Angle Inclination Effect on Vortex Breakdown Zone in Rotating Flow Inside a Vertical Conical Container

Numerous studies have focused on rotating flows in the last several years, driven by their industrial significance and application across various engineering domains. These flow patterns are encountered in various engineering applications, including viscometers, turbo machinery, and crystal production using the Czochralski growth method [1-4]. The first publication dedicated to such flow patterns appeared in 1921 and was presented by Von Karman [5], one of the eminent figures in fluid mechanics. He examined the problem of fluid flow induced by the rotation of an infinite disk, where the fluid far from the disk remains stationary. By applying the principle of similarity, he could transform the system of Navier-Stokes equations into a pair of nonlinear equations in axial coordinates. If we denote the fluid's rotation speed far from the disk as Ωf and the disk's rotation speed as Ωd, Von Karman investigated the case (Ωf=0, Ωd≠0). Subsequently, Bodewadt [6] delved into the complementary problem, examining the flow of a viscous and incompressible fluid in uniform rotation on a stationary disk (Ωf≠0, Ωd=0).

In his experiments, Vogel [7, 8] investigated the fluid flow in a vertical cylinder generated by the rotation of a bottom disk. He determined a curve illustrating the boundaries within which the vortex known as "breakdown" would appear, representing the stability limits in the (H/R, Re=ΩR2/ν) plane, where H is the cylinder's height, R is the radius, and Re is the rotational Reynolds number using the same geometric configuration as considered in Vogel's work, with H/R=1.59 and ΩR2/ν=1.58.

Using the laser-induced fluorescence visualization method, Escudier [9] observed the steady swirling flow of a fluid (glycerin/water) in a cylindrical chamber with a rotating bottom. The author highlighted the emergence of one, two, and three vortex breakdown bubbles. A stability map was developed by varying two parameters: the aspect ratio of the cylinder (H/R) and the rotational Reynolds number (ΩR2/ν).

In another study, Escudier et al. [10] presented numerical results for the flow within a truncated cone generated by the rotation of an end wall. Vortices were suppressed beyond a certain wall inclination angle for converging geometries (where the radius increases as one approaches the rotating end wall) and diverging geometries. Simultaneously, Moffat vortices of growing strength and extent appeared in the corner between the lateral and non-rotating end walls. In the case of diverging geometry, a recirculation zone emerged on the lateral wall and merged with the Moffat vortices. Streamline patterns facilitated the identification of coherent flow phenomena consistent with the calculated pressure variation around the domain's periphery.

Bessaih and colleagues [11-13] conducted a series of numerical studies in 1999, 2003, and 2009, investigating the stability of rotating liquid metal flows in cylindrical enclosures under magnetohydrodynamic (MHD) influence. Bessaih et al. [11] conducted a numerical study on the laminar MHD (Magneto Hydro Dynamic) flow of liquid metal within a cylindrical enclosure with an aspect ratio equal to 1. The upper wall of the enclosure is rotating, and the system is subjected to a vertical external magnetic field. The authors achieved a good agreement between asymptotic and numerical results, demonstrating that primary flow can be controlled by a careful selection of the electrical conductivity of the walls of the enclosure. Furthermore, Bessaïh et al. [12] conducted a numerical and analytical study of the stable laminar flow generated by a rotating disk at the top of a cylinder with an aspect ratio H/R equal to unity. The cylinder is filled with liquid metal and subjected to an axial magnetic field. The authors solved the governing equations of the flow using a finite volume method. They provided various analytical velocity formulas and demonstrated that, in the presence of a magnetic field, the analytical velocity profiles near the end disks, obtained for a high value of the magnetic interaction parameter N, exhibit an excellent agreement with those obtained from numerical simulations. Thus, they highlighted the significance of the electrical conductivity of the container walls (both disks and the side wall) in controlling the flow. Their research also explored how magnetic fields and wall conductivity affected flow patterns and heat transfer, especially for high Prandtl number flows [13]. Without a magnetic field, heat transfer behaved like a cylindrical tank with a propeller. In the presence of a magnetic field, significant heat transfer reduction occurred. Using magnetic fields to suppress convection during crystal growth is a well-established technique, with ongoing discussions involving experiments and numerical simulations.

de Moro Martins et al. [14] conducted a numerical simulation investigating vortex breakdown in constrained rotating flows. They examined two enclosed geometries, cylindrical and conical-cylindrical, with flow rotation initiated by a continuously rotating bottom wall. The study initially validated their computational code by comparing numerical results with experimental data. Subsequent simulations explored the transitional regime and vortex breakdown formation by varying parameters like the Reynolds number and aspect ratio. The investigation encompassed steady and transient scenarios, considering turbulence and non-turbulence models. Interestingly, distinctions emerged in the vortex breakdown process within conical-cylindrical geometries compared to cylindrical geometries.

Mahfoud et al. [15] carried out a numerical investigation into the flow resulting from the counter-rotation of the upper and lower disks inside a truncated conical enclosure filled with liquid metal under the influence of an axial magnetic field. The upper and lower disks rotated in opposite directions at constant angular velocities Ω. Their research uncovered that the external magnetic field effectively restrained fluid motion, prompting a shift to axisymmetric flow. Furthermore, they noted that the critical thresholds (Recr) corresponding to m=1 modes increased with higher Hartmann numbers (Ha). Lastly, they constructed a diagram illustrating the stability (Re-Ha) based on the numerical insights from this study.

In this research, we conduct a 3-D numerical analysis of the rotating flow within a vertical conical cavity. The container contains a liquid metal with a low Prandtl number (Pr=0.015). The upper disc is subjected to a fixed angular velocity, while the lower disc remains stationary. The container's sidewall is rigid and inclined. The study encompasses a range of Reynolds numbers (1000≤Re≤2500) and various aspect ratios (1≤H/R≤2.4).

Moreover, we examine three distinct angles of inclination (α=5°, 10°, and 15°). The primary objective of our study is to explore how these variables influence the emergence and suppression of vortex breakdown. Additionally, we aim to identify the critical Reynolds number (Rec) at which vortex breakdown occurs for different aspect ratios (A=H/R) and various inclination angles (α).

The equations described in references 1-2 are resolved by applying the finite volume method outlined by Bodewadt [6]. Specifically, the vector components (u, v, and w) are discretely stored at staggered locations, while the scalar variable (P) is situated at the central points of these discrete volumes. To handle the convection term, a second-order central difference scheme is employed, as suggested in reference 20. The SIMPLER algorithm, detailed in the study of Bodewadt [6], also manages the coupling between pressure and velocity. The convergence criteria is declared met for a given time step when the maximum difference between consecutive iterations becomes less than 10e-6 at that particular time step.

The findings from numerical simulations are showcased for the stable flow generated by the rotation of the upper disk within a conical vertical cylinder containing a viscous fluid. Several aspect ratios ranging from 1 to 2.4, along with three different inclination angles (α=5°, 10°, and 15°), have been taken into account. It has been observed that, for aspect ratios 1.2≤H/R≤1.6, the inclination of the wall leads to the earlier onset of vortex breakdown, whereas for aspect ratios 1.8≤H/R≤2.2, the inclination of the wall leads to the suppression of vortex breakdown. It is important to emphasize that the fundamental flow characteristics were initially examined by Von Kármán [5]. In his analysis, the bottom disk, which rotates, functions as a pump by drawing fluid axially and expelling it outward in a spiral mode; within a confined cylinder, this fluid follows a spiral path along the cylindrical wall as it enters through the stationary top wall, subsequently changing its direction to the axial flow as within the Reynolds number range associated with it approaches the rotating bottom wall. This spiral motion initially leads to increased swirl velocity due to the conservation of angular momentum, ultimately forming a concentrated vortex. Additionally, the results are presented the appearance of the recirculation bubble. The calculations were conducted for various inclination angles to investigate their influence on the position and disappearance of vortex breakdown.

5.1 Inclination effect of sidewall

To demonstrate the influence of the conical shape of the cylinder (the inclination of the cylinder's sidewall) on the flow streamlines and the occurrence or suppression of vortex breakdown, we present in Figure 4 the evolution of streamlines plotted in meridian planes (XZ) for a fixed value of Reynolds number (Re) of 2000 and three different conical.

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Figure 4. The evolution of streamlines in meridian planes (XZ) at Re=2000 for three different conical shapes with α=5°, 10°, and 15°, and various aspect ratios A=H/R: 1.4, 1.6, 1.8, and 2.0, respectively

Shapes characterized by the inclination angle α=5°, 10°, and 15°, along with aspect ratios A=H/R: 1.4, 1.6, 1.8, and 2.0, respectively. It is observed that for the case of an aspect ratio A=1.4 when the inclination angle increases from α=10° to 15°, the vortex breakdown appears, as illustrated in the second row of Figure 4. In the case of an aspect ratio A=1.6, an increase in the inclination angle from α=10° to 15° reveals a second set of vortex breakdowns, as shown in the second row of Figure 4. Conversely, for both cases with aspect ratios A=1.8 and A=2, increasing the inclination angle leads to the disappearance of vortex breakdown, as illustrated in the third and fourth rows of Figure 4, respectively. It is also noticeable that, for inclination angles α=5° and α=10°, increasing the aspect ratio leads to the emergence of vortex breakdown. However, at an inclination angle of α=15°, increasing the aspect ratio to A=1.6 results in the appearance of a second set of vortex breakdowns. However, the vortex breakdown disappears when the aspect ratio reaches A=1.8 and beyond. In summary, for aspect ratios A<1.6, an increase in the inclination of the sidewall leads to vortex breakdown. In contrast, for aspect ratios A>1.6, an increase in the inclination of the sidewall leads to the suppression of vortex break down. This can be explained by considering that, for aspect ratios A<1.6, the propagation or dissipation of kinetic energy within the fluid induced by the rotating disc becomes more significant as the inclination of the sidewall increases. Conversely, for aspect ratios A>1.6, the dissipation of kinetic energy within the fluid supplied by the rotating disc becomes less significant as the inclination of the sidewall increases.

5.2 Ratio aspect effect

To further illustrate the influence of aspect ratios A=H/R on the behaviors of the appearance, position, or disappearance of vortex breakdown in a convergent conical geometry (α=10°), we can take, for example, Re=2000 with a convergent conical shape characterized by α=10°. By increasing the aspect ratios A=H/R from 1.2 to 2.2, as presented in Figure 5, showing the evolution of streamlines in meridian planes (XZ), The flow exhibits vortex breakdown for aspect ratios within the range of 1.6≤A=H/R≤2. however, when H/R<1.6 and H/R>2, no vortex breakdown was observed, as evidenced by the streamlines presented in the figure. This pattern closely aligns qualitatively with the findings reported by Mahfoud et al. [16].

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Figure 5. The evolution of streamlines in meridian planes (XZ) at Re=2000 and α=10° for various aspect ratios A=H/R: 1.4, 1.6, 1.8, 2.0, and 2.2

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Figure 6. Spatial evolution of zero axial velocity for various aspect ratios at Re=2000

Figure 6 presents the spatial configuration of zero axial velocity values at varying aspect ratios under Re=2000 and α=10°. In this scenario, vortex breakdown becomes apparent; the recirculation bubble behaves like an obstacle against the axial flow when 1.6≤A=H/R≤2. Flow is redirected around the bubble, giving rise to a downstream wake region characterized by significantly reduced velocities. Within the vortex breakdown bubble, the impact of viscosity and inertia is negligible, resulting in zero axial velocity. It is worth noting that no such bubble is observed for A=H/R values below 1.6 or above 2.

5.3 Reynolds Re effect

5.3.1 Case A=H/R=1.4

For this low aspect ratio A=1.4, simulations were conducted for three different inclination angles, namely α=5°, α=10°, and α=15°, with a gradual increase in Reynolds numbers ranging from 1000 to 2000. The objective was to analyze the hydrodynamic evolution of the flow and determine the thresholds at which bifurcations in the flow structure occur. Figure 7 displays streamlines plotted in the cylinder's meridian plane (XZ).

For each scenario, as the Reynolds number (Re) increases, the deformation of streamlines becomes more pronounced, accompanied by the observation of secondary circulation within each half of the meridian planes. This phenomenon signifies fluid circulation induced by the Ekman pumping effect and the emergence of a pair of vortex breakdowns, where the axial flow is redirected around the recirculation region. This redirection is evident through the divergence of streamlines surrounding the bubble. With a further increase in Re, there is a transformation where the recirculation bubble collapses, forming a spiral column vortex along the axis. This transformation is illustrated in Figure 7 for two specific cases: Re=2000 and α=5°, as well as Re=2000 and α=10°.

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Figure 7. The evolution of streamlines in meridian planes (XZ) at aspect ratios A=H/R=1.4 for three different conical shapes with α=5°, 10°, and 15°, and various values of Re

The graph presented in Figure 8 illustrates the critical Reynolds values, denoted as Rec, at which vortex breakdown occurs for each inclination angle α, specifically α=5°, α=10°, and α=15°. According to the diagram of Figure 8, Rec increases with an increase in the inclination angle α. The critical Reynolds values, denoted as Rec, marking the onset of vortex breakdown are 1140, 1210, and 1370 for α=5°, α=10°, and α=15°, respectively. These critical thresholds were established by incrementally raising the Reynolds number.

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Figure 8. Critical Reynolds $R{{e}_{c}}$ thresholds at which vortex breakdown occurs for A=H/R=1.4 and various value of angle of inclination α=0°, α=5°, α=10°

5.3.2 Case A=H/R=1.8

For this medium aspect ratio A=1.8, simulations were conducted for three different inclination angles, namely α=5°, α=10°, and α=15°, with a gradual increase in Reynolds numbers ranging from 1000 to 2500. However, for the case of α=15°, calculations were stopped at Re=2300 because the flow became unstable beyond this value, and convergence could not be achieved.

Figure 9 illustrates the path lines of surface streamlines in a meridian plane XZ for A=1.8. In the cases of α=5° and α=10°, an increase in the Reynolds number initially leads to the appearance of a first pair of vortex breakdown, and with further increases in Reynolds number Re, a second pair of vortex breakdown emerges, as depicted in Figure 9 for α=5° and Re=2000, and α=10° and Re=2000.

The second recirculation bubble takes shape further downstream of the initial one, featuring a greater axial span yet a reduced diameter compared to the first bubble in the scenario where α equals 5 degrees, and the Reynolds number (Re) is elevated to 2500, these recirculation bubbles dissipate. Conversely, when α is 10 degrees, the downstream recirculation bubble draws nearer to the upstream counterpart, leading to their fusion into a unified recirculation bubble comprised of two separate segments, as illustrated in Figure 9 for Re=2500 and α=10°.

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Figure 9. The evolution of streamlines in meridian planes (XZ) at aspect ratios A=H/R=1.8 for three different conical shapes with α=5°, 10°, and 15°, and various values of Re

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Figure 10. Critical Reynolds $R{{e}_{c}}$ thresholds at which vortex breakdown occurs for A=H/R=1.8 and various value of angle of inclination α=0°, α=5°, α=10°

In the case of α=15°, it's noteworthy that a single pair of vortex breakdown appears at a significant Reynolds value of 2300.

Figure 10 presents the stability diagram in a Rec-α plane, highlighting the thresholds at which vortex breakdown occurs. It is observed that Rec increases progressively with the inclination angle α. The critical Reynolds values Rec for the appearance of vortex breakdown are 1470, 1650, and 2300 for α=5°, α=10°, and α=15°, respectively.

5.3.3 Case A=H/R=1.4

For this higher aspect ratio A=2.2, simulations were conducted at three distinct inclination angles: α=5°, α=10°, and α=15°. The Reynolds numbers were gradually increased from 1000 to 2500. However, in the case of α=15°, the calculations were halted at Re=2200 due to the onset of flow instability, making convergence unattainable.

Figure 11 depicts the trajectories of surface streamlines in the meridian XZ plane for A=2.2. In the case of α=5°, an initial increase in the Reynolds number results in the appearance of a primary pair of vortex breakdowns.

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Figure 11. The evolution of streamlines in meridian planes (XZ) at aspect ratios A=H/R=2.2 for three different conical shapes with α=5°, 10°, and 15°, and various values of Re

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Figure 12. Critical Reynolds $R{{e}_{c}}$ thresholds at which vortex breakdown occurs for A=H/R=2.2 and various value of angle of inclination α=0°, α=5°, α=10°

Subsequently, with further increments in the Reynolds number Re, a secondary pair of vortex breakdowns emerges, as illustrated in Figure 11 for α=5° and Re=2000, as well as α=5° and Re=2300. The second recirculation bubble develops downstream of the first, exhibiting a longer axial extent and a larger diameter than the initial recirculation bubble. Notably, even with the increase in Reynolds number to Re=2500, the flow structure had no significant alteration.

In the case of α=10°, an increase in Re-up to 2500 results in the appearance of a single recirculation bubble. Conversely, no vortex breakdown is observed for the case of α=15°.

Figure 12 presents a stability diagram in the Rec-α plane, highlighting the critical thresholds at which vortex breakdown occurs. It is also observed that Rec increases progressively with the inclination angle α. The critical Reynolds values Rec for the appearance of vortex breakdown are 2000 and 2300 for α=5° and α=10°, respectively.