Counterflow slip in a two-fluid whirlpool Free

Swirling two-fluid flows attract the attention of researchers due to applications in aerial vortex bioreactors, wherein a rotating air drives a water flow.^1–4 The air transports oxygen (O₂), required for tissue growth, to the interface.¹ O₂ diffuses through the interface and dissolves in water. The meridional circulation and rotation of water help in mixing the dissolved O₂ and other ingredients. Thus, the aerial vortex bioreactor provides nonintrusive and fine mixing, facilitating the growth of biological tissue cultures.³ 

A convenient model of an aerial vortex bioreactor is a sealed vertical cylindrical container (Fig. 1), whose one end disk rotates, generating a swirling flow, while other walls are stationary.³ The simple geometry and the confined motion ease both experimental and computational detailed studies. Comparison of their results helps understanding the flow physics, which appears nontrivial. Recent works revealed that swirling two-fluid flows have a few paradoxical features of fundamental and practical interest.

One important phenomenon 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 densities of oil and water are close. Fujimoto and Takeda first performed such a study.⁵ 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.”

The second important phenomenon is vortex breakdown (VB); first observed by Tsai et al.⁶ in a flow of soybean oil and water when a new circulation cell emerged near the center of the oil flow. The experimental study⁷ showed in detail how and why a VB “bubble” appears and disappears in the upper fluid. Naumov et al.⁸ detected VB in the lower fluid and tracked the flow evolution, as the rotation speeds up.

The third curious phenomenon is a dual VB almost simultaneously developing in both upper and lower fluids.^9,10 The fourth interesting phenomenon is the formation of a thin circulation layer (TCL), separating centrifugal circulations (CCs) of the upper and lower fluids. The counter–centrifugal circulation in the layer serves as a liquid bearing.^7,10 The fifth nonlinear-physics phenomenon is the hysteretic growth and decay of a waterspout column.¹¹ 

The most striking and important phenomenon is a slip at the interface where the radial velocity has a jump that changes the velocity, magnitude, and even its direction. This phenomenon was first detected in the flow of sunflower oil (upper fluid) and water (78%)–glycerol (22%) solution (lower fluid), driven by the rotation of the top disk.¹² Further studies revealed that the counterflow due to the slip only occurs around a part of the interface and disappears, replaced by the TCL in this flow, as the rotation intensifies.^7,10 An important and paradoxical experimental fact is that a spiral flow of a lighter upper fluid converging above the interface creates a diverging (!) spiral motion of the denser lower fluid under the interface due to the slip.

The slip manifests in different scenarios of flow topological transformations; as the rotation increases, a new cell of centrifugal circulation develops near the axis intersection with the interface (bottom) and expands downward (upward) in the experiment (numerical simulations). The simulations performed using the condition of velocity continuity at the interface agree well with the experiment in the upper fluid but totally disagree in the lower fluid.¹² This disagreement indicates that the continuity condition is not satisfied in this case. One tested alternative condition was that the slip velocity is proportional to the density difference multiplied by the centrifugal acceleration. Under such condition, the simulation results are closer to experimental results, but still there is no satisfactory agreement for the lower-fluid flow.^10,12 What condition should replace the continuity condition? It is an open problem requiring further research.

To this end and to test whether the slip is a generic phenomenon, we investigate here a two-fluid whirlpool: a flow in a sealed vertical cylindrical container driven by the rotating bottom disk and focusing on the early topological metamorphoses occurring in this flow as the rotation intensifies.

In this case, we change the vortex generation of the flow, which is relative to the direction of gravity (Fig. 1). Now the angular momentum is transmitted from a denser lower fluid to lighter upper fluid in contrast to the geometry presented in works.^8–13 The results appear even more intriguing than expected ones.

Figure 1 shows a photo of the experimental setup and a schematic representation of the problem. The bottom disk of a sealed vertical cylindrical container, of radius R and height h, rotates with angular velocity ω, while the other walls are stationary. We placed the entire cylindrical container inside a rectangular transparent box made of glass and filled it with water to minimize thermal changes and optical aberrations that take place during measurements in a cylindrical surface.

The axial extents at rest of the lower and upper fluids are h_g and h_o, respectively, and g is the gravitational acceleration. The dimensions are: R = 73 mm, h = 2.5R, h_g = R, and h_o = 1.5R. The lower fluid is a solution of 65% water and 35% glycerol (hereafter referred to as glycerol) of density ρ_g = 1070 kg/m³ and kinematic viscosity ν_g = 70 mm²/s. The upper fluid is polydimethylsiloxane oil of density ρ_o = 920 kg/m³ and viscosity ν_o = 8 mm²/s. The flow is kept isothermal at 22.6 °C. The strength of rotation is characterized by the Reynolds number Re = ωR²/ν_g.

The velocity fields in vertical and horizontal cross sections were measured using the particle image velocimetry (PIV). We performed the measurements in the vertical cross section, which contains the geometric center of the container [Fig. 1(c) left]. We performed the horizontal measurements in the two cross sections: (i) at 3 mm above and (ii) 3 mm below the interface [Fig. 1(c) right]. These 3-mm shifts help us explore the flow transformations across the interface avoiding effects of the interface deformation.

Distortions due to the cylindrical sidewall do not significantly affect velocities near the axis, though sometimes cause errors in the radial velocity at the periphery in PIV diagnostics and visualization. However, these errors do not significantly affect the velocity profiles presented below. Therefore, to estimate the distortion of the vector field, the PIV measurements of the velocity vector radial component profile in two orthogonal sections were performed: (a) in the diametric section passing through the semicircular wall of the cylinder and (b) in the horizontal section through the flat top or bottom.^14,15 It is seen that distortions due to the cylindrical side surface of the container uniformly add to the velocity radial component from the center to the periphery up to 7%–10% in the vertical section compared with the component measured in the section through the flat bottom or top, where distortions due to the wall curvature are absent.

Polyamide beads, of density 1030 kg/m³ and diameter around 10 μm for the lower fluid and polyethylene microspheres of density 950 kg/m³ and diameter around 5 μm for the upper fluid (to minimize the buoyancy effects), were employed as seeding light-scattering particles for both PIV measurements and adaptive visualization of a flow pattern (more details are in Ref. 10). The previous studies^7–13 showed that the experimental (with tracers) and numerical (with no tracer) velocity profiles and flow patterns agree well with the fluid contacting the rotating disk. This agreement indicates that the presence of the tracer particles does not significantly disturb the flow. The number of particles in the oil (plus the averaging procedure in tracking visualization) was sufficient for detecting the slip.

The parameters h_o and h_g are important for the transfer of angular momentum across the interface. These parameters control the Reynolds number range, where the radial velocity changes its direction in the fluid contacting the rotating disk.¹⁶ For this reason, here, h_o and h_g are chosen to be similar to those in the case where the top disk rotates^10,12 for proper comparison of the results.

The PIV system consists of a double-pulsed Nd:YAG Beamtech Vlite-200 laser (wavelength 532 nm, repetition rate 15 Hz, pulse duration 10 ns, and pulse energy 200 mJ), a CCD-camera IMPERX IGV-B2020 (eight bits per pixel, matrix resolution 2056 × 2060 pixels, and with a 1.3″ optical format equipped with Nikon SIGMA 50 mm f/2.8D lens), and a synchronizing processor. We calculated two-dimensional velocity fields using the commercial software ActualFlow, Version 1.18.8.0. The thickness of the laser light sheet, formed by a cylindrical lens to illuminate tracer particles, is about 0.8 mm in the measurement plane. The distance between the camera and the laser sheet equals 800 and 720 mm for the horizontal and vertical planes, respectively.

For every set of experimental conditions, we accumulated 500 images and averaged them to increase the signal-to-noise ratio. Time delay between the two images is varied from 100 to 800 ms, depending on the Re and fluid type. 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. In order to have a relatively large dynamic range (the span between the maximal and minimal velocity values), the size of the interrogation window initially was 64 × 64 pixels. In addition, during the correlation step, the local particle image concentration was accounted for the correlation analysis only in those interrogation areas where the particle concentration level is above the given number of particles. The threshold value for seeding particles' concentration is 5–8 tracers per 64 × 64 pixels area. The sub-pixel interpolation of a cross-correlation peak was performed over three points using one-dimensional approximation by the Gaussian function. The inaccuracy of position determination does not exceed 0.1 of pixel. Thus, the velocity measurement error estimates are 1% and 4% for tracer sizes of eight and two pixels, respectively. We validated the obtained instantaneous velocity vector fields with the two subsequent procedures: (i) the peak validation with the threshold 2.0 and (ii) the adaptive median filter over 8 × 8 nodes. Resulting spatial resolution of the velocity fields was about one vector per 1.4 mm.

The centrifugal force, generated by the rotation of the bottom disk, pushes the lower fluid from the axis (shown by the dotted-dashed lines in Figs. 1 and 2) to the sidewall, thus driving the centrifugal circulation (CC); the glycerol ascends near the sidewall up to the interface and then converges to the axis where it descends to the bottom thus closing the loop.

At a slow rotation, this CC drives the anti-centrifugal circulation (AC); the oil converges to the axis near the interface, ascends near the axis, diverges near the top disk, and descends near the sidewall, as schematically shown in Figs. 1(b) and 2 at Re = 150. Subscripts l and h denote the light and heavy fluids, respectively. As Re increases from 0 up to 250, the topology of the light-fluid flow changes thrice while the topology of the heavy-fluid flow remains invariant (Fig. 2). No visible deformation of the interface occurs for 0 < Re < 250. Taking into account these features and for better optical resolution, we only describe our PIV measurements of streamlines (Fig. 3) and velocity (Fig. 4) distributions in the oil flow. The scenario, shown in Fig. 2, follows from both visual observations and PIV measurements (Figs. 3 and 4).

The first change in the flow topology is the emergence of a new cell, CC_l, near the interface-axis intersection as Fig. 2 schematically shows at Re = 175. This change means that a counterflow initially develops near the axis where the lower fluid converges to the axis while the upper fluid diverges from the axis, as the arrows indicate in Fig. 2. It seems paradoxical that the converging flow of the lower fluid generates the diverging flow of the upper fluid. Figure 4 provides some quantitative data on velocity distribution at the rotation axis. In the new cell, CC_l, the velocity at the axis is negative (directed downwards) in contrast with the AC_l, where the velocity at the axis is positive (directed upwards), compare Figs. 2–4 at Re = 175. Figure 4 illustrates how velocity V_z at the axis (at r = 0) decreases and changes its sign as the rotation intensifies.

The second change in the flow topology is the separation of AC_l from the interface, which is forced by the radial expansion of CC_l, compare Figs. 2–3 at Re = 200 and Re = 225 at Fig. 4. The third change in the flow topology is the separation of AC_l from the axis forced by the vertical expansion of CC_l, which is maximal at the axis, compare Figs. 2–4 at Re = 250.

Figure 5 shows the r-profiled of swirl (V_a) and radial (V_r) velocities at cross sections, z = constant, with 3 mm distance from the interface in the lower (L) and upper (U) fluids for Re = 150 (left plot) and 250 (right plot). At Re = 150, the maximal swirl velocity is 0.162ωR in the lower fluid and 0.061ωR in the upper fluid, i.e., the fluid rotation remarkably decays from the bottom, where the maximal swirl velocity is ωR, to the interface vicinity and then across the interface. Nevertheless, the swirl velocity remains significantly larger than the radial (Fig. 5) and axial (Fig. 4) velocities.

The maximal magnitude of radial velocity decreases dramatically from 0.04ωR below the interface to 0.005ωR above the interface. The radial velocity is negligibly small near axis both in the lower and upper fluids. This feature helps us understand why the slip and counterflow initially develop near the axis, where the centrifugal force dominates the radial friction force and pushes outward the lower fluid. At Re = 250, the radial velocity is positive at the entire range of r and its magnitude is significantly larger than that at Re = 150.

Figure 6 depicts streamline patterns at horizontal cross sections z = constant in the lower fluid (3 mm below the interface, row L) and in the upper fluid (3 mm above the interface, row U) at Re = 150 (left column) and 250 (right column). At Re = 150, the spirally converging lower fluid drives the spirally converging upper fluid that is intuitively clear. In contrast, at Re = 250, the spirally converging lower fluid drives the spirally diverging upper fluid that looks counterintuitive and indicates that the centrifugal force dominates the radial friction force above the entire interface.

It is instructive to compare the topological changes in the flow driven by the rotating lower disk (case LD) with those occurring in the flow driven by the rotating upper disk (case UD, see Fig. 6 in Ref. 10). In both cases, (a) the new CC cell emerges in the fluid, which is separated from the rotating disk, and (b) the CC cell expands and occupies almost the entire fluid volume while the AC cell shrinks into a thin ring located near the sidewall.

The important qualitative differences are: (a) the AC ring is located near the still disk in case LD while near the interface in case UD, (b) the counterflow occurs around the entire interface in case LD while around only a part of the interface in case UD, and (c) the Reynolds numbers, where the topological changes occur, are larger in case LD than those in case UD. These differences occur because the heavy fluid drives the light fluid in case LD, while the light fluid drives the heavy fluid in case UD.

A possible physical mechanism of the slip is the competing of two factors driving the fluid separated from the rotating disk: (A) the convergence of the driving fluid near the interface and (B) the rotation of the driving fluid near the interface. Factor A tends to entrain the separated fluid to converge to the axis while in contrast, factor B tends to drive it outward. At small Re, the fluid rotation rapidly decays away from the rotating disk, while the meridional motion decays not so rapidly. Therefore, factor A dominates factor B at the interface and drives AC in the separated fluid.

As Re increases, factor B becomes dominating factor A. Initially, this dominance only occurs near the axis where the centrifugal force generates the local CC cell. Next, factor B becomes dominating factor A at the entire interface in case LD, widening the CC up to the sidewall and shrinking AC to the ring. In other words, the growing centrifugal force overcomes the friction force (resulting in the counterflow slip development) at the entire interface. Therefore, the slip is more prominent in case LD (where the bottom disk rotates) in comparison with case UD (where the upper disk rotates).

We now add some speculations on what condition should replace the continuity condition. Since the upper and lower fluids have different densities and viscosities, i.e., ρ and ν have jumps at the interface, the azimuthal V_ϕ and radial V_r velocities can also have jumps there. Suppose that the angular momentum ρrV_ϕ conserves across the interface. This yields that V_ϕo = V_ϕgρ_g/ρ_o, i.e., the azimuthal velocity also has a jump that is supported by the experiments.¹⁵ This jump only modestly increases V_ϕo compared to V_ϕg because ρ_g is only slightly larger than ρ_o. In contrast, the radial velocity changes even its direction. This difference can be due to the centrifugal force, f = ρV_ϕ²/r, which also has a jump at the interface and f_o > f_g. The relation between the difference, df = f_o − f_g, and the slip velocity, V_s = V_ro − V_rg, can be similar to the relation between Friction force (replaced by df) and Normalized velocity (replaced by V_s) for lubricated solids,¹⁷ as shown in Fig. 7.

As the driving force increases from zero up to some threshold value, no slip occurs. As slip emerges, the driving force initially decays, reaches its local minimum, and then grows as the slip velocity increases (Fig. 7). The reversed relation—V_s(df)—has a hysteresis loop since there is the range of df where three different values of V_s exist at each value of df. A similar hysteresis was experimentally observed in the oil–water swirling flow.¹¹ Note that the interface can behave as solid, because surface active impurities could be present in liquids and concentrate at the interface.

The centrifugal mechanism of counterflow slip differs from known mechanisms of slip caused by rough boundaries,¹⁸ hydrophobic surfaces,¹⁹ and surfactants.²⁰ A condition at the interface, which should replace the continuity of velocity, is a subject of further research. For the surfactant slip, the molecular dynamics technique is used to find the interface condition.^21–23 Unfortunately, these works only focus on the plane Couette flow, where the centrifugal force is not involved.

Our concluding statement is that the counterflow slip at the interface due to the centrifugal force is a generic phenomenon for swirling two-fluid flows. The slip occurs when either the upper or lower disk rotates in a vertical cylindrical container. The slip is more prominent when the lower disk rotates where the slip occurs on the entire interface and for a wide range of the Reynolds number. Therefore, this flow can serve as a benchmark for further studying the interface physics and boundary conditions.