Showing papers on "Reynolds number published in 1973"

Numerical study of slightly viscous flow

Abstract: A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.

Incipient Motion and Sediment Transport

Abstract: A review of existing literature reveals some disadvantages of using Shields diagram as the criterion for incipient motion of sediment particles on an alluvial bed. A new criterion based on average flow velocity, fall velocity, and shear velocity Reynolds number is proposed herein with the supporting data collected by different investigators. This new criterion is used to calculate the dimensionless critical unit stream power in a dimensionless unit stream power equation for sediment transport. The dimensionless unit stream power is the ratio of the time rate of potential energy expenditure per unit weight of water and the terminal fall velocity of the sediment. More than 1,000 sets of data from both laboratory flumes and natural streams published by different authors are used to support this dimensionless equation for sediment transport.

On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug

Abstract: Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally at Re > 5 × 104. Transition could be initiated at lower Re by introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.Mixed laminar and turbulent flows were observed far downstream for \[ 2000 < Re < 2700 \] when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lower Re. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.

The interaction between a pair of circular cylinders normal to a stream

Abstract: This paper describes how the flows around two circular cylinders, displaced in a plane normal to the free stream, interact as the two bodies are brought close together. Surface pressure measurements at a Reynolds number of 2·5 × 104, based on the diameter of a single cylinder, show the presence of a mean repulsive force between the cylinders. An instability of the flow was found when the gap between the cylinders was in the range between one diameter and about 0·1 of a diameter. Correlation measurements of hot-wire outputs indicate how mutual interference influences the formation of vortex streets from the two cylinders. Spanwise correlation measurements show that the correlation length doubles as the cylinders are brought into contact.

On optimum profiles in Stokes flow

Abstract: In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.

Stability of a circular cylinder oscillating in uniform flow or in a wake

Abstract: The lift and drag forces were measured on both a single circular cylinder and tandem circular cylinders in uniform flow at Reynolds numbers from 40 to 104, to investigate the stability of an oscillating cylinder A cylinder (the downstream one in the tandem case) was made to oscillate in either the transverse or longitudinal direction (perpendicular or parallel to the stream) In the case of a single cylinder, its oscillation causes the so-called synchronization in a frequency range around the Strouhal frequency (transverse mode) or double the Strouhal frequency (longitudinal mode) The aerodynamic damping for transverse oscillation becomes negative in the synchronization range In the case of tandem cylinders, at low Reynolds numbers in the pure Karman range synchronization was observed to occur only when the downstream cylinder oscillated inside the vortex-formation region of the upstream one, and at high (low subcritical) Reynolds numbers synchronization occurred irrespective of the cylinder spacing in either oscillating mode In the tandem case, too, the transverse oscillation of the downstream cylinder becomes unstable in the range of synchronization

Low Reynolds number flow past a porous spherical shell

Abstract: This paper is concerned with the flow of viscous fluids around and through porous bodies. Previous boundary conditions that have been used are discussed and a generalization of a boundary condition adopted by Beavers and Joseph, for plane boundaries, is proposed for curved surfaces. Using this condition the problem of slow viscous flow past a spherical shell is solved and several special limiting cases are considered.

Flow past an impulsively started circular cylinder

Abstract: An accurate method is described for integrating the Navier-Stokes equations numerically for the time-dependent flow past an impulsively started circular cylinder. Results of integrations over the range of Reynolds numbers, based on the diameter of the cylinder, from 5 to ∞ are presented and compared with previous numerical, theoretical and experimental results. In particular, the growth of the length of the separated wake behind the cylinder has been calculated for R = 40, 100 and 200 and is found to be in very good agreement with the results of recent experimental measurements. The calculated pressure distribution over the surface of the cylinder for R = 500 is also found to be in reasonable agreement with experimental measurements for the case R = 560.For Reynolds numbers up to 100 the equations were integrated until most of the features of the flow showed a close approximation to steady-state conditions. The results obtained are in good agreement with previous calculations of the steady flow past a circular cylinder. For R > 100 the integrations were continued until the implicit method of integration broke down by reason of its failure to converge. A secondary vortex appeared on the surface of the cylinder in the case R = 500, but for higher Reynolds numbers, including the case R = ∞, the procedure broke down before the appearance of a secondary vortex. In all cases the flow was assumed to remain symmetrical.

A new model for granular porous media: Part I. Model formulation

Abstract: A new model for porous media comprised of monosized, or nearly monosized grains, is developed. In applying this model to a packed bed, the bed is assumed to consist of a series of statistically identical unit bed elements each of which in turn consists of a number of unit cells connected in parallel. Each unit cell resembles a piece of constricted tube with dimensions which are random variables. The problem of flow through each unit cell is reduced, subject to reasonable assumptions, to the determination of the flow in an infinitely long periodically constricted tube. The solution of this flow problem is given in a companion publication. This model, together with the solution of the flow through it, can be used for the modeling of processes which take place in the void space of a bed. As a preliminary test, theoretical friction factor values, based on the proposed model, were compared with experimental ones for two different beds and found to be in good agreement even in the region of high Reynolds numbers where the nonlinear inertia terms are significant.

A modified wall wake velocity profile for turbulent compressible boundary layers.

Abstract: The law of the wake and the law of the wall in incompressible turbulent boundary layers formulated by Coles (1956) and their use by Mathews et al. (1970) in the development of a wall-wake representation of the velocity profile in a form applicable for isoenergetic compressible boundary layers are extended to a modified wall-wake velocity profile for turbulent compressible boundary layers. The modified wall-wake profile is shown to provide good representations of experimental velocity distributions.

Pressure distributions on circular cylinders at critical Reynolds numbers

Abstract: Measurements have been made of the mean and fluctuating pressure distributions on long circular cylinders, having smooth and rough surfaces, at Reynolds numbers of 1·11 × 105 and 2·35 × 105 in both uniform and turbulent streams. The presence of free-stream turbulence a t these Reynolds numbers was found to suppress coherent vortex shedding on the smooth cylinder and give rise to a complex pressure field in which the mean pressure distribution was almost independent of Reynolds number over the small range of Reynolds numbers tested. The pressure distributions on the rough cylinder were found to be completely different in uniform and turbulent streams; the presence of turbulence gave rise to an increase in the level of vortex shedding energy, and produced mean pressure distributions similar to those obtained on smooth cylinders at Reynolds numbers of the order of 107.

Theoretical Collision Efficiencies of Cloud Droplets at Small Reynolds Numbers

Abstract: An outline is presented of a new solution of the hydrodynamic equations for the motion of two spherical drops falling in a viscous fluid at small but non-zero Reynolds numbers. In this solution the flow of the medium is assumed to be governed by a modified form of the Oseen equations, and the boundary conditions at the two drops are simultaneously satisfied approximately. Although the equations allow partially for the effects of fluid inertia, a factor omitted in previous analyses based upon Stokesian hydrodynamics, the method of solution requires use of approximations to a larger extent than Stokesian solutions used by Davis and Sartor and by Hocking and Jonas. Collision efficiencies are presented for larger drops in the range 10–70μ radius. The new results tend to be larger than most previous theoretical computations, especially for drops of comparable size. For small size ratios, for which the coalescence efficiency should he close to unity, the present results appear to be consistent with col...

Motion of particles entrained in a plasma jet

Abstract: The motion of small particles (glass microspheres, 30 to 140 microns in diameter) entrained in a free argon plasma jet was studied by means of high-speed cine streak photography. Radial temperature and velocity profiles as well as axial profiles of temperature, velocity, and argon concentration in the jet were experimentally determined by means of a plasma calorimetric probe. The system was found to be characterized by low relative Reynolds numbers (0.2 to 20) and extremely high deceleration rates (about –2,000 g). Under these conditions, an increase of drag coefficient over that predicted by the standard curve was experimentally observed. This increase was attributed to the nonsteady flow field around the particle (the so-called “history term” in the equation of motion). A general computer program has been proposed which predicts the particle velocity, acceleration and temperature along its trajectory.

Rainfall effect on sheet flow over smooth surface

Abstract: The variation of friction factor for different flows and rainfall conditions is established by statistical analysis. Boundary shear stress was directly measured by hot-film anemometry. The friction factor is found to be a function of both the flow Reynolds number, R , and the rainfall intensity, I , for a flow Reynolds number of less than 900. For a flow Reynolds number greater than 2,000, the friction factor is only a function of Reynolds number. A numerical model and a simplified procedure to predict the water surface profiles and boundary shear stresses for sheet flow with rainfall are presented. The computations of flow depths and boundary shear stresses are not too sensitive to the effect of uncertainties in selecting friction factor.

Low speed aerodynamic characteristics of a 17 percent thick airfoil section designed for general aviation applications

Abstract: Wind-tunnel tests have been conducted to determine the low-speed two-dimensional aerodynamic characteristics of a 17-percent-thick airfoil designed for general aviation applications (GA(W)-1). The results were compared with predictions based on a theoretical method for calculating the viscous flow about the airfoil. The tests were conducted over a Mach number range from 0.10 to 0.28. Reynolds numbers based on airfoil chord varied from 2.0 million to 20.0 million. Maximum section lift coefficients greater than 2.0 were obtained and section lift-drag ratio at a lift coefficient of 1.0 (climb condition) varied from about 65 to 85 as the Reynolds number increased from about 2.0 million to 6.0 million.

A new model for granular porous media: Part II. Numerical solution of steady state incompressible Newtonian flow through periodically constricted tubes

Abstract: A numerical method for the solution of the problem of steady state, incompressible Newtonian flow through periodically constricted tubes is developed. All terms of the Navier-Stokes equation are retained, including the nonlinear inertia terms. Sample calculations for a uniform periodically constricted tube, the geometry of which is connected with the modeling of a packed bed of sand are given, including streamlines, axial and radial velocity profiles, pressure profiles, and the dimensionless pressure drop versus Reynolds number relation. The effect of some geometric characteristics of periodically constricted tubes on their friction factor is investigated numerically, and comparison of some existing experimental data with calculated ones is made.

Heat transfer from a sphere at low Reynolds numbers

Abstract: The heat transfer due to forced convection from an isothermal sphere in a steady stream of viscous incompressible fluid is calculated for low values of the Reynolds number and Prandtl numbers of O(1) The mean Nusselt number is compared with the results of experimental measurements At very low Reynolds numbers, both the local and mean Nusselt numbers are compared with the results obtained from the theory of matched asymptotic expansions

On the effective conductivity of a dilute suspension of spherical drops in the limit of low particle peclet number

Abstract: In this paper we consider the effective conductivity of a dilute suspension of neutrally bouyant spherical drops which is undergoing a simple shear flow. The thermal conductivity, viscosity and specific heat capacity of the drops are assumed to be different from those of the suspending fluid, though it is assumed that the local Peclet and Reynolds numbers are small both inside and outside the drop. The analysis consists of three parts: a derivation of the relationship between bulk heat flux on the one hand and the thermal and momentum fields at the microscale of the suspended particles on the other; a calculation of the local temperature field near a single neutrally buoyant spherical drop in shear flow with an imposed transverse temperature gradient at large distances; and a synthesis of the general relationship for bulk heat flux and the calculated local temperature field to determine an effective conductivity for a dilute suspension of spherical drops.

Analysis of turbulent pipe and channel flows at moderately large Reynolds number

Abstract: Effects of moderately large Reynolds numbers R are studied by considering higher order terms in the expansions for turbulent pipe and channel flows for R [rightward arrow] [infty infinity]. Matched asymptotic expansions using two length scales are employed to emphasize the two-layer structure of turbulent shear flows near solid walls. The effects appear as additional terms in extended forms of the law of the wall, the logarithmic velocity law, the velocity defect law and the logarithmic skinfriction law. These generalizations are critically compared with experimental results for pipe flows of Patel & Head and extremely good agreement is obtained. Also, possible applications are discussed for extending the range of skin-friction and heat-transfer devices which are based on wall similarity.

Nonlinear streaming effects associated with oscillating cylinders

Abstract: This paper deals with nonlinear streaming effects associated with oscillatory motion in a viscous fluid. A previous theory by Holtsmark et al. (1954) for the streaming near a circular cylinder in an incompressible fluid of infinite extent is reconsidered and used to obtain new numerical results, which are compared with earlier observations. The regime of validity of this theory is considered. The condition to be satisfied by the Reynolds number is found to be less stringent than was previously supposed.The more recent theory by Wang (1968) based on the outer–inner expansion technique is discussed and corrected with the Stokes drift.The case of an incompressible fluid enclosed between two coaxial cylinders, one of which is oscillating, is considered in detail. New theoretical and experimental results are given for various values of the parameters involved (Reynolds number, amplitude and cylinder radii).

An experimental and numerical study of the viscous stability of a round laminar vertical jet with and without thermal buoyancy for symmetric and asymmetric disturbances

Abstract: The fully viscous hydrodynamic stability equations for a round laminar vertical jet have been numerically solved using the proper boundary-layer base-flow velocity profile and for both symmetric and asymmetric disturbances. The symmetric mode is found to be unconditionally stable. The first asymmetric mode is found to be unstable and characteristics are compared with previous calculations. The computed critical Reynolds number for this mode is 9·4, which agrees with the calculations of Burridge (1968). Disturbance amplitude-ratio contours are also calculated and related to the convection of disturbances in the flow.The effect of thermal buoyancy on jet stability is assessed by solving the fully viscous equations, coupled through buoyancy, and using a buoyancy-perturbed jet flow. A Prandtl number σ of 6·7 is used, and positive thermal buoyancy is found to have a destabilizing effect. Stability characteristics for the limiting case of buoyancy, a purely thermal point-source plume, are determined for a Prandtl number of 2.Finally, an experiment was performed using water jets in water (σ ≈ 4·52–5·89) and a new method of jet production is described. The effect of varying amounts of thermal buoyancy on the laminar length of a jet undergoing naturally occurring transition was determined experimentally. These experiments confirm the calculated destabilizing effect of buoyancy. An empirical correlation is presented for the laminar length of a jet. Also, the effect of both symmetric and asymmetric artificially induced disturbances was determined experimentally. The disturbance amplitude ratio at which transition to turbulence takes place is found to be much less than for buoyant flows adjacent to a wall. The effects of frequency and amplitude of the artificial disturbances were experimentally determined and the trends are found to be consistent with the results of small disturbance theory.The principal new result is that positive thermal buoyancy destabilizes jet flow, and consequently calls into question earlier experimental studies wherein jet flows were observed by density differences. Another new result is the calculation. of amplitude-ratio contours for a non-buoyant jet and the quantitative description of jet stability in terms of these contours along paths of constant physical frequency. A comparison of the non-buoyant theory with experimental jets containing varying amounts of thermal buoyancy indicates that transition did not occur at a well-defined value of the amplitude ratio. Perhaps experiments with truly non-buoyant jets and/or more detailed buoyant calculations could explain this remaining question.