Dynamics of a “collective” bubble in a magma melt flow behind the decompression wave front

Specific features of the dynamics of the wave field structure and growth of a “collective” bubble behind the decompression wave front in the “Lagrangian” section of the formed cavitation zone are numerically analyzed. Two cases are considered: with no diffusion of the dissolved gas from the melt to cavitation nuclei and with the diffusion flux providing an increase in the gas mass in the bubbles. In the first case, it is shown that an almost smooth decompression wave front approximately 100 m wide is formed, with minor perturbations that appear when the front of saturation of the cavitation zone with nuclei is passed. In the case of the diffusion process, the melt state behind the saturation front is principally different: jumps in mass velocity and viscosity are observed in the vicinity of the free surface, and the pressure in the “collective” cavitation bubble remains unchanged for a sufficiently long time interval, despite the bubble growth and intense diffusion of the gas from the melt. It is assumed that the diffusion process (and, therefore, viscosity) actually become factors determining the dynamics of growth of cavitation bubbles beginning from this time interval. A pressure jump is demonstrated to form near the free surface.     

Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

Forced nonlinear oscillator with nonsymmetric dry friction

In this paper we discuss an approximately steady motion of an oscillator as a single whole with a constant “on the average” velocity. For that purpose we analyze the position and stability of some special points of the phase portrait. In the presence of internal excitation and nonsymmetric Coulomb dry friction, a motion of the oscillator with a constant “on the average” velocity is possible. The algebraic equation for this constant velocity is found. For different parameters of the model there exist at most three regimes of motion with a constant velocity, but only one or two of them are stable. The theoretical results obtained can be used for the design of worm-like moving robots.     

Measurements of mean and fluctuating wall shear stress beneath spanwise-invariant separation bubbles

 Pulsed-wire measurements of wall shear stress have been made beneath two separation bubbles. In one a cross flow was generated by means of a (25°) swept separation line. Fluctuating stresses in orthogonal “streamwise” and cross-flow directions are very nearly equal and independent of at least moderate cross flow velocity. These fluctuations are largely determined by large-scale motions in the outer flow, whereas the mean shear stresses are not. The pdf of the “streamwise” fluctuations is unchanged by the cross flow. When a cross flow is present the pdf of the cross-flow stresses is similar to the “streamwise” pdf. Dependence on Reynolds number is the same in both flows. Received: 10 April 1998/Accepted: 17 July 1998     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

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.     

Dynamic analysis of microrobots with Coulomb friction using harmonic balance method

In this paper, we investigate the dynamic analysis of a strongly nonlinear microrobot using a three-term harmonic balance method. The employed locomotion concept, namely “friction drive principle,” is based on the superposition of a horizontal vibration at the interface between the robot and work floor and an active variation of friction force, obtained by the vertical vibration of the base at the same interface. The equation of motion for the system reveals a parametrically excited oscillator with discontinuity for which the elastic force term is proportional to a signum function. The obtained periodic solution not only is of high accuracy, but also can predict the contribution of the friction coefficient in the average velocity of the slider. Results show that the velocity and the step efficiency of motion depend almost sinusoidally on the phase shift between the horizontal and the vertical vibration. Unlike traditional analytical techniques and in agreement with both numerical simulations and experimental results reported in the literature, the utilized method demonstrates that the maximum average velocity occurs at a phase shift that varies with respect to system’s configuration parameters. Besides, the effect of variation of different configuration parameters on the behavior of this type of microrobots has been studied and the maximum achievable performance in terms of velocity and the step efficiency has been evaluated.     

Gas-dynamic signs of explosive eruptions of volcanoes. 2. Model of homogeneous-heterogeneous nucleation. Specific features of destruction of the cavitating magma

The dynamics of state of the crystallite-containing magma is studied within the framework of the gas-dynamic model of bubble cavitation. The effect of crystallites on flow evolution is considered for two cases: where the crystallites are cavitation nuclei (homogeneous-heterogeneous nucleation model) and where large clusters of crystallites are formed in the magma in the period between eruptions. In the first case, decompression jumps are demonstrated to arise as early as in the wave precursor; the intensity of these jumps turns out to be sufficient to form a series of discrete zones of nucleation ahead of the front of the main decompression wave. Results of experimental modeling of an explosive eruption with ejection of crystallite clusters (magmatic “bombs”) suggest that a cocurrent flow of the cavitating magma with dynamically varying properties (mean density and viscosity) transforms to an independent unsteady flow whose velocity is greater than the magma flow velocity. Experimental results on modeling the flow structure during the eruption show that coalescence of bubbles in the flow leads to the formation of spatial “slugs” consisting of the gas and particles. This process is analyzed within a combined nucleation model including the two-phase Iordansky-Kogarko-van Wijngaarden model and the model of the “frozen” field of mass velocities in the cavitation zone. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 167–177, March–April, 2009.     

Nucleation and growth of a gas bubble in magma

The dynamics of a “collective” gas bubble in the magma melt during its decompression was numerically studied on the basis of a complete mathematical models of an explosive volcanic eruption. It is shown that the bubble size distribution obtained for the nucleation process has one peak, which allows considering a “collective” bubble. The main stages of bubble growth due to gas diffusion and changes in the viscosity of the medium are determined. It is shown that the high viscosity of the melt makes possible the transition from the Rayleigh equation to a simpler relation for the radial velocity of the bubble.     

Influence of wall proximity on the equilibrium temperature of curved interfaces

Equilibrium conditions of a single-component two-phase-system having a plane or a concave interface interacting with a solid wall are the major focus of the paper. The concave interface is termed “closed”, if it forms a vapour bubble, and “opened”, in the case of a common liquid meniscus. The equations derived describe the equilibrium temperature in dependence of the wall distance and the interfacial curvature. They show that an attraction between the vapour-liquid interface and the wall rises the equilibrium temperature. At comparable conditions, the equilibrium temperature is higher for the closed than for the opened interface. Received on 18 December 1997     

A numerical strategy for the direct 3D simulation of the expansion of bubbles into a molten polymer during a foaming process

In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo‐plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity–pressure‐mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first‐order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space–time discontinuous‐Galerkin method, coupled with an r‐adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D. Copyright © 2007 John Wiley & Sons, Ltd.     

On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

Measurements of local velocities inside thin liquid films in horizontal two-phase flow

A new experimental facility based on Laser Doppler anemometry permits accurate local measurements in a horizontal pipe. Measurements of the axial velocity component in the liquid layer of the atomization/stratified flow regime are reported. The new information includes time-averaged local velocities, RMS values, probability density distributions, and power spectra. Elimination of velocity bias and calculation of velocity spectra is accomplished by a recently developed “signal reconstruction” algorithm. The data suggest that only in the vicinity of the solid surface (sublayer) does the liquid motion resemble the well-known behavior of single phase flow. Beyond that, the flow field is strongly influenced by the wavy gas/liquid interface and by the apparently intensive energy transfer from the very fast moving gas to the liquid layer.     

Shape Oscillations of Rising Bubbles

The paper details results from an experimental study on bubbles rising in still tap water. Shape and motion parameters of the bubbles were measured using a combination of high speed cinematography and digital image processing. The Reynolds numbers of the bubbles studied ranged from about 700 to 1300, with the bubbles exhibiting all the familiar shape and motion characteristics: oblate spheroids becoming wobbly, and spiralling or zig-zagging motion becoming rocking as the bubble size increased. Time series of the bubble major axes revealed regular oscillations in the bubble shape. In most cases three frequencies could be readily identified, corresponding to those of vortex shedding from the bubble and two modes of ellipsoidal harmonics (modes 2,0 and 2,2). Comparison of time series of bubble shape and motion indicated a strong interaction between the shape oscillations of mode 2,0 and bubble motion. As the bubble size increased the frequency of both shape oscillation modes approached that of the vortex shedding, which remained constant at about 12 Hz for all of our experiments. The frequencies become equal for bubbles larger than in our study, at a Reynolds number of about 3000. Using data from the literature we found that the vortex shedding appears to become locked-in on the mode 2,0 shape oscillation.     

Motion of two interconnected mass points under action of non-symmetric viscous friction

This paper deals with the analysis of a one-dimensional motion of two mass points in a resistive medium. The force of resistance is described by small non-symmetric viscous friction acting on each mass point. The magnitude of this force depends on the direction of motion. The mass points are interconnected with a kinematic constraint or with an elastic element. Using the averaging method the expressions for the stationary “on the average” velocity of the systems’s motion as a single whole is found. In case of a small degree of non-symmetry an explicit expression for the stationary “on the average” velocity of the system is derived. For the other case we obtained algebraic equations for the corresponding stationary velocity.     

Flow measurement on an oscillating pipe flow near the entrance using the UVP method

 The authors have carried out a study to investigate and clarify the characteristics of purely oscillating pipe flows over the developing region. The main objective of this study is to establish the method of time-dependent velocity profiles obtained by the ultrasonic velocity profile (UVP) measurement method. First, the relationship between the test fluids and the microparticles, as reflectors of ultrasonic pulses, was investigated. In addition, the relationship between the sound speeds of the test fluids and the wall materials was studied. Second, the UVP was used to obtain the instantaneous velocity profiles in oscillating pipe flows, and the developing characteristics of the flows were analyzed. Finally, the “entrance length” (by analogy with a unidirectional pipe flow) required for oscillating pipe flows was analyzed by examining the amplitude of the harmonic spectral components of the oscillating frequency. A fast Fourier transform (FFT) is proposed as the applicable method to estimate the “entrance length”. From the Fourier transform of the velocity on the centerline, nonlinear oscillation of fluid occurs in the “entrance length” of the oscillating flows, and the viscous dissipation of the higher-order velocity harmoncis determines the entrance region. The “entrance length” can be obtained from the dissipation length of the third-order harmonic. These results prove that the UVP method is highly applicable to carry out the flow measurement in the “entrance length” of oscillating pipe flow. Received: 20 March 2000 / Accepted: 10 August 2001     

Thermal Growth of a Vapor Bubble Moving in a Superheated Liquid

The problem of thermal growth of a vapor bubble moving in a superheated liquid is solved for two models of the phase interface, namely: a rigid (no-slip condition) and a pliable (slip condition) spherical surface. The second self-similar solution of the problem of the motion of a vapor bubble with a pliable surface is found. On the basis of this solution, an approximate dependence of the nondimensional heat flux into the bubble on the Jacob and Péclet numbers is constructed. For two limiting cases, namely, for a bubble growing at rest and a moving bubble of constant radius, this dependence coincides with the known solutions. The calculation results are compared with the experimental data obtained for vapor bubbles rising in a superheated liquid.     

Measurement of multiple gas-bubble velocities in gas–liquid flows using hot-film anemometry

A dual-probe hot-film anemometry technique has been developed to measure multiple gas-bubble velocities corresponding to different gas-bubble size groups in air–water flows. A data reduction scheme using wavelet analysis combined with a phase-detection technique is used to discriminate the hot-film anemometer output signals into signals corresponding to different bubble size groups. The phase and bubble size discrimination is based on the magnitude and time derivative of the signal, and the streamwise length of the gas bubbles. A cross-correlation between the discriminated signals from the two probes yields an average time difference of arrival of the gas bubbles at the two sensor locations. The velocities are estimated from the distance between the sensors and the time difference of arrival. The mean bubble size is estimated from the chord length distribution. Measurements performed in vertical-up air–water slug flow show the technique to be a viable method for obtaining bubble velocity and size information. The velocity measurements from the hot-film anemometry are corroborated using a high-speed quantitative flow visualization system. Received: 22 December 1999/Accepted: 8 May 2001     

The behavior of viscoelastic squeeze films subject to normal oscillations, including the effect of fluid inertia

The problem of the squeeze film flow of a viscoelastic fluid between parallel, circular disks is analyzed. The upper disk is subject to small, axial oscillations. Lodge's “rubber-like liquid” is used as the viscoelastic fluid model, and fluid inertia forces are included. An exact solution to the equations of motion is obtained involving in-phase and out-of-phase components of velocity field and load, with respect to the plate velocity. Peculiar resonance phenomena in the load amplitude are exhibited at high Deborah number. At certain combinations of Reynolds number and Deborah number, the in-phase and/or out-of-phase velocity field components may attain an unusual circulating type of motion in which the flow reverses direction across the film. In the low Deborah number limit, and in the low Reynolds number limit, the results of this study reduce to those obtained by other workers.