Wave regimes of ascending flow of a thin layer of a viscous liquid in contact with a gas

The flow of a liquid in thin layers is one of the hydrodynamic problems of chemistry and heat engineering. The large surface area of films and their small thickness make it possible to accelerate thermal, diffusive, and chemical processes at the gas-liquid boundary.Theoretical studies of liquid flow in a vertical descending thin layer are presented in [1–4]. In this paper we study ascending wave flows of a liquid in a thin vertical layer in contact with a gas, i.e., flows in the direction opposite the action of the force due to gravity, with account for the action of the gas on the liquid surface. Such motions are encountered when oil is extracted from strata that are saturated with gas. At some distance from the stratum the oil and gas separate: the gas travels at high velocity inside the pipe, occupying a considerable portion of the pipe, and the liquid is displaced toward the pipe walls, forming a thin film. In certain cases a wave-like interface develops between the oil and gas that travels with a velocity greater than that of the liquid but less than the average gas velocity. Similar phenomena are observed in high velocity mass exchangers.We examine the effect of the gas for both laminar and turbulent flow.Studies that neglect the effect of the gas flow on the liquid show that for waves on the film surface whose lengths are considerably longer than the average thickness of the layer, the liquid motion in the film is described by boundary layer equations in which account is taken of the mass force, i.e., the force due to gravity. With some approximation, we can assume that in accounting for the effect of the gas on the liquid the liquid flow is described by these same equations.     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.     

Wave flow regimes of a thin layer of viscous fluid subject to gravity

The studies of Kapitsa initiated the detailed experimental and theoretical study of the flow of a thin layer of viscous liquid (liquid film) over a solid surface [1–2]. Extensive experimental data on this question have now been accumulated. As a rule, the existing theories are based on linearization of the problem and diverge considerably from the experimental results. The present paper is also addressed to the theoretical solution of this problem. The solution method used enables consideration of the wave flow of the liquid as a nonlinear problem and on this basis permits determining all the parameters of the wave regime-amplitude, wavelength, wave propagation speed, frequency.     

Instability of a film of viscous liquid under the influence of an adjacent gas flow

A study is made of the instability of a film of viscous liquid adjacent to a gas flow. Despite a number of investigations, there is no unified theory of this problem capable of explaining the experimental results of different authors. The present paper gives a solution of the problem that is valid for a large class of flows of liquid films in the case of laminar and turbulent flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–36, March–April, 1979.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

Prediction of amount of entrained droplets in vertical annular two-phase flow

Prediction of amount of entrained droplets or entrainment fraction in annular two-phase flow is essential for the estimation of dryout condition and analysis of post dryout heat transfer in light water nuclear reactors and steam boilers. In this study, air–water and organic fluid (Freon-113) annular flow entrainment experiments have been carried out in 9.4 and 10.2 mm diameter test sections, respectively. Both the experiments covered three distinct pressure conditions and wide range of liquid and gas flow conditions. The organic fluid experiments simulated high pressure steam–water annular flow conditions. In each experiment, measurements of entrainment fraction, droplet entrainment rate and droplet deposition rate have been performed by using the liquid film extraction method. A simple, explicit and non-dimensional correlation developed by Sawant [Sawant, P.H., Ishii, M., Mori, M., 2008. Droplet entrainment correlation in vertical upward co-current annular two-phase flow. Nucl. Eng. Des. 238 (6), 1342–1352] for the prediction of entrainment fraction is further improved in this study in order to account for the existence of critical gas and liquid flow rates below which no entrainment is possible.Additionally, a new correlation is proposed for the estimation of minimum liquid film flow rate at the maximum entrainment fraction condition. The improved correlation successfully predicted the newly collected air–water and Freon-113 entrainment fraction data. Furthermore, the correlations satisfactorily compared with the air–water, helium–water and air–genklene experimental data measured by Willetts [Willetts, I.P., 1987. Non-aqueous annular two-phase flow. D.Phil. Thesis, University of Oxford]. However, comparison of the correlations with the steam–water data available in literature showed significant discrepancies. It is proposed that these discrepancies might have been caused due to the inadequacy of the liquid film extraction method used to measure the entrainment fraction or due to the change in mechanism of entrainment under high liquid flow conditions.     

Pressure wave breakdown of a gas film in a liquid-filled slit

In [1–4] the results of investigating the breakdown of gas bubbles by medium-intensity pressure waves are presented and various bubble breakdown mechanisms are proposed. It is shown that breakdown may occur as a result of the formation of a cumulative jet on the boundary of the bubble or as a result of instability due to the relative motion of the bubble in the wave. In [5] experimental data on the pressure wave breakdown of a gas film in a liquid on a solid wall are reported. It is shown that at wave amplitudes p/p₀1 a liquid jet is formed at the edge of the gas film. The jet, traveling along the wall, strips off the film and carries it into the surrounding liquid. Below we investigate the pressure wave behavior of a gas film in a liquid-filled slit.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 175–178, September–October, 1992.     

Nonlinearwaves in a liquid film-gas flow system

The instability and regular nonlinear waves in the film of a heavy viscous liquid flowing along the wall of a round tube and interacting with a gas flow are investigated. The solutions for the wave film flows are numerically obtained in the regimes from free flow-down in a counter-current gas stream to cocurrent upward flow of the film and the gas at fairly large gas velocities. Continuous transition from the counter-current to the cocurrent flow via the state with a maximum amplitude of nonlinear waves and zero values of the liquid flow rate and the phase velocity is investigated. The Kapitsa-Shkadov method is used to reduce a boundary value problem to a system of evolutionary equations for the local values of the layer thickness and the liquid flow rate.     

Effects of a countercurrent gas flow on falling-film mode transitions between horizontal tubes

A liquid film falling between horizontal tubes is known to take the form of droplets, jets or sheets, depending on the liquid flow rate; the form of the flow is the so-called “falling-film mode”. Although previously neglected in studies of mode transition, a countercurrent gas flow often exists in falling-film heat exchangers, and its effect on the liquid flow might be important: it could impact the flow regime, lead to local “dryout,” and decrease the heat transfer rate. Experiments are conducted to explore the effects of a countercurrent gas flow and liquid feeding length on falling-film mode transitions for a liquid flowing over horizontal tubes. The effects on mode transition are shown to depend on fluid properties and are explained in terms of unsteadiness and film thickness. In general, transition hysteresis is reduced with an increasing gas velocity. A correlation is developed to predict the countercurrent gas flow effects on falling-film mode transitions. The liquid feeding length can affect mode transitions in quiescent surroundings and when a countercurrent gas flow imposed.     

The problem of plane flow of ideal liquid around a shell filled with gas

The plane flow of an ideal liquid around a certain volume of gas bounded by a rigid rectilinear plate perpendicular to the incident flow and by an absolutely elastic film fastened at the ends of the plate is examined. The existence of a solution is demonstrated in a certain range of variation of the parameters, and a method of finding it is indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–117, May–June, 1971.     

Wavy flow of a liquid film in the presence of a cocurrent turbulent gas flow

Wavy downflow of viscous liquid films in the presence of a cocurrent turbulent gas flow is analyzed theoretically. The parameters of two-dimensional steady-state traveling waves are calculated for wide ranges of liquid Reynolds number and gas flow velocity. The hydrodynamic characteristics of the liquid flow are computed using the full Navier-Stokes equations. The wavy interface is regarded as a small perturbation, and the equations for the gas are linearized in the vicinity of the main turbulent flow. Various optimal film flow regimes are obtained for the calculated nonlinear waves branching from the plane-parallel flow. It is shown that for high velocities of the cocurrent gas flow, the calculated wave characteristics correspond to those of ripple waves observed in experiments.     

Stability of a laminar film flow

The problem of the wave motion of a liquid layer was first investigated by Kapitsa [1, 2], who gave an approximate analysis of the free flow and flow in contact with gas stream, and evaluated the influence of the heat transfer processes on the flow. The problem of the stability of such a flow was studied in detail by Benjamin [3] and Yih [4, 5], These authors proposed seeking the solution of the resulting Orr-Sommerfeld equation in the form of a series in a small parameter and developed a corresponding method of successive approximations. As the small parameter [3–5], they made use of the product of the disturbance wave number and the Reynolds number. In these studies, the tangential stress on the free surface was taken equal to zero, and the fluid film was always considered essentially plane. At the same time, there are certain types of problems of considerable interest in which neither of these assumptions is satisfied. A good example might be the problem on the stability of the annular regime of two-phase flow in pipes and capillaries, when the basic stream of one fluid is separated from the pipe walls by an annular layer of another fluid. In this case, the interface has a finite radius of curvature and the tangential stress on the interface may be significantly different from zero.In the present paper, the problem of the flow stability of a fluid layer with respect to small disturbances of the boundary surface is considered with account for both the finite radius of curvature of the boundary surface and the nonzero hydrodynamic friction at the boundary. The film is assumed to be quite thin. This enables us, firstly, to consider the Reynolds number small, to use the general method of [5], and, second ly, to consider the film thickness sufficiently small in comparison with the radius of curvature of the substrate on which the film lies. Furthermore, for evaluating the stability of the laminar flow of the curved film we can use the results obtained for a plane film with account for the terms which depend on the curvature of the substrate.As a rule, previous studies have considered only one-dimensional disturbances of the boundary surface. In the present paper, in the first approximation, the stability is examined in relation to two-dimensional disturbances of this surface, corresponding to three-dimensional flow disturbances.As an example, the results obtained are applied to the investigation of the stability of the free flow of a layer of fluid over an inclined plane under the sole influence of gravity.     

Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow

Flow characteristics of liquid films vertically falling along the outer wall of a circular tube without concurrent gas flow are experimentally studied, and attention is given to the longitudinally developing liquid film flow in the flow direction. Flow measurements are carried out by the methods of needle contact and electric capacity, and the obtained data are statistically processed.There exists a definite difference in flow characteristics such as wave motion patterns, film thicknesses, critical Reynolds number, and so on, depending strongly on the longitudinal distance in the flow direction as well as the liquid film Reynolds number. Measured probability distributions of interfacial waves can be well expressed by the functions of probability distribution statistically well-known as normal, logarithmic normal and gamma distributions. In terms of these functions, interfacial wave patterns are definitely classified over the whole experimental flow regime. As a rule, interfacial wave motion proceeds vigorously with increases of the longitudinal distance and Reynolds number; however, there exists a flow condition that wave fluctuation never grows up but declines regardless of an increase of Reynolds number.     

Subsonic gas-liquid cavitation flow past a disk

The problem of axisymmetric subsonic gas-liquid cavitation flow past a disk in accordance with the Riabouchinsky scheme is solved using the method of [1]. Formulas relating the main flow parameters with the cavitation number, the Mach number on the free boundary and the gas/liquid volume ratio under stagnation conditions are presented.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 202–206, March–April, 1996.     

Stability of non isothermal flow of a film of viscous liquid on an inclined plane

A study is made of the stability of nonisothermal flow of a film of viscous liquid down an inclined plane under the influence of gravity with allowance for dissipation of energy in the flow. It is assumed that the liquid is incompressible, and that its physical properties do not depend on the temperature. On the free surface of the film, allowance is made for evaporation and condensation effects. The treatment is in the long-wavelength approximation of the method proposed by Yih Chia-shun [1]. The expression obtained for the critical Reynolds number at which the flow becomes unstable indicates that viscous dissipation plays a destabilizing part in a nonisothermal flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 145–148, July–August, 1979.     

Supersonic two-phase flow around a wedge

Supersonic two-phase flow around bodies is encountered in calculating the flow around the last stages of blades of condensing turbines, in studying the motion of airplanes under cloudy conditions, etc. In the latter case, there is, along with erosion of the forward edges of the wing profiles, a change in the wave structure and interference situation in the flow about the airplane, leading to off-design regimes of motion. Supersonic flow of a two-phase mixture around a wedge, without taking account of the influence of the particles on the flow, was investigated in [1–3]. In [4], also in this kind of simplified setting, a study was made of the interaction of particles with the surface of a wedge in which reflection of the particles from the wall was taken into account. Morganthaler [5] made an experimental study of the flow of a mixture of air and aluminum oxide particles around a wedge. In [6] a theoretical study was made of a supersonic two-phase flow around thin flat axially-symmetric bodies. In particular, for the flow around a wedge, closed form solutions were obtained for the form of the shock wave, the gas streamlines and particle paths, and the distribution of all the parameters along the surface of the wedge. On the basis of the equations given in [7] and the method of characteristics, which were developed for flows consisting of a mixture of a gas and heterogeneous particles in nozzles [8,9], we present below a study of a supersonic two-phase flow around a wedge.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 83–88, March–April, 1972.     

Stability of a flow in a circular tube

Many studies, both theoretical and experimental, have been dedicated to the stability of flow in a circular tube (see, for example, review [1]). In every case mathematical investigation has not succeeded in obtaining an expression for hydrodynamic instability of such a flow for disturbances of sufficiently low amplitude. (An exception is [2].) Experiment also indicates the stability of such a flow [3], with a laminar mode being extended to Reynolds numbers of the order of tens of thousands. These facts are the basis for the assumption that the flow of a viscous incompressible liquid in a circular tube is stable for small perturbations. However, there is no analytical or even numerical proof of this hypothesis. Moreover, some studies, for example [2], indicate the instability of such a flow in relation to three-dimensional nonaxiosymmetric perturbations. The analysis of hydrodynamic stability with respect to three-dimensional disturbances of flow within a circular tube conducted in this study showed the stability of the flow over a wide range of wave numbers and Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 20–24, January–February, 1973.     

Linear and nonlinear stability of combined plane-parallel flow of a film of liquid and gas

The problem of the linear stability of a layer of liquid entrained by a gas has been investigated for some special cases in [1–7]. In [8], the linear problem was solved numerically and the solution compared with some analytic solutions for special cases of the flow. In the present paper, the results of linear analysis are presented more comprehensively; the problem of finite-amplitude stability of the film is posed and solved numerically; the results of the linear and nonlinear analysis are compared with data of an experiment performed by the authors and by other experimentalists.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–42, January–February, 1979.We are grateful to V. Ya. Shkadov for supervising the work, to all the participants of G. I. Petrov's seminar for helpful discussion, and also to E. L. Kokon for assistance in evaluating the experimental data.     

Calculation of supersonic gas flow over a ducted body in regimes with a detached shock wave

Calculations were made of the supersonic flow of an inviscid gas which does not conduct heat over two-dimensional and axisymmetric ducted bodies in regimes with a detached shock wave and completely subsonic gas velocity in the cylindrical duct. The investigated bodies have a pointed leading edge. The flow rate of the gas through the duct is assumed to be given. This corresponds to the presence in the exit section of the duct of a throttle or an impermeable barrier (in which case the flow rate is zero). The numerical algorithm used in the calculations is based on stabilization in time and Godunov's difference scheme [1] with separation of the shock wave. The integrated flow characteristics are given. The values of the wave resistance coefficient obtained in the calculations are compared with the values found using Taganov's approximate approach.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–163, May–June, 1981.I thank A. N. Kraiko for regular consultations, Yu. B. Lifshits for a helpful comment, and V. A. Vostretsov for assisting in the work.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.