The behavior of gas-liquid interfaces in vertical tubes

Experimental studies of interface behavior when a gas flow, confined in a vertical tube, flows past a stationary body of liquid are presented. Critical conditions necessary for the interface to become unstable or break up are investigated. Specific phenomena studied include: penetration of liquid from a reservoir into the top open end of a vertical tube from which gas is emerging, flow of gas past a liquid ring maintained on the inside wall of the tube, conditions for the support of a “hanging film” on the tube wall, formation of droplets and establishment of a continuous upwards-flowing liquid film. A general mathematical formulation of this problem is presented and used to derive the set of relevant dimensionless parameters. Solutions are obtained to certain simple cases and are shown to be consistent with experiment in the limits in which one or more of the variables exerts negligible influence.     

Turbulent film condensation on a horizontal elliptical tube

An investigation has been made of turbulent film condensation on a horizontal elliptical tube. The present study is based on Colburn analogy [1] and potential flow theory to determine the high tangential velocity of vapor flow at the boundary layer and to define the local interfacial shear owing to high velocity vapor flow across the tube surface. The condensate film flow and local/or mean heat transfer characteristics from a horizontal elliptical tube with variable ellipticities, e, under the influence of Froude number, sub-cooling parameter and system pressure have been performed. The present result for dimensionless mean heat transfer coefficient reduces to the same result obtained by Sarma et al.s [2] e=0 (circular tube). Compared with laminar model by Yang and Hsu [3], the present turbulent model shows in better agreement with Michaels experimental data [4] (for e=0). The dependence of mean Nusselt coefficient on the effect of n (power of Reynolds) [1] is also discussed.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

Construction of liquid flow past a solid body with an air layer

It is known that the friction drag for a body moving in a liquid may be reduced by varying the physical properties of the liquid at the body surface and, in particular, by creating an artificial air cavity on the body surface.The question of the magnitude of the drag gain in the case of the presence of a wall fluid layer with different physical constants was considered in [1–3].In [4, 5] results are presented of theoretical and experimental studies on the question of determining the parameters of artificial air cavities created on the lower surface of a horizontal wall in order to reduce the drag.In the following an attempt is made to construct the three-dimensional flow of an ideal liquid past a body which is enveloped entirely in an air cavity.     

Parametric Excitation of a Secondary Flow in a Vertical Layer of a Fluid in the Presence of Small Solid Particles

Thermal convection is studied in an inhomogeneous medium consisting of a fluid and a solid admixture under conditions of finite–frequency vibrations. Convection equations are derived within the framework of the generalized Boussinesq approximation, and the problem of flow stability in a vertical layer of a viscous fluid with horizontal oscillations along the layer to infinitely small perturbations is considered. A comparison with experimental data is made.     

A study of the vibration of a horizontal U-bend subjected to an internal upwards flowing air–water mixture

U-bends are a common geometry in heat exchangers. In this paper, a U-bend in the vertical plane connected to horizontal straight pipes is considered. An initially stratified water/air flow moves upwards against gravity. The aim of this research is to investigate the internal flow profile and resulting force when the U-bend is subjected to a stratified air–water flow at the inlet. This is done numerically, i.e. by solving the unsteady Reynolds-averaged Navier–Stokes equations. For low mass flow rates, large gas bubbles are naturally formed at the entrance of the bend. The transient force on the tube allows to determine precisely the time instants of bubble initiation and thus to quantify the bubble frequency. Firstly, the tube is assumed to be rigid and the dependence of force oscillation on the inlet conditions is investigated. Secondly, the influence of the viscosity, wall wetting and the mass flow rate is analyzed. Finally, a fluid–structure interaction calculation is performed in order to quantify the vibration characteristics of the tube.     

Plane capillary flow of a viscous fluid with mutiply connected boundary in the Stokes approximation

A study is made of the process of relaxation to the equilibrium configuration of an isolated volume of a viscous incompressible Newtonian fluid under the influence of capillary forces. The fluid has the form of an infinite cylinder of arbitrary shape with a smooth compact and, in general, multiply connected boundary. In the course of relaxation, internal cavities collapse, and the cylinder acquires asymptotically a circular configuration. The quasisteady Stokes approximation [1] is used to describe the flow. First proposed by Frenkel' [2], this approximation has been used in the calculation of a dynamic boundary angle [3], the collapse of a circular cylinder [4], and the collapse of a hollow cylinder [5]. The analogy between the hydrodynamic equations in the Stokes approximation and the equations of elasticity theory made it possible [6] to describe the relaxation of a simply connected cylinder by a method close to the one employed by Muskheleshvili [7]. In the present paper, the approach of the author [8] based on work of Grinberg [9] and Vekua [10] is developed. It is shown that the true pressure distribution gives a minimum of the integral of the square of the pressure over the region for fixed integral of the pressure over the boundary. An explicit expression for the pressure is obtained in the form of the projection of a generalized function with support on the boundary onto the subspace of harmonic functions. The velocity field on the boundary of the region is calculated. An upper bound is found for the law of decrease of the perimeter of the region and for the time during which the number of connected components of the boundary remains unchanged.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–122, January–February, 1992.     

Investigation of the flow and heat transfer of viscous reacting fluids in long tubes

Heat transfer and resistance in the case of laminar flow of inert gases and liquids in a circular tube were considered in [1–4], the justification of the use of boundary-layer type equations for investigating two-dimensional flows in tubes being provided in [4]. The flow of strongly viscous, chemically reacting fluids in an infinite tube has been investigated analytically and numerically in the case of a constant pressure gradient or constant flow rate of the fluid [5–8]. An analytic analysis of the flow of viscous reacting fluids in tubes of finite length was made in [9, 10]. However, by virtue of the averaging of the unknown functions over the volume of the tube in these investigations, the allowance for the finite length of the tube reduced to an analysis of the influence of the time the fluid remains in the tube on the thermal regime of the flow, and the details of the flow and the heat transfer in the initial section of the tube were not taken into account. In [11], the development of chemical reactions in displacement reactors were studied under the condition that a Poiseuille velocity profile is realized and the viscosity does not depend on the temperature or the concentration of the reactant; in [12], a study was made of the regimes of an adiabatic reactor of finite length, and in [13] of the flow regimes of reacting fluids in long tubes in the case of a constant flow rate. The aim of the present paper is to analyze analytically and numerically in the two-dimensional formulation the approach to the regimes of thermal and hydrodynamic stabilization in the case of the flow of viscous inert fluids and details of the flow of strongly viscous reacting fluids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–25, January–February, 1930.     

Simultaneous heat and mass transfer inside a vertical tube in evaporating a heated falling alcohols liquid film into a stream of dry air

A numerical study of the evaporation in mixed convection of a pure alcohol liquid film: ethanol and methanol was investigated. It is a turbulent liquid film falling on the internal face of a vertical tube. A laminar flow of dry air enters the vertical tube at constant temperature in the downward direction. The wall of the tube is subjected to a constant and uniform heat flux. The model solves the coupled parabolic governing equations in both phases including turbulent liquid film together with the boundary and interfacial conditions. The systems of equations obtained by using an implicit finite difference method are solved by TDMA method. A Van Driest model is adopted to simulate the turbulent liquid film flow. The influence of the inlet liquid flow, Reynolds number in the gas flow and the wall heat flux on the intensity of heat and mass transfers are examined. A comparison between the results obtained for studied alcohols and water in the same conditions is made.     

Magnetic control of natural convection in the horizontal Bridgman configuration using a spectral method: transversal plan

The present study investigates the electromagnetic braking of buoyancy convective flows occurring in differentially heated cavities, filled with low Prandtl, dilute, incompressible and electrically conducting alloys, and subjected to a constant horizontal temperature gradient. In practice, such flows known as ‘Hadley circulation’ are relevant in material processing technologies, such as the horizontal Bridgman configuration. A collocation spectral numerical method is developed to solve the two-dimensional Navier–Stokes equations, modelling the flow phenomena occurring in such configurations, using a vorticity–stream function formulation. The two components of the velocity are deduced from the stream function and the temperature distribution is obtained through the resolution of the energy conservation equation. The results in terms of velocity and temperature distributions for a given Grashof number are obtained for various Hartmann numbers and show that as the Hartmann number increases, the electromagnetic braking of the flow is observed. Moreover, the results illustrate the changes affecting the flow structure which becomes quasi-parallel in the core region of the cavity for sufficiently high values of Ha and the onset of the Hartmann and parallel layers along the boundaries. Also, with increasing Ha, the isotherms are less affected by the convective flow and become parallel to the vertical walls indicating that heat transfer is mainly achieved by conduction.     

Convective flow with uniform vorticity in a vertical layer

The free convective circulation of liquid in plane vertical slits of circular and square cross section with a longitudinal horizontal temperature gradient at the boundaries was investigated experimentally. It was found that under such heating conditions there is a uniform-vorticity flow with a region of quasirigid rotation, which has the shape of a disk in a circular slit and the shape of a cross in a square slit; in each longitudinal section of this zone the liquid moves along concentric trajectories with constant angular velocity. Dimensionless numbers for the problem were established by tests with various liquids and cavities of different dimensions. In dimensionless form, the angular velocity of the vortex and the temperature gradient in it depend linearly on the temperature difference at the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–165, May–June, 1984.     

Frictional pressure drop during vapour–liquid flow in minichannels: Modelling and experimental evaluation

Condensation in minichannels is widely used in air-cooled condensers for the automotive and air-conditioning industry, in heat pipes and other applications for system thermal control. The knowledge of pressure drops in such small channels is important in order to optimize heat transfer surfaces. This paper presents a model for calculation of the frictional pressure gradient during condensation or adiabatic liquid–gas flow inside minichannels with different surface roughness. In order to account for the effects of surface roughness, new experimental frictional pressure gradient data associated to single-phase flow and adiabatic two-phase flow of R134a inside a single horizontal mini tube with rough wall has been used in the modelling. It is a Friedel (1979) [Friedel, L., 1979. Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow. In: Proceedings of the European Two-Phase Flow Group Meeting, Ispra, Paper E2] based model and it takes into account mass velocity, vapor quality, fluid properties, reduced pressure, tube diameter, entrainment ratio and surface roughness. With respect to the flow pattern prediction capability, it has been built for shear dominated flow regimes inside pipes, thus, annular, annular-mist and mist flow are here predicted. However, the suggested procedure is extended to the intermittent flow in minichannels and it is also applied with success to horizontal macro tubes.     

A semi-implicit lagrangian scheme for 3D shallow water flow with a two-layer turbulence model

A semi-implicit Lagrangian finite difference scheme for 3D shallow water flow has been developed to include an eddy viscosity model for turbulent mixing in the vertical direction. The α-co-ordinate system for the vertical direction has been introduced to give accurate definition of bed and surface boundary conditions. The simple two-layer mixing length model for rough surfaces is used with the standard assumption that the shear stress across the wall region at a given horizontal location is constant. The bed condition is thus defined only by its roughness height (avoiding the need for a friction formula relating to depth-averaged flow, e.g. Chezy, used previously). The method is shown to be efficient and stable with an explicit Lagrangian formulation for convective terms and terms for surface elevation and vertical mixing handled implicitly. The method is applied to current flow around a circular island with gently sloping sides which produce periodic recirculation zones (vortex shedding). Comparisons are made with experimental measurements of velocity using laser Doppler anemometry (time histories at specific points) and surface particle-tracking velocimetry.     

Radiative cooling of plasma behind a reflected shock front

Calculation of gas flow in a shock tube on the basis of ideal theory [1] leads to results that differ from the real picture. In particular, the calculated velocity of the reflected shock wave exceeds the experimentally measured velocity [2] by about 20%. The calculated parameters of shock-heated gas agree well with the experimental results only directly behind the shock front [3]. The present paper reports a theoretical and experimental investigation of the variation of the plasma parameters behind the front of a reflected shock wave in argon. A picture of the gas-dynamic processes taking place after reflection of the incident shock wave by the end of the shock tube is determined. A method is developed for approximate analytic calculation, this making it possible to determine not only the parameters of the gas directly behind the front of the reflected shock wave for different positions of the wave relative to the end of the shock tube but also the variation of these parameters in other regions behind the reflected shock wave. The calculation takes into account the influence of the boundary layer and radiative cooling in the approximation of a low degree of ionization of the plasma and persistence of equilibrium conditions in the entire region behind the reflected shock wave. The experimental and theoretical profiles of the radiation behind the reflected shock wave are compared.     

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

The influence of a magnetic field on the mechanical behavior of a fluid interface

This work focuses on a theoretical investigation of the shape and equilibrium height of a magnetic liquid–liquid interface formed between two vertical flat plates in response to vertical magnetic fields. The formulation is based on an extension of the so called Young–Laplace equation for an incompressible magnetic fluid forming a two-dimensional free interface. A first order dependence of the fluid susceptibility with respect to the magnetic field is considered. The formulation results in a hydrodynamic-magnetic coupled problem governed by a nonlinear second order differential equation that describes the liquid–liquid meniscus shape. According to this formulation, five relevant physical parameters are revealed in this fluid static problem. The standard gravitational Bond number, the contact angle and three new parameters related to magnetic effects in the present study: the magnetic Bond number, the magnetic susceptibility and its derivative with respect to the field. The nonlinear governing equation is integrated numerically using a fourth order Runge-Kutta method with a Newton–Raphson scheme, in order to accelerate the convergence of the solution. The influence of the relevant parameters on the rise and shape of the liquid–liquid interface is examined. The interface shape response in the presence of a magnetic field varying with characteristic wavenumbers is also explored. The numerical results are compared with asymptotic predictions also derived here for small values of the magnetic Bond number and constant susceptibility. A very good agreement is observed. In addition, all the parameters are varied in order to understand how the scales influence the meniscus shape. Finally, we discuss how to control the shape of the meniscus by applying a magnetic field.     

The effect of hydrodynamic friction on the stability of a wall layer in annular two-phase flow

This note is concerned with annulat flow in a vertical tube when the gas stream in the central portion of the tube is separated from the wall by an annular layer of liquid. The friction at the interface may have either a stabilizing or destabilizing effect on such a flow regime with respect to small disturbances of the interface. The liquid-layer thickness is assumed small, which permits direct use of the results of [1] in studying the stability. The analysis is applicable to laminar and developed turbulent gas motions; in both cases the motion in the liquid layer is assumed laminar.