Fluctuating flow at small Reynolds number

Zusammenfassung Für den Widerstand von Kugeln und Kreiszylindern in periodisch schwankendem Parallelstrom werden Formeln angegeben, gültig für kleine Reynolds-Zahlen.     

Flow past a porous sphere at small Reynolds number

low of an incompressible viscous fluid past a porous sphere has been discussed. The flow has been divided in three regions. The Region-I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear fluid. In Regions II and III clear fluid flows and Stokes and Oseen solutions are respectively valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Stream function of Region II is matched with that of Region I at the surface of the sphere by the conditions suggested by Ochao-Tapia and Whitaker and it is matched with that of Oseen’s solutions far away from the sphere. It is found that the drag on the sphere reduces significantly when it is porous and it decreases with the increase of permeability of the medium.Received: February 7, 2002; revised: April 8, 2003 / June 9, 2004     

The transverse force on a spinning porous sphere at small Reynolds number

This paper deals with the problem of the flow about a spinning porous sphere moving in an incompressible viscous fluid. The effect of permeability on the flow has been studied analytically at low Reynolds number by the method of matched asymptotic expansions. It turns out that the permeability on the drag force up to 0(R) reduces the effective radius by a factor (2+K/a ²)^–1,K being the permeability associated with the porosity of the reduced radius of the sphere. Further, it is found that the permeability affects the lift forceF _L , which is orthogonal to its direction of motion. The ratio of the effective angular velocity and actual angular velocity decreases forK>0.25. ForK=0.25 there is no spinning effect on the sphere.

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Zusammenfassung Diese Arbeit behandelt die Strömung um eine rotierende poröse Kugel in einer zähen inkompressiblen Flüssigkeit. Der Einfluss der Durchlässigkeit wurde für kleine Reynolds-Zahlen mit Hilfe von angepassten asymptotischen Entwicklungen bestimmt. Es wurde gefunden, dass die Durchlässigkeit den Widerstand O(R) vermindert, und die Auftriebskraft senkrecht zur Bewegungsrichtung beeinflusst.

     

Heat transfer from a porous sphere in low Reynolds number flow

This paper deals with the problem of the effect of permeability in the steady state heat transfer from a single porous sphere at low Reynolds number with the assumptionsPr=0(1) andR<1. The solution is sought by the method of matched asymptotic expansions. The Nusselt number,Nu, for isothermal sphere has been calculated for different values ofk, wherek being the permeability associated with the porosity of the sphere. We notice that the Nusselt number increases as the permeability increases in the range 0<R<0.35 with the Prandtl numberPr=0.7, the rate of increase is very small, while for the Reynolds numberR0.35 Nusselt number decreases ask increases and the rate of decrease is large.

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Zusammenfassung In dieser Arbeit wird der Einfluss der Durchlässigkeit von einer porösen Kugel auf den Wärmeübergang behandelt, bei kleiner Reynolds-ZahlR undPr=0(1). Die Lösung wird mit der Methode der angepassten asymptotischen Entwicklung gewonnen. Die Nusselt'sche ZahlNu für die isotherme Kugel wurde für verschiedene Werte des Durchlässigkeits-Koeffizientenk ausgerechnet. Man findet, dass die Nusselt'sche Zahl (schwach) zunimmt mit der Durchlässigkeit für 0<R<0.35 mitPr=0.7, wogegen fürR0.35Nu (stark) abnimmt mit wachsendemk.

     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Exact and approximate expressions for resistance coefficients of a colloidal sphere approaching a solid surface at intermediate Reynolds numbers

A mathematical model has been developed to describe the force of liquid flow acting on a colloidal spherical particle as it approaches a solid surface at intermediate-Reynolds-number-flow regime. The model has incorporated bispherical coordinates to determine a stream function for the flow disturbed by the sphere. The stream function was then used to derive the flow force on the particle as a function of the inter-surface separation distance. The force equation was related to the modified Stokes equation to obtain an exact analytical expression for the correction factor to the Stokes law. Finally, a rational approximation is presented, which is in good agreement with the exact numerical result, and can be readily applied to more general particle–surface interactions involving short-range hydrodynamics associated with colloidal particles in the near vicinity of a large solid collector surface at intermediate Reynolds number of the supporting flow.     

Unsteady heat transfer from a circular cylinder in a low Reynolds number flow

This paper examines the temperature distribution in a low Reynolds number flow past a circular cylinder following a sudden change in the temperature of the cylinder. Solutions describing the initial development of the temperature field and the final approach to the eventual steady state are obtained. Expressions for the Nusselt number for small and large values of time are determined.     

On the steady flow of a viscous fluid past a sphere at small reynolds numbers using Oseen's approximation

Sommaire On a obtenu une expression générale pour la fonction de courant deStokes pour l'écoulement d'un liquide visqueux autour d'une sphère. Les lignes de courant ont été tracés pour les nombres deReynolds=1, 4 et 10. Le décollement du courant fluide n'a pas lieu pour les N. R.=1 et 4, tandis que pour le N. R.=10 les tourbillons attachés se forment dans le sillage.     

Hartmann duct flow at moderate magnetic Reynolds numbers

The application of a uniform external magnetic field on the turbulent duct flow of an electrically conducting fluid leads to several interesting changes in the structure and the mean charateristics of the flow. This is fairly well understood from the existing studies of duct flows in the low magnetic Reynolds number (R_m) limit. In this paper, we present the results for magnetohydrodynamic duct flow at moderate R_m obtained from direct numerical simulations (DNS). Several differences are observed to occur in this case as compared to low R_m flows, such as increased Hartmann layer thickness and enhanced large scale turbulence in the core region of the duct cross-section due to partial expulsion of magnetic flux. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boiling heat transfer on a sphere with turbulent vapour film

The theoretical study investigates turbulent film boiling on a sphere immersed in the stagnant liquid. It begins by assuming the surface temperature of the sphere is isothermal. The result shows that the boiling heat transfer under the turbulent vapour shows that both the temperature and the velocity present a non-linear distribution. Besides, the increase of the thermal radiation, wall temperature and buoyancy effects can enhance the heat transfer efficiency. Furthermore, a comparison between the results of the present study and those reported in a previous theoretical study of laminar film boiling is provided. It is found that turbulent film boiling with higher Rayleigh number will have higher Nusselt number.     

Natural convection from a vertical cylinder at very small Prandtl numbers

Summary The flow arising from the natural convection from an isothermal vertical cylinder, valid for very small Prandtl numbers, is determined.

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Zusammenfassung Das Problem der freien konvektiven Grenzschichtströmung entlang eines vertikalen isothermen Zylinders wird für den Fall gelöst, dass die Prandtl'sche Zahl sehr klein ist, und die Grenzschicht dick im Vergleich zum Radius des Zylinders.

     

Dynamics of regularized cavity flow at high Reynolds numbers

A numerical simulation of the unsteady incompressible flows in the unit cavity is performed by using a Chebychev-Tau approximation for the space variables. The high accuracy of the spectral methods and the condensed distribution of the Chebychev-collocation points near the boundary enable us to obtain reliable results for high Reynolds numbers with a moderate number of modes. It is found that a stationary solution always exists for Reynolds numbers up to 10000. For Reynolds numbers larger than a critical value which is between 10000 and 12000, no more steady solution is found, instead, there is a persistent oscillation indicating a Hopf bifurcation.     

On the Lagerstrom model for flow at low Reynolds numbers

We show that a model proposed by Lagerstrom for viscous incompressible flow at low Reynolds numbers has a solution. The proof is constructive in that we obtain lower and upper bounds for the solution. The dependence of the initial slope of the solution on a small parameter is also made explicit.     

Flow separation behind ellipses at Reynolds numbers less than 10

Flow separation behind two-dimensional ellipses with aspect ratios ranging from 0, a flat plate, to 1, a circular cylinder, were investigated for Reynolds numbers less than 10 using both a cellular automata model and a commercial computational fluid dynamics software program. The relationship between the critical aspect ratio for flow separation and Reynolds number was determined to be linear for Reynolds numbers greater than one. At slower velocities, the critical aspect ratio decreases more quickly as the Reynolds number approaches zero. The critical Reynolds numbers estimated for flow separation behind a flat plate and circular cylinder agree with extrapolations from experimental observations. Fluctuations in the values of the stream function for laminar flow behind the ellipses were found at combinations of Reynolds number and aspect ratio near the critical values for separation.     

Sedimentation in ordered emulsions of drops at low Reynolds numbers

We determine the sedimentation velocity of a periodic array of drops through a viscous Newtonian fluid by extending an earlier analysis for the rigid particles. The results, which are given for the complete range of volume fractionc and the viscosity ratioK for the simple cubic, bodycentered cubic and face-centered cubic arrays, are in excellent agreement with the previously known results for the two limiting cases of the rigid spheres (K=) and the spherical bubbles (K=0) for all values ofc and with an asymptotic expression forc1 for allK.

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Zusammenfassung Wir bestimmten die Ablagerungsgeschwindigkeit für eine periodische Anordnung von Tropfen in einer viskosen Newtonschen Flüssigkeit, indem wir eine frühere Analyse für feste Teilchen erweiterten. Die Ergebnisse, welche für den gesamten Bereich des Volumenbruchteilsc und den Viskositäts-ParameterK für einfache kubische, raumzentriert kubische und flächenzentriert kubische Anordnungen gegeben werden, sind in ausgezeichneter Übereinstimmung mit bekannten Ergebnissen für die beiden Grenzfälle der starren Kugel (K=) und der kugelförmigen Blase (K=0) für alle Werte vonc, und mit einem asymptotischen Ausdruck fürc1 und alleK-Werte.

     

Asymptotic analysis of compound liquid jets at low Reynolds numbers

Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations which govern the fluid dynamics of both hollow and solid, compound jets such as those employed in the manufacture of textile fibers, composite fibers and optical fibers. These fibers consist of an inner material which may be a round jet or an annular one and which, in turn, is surrounded by an annular jet in contact with ambient air. It is shown that the leading-order equations are one-dimensional and that it is possible to obtain analytical solutions for several flow regimes for steady, compound jets. These analytical solutions are presented here. The one-dimensional leading-order equations for the fluid dynamics of annular liquid jets at low Reynolds numbers are also derived here.     

Turbulent drag reduction at moderate Reynolds numbers via spanwise velocity waves

The question whether large turbulent drag reduction can be achieved at the high values of Re typical of applications is addressed. Answering such question, either by experiments or DNS, is obviously challenging. For DNS, the problem lies in the tremendous increase of the computational cost with Re, that has to be appreciated in view of the need of carrying out an entire parametric study at every Re, owing to the unknown location of the optimal forcing parameters. In this paper we limit ourselves to considering an open-loop technique based on spanwise forcing, the streamwise-traveling waves introduced by [1], and explore via Direct Numerical Simulations (DNS) how the drag reduction varies when the friction Reynolds number is increased from Re_τ = 200 to Re_τ = 2000. To achieve high Re while keeping the computational cost affordable, computational domains of reduced size are employed. We adopted special care to interpret results that are indeed still box-size dependent, as well as strategies to compute the random errors and give the results an error bar. Our results indicate that still R = 0.29 can be obtained at Re_τ = 2000 in the partial region of the parameter space studied. The maximum R is found to decrease as R ˜ Re_τ^−0.22 in the Reynolds range investigated. As most important outcome, we find that the sensitivity of R to Re becomes smaller when far from the low-Re optimum parameters: in this region, we suggest R ˜ Re_τ^−0.08. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Separation of streamlines for spatially periodic flow at non-zero Reynolds numbers

The flow generated by a small rotating circular cylinder at the center of a corrugated outer cylinder is considered. By using a Stokes expansion, the first order correction in the Reynolds numberR is found for the creeping flow solution. An approximate critical Reynolds numberR _c is found at which separation appears, and it is expressed in terms of the boundary parameters. Separation is found to occur in the concave regions of the boundary skewed opposite to the direction of rotation of the inner cylinder. By partially solving for the second order correction in the Stokes expansion, it is found that an increase inR causes an increase in the torque exerted on the outer boundary.This work was supported in part by a grant from NSERC.