The effects of bubbles on the volume fluxes and the pressure gradients in unsteady and non-uniform flow of liquids

Kowe, R; Hunt, J. C. R.; Hunt, A.; Couët, Benoı̂t; Bradbury, L. J. S.

doi:10.1016/0301-9322(88)90060-2

Cited by 51 publications

(24 citation statements)

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“…see Kowe et al 1988) so that the average pressure in a homogeneous distribution of bubbles rising in a channel is reduced. This is why in a layer of bubbles rising in a pipe, those at the top and bottom will tend to converge towards the middle, causing collisions and larger bubbles to be formed (figure 3a).…”

Section: Interstitial Velocity and Drift-flux Relations For Bubbly Flowsmentioning

confidence: 99%

Inviscid mean flow through and around groups of bodies

2004

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General estimates are derived for mean velocities through and around groups or arrays of fixed and moving bodies, in unbounded and bounded domains, which lie within a defined perimeter. Robust kinematic flow concepts are introduced, namely the Eulerian spatial mean velocity $\overline{u}_E$ in the fluid volume between the bodies, the Eulerian flow outside the group, ${\bm u}_E^{(0)}$, and the Lagrangian mean velocity of material surfaces or fluid particles as they pass through the group of bodies ($\overline{u}_L^{(S)}$, $\overline{u}_L^{(P)}$). The Eulerian mean velocity is related to the momentum in the fluid domain, and is mainly influenced by fast moving regions of the flow. The Lagrangian mean velocity weights slowly moving regions of flow and is related to how material sheets deform as they are advected through groups of bodies. When the bodies are well-separated, the interstitial Eulerian and Lagrangian mean velocities ($\overline{u}_E^{(I)}$, $\overline{u}_L^{(I)}$), are defined and calculated in terms of the far-field contributions from the velocity or displacement field within the group of bodies.In unbounded flow past well-separated bodies situated within a rectangular perimeter, the difference between the Eulerian and Lagrangian mean velocity is negligible (as the void fraction of the bodies, $\alpha\,{\rightarrow}\,0$). Within wide and short rectangular arrays, the Eulerian mean velocity is faster than the free-stream velocity $U$ because most of the incident flow passes through the array and $\overline{u}_E\,{=}\,U(1-\alpha)^{-1}$. Within long and thin rectangular arrays (and other cases where the reflux velocity is negligible), the Eulerian mean velocity, $\overline{u}_E\,{=}\,U(1-(1+C_m)\alpha)/(1-\alpha)$, is slower than the free-stream velocity, because most of the incident flow passes around the array. For a spherical or circular arrays of bodies, the particle Lagrangian mean velocity is $\overline{u}_L^{(P)}\,{=}\,U(1+C_m\alpha)^{-1}$ and differs from $\overline{u}_E$. These calculations are extended to examine the mean and interstitial flow through clouds of bodies in bounded channel flows.The new concepts are applied to calculate the mean flow and pressure between and outside clouds of bodies, the average velocity of bubbly flows as a function of void fraction, and the tendency of clouds of bubbles to be distorted depending on their shape.

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Section: Interstitial Velocity and Drift-flux Relations For Bubbly Flowsmentioning

confidence: 99%

Inviscid mean flow through and around groups of bodies

2004

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“…The deformation of a material surface initially spanning the channel, by a translating body, consists of two distinct components: a localized drift (positive displacement) component where the body passes, and a nonlocal reflux (negative displacement) component. The return flow or reflux created by rising bubbles is chiefly responsible for the hindered rise speed of bubbles and the slip velocity between the gas and liquid phases (Kowe et al 1988). Bush & Eames (1998) examined the fluid transport by 'planar' high Reynolds number bubbles rising in a Hele-Shaw cell, and observed good agreement between measured and predicted drift volumes.…”

Section: Introductionmentioning

confidence: 99%

Fluid displacement by Stokes flow past a spherical droplet

2003

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The concept of 'drift', which has been exploited in many high Reynolds number and inviscid flow problems, is here applied to examine transport by a spherical viscous droplet (of radius a) moving in a Stokes flow.In an unbounded flow, the velocity in the direction of translation of a spherical droplet is positive everywhere because streamlines, in the fluid frame of reference, 'close' at infinity. Fluid particles are displaced a positive distance, X, forward, which is expressed in terms of the initial distance from the stagnation streamline ρ 0 . Asymptotic expressions are developed for X in the limits of ρ 0 /a 1 and 1. The nature of the singularity of the centreline displacement changes from O(−a log(ρ 0 /a)) to O(a 2 /ρ 0 ) as the viscosity of the droplet, compared to the ambient fluid, increases. By employing a mass-conservation argument, asymptotic expressions are calculated for the partial drift volume, D p , associated with a circular material surface of radius ρ m which starts far in front of a droplet that translates a finite distance. Since the velocity perturbation decays slowly with distance from the droplet, D p tends to become unbounded as ρ m increases, in contrast to inviscid flows.The presence of a rigid wall ensures that the velocity perturbation decays sufficiently rapidly that fluid particles, which do not lie on the stagnation streamline, are displaced a finite distance away from the wall. The distortion of a material surface lying a distance h L above a wall, by the droplet, starting a distance h S from the wall and moving away, is studied. The volume transported away from the wall, calculated using a multipolar flow approximation, is D p = πh 2 L a(3λ + 2)/(λ + 1), and is weakly dependent on the starting position of the droplet, in accordance with numerical results. When the material surface is close to the wall (h L /a 1), the volume transported away from a wall is significantly smaller than for inviscid flows because the no-slip condition on the rigid wall tends to inhibit 'scouring'. When the material surface is far from the wall (h L /a 1), the viscously dominated flow transports a larger volume of fluid away from the wall because the flow decays slowly with distance from the droplet.These results can be generalized to arbitrarily shaped bodies, since the transport processes are dominated by the strength of the Stokeslet. The effect of boundaries and inertia on fluid transport processes is briefly discussed.

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“…This three-dimensional example can be applied to practical problems (e.g. the experiments of Bataille et al 1991;Kowe et al 1988).…”

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confidence: 99%

Drift, partial drift and Darwin's proposition

1994

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A body moves at uniform speed in an unbounded inviscid fluid. Initially, the body is infinitely far upstream of an infinite plane of marked fluid; later, the body moves through and distorts the plane and, finally, the body is infinitely far downstream of the marked plane. Darwin (1953) suggested that the volume between the initial and final positions of the surface of marked fluid (the drift volume) is equal to the volume of fluid associated with the 'added-mass' of the body.We re-examine Darwin's (1953) concept of drift and, as an illustration, we study flow around a sphere. Two lengthscales are introduced: pmax, the radius of a circular plane of marked particles; and XO, the initial separation of the sphere and plane. Numerical solutions and asymptotic expansions are derived for the horizontal Lagrangian displacement of fluid elements. These calculations show that depending on its initial position, the Lagrangian displacement of a fluid element can be either positive -a Lagrangian drift -or negative -a Lagrangian reflux. By contrast, previous investigators have found only a positive horizontal Lagrangian displacement, because they only considered the case of infinite XO. For finite XO, the volume between the initial and final positions of the plane of marked fluid is defined to be the 'partial drift volume', which is calculated using a combination of the numerical solutions and the asymptotic expansions. Our analysis shows that in the limit corresponding to Darwin's study, namely that both xo and pmax become infinite, the partial drift volume is not well-defined: the ordering of the limit processes is important. This explains the difficulties Darwin and others noted in trying to prove his proposition as a mathematical theorem and indicates practical, as well as theoretical, criteria that must be satisfied for Darwin's result to hold.We generalize our results for a sphere by re-considering the general expressions for Lagrangian displacement and partial drift volume. It is shown that there are two contributions to the partial drift volume. The first contribution arises from a reflux of fluid and is related to the momentum of the flow; this part is spread over a large area. It is well-known that evaluating the momentum of an unbounded fluid is problematic since the integrals do not converge; it is this first term which prevented Darwin from proving his proposition as a theorem. The second contribution to the partial drift volume is related to the kinetic energy of the flow caused by the body: this part is Darwin's concept of drift and is localized near the centreline. Expressions for partial drift volume are generalized for flow around arbitrary-shaped two-and three-dimensional bodies. The partial drift volume is shown to depend on the solid angles the body subtends with the initial and final positions of the plane of marked fluid. This result explains why the proof of Darwin's proposition depends on the ratio t Present address: Department of Meteorology, University of Reading, Reading, RG6 2AU, UK Pmax 1x0. 202u...

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The effects of bubbles on the volume fluxes and the pressure gradients in unsteady and non-uniform flow of liquids

Cited by 51 publications

References 10 publications

Inviscid mean flow through and around groups of bodies

Inviscid mean flow through and around groups of bodies

Fluid displacement by Stokes flow past a spherical droplet

Drift, partial drift and Darwin's proposition

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