Investigations of the Marangoni effect on the regular structures in heated wavy liquid films

Formation and development of quasi-regular metastable structures within laminar-wavy falling films were studied. These structures emerge within the residual layer between large waves and could be one reason for the break up of the falling film. The temperature field of the film surface was visualised using IR-thermography. The film thickness was obtained from point measurements with the chromatic confocal imaging method and converted into a film thickness field, based on a quasi-steady assumption and IR thermography images. The thermo-capillary nature (Marangoni effect) of the regular structures was proven experimentally.     

Temperature pulsations and thermocapillary effects on the surface of a wavy heated liquid film

The temperature pulsations and wave characteristics in water film flow along a vertical plate with a heater are investigated. Using an infrared scanner, the temperature field on the film surface is measured for various heat flux densities on the heater. Experimental data on the variation of the temperature with time on a local segment of the liquid film surface during wave transmission are obtained. In the absence of a heat flux the data obtained are in good agreement with the results of other researchers for an isothermal liquid film. When the down-flowing liquid is heated, the thermocapillary forces lead to the formation of rivulets and a thin film between them. It is shown that in the inter-rivulet zone the relative wave amplitude increases due to the action of the thermocapillary forces.     

Modeling heat and mass transfer in wavy and turbulent falling liquid films

A simplified model was developed for describing the heat and mass transfer through a gas-liquid or a solid-liquid interface of a wavy or turbulent falling liquid film. The model assumes that turbulent transport near a gas-liquid or a solid liquid interface is governed by eddies whose length and velocity scales can be characterized by bulk turbulence and that in a region near the interface, the turbulence is damped by viscosity and not surface tension. A comparison with literature correlations and data shows that the model can predict the general form of the equations used in correlating transfer coefficients in wavy and turbulent falling films with or without interfacial shear.

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Modellierung von Wärme- und Stoffübertragung in einem welligen und turbulenten Rieselfilm

Zusammenfassung Es wurde ein vereinfachtes Modell entwickelt, das den Wärme-und Stoffübergang über die Phasengrenze von gasförmig-flüssig oder fest-flüssig an einem welligen turbulenten Rieselfilm beschreibt. Das Modell nimmt an, daß der turbulente Transport in der Nähe der gasförmigen-flüssigen oder festenflüssigen Phasengrenze durch Wirbel bestimmt wird, deren Längen-und Geschwindigkeitsmaße durch den Turbulenzgrad in der Strömung charakterisiert sind und daß in unmittelbarer Nähe der Phasengrenze die Turbulenz durch die Viskosität und nicht durch die Oberflächenspannung gedämpft wird. Ein Vergleich mit Werten aus der Literatur und den gefundenen Daten zeigt, daß das Modell die generelle Form von Gleichungen vorhersagen kann, wie sie zur Berechnung von Transportkoeffizienten in welligem und turbulentem Rieselfilm mit oder ohne Schubspannung an der Phasengrenze benutzt werden.

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Nomenclature a proportionality constant in eddy diffusivity expression, s^–1 - C _p heat capacity at constant pressure, J/kg °K - d tube diameter, m - D molecular diffusivity of gas or solid in liquid, m²/s - g, g _c gravitational acceleration constant, m/s²; proportionality factor, kg m/N s² - h, h ^* heat transfer coefficient, W/m² °K; (h/k) (v²/g)^1/3, dimensionless - k thermal conductivity of liquid, W/m °K - k _L , k _L ^* liquid-side mass transfer coefficient, m/s; (k _L/D) (v²/g)^1/3, dimensionless - l _D , l _H eddy mixing length for mass; for heat, m - l ₀ eddy mixing length in bulk liquid, m - L tube length, m - m, n exponent - Pr Prandtl number,c _p/k - q heat flux, W/m² - Q volumetric flow rate, l/min - Re film Reynolds number, 4Q/ dv or 4 / - Sc Schmidt number,v/D - T _b ,T _w bulk liquid temperature; wall temperature - _y , _y y-component eddy fluctuating velocity;Y-component, m/s - v₀ eddy velocity in the bulk liquid, m/s - v ^* friction velocity, ( _wg_c/)^1/2, m/s - v _i ^* friction velocity at gas-liquid interface, ( _ig_c/)^1/2, m/s - y normal distance measured from the gas-liquid interface, m - Y normal distance measured from the solid-liquid interface, m - thermal diffusivity, m²/s - mass flow rate per unit periphery, kg/m s - film thickness, m - (^*, ⁺ dimensionless film thickness, /(v²/g)^1/3;v ^*/v - _D , _H eddy diffusivity for mass; for heat, m²/s - thickness of the zone of damped turbulence, m - surface tension, N/m - v kinematic viscosity, m²/s - liquid density, kg/m³ - _w , _i shear stress at wall; at interface, N/m² - _i ^* dimensionless interfacial shear, _i g _c / (vg) ^2/3     

Experimental investigation into three-dimensional wavy liquid films under the influence of electrostatic forces

Three-dimensional interfacial waves that develop on the free surface of falling liquid films are known to intensify heat and mass transfer. In this context, the present paper studies the effect of electrostatic forces applied to a falling film of dielectric liquid on its three-dimensional nonlinear wave dynamics. Therefore, measurements of the local film thickness using a confocal chromatic imaging method were taken, and the complex wave topology was characterized through photography. The experiments show a complex interaction between the electric field and the hydrodynamics of the falling film, whereby electrostatic forces were found to both increase and decrease wave peak height in different regions of the wave. Additionally, an electrically induced breakup of the three-dimensional wave fronts, which leads to a locally doubled frequency in streamwise direction, is found. The ability to influence the wave topology demonstrated here opens the possibility to optimize heat transfer processes in falling liquid films.     

Thermal imaging study on the surface wave of heated falling liquid films

By using thermal imaging technique and film thickness metering system, the surface wave and film thickness of the heated falling liquid film were experimentally investigated. Temperature variations of the heated film induce surface tension gradient and so-caused Marangoni flow that attempts to avoid the temperature variations. There are three kinds of Marangoni flow appearing in the heated falling liquid film. It is found that the lateral Marangoni flow (MF I) and the streamwise Marangoni flow (MF II) make the heated film thick, while the Marangoni flow in the surface wave (MF III) reinforces the wave and makes the heated film thin. The intensity of Marangoni flow is determined by the flow rate and the heating conditions. MF I and MF II are both enhanced with the increasing liquid flow rate. Moreover, MF III is prominent under moderate flow rates and is gradually weakened at high flow rates. The distance over which MF III starts, increases with a rise in flow rate, but is independent of the heating condition.     

Local film thickness and temperature distribution measurement in wavy liquid films with a laser-induced luminescence technique

Heat transfer in falling liquid film systems is enhanced by waviness. Comprehension of the underlying kinetic phenomena requires experimental data of the temperature field with high spatiotemporal resolution. Therefore a non-invasive measuring method based on luminescence indicators is developed. It is used to determine the temperature distribution and the local film thickness simultaneously. Results are presented for the temperature distribution measurement in a laminar-wavy water film with a liquid side Reynolds number of 126 flowing down a heated plane with an inclination angle of 2° at two positions in flow direction. The measured temperature distributions are used to calculate the local heat transfer coefficient for solitary waves at two positions in flow direction.     

An investigation of falling liquid films on a vertical heated/cooled plate

The temperature field and flow patterns of a liquid film flowing over a vertical uniformly heated surface have been experimentally investigated. Our experiments show that this film flow is sensitive to the heating conditions. When the film is cooled by the substrate, its surface area increases, and when it is heated its surface area decreases. The analysis attributed the changing properties of the flow to lateral Marangoni effect, i.e. to surface tension gradient transverse to the flow. The influence of the viscosity variations on the non-isothermal liquid film flow was also considered and compared with that of the surface tension variations. It was shown that the contraction or extension of the films was mainly caused by the lateral surface tension gradient that might be determined by the viscosity variations.     

Nonlinear resonance in viscous films on inclined wavy planes

We study nonlinear resonance in viscous gravity-driven films flowing over undulated substrates. Numerical solution of the full, steady Navier–Stokes equations is used to follow the emergence of the first few free-surface harmonics with increasing wall amplitude, and to study their parametric dependence on film thickness, inertia and capillarity. Bistable resonance is computed for steep enough bottom undulations. As an analytic approach, we apply the integral boundary-layer method and derive an asymptotic equation valid for rather thin films. The analysis recovers the key numerical findings and provides qualitative understanding. It shows that higher harmonics are generated by a nonlinear coupling of the wall with lower-order harmonics of the free surface. It also accounts for bistable resonance, and produces a minimum model whose solution is similar to that of the Duffing oscillator.     

Linear resonance in viscous films on inclined wavy planes

We study viscous gravity-driven films flowing over periodically undulated substrates. Linear analysis describes steady flow along small amplitude corrugations for films of arbitrary thickness. Solving the resulting system numerically, we demonstrate resonance (or, possibly, near resonance) and identify different behaviours for thin, intermediate and thick films. Approximating the leading-order velocity profile by the free surface value allows for an analytic solution, which – in the limit of high Reynolds numbers – recovers the different regimes and reveals the relevant physical mechanisms. Our results support the view that the resonance is associated with an interaction of the undulated film with capillary-gravity waves travelling against the mean flow direction. As a consequence, the resonance peak is attained under conditions that render the wave phase velocity equal to zero in the laboratory reference frame, and thus permit direct exchange of energy between the steadily deformed film and the free surface.     

Generation of the mean flow in the fluid layer with a nonuniformly heated wavy boundary

The problem of convection in the horizontal fluid layer with a wavy lower boundary is considered. It is shown that for the periodic temperature distribution with a certain phase shift given on the wavy boundary, in the fluid layer a unidirectional horizontal flow arises. The flow velocity linearly decreases with increase in attitude and depends on the relief distribution wavelength. There is an optimum wavelength (of the order of the layer thickness) at which the velocity reaches its maximum value.     

Influence of wavy surfaces on coherent structures in a turbulent flow

We describe how outer flow turbulence phenomena depend on the interaction with the wall. We investigate coherent structures in turbulent flows over different wavy surfaces and specify the influence of the different surface geometries on the coherent structures. The most important contribution to the turbulent momentum transport is attributed to these structures, therefore this flow configuration is of large engineering interest. In order to achieve a homogeneous and inhomogeneous reference flow situation two different types of surface geometries are considered: (1) three sinusoidal bottom wall profiles with different amplitude-to-wavelength ratios of α = 2a/Λ = 0.2 (Λ = 30 mm), α = 0.2 (Λ = 15 mm), and α = 0.1 (Λ = 30 mm); and (2) a profile consisting of two superimposed sinusoidal waves with α = 0.1 (Λ = 30 mm). Measurements are carried out in a wide water channel facility (aspect ratio 12:1). Digital particle image velocimetry (PIV) is performed to examine the spatial variation of the streamwise, spanwise and wall-normal velocity components in three measurement planes. Measurements are performed at a Reynolds number of 11,200, defined with the half channel height h and the bulk velocity U _B. We apply the method of snapshots and perform a proper orthogonal decomposition (POD) of the streamwise, spanwise, and wall-normal velocity components to extract the most dominant flow structures. The structure of the most dominant eigenmode is related to counter-rotating, streamwise-oriented vortices. A qualitative comparison of the eigenfunctions for different sinusoidal wall profiles shows similar structures and comparable characteristic spanwise scales Λ _z = 1.5 H in the spanwise direction for each mode. The scale is observed to be slightly smaller for α = 0.2 (Λ = 15 mm) and slightly larger for α = 0.2 (Λ = 30 mm). This scaling for the flow over the basic wave geometries indicates that the size of the largest structures is neither directly linked to the solid wave amplitude, nor to the wavelength. The characteristic spanwise scale of the dominant eigenmode for the developed flow over the surface consisting of two superimposed waves reduces to 0.85 H. However, a scale in the order of 1.3 H is identified for the second mode. The eigenvalue spectra for the superimposed waves is much broader, more modes contribute to the energy-containing range. The turbulent flow with increased complexity of the bottom surface is characterized by an increased number of dominant large-scale structures with different spanwise scales.     

Motion of gas bubbles in a nonuniformly heated liquid

The thermocapillary drift of air bubbles in water was studied experimentally. The linear connection between the drift velocity and the temperature gradient predicted by theory was confirmed. The experimental and theoretical results are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–57, September–October, 1979.We thank M. T. Sharov for assistance in setting up the experiment.     

Gas–liquid pressure drop in vertical internally wavy 90° bend

Experiments of air water two-phase flow pressure drop in vertical internally wavy 90° bend have been carried out. The tested bends are flexible and made of stainless steel with inner diameter of 50 mm and various curvature radiuses of 200, 300, 400 and 500 mm. The experiments were performed under the following conditions of two-phase parameters; mass flux from 350 to 750 kg/m² s. Gas quality from 1% to 50% and system pressure from 4 to 7.5 bar. The results demonstrate that the effect of the above-mentioned parameters is very significant at high ranges of mass flow quality. Due to the increasing of two-phase flow resistance, energy dissipations, friction losses and interaction of the two-phases in the vertical internally wavy 90° bend the total pressure drops are perceptible about 2–5 times grater than that in smooth bends. Based on the mass and energy balance as well as the presented experimental results, new empirical correlation has been developed to calculate the two-phase pressure drop and hence the two-phase friction factor of the tested bends. The correlation includes the relevant primary parameter, fit the data well, and is sufficiency accurate for engineering purposes.     

Model of a wavy flow in a falling film of a viscous liquid

A new model is developed for describing long-wave perturbations in a falling film of a viscous liquid. The model is based on an integral approach and an expansion of the velocity profile into a series in linearly independent basis functions of a boundary-value problem. A linear analysis of film flow stability is performed, and dispersion dependences are obtained. Results predicted by the new model are demonstrated to be in good agreement with available experimental data on the film flow over a gently sloping surface.     

Effects of variable fluid properties on natural convection of MHD fluid from a heated vertical wavy surface

In the present paper, the influence of temperature-dependent fluid properties, density, viscosity and thermal conductivity on MHD natural convection flow from a heated vertical wavy surface is studied. It is assumed that, the fluid density and the thermal conductivity vary as exponential and linear functions of temperature, respectively. However, the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature. The model analysis used here is more relevant to liquid flow. Using the appropriate variables, the wavy surface are transformed into a flat one. The transformed boundary layer equations are solved numerically, using implicit-Chebyshev pseudospectral method, for several sets of values of the physical parameters, namely, the temperature dependent fluid properties parameters, the magnetic parameter, the amplitude-wavelength ratio parameter, and the Prandtl number. The numerical values obtained for the velocity, temperature, shearing stress, and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of the physical parameters on the flow and heat transfer characteristics are discussed. The results were compared with numerical solutions of previous works. The present results are found to be in good agreement.     

Stability of liquid flow between heated rotating cylinders

The effect of a radial temperature gradient on stability of steady-state flow of a viscous liquid between two solid concentric cylinders both rotating in the same direction is considered. The linear stability problem is considered in the Boussinesq approximation. Sufficient stability and instability conditions for the flow relative to rotationally symmetric perturbation are obtained. Neutral curves are computed for a wide range of problem parameters.     

Evaporation of thin liquid droplets on heated surfaces

We carry out combined experimental and theoretical studies of liquid droplet evaporation on heated surfaces in a closed container filled with saturated vapor. The droplets are deposited on an electrically heated thin stainless steel foil. The evolution of droplet shapes is studied by optical methods simultaneously with high-resolution foil temperature measurements using thermochromic liquid crystals. A mathematical model is developed based on the assumptions that the droplet surface has uniform mean curvature and the contact line is pinned during evaporation. Both the dynamics of liquid–vapor interface and the temperature profiles at the foil are shown to be in good agreement with the experimental data.