Showing papers on "Open-channel flow published in 1970"

The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe

Abstract: Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold.

Initial Instability of Fine Bed Sand

Abstract: The initial entrainment characteristics of fine bed sand are objectively defined in terms of the measured distributions of critical instantaneous bed shear stress associated with the observed movement of individual surface grains. Critical flow conditions are then predicted by equating the lower extremes in these characteristic critical shear stress distributions to the upper extremes in the distribution of instantaneous bed shear stress produced by the particular type of background flow under consideration. The method is applied to the case of two-dimensional channel flow over a flat bed and yields sufficiently consistent results to suggest an extension to Shields' curve for small grain Reynolds numbers. The detailed experimental observations clearly illustrate the important role played by bed region turbulence in the interaction process between the fluid and the mobile bed grains constituting the deformable boundary.

Deposition of Particles in Turbulent Flow on Channel or Pipe Walls

Abstract: This paper describes a method for predicting the deposition of particles entrained in turbulent flow. The method predicts the deposition of particles ranging from molecular size to ∼100µ in diamete...

Entrainment and the structure of turbulent flow

Abstract: Although the entrainment of non-turbulent fluid into a turbulent flow occurs across sharply defined boundaries, its rate is not determined solely by the turbulent motion adjacent to the interface but depends on overall properties of the flow, in particular, on those that control the energy balance. In the first place, attention is directed to the many observations which show that the motion in many turbulent shear flows has a structure closely resembling that produced by a rapid, finite, plane shearing of initially isotropic turbulence. The basic reasons for the similarity are the stability and permanence of turbulent eddies and the finite distortions undergone by fluid parcels in free turbulent flows. Next, the existence of eddy similarity and the condition of overall balance of energy are used to account for the variation of entrainment rates within groups of broadly similar flows, in particular mixing layers between streams of different velocities and wall jets on curved surfaces. For some flows which satisfy the ordinary conditions for self-preserving development, no entrainment rate is consistent with the energy balance and self-preserving development is not possible. Examples are the axisymmetric, small-deficit wake and the distorted wake. Finally, the implications of an entrainment rate controlled by the general motion are discussed. It is concluded that the relatively rapid entrainment in a plane wake depends on an active instability of the interface, not present in a constant-pressure boundary layer whose slow rate of entrainment is from ‘passive’ distortion of the bounding surface by eddies of the main turbulent motion. Available observations tend to support this conclusion.

Vertical Transfer in Open Channel Flow

Abstract: Vertical turbulent mass transfer coefficients for suspended sediment particles and dye, and fall velocities of the sediment particles were determined experimentally in a large laboratory flume. Local values of the turbulent mass transfer coefficient, and turbulent fall velocities for the sand, were determined from an integrated form of the transfer equation. The results of the experiments indicate that: (1) Magnitude and distribution of the vertical turbulent mass transfer coefficient for dye and fine sediment are close to those of the momentum transfer coefficient, which lends added support to Reynolds' analogy for the equivalence of mass and momentum transfer in turbulent shear flow; (2) in comparison with dye, the mass transfer coefficient for medium sand is somewhat less in magnitude and its distribution indicates that the transfer coefficient is larger near the bed; and (3) the fall velocity of sediment particles is larger in turbulent channel flow than in quiescent water.

Bounds for turbulent shear flow

Abstract: Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.

Flume Studies of the Sediment Transfer Coefficient

Abstract: Flume experiments show that in open channel flow the sediment transfer coefficient increases with distance from the channel bed, reaching a maximum value at a distance from the bed equal to about 20% of the flow depth, and then tends to remain at a constant value up to the water surface. The experiments also yield evidence indicating that the sediment transfer coefficient is larger for larger particle sizes.

Onset of Convection in a Porous Channel with Net Through Flow

Abstract: The critical Rayleigh number for convective flow in a porous two‐dimensional channel is found when there is a net flow of fluid up through the channel.

Boundary Shear Distribution in Open Channel Flow

Abstract: BOUNDARY SHEAR DISTRIBUTION IN BOTH ROUGH AND SMOOTH OPEN CHANNELS OF RECTANGULAR AND TRAPEZOIDAL SECTIONS IS OBTAINED BY DIRECT MEASUREMENT OF SHEAR DRAG ON AN ISOLATED LENGTH OF THE TEST CHANNEL UTILIZING THE TECHNIQUE OF THREE POINT SUSPENSION SYSTEM SUGGESTED BY BAGNOLD. EXISTING SHEAR MEASUREMENT TECHNIQUES HAVE BEEN REVIEWED CRITICALLY. COMPARISON OF THE MEASURED DISTRIBUTION HAS BEEN MADE WITH OTHER INDIRECT ESTIMATES, FROM ISOVELS, AND PRESTON TUBE MEASUREMENTS, BASED ON KEULEGAN'S RESISTANCE LAWS. THE DISCREPANCIES BETWEEN THE DIRECT AND INDIRECT ESTIMATES ARE EXPLAINED. OF THE TWO INDIRECT ESTIMATES THE SURFACE PITOT TUBE TECHNIQUE IS FOUND TO BE MORE RELIABLE. THE INFLUENCE OF SECONDARY FLOW ON THE BOUNDARY SHEAR DISTRIBUTION COULD NOT BE ACCURATELY DEFINED IN THE ABSENCE OF A DEPENDABLE THEORY ON SECONDARY FLOW. /ASCE/

Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel

Abstract: The stability of the laminar flow in a rectangular channel with aspect ratio 1:8 was investigated experimentally, with and without artificial excitation. The critical Reynolds number based on the hydraulic diameter and the average velocity was found to be 2600. Behaviour of damped and growing waves, using artificial excitation, was examined in detail. In particular the progress of growing disturbances was followed. Breaking was found to be the ultimate fate of a growing wave. Spectra of growing and damped waves were also obtained. Measurements were made for wavelengths, wave speeds and amplification or damping rates. The neutral stability boundary in the αr, R plane was determined. In the damped region, comparison of several aspects of the behaviour of the measured disturbances with the plane Poiseuille theory for spatial decay yielded good agreement.Three-dimensionality and non-linear subcritical instability were briefly examined. Neutral subcritical waves at low Reynolds numbers appeared possible when the exciter amplitude was quadrupled.The possible bearings of the present study on the stability of plane Poiseuille flow are suggested.

Turbulence Structure in Plane Couette Flow

Abstract: Turbulent flow between two plane surfaces in relative shearing motion has been studied with air in a belt-type apparatus with one fixed surface. Mean velocities, turbulence intensities, scales and energy spectra measured in this flow indicate two regions of appreciably different characteristics. A wall region near the plane surfaces is found to occur with flow characteristics close to those found near walls of other turbulent shear flows when these are expressed in terms of the wall similarity parameters. A core region, of width at least three-eights the distance between the planes, appears away from these evidencing the characteristics of the homogeneous turbulence predicted by von Karman. The mean velocity has a linear gradient in this region and the turbulence intensity and scales are essentially constant; in fact, homogeneity is strongly suggested. For a more than two-fold range in flow Reynolds numbers (center line velocity 15 fps to 35 fps) the microscale is found to be constant, the macroscale increases linearly and the relative intensity is constant.

The flow created by a sphere moving along the axis of a rotating, slightly-viscous fluid

Abstract: The title flow has been studied by measuring the drag force on, and by observing the flow field around, a sphere rising through a large, rotating tank of water. Long, almost stagnant, regions are formed up- and downstream within the shadow of the sphere and are surrounded by a thin annular region within which the velocity is larger than the mean velocity of the approach flow. Several regions are found within which vortex-jump phenomena occur and it is concluded that such features exert a controlling influence over the dynamics of the observed flow field.

Dynamic Equations for Steady Spatially Varied Flow

Abstract: Dynamic equations for steady spatially varied flow in uniform channels are derived separately based on momentum and energy principles. It is shown that the momentum equation is inherently different from the energy equation. In general, six different gradients are involved in steady spatially varied flow: the friction slope in the momentum equation, the dissipated energy gradient in the energy equation, the total head gradient, the gradient of the piezometric head, the free surface slope, and the channel slope. Conventional practice of using the Manning, Chezy, or Weisbach formulas to evaluate the friction slope or the dissipated energy gradient is only an approximation.

Free Surface, Velocity Gradient Flow Past Hemisphere

Abstract: The drag on a hemisphere in a finite-flow field with free surface effects is evaluated by successively studying six flow fields, which allows the study of one new variable at a time. These flow fields are: the semi-infinite uniform and nonuniform flow fields, the finite uniform and nonuniform flow fields without free surface effects, and the finite uniform and nonuniform flow fields with free surface effects. Three regimes of flow are defined for the hemisphere which are: (1) A regime of pronounced free surface effects, (2) a regime of moderate free surface effects, and (3) a regime of negligible free surface effects. The effect of these variables and range of conditions under which they are effective is considered.

Effects of density differences on lateral mixing in open-channel flows

Abstract: This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also. Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source. A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

On the turbulent flow over a wavy boundary

Abstract: Two hypotheses concerning the turbulent flow over an infinitesimal-amplitude travelling wave are investigated. One hypothesis, originally made by Miles, is that the wave does not affect the turbulence and therefore the turbulent Reynolds stresses are dependent only on height above the mean wave surface. Alternatively, the proposal that turbulent stresses are primarily dependent on height above the instantaneous wave surface is examined. Numerical solutions of the appropriate equations are compared with Stewart's recent experimental results and with the approximate solutions employed by Miles and others. No definite conclusion can be reached from comparison with experimental results since the predicted flows are quite sensitive to details of the mean velocity profile near the wave surface where no data was taken. It is found that the asymptotic results do not apply for the conditions investigated.

Pattern of Potential Flow in a Free Overfall

Abstract: A theory is presented to describe the two-dimensional flow of liquid in an open channel which terminates abruptly at a sharp drop A dimensionless diagram expressing constancy of total head on the upper free surface and similar to the conventional specific-head versus depth and discharge versus depth diagrams is used to trace the course of the flow in the overfall Together with the fact that the ratio of free-surface-velocity magnitude to average horizontal component of velocity in the cross-section is a decreasing function of distance in the direction of flow, an observation drawn from a subsequent, detailed examination of the flow, the diagram is used to show that the limiting case of the free overfall is at a Froude number of approach of unity A detailed description of the flow is given by an integral equation derived from conformal mapping and singularity distributions Comparisons are made between the computed profiles and theoretical and experimental results of other investigators

Nonequilibrium Transport of Suspended Sediment

Abstract: The nonequilibrium transport of suspended sediment in an open channel is studied at an abrupt change from a nonerodible to an erodible bed. The convective-diffusion equation is solved for the case of constant velocity distribution and parabolic eddy diffusion coefficient. Diffusion in the direction of flow is neglected. Graphs of the concentration distribution, the channel-depth average concentration and the scour rate are presented.

Distribution of tractive force in open channels

Abstract: The designs of alluvial channels by the tractive force method requires information on the distribution of wall shear stress over the wetted perimeter of the cross-section. The present study was undertaken in order to provide some details on actually measured shear distributions and, hence, to check the validity of currently available design information. The latter is entirely analytical in origin and is based either on the assumption of laminar flow or on over-simplified models of turbulent channel flow. The experiments were carried out in a smooth-walled laboratory flume at various aspect ratios of the rectangular cross-section. Wall shear stress was determined with Preston tubes which were calibrated by a method exploiting the logarithmic form of the inner law of velocity distribution. Results are presented which clearly suggest that none of the present analytical techniques can be counted upon to provide any precise details on tractive force distributions in turbulent channel flow.

Bed Topography Effect on Flow in a Meander

Abstract: Stabilized bed topography in a meandering channel with fixed walls and movable bed was measured at various width-depth ratios and Froude numbers. Velocity and bed-shear distributions were measured in a fixed-bed model whose configuration conformed to the stabilized topography of the movable-bed model. An approximate theoretical analysis of the bed configuration under fully developed flow conditions shows that the transverse bed slope is directly proportional to a parameter characterizing the relative importance of fluid inertia compared with the gravity acting on a particle. Good agreement has been observed between the theoretical and experimental results in the downstream half of the model channel. The effects of formation of the bed topography on the flow characteristics are considered.

Computation of Open-Channel Surges and Shocks

Abstract: Unsteady flows in fixed-bed open channels have been computed for a number of years on the basis of the same assumptions as those which had previously underlain calculation of steady, gradually varied flow, the key one being hydrostatic pressure distribution in every cross-section. The St. Venant equations comprising the equations of motion and continuity for this case commonly are solved either by the method of characteristics or by finite differences in a rectangular network in the x-t plane. Positive waves are characterized by converging characteristics; once these intersect a bore or shock forms, and many customary methods of solution fail. The equations of motion for flow with and without shocks are compared, the generation of bores is analyzed by exact solution of characteristic equations, and several numerical schemes are presented for computation of unsteady flows that may contain shock zones. Comparison is made between computed results and experimental data gathered for a variety of cases; corroboration is generally good, though computation based upon characteristics is inconsistent and generates the largest errors.

Incompressible Laminar Flow in the Entrance Region of a Rectangular Duct

Abstract: Incompressible laminar flow in entrance region of rectangular duct allowing direct computation of eigenvalues

Predicting Concentration Profiles in Open Channels

Abstract: A steady state two-dimensional conservation of mass equation is solved numerically in order to describe the process of vertical mass transfer for both neutrally buoyant liquid and suspended sediment particles in an open channel flow. The relevancy of the equation is illustrated by comparing its solution with laboratory data obtained from experiments conducted in a large flume. The solution is then used to graphically illustrate the effects of various hydraulic parameters on mass transfer. Parameters investigated include the velocity distribution, the magnitude and distribution of the turbulent mass transfer coefficient, the particle fall velocity and the boundary conditions at the channel bed. The solutions indicate that the fall velocity controls the rate of descent of the dispersant mass but has little effect on the rate of spread of the dispersant. Conversely, the transfer coefficient controls the rate of spread of the dispersant but has little effect on its rate of descent. The most striking conclusion drawn from this study is that the dispersant distribution is not very sensitive to the distribution of the turbulent mass transfer coefficient.