Great difference of flow depth and Manning's roughness coefficients in main channels and floodplains, generate a strong gradient of lateral velocity and hence, a lateral shear stress in the interface of the main channel and floodplain. This phenomenon increases the head loss in river system. Mathematical models which are currently used for water surface profile computations in rivers, e.g. HEC-RAS and MIKE11, neglect this mechanism. For taking into account this mechanism, the energy slope and energy correction factor should be modified in Gradually varied flow computations. In this paper, using exchange discharge method, the current procedure of Gradually varied flow computations were modified for compound channels. Comparison of this method and HEC-RAS results in an experimental flume with heterogeneous compound section in case of drawdown profile, M2, showed that the accuracy of these methods are 95 and 84.5%, respectively.

Analysis of flow through rockfill materials is usually carried out by solving the differential equation which is a combination of nonlinear [i=mvn] and continuity equations. However, solving this differential equation either by means of finite difference method or similar procedures is relatively time consuming, and with uncertainty involved in water surface profile this becomes even more difficult. To analyze flow through rockfill materials an alternative method which is based on Gradually varied flow theory may be applied that is comparatively simpler and less time consuming. To apply this alternative method it is necessary to examine the validity of assumptions on which Gradually varied flow method are based. Moreover, the involved parameters in Gradually varied flow equation should be reverted to porous media condition. In this research by conducting a vast number of experiments on various types of materials the acceptability of assumptions of Gradually varied flow method for analysis of flow through coarse porous media is investigated and the effect of physical characters of porous medium such as void ratio, uniformity coefficient Cu, gradation coefficient Cc and viscosity of fluid as well as certain features of flow such as velocity and hydraulic gradient are also examined. Findings indicate that: 1). by accepting some degrees of discrepancies, one may use the Gradually varied flow method for analysis of flow through coarse porous media. 2). the permeability parameters of coarse porous media may be successfully related to the physical characters of granular materials. 3). A Comparison between observed and computed flow profiles through media indicates accuracy and applicability of equations that are derived by authors of this paper.

The length of the steady Gradually varied flow (G.V.F) profile in a circular gravity pipe section is computed using the Graphical Integration Method. The equations used for the solution are: a) the dynamic equation of Gradually varied flow in a prismatic channels, b) the hydraulic exponents M and N equation derived by Chow [2], and c) the varied hydraulic exponents M (y/do) and N(y/do) equation modified by Zaghloul [15]. The results of the calculated G.V.F profile length using the modified hydraulic exponents M(y/do) and N(y/do) equation are closer to the G.V.F length calculated based on the exact formulation of the G.V.F dynamic equation. The percentage difference ranges from 0.67% to 8.72% for various bed slopes and G.V.F depth limits. The calculated G.V.F profile length using the Chow hydraulic exponents M and N resulted in wider values with percentage difference ranges from 0.16 to 25.59%. Hence, a remarkable improvement of the computation of G.V.F profile is achieved using the modified M(y/do) and N(y/do) hydraulic exponents.The unsteady Gradually varied flow wave propagation in circular gravity pipe section is simulated using the Explicit Method. The Extended Transport block (EXTRAN) of the latest Storm Water Management Model (PCSWMM2000) was used to route the wave through a circular gravity pipe section. The resulting routed hydrographs by the Explicit Method and the EXTRAN Block of the PCSWMM 2000, provided similar flow peak and lag time in both cases.A computer package was developed for the Steady G.V.F length calculation for gravity pipes using the Microsoft Excel spreadsheet. The results are plotted using the Excel graphics capabilities.A second computer package using Microsoft Excel was developed for the Unsteady G.V.F simulation based on the Explicit Method. The routed hydrographs arc plotted by the Excel graphics capabilities.The Excel packages for the Steady and Unsteady G.V.F are user friendly and are posted on the Internet Website location http://briefcase.yahoo.com/civil engineering2001.ARead me File is provided for each Excel package and is used as a users guide.

Background and Objectives Gradually varied flow is one of the common profiles of open channel flow which takes place in canals and natural conduits due to hydraulic structures and morphological causes. Spatial variations of flow characteristics is one of the most apparent properties of Gradually varied flow that its precise calculation through accurate solution of dynamic equation of Gradually varied flow has significant role. Three main approaches to solve dynamic equation of Gradually varied flow are analytical, numerical and graphical ones. Of these mentioned methods, direct integration of dynamic equation of Gradually varied flow is the most accurate method. Gaussian hyper-geometric functions (GHF), which has been considered in this research work, is one of the approaches to integrate directly of dynamic equation of Gradually varied flow. Creation of dimensionless form of dynamic equation of Gradually varied flow is the basis of the GHF method. This is done using normal depth of flow (yn-based method) or critical depth of flow (yc-based method). At the end of the paper, a comparison has been performed between GHF method and Rong-Kutta numeric method to predict GVF profile in a rectangular laboratory flume for three M1, S2 and C3 profiles. Methodology The GHF solver (F2 1(a, b; c; z)=Γ (c)Γ (a)Γ (b)Σ Γ (a+k)Γ (b+k)Γ (c+k)k! zk∞ k=0) has been implemented to integrate differential dynamic equation of GVF [dydx=So-Sfcosθ-α Q2TgA3]for five channel slops including Mild (M), Steep (S), Horizontal (H), Critical (C) and Adverse (A). Dimensionless form of GVF equation makes it easy to integrate using GHF solver. Due to the absence of normal depth for horizontal and adverse slopes, yc was used to produce dimensionless form of GVF equation for H and A slopes. Other three slopes were made dimensionless using yn. For the first scenario, two dimensionless parameters were defined as v=yyc and x#=xSc*yc. The correspondes parameters for the second scenario were u=yyn and x*=xSo*yn. The final integrated forms of the GVF equation include the hydraulic exponents as M and N which can be solved for certein values of M and N. To compar the results between numerical (Runge-Kutta 4th order method) and analytical (GHF) solvers, lobaratory GVF data measured from a rectangular flume of length 7 m, width 0. 1 m, height 0. 3 m and roughness 0. 011 were used. Three profiles as M1, S2 and C3 were formed experimentally to measure the GVF characteristics. Also, three performance assessment indices as RMSE, R2 and E were applied to compare the solvers accuracy. Findings Two analytical answers (overall 10 answers) were obtained to solve Gradually varied flow dynamic equation. The first group belongs to the channels of slope types namely M, S and C. The second one can be used for the slopes of types H and A. These two mentioned group answers should be used according to the profile name and corresponding zone. Regarding to the laboratory measured data of GVF profiles, results showed that application of GHF not only resolve discretization selection for numerical methods, but also predicts GVF profile characteristics more accurate than to the numerical Rong-Kutta 4th order solver. The amount of the (RMSE, R2, E) for the M1, S2 and C3 profiles for GHF were (0. 0173, 0. 9986, 1. 11), (0. 0167, 0. 9984, 1. 12) and (0. 0204, 0. 9988, 0. 985), respectively; while the corresponding values of numerical solver were calculated as (0. 0458, 0. 9864, 4. 05), (0. 0259, 0. 991, 2. 38) and (0. 0327, 0. 985, 3. 65). These values prove the superiority of GHF predictor. Conclusion Gradually varied flow profiles are common at the flow conduits with hydraulic structures. Predicting flow characteristics of GVF can be achievable through solving differential dynamic equation. In this paper, GHF was applied to solve the equation analytically. The obtained analytical answers can be used for all zones of five channel slope types.

In this research a mathematical model was developed to study bed elevation variation of alluvial rivers. It utilizes two principal modules of hydraulics and sediment transport for simulation purposes. SDAR (Scour and Deposition model of Alluvial Rivers) is a new model with both one and semi-two dimensional (S-2D) computational schemes. It is regarded S-2D in a sence that lateral variation of velocity, hydraulic stresses, and geometrical specifications are achieved by dividing the main channel into serveral stream tubes. In order to overcome the existing limitations, a new idea of reachwise stream tube concept was also introduced. This allows to include branch connections and withdrawal points across the tube barriers. Sediment routing and bed variation calculations are accomplished along each river strip desigated by virtual interfaces of the tubes. Presently, quasi-steady Gradually varied flows are processed by the model. It should also be emphasised that this version is only valid for alluvial rivers composed of noncohesive bed material. To assess the model, several river cases and laboratory data base were used. During calibration runs, the ability of model in longitudinal and transversal bed profile simulation and armor layer development predection were especially detected. Results of simulation are also compared with the results of well-known models, e.g. HEC-6, GSTARS-2, and FLUVIAL-I2. It was found that the ability of model in simulating bed variation is noticeably increased when S-2D concept is introduced. Indeed, the comparative validity tests confirm SDAR"s promising functioning in facing with complex real engineering cases. Obviously more article discussions would bring oppurtunities to demonestrate it"s technical cappabilities profoundaly.

Manning roughness coefficient is one of the most important parameters in designing water conveyance structures. Unsuitable selection of this coefficient brings up some mistakes. This research aims to present a method to determine the Manning roughness coefficient based on a combination of optimization algorithm of simulated annealing (SA) with Gradually varied flow equations. Therefore, in a lab rectangular flume of 12 m, 60 cm and 65 cm in length, width and height with fixed channel bed slope of 0.0002, nine series of water level profiles were carried out. Then, an objective function based on observed and calculated water level gradient was defined to decide on manning roughness coefficient while it was minimized with simulated annealing optimization method. The values of objective function parameters were discussed by sensitivity analysis and the most optimal objective function was obtained. To measure the accuracy of coefficient obtained, Statistics indices of R2, Root mean square error (RMSE), Mean bias error (MBE), d were used. The results showed that manning roughness coefficient has a great accuracy.

IntroductionNon-darcy flows into two categories: parallel flows (such as gravel dams, gabions, etc.) and radial flows (such as flows near wells drilled in coarse-grained alluvial beds, etc.) are divided. In the first category, streamlines are almost parallel so that there is no curvature or contraction of streamlines in the plan view. This type of flow is found in both pressurized and free-surface modes. Radial non-darcy flow analysis has many applications in the fields of civil engineering, geology, oil, and gas. The equations governing the radial non-darcy flow are solved using numerical methods of finite differences, finite elements and finite volumes. Solving these equations requires boundary conditions and a lot of data and is almost bulky, time consuming and costly. While, Gradually varied flow theory, requires much less data and is easier and less expensive. For this reason, in the present study, for the first time, using experimental data recorded in a large-scale (almost real) device, the application of the Gradually varied flow theory in radial non-darcy flows with free surface has been investigated. In other words, since the calculation of water surface profiles in a radial rockfill is of great importance. In the present study, using large-scale (almost real) experimental data and the Gradually varied flow theory, the water surface profile in radial non-darcy flow with free surface and in steady state has been investigated.MethodologyIn the present study, due to the compatibility of cylindrical coordinates and its adaptation to the physics of problems related to radial flows, a device has been constructed in the laboratory of Bu Ali Sina University in the form of a semi-cylinder with a diameter of 6 meters and a height of 3 meters. The dimensions of this device are made on a large scale and the effects limitations have practically no effect on the testing process. To measure piezometric pressure, piezometric grids have been used. The device has a volume of 14,000 liters and a capacity of materials weighing approximately 40 tons. Four pumps are installed in parallel at the top of the device to generate the required flow. Coarse-grained river materials with a diameter between 2 to 10 cm, a porosity of 40%, a Cu of 2.13, and a Cc of 1.016 have been used. To perform the tests, the model is first filled to a certain height (53, 60, 70, 85, 95, 110, 120, 140, 150, and 160 cm) by pumping operations. The flow rate created in these experiments is in the range of 49.94 to 53.16 L/s.Results and DiscussionOne-dimensional analysis of steady-non-darcy flow using Gradually varied flow theory and two-dimensional analysis using Parkin equation solution. Most research has been done in parallel flow rockfills. Also, solving the Parkin equation in both parallel and radial flows requires a lot of data such as boundary conditions upstream and downstream, as well as the boundary condition of the water surface profile, and the calculation process is complex and time-consuming. The Gradually varied flow theory requires much less data than solving the Parkin equation, and the water surface profile obtained from it is also used as the main boundary condition in solving the Parkin equation. In other words, calculating the water surface profile in a radial rockfill is very important to studying the movement of water. Also, the water surface profile is the main boundary condition in the two-dimensional analysis of steady flow (solving the Parkin equation), and with it, upstream and downstream boundary conditions will be practically available. For this reason, in the present study, using large-scale (almost real) experimental data and the Gradually varied flow theory, the water surface profile in the case of radial non-darcy flow has been calculated. To calculate the flow depth at different points (water surface profile) using the Gradually varied flow theory, the amount of flow depth at one point and the coefficients m and n must be available. Since the flow depth measurement in the well (downstream of the desired interval) can be measured, in the present study, the calculations started from the downstream (depth of flow in the well).ConclusionIf the Gradually varied flow theory is used to calculate the water surface profile in the case of radial non-darcy flow with a free surface, the mean relative error in the case of pumped heights is 53, 60, 70, 85, 95, 110, 120, 140, 150 and 160 cm are equal to 1.56, 0.96, 0.61, 0.45, 0.28, 0.19, 0.13, 0.16, 0.11 and 0.05 are calculated, respectively. In other words, the average mean relative error (MRE) of calculating the water surface profile for different heights of pumped water is equal to 0.45%. Also, according to the obtained results, the greater the depth of water pumped upstream, the higher accuracy of the Gradually varied flow theory.KeywordsRadial Non-Darcy flow, Steady flow, One-Dimensional Analysis, Gradually varied flow Theory.

The basic equation of Gradually varied flow (GVF) describes the variation of water depth with flow process. Several methods have been developed for numerical solution of the water surface profile in GVF and one of the challenges of numerical integration is determining the appropriate integration spatial step size. In this paper a novel Adaptive Newton-Raphson method is developed for numerical solution of the GVF equation. In this method the spatial steps are determined by using error estimation during calculation, this procedure is smart and increases accuracy and speed of computation the water surface profile in GVF. Several examples were analyzed using the proposed method and compared with the results of previous researches and the accuracy of the proposed method was evaluated. The results indicate good accuracy of the proposed method in comparison with other methods. As shown in one of examples presented in the paper, the obtained results from 10 step of developed Adaptive Newton-Raphson method approximately equal with 90 step of standard direct step method.

The non-uniform flow in a prismatic channel with gradual changes in the free water surface level is called the Gradually varied flow (GVF). Calculation of the GVF profiles over the last century has become a significant topic for the researchers in the relevant fields.  To obtain this profile, the nonlinear ordinary differential equation of the GVF needs to be solved. This can be carried out either numerically or analytically. Although several studies have been conducted on the GVF in open channels in various forms (Jan & Chen, 2013; Vatankhah, 2010, 2015; Homayoon & Abedini, 2019), the number of semi-analytical studies in the field of gradual variable flow in trapezoidal and triangular channels is limited, which requires further investigation. In this research, the Adomian Decomposition Method (ADM) is used to find a semi-analytical solution for solving the GVF equation in the triangular and trapezoidal prismatic channels. In this method, the Manning equation is used as the resistance equation. Moreover, for the aim of verifying the semi-analytical solutions, the ADM results are compared with the finite difference method (FDM). The presented semi-analytical solutions in this paper can be used to validate other numerical methods in similar studies.

Energy and momentum approaches can lead to different results in analysis of flow through compound channels. This study is an attempt to shed more light on the validity of these approaches for analysis of Gradually-varied flow in compound channels. At first the energy and momentum equations and a generalized form of steady Gradually-varied governing equation are introduced. This G.E. provides a framework to highlight the differences between the two approaches using some numerical examples. Then, experimental data available in the literature and those collected in this study are used to compare different numerical water surface profiles with laboratory measurements. This study indicates that the momentum principle can perform better for water surface profile computation in compound channels, at least for the investigated experimental data. Error analysis also supports this conclusion. Two dimensional Shiono and Knight method for velocity distribution is employed to study more on the subject. In addition, two dimensional analyses introduces a procedure by which one can decide on the superiority of one approach over the other, based on geometrical and hydraulic characteristics of a compound channel, with no need for experimental work.