Stability of a viscous liquid film flowing down a periodic surface

The paper is devoted to a theoretical analysis of linear stability of the viscous liquid film flowing down a wavy surface. The study is based on the Navier–Stokes equations in their full statement. The developed numerical algorithm allows us to obtain pioneer results in the stability of the film flow down a corrugated surface without asymptotic approximations in a wide range over Reynolds and Kapitsa’s numbers. It is shown that in the case of moderate Reynolds numbers there is a region of the corrugation parameters (amplitude and period) where all disturbances decay in time and the wall corrugation demonstrates a stabilizing effect. At the same time, there exist corrugation parameters at which the steady-state solution is unstable with respect to perturbations of the same period as the period of corrugation. In this case the waveless solution cannot be observed in reality and the wall corrugation demonstrates a destabilizing effect.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

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.     

Measurement of water falling film thickness to flat plate using confocal chromatic sensoring technique

An experimental investigation of wavy water film falling down a flat plate has been carried out using confocal chromatic sensoring technique to determine the instantaneous and statistical characteristics of the film. The experiments involved three parameters: liquid feed mode, Reynolds number and plate inclination angle. The present time–average film thickness data is compared with the previous experimental and theoretical results showing a good agreement. A new correlation for the average film thickness is suggested. Our results show that the liquid feed mode has a vital influence on the film thickness and that the film thickness increases with Reynolds number and decreased plate inclination angle. The root–mean–square value of the film thickness fluctuations depends non-monotonically on the film Reynolds number. The corresponding mechanisms are analyzed.     

Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations

Two-dimensional gravity-driven film flows along a substrate with rectangular corrugations are studied numerically by using Finite Volume Method. The volume of fluid (VOF) method is utilized to capture the evolution of free surfaces. The film flows down an inclined plate are simulated to validate the numerical implementation of the present study. Results obtained indicate that the phase shift between the surface wave and the wall corrugation increases as the Reynolds number. The parametric studies on the interesting resonant phenomenon indicate that the peak Reynolds numbers increase as the raise of the wall depth or the decline of the inclination angle. The dependence of the flow fields is analyzed on the Reynolds numbers and wall depth in details. It is found that the vortical structures in the steady flows, either produced by the interaction between capillary wrinkling and inertia, or by the rectangular geometry, are closely related to the remarkable deformation of the free surfaces. This conclusion is also confirmed by the transient flow development of two typical simulations, i.e., flows in capillary–inertial regime and in inertial regime.     

Experiments on the linear instability of flow in a wavy channel

The effects of wall corrugation on the stability of wall-bounded shear flows have been examined experimentally in plane channel flows. One of the channel walls has been modified by introduction of the wavy wall model with the amplitude of 4% of the channel half height and the wave number of 1.02. The experiment is focused on the two-dimensional travelling wave instability and the results are compared with the theory [J.M. Floryan, Two-dimensional instability of flow in a rough channel, Phys. Fluids 17 (2005) 044101 (also: Rept. ESFD-1/2003, Dept. of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada, 2003)]. It is shown that the flow is destabilized by the wall corrugation at subcritical Reynolds numbers below 5772, as predicted by the theory. For the present corrugation geometry, the critical Reynolds number is decreased down to about 4000. The spatial growth rates, the disturbance wave numbers and the distribution of disturbance amplitude measured over such wavy wall also agree well with the theoretical results.     

A study of thin water film flow down an inclined plate without and with countercurrent air flow

Characteristics of thin water film flow down an inclined plane surface without as well as with superposed countercurrent air flow were studied experimentally. Three different angles of inclination of the channel with horizontal were investigated. At each inclination angle, five film Reynolds numbers were studied. Experiments were performed beginning at zero air flow and then increasing the air flow rate in steps until water entrainment occurred. Visual observation of the film surface was carried out as were film thickness measurements by means of capacitance probes. Results presented include mean film thickness and distribution, frequency spectra and propagation velocities of interfacial disturbances, and the incipience condition for entrainment of water from the film into the air stream.     

Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel

We study the flow of a liquid down an inclined channel with a sinusoidal bottom profile. We show how wavy bottom variations, which are long compared with the film thickness or the amplitude, modify the flow with respect to that down a flat inclined channel. We consider different perturbation analyses. Their results are compared with experimental data on the velocity profiles and on the film thickness. We discuss the effect of waviness, inclination angle, film thickness, and Reynolds number.     

Instabilities of a gas-liquid flow between two inclined plates analyzed using the Navier–Stokes equations

The paper is devoted to a theoretical analysis of a counter-current gas-liquid flow between two inclined plates. We linearized the Navier–Stokes equations and carried out a stability analysis of the basic steady-state solution over a wide variation of the liquid Reynolds number and the gas superficial velocity. As a result, we found two modes of the unstable disturbances and computed the wavelength and phase velocity of their neutral disturbances varying the liquid and gas Reynolds number. The first mode is a “surface mode” that corresponds to the Kapitza's waves at small values of the gas superficial velocity. We found that the dependence of the neutral disturbance wavelength on the liquid Reynolds number strongly depends on the gas superficial velocity, the distance between the plates and the channel inclination angle for this mode. The second mode of the unstable disturbances corresponds to the transition to a turbulent flow in the gas phase and there is a critical value of the gas Reynolds number for this mode. We obtained that this critical Reynolds number weakly depends on both the channel inclination angle, the distance between the plates and the liquid flow parameters for the conditions considered in the paper. Despite a thorough search, we did not find the unstable modes that may correspond to the instability in frame of the viscous (or inviscid) Kelvin–Helmholtz heuristic analysis.     

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.     

Heat transfer and interfacial drag in countercurrent steam-water stratified flow

Local condensation heat transfer coefficients and interfacial shear stresses have been measured for countercurrent stratified flow of steam and subcooled water in rectangular channels over a wide range of inclination angles (4–87°) at two aspect ratios. Dimensionless correlations for the interfacial friction factor have been developed that show that it is a function of the liquid Reynolds number only. Empirical correlations of the heat transfer coefficient, based upon the bulk flow properties, have also been set up for the whole body of data encompassing the different inclination angles and aspect ratios. These indicate that the Froude number as a dimensionless gas velocity is a better correlating parameter than the gas Reynolds number. As an alternative approach, a simple dimensionless relationship for the beat transfer coefficient was obtained by analogy between heat and momentum transfer through the interface. Finally, a turbulence-centered model has been modified by using measured interfacial parameters for the turbulent velocity and length scales, resulting in good agreement with the data.     

Turbulent flow structure and heat transfer in an inclined bubbly flow. Experimental and numerical investigation

The effect of channel inclination on the variation in the wall shear stress and the heat transfer in a two-phase bubbly flow in a rectangular channel is experimentally and numerically investigated. The wall friction was measured using the electrodiffusion method and the temperature was measured by tiny platinum resistance thermometers. The model is based on the system of RANS equations with account for the back influence of the bubbles on the flow characteristics. Flow turbulence is calculated according to the model of transport of the Reynolds stress tensor components. It is shown that in the gas-liquid flow the angle of the channel inclination to the horizon can have a considerable effect on the friction and the heat transfer. The greatest friction and heat transfer values correspond to the angles of channel inclination ranging from 30 to 50°. In the inclined two-phase bubbly flow the shear stress enhancement on the wall amounts to 30% and that of the heat transfer to 15%. A friction and heat transfer reduction to 10 and 25%, respectively, is noticed in near-horizontal flows.     

Development of a PLIC-VOF method for the dynamic simulation of entry region flow in a laminar falling film

The present work describes a numerical procedure to simulate the development of hydrodynamic entry region in a gravity-driven laminar liquid film flow over an inclined plane. It provides a better insight into the physics of developing film in entry region. A novel numerical approach is proposed which has the potential to provide solutions for the complex physics of liquid film spreading on solid walls. The method employs an incompressible flow algorithm to solve the governing equations, a PLIC-VOF method to capture the free surface evolution and a continuum surface force (CSF) model to include the effect of surface tension. To account for the moving contact line on the solid substrate, a precursor film model based wall treatment is implemented. Liquid film flow has been simulated for the Reynolds number range of 5 ≤ Re ≤ 37.5, and the predicted results are found to agree well with the available analytical and experimental data.     

Waves on down-flowing fluid films: Calculation of resistance to arbitrary two-dimensional perturbations and optimal downflow conditions

Wavy downflow of viscous fluid films is studied. The full Navier-Stokes equations are used to calculate the hydrodynamic characteristics of the flow. The stability of calculated nonlinear waves to arbitrary two-dimensional perturbations is considered within the framework of the Floquet theory. It is shown that, for small values of the Kapitza number, the waves are stable over a wide range of wavelengths and values of the Reynolds number. It is found that, as the Kapitza number increases, the parameter range where nonlinear waves are calculated is divided into a series of alternating zones of stable and unstable solutions. A large number of narrow zones where the solutions are stable are revealed on the wavelength-Reynolds number parameter plane for large values of the Kapitza number. Optimal regimes of film downflow that correspond to the minimum value of average film thickness for nonlinear waves with different wavelengths are determined. The basic characteristics of these waves are calculated in a wide range of Reynolds and Kapitza numbers.     

Jet Formation in Gravitational Flow of a Heated Wavy Liquid Film

Jet formation was studied in the region of two-dimensional and three-dimensional waves in a heated liquid film flowing down a vertical surface. Jet-to-jet spacings were measured versus the film Reynolds number and the heat flow density. Three-dimensional waves on the film surface were formed naturally or by artificial perturbations. In addition to the thermocapillary mechanism of jet formation, a thermocapillary–wavy mechanism was found to exist.     

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.     

Multi-layer film flow down an inclined plane: experimental investigation

We report the results from an experimental study of the flow of a film down an inclined plane where the film itself is comprised of up to three layers of different liquids. By measuring the total film thickness for a broad range of parameters including flow rates and liquid physical properties, we provide a thorough and systematic test of the single-layer approximation for multi-layer films for Reynolds numbers \(Re = \rho Q/\mu \approx 0.03 - 60\) . In addition, we also measure the change in film thickness of individual layers as a function of flow rates for a variety of experimental configurations. With the aid of high-speed particle tracking, we derive the velocity fields and free-surface velocities to compare to the single-layer approximation. Furthermore, we provide experimental evidence of small capillary ridge formations close to the point where two layers merge and compare our experimental parameter range for the occurrence of this phenomenon to those previously reported.     

Application of PIV to velocity measurements in a liquid film flowing down an inclined cylinder

PIV technique is applied for measurements of instant velocity distributions in a liquid film flowing down an inclined tube in the form of a wavy rivulet. An application of special optical calibration is applied to correct distortion effects caused by the curvature of the interface. A vortex flow of liquid is observed inside a wave hump in the reference system moving with wave phase velocity. Conditionally averaged profiles of longitudinal and transverse components of liquid velocity are obtained for different cross-sections of developed non-linear waves. It is shown that the increase in wave amplitude slightly changes the location of the vortex center. The analysis of modification of vortex motion character due to wavy flow conditions, such as tube inclination angle, film Reynolds number, wave excitation frequency, is fulfilled.