibrate and test the theoretical approach. In particular, 46 measurements were used to calibrate the relationship between the scale factor Γ of the velocity profile, the Froude number, and the channel slope. This relationship was calibrated using the entire available dataset or varying the scaling coefficient a with the investigated vegetation type. The measured values of the Darcy-Weisbach friction factor, obtained by the measured flow velocity, water depth and slope values, were compared with those calculated by the theoretical flow resistance law, coupled with the relationship for estimating the Γ function having a scaling coefficient different for each investigated vegetation type. This comparison allowed to demonstrate that an accurate estimate of the Darcy-Weisbach friction factor (errors less than or equal to ±10% for 87% of the investigated cases) can be obtained. However, for the investigated vegetation species, that are characterized by a large range of bending stiffness, also a mean value of the scaling coefficient a equal to 0.3283 allows an accurate estimate of the Darcy-Weisbach friction factor.     

Free over-fall in exponential channel cross-sections based on free vortex theory in supercritical flow regimes

Free over-fall can be used as a flow metering hydraulic structure by a single measurement of an end depth. Many theoretical and experimental researches carried out on free over-fall with various approaches for different cross-sectional shapes. This paper presents a theoretical method to compute the end depth ratio (EDR) and the end depth discharge (EDD) relationships in steep sloping channel for the exponential cross-section. The exponential cross-section is a general section which can reduce to rectangular, wide rectangular, parabolic, semi-parabolic, triangular, and semi-triangular channels. Applying the momentum equation based on the free vortex theory, a theoretical approach is presented to obtain the EDR for the exponential channel cross-section in supercritical flow regime. Experimental and theoretical studies are then utilized to verify the proposed EDR and EDD relationships. The computed results are in acceptable agreement with the relevant experimental and theoretical studies. Direct solutions of the discharge for the known end depth for each cross-section are provided in tabular forms where two empirical discharge expressions with their relevant range of applications for each channel cross-section are detailed as the main outcome of this study.     

Structure and dynamics of the turbulent flow through a central baffle

Central baffle flume (CBF) can be utilized as a control structure to measure flow discharge in irrigation channels under free and submerged flow conditions. Stage-discharge relationship has been extensively studied for various geometrical parameters and flow conditions, whereas internal structure of the flow around a baffle has not been investigated in the literature. To address this need, the present work investigates the turbulent flow around a central baffle through high-resolution numerical simulations using an open source computational model. Velocity measurements were conducted in a laboratory flume to setup and validate the numerical model. Comparison of the numerical results with the experimental measurements proves that the present numerical model can predict water depth and velocity field. Longitudinal distance from the apex to the intersection point of water and critical depths can be estimated as L_xc = 2L_e, where L_e is the longitudinal length of the guide walls. A horseshoe vortex system identified in front of the baffle produces a significant bump on the free-surface and rib vortices generated from the baffle extend up to the sidewalls of the channel. The vertical separation layer observed downstream of the baffle results in a reverse flow and a vortex pair is formed by the impingement of the resulting reverse flow on the back of the baffle. Reverse flow, plunging flow structure, splash and rebounding wave events observed at the downstream produce substantial hydrodynamic effects on the baffle. Geometry of the central baffle was modified to suppress recirculation effects based on the insights into the complete flow structure around the baffle. Eventually, vortex structures were suppressed and the length of the recirculation zone was reduced by 76%.     

Water surface profile along a side weir in a parabolic channel

A side weir is an overflow structure set into the side of a channel. This structure is used for water level control in channels, diverting excess water from a main channel into a side channel and as storm overflows from urban sewage systems. Computation of water surface profile over the side weirs is essential to determine the flow rate of the side weir. Discharge estimation of the side weir is still an important research subject. Most previous research works for the side weir were carried out in main channels with rectangular, triangular, trapezoidal and circular cross sections. Analytical solutions for water surface profile along a rectangular side weir are available only for the special cases of rectangular and trapezoidal main channels on the basis of a constant specific energy. No analytical solution is available for a rectangular side weir located in a parabolic channel. This research presents an elegant analytical solution for establishing the water surface profile along a side weir located in a parabolic channel which involves the use of incomplete elliptic integrals. The solution, which yields a direct computation, should be a useful computational tool for evaluation and design of rectangular side weirs in parabolic channels.