Terminal velocity of a Taylor drop in a vertical pipe

A scaling analysis based on the field equations for two phases and the jump conditions at the interface is carried out to deduce a balance of forces acting on a Taylor drop rising through stagnant liquid in a vertical pipe. The force balance is utilized to deduce a functional form of an empirical correlation of terminal velocity of the Taylor drop. Undetermined coefficients in the correlation are evaluated by making use of available correlations for two limiting cases, i.e. extremely high and low Reynolds number Taylor bubbles in large pipes. Terminal velocity data obtained by interface tracking simulations are also used to determine the coefficients. The proposed correlation expresses the Froude number Fr as a function of the drop Reynolds number Re_D, the Eötvös number Eo_D and the viscosity ratio μ^*. Comparisons between the correlation, simulations and experimental data confirm that the proposed correlation is applicable to Taylor drops under various conditions, i.e., 0.002 < Re_D < 4960, 4.8 < Eo_D < 228, 0 ? μ^* ? 70, 1 < N < 14700, −12 < log M < 4, and d/D < 1.6, where N is the inverse viscosity number, M the Morton number, d the sphere-volume equivalent drop diameter and D the pipe diameter.     

Gravity-driven motion of a deformable drop or bubble near an inclined plane at low Reynolds number

The creeping motion of a three-dimensional deformable drop or bubble in the vicinity of an inclined wall is investigated by dynamical simulations using a boundary-integral method. We examine the transient and steady velocities, shapes, and positions of a freely-suspended, non-wetting drop moving due to gravity as a function of the drop-to-medium viscosity ratio, λ, the wall inclination angle from horizontal, θ, and Bond number, B, the latter which gives the relative magnitude of the buoyancy to capillary forces. For fixed λ and θ, drops and bubbles show increasingly pronounced deformation in steady motion with increasing Bond number, and a continued elongation and the possible onset of breakup are observed for sufficiently large Bond numbers. Unexpectedly, viscous drops maintain smaller separations and deform more than bubbles in steady motion at fixed Bond number over a large range of inclination angles. The steady velocities of drops (made dimensionless by the settling velocity of an isolated spherical drop) increase with increasing Bond number for intermediate-to-large inclination angles (i.e. 45° ? θ ? 75°). However, the steady drop velocity is not always an increasing function of Bond number for viscous drops at smaller inclination angles.     

Experimental study of sedimentation characteristics of spheroidal particles

The drag of non-spherical particles is a basic, important parameter for multi-phase flow. As the first step in research in this area, the terminal velocities, Ut, of hemispherical and spherical segment particles with maximal diameters of 6-21 mm were measured in static fluids by using a high-speed video camera. The drag coefficient, CD, measured for Reynolds number, Re of 10^1-10^5, has been obtained and compared with those for a sphere. The Re based on the terminal velocity has a logarithmic linear relationship with Ar number for both the facet facing upwards or downwards for the two experimental spheroidal particles, and their Co values are greater than those of spheres. A shape function that depends on the initial orientation of the particle facet is presented to correct for the shape effects.     

Effects of incompressible surfactant on thermocapillary interactions of spherical drops

Collision efficiencies are determined for two surfactant-covered spherical drops in the limit of nearly uniform surface coverage in thermocapillary motion. The problem is linearized by assuming dilute surfactant concentration, with the effect of surfactant controlled by a single retardation parameter A  . The mobility function _LA$L_A$ along the drops’ line of centers is much less than zero over a wide range of parameters, so that the smaller drop can move faster than the larger one at moderate to large separations. At surface Péclet numbers less than 10, the incompressible surfactant model agrees well with solution of the full convective-diffusion equation for the minimum separation between drops. With the exception of non-conducting drops, the collision efficiencies become zero at moderate values of A. A model system of contaminated ethyl salicylate (ES) drops in diethylene glycol (DEG) is studied in thermocapillary motion. Population dynamics simulations confirm the coalescence-inhibiting effect of incompressible surfactant on the evolution of the ES/DEG drop-size distribution.     

Time-dependent drag reduction and ageing in aqueous solutions of a cationic surfactant

The turbulent pipe flow of a highly dilute aqueous cationic surfactant solution is investigated by means of a pulsed ultrasound Doppler method with special emphasis on the wall boundary layer. The velocity profiles are recorded for several Reynolds numbers at varying ages of the solution. The wall shear stress velocities u _τ used for the normalization of the velocity profiles are determined by fitting the measured profiles to the universal linear velocity profile in the viscous sublayer. The theoretical pressure loss is then calculated from the numerical values of u _τ and compared to the experimental values. Two different scaling methods are discussed for the velocity fluctuations concerning the correlation of the root-mean square values with the effect and the amount of drag reduction. It is shown that outer scaling with the mean velocity is appropriate for the detection of drag reduction in surfactant solutions, rather than inner scaling with the wall shear stress velocity, which is common practice in investigations of 'usual' turbulent flows.     

Thin Shear Layers in High Reynolds Number Turbulence—DNS Results

Using direct numerical simulation of turbulence in a periodic box driven by homogeneous forcing, with a maximum of 4096³ grid points and Taylor micro-scale Reynolds numbers R _λ up to 1131, it is shown that there is a transition in the forms of the significant, high vorticity, intermittent structures, from isolated vortices when R _λ is less than 10² to complex thin-shear layers when R _λ exceeds about 10³. Both the distance between the layers and their widths are comparable with the integral length scale. The thickness of each of the layers is of the order of the Taylor micro-scale λ. Across the layers the velocity ‘jumps’ are of the order of the rms velocity u _o of the whole flow. Within the significant layers, elongated vortical eddies are generated, with microscale thickness ? _v ~10η???λ, with associated peak values of vorticity as large as 35ω _rms and with velocity jumps as large as 3.4u _o , where η is the Kolmogorov micro scale and ω _rms the rms vorticity. The dominant vortical eddies in the layers, which are approximately parallel to the vorticity averaged over the layers, are separated by distances of order ? _v . The close packing leads to high average energy dissipation inside the layer, as large as ten times the mean rate of energy dissipation over the whole flow. The interfaces of the layers act partly as a barrier to the fluctuations outside the layer. However, there is a net energy flux into the small scale eddies within the thin layers from the larger scale motions outside the layer.     

Turbulence production in flow over a wavy wall

Measurements of the spatial and time variation of two components of the velocity have been made over a sinusoidal solid wavy boundary with a height to length ratio of 2a/λ = 0.10 and with a dimensionless wave number of α⁺ = (2π/λ)(v/u ^?) = 0.02. For these conditions, both intermittent and time-mean flow reversals are observed near the troughs of the waves. Statistical quantities that are determined are the mean streamwise and normal velocities, the root-meansquare of the fluctuations of the streamwise and normal velocities, and the Reynolds shear stresses. Turbulence production is calculated from these measurements. The flow is characterized by an outer flow and by an inner flow extending to a distance of about α^?1 from the mean level of the surface. Turbulence production in the inner region is fundamentally different from flow over a flat surface in that it is mainly associated with a shear layer that separates from the back of the wave. Flow close to the surface is best described by an interaction between the shear layer and the wall, which produces a retarded zone and a boundary-layer with large wall shear stresses. Measurements of the outer flow compare favorably with measurements over a flat wall if velocities are made dimensionless by a friction velocity defined with a shear stress obtained by extrapolating measurements of the Reynolds stress to the mean levels of the surface (rather than from the drag on the wall).     

Reynolds Number Dependence of Mixed Structure Functions of Temperature and Longitudinal Velocity

Reynolds number dependence of mixed structure functions of longitudinal velocity u and temperature Θ is examined over a R _λ (Taylor microscale Reynolds number) range of 180-5950 for four flows. It is found that the mixed structure functions exhibit some behaviours similar to those of individual ones of the velocity and temperature. The prediction of the bivariate lognormal model for the R _λ dependence of the mixed structure functions is approached by the present measurements only at very high R _λ. At values of the longitudinal separation r close to the integral length scale as well as Taylor microscale, the velocity and temperature increments are not statistically independent. Several methods have been used to estimate the intermittency exponents μ and μ_Θ of the velocity and temperature in the lognormal model. Each method yields a different value of μ and μ_Θ, which also depend on R _λ and the type of flows. The present measurements suggest that the best estimates for μ and μ_Θ are 0.25 ± 0.05 and 0.30 ± 0.05, respectively.     

Low-Reynolds-number motion of a deformable drop between two parallel plane walls

The motion of a three-dimensional deformable drop between two parallel plane walls in a low-Reynolds-number Poiseuille flow is examined using a boundary-integral algorithm that employs the Green’s function for the domain between two infinite plane walls, which incorporates the wall effects without discretization of the walls. We have developed an economical calculation scheme that allows long-time dynamical simulations, so that both transient and steady-state shapes and velocities are obtained. Results are presented for neutrally buoyant drops having various viscosity, size, deformability, and channel position. For nearly spherical drops, the decrease in translational velocity relative to the undisturbed fluid velocity at the drop center increases with drop size, proximity of the drop to one or both walls, and drop-to-medium viscosity ratio. When deformable drops are initially placed off the centerline of flow, lateral migration towards the channel center is observed, where the drops obtain steady shapes and translational velocities for subcritical capillary numbers. With increasing capillary number, the drops become more deformed and have larger steady velocities due to larger drop-to-wall clearances. Non-monotonic behavior for the lateral migration velocities with increasing viscosity ratio is observed. Simulation results for large drops with non-deformed spherical diameters exceeding the channel height are also presented.     

A Taylor drop rising in a liquid co-current flow

The present work presents a numerical study on the behavior of isolated liquid Taylor drops rising in vertical tubes with co-current heavier continuous phase. Numerical simulations were performed with a previously validated model, implementing Volume of Fluid method in an axisymmetric geometry. Detailed flow patterns and drop shapes are provided and discussed for several conditions. The balance between gravity effect and velocity of the continuous phase flow was found to have a great influence in the flow patterns observed. The increase of inertial effects, due to the increase of Eo number and the co-current velocity, leads to the occurrence of closed recirculations below the drops. Furthermore, the continuous phase stabilization distance below the drop is a function of the drop Reynolds number. Drop and continuous phase velocities relationship was studied. A viscosity ratio related term was appended to a pre-existing correlation. The flow in the absence of gravity was also studied. Results demonstrate that micro-scale flow is a lower limit to the cases studied in the present work and suggest that the viscosity ratio affects mainly the inertial part of the drop velocity.     

Evaporation of single liquid drops in an immiscible liquid at elevated pressures: equipment and preliminary results

This paper describes a piece of equipment which has been specially designed to permit the study of the evaporation of single liquid drops in an immiscible liquid at elevated pressures up to the order of 1 MPa. It is equipped with a piezoelectric discharger for yielding nucleation in each drop and with a dilatometer for detecting the change in volume of the drop in the course of evaporation succeeding the nucleation. Some preliminary results, obtained with the equipment, for n-pentane drops evaporating in a water medium at pressures up to 0.5 MPa (5 atm) are also presented.List of symbols C _D drag coefficient - D ₀ initial, equivalent spherical diameter of drop (= (6 V ₀/)^1/3) - H vertical position of two-phase bubble measured from the nozzle tip - H _t column height required for drop to vaporize completely - Nu Nusselt number related to instantaneous two-phase bubble diameter and thermal conductivity of the continuous phase - p ^* pressure at the nozzle tip - Pe Peclet number related to instantaneous diameter and rise velocity of two-phase bubble and to thermal diffusivity of the continuous phase - Re Reynolds number related to instantaneous diameter and rise velocity of two-phase bubble and to viscosity of the continuous phase - t time - t _v time required for drop to vaporize completely - T temperature in the bulk of the medium - T excess, assumed in theoretical model, of T above the temperature of two-phase bubble - T ₁ ^* excess of T above the saturation temperature of the dispersed-phase fluid corresponding to p ^* - T ₂ ^* excess of T above the temperature at which the sum of the saturated vapor pressures of the dispersed- and the continuous-phase fluids is equal to p ^* - V volume of two-phase bubble - V ₀ initial value of V     

A novel tethered-sphere add-on to enhance grid turbulence

The new turbulence generator consists of a standard uniform grid with tethered spheres attached to its nodes and is capable of producing approximately twice the turbulence energy per unit pressure drop coefficient C _p than the same bare grid without the spheres. At the same time, the Reynolds number Re_λ based on the Taylor microscale is also amplified by a factor of roughly 2, and the turbulence anisotropy is reduced to a constant level of 10% at all downstream distances without further flow conditioning after the grid. The new grid’s simple design makes it suitable for a variety of fluid-flow facilities, in particular smaller water tunnels. Its performance in comparison with the plain grid is documented by measurements of the streamwise decay of turbulence energy and velocity spectra in the Re_λ range of 50–100.     

Experimental evaluation of Marangoni stress and surfactant concentration at interface of contaminated single spherical drop using spatiotemporal filter velocimetry

Spatiotemporal filter velocimetry (SFV) was extended to Lagrangian measurements with boundary-fitted measurement areas, and was applied to flows about single spherical drops of glycerol-water solution falling in stagnant silicon oil under clean and contaminated conditions to examine its applicability to the estimation of the Marangoni stress and surfactant concentration at a moving interface. Effects of bulk concentration of surfactant on the velocity field, the Marangoni stress and the surface concentration of surfactant were discussed from the measured data. As a result, we confirmed that accurate velocity distribution in the vicinity of the interface measured by SFV enables us to evaluate interfacial velocity and interfacial shear stresses and to estimate the Marangoni stress, interfacial tension and surfactant concentration at the interface with the assumption of negligible surface viscosity. The flow inside the drop and the interfacial velocity become weak due to the Marangoni stress caused by the gradient of surfactant concentration at the interface as the bulk concentration of surfactant increases. These results demonstrate that SFV is of great use in experimental analysis of adsorption and desorption kinetics at a moving interface.     

Computations of breakup modes in laminar compound liquid jets in a coflowing fluid

We present a numerical investigation of breakup modes of an axisymmetric, laminar compound jet of immiscible fluids, which flows in a coflowing immiscible outer fluid. We use a front-tracking/finite difference method to track the unsteady evolution and breakup of the compound jet, which is governed by the Navier–Stokes equations for incompressible Newtonian fluids. Numerical results show that depending on parameters such as the Reynolds number Re (in the range of 5–30) and Weber Number We (in the range of 0.1–0.7), based on the inner jet radius and inner fluid properties, the compound jet can break up into drops in various modes: inner dripping–outer dripping (dripping), inner jetting–outer jetting (jetting), and mixed dripping–jetting. Decreasing Re or increasing We promotes the jetting mode. The transition from dripping to jetting is also strongly affected by the velocity ratios, U₂₁ (intermediate to inner velocities) and U₃₁ (outer to inner velocities). Increasing U₂₁ makes the inner jet thinner and stretches the outer jet and thus promotes jetting. In contrast, increasing U₃₁ thins the outer jet, and thus, when the inner jet is dripping, the outer jet can break up into drops in the mixed dripping–jetting mode. Continuously increasing U₃₁ results in thinning both inner and outer jets and thus produces small drops in the jetting mode. In addition, starting from dripping, a decrease in the interfacial tension ratio of the outer to inner interfaces results in the mixed dripping–jetting and jetting modes. These modes produce various types of drops: simple drops, and compound drops with a single inner drop (single-core compound drops) or a few inner drops (multi-core compound drops).     

Blade manipulators in turbulent channel flow

We report here the results of a series of careful experiments in turbulent channel flow, using various configurations of blade manipulators suggested as optimal in earlier boundary layer studies. The mass flow in the channel could be held constant to better than 0.1%, and the uncertainties in pressure loss measurements were less than 0.1 mm of water; it was therefore possible to make accurate estimates of the global effects of blade manipulation of a kind that are difficult in boundary layer flows. The flow was fully developed at the station where the blades were mounted, and always relaxed to the same state sufficiently far downstream. It is found that, for a given mass flow, the pressure drop to any station downstream is always higher in the manipulated than in the unmanipulated flow, demonstrating that none of the blade manipulators tried reduces net duct losses. However the net increase in duct losses is less than the drag of the blade even in laminar flow, showing that there is a net reduction in the total skin friction drag experienced by the duct, but this relief is only about 20% of the manipulator drag at most.List of symbols A, A log law constants - c chord length of manipulator - D drag of the manipulator - dp/dx pressure gradient in the channel - h half height of the channel - H height of the channel (2h) - K log law constant - L length of the channel - L.E. leading edge of the manipulator - P static pressure - P _x static pressure at a location x on the channel - P _xm static pressure at the location x in the presence of manipulator - p _ref static pressure at any reference location x upstream of the manipulator - Re Reynolds number - t thickness of the manipulator - T.E. trailing edge of the manipulator - u velocity in the channel - U friction velocity - U ^* average velocity in the channel - u _c centre-line velocity in the channel - U ₊ U/U _* - u _m velocities in the channel downstream of the manipulators - u _ref velocities in the channel at reference location upstream of the manipulators - w Coles's wake function - W width of channel Also National Aeronautical Laboratory, Bangalore 560 017, India     

Numerical simulations of large falling drops

The numerical simulation in a two‐phase medium of falling drops allows their velocities and shapes during the fall to be calculated. The terminal velocities and shapes for bromoform and chlorobenzene drops falling into water have been obtained. Although the method used calculates the flow inside and around the drop, it has not been possible to give results independent of the spatial discretization and the boundary effects. However, taking these influences into account, the numerical results agree with the experimental data given and the study consists of a good validation of the SURFER code used. Copyright © 2004 John Wiley & Sons, Ltd.     

Investigation of liquid–liquid drop coalescence using tomographic PIV

High-speed tomographic PIV was used to investigate the coalescence of drops placed on a liquid/liquid interface; the coalescence of a single drop and of a drop in the presence of an adjacent drop (side-by-side drops) was investigated. The viscosity ratio between the drop and surrounding fluids was 0.14, the Ohnesorge number (Oh = μ_d/(ρ_dσD)^1/2) was 0.011, and Bond numbers (Bo = (ρ _d  − ρ _s )gD ²/σ) were 3.1–7.5. Evolving volumetric velocity fields of the full coalescence process allowed for quantification of the velocity scales occurring over different time scales. For both single and side-by-side drops, the coalescence initiates with an off-axis film rupture and film retraction speeds an order of magnitude larger than the collapse speed of the drop fluid. This is followed by the formation and propagation of an outward surface wave along the coalescing interface with wavelength of approximately 2D. For side-by-side drops, the collapse of the first drop is asymmetric due to the presence of the second drop and associated interface deformation. Overall, tomographic PIV provides insight into the flow physics and inherent three-dimensionalities in the coalescence process that would not be achievable with flow visualization or planar PIV only.     

不同展长比旗帜在均匀来流中摆动的实验研究

为研究展长比对旗帜摆动特性的影响,在低速风洞内开展了质量比M*为0.6,展长比H*为0.2～1.5的柔性旗帜在不同来流速度下的失稳实验.利用高速摄影机和图像处理技术分析了无量纲速度U*和展长比H*对旗帜运动学特征(频率和振幅)的影响;利用自行研制的天平测量了旗帜受到的阻力,首次给出了旗帜的动力学特征.结果发现,保持展长...     

On two distinct types of drag-reducing fluids,diameter scaling,and turbulent profiles

Two distinct scaling procedures were found to predict the diameter effect for different types of drag-reducing fluids. The first one, which correlates the relative drag reduction (DR) with flow bulk velocity (V), appears applicable to fluids that comply with the 3-layers velocity profile model. This model has been applied to many polymer solutions; but the drag reduction versus V scaling procedure was successfully tested here for some surfactant solutions as well. This feature, together with our temperature profile measurements, suggest that these surfactant solutions may also show this type of 3-layers velocity profiles (3L-type fluids).The second scaling procedure is based on a correlation of τ_w versus V, which is found to be applicable to some surfactant solutions but appears to be applicable to some polymer solutions as well. The distinction between the two procedures is therefore not simply one between polymer and surfactants. It was also seen that the τ_w versus V correlation applies to fluids which show a stronger diameter effect than those scaling with the other procedure. Moreover, for fluids that scale according to the τ_w versus V procedure, the drag-reducing effects extend throughout the whole pipe cross section even at conditions close to the onset of drag reduction, in contrast to the behavior of 3L fluids. This was shown by our measurements of temperature profiles which exhibit a fan-type pattern for the τ_w versus V fluids (F-type), unlike the 3-layers profile for the fluids well correlated by drag reduction versus V. Finally, mechanically-degraded polymer solutions appeared to behave in a manner intermediate between the 3L and F fluids.Furthermore, we also showed that a given fluid in a given pipe may transition from a Type A drag reduction at low Reynolds number to a Type B at high Reynolds number, the two types apparently being more representative of different levels of fluid/flow interactions than of fundamentally different phenomena of drag reduction. After transition to the non-asymptotic Type B regime, our results suggest that, without degradation, the friction becomes independent of pipe diameter and that the drag reduction level becomes also approximately independent of the Reynolds number, in a strong analogy to Newtonian flow.     

Shapes of ellipsoidal bubbles in infinite stagnant liquids

Aspect ratios E of ellipsoidal bubbles in infinite stagnant clean liquids are measured for log M = −6.6, −5.5, −4.9 and −3.9, where M is the Morton number. An empirical correlation of E applicable to a wide range of the Morton number is proposed by making use of the present data and Sugihara's data at log M = −11 (2007). The aspect ratio in this correlation is expressed in terms of the combination of the Eötvös number and the bubble Reynolds number to account for the effects of the inertial, viscous, buoyant and surface tension forces on E. Terminal velocities of ellipsoidal bubbles are accurately predicted by using the proposed correlation and a drag correlation proposed by Rastello et al. (2011).