Fluid dynamics of slender,thin, annular liquid jets

Perturbation methods are used to obtain the one-dimensional, asymptotic equations that govern the fluid dynamics of slender, thin, inviscid, incompressible, axisymmetric, irrotational, annular liquid jets from the Euler equations. It is shown that, depending on the magnitude of the Weber number, two flow regimes are possible: an inertia-dominated one corresponding to large Weber numbers, and a capillary regime for Weber numbers of the order of unity. The steady equations governing these two regimes have analytical solutions for the liquid's axial velocity component and require a numerical integration to determine the jet's mean radius for inertia-dominated jets. The one-dimensional equations derived in this paper are shown to be particular cases of a hydraulic model for annular liquid jets, and this model is used to determine the effects of gravity modulation on the unsteady fluid dynamics of annular liquid jets in the absence of mass injection into the volume enclosed by the jet and mass absorption. It is shown that both the convergence length and the pressure coefficient are periodic functions of time which have the same period as that of the gravity modulation, but undergo large variations as the amplitude, frequency and width of gravitational pulses is varied.     

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.     

Spatial instability of a viscous liquid sheet

In the present study, the spatial instability for a two‐dimensional viscous liquid sheet, which is thinning with time, has been analysed. The study includes the derivation of a spatial dispersion equation, numerical solutions for the growth rate of sinuous disturbances, and parameter sensitivity studies. For a given wave number, the growth rate of the disturbance is essentially a function of Weber number, Reynolds number, and gas/liquid density ratio. The analysis indicates that the cut‐off wave number of the disturbance becomes larger with an increase in Weber number or gas/liquid density ratio. Thus, the liquid sheet should produce finer drops. When the Reynolds number decreases, the higher viscosity has a greater damping effect on shorter waves than longer waves. This could explain that only large drops and ligaments were observed in past measurements for the disintegration of a very viscous sheet. The spatial instability results of the present study were also compared with the temporal theory. The importance of spatial analysis was found and demonstrated for the cases of low Weber numbers. The temporal theory underestimates growth rates when the Weber number is less than 100. The discrepancy between the two theories increases as the Weber number further decreases. Copyright © 2005 John Wiley & Sons, Ltd.     

Bubbly jets in stagnant water

Air–water bubbly jets are studied experimentally in a relatively large water tank with a gas volume fraction, C_o, of up to 80% and nozzle Reynolds number, Re, ranging from 3500 to 17,700. Measurements of bubble properties and mean axial water velocity are obtained and two groups of experiments are identified, one with relatively uniform bubble sizes and another with large and irregular bubbles. For the first group, dimensionless relationships are obtained to describe bubble properties and mean liquid flow structure as functions of C_o and Re. Measurements of bubble slip velocity and estimates of the drag coefficient are also provided and compared to those for isolated bubbles from the literature. The study confirms the importance of bubble interactions to the dynamics of bubbly flows. Bubble breakup processes are also investigated for bubbly jets. It was found that a nozzle Reynolds number larger than 8000 is needed to cause breakup of larger bubbles into smaller bubbles and to produce a more uniform bubble size distribution. Moreover, the Weber number based on the mean water velocity appears to be a better criteria than the Weber number based on the bubble slip velocity to describe the onset of bubble breakup away from the nozzle, which occurs at a Weber number larger than 25.     

Numerical simulation of isothermal nonwetting

A numerical simulation of isothermal wetting suppression in the presence of shear is considered during which wetting may be prevented when a drop approaches a moving wall. Air is driven into the passage between the solid and liquid surfaces by viscous action, preventing wetting. Silicone‐oil and water drops are investigated for different wall velocities and wall distances. The droplet dimples at the upstream side and bulges at the downstream side when nonwetting occurs. The free‐surface deformation can be enlarged by either increasing the wall velocity or decreasing the wall distance. The low‐viscosity silicone‐oil used in these calculations is much more sensitive to shear wetting suppression than is water, because the Weber number of the silicone‐oil is larger than that of water. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical study of flow and heat transfer during a high-speed micro-drop impact on thin liquid films

Numerical simulation of high-speed micro-droplet impingement on thin liquid film covering a heated solid surface has been carried out. Effect of droplet Weber number and liquid film thickness on the characteristics of flow and heat transfer has been investigated using the coupled level set and volume of fluid method. The code is validated against both the experimental and numerical results from the literature. Results show that the crown dynamics is mostly affected by variations in the initial film thickness but is weakly influenced by changes in the Weber number. The liquid within the film can be categorized as three regions based on the heat transfer distribution: the static film region, the transition region, and the impact region. The transient local wall temperature shows three stages: first stage when the temperature decreases rapidly, followed by a second stage in which the temperature starts to rise and then becomes almost constant in the third stage. After drop impact, the local Nusselt number continuously increases until reaching a maximum value, and then decreases approaching the initial impact stage. Our analysis of the change in Weber number shows that larger Weber number contributes to intense temperature variation at the crater core relative to other radial locations. Lastly, the results reveal that the thinner liquid film leads to lower wall temperature and hence, higher average Nusselt number.     

A numerical study on droplet-particle collision dynamics

The impact of liquid droplets onto spherical stationary solid particles under isothermal conditions is simulated. The CFD model solves the Navier-Stokes equations in three dimensions and employs the Volume of Fluid Method (VOF) coupled with an adaptive local grid refinement technique able to track the liquid-gas interface. A fast-marching algorithm suitable for the quick computation of distance functions required during the grid refinement in large 3-D computational domains is proposed. The numerical model is validated against experimental data for the case of a water droplet impact onto a spherical particle at low We number and room temperature conditions. Following that, a parametric study is undertaken examining (a) the effect of Weber number (= ρu²D_o/σ) in the range of 8 to 80 and (b) the droplet to particle size ratio ranging in-between 0.31 and 1.24, on the impact outcome. This has resulted to the identification of two distinct regimes that form during droplet-particle collisions: the partial/full rebound and the coating regimes; the latter results to the disintegration of secondary satellite droplets from elongated expanding liquid ligaments forming behind the particle. Additionally, the temporal evolution of variables of interest, such as the maximum dimensionless liquid film thickness and the average wetting coverage of the solid particle by the liquid, have been quantified. The present study assists the understanding of the physical processes governing the impact of liquids onto solid spherical surfaces occurring in industrial applications, including fluid catalytic cracking (FCC) reactors.     

An experimental investigation on the effects of surface gravity waves on the water evaporation rate in different air flow regimes

Estimating rate of evaporation from undisturbed water surfaces to moving and quiet air has been the topic a vast number of research activities. The obvious presence of various shapes of gravity waves on the water body surfaces was the motivation of this experimental investigation. In this investigation experimental measurements have been done to quantify evaporation rate from wavy water surfaces in free, mixed and forced convection regimes. The effects of a wide range of surface gravity waves from low steepness, round shaped crest with slow celerity, to steep and very slight spilling crest waves, on the water evaporation rate have been investigated. A wide range of ${\text{Gr}}/{\text{Re}}^{2} (0.01 \le {\text{Gr}}/{\text{Re}}^{2} \le 100)$ was achieved by applying different air flow velocities on a large heated wave flume equipped with a wind tunnel. Results reveal that wave motion on the water surface increase the rate of evaporation for all air flow regimes. For free convection, due to the effect of wave motion for pumping rotational airflows at the wave troughs and the dominant effect of natural convection for the air flow advection, the maximum evaporation increment percentage from wavy water surface is about 70 %. For mixed and forced convection, water evaporation rate increment is more sensitive to the air flow velocity for the appearance of very slight spilling on the steep wave crests and the leeward air flow structures.     

Experimental study of air–water flow in downward sloping pipes

This paper presents results from seven experimental facilities on the co-current flow of air and water in downward sloping pipes. As a function of the air flow rate, pipe diameter and pipe slope, the required water discharge to prevent air accumulation was determined. In case the water discharge was less than the required water discharge, the air accumulation and additional gas pocket head loss were measured. Results show that volumetric air discharge as small as 0.1% of the water discharge accumulate in a downward sloping section. The experimental data cover all four flow regimes of water-driven air transport: stratified, blow-back, plug and dispersed bubble flow. The analysis of the experimental results shows that different dimensionless numbers characterise certain flow regimes. The pipe Froude number determines the transition from blow-back to plug flow. The gas pocket head loss in the blow-back flow regime follows a pipe Weber number scaling. A numerical model for the prediction of the air discharge as a function of the relevant system parameters is proposed. The novelty of this paper is the presentation of experimental data and a numerical model that cover all flow regimes on air transport by flowing water in downward inclined pipes.     

The motion of viscous liquid column with finite length in a vertical straight capillary tube

This paper deals with the motion of viscous liquid column with finite length and two free surfaces in a vertical straight capillary tube. It is assumed that fluid is Newtonian. Linearizing the boundary conditions, analytic expressions in the form of infinite series have been obtained for velocity, piessure and free surface at low Reynolds number. The numerical calculation is carried out for a set of cylinder’s length of water and blood. It has been revealed that there are considerable circulating currents at the upper and lower meniscuses. Its maximum velocity is about 57% of the average velocity of the mainstream. Iner-tial effect is also studied in this paper. Using the time-dependent method in finite difference techniques, numerical solution of the corresponding nonlinear equation at Re<24.5 is computed. Comparing it with analytic exact solution at low Reynolds number shows that inertial effect is negligible provided Re<24.5.     

Acoustically excited air-assisted liquid sheets

The effect of acoustic excitation on the disintegration characteristics of air-assisted liquid sheets, which utilize water at ambient temperature, and for velocities up to 1.8 m/s, is investigated. The study using high-speed imaging techniques revealed that optimum frequency modulation of the perturbation generator has a pronounced influence on the associated surface waves and the subsequent breakup of the liquid sheet. The analysis includes characterization of critical wave amplitude, breakup length, and breakup frequency, for Weber numbers in the range 0.30<We_abs<0.44, which are compared with flow features in the absence of acoustic excitation. The results show that acoustic perturbation can effectively suppress the dominance of gravitational and surface tension effects. As a consequence, for low Weber number flows, the interfacial waves exhibit regularity, and thus a better control of primary breakup processes of liquid sheet may be accomplished.     

A direct numerical simulation of axisymmetric liquid–gas two‐phase laminar flows in a pipe

An accurate finite‐volume‐based numerical method for the simulation of an isothermal two‐phase flow, consisting of a liquid slug translating in a non‐reacting gas in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed‐grid with a cut‐cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity and pressure fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, and large property jumps at the interface. Copyright © 2010 John Wiley & Sons, Ltd.     

Darcian mixed convection plumes along vertical adiabatic surfaces in a saturated porous medium

The mixed convection flow due to a line thermal source embedded at the leading edge of an adiabatic vertical plane surface immersed in a saturated porous medium has been studied. Both weakly and strongly buoyant plume regimes have been considered. The cases of buoyancy assisting and buoyancy opposing flow conditions have been incorporated in the analysis. The results are presented for the entire range of buoyancy parameter from the pure forced convection (ξ=0) to the pure free convection (ξ → ∞@#@) regimes. For buoyancy-assisting flow, the wall temperature and the velocity at the wall increase as the plume strength increases. However, they all decrease as the free-stream velocity increases. For buoyancyopposing flow, the temperature at the wall increases as the strength of the plume increases but velocity at the wall decreases.     

Experimental investigation of inclined liquid water jet flow onto vertically located superhydrophobic surfaces

In this study, the behaviour of an inclined water jet, which is impinged onto hydrophobic and superhydrophobic surfaces, has been investigated experimentally. Water jet was impinged with different inclination angles (15°–45°) onto five different hydrophobic surfaces made of rough polymer, which were held vertically. The water contact angles on these surfaces were measured as 102°, 112°, 123°, 145° and 167° showing that the last surface was superhydrophobic. Two different nozzles with 1.75 and 4 mm in diameters were used to create the water jet. Water jet velocity was within the range of 0.5–5 m/s, thus the Weber number varied from 5 to 650 and Reynolds number from 500 to 8,000 during the experiments. Hydrophobic surfaces reflected the liquid jet depending on the surface contact angle, jet inclination angle and the Weber number. The variation of the reflection angle with the Weber number showed a maximum value for a constant jet angle. The maximum value of the reflection angle was nearly equal to half of the jet angle. It was determined that the viscous drag decreases as the contact angle of the hydrophobic surface increases. The drag force on the wall is reduced dramatically with superhydrophobic surfaces. The amount of reduction of the average shear stress on the wall was about 40%, when the contact angle of the surface was increased from 145° to 167°. The area of the spreading water layer decreased as the contact angle of the surface increased and as the jet inclination angle, Weber number and Reynolds number decreased.     

Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop

The mechanisms of acceleration-induced breakup of liquid drops are reviewed briefly. Data on acceleration-induced fragmentation of liquid drops have been collected from the literature and are presented on a consistent basis. Included are critical Weber number data, breakup time data, velocity history data and fragment size data. A triangular relationship based on the concept of a critical Weber number, breakup time data and velocity history data is presented which permits prediction of the maximum size of stable fragments.     

Formation of radially expanding liquid sheet by impinging two round jets

A thin circular liquid sheet can be formed by impinging two identical round jets against each other. The liquid sheet expands to a certain critical radial distance and breaks. The unsteady process of the formation and breakup of the liquid sheet in the ambient gas is simulated numerically. Both liquid and gas are treated as incompressible Newtonian fluids. The flow considered is axisymmetric. The liquid-gas interface is modeled with a level set function. A finite difference scheme is used to solve the governing Navier-Stokes equations with physical boundary conditions. The numerical results show how a thin circular sheet can be formed and break at its circular edge in slow motion. The sheet continues to thin as it expands radially. Hence, the Weber number decreases radially. The Weber number is defined as ρu 2 h/σ, where ρ and σ are, respectively, the liquid density and the surface tension, and u and h are, respectively, the average velocity and the half sheet thickness at a local radial location in the liquid sheet. The numerical results show that the sheet indeed terminates at a radial location, where the Weber number reaches one as observed in experiments. The spatio-temporal linear theory predicts that the breakup is initiated by the sinuous mode at the critical Weber number We c =1, below which the absolute instability occurs. The other independent mode called the varicose mode grows more slowly than the sinuous mode according to the linear theory. However, our numerical results show that the varicose mode actually overtakes the sinuous mode during the nonlinear evolution, and is responsible for the final breakup. The linear theory predicts the nature of disturbance waves correctly only at the onset of the instability, but cannot predict the exact consequence of the instability.     

AN EULERIAN FINITE ELEMENT METHOD FOR TIME-DEPENDENT FREE SURFACE PROBLEMS IN HYDRODYNAMICS

A numerical method based on the finite element method is presented for simulating the two-dimensional transient motion of a viscous liquid with free surfaces. For ease of numerical treatment of the free surface expressed by a multiple-valued function, the marker particle method is employed. Numerous virtual particles are spread over all regions occupied by liquid. They move about on a fixed finite element mesh with the liquid velocity at their positions. These particles contribute nothing to the dynamics of the liquid and only serve as markers of liquid regions. The velocity field within liquid regions is calculated by solving the Navier– Stokes equations and the equation of continuity by the finite element method based on quadrilateral elements. A detailed discussion is given of the methodological problems arising in the implementation of the marker particle method on an unstructured finite element mesh and of the solutions to these problems. The proposed method is demonstrated on three sample problems: the broken dam problem, the impact of a falling liquid drop on a still liquid and the entry of a rigid block into water. Good agreement has been obtained in the comparison of the present numerical results with available experimental data.     

Simulation of viscous water column collapse using adapting hierarchical grids

An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at Re=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd.     

Experimental and numerical investigation a bubble-driven laminar flow in a axisymmetric vessel

Measurements have been obtained, by laser-Doppler anemometry (LDA), of the axisymetric, recirculating liquid flow caused by a column of air bubbles ($\text{5–6}\text{1}\text{2}\text{mm dia.}$) rising through caster oil in a cylindrical enclosure (100 mm dia.). The liquid velocities correspond to creeping flow. Axial and radial liquid velocity profiles are reported at eight axial stations and, close to within the bubble column, as a function of time. The maximum liquid velocity found outside the bubble column is about 0.5 of that of the bubbles and a very rapid radical decay from this value is noted. The temporal variation of the velocity field, due to the passage of the air bubbles, is undetectable at radial locations greater than about $\text{1}\text{1}\text{2}$ bubble radii from the centreline.The variation of bubble velocity with axial distance was aise measured by LDA for liquid height to enclosure diámeter ratios of 0.98 and 2.78. The maximum bubble velocities were about 0.1–0.2 higher than the Strokes law terminal velocity. The increase is due to the convection of the bubble column by the liquid flow. The maximum bubble velocity is established within approximately three bubble diameters of the air inlet.The motion of the liquid has been calculated by the numerical solution of the steady form of the equations of motion, with the inner boundary of the area of integration lying 1.3 bubble radii from the centerline. The boundary conditions at this surface are assumed to be steady and are taken from measurements of the time-averaged velocity components. The assumption of steady flow at this boundary is supported by experimental observation and results in calculations which are generally in close agreement with the measurements. Discrepancies are confined to the immediate vicinity of the bubble column near to the top and bottom of the enclosure. These are ascribed to a combination of small asymmetries in the experiment and inadequate numerical resolution in these regions.     

Film thickness measurements in liquid–liquid slug flow regimes

At present there is significant interest in the development of small scale medical diagnostic equipment. These devices offer faster processing times and require smaller sample volumes than equivalent macro scale systems. Although significant attention has been focused upon their outputs, little attention has been devoted to the detailed fluid mechanics that govern the flow mechanisms within these devices. Conventionally, the samples in these small scale devices are segmented into distinct discrete droplets or slugs which are suspended in an organic carrier phase. Separating these slugs from the channel wall is a very thin film of the organic carrier phase.The magnitude of this film is the focus of the present study and the effects of sample slug length and carrier phase fluidic properties on the film are examined over a range of Capillary numbers. A non-intrusive optical technique was used to capture images of the flow from which the magnitude of the film was determined.The experimental results show that the film is not constant along the length of the slug; however above a threshold value for slug length, a region of constant film thickness exists. When compared with existing correlations in the literature, the experimental data showed reasonable agreement with the Bretherton model when the Capillary number was calculated based on the mean two phase flow velocity. However, significant differences were observed when the Capillary number was redefined to account for the mean velocity at the liquid interface, i.e., the mean slug velocity.Analysis of the experimental data revealed that it fell into two distinct flow regimes; a visco-capillary regime and a visco-inertial regime. A modified Taylor expression is presented to estimate the magnitude of the film for flows in the visco-capillary regime while a new model is put forward, based on Capillary and Weber numbers, for flows in the visco-inertial regime. Overall, this study provides some novel insights into parameters, such as aqueous slug length and carrier phase fluidic properties, that affect the thickness of the film in liquid–liquid slug flow regimes.