Prismatic Channels Research Articles

Critical depth prediction in straight compound channels

Critical depth is the depth at which the flow in an open conduit undergoes a transition from subcritical to supercritical regimes. The prediction of critical depth in a prismatic channel can be achieved to a high degree of accuracy. This is not the case for two-stage channels (main channel with flanking floodplain), also commonly known as compound channels. Traditionally, critical depth is predicted based on minimum specific energy, and this paper investigates the validity of this and several other methods of critical depth prediction for compound channels. It has been found that flow in compound channels does not undergo a sharp critical flow transition as occurs in prismatic channels. Instead, the change between fully subcritical and fully supercritical occurs over a depth range, whereby different regions in the cross-section maybe either sub- or supercritical. This has been termed a ‘mixed-flow’ region. It has been found that the point of minimum specific energy does not coincide with either the beginning or the end of this mixed flow transition, rendering such predictions ineffectual. A new method for determining critical depth in a compound channel is presented and compared with other prediction techniques. Methods of incorporating the effects of the momentum interaction have been used to improve the prediction of the velocity profile within the channel and locate the mixed flow transition zone.

Critical depth prediction in straight compound channels

Critical depth is the depth at which the flow in an open conduit undergoes a transition from subcritical to supercritical regimes. The prediction of critical depth in a prismatic channel can be achieved to a high degree of accuracy. This is not the case for two-stage channels (main channel with flanking floodplain), also commonly known as compound channels. Traditionally, critical depth is predicted based on minimum specific energy, and this paper investigates the validity of this and several other methods of critical depth prediction for compound channels. It has been found that flow in compound channels does not undergo a sharp critical flow transition as occurs in prismatic channels. Instead, the change between fully subcritical and fully supercritical occurs over a depth range, whereby different regions in the cross-section maybe either sub- or supercritical. This has been termed a ‘mixed-flow’ region. It has been found that the point of minimum specific energy does not coincide with either the beginning or the end of this mixed flow transition, rendering such predictions ineffectual. A new method for determining critical depth in a compound channel is presented and compared with other prediction techniques. Methods of incorporating the effects of the momentum interaction have been used to improve the prediction of the velocity profile within the channel and locate the mixed flow transition zone.

Critical depth prediction in straight compound channels

Critical depth is the depth at which the flow in an open conduit undergoes a transition from subcritical to supercritical regimes. The prediction of critical depth in a prismatic channel can be achieved to a high degree of accuracy. This is not the case for two-stage channels (main channel with flanking floodplain), also commonly known as compound channels. Traditionally, critical depth is predicted based on minimum specific energy, and this paper investigates the validity of this and several other methods of critical depth prediction for compound channels. It has been found that flow in compound channels does not undergo a sharp critical flow transition as occurs in prismatic channels. Instead, the change between fully subcritical and fully supercritical occurs over a depth range, whereby different regions in the cross-section maybe either sub- or supercritical. This has been termed a ‘mixed-flow’ region. It has been found that the point of minimum specific energy does not coincide with either the beginning or the end of this mixed flow transition, rendering such predictions ineffectual. A new method for determining critical depth in a compound channel is presented and compared with other prediction techniques. Methods of incorporating the effects of the momentum interaction have been used to improve the prediction of the velocity profile within the channel and locate the mixed flow transition zone.

Analytical Solution to the Zero-Inertia Problem for Surge Flow Phenomena in Nonprismatic Channels

Surge flow phenomena. e.g.. as a consequence of a dam failure or a flash flood, represent free boundary problems. ne extending computational domain together with the discontinuities involved renders their numerical solution a cumbersome procedure. This contribution proposes an analytical solution to the problem, It is based on the slightly modified zero-inertia (ZI) differential equations for nonprismatic channels and uses exclusively physical parameters. Employing the concept of a momentum-representative cross section of the moving water body together with a specific relationship for describing the cross sectional geometry leads, after considerable mathematical calculus. to the analytical solution. The hydrodynamic analytical model is free of numerical troubles, easy to run, computationally efficient. and fully satisfies the law of volume conservation. In a first test series, the hydrodynamic analytical ZI model compares very favorably with a full hydrodynamic numerical model in respect to published results of surge flow simulations in different types of prismatic channels. In order to extend these considerations to natural rivers, the accuracy of the analytical model in describing an irregular cross section is investigated and tested successfully. A sensitivity and error analysis reveals the important impact of the hydraulic radius on the velocity of the surge, and this underlines the importance of an adequate description of the topography, The new approach is finally applied to simulate a surge propagating down the irregularly shaped Isar Valley in the Bavarian Alps after a hypothetical dam failure. The straightforward and fully stable computation of the flood hydrograph along the Isar Valley clearly reflects the impact of the strongly varying topographic characteristics on the How phenomenon. Apart from treating surge flow phenomena as a whole, the analytical solution also offers a rigorous alternative to both (a) the approximate Whitham solution, for generating initial values, and (b) the rough volume balance techniques used to model the wave tip in numerical surge flow computations.

Deflecting Velocity Rod for Flow Measurements in Small Channels

A hinged rod, when immersed in flowing water, deflects against its weight due to the hydrodynamic force of water. A relationship was derived to calculate the average flow velocity in small channels from the deflection of the rod, length and thickness of the rod, density of the rod's material, flow depth, and an experimentally determined rod's velocity coefficient. The rod's velocity coefficient was found to be invariant with flow velocity (V) and Reynolds number (R) for the range investigated in this study (V ⩽ 60 cm/s and 25,000 ⩽ R ⩽ 60,000). The calculated velocity compared very well with the measured velocity, with an average error of only 0.7%. Analysis suggested that a rectangular rod made of seasoned wood (density ⩾ 0.7) with length = 100 cm, width = 2 to 4 cm, and thickness = 3 to 5 cm will provide a simple but acceptable technique for determination of average flow velocity in small prismatic channels having a flow velocity ⩽1 m/s and flow depth ⩽45 cm with less than 5% error.

Integrating Equation of Gradually Varied Flow

This paper presents an improvement in the direct method originally proposed by Chow for integrating the equation of gradually varied flow in prismatic channels. The hydraulic parameters assumed to be constants at all depths by Chow are introduced in the integration as variable parameters. The differential equation of gradually varied flow is integrated, considering the effect of variable parameters. The results obtained by this improved method compare well with the results of the numerical method. The advantage of this method over the existing method is that acceptable accuracy is obtained in only one step for a wide range of cases.

I* Method of Complete Calculation of Turbulent Accelerated Smoothly Varying Flows in Prismatic Channels

I* Method of Complete Calculation of Turbulent Accelerated Smoothly Varying Flows in Prismatic Channels

Flip Bucket without and with Deflectors

Flip buckets are commonly used to discharge flow away from a hydraulic structure into a plunge pool to dissipate energy. In the past, flip buckets have often been designed in accordance with site-related hydraulic model studies. Consequently, limited generalized design guidelines are available. The present study considers flip buckets either in a prismatic rectangular channel or extended by a lateral deflector resulting in a curved jet trajectory. The main features of flip buckets are investigated, including scale effects in hydraulic models, bucket pressure distribution, and nappe trajectories with and without the presence of deflectors. An analysis is presented mainly involving the approach Froude number, the so-called bend number, and the bucket takeoff angle. It is demonstrated that the near field of a bucket-deflected jet follows the conventional parabola of a mass point, provided that the takeoff angle is correctly accounted for and that the flow is scaled using the Froude similarity law. Furthermor...

Limiting and Sill-Controlled Adverse-Slope Hydraulic Jump

This note analyzes, from a theoretical and experimental point of view, hydraulic jumps on adverse slopes in rectangular prismatic channels. The analysis is carried out for the classical adverse-slope hydraulic jump and the jump forced by the presence of a sill. Data collected in two series of experiments involving different equipment were added to available results to obtain a general relationship for the sequent depth ratio as a function of the upstream Froude number and adverse-slope angle. The presence of a sill stabilizes the jump.

Numerical simulation of A-type hydraulic jumps at positive steps

A numerical simulation of the A-type hydraulic jump at a positive step, which is an example of mixed supercritical-subcritical flow with a discontinuity at the channel bed, is given by using an integral approach. A gradually varied subcritical flow over a rectangular, horizontal, and prismatic channel with an abrupt bottom rise is considered as the initial condition. Then, the upstream depth is decreased to a value producing a supercritical flow and remaining unchanged during computations. The resulting unsteady flow is solved by using both the MacCormack and the dissipative two-four schemes for the one-dimensional, unsteady Saint-Venant equations. In the numerical simulation, the step is treated as an internal boundary. At the downstream and the internal boundaries, the method of characteristics is employed to compute the relevant parameters. The numerical simulation is verified by comparing the results with the available data and analytical methods.Key words: hydraulic jump, positive step, numerical simulation, internal boundary.

Nonkinematic Effects in Storm Hydrograph Routing

The propagation of flood waves down very long prismatic rectangular channels was numerically simulated using the full St. Venant equations in the full equations (FEQ) model. The importance of nonkinematic effects on the behavior of each flood wave was quantified by calculating the dimensionless width of the simulated looped rating curve, which is directly related to the sum of the nonkinematic terms in the equation of motion. Dimensionless loop widths, WL, were found to be closely related to the dimensionless wave period, τ, of Ponce et al. for a wide range of simulated bed slopes, channel widths, Manning n-values, and inflow hydrograph rise times. Since the variables defining τ can be estimated prior to detailed modeling, ranking of streams in order of decreasing values of estimated τ should roughly correspond to ranking in order of increasing nonkinematic effects. However, backwater and expansion/contraction effects are not captured by τ.

Computational-Reach Length for Prismatic Channels

In this paper, we present a methodology for the selection of reach length in unsteady, one-dimensional models of open channel flow. Using computational experiments, we determined an appropriate reach length for 425 combinations of channel and flow parameters. For each combination of parameters, we refined the finite difference grid until numerical error was reduced to 1.0% of a high resolution solution. We generated a set of dimensionless parameters and used regression analysis to relate channel and flood properties to an appropriate computational reach length.

Boundary Shear in Circular Pipes Running Partially Full

The distribution of boundary shear stress in circular conduits flowing partially full, with and without a smooth flat bed simulating deposited sediments, has been examined experimentally ranging from 0.375 < F < 1.96 (F = Froude number) and 65,000 < R < 342,000 (R = hydraulic radius), using the Preston tube technique. The invert level of the flat bed and the water depth have been varied to simulate a wide range of possible flow conditions that may occur in culverts, sewers, and hydropower tunnels. The distribution of boundary shear stress around the wetted perimeter is shown to be highly sensitive to changes in cross-sectional shape. The results have been analyzed in terms of the variation of local/global shear stress versus perimetric distance, and the percentage of the total shear force acting on the wall or bed of the conduit. The %SFw results (%SFw = the percentage of the average wall shear force of the overall shear force) have been shown to agree well with Knight's empirical formula for prismatic channels. The influence of secondary flows on the distribution of boundary shear stress and the implications of this for sediment transport have also been examined.

Distortedbcc Indium Cubes as Structural Motifs in Ca[TIn4] (T=Rh, Pd, Ir)

The title compounds were prepared from the elements by reactions in water-cooled glassy carbon crucibles under an argon atmosphere in a high-frequency furnace. CaPdIn4 crystallizes with the YNiAl4-type structure: Cmcm, a=446.7(3), b=1665(1), c=754.3(5) pm, wR2=0.0465 with 646 F2 values and 24 variables. The structure is built up from a complex three-dimensional [PdIn4] polyanion in which the calcium atoms occupy distorted pentagonal tubes formed by indium and palladium atoms. CaRhIn4 and CaIrIn4 adopt the LaCoAl4-type structure: Pmma, a=867.6(1), b=422.91(8), c=745.2(1) pm, wR2=0.0583 with 468 F2 values and 24 variables for CaRhIn4; a=869.5(1), b=424.11(6), c=746.4(1) pm, wR2= 0.0614 471 F2 values with 24 variables for CaIrIn4. This structure type, too, has a three-dimensional [RhIn4] polyanion which is related to the structure of binary RhIn3. The calcium atoms fill distorted pentagonal prismatic channels formed by indium atoms. Semi-empirical band structure calculations for Ca-RhIn4 and CaPdIn4 reveal strongly bonding In-In, Rh-In and Pd-In interactions but weaker Ca-Rh, Ca-Pd and Ca-In interactions. CaRhIn4 and Ca-PdIn4 are compared with other indium-rich compounds such as YCoIn5 and Y2CoIn8, and with elemental indium. Common structural motifs of the indium-rich compounds are distorted bcc-like indium cubes.

Structure, Chemical Bonding, and Properties of ZrIn2, IrIn2, and Ti3Rh2In3

ZrIn2, IrIn2, and Ti3Rh2In3 were prepared from the elements by reactions in sealed tantalum tubes. ZrIn2 and IrIn2 have previously been characterized only on the basis of X-ray powder data. Their structures were now refined from single-crystal X-ray diffractometer data: I41/amd, a=438.70(8) pm, c=2723.8(7) pm, wR2=0.0308, 561 F2 values, 14 variables for ZrIn2 (HfGa2-type); Fddd, a=980.1(1) pm, b=534.67(8) pm, c=1804.9(2) pm, wR2=0.0462, 522 F2 values, 17 variables for IrIn2 (Mg2Cu-type); and P62m, a=728.7(3) pm, c=306.7(1) pm, wR=0.0192, 261 F2 values, 12 variables for Ti3Rh2In3 (ordered Th3Pd5-type). ZrIn2 crystallizes with a superstructure of the cubic close-packing. Six different ordered fcc cells are stacked one upon each other. A group–subgroup scheme for various fcc superstructures is presented. Due to the distortions in the superstructure, the two crystallographically different indium atoms form separate networks: zig-zag chains by In1 and a square network by In2. The iridium atoms in IrIn2 form infinite chains with an Ir–Ir distance of 280 pm. Each iridium atom has a square antiprismatic indium coordination. Also in IrIn2 we observe two independent indium networks: the In1 atoms form a spiral-like three-dimensional network which is penetrated by planar two-dimensionally infinite layers of condensed enlongated In6 hexagons. LMTO electronic structure calculations revealed that the In–In interactions within the In networks of ZrIn2 and IrIn2 are slightly stronger than those between these networks. ZrIn2 and IrIn2 are Pauli paramagnetic and good metallic conductors. Ti3Rh2In3 is the first compound which adopts an ordered Th3Pd5-type structure. The rhodium and indium atoms form a two-dimensional [Rh2In3] network. The distorted pentagonal prismatic channels of the [Rh2In3] network are threaded by linear chains of titanium atoms with a Ti–Ti distance of 307 pm. According to LMTO electronic structure calculations, the strongest bonding interactions occur for the Ti–Rh contacts. Ti–Ti bonding in Ti3Rh3In3 is compared with Ti–Ti and Hf–Hf bonding in Ti2In5 and Hf2In5, respectively, and the hexagonal close-packed structures of titanium and hafnium.

EFFECT OF WALL SHEAR ON PARTICLE MOTION IN SEDIMENT LADEN OPEN CHANNEL FLOW

The paper deals with the concentration distributions of sediment particles in turbulent rectangular open channel flow. Special attention is paid on the effect of channel wall shear on the concentration profiles of sediment particles. Using the Reynolds strecc model, a mathematical model to describe the fluid and sic: din-mut motion in a prismatic open channel was presented. After confirming the validity of this model by laboratory experiments, sediment motions in a lateral cross section were discussed by this model. It was shown that the sediment particles flow downward with circularity trajectories like secondary current of fluid motion. Due to the settling of the sediment particles, the location of the center of the trajectory is somewhat near to the sidewall and to the bottom, compared with that of fluid motion. Experiments on sediment laden open channel flow, reported so far, were also reproduced numerically. In some cases of the reported paper, it was found that the circulatory motion of sediments spread almost over cross section. To the case of compound channels, the effect of relative height of the flood plane on sediment distributions were also investigated.

On gradually varied flow profiles in rectangular openchannels

This technical note presents the equation for one-dimensional gradually-varied steady flow profiles with constant discharge, in a rectangular-section prismatic channel of constant roughness and small longitudinal slope. The flow profile differential equation was transformed by means of a polynomial approximation and then, by an algebraic procedure followed by the integration of the equation, the solution was obtained. This solution was applied to a particular case which, through a comparative analysis, allowed to verify a satisfactory accuracy. The solution obtained here, though rather complex from an algebraic point of view, is easily programmable; its major merit consists in its contribution to the mathematical knowledge of flow profiles.

BOUNDARY SHEAR STRESS DISTRIBUTIONS IN SMOOTH RETANGULAR OPEN CHANNEL FLOWS.

Keywords NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPORE SHEAR, STRESS, DISTRIBUTION, BOUNDARIES, SMOOTH, RECTANGULAR, OPEN, CHANNELS, FLOW, METHODS, ANALYTICAL, FLUIDS, THREE DIMENSIONAL, PERIMETER, WETTED, STEADY, UNIFORM, ASPECT, RATIOS, MODELS, MATHEMATICAL, HYDRAULICS, HYDRODYNAMICS, RIVERS, THEORY, MECHANISMS, ENERGY, TRANSPORTATION, INFLUENCE, ROUGHNESS, DISSIPATION... Show All

New studies of nonuniform flows: Finite length and averaged hydraulic gradient of rapid flows in prismatic channels

New studies of nonuniform flows: Finite length and averaged hydraulic gradient of rapid flows in prismatic channels

Variable-Parameter Stage-Hydrograph Routing Method. I: Theory

An approach for developing a simplified variable-parameter stage-hydrograph routing method from the Saint Venant equations is proposed for routing floods in any shape of prismatic channel and flow following a generalized friction law. This approach enables one to relate the parameters of the routing equation to the channel and flow characteristics, and also enables the development of a theoretically based procedure for varying these parameters at every routing time level. Further, it allows the simultaneous computation of the discharge hydrographs corresponding to given input-stage and routed-stage hydrographs.