Analytical Model of Surge Flow in Nonprismatic Permeable Channels and Its Application in Arid Regions

A surge running down a dry wadi bed as a consequence of a controlled water release from a reservoir—e.g., for artificial groundwater recharge—represents a free boundary problem. After some time, when aiming for groundwater recharge, the infiltration equals inflow and thus forms a kind of “standing” wave. The numerical solution of such phenomena generally involves considerable problems. For avoiding the numerical inconvenience resulting from the complex interacting surface/subsurface flow, we present an analytical solution of the slightly modified zero-inertia (ZI) equations. The development introduces a momentum-representative cross section for portraying the transient development of momentum and refers to a channel with constant slope, irregular geometry, and a permeable channel bed with significant infiltration. Due to the structure of the solution, any arbitrary infiltration model can be used for quantifying the infiltration losses. For both synthetic prismatic and nonprismatic test channels, the robust and easy-to-use analytical ZI model shows an excellent match with the results of a comparative numerical simulation. Finally, the ZI model is employed for simulating a surge flow downstream of the Wadi Ahin groundwater recharge dam (Oman), in order to perform a scenario for artificial groundwater recharge in a natural wadi channel reach. This realistic application illustrates the potential of the new approach by even computing an almost standing wave and shows its applicability for an accurate and robust evaluation of release strategies.     

Overbank Flow in Compound Channels with Nonprismatic Floodplains

In this paper, the authors present experimental results of overbank flow in compound channels with nonprismatic floodplains and different convergence angles. The depth-averaged velocity, the local velocity distributions, and the boundary shear stress distributions were measured along the converging flume portion for different relative depths. The momentum balance is used to analyze the force acting on the flow in the main channel and for the whole cross section. Using the experimental data, various terms in the momentum equation are also calculated. The apparent shear forces on the vertical interface between the main channel and floodplains are evaluated for compound channels with nonprismatic floodplains, and the results are then compared with the prismatic cases. The energy balance in nonprismatic compound channels is also investigated by using the water surface elevation.     

Discretization of Integral Equations Describing Flow in Nonprismatic Channels with Uneven Beds

Application of the finite-volume method in one dimension for open channel flow predictions mandates the direct discretization of integral equations for mass conservation and momentum balance. The integral equations include source terms that account for the forces due to changes in bed elevation and channel width, and an exact expression for these source term integrals is presented for the case of a trapezoidal channel cross section whereby the bed elevation, bottom width, and inverse side slope are defined at cell faces and assumed to vary linearly and uniformly within each cell, consistent with a second-order accurate solution. The expressions may be used in the context of any second-order accurate finite-volume scheme with channel properties defined at cell faces, and it is used here in the context of the Monotone Upwind Scheme for Conservation Laws (MUSCL)-Hancock scheme which has been adopted by many researchers. Using these source term expressions, the MUSCL-Hancock scheme is shown to preserve stationarity, accurately converge to the steady state in a frictionless flow test problem, and perform well in field applications without the need for upwinding procedures previously reported in the literature. For most applications, an approximate, point-wise treatment of the bed slope and nonprismatic source terms can be used instead of the exact expression and, in contrast to reports on other finite-volume-based schemes, will not cause unphysical oscillations in the solution.     

Analytic Stage-Discharge Formulas for Flow in Straight Prismatic Channels

Three analytic stage-discharge formulas, suitable for hand calculation, have been derived for prismatic open channels. The three types of prismatic geometry investigated are: simple rectangular, symmetric, and asymmetric rectangular compound channels. The formulas include three key parameters that govern the local friction factor, eddy viscosity, and secondary flow. The discharge results and the corresponding analytic depth-averaged velocity and bed shear stress distributions show good agreement with the experimental data, even when constant parameter values are assumed. The influence of these three parameters on the stage-discharge relationships is explored.     

Systematic Surge Protection for Worst-Case Transient Loadings in Water Distribution Systems

Estimating appropriate water demands for the design of a distribution system is itself difficult, but the continuously fluctuating nature of these demands has the added potential of creating water hammer problems that might result in catastrophic pipeline or system failure. To first identify and then avoid these eventualities, this paper searches a predefined set of possible water hammer events in water distribution systems to identify the most severe transient loadings and then conducts a search for suitable surge protection strategies. Genetic algorithms and particle swarm optimization are combined with transient analysis first to identify a set of worst-case loads and then to seek an optimal protection strategy to cope with them. Case studies show that the worst case is not always obvious and cannot always be assumed a priori to correspond with high or low demand scenarios. Both the search for the worst-case loading and its associated optimal protection strategies are strongly sensitive to the characteristics of both the pipe system and the candidate transient events.     

Transverse Dispersion Caused by Secondary Flow in Curved Channels

A new theoretical equation is proposed to describe the streamwise variations of the transverse velocity along a curved channel with a constant curvature. Furthermore, based on this theoretical equation for the transverse velocity, a new equation for the transverse dispersion coefficient is developed to incorporate the effect of the secondary flow on the transverse dispersion in curved channels. The new equations for the transverse velocity and dispersion coefficient are verified with experimental data sets that were obtained from laboratory experiments conducted in two different channels. The results show that the proposed velocity equation properly describes the streamwise variations of the secondary flow developed in the curved channels. The reach-averaged values of the transverse dispersion coefficient calculated by the new equation are in relatively good agreement with the observed values from the laboratory channels. Sensitivity analysis reveals that both the secondary flow and the transverse dispersion coefficient are proportional to the roughness factor, and in inverse proportion to the aspect ratio of the channel.     

Analytical Approach to Calculate Rate of Bank Erosion

Bank erosion consists of two processes: basal erosion due to fluvial hydraulic force and bank failure under the influence of gravity. Because bank resistance force varies with the degree of saturation of bank material, the probability of bank failure is the probability of the driving force of bank failure being greater than the bank resistance force. The degree of saturation of bank material increases with river stage; therefore, the frequency of bank failure is correlated to the frequency of flooding. Consequently, the rate of bank erosion is due to both basal erosion and bank failure, and bank failure is a probabilistic phenomenon. In this paper, for cohesive bank material experiencing planar bank failure, a deterministic approach was adopted for basal erosion analysis, whereas the probability of bank failure was included in the analysis of bank failure. A method for calculating the rate of bank erosion was derived that integrates both basal erosion and bank failure processes, and accounts for the effects of hydraulic force, bank geometry, bank material properties, and probability of bank failure.     

Hydraulic Resistance of Flow in Channels with Cylindrical Roughness

A laboratory study on the hydraulics of flow in an open channel with circular cylindrical roughness is presented. The laboratory study consists of an extensive set of flume experiments for flows with emergent and submerged cylindrical stems of various sizes and concentrations. The results show that the flow resistance varies with flow depth, stem concentration, stem length, and stem diameter. The stem resistance experienced by the flow through the vegetation is best expressed in terms of the maximum depth-averaged velocity between the stems. Physically based formulas for flow resistance, the apparent channel velocity, and flow velocities in the roughness and surface layers are developed. The formulas are validated with the flume data from the present study as well as those from past studies. A method for calculating channel hydraulic conditions using these formulas is presented.     

Direct Solution to Problems of Open-Channel Transitions: Rectangular Channels

The specific energy equation has many applications in rectangular open-channel flow problems. The existing methods of solving this equation are as follows: trial-and-error solution, graphical solution, and design tables prepared from the specific energy equation expressed in dimensionless form. No direct solutions are available in the technical literature for this equation to date because it was presumed that finding roots of this equation requires a series of substitutions. In this technical note, the writer develops an analytical solution for transitions located in rectangular open channels: (1) to avoid the inconvenience in available solutions; (2) to derive a less time consuming method; and (3) to achieve more accurate results than those as a result of using existing methods.     

Flow through Trapezoidal and Rectangular Channels with Rigid Cylinders

A physical model study was performed in which wood dowels were used to model rigid vegetation. The dowel configurations used in the flume were intended to simulate the effects of willow post systems (i.e., collections of rigid cylinders placed along a streambank to reduce streambank erosion). In addition, an analytical model is presented for predicting depth-averaged velocity distributions in straight trapezoidal or rectangular channels with newly constructed willow post systems. The analytical model is founded on wake theory and is applicable to channels with submerged and unsubmerged rigid cylinders. Data from three independent physical model studies were used to validate the analytical model. Depth-averaged velocities predicted using the analytical model, Upred, were compared to velocities observed in the physical models, Uobsd, and yielded discrepancy ratios, Upred/Uobsd, that were typically between 0.80 and 1.20. Results from this study are that significant variables for reducing local velocities are cylinder height, diameter, and density (number of cylinders per unit area); and the arrangement of the cylinders (i.e., rectangular versus staggered grid) is inconsequential.     

Analytical Solution of the Burgers Equation for Simulating Translatory Waves in Conveyance Channels

A Burgers equation model (BEM) for simulating translatory waves in conveyance channels is extracted from the Saint-Venant equations for small perturbations in initial uniform flow. The present study improves upon the previous model and presents analytical solutions for the simulation of translatory waves occurring in conveyance channels. The BEM is reduced to the linear diffusion equation using the Cole–Hopf transformation and then solved by means of the Green’s function assuming an infinite domain. The simulation studies performed show that the BEM results are comparable to those of the Saint-Venant equations for small perturbations in the initial uniform flow conditions and for Froude numbers within the subcritical region. The BEM could be useful for flood routing and for simulating release of water from a reservoir into a conveyance channel when the flow perturbation is small.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.     

Flow Patterns in Compound Channels with Vegetated Floodplains

Understanding the hydraulics of flow in a compound channel with vegetated floodplains is very important for determining the stage-discharge curve and for supporting the management of fluvial processes. In this paper, the flow patterns over different types of vegetation, such as tree, shrub, and grass, are described, based on an experimental study. For vegetation on the floodplain, the authors choose plastic grass, duck feathers, and plastic straws as model grass, shrubs, and trees, respectively. A 3D acoustic Doppler velocimeter was used to measure the local flow velocities for different types of vegetation on the floodplain, and the total discharge and flume slope were measured independently. In the cases of nonvegetated floodplains, all measured streamwise velocity distributions followed the logarithmic distribution, but for vegetated floodplains, they followed an S-shaped profile, exhibiting three zones. For all cases, the fluctuating velocity followed a normal distribution. The influence of different types of vegetation on the distributions of the secondary currents, turbulence intensities, and Reynolds shear stresses were also analyzed.     

Steady and Unsteady Simulations of Turbulent Flow and Transport in Ultraviolet Disinfection Channels

The effects of unsteadiness in the turbulent flow through a staggered array of circular cylinders, modeling an ultraviolet disinfection system, are studied by means of solutions of the two-dimensional Reynolds-averaged Navier–Stokes equations incorporating the standard k–? turbulence model. Time averaging is applied to the unsteady solution, and the time-averaged characteristics are compared with a solution where a steady flow is a priori assumed, as well as with time-averaged measurements. Differences between the predictions of time-averaged and the steady-flow models are found to be largest in the entrance region of the array, and to decline in importance in the downstream direction. Comparison with measurements indicate that, while the time-averaged unsteady model predictions exhibited better agreement in some respects, the turbulent kinetic energy remained substantially underpredicted. Predictions of head losses through the array are also discussed.     

金属压力加工问题的流函数速度模式上限解

采用流函数法构造了塑性变形金属流动的完备速度模式,对Mises材料平面变形和轴对称变形问题,提出并推证了运动可能速度场应满足的力学边界条件一\     

Solution for the Closed-Loop Problem in Pressurized Multipipe Systems

The fundamental background of the solution for the steady-state flow in pressurized water closed-loop pipe systems, without any reservoirs in-between, is presented in this paper. The use of the steady-state mass balance and energy equations to calculate discharges and heads in this type of hydraulic system leads to an undetermined problem. The way to solve this indeterminacy is to consider an additional continuity equation associated with the difference between initial and final conditions, taking into account fluid compressibility and pipe-wall deformability. A complete formulation is derived considering pressure and temperature changes in the hydraulic system. Simplified formulae are presented for isothermal flows in simple systems and multiple closed-loops with pipes in series and in parallel. This problem can also be solved by a pseudotransient analysis technique applied to steady-state conditions. Proposed solutions for this problem are applied to steady-state flows and tested for different system configurations.     

Dam Break in Channels with 90° Bend

In practice, dam-break modeling is generally performed using a one-dimensional (1D) approach for its limited requirements in data and computation. However, for valleys with multiple sharp bends, such a 1D model may fail for predicting as well the maximum water level as the wave arrival time. This paper presents an experimental study of a dam-break flow in an initially dry channel with a 90° bend, with refined measurements of water level and velocity field. The measured data are compared to some numerical results computed with finite-volume schemes associated with Roe-type flux calculation. The 1D approach reveals the expected limits, while a full two-dimensional (2D) approach provides fine level prediction and rather satisfactory information about the arrival time. A hybrid approach is now proposed, mixing the 1D model for the straight reaches and local 2D models for the bends. The compatibility of the Roe fluxes at the interfaces requires a careful formulation, but the resulting scheme seems able to capture reflection and diffraction processes in such a way that the results are really good in what concerns the water level.     

Hydraulic Models of the Flow Distribution in a Four Branch Open Channel Junction with Supercritical Flow

Intense rainfall on urban areas can generate severe flooding in the city, and if the conditions are right, the flow in the streets can be supercritical. The redistribution of the flow in street intersections determines the flow rates and water levels in the street network. We have investigated the flow that occurs when two supercritical flows collide in a 90° junction formed by streets of identical cross section. Several flow configurations within the intersection are possible, depending on the position of the hydraulic jumps that form in and upstream of the intersection. Previous work has identified three flow types, with Type II flows being further classified into three subregimes. Hydraulic models have been developed, based on the principles of the conservation of flow and momentum flux in the intersection, which predict the angles at which the jumps will form. These models can be used to determine the flow type that will occur. Moreover, additional models have been developed for computing the outflow discharge distribution. For Type I flows, it has not been possible to develop such a hydraulic model for the discharge distribution, but some data are provided for one configuration to indicate the influence of different parameters. For Type II and Type III flows, such models are developed, and their predictions agree with data obtained from the channel intersection facility at the Laboratory of Fluid Mechanics and Acoustics in Lyon.     

Stress Solution of Multiple Elliptic Hole Problem in Plane Elasticity

Using Schwarz’s alternating method and Muskhelishvili’s complex variable function techniques, this paper presents an iterative algorithm method for the effective and accurate calculation of the stresses in an elastic solid of infinite extent containing multiple elliptic holes and subject to external loading at the infinity. The elliptic holes can have different dimensions and locate at any points while their axis orientations must be orthogonal. The proposed iterative algorithm method is based on the approximation of the resultant force vector on each elliptic hole boundary as a series of complex variable. As a result, exact closed-form analytical stress solutions can then be obtained for the solid with a single elliptic hole whose boundary is subject to the reverse resultant force vector in the forms of complex series. The numerical results presented in the paper show that the iterative solution converges quickly and stably. The proposed convergent criterion ensures the satisfaction of the required accuracy of the stress results. The stress concentration at elliptic holes can then be evaluated with high accuracy.