Numerical Investigation of the Effect of the Ratio of the Vortex-Finder

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Numerical Investigation of the Effect of the Ratio of the VortexFinder Diameter to the Spigot Diameter on the Steady State of the Air Core in a Hydrocyclone Yanxia Xu, Xingfu Song,* Ze Sun, Bo Tang, Ping Li, and Jianguo Yu* National Engineering Research Center for Integrated Utilization of Salt Lake Resources, East China University of Science and Technology, Shanghai, China ABSTRACT: The air core is an important phenomenon in a hydrocyclone. The steady state of the air core plays a key role in the performance and separation efficiency of a hydrocyclone. The unsteady structure of the air core reflects the unsteady flow field and exacerbates this unsteadiness, thus affecting the steadiness and homogeneity of the flow field, which will lead to lower separation efficiency. The effects of the ratio of the vortex-finder diameter to the spigot diameter (Do/Du) on the steady state of the air core were studied by computational fluid dynamics in this paper. In this approach, the Reynolds stress model used to describe the turbulent fluid flow and the volume-of-fluid multiphase model simulated the interface between the liquid and air core. The results show that the steady state of the air core is mainly decided byDo/Du. Different ranges of Do/Du lead to different steady states of the air core. No air core appeared inside the hydrocyclone with a Do/Du of 1.2. The discontinuous ones occur when Do/Du enlarges to 1.6 and 1.8. Through the continuous air core grown in the hydrocyclone with a Do/Du larger than 1.8, the rules of the steady state were different. In addition, the operation conditions of the inlet velocity did not change the trend of the steady state of the air core in a certain hydrocyclone.

1. INTRODUCTION The cyclone1−3 is a centrifugal separation and classification apparatus, being developed for different industrial needs. It is used for solid−solid and solid−liquid hydrocyclone separation4−8 in a liquid-phase continuum; by means of the tangential-fluid and solid-particle feeding, a strong rotational movement takes place inside the device because of the centrifugal force. Thus, in this field, the solid particles are suspended in the fluid and tend to move toward the walls. It is clear that the flow behavior9 in this apparatus is quite complex, contrary to its simple design and operation. The air core generated inside the hydrocyclone is one of the most important and complex internal structures.10,11 The pressure decreases until its value is smaller than the atmospheric pressure by the high tangential velocity of the fluid and a negative pressure is created in the central axis, resulting in air being driven from the spigot to the vortex finder. The air core is a unique phenomenon in the hydrocyclone. The air-core diameter reflects the region of the negative pressure inside the hydrocyclone, as well as the separation zone, which affect the performance of the hydrocyclone. At the same time, the formation of the air core marks the normal working conditions of the hydrocyclone. These present the importance of the air core inside the hydrocyclone. Some research studied the hydrocyclone by neglecting the air core.12−14 However, since the middle of the last century, more and more studies focused on the air core. Binnie and Hookings’ pioneer research15 focused on the air-core diameter in a swirling flow from fundamental principles. Subsequently, Smith16 assumed a free vortex in viscid flow to determine the air-core diameter. Fontein et al.17 demonstrated that the aircore diameter increases with a vortex-finder diameter increase, but the effect of the vortex-finder diameter on the air-core © 2013 American Chemical Society

diameter changing with time was not referred to. With the development of calculations, Pericleous and Rhodes 18 determined the air core by modeling the collection of air bubbles along the center line using the PHOENICS fluid flow code. Moreover, empirical or semiempirical formulas and theoretical approaches were employed to study the air-core diameter. For example, Davidson19 analyzed the air-core diameter in a hydrocyclone using the physics of uniform density and inviscid flow at each outlet and modified it by an empirical factor to account for the viscous effects. Dyakowski and Williams20 used a method to predict the size of the air core within a hydrocyclone based on calculation of the internal pressure distribution by solving a set of conservation equations. At the same time, they obtained the results for the development of empirical correlations to describe the air-core characteristics as a function of the operational variables and their effects on the separation efficiency using electrical resistance tomography.21 Steffens et al.22 extended this to include the prediction of the air-core diameter by Rietema-type simplifications of the Navier−Stokes equations. Concha et al.23 developed an equation for the air-core diameter in a hydrocyclone in terms of the geometrical and operational parameters based on a phenomenological equation. Evidence suggested that it was difficult to specify the form and location of the air-core surface. To simplify this problem, many researchers have chosen to model the interface that bounds the air core as a fixed cylindrical surface and developed theoretical models to approximate the air-core diameter.24 Received: Revised: Accepted: Published: 5470

August 3, 2012 December 23, 2012 March 21, 2013 March 21, 2013 dx.doi.org/10.1021/ie302081v | Ind. Eng. Chem. Res. 2013, 52, 5470−5478

Industrial & Engineering Chemistry Research

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Computational fluid dynamics (CFD)25−30 provides a means to study the internal flow field (especially the air core) inside the hydrocyclone and has been widely applied. For example, Wang et al.31,32 and Mousavian and Najafi33 successfully presented the air-core structure inside the gas−liquid−solid multiphase flow, especially the air-core development with time. It is demonstrated that the interface between the liquid and the air core could be modeled by the volume-of-fluid (VOF) model. Sripriya et al.34 also studied the performance and flow characterization in the presence and absence of an air core. Evan et al.35 studied the flow in a hydrocyclone operating with an air core and with an inserted metal rod. Gupta et al.36 studied the mechanism of the air core and vortex formation in a hydrocyclone. Large-eddy simulation (LES) has also been taken into consideration. 37−39 Furthermore, Hararah et al. 40 presented measurements of the air-core diameter in transparent hydrocyclones operated with a suspension of glass spheres and indicated the dependencies between the feed-solids content, air-core diameter, and spray angle of the underflow. Davaille et al.41 thought that the air core was possible to model by using a solid cylinder with free slip boundary conditions, despite the unsteady nature of the air-core profile. However, these experimental investigations and theoretical calculations only studied the average or instantaneous diameter of the air core, as well as its structure. Few concerned the steady state of the air core, which means that the structure of the air core changes with time. However, the formation of the air core indicates the normal working state of the hydrocyclone but not the steady state of the flow field. The unsteady air core reflects the unsteady state of the flow field and exacerbates this unsteadiness, thus affecting the steadiness and homogeneity of the flow field, which will lead to low separation efficiency. Therefore, it is necessary to study the state of the air core inside the hydrocyclone, as well as the factors that influence its steady state. In this paper, a threedimensional mathematical model was built to investigate the effects of the ratio of the vortex-finder diameter to the spigot diameter (Do/Du) on the steady state of the air core. Furthermore, the effects of operation conditions on the steady state of the air core in the hydrocyclone with different Do/Du were also investigated.

Figure 1. Schematic (a) and grid (b) representations of the hydrocyclone.

Table 1. Structure Parameters of the Hydrocyclone parameter

symbol

body diameter inlet diameter vortex-finder length cylindrical-part length conical-part length

Dc Di Lv Lc Lp

dimension (mm) 140 50 100 150 372

cyclone flow. The VOF-free surface model can describe the interface between the liquid and air core in a hydrocyclone. In the VOF model used, gas and liquid, the two fluids considered in this work, share a single set of momentum equations, and the volume fraction of each of the fluids is tracked throughout the computational domain. The interface between the air core and liquid in a hydrocyclone is then determined as a transient process. More details about the mathematics model are seen in ref 45. The simulations were conducted using Fluent 6.3 software. Second-order upwinding and the SIMPLE pressure−velocity coupling algorithm were used. The convergence strategy used the unsteady solver, and the time step was chosen as 10 −4−5 × 10−4 s. Trial tests show that there was no sensitivity of the results to the time step in this range. In this work, the time step was chosen as 5 × 10−4 s. 2.2. Simulation Conditions. “Velocity inlet” boundary conditions were used at the hydrocyclone inlet, and both outlets used “pressure outlet” conditions. The pressures at the two outlets (vortex finder and spigot) of the hydrocyclone were both 1.0 atm. The inlet velocity was 2.000 m/s. Figure 1b shows the computational domain of the model. The whole computational domain was divided by 142562 unstructured hexahedron grids. The coordinate origin of the hydrocyclone was set in the center on the interface of the cylinder and cone sections. Trial numerical results demonstrated that the solution is independent of the mesh size used. 2.3. Model Validation. Before being applied, the reliability and robustness of the numerical tool that was used for the present study were validated by comparing the predicted and measured experimental flow fields including the tangential and axial velocity distributions at different heights of the hydro-

2. MATHEMATICAL MODEL 2.1. Model Description. The hydrocyclone in this work is shown in Figure 1a. Also, its main structure parameters are given in Table 1. Do/Du values range from 1.2 to 4. Inside the hydrocyclone, a liquid-phase continuum, by means of the tangential-fluid and solid-particle feeding, a strong rotational movement takes place inside the device because of the centrifugal force. In spite of the simple geometry and operation of the hydrocyclone, the detailed flow fluid and mechanisms are extremely complicated to explain. The key component for the flow field inside the hydrocyclone is the turbulence closure model in the description of the fluid dynamics of the hydrocyclone. Therefore, an appropriate turbulent model must be applied for characterization of the rotating turbulent flow. A number of models have been tried in the past.42−44 In this work, the fluid flow is modeled as turbulent, described by the Reynolds stress model (RSM), and the interface between the liquid and air core is modeled by the VOF model. Although LES shows considerable potential, it is enormously computationally expansive. Also, the RSM model used has proven to be an appropriate model to describe the anisotropic turbulence of 5471

dx.doi.org/10.1021/ie302081v | Ind. Eng. Chem. Res. 2013, 52, 5470−5478


Article

Figure 2. Air-core formation in the hydrocyclone with Do/Du of (a) 1.2, (b) 1.6, (c) 1.8, (d) 2.0, (e) 2.2, (f) 2.6, (g) 3.2, and (h) 4.0.

cyclone in our previous study.45 The model and measurement data that were selected for the present study were adapted from the work of Hsieh46 in a 75 mm hydrocyclone by using laser Doppler velocimetry. The simulation results demonstrated the same trend with the experimental results, but still there were some errors in the precision, especially in the axial velocity. The results were in line with the previous studies31,47 and show that our model and calculation could predict the flow field in a hydrocyclone accurately. More details can be seen in ref 43.

3. RESULTS AND DISCUSSION 3.1. Air-Core Formation in a Hydrocyclone with Different Do/Du. In the hydrocyclone, the flow field was initiated as air-filled at atmospheric pressure. At a certain flow velocity, the fluid was introduced into the hydrocyclone. Because of the presence of centrifugal forces, a circumferential flow against the outer wall was established. The threedimensional motion of fluid flow in the hydrocyclone accelerated within a reduction in the radius, which converted 5472



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Figure 3. Air-core diameters through underflow to overflow of the hydrocyclone with a Do/Du of 1.6.






the static pressure of the flow to dynamic pressure except friction losses. When the surplus hydrostatic pressure had no sufficient energy to compensate for the energy loss and continued to maintain growth of the flow velocity simultaneously, a negative pressure zone was formed. Also, on the basis of the same pressure between the two outlets and the

atmosphere generally, the negative pressure zone was able to absorb outside air; thus, an air core was formed. Figure 2a is the flow-field development process at the surface of x = 0 mm inside the hydrocyclone with a Do/Du of 1.2 from 0 to 15.0 s. The contour presents the volume of air inside the hydrocyclone. It was filled with air marked in red at 0 s, while the liquid marked in blue occupied parts of the hydrocyclone at 5473



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with liquid, the air core formed in the center of the hydrocyclone at 2.0 s. However, after the flow field was steady, the air core only existed in the spigot and vortex finder. It was noticed that the air core formed inside the hydrocyclone through underflow to overflow when Do/Du was 1.8 after 2.0 s, as shown in Figure 2c. Also, the air core basically remained steady at 4.0, 5.0, 10.0, and 15.0 s, where the stabilities were much better than that in the hydrocyclone with a Do/Du of 1.6. The vortex-finder diameter continued to be enlarged to 50 mm. Though the air core in the hydrocyclone with a Do/Du of 2.0 did not form completely at 2.0 s, the diameter and stability of the air core were improved compared with that inside the hydrocyclone with a Do/Du was 1.8 after 4.0 s. It can be seen that when Do/Du was higher than 2.0 in Figure 2d−h, the air core formed continuously and relatively steadily inside the hydrocyclone through underflow to overflow. When a continuous air core formed inside the hydrocyclone, the diameter of the air core was observed to increase with an increase in Do/Du. 3.2. Steady State of the Air Core in the Hydrocyclone with Different Do/Du. From the above analysis, the air core formed inside the hydrocyclone with Do/Du higher than 1.2. Inside the hydrocyclone, the air core was determined where the volume of air was greater than 0.9. The diameter of the air core


0.5 s. After the liquid completely filled the device, there was no air inside it. The air core was not formed in the hydrocyclone with a Do/Du of 1.2. The distribution of the air and liquid in the hydrocyclone with a Do/Du of 1.6 in Figure 2b differed from that with a Do/Du of 1.2. When the hydrocyclone was filled

Figure 10. Air-core diameters through underflow to overflow of the hydrocyclone with a Do/Du of 1.6 at inlet velocities of (a) 1.333, (b) 1.556, (c) 1.778, and (d) 2.222 m/s. 5474



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Figure 11. Air-core diameters through underflow to overflow of the hydrocyclone with a Do/Du of 2.0 at inlet velocities of (a) 1.333, (b) 1.556, (c) 1.778, and (d) 2.222 m/s.

could be determined through calculation of the coordinate value of the air boundary. To quantitatively analyze these air cores, their diameters through underflow to overflow inside these hydrocyclones are presented. After the flow field was basically steady, the air-core diameter, located at different heights of the hydrocyclone with a Do/Du of 1.6, was constructed at different times (Figure 3). It is clear that the air core was assuredly discontinuous, the value of the diameter of which pointed to zero near the interface of the cylinder and cone sections. This result is the same as that in Figure 2b. Through the air-core diameters at different times in the same figure, it was found that the air core on the cylinder section and spigot did not change with time from 5.0 to 15.0 s, while the situation at 4.0 s was different. This may be caused by the air core, which was not fully developed. Thus, it is concluded that the flow field inside the hydrocyclone with a Do/Du of 1.6 became relatively steady. However, it still changed with time on the cone section. Similarly, the air-core diameter at different heights in the hydrocyclone with a Do/Du of 1.8 was obtained from 4.0 to 15.0 s (Figure 4). The continuity of the air core was improved in this hydrocyclone. The disconnection only existed at the height between z = −0.05 and +0.05 m. The same results are presented in Figure 3; that is, the air core on the cylinder

section and spigot did not change with time from 5.0 to 15.0 s. However, with an increase of Do/Du, the steady state of the flow field inside the hydrocyclone was enhanced because the air core at 4.0 s was much closer to that at the other time. The air core inside the hydrocyclone with a Do/Du of 2.0 was continuous, as shown in Figure 5. It is important that the air cores on the cylinder section and spigot were very steady, which did not vary with time from 4.0 to 15.0 s, while acute unsteadiness occurred on the cone section without any rules. The steady and unsteady states of the air core could be divided into three parts. In the first part, the air core was steady, mainly on the cylinder section, which did not vary with time. In the second part, the air core between the bottom of the vortex finder and the interface of the cylinder and cone sections was slightly unsteady. The unsteadiness of the air core was quarried from part of the conical section. Nevertheless, in the third part, the air core on the spigot was also steady, which did not change over time. In addition, the same results were accomplished in the hydrocyclone with a Do/Du of 2.2, as shown in Figure 6. As shown in Figure 7, when Do/Du of the hydrocyclone increased to 2.6, another significant phenomenon appeared; that is, the air core became extraordinarily stable. The air-core diameters at every height inside the hydrocyclone did not vary with time. Besides the cylinder section and spigot, the air cores 5475



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Figure 12. Air-core diameters through underflow to overflow of the hydrocyclone with a Do/Du of 4.0 at inlet velocities of (a) 1.333, (b) 1.556, (c) 1.778, and (d) 2.222 m/s.

trend of the steady state of the air core in a certain hydrocyclone. The air core in the center of the hydrocyclone with a Do/Du of 1.8 was discontinuous. When the Do/Du values were 2.0 and 2.6, the air core became continuous. Also, the high unsteadiness occurred on the cone section in the hydrocyclone with a Do/Du of 2.0. In the hydrocyclone with a Do/Du of 4.0, the air core was continuous and steady, which was no longer changing over time through the whole hydrocyclone at any inlet velocities. First, analysis of the steady state of the air core in the hydrocyclone with a Do/Du of 1.6 was illustrated in Figure 10. The steady state of the air core existed only in part of the cylinder section and spigot. Meanwhile, the air core in the cylinder section and spigot was almost not changing with time. However, at a velocity of 1.556 m/s, serious discontinuity of the air core at 14.0 and 15.0 s led to different phenomena. Figure 11 shows that the air-core diameters changed over time through underflow to overflow when Do/Du is 2.0. In this figure, the high unsteadiness of the air core existed on the cone section, but the diameters of the air core on the cylinder section and spigot were basically not changing over time. These phenomena appeared to be the same in four different inlet velocities (Figure 11a−d). The inlet velocity only changed the value of the air-core diameter and not its trend.

on the cone section also remained steady. To further prove this situation, Do/Du increased to 3.2 and 4.0, and the same results were received in Figures 8 and 9. From the analysis above, the flow field inside the hydrocyclone was steadier with the vortex-finder diameter increasing. The air core in the center of the hydrocyclone with Do/Du of 1.6 and 1.8 was discontinuous. However, the air core on the cylinder section and spigot did not change over time. When Do/Du values were 2.0 and 2.2, the air core became continuous, and the ones on the cylinder section and spigot still did not change over time. However, the high unsteadiness occurred on the cone section. The air core was still continuous inside the hydrocyclone with Do/Du greater than 2.6, and it remained steady over time. It is concluded that the steady state of the air core could be classified by Do/Du. 3.3. Effects of the Operation Conditions on the Steady State of the Air Core. To further study the effects of the operation conditions on the steady state of the air core, we investigated the steady state of the air core in the hydrocyclone with certain Do/Du values at different inlet velocities. Three representative Do/Du values, 1.6, 2.0, and 4.0, were selected for this study. The inlet velocities were 2.222, 1.778, 1.556, and 1.333 m/s. The same results with the above conclusions were shown. The inlet velocity did not change the 5476



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(6) Wang, S. G.; Wang, Z. B.; Chen, W. M. A Physical Description on Stochastic Characteristics of the Separation Process in Hydrocyclones. Chem. Equip. Technol. 2005, 6 (5), 16−21. (7) Bamrungsri, P.; Puprasert, C.; Guigui, C. Development of a Simple Experimental Method for the Determination of the Liquid Field Velocity in Conical and Cylindrical Hydrocyclones. Chem. Eng. Res. Des. 2008, 86, 1263−1270. (8) Doby, M. J.; Nowakowski, A. F.; Nowak, E.; Dyakowski, T. Numerical and Experimental Examination of Swirl Flow in a Cylindrical Container with Rotating Lids. Miner. Eng. 2007, 20, 361−367. (9) Bergström, J.; Vomhoff, H. Experimental Hydrocyclone Flow Field Studies. Sep. Purif. Technol. 2007, 53, 8−20. (10) Neesse, T.; Dueck, J. Air Core Formation in the Hydrocyclone. Miner. Eng. 2007, 20, 349−354. (11) Doby, M. J.; Nowakowski, A. F.; Yiu, I.; Dyakowski, T. Understanding Air Core Formation in Hydrocyclone by Studing Pressure Distribution as a Function of Viscosity. Int. J. Miner. Process. 2008, 86, 18−25. (12) Davidson, M. R. Numerical Calculations of Flow in a Hydrocyclone Operating without an Aircore. Appl. Math. Model. 1988, 12 (2), 119−128. (13) Chu, L. Y.; Yu, W.; Wang, G. J. Enhancement of Hydrocyclone Separation Performance by Eliminating the Air Core. Chem. Eng. Process. 2004, 43, 1441−1448. (14) Marins, L. P. M.; Duarte, D. G.; Loureiro, J. B. R.; Moraes, C. A. C.; Freire, A. P. S. LDA and PIV Characterization of the Flow in a Hydrocyclone without an Air-core. J. Pet. Sci. Eng. 2010, 70, 168−176. (15) Binnie, M.; Hookings, G. A. Laboratory Experiments on Whirlpools. Proc. R. Soc. London, A 1948, 194, 398−415. (16) Simth, J. L. An Analysis of the Vortex Flow in the Cyclone Separator. J. Basic Eng. 1962, 609−616. (17) Fontein, F. J.; Van Kooy, J. G.; Leniger, H. A. The Influence of Some Variables upon Hydrocyclone Performance. Br. Chem. Eng. 1962, 7 (1), 410−420. (18) Pericleous, K. A.; Rhodes, N. The Hydrocyclone Classifiera Numerical Approach. Int. J. Miner. Process. 1986, 17, 23−43. (19) Davidson, M. R. An Adaptive Method of Predicting the Air Core Diameter for Numerical Models of Hydrocyclone Flow. Int. J. Miner. Process. 1995, 43167−43177. (20) Dyakowski, T.; Williams, R. A. Prediction of Air Core Size and Shape in a Hydrocyclone. Int. J. Miner. Process. 1995, 43 (1−2), 1−14. (21) Williams, R. A.; IIyas, O. M.; Dyakowski, T.; Dickin, F. J.; Gutierrez, J. A.; Wang, M.; Beck, M. S.; Shah, C.; Rushton, A. Air Core Imaging in Cyclonic Separators: Implications for Separator Design and Modelling. Chem. Eng. J. Biochem. Eng. J. 1995, 56 (3), 135−141. (22) Steffens, P. R.; Whiten, W. J.; Appleby, S.; Hitchins, J. Prediction of Air Core Diameters for Hydrocyclones. Int. J. Miner. Process. 1993, 39, 167−177. (23) Concha, F.; Barrientos, A.; Montero, J.; Sampaio, R. Air Core and Roping in Hydrocyclones. Int. J. Miner. Process. 1996, 44−45, 743−749. (24) Hsieh, K. T.; Rajamani, R. K. Mathematical Model of the Hydrocyclone Based on Physics of Fluid Flow. AIChE J. 1991, 35 (5), 735−746. (25) Song, X. F.; Zhang, M. H.; Wang, J.; Li, P.; Yu, J. G. Optimization Design for DTB Industrial Crystallizer of Potassium Chloride. Ind. Eng. Chem. Res. 2010, 49, 10297−10302. (26) Cullivan, J. C.; Williams, R. A.; Cross, C. R. Understanding the Hydrocyclone Separator through Computational Fluid Dynamics. Trans. Inst. Chem. Eng. 2003, 81, 455−466. (27) Slack, M. D.; Porte, S. D.; Engelman, M. S. Designing Automated Computational Fluid Dynamics Modeling Tools for Hydrocyclone Design. Miner. Eng. 2004, 17, 705−711. (28) Nowakowski, A. F.; Cullivan, J. C.; Williams, R. A.; Dyakowski, T. Application of CFD to Modeling of the Flow in Hydrocyclone. Is This a Realizable Option or Still a Research Challenge? Miner. Eng. 2004, 17, 661−669.

The results of the air-core diameter inside the hydrocyclone with a Do/Du of 4.0 at different velocities are shown in Figure 12. From this figure, the diameters of the air core in the hydrocyclone were not changing over time. The curves of the air-core diameter with the height of the hydrocyclone at different time points are superposed in Figure 12a−d. These phenomena appeared to be the same at four different inlet velocities (Figure 12a−d). Different inlet velocities changed some values of the air-core diameter, but the trend of the steady state of the air-core diameter was not changed. In conclusion, when Do/Du was more than 2.6, the air-core diameter did not change with time after the air core had formed. The same conclusion could be observed at different velocities, while the steady state of the air core in the hydrocyclone could be decided by Do/Du.

4. CONCLUSIONS In a hydrocyclone, air core was an unavoidable phenomenon. The steady state of the air core was one of the most important factors for separation efficiency. We find that the diameter of the vortex finder played a key role for the steady state of the air core. (1) The structure and state of the air core were different in the hydrocyclone with different Do/Du. When the Do/Du values were 1.6 and 1.8, the air core were not continuous. In the range of 1.8−2.6, the high unsteadiness of the air core was quarried from part of the conical section. However, the air core, mainly on the cylinder section and spigot, was steady, which did not vary with time. When Do/Du of the hydrocyclone increased to 2.6, another significant phenomenon happened; that is, the air core became extraordinarily steady through the whole hydrocyclone. The air-core diameters at every height inside the hydrocyclone did not vary with time. (2) Operation conditions were demonstrated as having no effect on the steady state of the air core, but they changed the values of the air-core diameter. If Do/Du was fixed, the steady state of the air core remained the same under different inlet velocities.

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AUTHOR INFORMATION

Corresponding Author

*Tel: 86-21-64252170. E-mail: [email protected] (X.S.), [email protected] (J.Y.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the MOST Project of Personnel Service Corporate Actions (Grant 2009GJG20011).

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REFERENCES

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Numerical Investigation of the Effect of the Ratio of the Vortex-Finder

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