Channel Of Rectangular Cross-section Research Articles

Unstable Turbulent Flow in Open Channels

Controversy on the determination of the friction factor, f, for turbulent flow in open channels of rectangular cross section has existed among previous investigators. A number of them found that the Froude number affects the value of the friction factor, while others did not find such a dependence. In this study, the theoretical expression for the critical Froude number is first derived. Given a value of the Reynolds number, the flow becomes unstable above the critical Froude number. This critical Froude number, which appears to be strongly dependent upon the channel width, is then introduced in an equation by Rouse to calculate the values of the friction factor under unstable flow conditions. The influence of the Froude number on the friction factor appears to be important, but the channel width is a determinating factor as well.

Structures of three crystals containing approximately — linear chains of triiodide ions

Abstract The structures of the ordered crystals of tetra(n-butyl)ammonium triiodide [I: a = 15.791(8), b = 15.993(8), c = 9.578(5)Å, α = 74.48(8)°, β = 101.52(9), γ = 96.63(9)°, P[unk], Z =4)] and (benzamide)2 · HI3; [II: a = 20.824(10), b = 9.874(5), c = 9.620(5) Å, α = 95.58(9), β = 102.10(10), γ = 94.72(9)°, P[unk], Z=4] have been fully refined, while that of caffeine · H2O · HI3 [III: a = 14.043(5), b = 12.202(5), c = 9.701(5)Å, β = 106.5(1), P21/a, Z = 4] has been determined in outline because of unresolved problems of disorder. Intensity measurements were made on a four-circle diffractometer using graphite-monochromated MoKα. The numbers of reflections used in the final refinement cycle were: 15370, II5295, III 2971.I has cations of D 2 d symmetry (i.e. in the form of flattened crosses) which interleave to form channels of rectangular cross-section with axes along [001]; these channels contain single, almost-linear chains of triiodide ions, with weak I[unk] – I[unk] interactions within the chains. II is a pseudo-Type A basic salt, with pairs of benzamide molecules, (joined by a short, presumably symmetrical, proton bond between carbonyl groups) forming the cations; these cations are arranged so as to leave channels (axes along [001]) of rectangular cross-section which contain double, almost-linear chains of triiodide ions. In III the caffeine molecules are presumably protonated at N(9) and are hydrogen bonded via water molecules to form corrugated sheets, the molecular plane being in the sheets. These sheets are so disposed as to leave channels of rectangular cross-section which contain double chains of polyiodide ions. One of the three iodines is disordered along the chain direction and it was not possible to identify the nature of the chains ( –I[unk]

Flow Prediction in Rotating Ducts Using Coriolis-Modified Turbulence Models

A parabolic numerical analysis procedure has been used to predict the flow in a straight, radial rotating channel of rectangular cross-section, chosen as a simple model of an impeller passage. A two equation turbulence model was employed, with alternative modifications, to include the influence of Coriolis force on turbulent kinetic energy. Alternative Coriolis force terms were evaluated by comparisons in a high-aspect-ratio duct with measured velocity, wall shear stress and turbulent viscosity. Secondary velocity predictions were checked with data from a low-aspect-ratio duct where the Coriolis modification of turbulence was found less influential than the secondary flow in the modification of side wall shear stress.

Fonction de transfert d'un element de stockage par chaleur sensible fonctionnant en regime periodique

This paper deals with the extraction of the mean component of a periodic thermal power (solar energy for example). We show that such a filtering may be achieved by introducing a solid sensible heat thermal storage unit. We have defined a theoretical model for a specific heat storage configuration composed of a number of rectangular cross-section channels for the flowing fluid connected in parallel and separated by the solid storage material. Our model yields basic physical conclusions for the temperature fluctuations at the exit extremity and allows us to define an optimal geometry for the designs. The results are set explicitly for two different fluid-storage material couples: air-alumina and air-refractory brick

Ondes d'inertie dans un canal infini de section rectangulaire

Inertial waves in a channel of finite width consist of two different systems. In the surface layer a suddenly imposed wind produces the same system of Ekman currents and inertial waves as in the open ocean. Besides that the wind produces cross oscillations and a permanent deflection of the sea surface and due to this a geostrophic current. The onset of this geostrophic current is accompanied by a second system of inertial waves which is out of phase by 1800. In the vertically integrated equations the two systems cancel each other.

ON THE CALCULATION OF THE WIDTH AVERAGED FLOW DUE TO LONG WAVES IN AN OPEN CHANNEL

This paper describes the use of mean turbulence energy closure models for the calculation of the vertical structure of the width-averaged flow due to the propagation of long waves in an open channel of constant width and rectangular cross-section. Two turbulence models were considered which required the solution of one or both of the differential equations governing the turbulence energy and its dissipation rate. Numerical solutions were obtained for the motion induced by long waves in a laboratory channel and in the Humber Estuary. The computed values of surface velocity and bed shear stress in the laboratory channel were found to agree well with observed values. The calculation procedure also accurately reproduced the vertical structure of the tidal flow in the Humber Estuary. Existing theoretical relationships for the bed shear stress under long waves were verified by the numerical calculations and a distribution of the dissipation length scale, postulated on physical grounds, was also confirmed. It wa...

Gravity-capillary waves with edge constraints

This paper presents a theoretical and experimental investigation into a novel class of water-wave motions in narrow open channels. The distinctive condition on these motions is that the lines of contact between the free surface and the sides of the channel are fixed, which condition bears crucially on the hydrodynamic effects of surface tension. Most of the account concerns travelling waves in channels of rectangular cross-section that are exactly brimful, but the relevance of this prototype to other, more usual situations is explained with reference to the phenomenon of contact-angle hysteresis.In § 2 a linearized theory is developed which poses an eigenvalue problem of unusual kind. Unlike the familiar and much simpler problem corresponding to mobile lines of contact at which the free surface remains horizontal, the new problem has no explicit solution and the edge conditions are not automatically compatible with the kinematic conditions at the solid boundaries. Treatment by functional analytic methods is necessary to verify that solutions exist having physically appropriate properties, but this approach gives a final bonus in securing comparatively easy estimates for some of these properties. A variational characterization of the eigenvalues is used to settle questions of existence and the ordering of possible wave modes, and finally to establish approximate formulae relating wavelength to frequency.In § 3 experiments are reported which were performed with clean water filling three channels made of Perspex. Over continuous ranges of frequency, delimited so that only the fundamental progressive-wave mode was generated, wavelengths were measured by an electronic technique. The measurements agree well with the theoretical predictions, diverging markedly from behaviour to be expected in the absence of edge constraints.Appendix A outlines a supplementary theoretical argument proving that the first eigenvalue of the problem treated in § 2 is always simple. Appendix B reviews three generalizations of the theory.

FLOW IN A BEND IN A ROTATING OPEN CHANNEL

A constant flow down a circular bend in a rotating open channel of rectangular cross section is considered. By using the eddy viscosity concept combined with a finite slip velocity at the bed, simple analytical solutions of the problem are obtained. They show a helical motion induced by the combined effects of channel curvature and Coriolis force. For values of the inverse Rossby number close to zero we have the well known secondary flow away from the centre of curvature near the surface and towards it near the bottom. As the inverse Rossby number increases the secondary flow is strengthened or weakened depending on the relative direction of rotation of the basic flow. The basic flow is also shown to be strongly influenced by the inverse Rossby number.

Asymptotic drag formulas for rapidly rotating radial channels of rectangular cross section

Taking, as an example, the solution of the drag problem of a slitlike channel, the change in flow characteristics with increase in intensity of rotation is shown. The form of the asymptotic drag formulas is determined for rapidly rotating channels with sides in a finite ratio, at Rossby numbers of the order of unity. Good agreement with experimental data is found. The final form of the asymptotic solution, valid at small Rossby numbers, is found.

Flow of Liquid Metals in Curved Channels under a Transversely Applied Magnetic Field : 1st Report, Analysis of Laminar Flow

For obtaining the preliminary design data for MHD pumps, MHD power generators etc., the investigation of the flow of electrically conducting fluids in channels under an applied magnetic field is growing in importance. In this paper is reported an experimental and analytical study of flows in curved channels of rectangular cross section under a transversely applied magnetic field. Mercury is run in channels of relatively small radii of curvature where the secondary flow exerts a strong influence. The influence of the Reynolds number, the Hartmann number, the radius of curvature and the aspect ratio on the flow in curved channels is made clear, and an investigation with the aid of calculus of variations is also made.

Hydrodynamics and heat transfer in curvilinear channels of rectangular cross section

Experimental results of a study of hydraulic resistance and heat transfer in helical channels of rectangular cross section for laminar and turbulent flows in the isothermal and nonisothermal regimes are given.

Free vibration of clusters of cylinders in liquid-filled channels

This paper presents two methods for the calculation of the virtual masses of clusters of parallel cylinders in liquid contained by an outer channel. The first method is based on classical potential flow theory, and its applicability is limited to circular cylinders in circular cylindrical channels; the second, using fluid finite elements, may be applied to more general geometries. Illustrative examples compare the two methods, the classical one being shown to be computationally more efficient—whenever applicable. The eigenvalues and modal shapes are calculated for clusters in still fluid, in channels of circular and rectangular cross-section, showing the effect of proximity on the natural frequencies. Some distinctive patterns in the coupled motions of the cylinders are seen to occur, some not intuitively evident. The effect of viscous hydrodynamic coupling is also discussed.

Forces and Moments on a Ship Moving in a Shallow Channel

The method of matched asymptotic expansions is applied to the problem of a ship moving with constant velocity in a channel of rectangular cross section. The depth of the channel is presumed small so that a linearized shallow-water solution is developed. The ship is assumed to be slender and is replaced in the outer region by a distribution of sources and vortices. As in earlier theories for slender bodies translating in shallow water, the source strength is proportional to the change in cross-sectional area of the ship. An integral equation must be solved to find the vortex strength. Using the source and vortex strengths, expressions are developed for the side force, yaw moment, sinkage, and trim. Numerical results are presented which show good agreement with experimental results.

Motion of miscible and immiscible fluids in closed horizontal and vertical ducts

A theory of large lossless waves with two fluids in horizontal closed channels of arbitrary cross-section is developed. The dynamic conditions for infinitesimally small disturbances is derived from this theory and it is shown that the dynamic condition for waves of finite magnitude is in agreement with Long's (1956) first order estimate for small waves in channels of rectangular cross-section. It therefore appears probable that an adequately accurate dynamic condition is available for such waves of all sizes. Results from the theory are used to quantitatively explain experimental results by Wallis & Dobson (1973) for the onset of slugging in horizontal channels and by Leach & Thompson (1975) for the counter-current discharge of fluids along a horizontal pipes between closed tanks. In both cases an influence of the ratio of the densities of the phases upon the usually accepted Froude number, which already contains density correction factors, is found. Transfer of miscible unstably stratified fluids through each other in vertical ducts is found to be described by a turbulent diffusion process which is controlled by the rate of energy dissipation and the duct diameter. The process is therefore quite different to that in horizontal ducts.

Design Optimization of a Single Fluid, Solid Sensible Heat Storage Unit

The optimization of the design of a solid sensible heat storage unit initially at a uniform-temperature is presented. The storage unit is composed of a number of rectangular cross-sectional channels for the flowing fluid, connected in parallel and separated by the heat storage material. The complex method for constrained nonlinear optimization as presented by M. J. Box is utilized, with some modifications. The design optimization is based upon achieving maximum utilization of the heat storage or removal capabilities of the material for a given set of operating conditions. This is achieved by varying the storage unit’s geometry while placing constraints on the maximum and minimum length of the unit, fluid channel size, storage material thickness, maximum and minimum outlet fluid temperature, and the minimum amount of heat to be stored.

Velocity field of a liquid stream in a spiral channel of rectangular cross section

The equations for the velocity distribution of an ideal incompressible liquid in a rectangular tube twisted along a spiral line are established within the framework of the theory of a uniform spiral stream [1]. On the basis of the equations obtained, a numerical calculation is made of the velocity field in a hydroseparator [2] designed for the enrichment of valuable minerals. The results obtained are compared with the available experimental data, indicating their satisfactory qualitative agreement. Secondary flows develop during the flow of a liquid in curvilinear channels. The liquid near the upper and lower walls flows toward the inner side wall while the liquid in the central part of the stream flows from the inner to the outer walls. A qualitative explanation of this phenomenon is given in [3, 4]. Despite the fact that viscous forces play an important role in the formation of the secondary flow, a correct description of the flow structure can be obtained within the framework of the Gromeka equation. Good qualitative and quantitative agreement between experimental data and theoretical calculations of vortex flows made on the basis of the Gromeka equation has been obtained in many reports ([1, 5], for example).

Transient Response of Solid Sensible Heat Thermal Storage Units—Single Fluid

The transient response of a solid sensible heat storage unit which receives or supplies heat to a single flowing fluid is presented. The storage unit is composed of a number of rectangular cross-sectional channels for the flowing fluid, connected in parallel and separated by the heat storage material. The energy equation for the fluid and the transient conduction equation for the storage material are solved using finite difference techniques. The parameters which characterize the transient behavior of these units are identified. Results suitable for the prediction of the rate of heat storage and the outlet temperature of the fluid leaving the storage unit are presented as functions of the identified nondimensional parameters.

Slow flow of a conducting fluid in a curved channel of rectangular cross section

This paper presents an analysis of hydromagnetic slow motion in a curved channel of rectangular cross section in the presence of a magnetic field applied at right angles to the horizontal planes. The problem is considered in the cases when the flow takes place (i) due to a constant pressure gradient along the channel and (ii) due to rotation of the outer curved wall with a constant angular velocity. In both the cases, analytical solutions are obtained by finite Hankel transforms. The interaction between the velocity and magnetic fields is brought out with the help of numerically computed results.

Theory for Hinged Wavemakers of Finite Draft in Water of Constant Depth

Extending the theoretical work of Biesel, a study is made of a flap-type wavemaker hinged in the middle of a wall. The first-order linearized hydrodynamic equations of motion are solved to obtain the velocity potential, which is composed of a wave part and a spatially transitory part. Characteristics of generated waves far away from the wavemaker are analyzed as to the wave profile, flap stroke, water particle excursions, and power con- sumption of the wavemaker. The importance of the spatially transitory part is discussed with regard to the role played by the inertia pressure, which gives rise to added inertia of the flap. Surface elevations very near the flap are found. The hydrodynamically induced torque and the total hydrodynamic force on the flap are computed. The computed results seem to be well explained by the differing behaviors of the normal pressure and of the iner- tia pressure with increasing water depth. HE theoretical foundation of laboratory wave- generating apparatus was developed in detail by Biesel 1 and coworkers of Grenoble, pertaining to two-dimension al irrotational motions in prismatic channels of rectangular cross-section. Following the classical wavemaker theory of Havelock, Biesel solved the first-order linearized hydrodynamic equations of motion to find the resultant wave motions caused by a wavemaker at one end of a channel. As examples, he considered two simple cases, namely, a piston- type wavemaker, and a flap-type wavemaker hinged at the bottom of a wall. A brief comparison was made of the two types of wavemaker with regard to the quality of generated waves, wavemaker stroke, and the total hydrodynamic force applied at the wavemaker. The validity of the first-order wavemaker theory was experimentally verified by Ursell, Dean, and Yu,2 with particular reference to a piston-type wavemaker. For all practical purposes, however, flap-type wavemakers in common use today are fabricated with their lower ends hinged in the middle of the wall, unlike the simple flap-type wavemaker investigated by Biesel.'An article by Taniguchi and Shibata3 reported that agreement was satisfactory bet- ween the predictions of the first-order wavemaker theory and the measurements conducted in the Mitsubishi Nagasaki ex- perimental tank. The Nagasaki tank was equipped with a flap- type wavemaker, whose flap draft (the distance from the un- disturbed mean water surface to the lower hinges of the flap) was 1.77 m, and the constant water depth was 6.31 m. Their measurements dealt with flap frequency, flap stroke, wave profile, and the reaction force at the piston rod of the hydraulic drive. As was pointed out by Taniguchi and Shibata, published theses on wavemakers are rather scarce, and few scattered technical data for wavemaker performance are available in the literature. In most descriptive reports of existing towing tank facilities,4'5'6 wavemakers seem to be somewhat cursorily treated without providing much technical information. A unified account of a flap-type wavemaker, including detailed analysis of the added inertia effect of the flap, which was not previously discussed, is presented here. Relevant theoretical considerations, based on the first-order linearized hydrodynamic equations of motion, are given and possible analytical approaches are used. Treating the flap draft as a parameter, numerical computations are carried out to study the quality of generated waves and the hydrodynamically in- duced force and torque on the flap. The added inertia effect of the flap, which is a consequence of the spatially transitory part of fluid motion, is closely examined with respect to the flap torque augmentation and water particle displacements near the wavemaker. The analysis of a flap-type wavemaker is of practical usefulness in that it can be utilized for more general purposes, including the determination of the added mass and the damping coefficient of thin vertical bodies in water.

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.