Appearance of Nucleation Shocks in a Boiling Liquid

Depressurization of high-pressure vessels filled with a liquid coming to the boil with a decrease in pressure is investigated. After depressurization, which takes less than 1 ms, a rarefaction wave propagates into the vessel. Experiments [1–5] demonstrated that the pressure behind the wave goes over to a constant value, which is independent of the vessel diameter and lies between the saturation point and atmospheric pressure.The correspondence between the experiments and different bubble formation theories (bubble formation on the vessel walls, due to rupture of the bonds between the water molecules or boiling on foreign particles) is analyzed. A “ mechanical nucleation” theory is proposed, in which it is assumed that the liquid comes to the boil on a limited number of foreign particles, and the bubbles formed on the nucleation centers undergo multiple fragmentation due to the instability developing under the action of centrifugal accelerations of the bubble surface in the course of bubble growth.The calculations demonstrated that, after depressurization, bubble fragmentation occurs at the vessel outlet. Due to the growth of the phase interface in the course of fragmentation, the boiling intensity increases, the pressure grows, and a shock wave propagates into the vessel, following the rarefaction wave. Multiple fragmentation of the bubbles occurs on the shock front. This wave is followed by a series of waves with smaller amplitudes. The pressure in the vessel attains a stable level, without any shocks. This level is characterized by the metastability or superheating of the liquid, i.e. the difference between the liquid temperature and the saturation point. It is demonstrated that the resultant pressure in the vessel is independent of the number of initial boiling centers or the initial pressure in the vessel and is determined only by the initial temperature. For water, the dependence of the superheating of the liquid on the initial temperature is found and compared with experimental data.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 103–117.Original Russian Text Copyright © 2005 by Ivashnev and Smirnov.     

Specific features of the modeling of boiling-fluid flows

Experiments show that in low-and high-velocity flows the boiling process is fundamentally different: in the former, the fluid boils on the walls, and in the latter in the volume. In high-velocity flows, the boiling intensity is orders of magnitude greater. In modeling fast and slow flows, the number of bubbles, which is a free parameter of the model and must be specified, differs by orders of magnitude. When high-speed flows of different kinds are modeled (vessel depressurization, nozzle flows) the number of bubbles specified also differs by orders of magnitude. In this study, we formulate the hypothesis that in both kinds of flows the process of boiling starts similarly, namely, on the walls. However, in high-speed flows the number of bubbles increases by orders of magnitude due to bubble fragmentation. As a result of intense fragmentation, the system “forgets” the initial number of bubbles and the process becomes volume boiling. This approach makes it possible to construct a universal model of boiling. To test this hypothesis, we constructed a mathematical model which takes into account the possibility of bubble fragmentation due to the instability developing under the action of centrifugal accelerations of the bubble surface. This model was used to calculate the process of depressurization of a high-pressure vessel. The calculations demonstrated that, for any initial number of bubbles, 1 ms after depressurization the bubble number attains the same level. Bubble fragmentation takes place in “self-sustained detonation waves”. The stationary structure of detonation waves in a boiling fluid is investigated. A scheme of the wave structure according to which the wave consists of a shock wave and a relaxation zone is proposed. Calculations of a boiling-fluid flow through a Laval nozzle reveal the periodic appearance of detonation waves. Accordingly, nozzle flows should be accompanied by significant oscillations of the parameters.     

Boiling shocks in nozzles

Experiments on the depressurization of high-pressure vessels have shown the vaporization occurs mainly in “boiling shocks” moving with a velocity ∼ 10 m/s. This phenomenon was explained by proposing a boiling liquid model which takes into account the possibility of bubble fragmentation due to instability developing in the flow around the bubbles [1]. In the present study, this model is used for modeling the flow in a Laval nozzle. The flows from vessels and nozzle flows are described without variation of the free parameters, namely, the initial number of bubbles and the critical Weber number. The existence of self-oscillating regimes of boiling-liquid flow through a nozzle is detected. The origin of the oscillations is established.     

The nature of “slow” rarefaction waves accompanying the escape of a boiling liquid

The depressurization of a high-pressure container in the form of a tube closed at both ends and initially filled with water not heated to the boiling point is studied. At the zero time, one of the ends of the tube is opened, and the liquid begins to escape into the atmosphere. Since the atmospheric pressure is less than the saturation pressure of the liquid, its escape is accompanied by boiling. It is known from experiments [1, 2] that after depressurization a rarefaction wave travels into the channel with the speed of sound in the pure liquid. After it has passed, a two-phase mixture is formed in the container. A slow (or secondary) rarefaction wave (Fig. 1a) moves through this mixture with a velocity relative to the tube walls of order 10 m/sec, transforming the two-phase mixture to the equilibrium state. To explain the features of the escape process, we propose a new mathematical model of a boiling liquid that takes into account two mechanisms of vapor formation —boiling at nucleating centers present in the liquid and fragmentation of the formed bubbles. If the second mechanism is to be realized, a certain relationship must be established between the bubble size and the difference of the velocities of the phases. The slow rarefaction wave is described by means of the proposed model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–66, May–June, 1991.     

A model of critical boiling flow with account of wall boiling-up

A model for simulation of critical flows of boiling liquids is proposed, which takes into account the existence of two vapor phases: α-bubbles “attached” to the channel walls and β-bubbles moving in the flow. The model takes into account the possibility of breakdown of both groups of the bubbles due to both the Kelvin-Helmholtz instability caused by the flow around the bubbles and the Rayleigh-Taylor instability induced by the increase in the expansion rate of the vapor-water mixture. It is shown that the experiments on depressurization of high-pressure vessels can be explained by assuming that the initial boiling centers are located only on the walls, and in the flow core the bubbles appear only after the injection of the fragments of broken wall bubbles into the flow. The experimental oscillograms are compared with the calculated curves obtained using the model proposed and the models which take into account only one instability mode, i.e. the Kelvin-Helmholtz or Rayleigh-Taylor instability. The waves which accompany the vessel depressurization process are described.     

Formation conditions for bubble suspensions upon shock-wave loading of liquids

The conditions of development of bubble cavitation in liquid media upon shock-wave loading are found. It is s shown that, for the development of an unbounded cavitation, the bubbles should grow to certain critical sizes sufficient for their transition to a nonequilibrium state owing to the elastic energy transferred by a rarefaction wave to a liquid sample (at the stage of unloading). In contrast to low-viscosity liquids, in high-viscosity ones (such as glycerin) these conditions cannot be satisfied for any really attainable parameters of shock waves. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 53–63, March–April, 2000.     

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.      

A numerical study of weak shock wave propagation in a reactive bubbly liquid

A numerical study is presented on the response of a weakly shock compressed liquid column that contains reactive gas bubbles. Both the liquid and gas are considered compressible. Compressibility of the liquid allows calculation of shock and rarefaction waves in the pure liquid as well as in the gas/liquid mixture. A microscopic model for local bubble collapse is coupled with a macroscopic model of wave propagation through the gas/liquid mixture. In the particular cases presented here, the characteristic times of propagation of the shock wave and bubble collapse are of the same order of magnitude. Consequently, the coupling between various phenomena is very strong. The present model based on fundamental principles of two-phase fluid mechanics takes into account the coupling of localized bubble oscillations. This model is composed of a microscopic one in the scale of a bubble size, and a macroscopic one which is based on the mixture theory. The liquid under study is water, and the gas is a reactive mixture of argon, hydrogen and oxygen. Received 18 December 1995 / Accepted 2 June 1996     

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.     

Shock waves in water with Freon-12 bubbles and formation of gas hydrates

The evolution of a shock wave and its reflection from a wall in a gas-liquid medium with dissolution and hydration are experimentally investigated. Dissolution and hydration behind the front of a moderate-amplitude shock wave are demonstrated to be caused by fragmentation of gas bubbles, resulting in a drastic increase in the area of the interphase surface and in a decrease in size of gas inclusions. The mechanisms of hydration behind the wave front are examined. Hydration behind the front of a shock wave with a stepwise profile is theoretically analyzed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 58–75, May–June, 2007.     

A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.      

On Time Reversible Description of the Process of Coagulation and Fragmentation

The process of coagulation is associated with scalar conservation laws, where the adhesion particle dynamics results from shock waves. Conversely, the fragmentation of a massive particle into a number of smaller ones, or into a continuous (dust) distribution, is associated with rarefaction waves. It is generally agreed that a reversible solution of a conservation law can include neither shock waves nor the spontaneous emergence of rarefaction waves. The present paper is an attempt to demonstrate that both coagulation and fragmentation may coexist for a reversible solution, under a natural generalization of the system of conservation law. This is done by introducing an action principle which includes, in addition to the inertial (kinetic energy) term, also an appropriately defined internal energy. The above generalization of the system of conservation law appears as the Euler–Lagrange equations for this action.     

Evolution of bubble spectrum in the process of cavitational fragmentation by a shock wave reflected from a free liquid surface

Equations which describe the evolution of the bubble spectrum in the process of cavitational fragmentation by a shock wave reflected from a free liquid surface are formulated. As an example, the effect of artificial saturation of the initial fluid with large bubbles on the dispersity of a liquid-drop gas suspension focused by dispersion is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 51–58, May–June, 1992.     

Experimental investigation of steel spheres as a passive enhancement technique for flash boiling of HCFC-22

An experimental investigation was conducted to study the influence of a layer of 3.6-mm-diameter steel spheres on the mass flow during flash boiling in a glass pressure vessel. It was observed that the steel spheres added numerous heterogeneous nucleation sites within the liquid and promoted abundant vapor bubble growth during depressurization. The steel spheres were in contact with each other and with the interior of the glass vessel. The data from these experiments were compared with baseline experimental results primarily with regard to the mass flow. Each test was run for 60 s, using controlled variables of orifice diameters (1.59 and 5.56 mm), initial refrigerant amounts (0.23, 0.45, and 0.68 kg), initial pressures (575 and 840 kPa), and vessel geometries (665 and 1110 ml). Pressures, temperatures, and mass flow rates, along with calculated saturation temperatures, amount of superheat, mass flux, and total mass flashed, were used to compare the baseline experiments with the enhanced boiling method. Results showed an increase in the total mass flashed at each test condition, ranging from an average of 22% to 81% with respect to baseline experiments.     

Interference between a centered rarefaction wave and an inclined shock wave

The gas flow in the zone of interaction between an oblique shock and a centered isentropic rarefaction wave is studied using the direct statistical simulation method for solving the Boltzmann equation. The data of calculations of the shock and rarefaction wave structures, flow fields, and streamlines are given for the free-stream Mach number M_∞ = 6, 4 and 2. The formation of the interaction zone is simulated by a gas flow past a double-plane wedge in which the break of the generating line leads to formation of the centered isentropic rarefaction wave. The results of calculations of this flow in solving the Boltzmann equation are given in the Euler approximation.     

Shock wave structure in a mixture of gas,vapour and droplets

The structure of the relaxation zone behind a shock wave of moderate strength in a mixture of gas, vapour and droplets is analysed. A model is presented for shock induced evaporation, which is based on wet-bulb equilibrium and on the absence of relative motion between droplets and gas. Experimental and numerical data on heterogeneous condensation induced by an unsteady rarefaction wave and on re-evaporation due to shock wave passage are reported for a mixture of water vapour, nitrogen gas and condensation nuclei. Pressure, temperature, saturation ratio and droplet size are experimentally obtained and are very well predicted by a numerical simulation based on the non-linear quasisteady wet-bulb model for phase transition, as well for the expansion wave as for the shock wave. During expansion, droplet number density decays much faster than predicted, which is not yet satisfactorily explained. Shock induced droplet evaporation is studied for post-shock saturation ratios ranging from 5×10^–3 to 0.2, corresponding to shock Mach numbers of 1.2 to 1.9. The evaporation times are well predicted by the theoretical model. No evidence is found for droplet break-up for Weber numbers up to 13, and droplet radii of the order of 1m.On leave at Institute of Fluid Science, Shock Wave Research Center, Tohoku University, Sendai 980, JapanThis article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Interaction of a high-speed combustion front with a closely packed bed of spheres

The head-on collision of a combustion front with a closely packed bed of ceramic-oxide spheres was investigated in a vertical 76.2 mm diameter tube containing a nitrogen diluted stoichiometric ethylene–oxygen mixture. A layer of spherical beads in the diameter range of 3–12.7 mm was placed at the bottom of the tube and a flame was ignited at the top endplate. Four orifice plates spaced at one tube diameter were placed at the ignition end of the tube in order to accelerate the flame to either a “fast-flame” or a detonation wave before the bead layer face. The mixture reactivity was adjusted by varying the initial mixture pressure between 10 and 100 kPa absolute. The pressure before and within the bead layer was measured by flush wall-mounted pressure transducers. For initial pressures where a fast-flame interacts with the bead layer peak pressures recorded at the bead layer face were as high as five times the reflected Chapman–Jouget detonation pressure. The explosion resulting from the interaction developed by two distinct mechanisms; one due to the shock reflection off the bead layer face, and the other due to shock transmission and mixing of burned and unburned gas inside the bead layer. The measured explosion delay time (time after shock reflection from the bead layer face) was found to be independent of the incident shock velocity. As a result, the explosion initiation is not the direct result of the shock reflection process but instead is more likely due to the interaction of the reflected shock wave and the trailing flame. The bead layer was found to be very effective in attenuating the explosion front transmitted through the bead layer and thus isolating the tube endplate. This paper is based on work that was presented at the 21th International Colloquium on the Dynamics of Explosions and Reactive Systems, Poitiers, France, July 23–27, 2007.     

Detonation wave initiated by explosive condensation of supersaturated carbon vapor

An experimental study of the influence of condensation of supersaturated carbon vapor formed behind reflected shock waves on the process of propagation of a shock wave and formation of a detonation wave of condensation is carried out. Highly supersaturated carbon vapor was formed from thermal decay of unstable carbon suboxide C₃O₂ → C + 2CO behind a shock wave in mixtures containing 10–30% C₃O₂ in Ar. This reaction was followed by fast growth of condensed carbon particles, accompanied by heat release. Experiments have shown a considerable temperature and pressure increase in the narrow zone behind the wave front, resulting in shock wave amplification and transition to a detonation-like regime. An analysis of the kinetics and heat release in the given conditions and calculations based upon one-dimensional detonation theory have shown that in a mixture of 10% C₃O₂ + Ar, insufficient heat release resulted in a regime of “overdriven detonation”. In a mixture of 20% C₃O₂ + Ar a very good coincidence of measured values of pressure and wave velocity with calculated Chapman–Jouguet parameters is observed. In a 30% C₃O₂ + Ar mixture, an excess heat release caused a slow down of the effective condensation rate and a regime of “underdriven detonation” is observed.     

Fiber-optic probe measurements of void fraction and bubble size distributions beneath breaking waves

Experiments were performed to determine the accuracy of single-tip fiber-optic probes for making simultaneous measurements of the void fraction and bubble size distributions under breaking waves. Tests in a vertical bubble column showed that the normalized RMS error in the void fraction measurements was ∼10%. The relationship between bubble rise time and bubble velocity was investigated in a unidirectional flow cell. Similar to previous studies the rise time and bubble velocity were found to be related by a power law equation. The probes can provide mean bubble velocities accurate to ±10% when a minimum of ∼15 individual bubble velocities are averaged. The fiber-optic probes were deployed beneath a plunging breaking wave in a laboratory wave channel. The slope and shape of the bubble cord length size distribution measured with the probes was found to agree closely with the size distribution measured from digital video recordings. The probes were then positioned in the splash-up zone of a plunging breaker and the resulting cord length distribution had a shape and slope that was in agreement with previous measurements. These results demonstrate that single-tip fiber optic probes can provide accurate simultaneous measurements of the void fraction and bubble sizes inside the dense bubble clouds entrained by breaking waves.