Dispersion in Laminar Flow Through Tubes by Simultaneous Diffusion and Convection

1981 ◽  
Vol 48 (2) ◽  
pp. 217-223 ◽  
Author(s):  
J. S. Yu
Keyword(s):  
Laminar Flow ◽  
Molecular Diffusion ◽  
Solute Concentration ◽  
Dimensionless Time ◽  
Mean Flow ◽  
Round Tube ◽  
Flow Through ◽  
The Mean ◽  
Time Required ◽  
Mean Concentration

The dispersion of a small quantity of a solute initially injected into a round tube in which steady-state laminar flow exists is critically examined. It is shown that the mean solute concentration profile is far from being symmetric at small dimensionless times after injection. The mean concentration and the axial location at the peak of the profile are presented in detail as functions of time for flow with various Peclet numbers. It is suggested that such results may be useful for determining either the molecular diffusion coefficient or the mean flow velocity or both from experimental measurements. A previously established criterion in terms of the Peclet number for determining the minimum dimensionless time required for applying Taylor’s theory of dispersion is graphically illustrated. Although the complete generalized dispersion equation of Gill’s model is exact, the truncated two-term form of it with time-dependent coefficients is exact only asymptotically at large values of time; however, at small Peclet numbers, the two-term approximation is shown graphically to be reasonably satisfactory over all values of time. The exact series solution is compared with the solution of Tseng and Besant through the use of Fourier transform.

Dispersion of gases in laminar flow through a circular tube

1961 ◽  
Vol 261 (1305) ◽  
pp. 227-236 ◽  
Keyword(s):  
Molecular Weight ◽  
Laminar Flow ◽  
Molecular Diffusion ◽  
Circular Tube ◽  
Ultra Violet ◽  
Narrow Beam ◽  
Violet Light ◽  
Maximum Value ◽  
Flow Through ◽  
Mean Concentration

The longitudinal dispersion of a finite slug of gas has been measured at various velocities by using a gas (1,3-butadiene) that absorbs in the ultra-violet region and passing the dispersed slug through a narrow beam of ultra-violet light of wavelength 250mμ. To avoid effects of differences in gas density, the second gas (1-butyne) was chosen to have the same molecular weight as butadiene. The range of applicability of Sir Geoffrey Taylor’s virtual coefficient of diffusivity has been discussed. The experimental observation that the peak mean concentration passes through a maximum value with velocity has been explained by considering the relative rates of dispersion by convection, longitudinal molecular diffusion and radial molecular diffusion.

Extended Stokes series: laminar flow through a loosely coiled pipe

1978 ◽  
Vol 86 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Milton Van Dyke
Keyword(s):  
Laminar Flow ◽  
Flow Speed ◽  
Square Root ◽  
Mean Flow ◽  
Dean Number ◽  
Curvature Ratio ◽  
Wide Range ◽  
Square Root Singularity ◽  
Flow Through ◽  
The One

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than$\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.

Hot-Wire Measurements of Turbulence Correlations in a Triangular Duct

1962 ◽  
Vol 29 (4) ◽  
pp. 609-614 ◽  
Author(s):  
C. J. Cremers ◽  
E. R. G. Eckert
Keyword(s):  
Heat Transfer ◽  
Laminar Flow ◽  
Cross Section ◽  
Circular Tube ◽  
Turbulent Transport ◽  
Circular Cross Section ◽  
Hot Wire ◽  
Round Tube ◽  
Flow Through ◽  
Hot Wires

Previous studies by flow visualization have indicated that the flow through a duct of triangular cross section is in its characteristics quite different from flow through a duct with circular cross section. They revealed among others that purely laminar flow exists in the corners of the duct even though the bulk of the fluid moves in turbulent motion. Heat-transfer measurements in such a duct appear to indicate that the turbulent transport in the direction of the height of the duct is considerably smaller than expected from circular tube measurements. The present paper reports the measurements of turbulent correlations for turbulent flow through such a duct. These measurements have been made with hot wires of very small dimensions. They again reveal the existence of a laminar corner region. In the bulk of the fluid, the differences of the correlations to those in a round tube turned out to be smaller than originally suspected.

The dispersion of matter in turbulent flow through a pipe

1954 ◽  
Vol 223 (1155) ◽  
pp. 446-468 ◽  
Keyword(s):  
Turbulent Flow ◽  
Capillary Tube ◽  
Resistance Coefficient ◽  
Mean Flow ◽  
Combined Action ◽  
Flow Through ◽  
The Mean ◽  
Coefficient Of Diffusion ◽  
Good Agreement ◽  
Made In

The dispersion of soluble matter introduced into a slow stream of solvent in a capillary tube can be described by means of a virtual coefficient of diffusion (Taylor 1953 a ) which represents the combined action of variation of velocity over the cross-section of the tube and molecluar diffusion in a radial direction. The analogous problem of dispersion in turbulent flow can be solved in the same way. In that case the virtual coefficient of diffusion K is found to be 10∙1 av * or K = 7∙14 aU √ γ . Here a is the radius of the pipe, U is the mean flow velocity, γ is the resistance coefficient and v * ‘friction velocity’. Experiments are described in which brine was injected into a straight 3/8 in. pipe and the conductivity recorded at a point downstream. The theoretical prediction was verified with both smooth and very rough pipes. A small amount of curvature was found to increase the dispersion greatly. When a fluid is forced into a pipe already full of another fluid with which it can mix, the interface spreads through a length S as it passes down the pipe. When the interface has moved through a distance X , theory leads to the formula S 2 = 437 aX ( v * / U ). Good agreement is found when this prediction is compared with experiments made in long pipe lines in America.

A theoretical study of viscous effects in peristaltic pumping

1994 ◽  
Vol 279 ◽  
pp. 177-195 ◽  
Author(s):  
Alden M. Provost ◽  
W. H. Schwarz
Keyword(s):  
Wave Propagation ◽  
Pressure Gradient ◽  
Flow Rate ◽  
Mean Flow ◽  
Viscous Effects ◽  
Transverse Waves ◽  
Peristaltic Pumping ◽  
Mean Pressure ◽  
Flow Through ◽  
The Mean

Intuition and previous results suggest that a peristaltic wave tends to drive the mean flow in the direction of wave propagation. New theoretical results indicate that, when the viscosity of the transported fluid is shear-dependent, the direction of mean flow can oppose the direction of wave propagation even in the presence of a zero or favourable mean pressure gradient. The theory is based on an analysis of lubrication-type flow through an infinitely long, axisymmetric tube subjected to a periodic train of transverse waves. Sample calculations for a shear-thinning fluid illustrate that, for a given waveform, the sense of the mean flow can depend on the rheology of the fluid, and that the mean flow rate need not increase monotonically with wave speed and occlusion. We also show that, in the absence of a mean pressure gradient, positive mean flow is assured only for Newtonian fluids; any deviation from Newtonian behaviour allows one to find at least one non-trivial waveform for which the mean flow rate is zero or negative. Introduction of a class of waves dominated by long, straight sections facilitates the proof of this result and provides a simple tool for understanding viscous effects in peristaltic pumping.

Improved κ-ɛ Modelling of Impeller Flow Performance of a Mixed-Flow Pump under off-Design Operating States

1993 ◽  
Vol 207 (4) ◽  
pp. 219-229 ◽  
Author(s):  
S M Fraser ◽  
Y Zhang
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Navier Stokes ◽  
Mean Flow ◽  
Mixed Flow ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
The Mean ◽  
Flow Pump

Three-dimensional turbulent flow through the impeller passage of a model mixed-flow pump has been simulated by solving the Navier-Stokes equations with an improved κ-ɛ model. The standard κ-ɛ model was found to be unsatisfactory for solving the off-design impeller flow and a converged solution could not be obtained at 49 per cent design flowrate. After careful analysis, it was decided to modify the standard κ-ɛ model by including the extra rates of strain due to the acceleration of impeller rotation and geometrical curvature and removing the mathematical ill-posedness between the mean flow turbulence modelling and the logarithmic wall function.

Oscillatory forcing of flow through porous media. Part 2. Unsteady flow

2002 ◽  
Vol 465 ◽  
pp. 237-260 ◽  
Author(s):  
D. R. GRAHAM ◽  
J. J. L. HIGDON
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Pressure Gradient ◽  
Flow Rate ◽  
Flow Through Porous Media ◽  
Mean Flow ◽  
Two Fluids ◽  
Mean Pressure ◽  
Flow Through ◽  
The Mean

Numerical computations are employed to study the phenomenon of oscillatory forcing of flow through porous media. The Galerkin finite element method is used to solve the time-dependent Navier–Stokes equations to determine the unsteady velocity field and the mean flow rate subject to the combined action of a mean pressure gradient and an oscillatory body force. With strong forcing in the form of sinusoidal oscillations, the mean flow rate may be reduced to 40% of its unforced steady-state value. The effectiveness of the oscillatory forcing is a strong function of the dimensionless forcing level, which is inversely proportional to the square of the fluid viscosity. For a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing may be used to reduce the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. The temporal waveform of the oscillatory forcing function has a significant impact on the effectiveness of this technique. A spike/plateau waveform is found to be much more efficient than a simple sinusoidal profile. With strong forcing, the spike waveform can induce a mean axial flow in the absence of a mean pressure gradient. In the presence of a mean pressure gradient, the spike waveform may be employed to reverse the direction of flow and drive a fluid against the direction of the mean pressure gradient. Owing to the viscosity dependence of the dimensionless forcing level, this mechanism may be employed as an oscillatory filter to separate two fluids of different viscosities, driving them in opposite directions in the porous medium. Possible applications of these mechanisms in enhanced oil recovery processes are discussed.

Mercury fractionation in sediments of the Lower Vistula River (Poland)

2007 ◽  
Vol 36 (3) ◽  
Author(s):  
Leonard Boszke ◽  
Artur Kowalski
Keyword(s):  
Baltic Sea ◽  
Catchment Area ◽  
Total Mercury ◽  
The Baltic Sea ◽  
Vistula River ◽  
Mercury Fractionation ◽  
Flow Through ◽  
The Mean ◽  
The Baltic ◽  
Mean Concentration

Mercury fractionation in sediments of the Lower Vistula River (Poland)The Vistula is the second largest river in the Baltic Sea catchment area and provides one of the main inputs to the Baltic. The river and its tributaries flow through some of the major industrialized and urbanised regions of Poland, making it one of the most highly human-impacted rivers in Europe. Although the river status is monitored routinely, little is known about mercury forms in the sediments. This study examines mercury fractionation in the sediments of the lower part of the Vistula River. The results show that the cities along this stretch of river have a relatively low impact on both the mercury forms found in the sediment and its bioavailability in the floodplain soils. The mean concentration of total mercury in the sediments was 65 ± 14 ng g

The influence of mean shear on unsteady aperture flow, with application to acoustical diffraction and self-sustained cavity oscillations

1981 ◽  
Vol 109 ◽  
pp. 125-146 ◽  
Author(s):  
M. S. Howe
Keyword(s):  
Leading Edge ◽  
Vortex Sheet ◽  
Mean Flow ◽  
Kutta Condition ◽  
Sustained Oscillations ◽  
Linearized Theory ◽  
Flow Through ◽  
The Mean ◽  
Theoretical Results ◽  
Disturbed Motion

This paper discusses the linearized theory of unsteady flow through a two-dimensional aperture in a thin plate in the presence of a grazing mean flow on one side of the plate. The mean shear layer is modelled by a vortex sheet, and it is predicted that at low mean-flow Mach numbers there is a transfer of energy from the mean flow to the disturbed motion of the vortex sheet provided (i) the Kutta condition is imposed at the leading edge of the aperture, resulting in the unsteady shedding of vorticity from the edge, and (ii) the width of the aperture 2s satisfies ½ < 2s/λ < 1.1, where λ is the hydrodynamic wavelength of the disturbance on the vortex sheet within the aperture. The theory is used to examine the effect of mean shear on the diffraction of sound by a perforated screen, and to predict the spontaneous excitation and suppression of self-sustained oscillations in a wall-cavity beneath a nominally steady mean flow. In the latter case support for the proposed theory is provided by a favourable comparison of theoretical results with experimental data available in the literature.

Relaminarization in highly accelerated turbulent boundary layers

1973 ◽  
Vol 61 (3) ◽  
pp. 417-447 ◽  
Author(s):  
R. Narasimha ◽  
K. R. Sreenivasan
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Pressure Gradient ◽  
Laminar Boundary Layer ◽  
Outer Layer ◽  
Transitional Region ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Turbulence Decay ◽  
The Mean

The mean flow development in an initially turbulent boundary layer subjected to a large favourable pressure gradient beginning at a point x0 is examined through analyses expected a priori to be valid on either side of relaminarization. The ‘quasi-laminar’ flow in the later stages of reversion, where the Reynolds stresses have by definition no significant effect on the mean flow, is described by an asymptotic theory constructed for large values of a pressure-gradient parameter Λ, scaled on a characteristic Reynolds stress gradient. The limiting flow consists of an inner laminar boundary layer and a matching inviscid (but rotational) outer layer. There is consequently no entrainment to lowest order in Λ−1, and the boundary layer thins down to conserve outer vorticity. In fact, the predictions of the theory for the common measures of boundary-layer thickness are in excellent agreement with experimental results, almost all the way from x0. On the other hand the development of wall parameters like the skin friction suggests the presence of a short bubble-shaped reverse-transitional region on the wall, where neither turbulent nor quasi-laminar calculations are valid. The random velocity fluctuations inherited from the original turbulence decay with distance, in the inner layer, according to inverse-power laws characteristic of quasi-steady perturbations on a laminar flow. In the outer layer, there is evidence that the dominant physical mechanism is a rapid distortion of the turbulence, with viscous and inertia forces playing a secondary role. All the observations available suggest that final retransition to turbulence quickly follows the onset of instability in the inner layer.It is concluded that reversion in highly accelerated flows is essentially due to domination of pressure forces over the slowly responding Reynolds stresses in an originally turbulent flow, accompanied by the generation of a new laminar boundary layer stabilized by the favourable pressure gradient.

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