# Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{5}). The Large Eddy Simulation (LES) is adopted as the numerical approach, with a localized dynamic Subgrid-Scale Stresses (SGS) model including an energy equation. Since the flow obviously requires a surface tracking or capturing method, a volume-of-fluid (VOF) approach has been chosen based on the findings, where this method provided stable shapes in the ranges of parameters found in the corresponding experiments. Expected ellipse and triangle shapes are revealed and analyzed. A detailed character of the numerical results allows for an in-depth discussion and analysis of the mechanisms and features which accompany the characteristic shapes and their alterations. As a result, a unique insight into the polygon flow structures is provided.

## 1. Introduction

#### 1.1. Experimental Studies

^{2}/ν, where ω is the disk angular velocity, R is the cylinder radius and ν is the kinematic viscosity) were identified [8]. The results from the above studies indicate that the creation of the polygons, their shapes, stability and rotating velocity depend on the geometry of the experiment and on the Reynolds number. In an additional communication, the use of the Froude number, defined as Fr = ωr/(gr)

^{0.5}, where r is the bottom disk radius, was suggested in [12], which looks quite reasonable because free surface flows are considered. With the same experimental device and flow parameters as of [8], the free surface cross section transition by visualizing the flow pattern using anisotropic flakes was studied in [13]. The vertical cross section along the centerline of the cylinder was recorded. The orientation of the flakes, which were arranged along the principal strain in the flow, was used to observe the flow patterns. The measurements of the free surface level at the cylinder axis and the radial velocity profile, and the computation of the turbulent intensity as function of Re, were reported in [14].

#### 1.2. Analytical Modeling

#### 1.3. Numerical Modeling

^{5}, as described in detail above. However, to the best of our knowledge, only very few studies have started to go in that direction. Moreover, significant simplifications continue to be used. Experiments with small h/R ratio (see Table 1) were conducted in [9]. A polygonal flow structure was reported to be created by spiraling vortices in the corners of the polygon. Experiments reported in [9] were modeled in [43] with direct numerical simulation (DNS) at Re up to 27,000 without free surface distortion. The model predicted well the first symmetry breaking from the experimental work of [9] for large h/r, but large discrepancies were obtained for small h/r. This was first attributed to the assumption of the flat free surface, but as was shown in [44] adding the deformation of the free surface did not improve the results. In [43], it was suspected that surface tension might be the cause for this disagreement. Following the definition given in [45] for the critical Reynolds number (Re

_{NM}= VL/ν) that governs the instability, they showed that, for every Reynolds number, Re

_{NM}remains constant by taking the characteristic velocity, V, as maximum velocity of the disk and the characteristic length, L, as the thickness of the shear layer, estimated as L = (E

_{h}/4)

^{1/4}h, where E

_{h}= ν/(ωh

^{2}) is the Ekman number.

## 2. Numerical Modeling

#### 2.1. Choice of the Numerical Approach

^{5}, direct numerical simulation (DNS) is impractical. Employing Reynolds-averaged Navier-Stokes (RANS) equations, unsteady simulations have proven to be unrealistic as, in a preliminary study, they revealed no asymmetry in the shape of the free surface (these results are not shown for the sake of brevity). It was thus decided to use Large Eddy Simulation (LES). The essential specific details of the adopted approach are presented below.

^{4}and the Froude numbers Fr = 0.2 and 0.8 were reported in [62]. A Lagrangian dynamic subgrid-scale model, complemented with LSM for capturing the free surface, was employed. The mean interface level and the root mean square (r.m.s.) of interface fluctuations from the simulation were reported to be in excellent agreement with the experimental results. Along with VOF and LSM methods, a number of hybrid techniques were developed for interface capturing. The coupled level set/volume-of-fluid (CLSVOF) method, as developed in [63], is applied when the free surface is expected to be highly complex. Results of a very recent work which deals with mechanisms of air-core vortex evolution in an intake flow are reported in [64].

#### 2.2. Computational Details

_{ν}), introduced in [68], as the criterion. The LESIQ

_{ν}is a dimensionless number between zero and unity, reflecting the ratio between resolved and total turbulent kinetic energy. According to [68,69], based on [70], an index of quality greater than 0.8 is considered a good LES, while 0.95 and higher may be considered a DNS. Figure 2a,b show the contours of instantaneous values of LESIQ

_{ν}for the fastest case of the present investigation, namely, of 4.6 Hz. Figure 2a shows the values of LESIQ

_{ν}for the free surface of the liquid, while Figure 2b shows its values in the vertical cross-section of the flow. One can see from the figures that LESIQ

_{ν}is above 0.8 practically everywhere, and in most locations, it is close to or above 0.9. A parameter, which may be used for ensuring temporal resolution of the simulation, is the Courant number. One can see from Figure 2c, for the same complex case of Figure 2b, that the Courant number is smaller than unity practically all over the simulated flow field. The only exceptions are the “corners”, i.e., the locations where the rotating disk meets with the stationary wall, but this effect does not influence the outcome of the simulations. Calculation using a grid refined further by the factor of 1.5 in all the three directions (3.3 times global refinement) was done with no appreciable difference in the results.

^{−4}for the continuity equation, while the residuals were as low as about 10

^{−8}for the rest of the conservation equations solved.

## 3. Results and Discussion

#### 3.1. Flow Patterns

#### 3.2. Pattern Stability

#### 3.3. Velocity Field

#### 3.4. Fourier Analysis

_{0}is the average radius of the polygon; c

_{1}corresponds to the transition of the center of the shape; c

_{2}represents an ellipse and c

_{3}a triangle. In Figure 13 and Figure 14, the dominant and adjacent coefficients are presented for each rotating disk frequency. In order to juxtapose the values of the coefficients and the visual representation of the free surface shapes, Figure 13a and Figure 14a are accompanied with top views at certain times. The dashed lines show the time interval chosen for the free surface level contours presented in Figure 4 and Figure 5.

_{0}, c

_{2}and c

_{3}computed at the surface of the 3.4 Hz frequency, as function of time. It can be seen that c

_{0}, which corresponds to the average radius of the polygon, is the most dominant coefficient; c

_{2}is the next most dominant coefficient, suggesting that the polygon shape is closer to an ellipse; c

_{3,}which corresponds to a triangle, is noticeably smaller than c

_{2}. All other coefficients are smaller than c

_{3}and therefore are not presented. The ellipse shapes remain dominant in all computation times. This shows that the ellipse patterns are very stable. The dashed lines show the time interval selected for Figure 4, and first and last figures in Figure 4 are also presented in Figure 13a. Figure 13b shows the dependence of the height on the Fourier coefficients. At each height, the average coefficient in time (solid line) is plotted together with its standard deviation (dashed lines). Since c

_{0}is larger than c

_{2}and c

_{3}by about an order of magnitude, the left vertical axis corresponds to c

_{0}and the right vertical axis corresponds to c

_{2}and c

_{3}. It can be seen that as the height above the disk increases, the average radius of the shapes becomes larger and the ellipse coefficient, c

_{2}, becomes smaller. This corresponds to the results in Figure 6 where the ellipse transforms into a larger circle when the height increases. In addition, the standard deviations decrease with the height, indicating that the shapes are more stable as the height increases. Accordingly, the widths of the contours (“braids”) in Figure 6 decrease with the height.

_{3}become higher than c

_{2}. On the other hand, coefficient c

_{4}that corresponds to the next polygon (square) is much smaller than c

_{3}, too. This shows quantitatively that the dominant pattern is triangle, as it was shown in Figure 5. In addition, all coefficients remain around their certain values for all computation time. This shows again that the patterns are stable. The dash lines relate Figure 14 to the first and last shapes in Figure 5. Similar to Figure 13b, Figure 14b shows that the shape is getting larger with the height while c

_{3}, which is the dominant coefficient (except of course c

_{0}), is getting smaller.

_{0}and c

_{3}, obtained numerically in the present study for 4.6 Hz where the triangle polygon shape prevails, was previously reported in [21]—see Figure 4 in that work. The triangle polygon shapes reported in [21] for the initial water height of 40 mm and the rotating disk frequency of 2.4 Hz (Figure 3 in their paper) closely resemble the shapes presented in Figure 5 of the present study. This points at a strong similarity between the triangle shapes obtained experimentally and numerically.

_{2}(see Figure 13), the rotation rate of the polygon is computed by dividing the dominant frequency, 2.07 Hz, by the two polygon “corners” (N = 2), which results in a rotation rate of 1.035 Hz and a ratio of 0.304 between the rotation velocities of the polygon and the disk. The second dominant frequency, 4.11 Hz by the results of Figure 15, corresponds to the next polygon—a triangle. Then, additional peaks, corresponding to higher values of N, are observed. The relative magnitude of the second peak may indicate that the disk rotating frequency is close to a transition between an ellipse and a triangle, as indeed was demonstrated by [10] and can be seen from Figure 3. It may be noted that the computed ratio of frequencies in our case is also close to the value observed by [19].

_{0}) is c

_{3}, and its peak frequency from the computation, 4.13 Hz, see Figure 16, should be divided by three, resulting in the ratio of 0.299 between the polygon and the disk rotation velocities. Again, this is very close to the results of [10], where it is reported that the ratios should be within 1/4 to 1/3 [15,19] (see Figure 5 in [19]). It is curious that for the triangular shape, Figure 16, the next peak is smaller than that corresponding to the one at an adjacent higher frequency. It may be speculated that the circles observed in [10], between the triangles and squares, and also maybe those above the squares, are in fact a superposition of a number of polygon shapes. It should be stressed that this study has focused on the ellipse and triangle shapes, hence the higher-order Fourier coefficients (beyond c

_{3}) would be explored in a future study.

## 4. Conclusions

^{5}, were modeled numerically for the first time in the literature. The parameters and features of this work were chosen based on a scrupulous examination of the existing knowledge, which was extensively presented and analyzed in detail. Accordingly, the cylinder inner radius was 145 mm and the initial water height in it was 60 mm. Following some representative experiments from the literature, bottom disk rotation frequencies of 3.0, 3.4, 4.0 and 4.6 Hz, corresponding to ellipse and triangle shapes, were chosen for the present simulation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The system addressed in the present study: (

**a**) physical model and an example of a polygon created on the disk surface; (

**b**) computational domain and grid (front and top views).

**Figure 2.**Evaluation of Large Eddy Simulation (LES) quality for a typical instant flow field: (

**a**) LES index of quality (LESIQ

_{ν}) for a free distorted surface shown in top view; (

**b**) LESIQ

_{ν}for a vertical cross-section of the same flow field; (

**c**) the Courant number for the same cross-section.

**Figure 3.**Circumferentially and time-averaged water height, simulated in the present study, vs. measurements from the literature.

**Figure 4.**Top view of the free surface height above rotating disk for 3.4 Hz, from top left to bottom right, equally spaced within about 14 s of computation.

**Figure 5.**Top view contours of the free surface height above rotating disk for its rotation frequency of 4.6 Hz, from top left to bottom right, equally spaced within 14 s of computation.

**Figure 6.**Free surface contours at six heights for 3.4 Hz disk rotation frequency. Each contour is rotated to the contour of t = 0 s using maximum correlation between the shapes. For each height 1085 contours are plotted representing results for 108.5 s.

**Figure 7.**Free surface contours at six levels for 4.6 Hz disk rotation frequency. Each contour is rotated to the contour of t = 0 s using maximum correlation between the shapes. For each height 360 contours are plotted, representing results for 36 s.

**Figure 8.**Top view of the radial, axial (both in m/s) and normalized tangential average velocities at four heights (from top to bottom: 5, 10, 30 and 60 mm) above the 3.4 Hz rotating disk. Each contour is rotated to the contour of t = 0 s using maximum correlation between the shapes. Velocity values are averaged if the specific location was occupied by water more than 50% of the frames, otherwise white color is assigned.

**Figure 9.**Front view contours of the radial, axial (both in m/s) and normalized tangential velocities in the two vertical cross-sections corresponding to the main axes of the ellipse at the disk rotation frequency of 3.4 Hz. Each velocity field is averaged over 950 frames which represent 95 s of a transient.

**Figure 10.**Top view contours of the radial, axial (both in m/s) and normalized tangential velocities at four heights (from top to bottom: 5, 10, 30 and 60 mm) above the 4.6 Hz rotating disk. Each contour is rotated to the contour of t = 0 s using maximum correlation between the shapes. Velocity values are averaged if the specific location was occupied by water more than 50% of the frames, otherwise white color is assigned.

**Figure 11.**Front view contours of the radial, axial (both in m/s) and normalized tangential velocities for the vertical cross-section of the triangle, at the at the disk rotation frequency of 4.6 Hz. Each picture represents averaging of 1095 planes which are taken over the time span of 36.5 s (3 planes for every 0.1 s).

**Figure 12.**Instantaneous tangential velocity contours: (

**a**) elliptical polygon for 3.4 Hz, (

**b**) triangular polygon for 4.6 Hz.

**Figure 13.**The Fourier coefficients, c

_{0}to c

_{3}for 3.4 Hz rotating disk frequency: (

**a**) the Fourier coefficients as function of time at the surface of the rotating disk. (

**b**) The average and standard deviation of the Fourier coefficients for the time interval presented in (

**a**) at different heights above the rotating disk. The left vertical axis corresponds to c

_{0}and the right vertical axis corresponds to c

_{2}and c

_{3}.

**Figure 14.**The Fourier coefficients, c

_{0}to c

_{4}, for 4.6 Hz rotating disk frequency: (

**a**) the Fourier coefficients as function of time at the surface of the rotating disk. (

**b**) The average and standard deviation of the Fourier coefficients for the time interval presented in (

**a**) at different heights above the rotating disk. The left vertical axis corresponds to c

_{0}and the right vertical axis corresponds to c

_{2}and c

_{3}.

r mm | R mm | h mm | ω rad/s | ν × 10^{6} m ^{2}/s | Re = ωr^{2}/ν | Fr = ωr/(gh)^{0.5} | h/R | r/R | Detectable Polygons—N | |
---|---|---|---|---|---|---|---|---|---|---|

Vatistas [4] | 126 | 142.5 | 150 | 25.1 | 1 ^{a} | 3.99 × 10^{5} | 2.61 | 1.05 | 0.88 | No, periodic sloshing without surface exposed |

126 | 142.5 | 293 | 81.7 | 1.30 × 10^{6} | 6.07 | 2.06 | 0–2, periodic sloshing with surface exposed from circular to elliptical cross section | |||

101 ^{a} | 142.5 | 63 | 10.5–31.4 | 1.07 × 10^{5}–3.20 × 10^{5} | 1.35–4.04 | 0.44 | 0.71 | 0–6 | ||

69 | 1.29–3.86 | 0.48 | 0–5 | |||||||

Vatistas et al. [6] | 126 | 142 | 24, 27 | 5.2–25.1 | 1 | 8.31 × 10^{4}–4.16 × 10^{5} | 1.28–6.80 | 0.17–0.19 | 0.89 | 0–6 |

5 | 18.8–31.4 | 56.7 | 5.28 × 10^{3}–8.80 × 10^{3} | 10.72–17.87 | 0.04 | 5–11 | ||||

7 | 15.7–25.1 | 4.40 × 10^{3}–7.04 × 10^{3} | 7.55–12.08 | 0.05 | 5–10 | |||||

10 | 19.9–30.4 | 5.57 × 10^{3}–8.50 × 10^{3} | 8.00–12.22 | 0.07 | 4–8 | |||||

22 | 15.7–27.2 | 4.40 × 10^{3}–1.09 × 10^{5} | 0.61–7.38 | 0.15 | 0–8 | |||||

>100 | >83.8 | 640 | >2.08 × 10^{3} | >10.66 | 0.70 | 0 | ||||

Jansson et al. [7] | 144 ^{b} | 145^{b} | 25–100 | 3.1–44.0 | 1 ^{c} | 6.51 × 10^{4}–9.12 × 10^{5} | 0.46–12.79 | 0.17–0.69 | 0.99 | 0–6 |

15 | 4.34 × 10^{3}–6.08 × 10^{4} | 0–3 | ||||||||

193 | 194 | 25–95 | 3.1–31.4 | 1 ^{c} | 1.17 × 10^{5}–1.17 × 10^{6} | 0.61–12.24 | 0.13–0.52 | 0–5 | ||

Suzuki et al. [8] | 42 | 42.5 | 40 | <146.4 | 1 | <2.58 × 10^{5} | <9.82 | 0.94 | 0.99 | 0–2, surface exposed from circular to elliptical cross sections |

Poncet and Chauve [9] | 140 | 140.85 ± 0.05 | 2.5–15 | 0–6.2 | 1 | ≤1.21 × 10^{5} | 2.25–5.52 | ≤0.11 | 0.99 | 0–8 without 2 |

Bach et al. [10] | 144 | 145 | 30-80 | 2.5–40.8 | 1 | 5.21 × 10^{4} – 8.47 × 10^{5} | 0.41–10.48 | 0.21–0.55 | 0.99 | 0-5 |

Ait Abderrahmane et al. [11] | 140 | 142 | 20, 30, 40 | 7.9–28.0 | 1-22 ^{d} | 7.00 × 10^{3}–5.48 × 10^{5} | 1.76–8.84 | 0.14–0.28 | 0.99 | 0–6 |

Present study | 145 | 145 | 60 | 18.8, 21.4, 25.1, 28.9 | 1 | 3.96 × 10^{5}–6.08 × 10^{5} | 3.56–5.46 | 0.41 | 1 | 0–3 |

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**MDPI and ACS Style**

Rashkovan, A.; Amar, S.D.; Bieder, U.; Ziskind, G. Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom. *Appl. Sci.* **2021**, *11*, 1348.
https://doi.org/10.3390/app11031348

**AMA Style**

Rashkovan A, Amar SD, Bieder U, Ziskind G. Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom. *Applied Sciences*. 2021; 11(3):1348.
https://doi.org/10.3390/app11031348

**Chicago/Turabian Style**

Rashkovan, A., S.D. Amar, U. Bieder, and G. Ziskind. 2021. "Analysis of Polygonal Vortex Flows in a Cylinder with a Rotating Bottom" *Applied Sciences* 11, no. 3: 1348.
https://doi.org/10.3390/app11031348