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Showing papers on "Reynolds number published in 1969"


Journal ArticleDOI
TL;DR: In this article, the authors investigated the effect of peristaltic wave propagation on the flow of fluid in a tube and showed that the theoretical pressure rise per wavelength decreases linearly with increasing time-mean flow and that the percentage of reflux flow can be very high.
Abstract: Pumping by means of an infinite train of peristaltic waves is investigated under conditions for which the relevant Reynolds number is small enough for inertial effects to be negligible and the wavelength to diameter ratio is large enough for the pressure to be considered uniform over the cross-section. Theoretical results are presented for both plane and axisymmetric geometries, and for amplitude ratios ranging from zero to full occlusion. For a given amplitude ratio, the theoretical pressure rise per wavelength decreases linearly with increasing time-mean flow. An experiment with a quasi-two-dimensional apparatus confirmed the theoretical values.Calculations of the detailed fluid motions reveal that under many conditions of operation the net time-mean flow is the algebraic difference between a forward time-mean flow in the core of the tube and a backward (‘reflux’) time-mean flow near the periphery. The percentage of reflux flow can be very high. This reflux phenomenon is probably of physiologic significance in the functioning of the ureter and the gastro-intestinal system. A second fluid-mechanical peculiarity with physiological implications is that of ‘trapping’: under certain conditions an internally circulating bolus of fluid, lying about the axis, is transported with the wave speed as though it were trapped by the wave.

1,298 citations


Journal ArticleDOI
TL;DR: In this article, a detailed experimental study of turbulent boundary-layer development over rough walls in both zero and adverse pressure gradients is described, where the skin friction was determined by pressure tapping the roughness elements and measuring their form drag.
Abstract: This paper describes a detailed experimental study of turbulent boundary-layer development over rough walls in both zero and adverse pressure gradients. In contrast to previous work on this problem the skin friction was determined by pressure tapping the roughness elements and measuring their form drag.Two wall roughness geometries were chosen each giving a different law of behaviour; they were selected on the basis of their reported behaviour in pipe flow experiments. One type gives a Clauser type roughness function which depends on a Reynolds number based on the shear velocity and on a length associated with the size of the roughness. The other type of roughness (typified by a smooth wall containing a pattern of narrow cavities) has been tested in pipes and it is shown here that these pipe results indicate that the corresponding roughness function does not depend on roughness scale but depends instead on the pipe diameter. In boundary-layer flow the first type of roughness gives a roughness function identical to pipe flow as given by Clauser and verified by Hama and Perry & Joubert. The emphasis of this work is on the second type of roughness in boundary-layer flow. No external length scale associated with the boundary layer that is analogous to pipe diameter has been found, except perhaps for the zero pressure gradient case. However, it has been found that results for both types of roughness correlate with a Reynolds number based on the wall shear velocity and on the distance below the crests of the elements from where the logarithmic distribution of velocity is measured. One important implication of this is that a zero pressure gradient boundary layer with a cavity type rough wall conforms to Rotta's condition of precise self preserving flow. Some other implications of this are also discussed.

548 citations



Journal ArticleDOI
TL;DR: In this paper, skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000, and observations of hot-wire signals indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term "fully developed turbulent flow".
Abstract: Measurements of skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000 These measurements, and observations of hot-wire signals, indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term ‘fully developed turbulent flow’

425 citations


Journal ArticleDOI
TL;DR: In this article, the flow around a circular cylinder has been examined over the Reynolds number range 105 to 7·5 × 105, Reynolds number being based on cylinder diameter, and Narrow-band vortex shedding has been observed up to a Reynolds number of 5·5× 105, i.e. well into the critical regime.
Abstract: The flow around a circular cylinder has been examined over the Reynolds number range 105 to 7·5 × 105, Reynolds number being based on cylinder diameter. Narrow-band vortex shedding has been observed up to a Reynolds number of 5·5 × 105, i.e. well into the critical regime. At this Reynolds number the Strouhal number reached the unusually high value of 0·46. Spectra of the velocity fluctuations measured in the wake are presented for several values of Reynolds number.

377 citations


Journal ArticleDOI
TL;DR: In this article, a model for the drag exerted on a cloud of spherical particles of a given particle size distribution in low Reynolds number flow is derived, where the drag experienced by a particle depends only on the first three moments of the distribution function.
Abstract: A formula for the drag exerted on a cloud of spherical particles of a given particle size distribution in low Reynolds number flow is derived. It is found that the drag experienced by a particle depends only on the first three moments of the distribution function. A treatment of viscous interaction between N particles to the lowest order is carried out systematically. By appealing to the concept of ‘randomness’ of the particle cloud, equations describing the averaged properties of the fluid motion are derived. The averages are formed over a statistical ensemble of particle configurations. These mean flow equations so obtained are in a form resembling a generalized version of Darcy's empirical equation for the motion of fluid in a porous medium. The physical meaning of these equations is discussed.

324 citations


Journal ArticleDOI
TL;DR: In this paper, an incompressible fluid of density p and kinematic viscosity v in an infinite two-dimensional domain is assumed to have a uniform velocity U∞, in the direction of the positive x*-axis of a rectangular Cartesian coordinate system Ox* y*, at large distances from a fixed flat plate of zero thickness which occupies the interval −l < x* < 0 of the line Ox*.
Abstract: Consider an incompressible fluid of density p and kinematic viscosity v in an infinite two-dimensional domain. We assume that the fluid has a uniform velocity U∞, in the direction of the positive x*-axis of a rectangular Cartesian coordinate system Ox* y*, at large distances from a fixed flat plate of zero thickness which occupies the interval −l < x* < 0 of the line Ox*. Of special interest here is the structure of the flow when e ≪ 1 whereRe being the Reynolds number of the flow. A first examination was made by Blasius (1908), using Prandtl's theory of the boundary layer, who found inter alia that the leading term of the drag D on one side of the plate is given bythe numerical factor being determined by Goldstein (1930).

290 citations


01 Mar 1969
TL;DR: In this paper, a review report represents an attempt to evaluate critically the available data on high-speed boundary layer transition to turbulence and to interpret the apparent agreements and contradictions within some rational framework.
Abstract: : The review report represents an attempt to evaluate critically the available data on high-speed boundary layer transition to turbulence and to interpret the apparent agreements and contradictions within some rational framework. Special attention was paid to the more documentable discrepancies between reported results as touchstones of conceptual models and instability theories. Experiments with 'microscopic' information are used as backbone of conceptual models, both linear and nonlinear. Linear instability results are used as a point of departure for the examination of current controversial questions of transition reversal with cooling, unit Reynolds number effect, effect of aerodynamic noise in supersonic wind tunnels, et.

265 citations


Journal ArticleDOI
TL;DR: In this article, an experimental system which allows the ready acquisition of heat transfer data necessary for thermal design of turbine airfoils is described, and the measurement of local heat-transfer coefficients on a full-size model is accomplished by considering Joulean dissipation in very thin platinum strips bonded to the model.
Abstract: Although internal impingement cooling of the leading edge of gas-turbine airfoils has been shown to be effective, previously available heat-transfer data are not generally applicable to present-day turbine designs because of the unique geometry requirements. An experimental system which allows the ready acquisition of heat-transfer data necessary for thermal design of turbine airfoils is described. A cold-flow model is developed, and the measurement of local heat-transfer coefficients on a full-size model is accomplished by considering Joulean dissipation in very thin platinum strips bonded to the model. Heattransfer results are given which show the dependence of Nusselt number on Reynolds number, geometry, and chordwise location on the inside leading-edge region of the airfoil. Dimensionless correlations are presented which allow the designer to predict heat transfer for impingement cooling in these geometries for the range of parameters tested.

202 citations


Journal ArticleDOI
TL;DR: In this article, the Navier-Stokes equations are replaced by a set of finite difference equations and the numerical solution is obtained by means of an iteration method, where the flow is assumed to be uniform at infinity upstream and the range of Reynolds numbers extends from 1 to 60.
Abstract: Numerical solutions of the steady Navier‐Stokes equations are presented for two‐dimensional flows past a circular cylinder in an infinite domain. The flow is assumed to be uniform at infinity upstream and the range of Reynolds numbers extends from 1 to 60. The Navier‐Stokes equations are replaced by a set of finite difference equations and the numerical solution is obtained by means of an iteration method. Conditions at “infinity” are applied by matching to Imai's asymptotic solution. The results are compared with those of other analytical and numerical computations as well as with experiments. In particular, the discussion is concerned with the drag, the base pressure, the shape of the standing vortices, and some formulas suggested for large Reynolds numbers. Excellent agreement with recent experiments of Acrivos, Leal, Snowden, and Pan is obtained.

196 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a steady laminar model of salt fingers and showed that the latter become unstable with respect to internal gravity waves when the finger Reynolds number exceeds a critical value.
Abstract: We first consider a steady laminar model of salt fingers and show that the latter become unstable with respect to internal gravity waves when the finger Reynolds number exceeds a critical value. The criterion is then used in speculations about the statistically steady state in a fully developed similarity model where horizontally averaged temperature and salinity gradients are constant at all depths. Dimensional reasoning is used to obtain the asymptotic dependence of the turbulent flux on the molecular salt diffusivity. From this and other relationships order-of-magnitude estimates are obtained and compared with laboratory experiments and ocean observations.

Journal ArticleDOI
TL;DR: In this article, a mathematical analogue of the wake, which helps to explain these features of the flow, has been constructed and the pattern of vortices in the wake of a cone suggested by this model is in reasonable agreement with the photographic evidence.
Abstract: Water-tunnel experiments on two cones of taper ratio 36:1 and 18:1 have shown that the frequency of vortex shedding is controlled by the local diameter and has a value which is slightly lower than that on a parallel cylinder of the same size. The shedding process is modulated by a low-frequency oscillation which has a frequency dependent only on U2/v and is independent of any physical dimension of the model. A mathematical analogue of the wake, which helps to explain these features of the flow, has been constructed. The pattern of vortices in the wake of a cone suggested by this model is in reasonable agreement with the photographic evidence.

Journal ArticleDOI
TL;DR: Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for uniform axial translation of the spheroids and flow past a line of stationary spheriods and under an imposed pressure gradient as mentioned in this paper.
Abstract: Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used.

Journal ArticleDOI
TL;DR: In this article, the uniform approach of a solid sphere of radius a towards a stationary solid sphere through a viscous fluid is considered when the fluid motion may be regarded as essentially steady with Reynolds number sufficiently small to permit the linearization of the Navier-Stokes equations by neglect of the inertia terms.
Abstract: The uniform approach of a solid sphere of radius a towards a stationary solid sphere through a viscous fluid is considered when the fluid motion may be regarded as essentially steady with Reynolds number sufficiently small to permit the linearization of the Navier–Stokes equations by neglect of the inertia terms. Discussion is given when the minimum clearance ea between the spheres is arbitrary and a detailed analysis is made of the asymptotic behaviour of the solution as the minimum clearance tends to zero by constructing matched asymptotic expansions. The forces on the spheres are shown to be of the form as a0e−1 + b0loge + c0 + O(eloge) when e is small where a0, b0 and c0 are constants. The values of a0 and b0 are determined explicitly in the general case of arbitrary sized spheres and c0 is also determined explicitly when the stationary sphere degenerates into a plane. For this case, it is shown that the asymptotic expression for the force is useful in practice even when e is not very small, the error being less than 3% when e ∼ 0·5.

Journal ArticleDOI
TL;DR: In this paper, the stability of a buoyancy-driven system, recently called a ‘buoyancy layer’, is examined for the whole range of Prandtl numbers, s. In this limit, the system is neutrally stable to first order, but second-order effects render the flow unstable at a Reynolds number of order σ-½.
Abstract: A buoyancy-driven system can be unstable due to two different mechanisms—one mechanical and the other involving buoyancy forces. The mechanical instability is of the type normally studied in connexion with the Orr-Sommerfeld equation. The buoyancy-driven instability is rather different and is related to the ‘Coriolis’-driven instability of rotating fluids. In this paper, the stability of a buoyancy-driven system, recently called a ‘buoyancy layer’, is examined for the whole range of Prandtl numbers, s. The buoyancy-driven instability becomes increasingly important as the Prandtl number is increased and so particular interest is attached to the limit in which the Prandtl number tends to infinity. In this limit, the system is neutrally stable to first order, but second-order effects render the flow unstable at a Reynolds number of order σ-½. Consequences of the results for the stability of convection in a vertical slot are examined.

Journal ArticleDOI
TL;DR: In this paper, a finite difference approximation to the vorticity transport equation on an expanding spherical polar grid system is obtained by constructing a finite-difference approximation of the transport equation with second order accuracy both in time and space.
Abstract: Numerical solutions of the transient uniform flow around a sphere are obtained. The transition takes place between an initial potential flow and a fully developed viscous field. The fluid is incompressible, homogeneous, and its flow is governed by the complete Navier‐Stokes equations. The range of Reynolds number studied is Re = 1–1000 where a recirculatory wake appears and the nonlinear terms are essential, that is, they cannot be neglected or approximated. The flow is assumed to be axisymmetric throughout this range. A time‐dependent stream function‐vorticity formulation is adopted. The solution is obtained by constructing a finite difference approximation to the vorticity transport equation on an expanding spherical polar grid system. Central differencing of second‐order accuracy both in time (Dufort‐Frankel) and space is utilized. Experiments with numerical stability show an appreciable deviation from linearized stability analysis due to the large gradients of vorticity in the field. Quantitative physical results are obtained. The geometrical parameters characterizing the recirculatory wake compare favorably with those recorded in physical experiments. The detailed distribution of the vorticity on the sphere agrees with results obtained via the steady‐state approach at Re = 10, 40, and 100. The computed drag coefficient CD agrees well with the standard drag curve over the range of Reynolds numbers investigated.

Journal ArticleDOI
TL;DR: In this paper, a finite difference solution for the time dependent equations of motion has been carried out in order to extend the range of available data on steady flow around a cylinder to larger Reynolds numbers.
Abstract: Finite difference solutions for the time dependent equations of motion have been carried out in order to extend the range of available data on steady flow around a cylinder to larger Reynolds numbers. At the termination of the calculations for R = 40 and 200, the separation angle, the drag coefficient and the pressure and vorticity distributions around the surface of the cylinder were very close to their steady-state values. For R = 500 the separation angle and drag coefficient were very close to their steady-state values but the pressure distribution and vorticity distribution at the rear of the cylinder were still changing slightly. The results at R = 500 were found to be quite different from those at R = 200 so it is not clear how closely we approximated the steady solution for R → ∞. The forces on the cylinder due to viscous drag and due to pressure drag are found to be smaller for steady flow than for laboratory experiments where the wake is unsteady.

Journal ArticleDOI
TL;DR: In this paper, the velocity gradients at the surface of a cylinder for Reynolds numbers from 5 × 103 to 105 were measured using a specially designed sandwich electrode, which enabled the direction of the velocity gradient as well as its magnitude to be measured.
Abstract: Electrochemical techniques have been used to measure the velocity gradients at the surface of a cylinder for Reynolds numbers from 5 × 103 to 105. This is a companion study to that already reported by Dimopoulos & Hanratty (1968) for a Reynolds number range of 60–360. The use of a specially designed sandwich electrode enabled the direction of the velocity gradient as well as its magnitude to be measured. Of particular interest is the region of definite length after separation where the velocity gradient is negative, followed by an ill-defined region where the flow moves in the positive direction. Still farther downstream the direction of flow changes with time in an irregular fashion. The measured velocity gradients prior to separation are described satisfactorily by boundary-layer theory. The presence of a splitter plate in the rear of the cylinder eliminates periodic fluctuations in the wake and has a significant effect on the boundary layer prior to separation.

Journal ArticleDOI
TL;DR: In this paper, a finite difference method is extended to high Reynolds number flow about circular cylinders with particular emphasis given to the quantitative description of fine flow features, and a cell structure is chosen which provides local cell dimensions consistent with the structure of solutions expected.
Abstract: A finite difference method is extended to high Reynolds number flow about circular cylinders with particular emphasis given to the quantitative description of fine flow features. The method is of the explicit type and includes a directional difference scheme for the nonlinear terms which enhances calculational stability at high Reynolds numbers. A cell structure is chosen which provides local cell dimensions consistent with the structure of solutions expected. Solutions are presented for a range of Reynolds numbers from 1 to 3 × 105 in which the flow is started impulsively from rest, and the development is studied up to the approach of the steady‐state or the limit cycle condition, whichever is appropriate to the particular Reynolds number.

Journal ArticleDOI
TL;DR: In this article, the entrance length to Reynolds number for pipe and channel geometries with a flat velocity profile as the initial condition is given. And the equation for the pipe entrance is shown to be in good agreement with experimental data.
Abstract: Equations are given that relate the entrance length to Reynolds number for pipe and channel geometries with a flat velocity profile as the initial condition. These equations are linear combinations of the creeping flow and boundary layer solutions. The former is obtained by minimization of the viscous dissipation using the finite element method. The equation for the pipe entrance is shown to be in good agreement with experimental data.

Journal ArticleDOI
TL;DR: In this article, it has been shown that the thermocline microstructure has a predominant vertical scale of a few decimeters, the sheet waves have wavelengths in the range 5-10 meters, and the turbulence occurs at a Reynolds number of 200-1000.
Abstract: Dye-tracer experiments in the seasonal oceanic theormocline have revealed a microstructure with a significant fluctuation of shear and density gradient within a vertical distance of 1 cm. It has proved possible to relate the occurrence of localized spots of turbulence with this structure and with internal waves traveling along especially (statically) stable ‘sheets’ within the thermocline. The thermocline microstructure has a predominant vertical scale of a few decimeters, the sheet waves have wavelengths in the range 5–10 meters, and the turbulence occurs at a Reynolds number in the range 200–1000. Similar features are found on a very much larger scale in the atmosphere, although the methods of investigation (mainly instrumented aircraft) used here are incapable of yielding the detailed information obtained by divers working in the thermocline. It seems useful, therefore, to treat the observations of microstructure, waves, and turbulence in the thermocline as a possible model for the corresponding features in the atmosphere. This leads to some predictions concerning the atmosphere microstructure and also to suggestions for the planning of experimental investigations. The available evidence from the atmosphere appears to support the predictions that (1) the critical gradient Richardson number for reverse transition occurs at about unity regardless of the Reynolds number of the turbulence, and (2) that most of the atmosphere (with the exception of clouds and the boundary layer) is in laminar flow.

Journal ArticleDOI
TL;DR: In this article, the authors distinguish rigorously between wave motion and turbulence in a stratified fluid, although useful approximations seem feasible, although the task is made more difficult by the fact that properties of turbulence are very little like those described in most theories of turbulence, even when the Reynolds number is quite high.
Abstract: To distinguish rigorously between wave motion and turbulence in a stratified fluid seems impossible, although useful approximations seem feasible. The task is made more difficult by the fact that properties of turbulence are very little like those described in most theories of turbulence, even when the Reynolds number is quite high.

01 Feb 1969
TL;DR: Transonic wind tunnel tests of aerodynamic drag and lift forces on circular cylinder in two dimensional flow at high Reynolds number were conducted in this article, showing that the results showed that the circular cylinder had high Reynolds numbers.
Abstract: Transonic wind tunnel tests of aerodynamic drag and lift forces on circular cylinder in two dimensional flow at high Reynolds number

Journal ArticleDOI
TL;DR: In this paper, the authors measured mean velocity profiles, longitudinal velocity fluctuations and the wall shearing stress (TW) in two-dimensional accelerated incompressible turbulent boundary layers and found that the wall region adjusts itself to laminar conditions earlier than the outer region.
Abstract: Experiments on reverse transition were conducted in two-dimensional accelerated incompressible turbulent boundary layers. Mean velocity profiles, longitudinal velocity fluctuations and the wall-shearing stress (TW) were measured. The mean velocity profiles show that the wall region adjusts itself to laminar conditions earlier than the outer region. During the reverse transition process, increases in the shape parameter (H) are accompanied by a decrease in the skin friction coefficient (Cf). Profiles of turbulent intensity (u’2) exhibit near similarity in the turbulence decay region. The breakdown of the law of the wall is characterized by the parameter \[ \Delta_p (= u[dP/dx]/\rho U^{*3}) = - 0.02, \] where U* is the friction velocity. Downstream of this region the decay of fluctuations occurred when the momentum thickness Reynolds number (R) decreased roughly below 400.

Journal ArticleDOI
TL;DR: In this paper, numerical solutions for incompressible Newtonian flow around a circular cylinder for Reynolds numbers of 1, 2, 4, 10, 15, 30, 50, 100, and 500 were obtained in the form of vorticity and stream function distributions.
Abstract: Numerical solutions have been obtained for incompressible Newtonian flow around a circular cylinder for Reynolds numbers of 1, 2, 4, 10, 15, 30, 50, 100, and 500 The results are presented in the form of vorticity and stream function distributions The solutions allow detailed characterization of the vortex rings Drag coefficients, pressure distributions, and vortex dimensions are compared with experimental data and with available theoretical predictions for Reynolds numbers up to 50 The agreement with both experimental data and previous theoretical solutions is considered to be excellent

Journal ArticleDOI
TL;DR: In this paper, the relationship between aerodynamic noise disturbances and boundary-layer transition Reynolds numbers in high-speed wind tunnels was defined and extended to include axisymmetric models, showing a significant increase in the Reynolds numbers with increasing tunnel size.
Abstract: : An experimental investigation of laminar boundary-layer transition on a sharp, 10-deg total-angle, insulated cone at zero yaw was conducted in 12- and 40-in. supersonic wind tunnels at free-stream Mach numbers from 3 to 6. This research was directed toward defining the relationship between the aerodynamic noise disturbances and boundary-layer transition Reynolds numbers in high-speed wind tunnels and extended previously published planar results to include axisymmetric models. A significant increase in the Reynolds numbers with increasing tunnel size (similar to the planar results) is shown to exist. Sharp cone transition Reynolds numbers from ten facilities (12- to 54-in.) for free- stream Mach numbers from 3 to 14 and a unit Reynolds number per inch range from 100,000 to 1,200,000 were correlated using aerodynamic-noise-transition parameters. A quantitative correlation of the ratio between cone and planar Reynolds numbers values was developed which demonstrates a strong Mach number dependence and also indicates a variation with tunnel size and unit Reynolds number.


Journal ArticleDOI
TL;DR: In this article, it was shown that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers and linked with known asymptotic results.
Abstract: Numerical calculations show that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers These calculations are linked with known asymptotic results

Journal ArticleDOI
TL;DR: In this article, the authors investigated the nature of a shear flow over a wavy boundary of small amplitude and found that the viscosity of the flow is highly dependent on the wave amplitude.
Abstract: The nature of a shear flow over a wavy boundary of small amplitude is investigated. It is found that if the viscosity is small, the nature of the flow is highly dependent on the wave amplitude. If the wave amplitude is truly infinitesimal, the flow is described by the Orr-Sommerfeld equation and in the neighbourhood of the critical layer viscous stresses are important even in the limit of vanishing viscosity. However, if the wave is sufficiently large, viscous stresses may be neglected even in the critical layer. An approximate solution of the inviscid equations of motion is obtained to describe the flow over a small but finite wave in the limit of infinite Reynolds number.

Journal ArticleDOI
TL;DR: In this paper, the particle transport mechanism operating during the filtration of dilute suspensions through deep granular beds has been considered on a dimensionless unit cell basis, and it has been shown that the mechanism can be completely described by parameters representing the diffusional, gravitational, interception and impaction mechanism and by the Reynolds Number.