Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.      

Interaction of an ultrasonic wave with a bubbly mixture

The interaction of an ultrasonic wave with a bubbly two-phase flow is studied both experimentally and theoretically. Brief theoretical reviews of acoustic wave generation by a piston and of the interaction of a plane wave with a single bubble are given. A theory relating ultrasonic wave transmission through a bubbly flow with two-phase flow parameters, notably the bubble size and the volumetric interfacial area, is derived and compared with preliminary data. The theoretical and experimental limitations concerning the application of ultrasonic transmission measurements to the study of bubbly flow are discussed in detail, and recommendations for future work in this area are made.Nomenclature A projected cross-sectional area of a bubble - a bubble radius - a _T emitter radius - c speed of sound - d equivalent bubble diameter - I intensity - I ₀ incident intensity - J ₀ Bessel function of zero order - J ₀ Bessel function of first order - j ₁ spherical Bessel function of the first kind and of order l - k wave number - n _l spherical Bessel function of the second kind (Nishi's notation) - n number of bubbles per unit volume or per unit area - P ₀ pressure amplitude at the emitter; equilibrium pressure in the liquid - p pressure perturbation - r spherical coordinate, radial distance to the x-axis - S total scattering cross-sectional area, surface of the piston - S _i scattering cross-sectional area of the i-th bubble - T transmittance - t time - U complex source strength divided by the source area - W ratio of radial distance from the axis on the emitter surface to the radius of the emitter - x axis coinciding with the direction of propagation of the plane wave; distance between the transducers - void fraction - interfacial area per unit volume - spherical coordinate - ultrasonic wavelength - density - angular frequency     

Numerical simulation of shock propagation in a polydisperse bubbly liquid

The effect of distributed bubble nuclei sizes on shock propagation in a bubbly liquid is numerically investigated. An ensemble-averaged technique is employed to derive the statistically averaged conservation laws for polydisperse bubbly flows. A finite-volume method is developed to solve the continuum bubbly flow equations coupled to a single-bubble-dynamic equation that incorporates the effects of heat transfer, liquid viscosity and compressibility. The one-dimensional shock computations reveal that the distribution of equilibrium bubble sizes leads to an apparent damping of the averaged shock dynamics due to phase cancellations in oscillations of the different-sized bubbles. If the distribution is sufficiently broad, the phase cancellation effect can dominate over the single-bubble-dynamic dissipation and the averaged shock profile is smoothed out.     

Turbulence kinetic energy budget in bubbly flows in a vertical duct

Understanding turbulence kinetic energy (TKE) budget in gas–liquid two-phase bubbly flows is indispensable to develop and improve turbulence models for the bubbly flows. In this study, a molecular tagging velocimetry based on photobleaching reaction was applied to turbulent bubbly flows with sub-millimeter bubbles in a vertical square duct to examine the applicability of the k–ε models to the bubbly flows. Effects of bubbles on TKE budget are discussed and a priori tests of the standard and low Reynolds number k–ε models are carried out to examine the applicability of these models to the bubbly flows. The conclusions obtained are as follows: (1) The photobleaching molecular tagging velocimetry is of use for validating turbulence models. (2) The bubbles increase the liquid velocity gradient in the near wall region, and therefore, enhance the production and dissipation rates of TKE. (3) The k–ε models can reasonably evaluate the production rate of TKE in the bubbly flows. (4) The modulations of diffusion due to the bubbles have different characteristics from the diffusion enhancement due to shear-induced turbulence. Hence, the k–ε models fail in evaluating the diffusion rate in the near wall region in the bubbly flows. (5) The k–ε models represent the trends of the production, dissipation, and diffusion rates of ε in the bubbly flow, although more accurate experimental data are required for quantitative validation of the ε equation.     

Stability of flexural-gravity waves and quadratic interactions

The processes leading to strong distortion of the amplitude of a periodic flexural-gravity wave propagating in a fluid layer of finite depth beneath an infinite thin elastic plate are investigated. It is shown that three-wave resonance with noise harmonics of the flexural-gravity waves may lead to instability of a wave with the wave vector k _w , where |k _w |<k _min. Depending on |k _w |, either two copropagating noise harmonics or two noise harmonics whose wave vectors make a nonzero angle with the vector k _w may be most strongly amplified during the initial instants of time. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–101, January–February, 1999. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-010-1746) and the International Association for Assistance and Cooperation with Scientists from the Independent States of the former Soviet Union and the Russian Foundation for Basic Research (INTAS-RFBR 95-0435).     

Formation of shock waves in gas-liquid foams

The purpose of the present investigation is to analyze the phenomenon of shock wave formation in gas-liquid foams and to explain the qualitative differences which are found when comparing results from shock tube experiments performed with foams and bubbly liquids. It is well known that oscillatory pressure waves in bubbly liquids may reach an amplitude twice as large as that of the original pressure impulse. However, experiments showed that pressure disturbances in foams always attenuate without significant change in the wave pressure profile. In the present study this behavior is explained by analyzing shock wave formation using the Burgers equation which is derived from the conservation laws for a bubbly liquid. It is shown that the parameter of non linearity in the Burgers equation describing wave propagation in bubbly liquids is about 40 times higher than in foams. At the same time coefficient of bulk viscosity of a foam is about 10³ times greater than that of a bubbly liquid. This explains why in shock tube experiments with foams shock waves are not detected while they are easily observed when bubbly liquids are used under similar conditions.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Possibility of reaching higher contents of a gas in a liquid using shock waves

The effect of shock waves on gas absorption by liquid in bubbly media with different degrees of gas solubility is studied. It is shown that a shock wave acting on a gas-liquid medium can significantly enhance the mass transfer between the gas and the liquid. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 64–70, March–April, 2000.     

On the application of LDA to bubbly flow in the wobbling regime

 Considerations for applying LDA to bubbly flows with bubbles about 3 to 4 mm in diameter were investigated by means of detailed experiments in the model geometry of a train of bubbles. Both forward scatter and backscatter LDA were studied. The validity of phase discrimination via burst amplitude was tested and special attention was paid to the impact of bubble interface response to the laser beams. Forward and backscatter measurements can be compared well. In both configurations, predominantly the liquid phase is “seen” by LDA. A bubble itself only leads to a velocity realization in special conditions. In those cases the Doppler shift is determined by the motion of the bubble interface which consists of the motion of the center of gravity of the bubble as well as shape oscillations. In backscatter bubbles only give velocity realizations when their “cheeks” pass through the measuring volume virtually perpendicularly. It is shown that the bubble-caused velocity realization frequency is very low for bubbles of the size used. Phase discrimination on burst amplitude does not hold. In ambient cases such as bubble columns one can assume that only the liquid phase is being studied. Received: 4 May 1998/Accepted: 30 September 1998     

Characterization of wettability effects on pressure drop of two-phase flow in microchannel

The Characterization of the effects of surface wettability and geometry on pressure drop of slug flow in isothermal horizontal microchannels is investigated for circular and square channels with hydraulic diameter (D _h ) of 700 μm. Flow visualization is employed to characterize the bubble in slug flow established in microchannels of various surface wettabilities. Pressure drop increases with decrease in surface wettability, while the channel geometry influences slug frequency. It is observed that the gas–liquid contact line in advancing and receding interfaces of bubble change with surface wettability in slug flows. Flow resistance, where capillary force is important, is estimated using Laplace–Young equation considering the change of dynamic contact angles of bubble. The experimental study also demonstrates that the liquid film presence elucidates the pressure drop variation of slug flows at various surface wettabilities due to diminishing capillary effect.     

Distinctive Features of the Dynamics of a Bubble Detonation Wave in Tubular Clusters

The distinctive features of the passage of a detonation wave from a tubular body of a bubbly liquid confined by a layer of a “pure” liquid into a region of a homogeneous bubbly liquid are investigated. It is shown that the detonation wave propagating in the tubular bubble cluster can break away on the passage through a region, where the cluster radius increases jumpwise, due to the two-dimensional wave scattering in the expansion zone, despite an increase in the system power-intensity.     

Forced nonlinear oscillations of a gas bubble in a large spherical flask (resonator) filled with a fluid

Radial oscillations of a gas bubble in a large spherical flask filled with a fluid are considered. We derive an equation of the change of the bubble radius by the known law of pressure variation at the boundary of the liquid volume (the law of motion of the piston) for a period of time during which, repeatedly reflected from the piston, the leading front of the reflected-from-the bubble perturbations reaches the bubble. For further calculations of the change of the bubble radius, recurrent relations which include the wave reflected from the bubble in the previous cycle and its subsequent reflection from the piston are obtained. Under harmonic action of the piston on the fluid-bubble system, a certain periodic regime with a package of bubble oscillations is established. Institute of Mechanics, Ural Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 77–87, September–October, 1998.     

Dynamics of a bubble in a liquid under laser pulse action

Simulation was performed of the behavior of a vapor bubble in a liquid under laser irradiation in laboratory experiments. A mathematical model was developed to analyze the effect of heat conduction, diffusion, and mass transfer on the bubble dynamics under compression and expansion. It is found that at the stage of collapse, intense condensation occurs on the bubble wall, which results in a significant (more than 15fold) decrease in bubble mass and an increase in pressure (to 10⁵ atm) and temperature (to 10⁴ K(. Results of numerical calculations of the radius of the first rebound and the amplitude of the divergent shock wave in water are compared with experimental data. It is shown that small (:about 1%) additives of an incondensable gas lead to a considerable decrease in mass transfer on the bubble wall.     

Penetration of Cylindrical Projectiles into Saturated Sandy Media

The time and depth of vertical one-dimensional projectile penetration into sandy media in the near shore region are derived. A precise definition for the physical properties and for the behavior of the sandy medium following the projectile impact are evaluated. Three separate time intervals following projectile impact are identified. During the first 3 ms of penetration, the deviatoric friction stress is shown to be negligible and the integrated Mie–Grüneisen equation of state (or, equivalently, the Hugoniot-adiabat) may be applied to compute the normal penetration resistance force from the sand pressure. In order to compute sand pressure as a function of the sand density D by the integrated Mie–Grüneisen equation of state, the Mie–Grüneisen dimensionless constants γ₀ and s and the dimensional speed of sound C ₀ in the sandy medium are required. In order to illustrate the one-dimensional shock wave propagation in both wet and dry sands, Hugoniot data for wet and dry silica sands are evaluated by a three degrees of freedom algorithm to compute these required constants. The numerical results demonstrate that the amplitude of the shock wave pressure in the wet silica sand (41% porosity) is approximately one-third of the shock wave pressure amplitudes in the dry silica sands (22% and 41% porosity). In addition, the shock wave pressure dampens quicker in the wet sand than in the dry sands.     

On the liquid turbulence energy spectra in two-phase bubbly flow in a large diameter vertical pipe

The liquid turbulence structure of air–water bubbly flow in a 200 mm diameter vertical pipe was experimentally investigated. A dual optical probe was used to measure the bubble characteristics, while the liquid turbulence was measured using hot-film anemometry. Experiments were performed at two liquid superficial velocities of 0.2 and 0.68 m/s for gas superficial velocities in the range of 0–0.18 m/s, corresponding to an area averaged void fraction up to 13.6%. In general, there is an increase in the liquid turbulence energy when the bubbles are introduced into the liquid flow. The increase in the energy mainly occurs over a range of length scales that are on the order of the bubble diameter. A suppression of the turbulence was observed close to the wall at very low void fraction flows. Initially, the suppression occurs in the low wave number range and then extends to higher wave numbers as the suppression is increased.     

Dissociation rate constants of diatomic molecules under thermal equilibrium conditions

By means of an analysis of currently available experimental data and theoretical models, expressions for the dissociation rate constantk ₀ are obtained for thermal equilibrium conditions. These expressions are necessary for describing the molecular dissociation process under both thermal equilibrium and non-equilibrium conditions in the gas.k ₀ values are presented for the O₂, N₂, NO, CO, CN, and C₂ molecules at temperatures from 300 to 40,000 K, and their errors are estimated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–186, January–February, 1999.     

Experimental study on interfacial area transport of a vertical downward bubbly flow

In relation to the development of the interfacial area transport equation, local flow measurements of vertical downward air–water flows in a pipe with an inner diameter of 50.8 mm were performed at three axial locations of z/D=6.50, 34.0, and 66.5 as well as ten radial locations from r/R=0 to r/R=0.9 using a multi-sensor probe. In the experiment, the superficial liquid velocity and the void fraction ranged from –0.620 m/s to –2.49 m/s and from 0.21% to 8.4%, respectively. The dependence of the interfacial area transport on the liquid velocity, void fraction, and bubble size is discussed in detail.     

Diffusion stability of bubbles in a cluster

The diffusion stability of gas bubbles in one-fraction and two-fraction clusters subjected to an acoustic field is studied. For a one-fraction cluster, numerical values were obtained for the initial gas concentrations in the liquid at which the bubble tends to one of two equilibrium states because of diffusion processes between the bubble and the ambient liquid. It is found that a two-fraction cluster tends to become a one-fraction cluster. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 40–48, July–August, 2007.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.