Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

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.     

Motion of singles bubbles moving under a slightly inclined surface through stationary liquids

The influence of the liquid properties on the dynamical bubble shape and on the bubble motion has been investigated for bubbles moving under a downward facing inclined surface. The Morton number Mo varied from 2.59 × 10⁻¹¹ to 2.52 × 10⁺⁰¹. The Bond number Bo covered the range from 10 to 150 and the surface inclination angle θ was varied from 2° to 6°. To cover the wide range of Mo, several liquids such as glycerine, propanediol, water and isopropanol were used. The results have shown that the relation Fr = Fr(Bo, Mo, θ) is not adequate to describe the bubble motion, where Fr is the terminal Froude number. The choice of the terminal Reynolds number Re as the dependent parameter, allowed the clarification of the role of the Morton number on the bubble motion. At a given Bond number, the bubble Reynolds number decreases monotonously with the Morton number. Furthermore, an empirical correlation Re = Re(Bo, Mo, θ) is given that can be readily used in the mathematical modelling of bubble laden flows under solids.     

Enhanced nucleate boiling on copper micro-porous surfaces

Saturation boiling of PF-5060 dielectric liquid on Cu micro-porous surface layers (95, 139, 171, 197 and 220-μm thick) is investigated. These layers are deposited on 10 × 10 mm Cu substrates using two-stage electrochemical process. The basic micro-structure, obtained in the first stage using current density of 3 A/cm² for 15–44 s, depending on thickness, is strengthened by continuing electrochemical deposition using much lower current density for 10’s of minutes. For conditioned surface layers, after a few successive boiling tests, the pool boiling curves are reproducible and the temperature excursion prior to boiling incipience is either eliminated or reduced <7 K. Present nucleate boiling results are markedly better than those reported for dielectric liquids on micro- and macro-structured surfaces. Present values of CHF (22.7–27.8 W/cm²) and h_MNB (2.05–13.5 W/cm² K) are ∼40–70% higher than and >17 times those reported on plane surfaces (<16 W/cm² and ∼0.8 W/cm² K). Best results are those of the 171-μm thick layer: CHF of 27.8 W/cm² occurs at ΔT_sat of only 2.1 K and h_MNB of 13.5 W/cm² K occurs at ΔT_sat = 2.0 K.     

Evaluation analysis of prediction methods for two-phase flow pressure drop in mini-channels

Two-thousand and ninety-two data of two-phase flow pressure drop were collected from 18 published papers of which the working fluids include R123, R134a, R22, R236ea, R245fa, R404a, R407C, R410a, R507, CO₂, water and air. The hydraulic diameter ranges from 0.506 to 12 mm; Re_l from 10 to 37,000, and Re_g from 3 to 4 × 10⁵. Eleven correlations and models for calculating the two-phase frictional pressure drop were evaluated based upon these data. The results show that the accuracy of the Lockhart–Martinelli method, Mishima and Hibiki correlation, Zhang and Mishima correlation and Lee and Mudawar correlation in the laminar region is very close to each other, while the Muller-Steinhagen and Heck correlation is the best among the evaluated correlations in the turbulent region. A modified Chisholm correlation was proposed, which is better than all of the evaluated correlations in the turbulent region and its mean relative error is about 29%. For refrigerants only, the new correlation and Muller-Steinhagen and Heck correlation are very close to each other and give better agreement than the other evaluated correlations.     

Flow patterns of n-hexadecane–CO2 liquid–liquid two-phase flow in vertical pipes under high pressure

To promote a better understanding of liquid–liquid two-phase flow behavior, particularly under high pressure, flow patterns of n-hexadecane–CO₂ liquid–liquid two-phase upward flow in vertical stainless steel pipes were experimentally investigated. Observations were made in two 0.0015 m I.D. pipes of different lengths (0.068 m and 0.5 m) under high pressure varying from 10.3 to 29.6 MPa using a high pressure visualization system. The total flow rate was fixed at 2.0 × 10⁻⁶ m³/min, while the flow rate ratio (φ) varied from 0.05 to 19. Bubbly flow, plug flow, slug flow, annular flow, and near-one-phase flow regions were found in both pipes, while stratified flow was observed only in the 0.068 m pipe. Flow pattern maps were constructed in the flow rate ratio versus pressure graph, which demonstrates significant impacts of flow rate ratio, pipe length, and pressure on flow patterns. These impacts are discussed in detail. To the authors’ best knowledge, this work is the first attempt to observe complex liquid–liquid two-phase flow behavior with flow pattern transitions under high pressure, and contributes to a better understanding of liquid–liquid two-phase flow behavior.     

Two-phase flow laden with spherical particles in a microcapillary

Solid–liquid two-phase flow in a finite Reynolds number range (2 < Re < 12), transporting neutrally-buoyant microspheres with diameters of 6, 10, and 16 μm through a 260-μm microcapillary, is investigated. A standard microparticle-tracking velocimetry (μ-PTV) that consists of a double-pulsed Nd:YAG laser, an epi-fluorescent microscope, and a cooled-CCD camera is used to examine the flow. The solid particles are visualized in view of their spatial distributions. We observe a strong radial migration of the particles across the flow streamlines at substantially small Re. The degree of particle migration is presented in terms of probability density function. Some applications based on this radial migration phenomena are discussed in conjunction with particle separation/concentration in microfluidic devices, where the spatial distribution of particles is of great importance. In doing so, we propose a particle-trajectory function to empirically construct the spatial distribution of solid particles, which is well correlated with our experimental data. It is believed that this function provides a simple method for estimating the spatial distribution of particles undergoing radial migration in solid–liquid two-phase flows.     

Experimental evaluation of pressure drop in round tubes provided with physically separated, multiple, short-length twisted tapes

Isothermal pressure drop tests were performed on horizontal round tubes each containing identical, physically separated, equally spaced, short-length twisted tapes (TTs). The tests investigated the dependence of the Darcy friction factor, f, on the empty tube Reynolds number Re, TT twist ratio y, and TT spacing s. The variation of f across TTs belonging to the same test section was also examined. Ranges of the experimental variables examined were: 10,000 ? Re ? 90,000, 1.5 ? y ? 6, and s = 30, 40 and 50. The number of 360° revolutions was held constant for all the tapes and equal to 1.5. Tap water at room temperature and nearly atmospheric pressure was used as the working fluid. A correlation for the Darcy friction factor, in the form f = f (Re, y, s), was developed from the collected experimental data, with excellent accuracy.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

Determination of thermal conductivity and interfacial energy of solid Zn solution in the Zn-Al-Bi eutectic system

The equilibrated grain boundary groove shapes for solid Zn solution (Zn-3.0 at.% Al-0.3 at.% Bi) in equilibrium with the Zn-Al-Bi eutectic liquid (Zn-12.7 at.% Al-1.6 at.% Bi) have been observed from quenched sample with a radial heat flow apparatus. Gibbs-Thomson coefficient, solid-liquid interfacial energy and grain boundary energy for solid Zn solution in equilibrium with Al-Bi-Zn eutectic liquid have been determined to be (5.1 ± 0.4) × 10⁻⁸ K m, (80.1 ± 9.6) × 10⁻³ and (158.6 ± 20.6) × 10⁻³ J m⁻² from the observed grain boundary groove shapes, respectively. The thermal conductivity variation with temperature for solid Zn solution has been measure with radial heat flow apparatus and the value of thermal conductivity for solid Zn solution has been determined to be 135.68 W/km at the eutectic melting temperature. The thermal conductivity ratio of equilibrated eutectic liquid to solid Zn solution, R = K_L(Zn)/K_S(Zn) has also been measured to be 0.85 with Bridgman type solidification apparatus.     

Non-linear dynamics of semi-dilute polydisperse polymer solutions in microfluidics: A study of a benchmark flow problem

The complex flow behaviour of semi-dilute (15 < c/c^* < 22.5) polydisperse polyethylene oxide (PEO) aqueous solutions flowing through a planar microfluidic geometry with an 8:1:8 contraction-expansion is systematically studied. The molecular weight and distribution of the PEO samples are analysed by Gel Permeation Chromatography (GPC). Full rheometric characterizations using various techniques including piezoelectric axial vibrator (PAV) measurements at frequencies as high as 6700 Hz are carried out for one semi-dilute PEO solution. Complex flows over a wide range of elasticity numbers (20 ? El ? 120), Weissenberg numbers (7 ? Wi ? 121) and Reynolds numbers (0.08 ? Re ? 4.5) are characterized using micro-particle image velocimetry (μ-PIV) and pressure drop measurements. The evolution of vortex formation and dynamics has been visualized through a step-flow-rate experiment. The effect of El on vortex stability has been studied. Various flow dynamics regimes have been quantified and are presented in a Wi-Re diagram. The experimental results reveal that the elastic behaviour of polymer solutions is very sensitive to high molecular weight polymer in the polydisperse polymer samples, and the contraction ratio and the aspect ratio of flow geometry are the important design parameters in controlling the non-linear dynamics of semi-dilute polymer solutions in microfluidics.     

Study of Near Wall Coherent Flow Structures on Dimpled Surfaces Using Unsteady Pressure Measurements

Unsteady pressure measurements have been performed inside a spherical dimple in a narrow channel for turbulent flow at Re _D ?=?40,000 with the aim to study coherent vortex structures and to get a deep insight into flow physics. Results confirm the formation of asymmetric coherent vortex structures switching between two extreme positions. Analysis of the pressure temporal distributions and correlation functions shows the presence of the anti-phase motion inside the dimple. Typical power laws of the pressure fluctuation energy spectrum ω ^???1 and ω ^???7/3 are reproduced.     

Falling film flow along steep two-dimensional topography: The effect of inertia

Steady film flow along a vertical wall with isolated step changes is studied numerically for Reynolds numbers Re ∼ O(10⁻³–10²) and capillary numbers Ca ∼ O(10⁻²–10¹). The lengthscale of free surface capillary features upstream of a step-in or step-out decreases uniformly with Re and switches from a −1/3 to a −1/2 power-law dependence on Ca. The height of the capillary features first grows with Re, but eventually diminishes when inertia forces overpower capillary forces. Simultaneously, the key dynamics move from upstream to downstream of the step, and switch from capillary arrest to inertial re-directioning of the falling liquid. The latter mechanism involves a low-pressure region originating from the edge of the step. At a step-out, a new free surface feature appears with increasing Re, which is caused by liquid overshoot in the horizontal direction and is restrained initially by capillary and subsequently by inertial forces. Simple scaling arguments are shown to predict many of the above characteristics.     

Segmented gas–liquid flow characterization in rectangular microchannels

We used optical methods such as Laser Induced Fluorescence (LIF) and confocal Laser Scanning Microscopy (LSM) to characterize gas–liquid phase distribution in rectangular microchannels. Using a 2 m long microchannel with a hydraulic diameter of 200 μm enables the precise measurement of important parameters such as liquid slug length, bubble length, pressure drop and film thickness at the wall as well as in the corner of the microchannel for low Capillary numbers (Ca) ranging from 2 × 10⁻⁴ to 1 × 10⁻². This range of Ca was obtained by using different fluid pairs such as ethanol, water and different concentrated aqueous solutions of glycerol in combination with nitrogen.     

Vortex in a Supersonic Flow and its Influence on Blunt Body Flow and Heat Transfer

The dissipation of a tip vortex generated by a rectangular wing and the stagnation pressure and temperature distributions within the vortex are experimentally investigated. The interaction of the vortex with the bow shock ahead of a sphere and the heat flux distribution over the spherical surface are studied. The experiments were carried out at the Mach number M=3 and the unit Reynolds numbers Re₁=1.1·10⁷ and 3.7·10⁷ 1/m.     

Counter-current gas–liquid flow in a vertical narrow channel—Liquid film characteristics and flooding phenomena

Results are reported of an experimental investigation of gas–liquid counter-current flow in a vertical rectangular channel with 10 mm gap, at rather short distances from liquid entry. Flooding experiments are carried out using air and various liquids (i.e., water, 1.5% and 2.5% aqueous butanol solutions) at liquid Reynolds numbers Re_L < 350. Visual observations and fast recordings suggest that the onset of flooding at low Re_L (<250) is associated with liquid entrainment from isolated waves, whereas “local bridging” is dominant at the higher Re_L examined in this study. Significant reduction of flooding velocities is observed with decreasing interfacial tension, as expected. Instantaneous film thickness measurements show that under conditions approaching flooding, a sharp increase of the mean film thickness, of mean wave amplitude and of the corresponding RMS values takes place. Film thickness power spectra provide evidence that by increasing gas flow the wave structure is significantly affected; e.g., the dominant wave frequency is drastically reduced. These data are complemented by similar statistical information from instantaneous wall shear stress measurements made with an electrochemical technique. Power spectra of film thickness and of shear stress display similarities indicative of the strong effect of waves on wall stress; additional evidence of the drastic changes in the liquid flow field near the wall due to the imposed gas flow, even at conditions below flooding, is provided by the RMS values of the wall stress. A simple model is presented for predicting the mean film thickness and mean wall shear stress under counter-current gas–liquid flow, below critical flooding velocities.     

On the particle inertia-free collision with a partially contaminated spherical bubble

The collision between a contaminated spherical bubble and fine particles in suspension is considered for r_p/r_b ? 1 (r_p being the radius of the particles in suspension and r_b the radius of the bubble). The collision probability or efficiency is defined as the number of particles colliding the bubble surface to the number of particles initially present in the volume swept out by the bubble. In this note we show that the collision probability can be expressed as P_c(r_p/r_b,Re) = g(r_p/r_b)f(Re) for both mobile and immobile interfaces. For partially contaminated bubbles a linear or quadratic dependency in r_p/r_b is found depending on the level of contamination and the value of r_p/r_b. These behaviors are given by the flux of particles near the surface which is controlled by the tangential velocity for mobile interfaces and by the velocity gradient for immobile interfaces. The threshold value (r_p/r_b)_th between the r_p/r_b and (r_p/r_b)² evolution is shown to vary as sin^n(Re)(θ_clean/n(Re))sin(3θ_clean/4), θ_clean being the angle describing the front clean part of the bubble and n(Re) varying from n = 2 to n = 1 from small to large Reynolds number.     

Free-surface fluctuations and turbulence in hydraulic jumps

A hydraulic jump is the highly turbulent transition between a high-velocity impinging flow and a turbulent roller. The jump flow is characterised by some substantial air bubble entrainment, spray and splashing. In the present study, the free-surface fluctuations and air-water properties of the hydraulic jump roller were investigated physically for relatively small Froude numbers (2.4 < Fr₁ < 5.1) and relatively large Reynolds numbers (6.6 × 10⁴ < Re < 1.3 × 10⁵). The shape of the mean free surface profile was well defined, and the time-averaged free-surface elevation corresponded to the upper free-surface, with the quantitative values being close to the equivalent clear-water depth. The turbulent fluctuation profiles exhibited a maximum in the first part of the hydraulic jump roller. The free-surface fluctuations presented some characteristic frequencies between 1.4 and 4 Hz. Some simultaneous free-surface measurements at a series of two closely located points yielded the free-surface length and time scales of free-surface fluctuations in terms of both longitudinal and transverse directions. The length scale data seemed to depend upon the inflow Froude number, while the time scale data showed no definite trend. Some simultaneous measurements of instantaneous void fraction and free-surface fluctuations exhibited different features depending upon the phase-detection probe sensor location in the different regions of the roller.     

Carbon dioxide flow boiling in a single microchannel - Part II: Heat transfer

Flow boiling heat transfer coefficients of CO₂ have been measured in a single microchannel. Experiments were carried out in a horizontal stainless steel tube of 0.529 mm inner diameter, for three temperatures (−10, −5 and 0 °C), with the mass flux ranging from 200 to 1200 kg/m² s and the heat flux varying from 10 to 30 kW/m². The investigation covered qualities from zero to the dryout inception, i.e. pre-dryout conditions. Compared to larger microchannels and positive temperatures, a higher contribution of convective boiling was found, with a larger heat transfer coefficient than for pure nucleate boiling. Mainly two heat transfer regimes were found, depending on the boiling number (Bo). For Bo > 1.1 × 10⁻⁴, the heat transfer coefficient was highly dependent on the heat flux and moderately influenced by the quality and the mass flux. For Bo < 1.1 × 10⁻⁴, the heat transfer coefficient was hardly affected by the heat flux but strongly influenced by the quality and the mass flux. In addition, dryout results were reported. The effect of the mass flux on the dryout inception quality was found to be highly dependent on the heat flux and the saturation temperature.