Considerations on bubble fragmentation models

n this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22 f(D; D0) = D2 f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3 f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3 D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension

The necking time of gas bubbles in liquids of arbitrary viscosity

We report an experimental and theoretical study of the collapse time of a gas bubble injected into an otherwise stagnant liquid under quasi-static conditions and for a wide range of liquid viscosities.This work has been supported by the Spanish MINECO (Subdirección General de Gestión de Ayudas a la Investigación), Junta de Andalucía, and European Funds under Project Nos. DPI2014-59292-C3-1-P, DPI2014-59292-C3-3-P, and P11-TEP7495. Financial support from the University of Jaén, Project No. UJA2013/08/05, is also acknowledged

Modeling of the bubbling process in a planar co-flow configuration

This work presents an analytical model developed to describe the bubbling regime resulting from the injection of an air sheet of thickness 2H(o) with a mean velocity u(a) between two water streams of thickness H-w - H-o, moving at a uniform velocity u(w). Based on previous experimental and numerical characterizations of this flow, the gas stream is modeled as a two-dimensional sheet divided into three different parts in the streamwise direction: a neck that moves downstream at the water velocity, a gas ligament attached to the injector upstream of the neck, and a forming bubble downstream of the neck, whose uniform dimensionless half-thicknesses are eta(n)(tau), eta(l)(tau), eta(b)(tau) respectively, and the corresponding pressures are given by Pi(n)(tau), Pi(l)(tau), and Pi(b)(tau) Pi(n)(tau). Lengths are made dimensionless with H-o, and pressures with rho(a)u(a)(2) where rho(a) is the air density.This work has been supported by the Spanish MINECO (Subdi-rección General de Gestión de Ayudas a la Investigación), Junta de Andalucía, and European Funds under projects numbers DPI2014-59292-C3-1-P and DPI2014-59292-C3-3-P, and P11-TEP7495. Financial support from the University of Jaén, project UJA2013/08/05, is also acknowledged

On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size PDF of the resulting daughter bubbles

Based on energy principles, we propose a statistical model to describe the bubble size probability density function of the daughter bubbles resulting from the shattering of a mother bubble of size D0 immersed in a fully developed turbulent water flow. The model shows that the bubble size p.d.f. depends not only on D0, but also on the value of the dissipation rate of turbulent kinetic energy of the underlying turbulence of the water, [epsilon]. The phenomenological model is simple, yet it predicts detailed experimental measurements of the transient bubble size p.d.f.s performed over a range of bubble sizes and dissipation rates [epsilon] in a very consistent manner. The agreement between the model and the experiments is particularly good for low and moderate bubble turbulent Weber numbers, Wet = [rho][Delta]u2(D0)D0/[sigma] where the assumption of the binary breakup is shown to be consistent with the experimental observations. At larger values of Wet, it was found that the most probable number of daughter bubbles increases and the assumption of tertiary breakup is shown to lead to a better fit of the experimental measurements

On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency

The transient evolution of the bubble-size probability density functions resulting from the breakup of an air bubble injected into a fully developed turbulent water ow has been measured experimentally using phase Doppler particle sizing (PDPA) and image processing techniques. These measurements were used to determine the breakup frequency of the bubbles as a function of their size and of the critical diameter Dc defined as Dc = 1.26 ([sigma]/[rho])3/5[epsilon][minus sign]2/5, where [epsilon] is the rate of dissipation per unit mass and per unit time of the underlying turbulence. A phenomenological model is proposed showing the existence of two distinct bubble size regimes. For bubbles of sizes comparable to Dc, the breakup frequency is shown to increase as ([sigma]/[rho])[minus sign]2/5[epsilon][minus sign]3/5 [surd radical]D/Dc[minus sign]1, while for large bubbles whose sizes are greater than 1.63Dc, it decreases with the bubble size as [epsilon]1/3D[minus sign]2/3. The model is shown to be in good agreement with measurements performed over a wide range of bubble sizes and turbulence intensitie

Experimental characterization of starting jet dynamics

The dynamics of a laminar starting jet are explored in a series of laboratory experiments and numerical simulations. We identify new, objective methods for characterizing the leading vortex ring, enabling robust comparisons with results from a numerical model. Observations of circulation in the initial vortex ring and for the total jet are reported along with strain rate at the leading stagnation point. Growth and pairing of shear instabilities trailing the leading vortex ring is observed. Development of these secondary vortices and their subsequent interactions with the leading vortex has significant effects on the characteristics of the primary vortex ring. Strong fluctuations in strain rate at the leading edge are associated with the pairing of the initial vortex ring with a trailing secondary ringSupport for this research was provided by the Spanish MEC and European Union under Projects # ENE2005-08580-C02-01 and DPI2005-08654-C04-01Publicad

Global mode analysis of axisymmetric bluff-body wakes: Stabilization by base bleed

International audienceThe flow around a slender body with a blunt trailing edge is unstable in most situations of interest. Usually the flow instabilities are generated within the wake behind the bluff body, inducing fluctuating forces and introducing the possibility of resonance mechanisms with modes of the structure. Base bleed is a simple and well-known means of stabilizing the wake. In the present research, we investigate the global instability properties of the laminar-incompressible flow that develops behind a cylinder with sharp edges and axis aligned with the free stream using a spectral domain decomposition method. In particular, we describe the flow instability characteristics as a function of the Reynolds number, Re=?W8D/µ, and the bleed coefficient, defined as the bleed-to-free-stream velocity ratio, Cb=Wb/W8, where D is the diameter of the body and ? and µ the density and viscosity of the free stream, respectively. For a truncated cylinder of aspect ratio L/D=5, where L is the length of the body, our calculations reveal the presence of a first steady bifurcation in the wake at Re?391, as well as a second oscillatory one at Re?715 with an associated Strouhal number St?0.0905 for the most unstable azimuthal mode {pipe}m{pipe}=1. In addition, we report the existence of two critical values of the bleed coefficient Cb1*(Re,{pipe}m{pipe}) and Cb2*(Re,{pipe}m{pipe}) < Cb1*, which vary with the aspect ratio of the body, needed to stabilize both the first and second bifurcations in the range of Reynolds numbers under study, 0=Re=2200. Finally, the numerical results for the oscillatory mode obtained for a bulletlike body of aspect ratio L/D=2 without base bleed are compared with experiments performed in a wind tunnel using hot-wire anemometry, showing the limitations of using an axisymmetric basic flow at Reynolds numbers higher than the critical one corresponding to the first steady bifurcation in the global stability analysis. © 2009 American Institute of Physics