Dual vortex breakdown in a two-fluid confined flow

The core of a vortex can abruptly expand and the axial velocity can reverse. This phenomenon, referred to as vortex breakdown (VB), has been studied in more than one thousand papers starting with the report by Peckham and Atkinson.¹ The lasting attention is due to (a) important technological applications and (b) the apparently enigmatic mechanism of VB. The VB affects the lift and drag forces of a delta-wing aircraft, stabilizes flames in combustion chambers, helps mixing ingredients in chemical and biological reactors, and weakens tornadoes.² Following Vogel³ and Escudier,⁴ many studies of VB have been performed in a sealed cylindrical container. Its simple geometry and absence of ambient disturbances ease both experimental and numerical studies that help understand the VB physics.

Two-fluid flows recently attracted the attention of researchers due to applications in aerial vortex bioreactors.^5–8 The air flow transports oxygen, required for tissue growth, to the interface.⁵ The oxygen (O₂) diffuses through the interface and is dissolved in water. The meridional circulation of water helps mix the dissolved O₂ and other ingredients. Thus, the aerial vortex bioreactor provides the nonintrusive and fine mixing required for efficient growth of tissue cultures.⁶ A proper model of an aerial vortex bioreactor is a sealed vertical cylindrical container (Fig. 1) whose lid rotates while other walls are stationary.⁷ Recent studies revealed that such two-fluid flows have paradoxical features of fundamental and practical interest.

One important feature is a curious deformation of the interface. Its shape can significantly enlarge the interface area, thus enhancing diffusion of O₂. To observe deformations of the interface in a laboratory, oil–water systems are suitable because the densities of oil and water are close. Fujimoto and Takeda first performed such a study.⁹ Their experiments revealed amazing bends of the interface in a flow of silicon oil and water. As the rotation intensifies, the interface takes shapes named, by Fujimoto and Takeda, “hump,” “cusp,” “Mt. Fuji,” and “bell.”

Another important phenomenon is vortex breakdown in the upper fluid first observed by Tsai et al. in a flow of soybean oil and water.¹⁰ The experimental study¹¹ showed in detail how VB appears and disappears in the upper fluid. The numerical and experimental work¹² revealed that the VB mechanism is similar in one-fluid and two-fluid confined flows. The experimental study¹³ detected VB in the lower fluid and tracked its evolution process as the rotation speeds up.

The main difficulty in the experimental investigation of simultaneous VB development in both upper and lower fluids is that, under steady-flow conditions, the velocity of the lower fluid is very small to be precisely measured when VB forms in the upper fluid.

Our study overcame this difficulty by (a) selecting volumes and properties of the lower and upper fluids, (b) using enhanced laser illumination and the technique of tracer visualization with the floating averaging window, and (c) averaging 500 images of instantaneous velocity fields. This advance in the experimental technique helped discover a new spectacular and important fluid-mechanics phenomenon—double vortex breakdown—simultaneously developing in both fluids as the rotation intensifies.

Figure 1 shows a schematic of the device. The lid of a sealed vertical cylindrical container, of radius R and height h, rotates with the angular velocity ω, while the other walls are stationary. The axial extents at rest of the lower and upper fluids are h_w+ and h_o, respectively; g is the gravitational acceleration. The dimensions are R = 45 mm, h = 2.5R, h_w+ = 1.5R, and h_o = R. The lower fluid is a solution of 67% water and 33% glycerol (hereafter referred to as water+) of density ρ_w+ = 1070 kg/m³ and kinematic viscosity ν_w+ = 3 mm²/s. The upper fluid is sunflower oil of density ρ_o = 920 kg/m³ and viscosity ν_o = 49 mm²/s. The flow is kept isothermal at 22.6 °C.

The strength of rotation is characterized by the Reynolds number Re = ωR²/ν_o. The velocity fields in the vertical cross sections were measured using planar Particle Image Velocimetry (PIV). Polyamide beads, of density 1030 kg/m³ and diameter around 10 μm, were employed as seeding light-scattering particles for both PIV measurements and visualization of the flow pattern. The previous studies^11,12 showed that the experimental (with tracers) and numerical (with no tracer) velocity profiles and flow patterns agree well in the upper fluid (oil). This agreement indicates that the presence of the tracer particles does not significantly disturb the flow. The amount of particles in the oil (plus the averaging procedure in tracking visualization) was sufficient for detecting dual VB.

The PIV system consists of a double-pulsed Nd:YAG Quantel EVG00200 laser (wavelength 532 nm, repetition rate 15 Hz, pulse duration 10 ns, and pulse energy 200 mJ), a CCD-camera (8 bits per pixel, matrix resolution 1360 × 1024 pixels) equipped with a Nikon SIGMA 50 mm f/2.8D lens, and a synchronizing processor. Since the flow is steady and the errors (due to light reflection from the interface) are random, we increase the signal-to-noise ratio by averaging 500 instantaneous velocity fields. We calculated the velocity fields using the iterative cross correlation algorithm with a continuous window shift and deformation and 75% overlap of the interrogation windows. The threshold value for seeding particle concentration is 3–8 tracers per 64 × 64 pixel area. Distortions due to the cylindrical sidewall do not significantly affect velocities near the axis where vortex breakdown bubbles (VBBs) were detected while sometimes causing errors up to 5% in the radial velocity at the periphery. However, these errors do not significantly affect the velocity profiles presented below. The experimental technique is described in more detail in Ref. 13.

Figure 1 also shows a schematic of the steady axisymmetric meridional multicellular motion resembling a one-on-one dice of domino. We use cylindrical coordinates (r, θ, z). All measurements are performed in the plane of the meridional cross section where θ = constant. The centrifugal force, induced by the lid rotation, drives the bulk meridional circulation of oil shown in the top part of Fig. 1. This centrifugal circulation drives the counter-circulation of water+ (the middle part of Fig. 1). The oil rotation generates the bulk centrifugal circulation of water+ (the lower part of Fig. 1). The near-axis small circles in oil and water+ depict circulation cells, referred to as vortex breakdown bubbles (VBBs).

Figure 2 shows photos of this two-fluid flow at a few characteristic values of Re, illustrating how VBBs emerge and disappear. These photos were extracted from the attached video clip where the floating averaging window was used. The technique of tracer visualization has been improved and adapted to the experimental conditions for the optical diagnostics of vortex flow with the development of bubble-type vortex breakdown. This technique appears efficient for investigations of slow processes, where the displacement of tracers in the flow is extremely small and it is necessary to increase the exposure time to catch a vortex flow pattern. The main problem is the noise accumulation while drawing a track for a long exposure time. To overcome this problem, we use a short exposure time, subtract a pre-measured average intensity value, and apply the floating averaging window in order to obtain a sufficient track length and optimal contrast for the upper and lower liquids.

At Re = 500, the pattern of flow is similar to that depicted in Fig. 1, but with no VBB. Another difference is that the interface rises near the axis forming its hump, which looks as a white spot in the photo. At Re = 600, the hump expands, one VBB appears in the center of the oil domain, and another VBB starts developing in the center of the water+ domain. At Re = 700, both VBBs are well visible (see also the attached video clip). The upper VBB almost touches the enlarged hump.

At Re = 800, the VBB in oil becomes located just above the interface, which now has the Mt. Fuji shape. At Re = 900, the VBB in oil diminished being pressed to the interface near the axis. At Re = 1050, both VBBs disappear, and the interface takes the bell shape.

Figure 3 shows profiles of velocity V_z on the axis r = 0 for the same values of Re, as those in Fig. 2. This helps better understand how the VBBs appear and disappear. The square (circle) symbols represent V_z in the oil (water+). At the interface, both symbols are shown in Fig. 3. The profiles of the velocity were extracted from PIV data after averaging 500 instantaneous velocity fields. The flow is steady and axisymmetric for all Re values considered. The averaging was performed in order to increase the signal-to-noise ratio and to better detect the boundaries separating flow cells.

At Re = 500, there is no VBB, and oil rises (V_z > 0) along the axis. Water+ also rises except in a thin circulation layer (TCL), 1.52 < z/R < 1.56, where V_z < 0. The function V_z(z) has local minima in water+ near z = 1.09 and in oil near z = 2.12. As Re grows, the minimal velocities decrease and become negative.

At Re = 600, the range 0.751 < z/R < 0.862, where V_z < 0, corresponds to the VBB in water+ and the range 1.959 < z/R < 2.073, where V_z < 0, corresponds to the VBB in oil. This agrees with the photo for Re = 600 in Fig. 2. At Re = 700, the VBB ranges, where V_z < 0, expand and shift into 0.557 < z/R < 0.834 in water+ and 1.798 < z/R < 2.137 in oil. This agrees with the photo for Re = 700 in Fig. 2.

At Re = 800, the VBB range in water+ shrinks into 0.420 < z/R < 0.598. In contrast, the VBB range in oil further expands into 1.703 < z/R < 2.143 and touches the interface. There is no ascending near-axis flow of oil adjacent to the interface at Re = 800 contrary to that at Re = 700. This agrees with the photo for Re = 800 in Fig. 2.

At Re = 900, the VBB range in water+ becomes very small: 0.384 < z/R < 0.450. The VBB range in oil decreases to 1.704 < z/R < 2.112 and remains attached to the interface. This agrees with the photo for Re = 900 in Fig. 2. At Re = 1050, there is no VBB range in both fluids. The pattern at Re = 1050 is topologically identical to that at Re = 500: there are bulk centrifugal circulations of water+ and oil, separated by the TCL of water+. The TCL is invisible in Fig. 2 while well observed in Fig. 3.

Thus, the VBBs appear in both upper and lower fluid flows at Re around 600. It is instructive to compare this value with that in Ref. 13, where the VBB emerges in the lower fluid flow at Re around 300. This difference is due to the fact that the lower fluid is water in Ref. 13, while here the lower fluid is water+ whose viscosity is three times the water viscosity.

Figure 4 shows a photo (left) and a PIV velocity vector field (right) at Re = 750 illustrating the coexisting VBBs in the upper and lower fluids. To conveniently observe the PIV pattern, the arrows show velocity vectors normalized by their magnitude, i.e., only the direction of meridional velocity. The vector field visualizes circulations in both VBBs and the counterflow near the interface where the TCL is located. The TCL bends upward near the axis that agrees well with the V_z measurements, see Fig. 3.

In a steady axisymmetric state, recognizing patterns of flow and measuring its velocity are very challenging because the lower-fluid motion is very slow compared with the upper-fluid motion. This explains why the double vortex breakdown (VB) has not been observed and reported in the literature. To overcome this difficulty, we significantly advanced the experimental technique and carefully selected properties of the lower and upper fluids.

This advance allowed for discovering the appearance and disappearance of vortex breakdown bubbles in both fluids, as rotation intensifies. Near the interface, the oil (a) converges to the axis and (b) rotates. These two factors generate both (a) anti-centrifugal and (b) centrifugal circulations of the lower fluid, resulting in the flow pattern, explained in Ref. 11, schematically shown here in Fig. 1 (without VBBs), and observed in Figs. 2 and 3 at Re = 500.

To better understand why the VBBs simultaneously emerge in both fluids, we introduce the Reynolds number Re_l = V_w+R/ν_w+, where V_w+ is the maximal velocity of water+. Re_l characterizes the strength of the lower-fluid flow. V_w+ is significantly smaller than ωR because the motion decays downward. However, ν_w+ also can be significantly smaller than ν_o. By choosing a proper water–glycerol solution, we found such a ν_w+ value that makes Re and Re_l being almost equal, which results in the simultaneous vortex breakdown in both fluids.

As Re increases, the convergence to the axis, of (a) the upper fluid near the interface and (b) the lower fluid near the bottom, results in the development of tornado-like ascending near-axis swirling jets in both fluids. The convergence also generates local maxima of angular velocity and local minima of pressure near the intersections of the axis with (a) the interface and (b) the bottom. The minima of pressure suck the downstream fluid, cause local reversals of the axial velocity, and thus produce circulation cells (VBBs) in the upper and lower fluid flows, schematically shown in Fig. 1 and well observed in Figs. 2 and 3 at Re = 700 and in Fig. 4 at Re = 750. As Re further increases, the cells move downward, to the interface in oil and to the bottom in water+. These shifts reduce the convergence of the upper and lower fluids to the axis and thus kill both local circulation cells, as shown in Figs. 2 and 3 at Re = 1050.

The growing centrifugal force concentrates the meridional circulation of both fluids near the sidewall. This concentration prevents the development of vortex breakdown, as Re further grows. These results indicate that the VB scenario, observed here, is similar to the VB scenarios in one-fluid flows,² in the upper fluid of two-fluid flows,^10–12 and in the lower-fluid flow¹³ where the physical mechanism of VB is discussed in more detail.

The revealed dual-VBB pattern looks unusual and impressive, e.g., in Fig. 4 and in the supplementary material. The multiple circulation cells of this flow should provide efficient mixing. As known, single-VB flows help mix fuel, oxidizer, and flue gases in vortex combustors, thus stabilizing flame and minimizing pollution.² Therefore, there is a hope that the revealed dual-VB flow can lead to the development of “rotating two-fluid” technology for chemical and biological processes where fine, gentle, and nonintrusive mixing is required.

See supplementary material, VBB_Domino.mp4, which shows the evolution of VBBs in both upper and lower fluids as Re increases.