Upward and downward inclination oil–water flows

The effect of upward (+5°, +10°) and downward (−5°) pipe inclinations on the flow patterns, hold up and pressure gradient during two-liquid phase flows was investigated experimentally for mixture velocities between 0.7 and 2.5 m/s and phase fractions between 10% and 90%. The investigations were performed in a 38 mm ID stainless steel test pipe with water and oil as test fluids. High-speed video recording and local impedance and conductivity probes were used to precisely identify the different flow patterns. In both positive and negative inclinations the dispersed oil-in-water regime extended to lower mixture velocities and higher oil fractions compared to horizontal flow. A new flow pattern, oil plug flow, appeared at both +5° and +10° inclination while the stratified wavy pattern disappeared at −5° inclination. The oil to water velocity ratio was higher for the upward than for the downward flows but in the majority of cases and all inclinations oil was flowing faster than water. At low mixture velocities the velocity ratio increased with oil fraction while it decreased at high velocities. The increase became more significant as the degree of inclination increased. The frictional pressure gradient in both upward and downward flows was in general lower than in horizontal flows while a minimum occurred at all inclinations at high mixture velocities during the transition from dispersed water-in-oil to dual continuous flow.     

Influence of the aspect ratio on boiling flows in rectangular mini-channels

This article presents experiments conducted with two single rectangular mini-channels of same hydraulic diameter (1.4 mm) and different aspect ratios for conditions of horizontal boiling flow. The Forane® 365 HX used was subcooled (ΔT_sub = 15 °C) for all the boiling curves presented in the paper. Local heat transfer coefficients were measured for heat flux ranging from 25 to 62 kW m⁻² and mass flux from 200 kg m⁻² s⁻¹ to 400 kg m⁻² s⁻¹. The boiling flows were observed with two different cameras (depending on the flow velocity) through a visualization window. The flow patterns in the two channels were compared for similar conditions. The results show that the boiling heat transfer coefficient and the pressure drop values are different for the two single mini-channels. For low heat flux condition, the channel with lowest aspect ratio (H/W = 0.143) has a higher heat transfer coefficient. On the other hand, for high heat flux condition, the opposite situation occurs, namely the heat transfer coefficient becomes higher for the channel with highest aspect ratio (H/W = 0.43). This is probably due to the earlier onset of dryout in the channel with lowest aspect ratio. For the two cases of heating, the pressure drop for the two-phase flow remains lower for the channel with lowest aspect ratio. These results show that the aspect ratio plays a substantial role for boiling flows in rectangular channels. As for single-phase flows, the heat transfer characteristics are significantly influenced (even though the hydraulic diameter remains the same) by this parameter.     

Numerical modeling and experimental investigation of gas–liquid slug formation in a microchannel T-junction

Gas–liquid two-phase flow in a microfluidic T-junction with nearly square microchannels of 113 μm hydraulic diameter was investigated experimentally and numerically. Air and water superficial velocities were 0.018–0.791 m/s and 0.042–0.757 m/s, respectively. Three-dimensional modeling was performed with computational fluid dynamics (CFD) software FLUENT and the volume of fluid (VOF) model. Slug flow (snapping/breaking/jetting) and stratified flow were observed experimentally. Numerically predicted void fraction followed a linear relationship with the homogeneous void fraction, while experimental values depended on the superficial velocity ratio U_G/U_L. Higher experimental velocity slip caused by gas inlet pressure build-up and oscillation caused deviation from numerical predictions. Velocity slip was found to depend on the cross-sectional area coverage of the gas slug, the formation of a liquid film and the presence of liquid at the channel corners. Numerical modeling was found to require improvement to treat the contact angle and contact line slip, and could benefit from the use of a dynamic boundary condition to simulate the compressible gas phase inlet reservoir.     

Experimental study on axial development of liquid film in vertical upward annular two-phase flow

Accurate measurements of the interfacial wave structure of upward annular two-phase flow in a vertical pipe were performed using a laser focus displacement meter (LFD). The purpose of this study was to clarify the effectiveness of the LFD for obtaining detailed information on the interfacial displacement of a liquid film in annular two-phase flow and to investigate the effect of axial distance from the air–water inlet on the phenomena. Adiabatic upward annular air–water flow experiments were conducted using a 3 m long, 11 mm ID pipe. Measurements of interfacial waves were conducted at 21 axial locations, spaced 110 mm apart in the pipe. The axial distances from the inlet (z) normalized by the pipe diameter (D) varied over z/D = 50–250. Data were collected for predetermined gas and liquid flow conditions and for Reynolds numbers ranging from Re_G = 31,800 to 98,300 for the gas phase and Re_L = 1050 to 9430 for the liquid phase. Using the LFD, we obtained such local properties as the minimum thickness, maximum thickness, and passing frequency of the waves. The maximum film thickness and passing frequency of disturbance waves decreased gradually, with some oscillations, as flow developed. The flow development, i.e., decreasing film thickness and passing frequency, persisted until the end of the pipe, which means that the flow might never reach the fully developed state. The minimum film thickness decreased with flow development and with increasing gas flow rate. These results are discussed, taking into account the buffer layer calculated from Karman’s three-layer model. A correlation is proposed between the minimum film thickness obtained in relation to the interfacial shear stress and the Reynolds number of the liquid.     

Flow regime development analysis in adiabatic upward two-phase flow in a vertical annulus

In this work radial and axial flow regime development in adiabatic upward air-water two-phase flow in a vertical annulus has been investigated. Local flow regimes have been identified using conductivity probes and neural networks techniques. The inner and outer diameters of the annulus are 19.1 mm and 38.1 mm, respectively. The equivalent hydraulic diameter of the flow channel, D_H, is 19.0 mm and the total length is 4.37 m. The flow regime map includes 1080 local flow regimes identifications in 72 flow conditions within a range of 0.01 m/s < 〈j_g〉 < 30 m/s and 0.2 m/s < 〈j_f〉 < 3.5 m/s where 〈j_g〉 and 〈j_f〉 are, respectively, superficial gas and liquid velocities. The local flow regime has been classified into four categories: bubbly, cap-slug, churn-turbulent and annular flows. In order to study the radial and axial development of flow regime the measurements have been performed at five radial locations. The three axial positions correspond to z/D_H = 52, 149 and 230, where z represents the axial position. The flow regime indicator has been chosen as some statistical parameters of local bubble chord length distributions and self-organized neural networks have been used as mapping system. This information has been also used to compare the results given by the existing flow regime transition models. The local flow regime is characterized basically by the void fraction and bubble chord length. The radial development of flow regime shows partial and complete local flow regime combinations. The radial development is controlled by axial location and superficial liquid velocity. The radial flow regime transition is always initiated in the center of the flow channel and it is propagated towards the channel boundaries. The axial development of flow regime is observed in all the flow maps and it is governed by superficial liquid velocity and radial location. The prediction results of the models are compared for each flow regime transition.     

Evaporation heat transfer of R-134a inside a microfin tube with different tube inclinations

An experimental investigation has been carried out to study the heat transfer characteristics during evaporation of R-134a inside a single helical microfin tube. The microfin tube has been provided with different tube inclination angles of the direction of fluid flow from horizontal, α. The experiments were performed for seven different tube inclinations, α, in a range of −90° to +90° and four mass velocities of 53, 80, 107 and 136 kg/m² s for each tube inclination angle during evaporation of R-134a. The results demonstrate that the tube inclination angle, α, affects the boiling heat-transfer coefficient in a significant manner. For all refrigerant mass velocities, the best performing tube is that having inclination angle of α = +90°. The effect of tube inclination angle, α, on heat-transfer coefficient, h, is more prominent at low vapor quality and mass velocity. An empirical correlation has also been developed to predict the heat-transfer coefficient during flow boiling inside a microfin tube with different tube inclinations.     

Onset of entrainment and degree of dispersion in dual continuous horizontal oil–water flows

The transition from stratified to dual continuous oil–water flow (where each phase retains its continuity but there is dispersion of one phase into the other) as well as the dispersed phase fractions in the layers of the dual continuous pattern, were studied experimentally. Transition to this pattern from stratified flow occurs when drops of one phase appear into the other (onset of entrainment). The studies were carried out in a 38 mm ID horizontal stainless steel test section using two different inlet geometries, a T- and a Y-junction. The patterns were visualized through a transparent acrylic section located at 7 m from the inlet using a high speed video camera. Phase distribution measurements in a pipe cross section were obtained just before the acrylic section with a local impedance probe and the results were used to calculate the volume fraction of each phase entrained into the other. The onset of entrainment was found to occur at lower superficial water velocities as the oil superficial velocities increased. However, the inlet geometry did not affect significantly the transition line. During dual continuous flow, the dispersion of one phase into the opposite was found to extend further away from the interface with increasing water superficial velocity for a certain oil superficial velocity. An increase in the superficial water velocity increased the entrained fraction of water in oil (E_w/o) but there was no trend with the oil velocity. Similarly, an increase in the superficial oil velocity increased the fraction of oil drops in water (E_o/w) but the water velocity had no clear effect. The entrainment fractions were affected by the inlet geometry, with the T-inlet resulting in higher entrainment than the Y-inlet, perhaps because of the increased mixing induced by the T-inlet. The difference between the two inlets increased as the oil and water velocities increased.     

Comment on “Correlation of entrainment for annular flow in horizontal pipes”, by Pan,L., Hanratty,T.J., Int. J. Multiphase flow, 28(3), (2002), pp. 385–408

The above referenced paper, published in International Journal of Multiphase Flow (Pan and Hanratty, 2002), proposed an entrainment fraction correlation for annular flow in horizontal pipes. The entrainment fraction in annular flow is defined as the ratio of the mass flow rate of the liquid droplets in the gas to the total mass flow of liquid, F_E = W_LE/W_L. The proposed correlation was verified with experimental data for liquids with viscosities close to that of water. The proposed entrainment fraction correlation includes another correlation for the critical film flow rate, W_F,cr to estimate a maximum entrainment fraction F_E,max. It is shown that the critical film flow rate correlation can result in negative maximum entrainment fraction values, for low liquid flow rates.     

Purely elastic flow asymmetries in flow-focusing devices

The flow of a viscoelastic fluid through a microfluidic flow-focusing device is investigated numerically with a finite-volume code using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models. The conceived device is shaped much like a conventional planar “cross-slot” except for comprising three inlets and one exit arm. Strong viscoelastic effects are observed as a consequence of the high deformation rates. In fact, purely elastic instabilities that are entirely absent in the corresponding Newtonian fluid flow are seen to occur as the Deborah number (De) is increased above a critical threshold. From two-dimensional numerical simulations we are able to distinguish two types of instability, one in which the flow becomes asymmetric but remains steady, and a subsequent instability at higher De in which the flow becomes unsteady, oscillating in time. For the UCM model, the effects of the geometric parameters of the device (e.g. the relative width of the entrance branches, WR) and of the ratio of inlet average velocities (VR) on the onset of asymmetry are systematically examined. We observe that for high velocity ratios, the critical Deborah number is independent of VR (e.g. De_c ≈ 0.33 for WR = 1), but depends non-monotonically on the relative width of the entrance branches. Using the PTT model we are able to demonstrate that the extensional viscosity and the corresponding very large stresses are decisive for the onset of the steady-flow asymmetry.     

On two-phase flow patterns and transition criteria in aqueous methanol and CO2 mixtures in adiabatic,rectangular microchannels

This paper presents flow map investigations of adiabatic two-phase flow in square cross-sectioned, 200 μm deep microchannels fabricated in silicon, employing laser induced fluorescence microscopy. The influence of surface tension and nozzle geometry on the flow pattern transition was investigated using two nozzle widths (orifices of 30 μm and 50 μm, respectively) and methanol–water solutions with CO₂ as the gas phase. It was found and quantified that smaller nozzle geometries and smaller liquid surface tension promote the propagation of capillary gas bubbles at lower superficial gas and liquid velocities. Within the measurement domain of superficial gas (0.01–0.625 m/s) and liquid (0.0005–0.5000 m/s) velocities, we observed dispersed bubbly, regularly ordered bubbly, wedging, slug and annular flows, thus extending the experimental knowledge base to smaller superficial liquid velocities by almost two orders of magnitude. With the help of the flow maps presented herein, we were able to characterize the observed regularly ordered bubbly flow as the transition regime between dispersed bubbly and wedging flow. The results of the present investigation are of direct relevance to the operation of small-scale direct methanol fuel cells.     

Single-beam gamma densitometry measurements of oil–water flow in horizontal and slightly inclined pipes

Gamma densitometry is a frequently used non-intrusive method for measuring component volume fractions in multiphase flow systems. The application of a single-beam gamma densitometer to investigate oil–water flow in horizontal and slightly inclined pipes is presented. The experiments are performed in a 15 m long, 56 mm diameter, inclinable stainless steel pipe using Exxsol D60 oil (viscosity 1.64 mPa s, density 790 kg/m³) and water (viscosity 1.0 mPa s, density 996 kg/m³) as test fluids. The test pipe inclination is changed in the range from 5° upward to 5° downward. Experimental measurements are reported at three different mixture velocities, 0.25, 0.50 and 1.00 m/s, and the inlet water cut is varied from 0 to 1. The gamma densitometer is composed of radioactive isotope of Am-241 with the emission energy of 59.5 keV, scintillation detector [NaI(Tl)] and signal processing system. The time averaged cross-sectional distributions of oil and water phases are measured by traversing the gamma densitometer along the vertical pipe diameter. Based on water volume fraction measurements, water hold-up and slip ratio are estimated. The total pressure drop over the test section is measured and frictional pressure drop is estimated based on water hold-up measurements. The measurement uncertainties associated with gamma densitometry are also discussed. The measured water hold-up and slip ratio profiles are strongly dependent on pipe inclination. In general, higher water hold-up values are observed in upwardly inclined pipes compared to the horizontal and downwardly inclined pipes. At low mixture velocities, the slip ratio decreases as the water cut increases. The decrease is more significant as the degree of inclination increases. The frictional pressure drop for upward flow is slightly higher than the horizontal flow. In general, there is a marginal difference in frictional pressure drop values for horizontal and downwardly inclined flows.     

An inverse dispersed multiphase flow model for liquid production rate determination

The aim of this study is to develop a model for the determination of the superficial velocities in horizontal and slightly inclined oil–water pipe flow conditions by using pressure gradient and mixture density information. In this article an inverse model is suggested for a dispersion of oil in water and of water in oil. This approach permits to select dispersed flow conditions from a set of experimental data, and uses a new hybrid model for the effective viscosity. A set of 310 oil–water experimental data points collected on an experimental set-up of length L = 15 m and diameter D = 8.28 cm at various (slight) orientations is used to validate the inverse method. The comparison between model reconstructions and measured flow velocities show a reasonable agreement.     

Effect of drag reducing polymer on air–water annular flow in an inclined pipe

Drag reduction (DR) for air and water flowing in an inclined 0.0127 m diameter pipe was investigated experimentally. The fluids had an annular configuration and the pipe is inclined upward. The injection of drag reducing polymer (DRP) solution produced drag reductions as high as 71% with concentration of 100 ppm in the pipeline. A maximum drag reduction that is accompanied (in most cases) by a change to a stratified or annular-stratified pattern. The drag reduction is sensitive to the gas and liquid superficial velocities and the pipe inclination. Maximum drag reduction was achieved in the case of pipe inclination of 1.28° at the lowest superficial gas velocity and the highest superficial liquid velocity. For the first time in literature, the drag reduction variations with the square root of the superficial velocities ration for flows with the same final flow patterns have self-similar behaviors.     

Saturated critical heat flux in a multi-microchannel heat sink fed by a split flow system

An extensive experimental campaign has been carried out for the measurement of saturated critical heat flux in a multi-microchannel copper heat sink. The heat sink was formed by 29 parallel channels that were 199 μm wide and 756 μm deep. In order to increase the critical heat flux and reduce the two-phase pressure drop, a split flow system was implemented with one central inlet at the middle of the channels and two outlets at either end. The base critical heat flux was measured using three HFC Refrigerants (R134a, R236fa and R245fa) for mass fluxes ranging from 250 to 1500 kg/m² s, inlet subcoolings from ?25 to ?5 K and saturation temperatures from 20 to 50 °C. The parametric effects of mass velocity, saturation temperature and inlet subcooling were investigated. The analysis showed that significantly higher CHF was obtainable with the split flow system (one inlet–two outlets) compared to the single inlet–single outlet system, providing also a much lower pressure drop. Notably several existing predictive methods matched the experimental data quite well and quantitatively predicted the benefit of higher CHF of the split flow.     

Numerical modeling of fluid flow and heat transfer in a narrow Taylor-Couette-Poiseuille system

We consider turbulent flows in a differentially heated Taylor-Couette system with an axial Poiseuille flow. The numerical approach is based on the Reynolds Stress Modeling (RSM) of [Elena and Schiestel, 1996] and [Schiestel and Elena, 1997] widely validated in various rotor-stator cavities with throughflow ( [Poncet, 2005], [Poncet et al., 2005] and [Haddadi and Poncet, 2008]) and heat transfer (Poncet and Schiestel, 2007). To show the capability of the present code, our numerical predictions are compared very favorably to the velocity measurements of Escudier and Gouldson (1995) in the isothermal case, for both the mean and turbulent fields. The RSM model improves, in particular, the predictions of the k-ε model of Naser (1997). Then, the second order model is applied for a large range of rotational Reynolds (3744 ? Re_i ? 37,443) and Prandtl numbers (0.01 ? Pr ? 12), flow rate coefficient (0 ? C_w ? 30,000) in a very narrow cavity of radius ratio s = R_i/R_o = 0.961 and aspect ratio L = (R_o − R_i)/h = 0.013, where R_i and R_o are the radii of the inner and outer cylinders respectively and h is the cavity height. Temperature gradients are imposed between the incoming fluid and the inner and outer cylinders. The mean hydrodynamic and thermal fields reveal three distinct regions across the radial gap with a central region of almost constant axial and tangential mean velocities and constant mean temperature. Turbulence, which is weakly anisotropic, is mainly concentrated in that region and vanishes towards the cylinders. The mean velocity distributions are not clearly affected by the rotational Reynolds number and the flow rate coefficient. The effects of the flow parameters on the thermal field are more noticeable and considered in details. Correlations for the averaged Nusselt numbers along both cylinders are finally provided according to the flow control parameters Re_i, C_w, and Pr.     

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.     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

Characteristics of developing vertical bubbly flow under normal and microgravity conditions

The axial development of the void fraction profile, interfacial area concentration and Sauter mean bubble diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured using stereo image processing under normal and microgravity conditions. The flow measurements were performed at four axial locations (axial distance from the inlet, z normalized by the pipe diameter, D, z/D = 5, 20, 40 and 60) and with various flows: superficial gas velocity of 0.00840-0.0298 m/s, and superficial liquid velocity of 0.138-0.914 m/s. The effect of gravity on radial distribution of bubbles and the axial development of two-phase flow parameters is discussed in detail based on the obtained database and visual observation. Following Serizawa-Kataoka’s phase distribution pattern criteria under normal gravity conditions, the phase distribution pattern map was developed. Similar to normal gravity two-phase flows, wall, core and intermediate void peak patterns are observed under microgravity conditions but a transition void distribution pattern is not observed in the current experimental conditions. The data obtained in the current experiment are expected to contribute to the benchmarking of CFD simulation of phase distribution pattern and interfacial area concentration in forced convective pipe flow under microgravity conditions.     

Experimental study on the transition between stratified and non-stratified horizontal oil-water flow

The effect of oil and water velocities, pipe diameter and oil viscosity on the transition from stratified to non-stratified patterns was studied experimentally in horizontal oil-water flow. The investigations were carried out in a horizontal acrylic test section with 25.4 and 19 mm ID with water and two oil viscosities (6.4 and 12 cP) as test fluids. A high-speed video camera was used to study the flow structures and the transition. At certain oil velocity, stratified flow was found to transform into bubbly and dual continuous flows as superficial water velocity increased for both pipe diameters using the 12 cP oil viscosity. The transition to bubbly flow was found to disappear when the 6.4 cP oil viscosity was used in the 25.4 mm pipe. This was due to the low E?tv?s number. Transition to dual continuous flow occurred at lower water velocity for oil velocity up 0.21 m/s when 6.4 cP oil was used in the 25.4 mm ID pipe, while for U_so > 0.21 m/s, the transition appeared at lower water velocity with the 12 cP oil.The effect of pipe diameter was also found to influence the transition between stratified and non-stratified flows. At certain superficial oil velocity, the water velocity required to form bubbly flow increased as the pipe diameter increased while the water velocity required for drop formation decreased as the pipe diameter increased. The maximum wave amplitude was found to grow exponentially with respect to the mixture velocity. The experimental maximum amplitudes at the transition to non-stratified flow agreed reasonably well with the critical amplitude model. Finally, it was found that none of the available models were able to predict the present experimental data at the transition from stratified to non-stratified flow.     

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.