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Effects of the distance between a vibrating cantilever pair
European Journal of Mechanics B/Fluids 43 (2014) 154–165
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Effects of the distance between a vibrating cantilever pair Minsuk Choi a , Christian Cierpka b , Yong-Hwan Kim c,∗ a
Department of Mechanical Engineering, Myongji University, Yongin 449-728, Republic of Korea
b
Institute for Fluidmechanics and Aerodynamics, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
c
Green Energy Group, POSCO E&C, 36 Songdo-Dong, Yeonsu-Gu, Incheon 406-732, Republic of Korea
article
info
Article history: Received 20 March 2013 Received in revised form 19 June 2013 Accepted 15 August 2013 Available online 22 August 2013 Keywords: Vibrating cantilever Counter-rotating vortex Piezoelectric fan Unsteady simulation
abstract Two dimensional unsteady numerical simulations were conducted using a commercial code with a userdefined-function to investigate the effect of the distance between two cantilevers vibrating in counterphase or in phase. The performance of the cantilevers with different distances was mainly evaluated by the time-averaged axial velocity and the mass flow rate. It is evident that there is no interaction between the vortices by two cantilevers if they are too far apart. However, if two cantilevers are too close, they hinder each other in vortex generation. In particular, the interaction between two inner vortices generates a reversed flow which has a negative effect on the performance. Unless the distance is too close, the performance of the cantilever pair vibrating in counter-phase is always superior to the cantilever pair vibrating in phase. The optimal distance between two cantilevers in counter-phase is approximately equal to twice the size of a fully-grown vortex generated by the single cantilever, while there is no distinct optimal distance for a cantilever pair vibrating in phase. In case the distance is larger than three times the vortex size, the flow field generated by each cantilever is similar to the flow field of a single cantilever, which implies that two cantilevers work independently of each other. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction Rotary type fans with high efficiency and performance have been widely used for thermal and flow control in conventional large-sized devices. However, it is difficult to make the components of rotary fans, such as the rotor, bearings, motor and shaft, smaller than a critical size for small electronics. This situation forces designers to develop new types of cooling devices with a small size and a simplified structure. An alternative device of generating air flow is a hand-held fan, whose advantage is its simple structure, specifically, a vibrating flat plate. Any flat plate with a simple harmonic oscillation can induce an airflow, and consequently can be used as a cooling device. To obtain sufficient airflow, a fast vibrating motion is required and it can be generated using a piezoelectric material. When AC power is applied to the piezoelectric fan, the plate moves back and forth, because of the electrical potential difference across a piezoelectric material deposited on the plate, and consequently the device generates an airflow. In spite of the simple structure of piezoelectric fans, the airflow generated by the fan has not been fully understood. Toda [1,2] was one of the earliest researchers to perform experiments on the cooling effectiveness of a piezoelectric fan and confirmed that the
∗
Corresponding author. Tel.: +82 32 748 1859; fax: +82 32 748 4022. E-mail addresses: [email protected] (M. Choi), [email protected] (C. Cierpka), [email protected] (Y.-H. Kim). 0997-7546/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechflu.2013.08.006
fan can be used as a cooling device for a transistor. To characterize the flow structure around vibrating flat plates, Watanabe et al. [3], Ihara and Watanabe [4], Tsutsui et al. [5] and Takato et al. [6] investigated the airflow using LDV (Laser Doppler Velocimetry) measurements. However, the low spatial resolution of LDV was insufficient to clearly observe the vortex motion generated by the oscillating plates. Thereafter, a few attempts to characterize a small cantilever fan for electric devices were made by Burmann et al. [7], Acikalin et al. [8–10], Wait et al. [11] and Kimber and Garimella [12] using analytical, computational and experimental methods. In particular, Kimber et al. [13,14] investigated the heat transfer enhancement and the aerodynamic damping in a vibrating cantilever pair. Lin [15,16] found that the piezoelectric fan, based on his experimental and numerical results, can enhance the heat transfer coefficient significantly in comparison to natural convection. With the flow measurements around a vibrating cantilever, Kim et al. [17,18] were able to identify the cyclic generation of counter-rotating vortices induced by the cantilever in detail. They analyzed the vortex motion in a quantitative manner with high-resolution PIV (Particle Image Velocimetry) measurements and wavelet analysis. Eastman et al. [19] measured the thrust generated by a single slender cantilever and investigated the 3-D structure of the flow field around it. Recently, Choi et al. [20] have analyzed numerical simulations thoroughly and found that the static pressure difference across the cantilever tip plays an important role in the formation and development of counter-rotating vortices.
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Nomenclature c CCW CW d f0 h0 l md ms p VT Vxd Vxs
ϕ ω
Effective length of cantilever Counter-clockwise Clockwise Distance between two cantilevers Vibrating frequency Maximum tip deflection amplitude Actual length of cantilever Mass flow rate generated by a pair of cantilevers Mass flow rate generated by a single cantilever Static pressure Maximum tip speed Maximum axial velocity induced by a pair of cantilevers Maximum axial velocity induced by a single cantilever Phase angle Vorticity
To increase the cooled area or the performance of the system, it is beneficial to use multiple piezoelectric fans. Although many researchers have experimented on designing a piezoelectric fan, testing its performance and investigating the flow field around it, only a few papers have been published regarding the flow around two oscillating plates. Ihara and Watanabe [4] have performed numerical simulations using the discrete vortex method and applied smoke visualization to investigate the flow field around two plates oscillating in-phase and in counter-phase. However, the data from their simulation and experiment were insufficient to resolve the flow field in detail. Kimber and Garimella [13] tested the cooling effectiveness of a vibrating cantilever pair in phase on the heated surface and found it can increase the heat transfer coefficient in comparison to the single cantilever. Kimber et al. [14] found that the air damping decreases significantly with in-phase vibration while it increases with counter-phase vibration. Recently, Choi et al. [21] have simulated a 2D flow field generated by oscillating cantilever pair with different phase angles and reported that the interaction of two vibrating cantilevers changes the flow field significantly depending on the phase angles. In this study, the researchers found that the performance of the cantilever pair reaches its maximum when two plates are vibrating 180° out of phase. In addition, Ihara and Watanabe [4] observed that the distance between two vibrating plates in counter-phase has a significant effect on the induced flow field but could not explain this phenomenon in detail due to insufficient data. Therefore, the present study investigates the effect of the distance between two vibrating cantilevers on the performance and focuses on finding the underlying mechanisms of the performance variation. 2. Validation for a single cantilever Prior to the validation of the numerical results, it is necessary to define the phase angle. Fig. 1 shows the relationship of the phase angles to the deflections of the cantilever. In the previous paper of Choi et al. [20], the numerical results for a single cantilever were compared with the experimental velocity data measured by Kim et al. [17]. A brief comparison, therefore, between the computational results and the experimental data is shown in Fig. 2. In the experiment, the unsteady flow around a vibrating cantilever was visualized by smoke particles and the captured particle images showed the process of the vortex formation, as presented in the left of Fig. 2. The PIV technique has then been applied to obtain the quantitative flow field from the particle images and the results
Fig. 1. Definition of the phase angle.
are shown in the middle of Fig. 2. Finally, the computed flow field in the right of Fig. 2 was compared with the measured flow field. The process of the vortex formation matches well between experimental and computational data. In addition to the flow field, in the previous paper of Choi et al. [20], the deflection shape of the modeled cantilever was matched accurately to the experimentally measured vibration shape at each phase angle. In order to check the accuracy of the computed flow fields, the size of vortices and their trajectories were compared with the experimental data quantitatively. These comparisons confirmed that the computation can capture the overall features of the vortices observed in the experiment during a period and is thus well suited to perform a parameter test. 3. Computational method 3.1. Geometry of a vibrating cantilever pair This part begins with a brief description of the shape of the cantilever, which is the same as that in the experiment of Kim et al. [17]. The piezoelectric fan is a thin metal plate fixed to the apparatus in a cantilevered manner with its bottom end and its top end is free as shown in Fig. 3, the bottom part of which is coated with a piezoelectric material. The actual length (l) of the cantilever is 31 mm but its moving part, referred to as the effective length (c), is 25.4 mm. The plate with the thickness of 0.13 mm vibrates at its fundamental natural frequency (f0 ) of 180 Hz. The tip deflection is fixed at h0 /c = 0.054, where h0 is the maximum tip deflection amplitude, and the corresponding maximum tip speed (VT ) is 1.54 m/s. More details on the experimental setup and the measurement can be found in Kim et al. [17]. Choi et al. [21] simulated the flow around two vibrating cantilevers with eight different phase angles and a fixed distance between both cantilevers. They found that the cantilever pair vibrating in counter-phase is more effective in generating the airflow than the cantilever pair vibrating in phase. The distance between two cantilevers (d) was fixed at 8h0 and this value was chosen based on the size of the counter-rotating vortices generated by a single vibrating cantilever. As shown in Fig. 2, two vortices are generated alternately by the cantilever and they move downstream within −3h0 and 3h0 on the x-axis. Two cantilevers could work as obstacles in the vortex generation process if they are too close, while there would be no interaction between them if they are too far from each other. In the present study, two vibrating cantilevers with different distances between them were tested in phase or in counter-phase in order to find an optimal distance for the highest performance. The distance between the cantilevers was changed from 4h0 to 20h0 .
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Fig. 2. Flow field around vibrating cantilever: smoke visualization (left), PIV measurement (middle) and computational results (right). Source: Reproduced with data of Kim et al. [17] and Choi et al. [20].
3.2. Numerical methods and boundary conditions Two dimensional flow fields around a vibrating cantilever pair were simulated using a commercially available flow solver, Fluent 6.3. The 2D incompressible Reynolds-averaged Navier–Stokes equations, including continuity and momentum equations, were solved with SIMPLE algorithm. A second-order upwind scheme was used for the convection and diffusion terms and an implicit time marching was used to get unsteady flow fields. The flow field around the cantilever was assumed to be turbulent and the turbulent viscosity was obtained by the standard k–ε model. The movement of the cantilever in the computation was implemented using a UDF (user-defined-function), which describes the measured shapes of the deflected cantilever at eight instants in the experiment of Kim et al. [17]. The modeled cantilever was matched well to the experimentally measured vibration shape as shown in the previous study of Choi et al. [20]. The computational domain was divided into two sub-domains, stationary and deformed parts, owing to the motion of the cantilevers. The dynamic mesh
in the deformed part was implemented by using the spring-based smoothing method combined with the local re-meshing method. The mesh in the stationary part does not change during computations. The size of the computational domain was set to be much larger than the cantilever to allow the flow to develop with the lowest possible effects from the boundaries. The inlet and outlet boundaries were taken to be at a distance of 110h0 and 260h0 from the tip of the cantilever, where the effect of the vortices generated by the cantilevers could not be felt, and only the standard atmospheric static pressure was specified on the boundaries. There are two side walls at 100h0 away from the cantilever, and these walls were also present for the experiment. In order to measure the effects of the cantilever only, the no-slip condition was applied on the surface of the cantilever, while the slip (symmetric) condition was used on the side-walls and stiffener. The accuracy of these numerical methods and boundary conditions has already been validated in the previous study and the interested reader is referred to Choi et al. [20].
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Fig. 3. Schematic diagram of the computational geometry.
3.3. Vortex detection method
4.1. Cantilever pair vibrating in counter-phase
To extract the trajectories of the vortices, i.e., the locations of the vortex cores, an automatic detection algorithm, based on the Mexican hat wavelet, was applied to the absolute vorticity field. The same method was already applied to the experimental data of Kim et al. [18] and the computational results of Choi et al. [20] for tracking the vortices. For the current data processing, the numerical velocity fields were interpolated on an equidistant grid with a spacing of 1x = 1y = 0.2h0 . Based on this representation for each instant, the vorticity was calculated using second-order central differences. The λ2 -criterion was then used to filter high vorticity regions in the shear layer developed along the cantilever. Furthermore, the position of the two cantilevers was extracted to remove data points that lie too close to the cantilever surface. The algorithm now correlates the vorticity field with wavelets of different scales. This paper is not intended to discuss all the details of the method, but the interested reader is referred to Cierpka et al. [22] for a deeper treatment of the method. The main benefit of the wavelet method is that the procedure can be calibrated using the analytical Lamb–Oseen vortex model. In the first step, the possible size of structures is identified and the wavelet scales are adapted by the algorithm automatically, which provides the best spatial resolution. In the next step, maxima for the correlations are identified. The position of these maxima in the correlation plane between the wavelet and vorticity fields indicates the center of a vortex and the wavelet scale provides a direct measure of the vortex core size. Additional information such as the center velocity or the vorticity can now be extracted.
It is important for two vibrating cantilevers to generate sufficient air flow continuously for cooling small devices. A total of 32 data sets were captured for the computational results during a cycle and the corresponding flow fields were averaged to obtain a time-averaged flow field, which is suited to quantify the cooling effectiveness. Fig. 4 shows the time-averaged axial velocity distribution along the x-direction at three different ypositions, y/h0 = 1, 5 and 10. In all cases, the maximum velocity appears at y/h0 = 1 just downstream of the cantilever tip. As the vortices move downstream of the cantilever, the maximum axial velocity decreases and the width of the vortex-affected area increases gradually due to the diffusion. For the single cantilever, the averaged axial velocity is symmetric to the cantilever center and its maximum value is 3.8 m/s at y/h0 = 1. With two vibrating cantilevers in counter-phase, the axial velocity has a symmetric pattern to the center (x/h0 = 0) at all three y-positions and it is locally symmetric to each cantilever at y/h0 = 1. When two cantilevers are too close within 4h0 , the maximum axial velocity at y/h0 = 1 is much smaller than that observed using a single cantilever, because of the strong interaction between the vortices from each cantilever. However, the maximum velocity and total flow rate further downstream of the cantilever, y/h0 = 5 and 10, are much larger than those in the case of a single cantilever. When the distance between the cantilevers increases above 6h0 , the maximum axial velocity at y/h0 = 1 is larger than or similar to the single cantilever. In addition, a region with a negative axial velocity is clearly observed between two cantilevers at y/h0 = 1 in the cases of d = 6h0 and 8h0 . The axial velocity distribution at y/h0 = 10 has one peak if the distance is smaller than 8h0 but it has two peaks if the distance is larger than 10h0 . As the distance increases above 12h0 , the axial velocity distribution at the three different y-positions clearly shows that the interaction of the vortices from each cantilever is very weak and it looks like that the cantilevers work independently of each other. To quantify the effect of the distance between the cantilevers on the performance of them, the maximum axial velocity and the flow rate were obtained from the time-averaged axial velocity at y/h0 = 10. Each variable was normalized by the value of the single cantilever and the results are shown in Fig. 5. In particular, the latter was calculated by integrating the time-averaged axial
4. Computational results Checking the grid-independence of the numerical results was already conducted for a cantilever pair and five grids with different resolutions were tested in the previous study of Choi et al. [21]. The same grid resolution, which had obtained the grid-independent solution in the previous study, was applied to the present study. The computational grid was composed of triangular cells in the deformed part around the cantilevers and quadrilateral cells in the stationary part far from the cantilevers. Total cell number was about 74,000.
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(a) Single cantilever.
(b) d = 4h0 .
(c) d = 6h0 .
(d) d = 8h0 .
(e) d = 10h0 .
(f) d = 12h0 .
(g) d = 14h0 .
(h) d = 16h0 . Fig. 4. Time-averaged axial velocity distribution downstream of the cantilevers in counter-phase.
velocity from −20 to 20 of x/h0 . The maximum axial velocity is observed at a distance of d = 4h0 and, as the distance between two cantilevers increases, the maximum axial velocity decreases gradually to become 1.1 times the value of the single cantilever. As the distance increases, the mass flow rate increases to have its maximum value at d = 14h0 and decreases slowly thereafter. In the case of d < 8h0 , as shown in Fig. 4, the strong interaction between the counter-rotating vortices from each cantilever generates a smooth axial velocity distribution with a peak at the center (x/h0 = 0). Although the maximum axial
velocity is larger than in the other cases, the mass flow rate is smaller because of the large mixing loss caused by the vortex interaction. In the case of d > 8h0 , however, the interaction of the vortices is weaker and the axial velocity distribution has two peaks symmetrical to the center. As the distance increases further, the maximum axial velocity would converge to the value of the single cantilever and the mass flow rate doubles in comparison to the single cantilever. Here, there are some important points to determine which distance is optimal for the cantilever pair vibrating in counter-phase. It is evident that a larger value in the
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(i) d = 18h0 .
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(j) d = 20h0 . Fig. 4. (continued)
Fig. 5. Maximum time-averaged axial velocity and normalized mass flow rate in counter-phase at y/h0 = 10.
axial velocity and mass flow rate is better for cooling. The location of the maximum axial velocity is another important factor and it is chosen to be optimal at the center. Therefore, the authors propose the distance between the cantilevers of 6h0 and 8h0 best suited for the cooling of small electric devices, which is also equal to twice the size of the fully-grown vortex generated by the single cantilever. To investigate the interaction of the counter-rotating vortices from each cantilever in detail, velocity vectors and vorticity distribution were calculated and the results are shown in Fig. 6, where the phase angle is based on the movement of the right cantilever. When two cantilevers vibrate in counter-phase, the large and strong vortices from each cantilever are symmetric to the center line. In the case of ϕ = 0° and d = 4h0 , the vortices from the left cantilever (CW1 and CCW1) push the flow horizontally to the right while the vortices from the right cantilever (CW2 and CCW2) push the flow to the left. CW1 and CCW2 interact violently with each other and generate a strong reversed flow between them. CCW1 and CW2 also generate a flow in the downstream direction but its intensity is much smaller than the reversed flow. Consequently, a saddle point is clearly visible between the four vortices. At ϕ = 180° and d = 4h0 , the interaction between the vortices from the same cantilever is stronger than that between the vortices from the different cantilevers, generating a flow in the left-up or right-up direction. Because CW1 and CCW2 at this instant are too weak to generate the reversed flow, the resulting flow moves downstream. At ϕ = 0° and d = 8h0 , the reversed flow still exists between CW1 and CCW2 but its intensity decreases significantly in comparison to the case of ϕ = 0° and d = 4h0 . The interaction between CW1 and CCW1 or between CW2 and CCW2 pushes the flow in the downstream direction. At ϕ = 180° and d = 8h0 , CW1 and CCW2 are still strong enough to interact with CCW1 and CW2 respectively, generating a flow in the downstream direction, while the interaction between CW1 and CCW2 weakens because of the increased distance between two cantilevers. The distance between two cantilevers at d = 12h0 is larger than the
size of the fully-grown vortex, so that the interaction between CW1 and CCW2 is very weak even at ϕ = 0°. The vortices from each cantilever show similar patterns to the single cantilever, and it seems that the cantilevers generate the counter-rotating vortices independently of each other. The interaction between the vortices from the same cantilever is always stronger than that from different cantilevers. In the three cases shown in Fig. 6, the reversed flow between two cantilevers always exists at ϕ = 0° and causes the inner vortices, CW1 and CCW2, to lag behind the outer vortices, CCW1 and CW2. It is evident that the vortices from the different cantilevers interact violently with each other and cause large momentum loss if two cantilevers are too close, whereas the cantilever works independently of each other if the distance is large enough. Two numerical pressure sensors were installed on each side of the cantilever at 0.5h0 away from the tip and they measured the static pressure on both sides at the 16 time instants. The static pressure difference (1p) is calculated by subtracting the static pressure on the right side from the static pressure on the left side and the results are shown in Fig. 7. The static pressure difference on the left cantilever is symmetric of that on the right cantilever with respect to the zero line, so it is not shown here. In the previous study of Choi et al. [20], it was found that the static pressure difference is a major driving force in vortex formation and its value changes periodically during a cycle. Due to the method for calculating the static pressure difference, the clockwise vortex is generated with a positive pressure difference and the counter-clock-wise vortex is generated with a negative pressure difference. In addition, when the static pressure difference crosses the zero line, an already full-grown vortex separates from the tip and a new vortex with the opposite rotational direction is initiated. The absolute value of the maximum and minimum static pressure differences in all cases is larger than those for the single cantilever, implying that the counter-rotating vortices generated by the cantilever pair in counter-phase are stronger than the vortices generated by the single cantilever. At d = 4h0 , in particular, the vortex formed at the inner part of the cantilevers (CCW2 in Fig. 6(a)) is much stronger than at the outer part (CW2 in Fig. 6(b)), which is also observed in the static pressure difference. The pressure difference variation on the right cantilever converges to the value on the single cantilever as the distance between two cantilevers increases, but the fluctuating amplitude of the static pressure difference is still larger than the single cantilever even at d = 20h0 . This is the main reason why the maximum axial velocity induced by the cantilever pair at y/h0 = 1 is larger than that by the single cantilever. Fig. 8 shows the trajectories of the counter-rotating vortex pair obtained by the vortex detection method described above. The vortex center is blanked for clarity if the normalized vorticity, i.e. ωh0 /VT , is smaller than 3. For the single cantilever, the
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Fig. 6. Instantaneous velocity fields (vectors) and vorticity distribution (contours) for the cantilever pair vibrating in counter-phase.
trajectories of two counter-rotating vortices are symmetric to the cantilever. Two vortices are alternately generated by the single cantilever and the induced flow by the vortices moves in the downstream direction. When two cantilevers are sufficiently close, all trajectories are symmetric to the center but asymmetric to each cantilever. At d = 4h0 , two inner vortices cannot propagate to the downstream region because of the reversed flow between them, while two outer vortices are forced to move toward the inside of two cantilevers due to continuity. As the distance between two cantilevers increases, the reversed flow becomes weaker and the interaction between two outer vortices is negligible. This is why the trajectories of four vortices at d = 8h0 are more straight than those at d = 4h0 . At d = 12h0 , the trajectories are also symmetric to each cantilever and look similar to those for the single cantilever.
4.2. Cantilever pair vibrating in phase In the previous study of Choi et al. [21], the performance of the cantilever pair vibrating in phase was lower than the cantilever pair vibrating in counter-phase. Fig. 9 shows the time-averaged axial velocity distribution along the x-direction. At d = 4h0 , the maximum axial velocity at three different y-positions is much smaller than that of the single cantilever. Moreover, a strong reversed flow appears near the center line between two cantilevers, owing to the strong interaction between the counter-rotating vortices from each cantilever. As the distance increases, the maximum velocity at all three y-positions increases and the reversed flow region disappears. In the cases of d > 12h0 , the time-averaged velocity distribution splits into two distinctive distributions, indicating that
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Fig. 7. Static pressure difference across the tip of the right cantilever in counterphase.
the interaction between two cantilevers is very weak. It should be noted that the maximum axial velocity at y/h0 = 1 induced by the cantilever pair vibrating in phase is always smaller than the single cantilever. The maximum axial velocity and the flow rate at y/h0 = 10 were again calculated from the time-averaged flow field and the results are shown in Fig. 10. The maximum axial velocity has the smallest value at d = 4h0 and monotonically increases with the increase of the distance. The mass flow rate induced by the vibrating cantilevers in phase has the same trend as the maximum axial velocity. However, the mass flow rate is always smaller than twice the value of the single cantilever. Moreover, the maximum axial velocity is also smaller than the value for the single cantilever case. As the distance increases further, the maximum axial velocity would converge to the value of the single cantilever and the mass flow rate would double in comparison to the single cantilever, like the cantilever pair in counter-phase. When two cantilevers vibrate in phase, the flow pattern generated by the cantilevers is similar and independent of the distance
(a) Single cantilever.
(c) d = 8h0 .
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between them as shown in Fig. 11. The counter-rotating vortices from the same cantilever generate the flow in the left-up and the right-up directions at ϕ = 0° and ϕ = 180° respectively and these interactions are favorable in the cantilever performance. The reversed flow in the right-down and the left-down directions is also induced by the vortices from the different cantilevers (CW1 and CCW2) and it has a negative effect on the cantilever performance. If two cantilevers are too close to each other, as shown in Fig. 11(a) and (b), the negative effect of the vortex interaction increases and degrades the cantilever performance. In addition, the violent vortex interaction weakens the vortices after separating from the cantilever. As the distance between the cantilevers increases, the interaction between the vortices from the different cantilevers becomes weaker, while the interaction between the vortices from the same cantilever remains similar or becomes stronger. If d > 12h0 , the flow fields indicate that both cantilevers work independently of each other. The static pressure difference across the tip of the right cantilever shows the similar pattern to the single cantilever as shown in Fig. 12, but the absolute value of the maximum and minimum static pressure difference is always smaller than that in the single cantilever. This indicates that the counter-rotating vortices generated by the cantilever pair in phase are weaker than those by the single cantilever. Although the fluctuating amplitude of the pressure difference increases with the increase of the distance, the fluctuating amplitude even at d = 20h0 is still smaller than the single cantilever. If the distance increases further, the pressure difference at each instant converges to the value of the single cantilever. When two cantilevers are vibrating in phase, as shown in Fig. 13, the trajectories of the vortices are symmetric to each cantilever as well as to the center. However, the four vortices are initially weak due to small pressure differences across the cantilever tip and the induced flow is also weak in comparison to the single cantilever. The reversed flow generated by the interaction between two inner vortices also has a negative effect on the movement of the vortices. In particular, the vortices at d = 4h0 hardly move in the downstream direction and become weaker very quickly
(b) d = 4h0 .
(d) d = 12h0 .
Fig. 8. Trajectories of the center of counter-rotating vortices generated by the cantilever pair vibrating in counter-phase.
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(a) Single cantilever.
(b) d = 4h0 .
(c) d = 6h0 .
(d) d = 8h0 .
(e) d = 10h0 .
(f) d = 12h0 .
(g) d = 14h0 .
(h) d = 16h0 .
(i) d = 18h0 .
(j) d = 20h0 . Fig. 9. Time-averaged axial velocity distribution downstream of the cantilevers in phase.
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near the cantilever tip. As the distance between two cantilevers increases, the vortices propagate further downstream, because larger pressure difference induces stronger vortices. The vortex centers at d = 12h0 follow similar trajectories to the single cantilever up to y/h0 = 4. 4.3. Comparison with other studies
Fig. 10. Maximum time-averaged axial velocity and normalized mass flow rate in phase at y/h0 = 10.
Three research papers in the reference lists were considering a cantilever pair. Ihara and Watanabe [4] investigated numerically the flow field induced by two cantilevers with different distances between them. They simulated the flow field using the discrete vortex method combined with the singularity method and compared their numerical results with the experimental visualization
Fig. 11. Instantaneous velocity fields (vectors) and vorticity distributions (contours) for the cantilever pair vibrating in phase.
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Fig. 12. Static pressure difference across the tip of the right cantilever in phase.
qualitatively. In their numerical results where the distance is below 4h0 , the averaged axial velocity induced by the cantilever pair was the same as for the single cantilever for in-phase vibration but it was smaller than for the single cantilever for counter-phase vibration. In the present study, the cantilever pair in counter-phase shows better performance than the cantilevers vibrating in phase for the flow generation at d = 4h0 . The main reason for the difference is the existence of the reversed flow between the two cantilevers. In the time-averaged flow field of Ihara and Watanabe [4], no reversed flow was obtained between the two cantilevers vibrating in phase using the discrete vortex method. However, the reversed flow can be seen between two cantilevers vibrating in phase at d = 4h0 in Fig. 9(b) of the present study. Kimber and Garimella [13] measured the temperature distribution using an infrared camera on a heated surface, which was installed normal to a cantilever pair, and calculated the heat transfer coefficients in order to analyze the cooling effectiveness of the cantilevers. In their experiments, the distance between the two cantilevers vibrating in phase was changed from 0.5h0 to 4h0 . The
(a) Single cantilever.
(c) d = 8h0 .
cantilever pair is more effective than the single cantilever when the distance is below 1.5h0 , but less effective in cooling the heated surface when the distance is between 2h0 and 4h0 with respect to the area-averaged Nusselt number at the center of the cantilever width. Due to the existence of the front wall and the 3D effects, the flow pattern induced by the cantilever pair in their study is slightly different from the flow obtained in the present study and the direct comparison is difficult. However, it is possible to conjecture the performance of the cantilever pair used in the present study when the distance is below 4h0 . In Fig. 10, the maximum time-averaged axial velocity and normalized mass flow rate decrease monotonically as the distance between two cantilevers vibrating in phase decreases from 20h0 to 4h0 , implying that the distance reduction causes the cantilever pair to be less effective in cooling. In particular, the predicted cooling effect of the cantilever pair based on the maximum time-averaged axial velocity and normalized mass flow rate is smaller than that of the single cantilever at d = 4h0 , which is coincident with the result of Kimber and Garimella [13]. Therefore, the authors conjecture that the maximum time-averaged axial velocity and normalized mass flow rate might decrease continuously till d = 2h0 but increase around d = 1.5h0 to form a local peak when two cantilevers vibrate in phase. Kimber et al. [14] found the air damping has a large effect on the vibrating amplitude of an array of cantilevers when they are close to each other in the face-to-face configuration. The vibrating amplitude significantly increases with the cantilevers vibrating in phase, while it significantly decreases with the cantilevers vibrating in counter-phase. This implies that the air damping has a positive effect on the cantilever pair in phase but it has a negative effect on the cantilever pair in counter-phase. However, the present study was intended to investigate the effect of the distance between the cantilevers on the flow generation. Therefore, all simulations were performed for a specific amplitude which was the same as the single cantilever, rather than for a specific applied power. In a viewpoint of the applied power on the cantilevers, it is possible that the optimal distance between the vibrating cantilevers would change. Another important finding of Kimber
(b) d = 4h0 .
(d) d = 12h0 . Fig. 13. Trajectories of the center of counter-rotating vortices generated by the cantilever pair vibrating in phase.
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et al. [14] was that the effect of the air damping is negligible and the two cantilevers behave independently when the distance is above 12h0 , which is the same value as found in the present study. 5. Conclusions In the present study, the performance of a cantilever pair vibrating in counter-phase or in phase was investigated numerically for several distances between the two cantilevers. The performance was mainly evaluated by the time-averaged axial velocity and the mass flow rate downstream of the cantilevers. The following conclusions could be made from the numerical results. 1. The cantilever pair vibrating in counter-phase is superior to the single cantilever in generating the flow, unless the distance between the two cantilevers is too close. However, the cantilever pair vibrating in phase is inferior to the single cantilever in the maximum velocity regardless of the distance. Therefore, it is required to operate two cantilevers in counterphase with a proper distance between them. 2. When two cantilevers are vibrating in counter-phase, it is possible to find the optimal distance between them, at which the counter-rotating vortices from each cantilever interact favorably without interruption between each other. The optimal distance between two cantilevers is equal to twice the size of the fully-grown vortex generated by a single cantilever. The value for the optimal distance found in the present study is 6h0 ∼ 8h0 . 3. When the distance is larger than three times the vortex size, i.e. 12h0 , the interaction between the vortices from each cantilever is negligible and the two cantilevers seem to work independently from each other. References [1] M. Toda, Theory of air flow generation by a resonant type PVF2 bimorph cantilever vibrator, Ferroelectrics 22 (1979) 911–918. [2] M. Toda, Voltage-induced large amplitude bending device—PVF2 bimorph—its properties and applications, Ferroelectrics 32 (1981) 127–133.
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[3] H. Watanabe, A. Ihara, H. Hirama, S. Hayashizaki, Measurement of the three dimensional flow field around an oscillating flat plate with LDV, Trans. JSME B 56 (1990) 3708–3715. [4] A. Ihara, H. Watanabe, On the flow around flexible plates, oscillating with large amplitude, J. Fluids Struct. 8 (1994) 601–619. [5] T. Tsutsui, M. Akiyama, H. Sugiyanma, K. Shimanaka, Heat transfer enhancement around a rectangular cylinder set in near wake generated by elastically vibrating flat plate, Trans. JSME B 62 (1996) 3667–3674. [6] K. Takato, T. Tsutsui, M. Akiyama, H. Sugiyama, Numerical analysis of flow around vibrating elastic plate, Trans. JSME B 64 (1997) 3720–3728. [7] P. Burmann, A. Raman, S.V. Garimella, Dynamics and topology optimization of piezoelectric fans, IEEE Trans. Compon. Packag. Technol. 25 (2002) 592–600. [8] T. Acikalin, A. Raman, S.V. Garimella, Two-dimensional streaming flow induced by resonating thin beams, J. Acoust. Soc. Am. 114 (2003) 1785–1795. [9] T. Acikalin, S.M. Wait, S.V. Garimella, A. Raman, Experimental investigation of the thermal performance of piezoelectric fans, Heat Transf. Eng. 25 (2004) 4–14. [10] T. Acikalin, S.V. Garimella, Analysis and prediction of the thermal performance of piezoelectrically actuated fans, Heat Transf. Eng. 30 (2009) 487–498. [11] S.M. Wait, S. Basak, S.V. Garimella, A. Raman, Piezoelectric fans using higher flexural modes for electronics cooling applications, IEEE Trans. Compon. Packag. Technol. 30 (2007) 119–128. [12] M. Kimber, S.V. Garimella, Measurement and prediction of the cooling characteristics of a generalized vibrating piezoelectric fan, Int. J. Heat Mass Transfer 52 (2009) 4470–4478. [13] M. Kimber, S.V. Garimella, Cooling performance of arrays of vibrating cantilevers, J. Heat Transfer 131 (2009) 111401. [14] M. Kimber, R. Lonergan, S.V. Garimella, Experimental study of aerodynamic damping in arrays of vibrating cantilevers, J. Fluids Struct. 25 (2009) 1334–1347. [15] C.-N. Lin, Analysis of three-dimensional heat and fluid flow induced by piezoelectric fan, Int. J. Heat Mass Transfer 55 (2012) 3043–3053. [16] C.-N. Lin, Heat transfer enhancement analysis of a cylindrical surface by a piezoelectric fan, Appl. Therm. Eng. 50 (2013) 693–703. [17] Y.-H. Kim, S.T. Wereley, C.-H. Chun, Phase-resolved flow field produced by a vibrating cantilever plate between two endplates, Phys. Fluids 16 (2004) 145–162. [18] Y.-H. Kim, C. Cierpka, S.T. Wereley, Flow field around a vibrating cantilever: coherent structure eduction by continuous wavelet transform and proper orthogonal decomposition, J. Fluid Mech. 669 (2011) 584–606. [19] A. Eastman, J. Kiefer, M. Kimber, Thrust measurements and flow field analysis of a piezoelectrically actuated oscillating cantilever, Exp. Fluids 53 (2012) 1533–1543. [20] M. Choi, C. Cierpka, Y.-H. Kim, Vortex formation by a vibrating cantilever, J. Fluids Struct. 31 (2012) 67–78. [21] M. Choi, S.-Y. Lee, Y.-H. Kim, On the flow around a vibrating cantilever pair with different phase angles, Eur. J. Mech. B Fluids 34 (2012) 146–157. [22] C. Cierpka, T. Weier, G. Gerbeth, Evolution of vortex structures in an electromagnetically excited separated flow, Exp. Fluids 45 (2008) 943–953.
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European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Effects of the distance between a vibrating cantilever pair Minsuk Choi a , Christian Cierpka b , Yong-Hwan Kim c,∗ a
Department of Mechanical Engineering, Myongji University, Yongin 449-728, Republic of Korea
b
Institute for Fluidmechanics and Aerodynamics, Bundeswehr University Munich, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
c
Green Energy Group, POSCO E&C, 36 Songdo-Dong, Yeonsu-Gu, Incheon 406-732, Republic of Korea
article
info
Article history: Received 20 March 2013 Received in revised form 19 June 2013 Accepted 15 August 2013 Available online 22 August 2013 Keywords: Vibrating cantilever Counter-rotating vortex Piezoelectric fan Unsteady simulation
abstract Two dimensional unsteady numerical simulations were conducted using a commercial code with a userdefined-function to investigate the effect of the distance between two cantilevers vibrating in counterphase or in phase. The performance of the cantilevers with different distances was mainly evaluated by the time-averaged axial velocity and the mass flow rate. It is evident that there is no interaction between the vortices by two cantilevers if they are too far apart. However, if two cantilevers are too close, they hinder each other in vortex generation. In particular, the interaction between two inner vortices generates a reversed flow which has a negative effect on the performance. Unless the distance is too close, the performance of the cantilever pair vibrating in counter-phase is always superior to the cantilever pair vibrating in phase. The optimal distance between two cantilevers in counter-phase is approximately equal to twice the size of a fully-grown vortex generated by the single cantilever, while there is no distinct optimal distance for a cantilever pair vibrating in phase. In case the distance is larger than three times the vortex size, the flow field generated by each cantilever is similar to the flow field of a single cantilever, which implies that two cantilevers work independently of each other. © 2013 Elsevier Masson SAS. All rights reserved.
1. Introduction Rotary type fans with high efficiency and performance have been widely used for thermal and flow control in conventional large-sized devices. However, it is difficult to make the components of rotary fans, such as the rotor, bearings, motor and shaft, smaller than a critical size for small electronics. This situation forces designers to develop new types of cooling devices with a small size and a simplified structure. An alternative device of generating air flow is a hand-held fan, whose advantage is its simple structure, specifically, a vibrating flat plate. Any flat plate with a simple harmonic oscillation can induce an airflow, and consequently can be used as a cooling device. To obtain sufficient airflow, a fast vibrating motion is required and it can be generated using a piezoelectric material. When AC power is applied to the piezoelectric fan, the plate moves back and forth, because of the electrical potential difference across a piezoelectric material deposited on the plate, and consequently the device generates an airflow. In spite of the simple structure of piezoelectric fans, the airflow generated by the fan has not been fully understood. Toda [1,2] was one of the earliest researchers to perform experiments on the cooling effectiveness of a piezoelectric fan and confirmed that the
∗
Corresponding author. Tel.: +82 32 748 1859; fax: +82 32 748 4022. E-mail addresses: [email protected] (M. Choi), [email protected] (C. Cierpka), [email protected] (Y.-H. Kim). 0997-7546/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechflu.2013.08.006
fan can be used as a cooling device for a transistor. To characterize the flow structure around vibrating flat plates, Watanabe et al. [3], Ihara and Watanabe [4], Tsutsui et al. [5] and Takato et al. [6] investigated the airflow using LDV (Laser Doppler Velocimetry) measurements. However, the low spatial resolution of LDV was insufficient to clearly observe the vortex motion generated by the oscillating plates. Thereafter, a few attempts to characterize a small cantilever fan for electric devices were made by Burmann et al. [7], Acikalin et al. [8–10], Wait et al. [11] and Kimber and Garimella [12] using analytical, computational and experimental methods. In particular, Kimber et al. [13,14] investigated the heat transfer enhancement and the aerodynamic damping in a vibrating cantilever pair. Lin [15,16] found that the piezoelectric fan, based on his experimental and numerical results, can enhance the heat transfer coefficient significantly in comparison to natural convection. With the flow measurements around a vibrating cantilever, Kim et al. [17,18] were able to identify the cyclic generation of counter-rotating vortices induced by the cantilever in detail. They analyzed the vortex motion in a quantitative manner with high-resolution PIV (Particle Image Velocimetry) measurements and wavelet analysis. Eastman et al. [19] measured the thrust generated by a single slender cantilever and investigated the 3-D structure of the flow field around it. Recently, Choi et al. [20] have analyzed numerical simulations thoroughly and found that the static pressure difference across the cantilever tip plays an important role in the formation and development of counter-rotating vortices.
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Nomenclature c CCW CW d f0 h0 l md ms p VT Vxd Vxs
ϕ ω
Effective length of cantilever Counter-clockwise Clockwise Distance between two cantilevers Vibrating frequency Maximum tip deflection amplitude Actual length of cantilever Mass flow rate generated by a pair of cantilevers Mass flow rate generated by a single cantilever Static pressure Maximum tip speed Maximum axial velocity induced by a pair of cantilevers Maximum axial velocity induced by a single cantilever Phase angle Vorticity
To increase the cooled area or the performance of the system, it is beneficial to use multiple piezoelectric fans. Although many researchers have experimented on designing a piezoelectric fan, testing its performance and investigating the flow field around it, only a few papers have been published regarding the flow around two oscillating plates. Ihara and Watanabe [4] have performed numerical simulations using the discrete vortex method and applied smoke visualization to investigate the flow field around two plates oscillating in-phase and in counter-phase. However, the data from their simulation and experiment were insufficient to resolve the flow field in detail. Kimber and Garimella [13] tested the cooling effectiveness of a vibrating cantilever pair in phase on the heated surface and found it can increase the heat transfer coefficient in comparison to the single cantilever. Kimber et al. [14] found that the air damping decreases significantly with in-phase vibration while it increases with counter-phase vibration. Recently, Choi et al. [21] have simulated a 2D flow field generated by oscillating cantilever pair with different phase angles and reported that the interaction of two vibrating cantilevers changes the flow field significantly depending on the phase angles. In this study, the researchers found that the performance of the cantilever pair reaches its maximum when two plates are vibrating 180° out of phase. In addition, Ihara and Watanabe [4] observed that the distance between two vibrating plates in counter-phase has a significant effect on the induced flow field but could not explain this phenomenon in detail due to insufficient data. Therefore, the present study investigates the effect of the distance between two vibrating cantilevers on the performance and focuses on finding the underlying mechanisms of the performance variation. 2. Validation for a single cantilever Prior to the validation of the numerical results, it is necessary to define the phase angle. Fig. 1 shows the relationship of the phase angles to the deflections of the cantilever. In the previous paper of Choi et al. [20], the numerical results for a single cantilever were compared with the experimental velocity data measured by Kim et al. [17]. A brief comparison, therefore, between the computational results and the experimental data is shown in Fig. 2. In the experiment, the unsteady flow around a vibrating cantilever was visualized by smoke particles and the captured particle images showed the process of the vortex formation, as presented in the left of Fig. 2. The PIV technique has then been applied to obtain the quantitative flow field from the particle images and the results
Fig. 1. Definition of the phase angle.
are shown in the middle of Fig. 2. Finally, the computed flow field in the right of Fig. 2 was compared with the measured flow field. The process of the vortex formation matches well between experimental and computational data. In addition to the flow field, in the previous paper of Choi et al. [20], the deflection shape of the modeled cantilever was matched accurately to the experimentally measured vibration shape at each phase angle. In order to check the accuracy of the computed flow fields, the size of vortices and their trajectories were compared with the experimental data quantitatively. These comparisons confirmed that the computation can capture the overall features of the vortices observed in the experiment during a period and is thus well suited to perform a parameter test. 3. Computational method 3.1. Geometry of a vibrating cantilever pair This part begins with a brief description of the shape of the cantilever, which is the same as that in the experiment of Kim et al. [17]. The piezoelectric fan is a thin metal plate fixed to the apparatus in a cantilevered manner with its bottom end and its top end is free as shown in Fig. 3, the bottom part of which is coated with a piezoelectric material. The actual length (l) of the cantilever is 31 mm but its moving part, referred to as the effective length (c), is 25.4 mm. The plate with the thickness of 0.13 mm vibrates at its fundamental natural frequency (f0 ) of 180 Hz. The tip deflection is fixed at h0 /c = 0.054, where h0 is the maximum tip deflection amplitude, and the corresponding maximum tip speed (VT ) is 1.54 m/s. More details on the experimental setup and the measurement can be found in Kim et al. [17]. Choi et al. [21] simulated the flow around two vibrating cantilevers with eight different phase angles and a fixed distance between both cantilevers. They found that the cantilever pair vibrating in counter-phase is more effective in generating the airflow than the cantilever pair vibrating in phase. The distance between two cantilevers (d) was fixed at 8h0 and this value was chosen based on the size of the counter-rotating vortices generated by a single vibrating cantilever. As shown in Fig. 2, two vortices are generated alternately by the cantilever and they move downstream within −3h0 and 3h0 on the x-axis. Two cantilevers could work as obstacles in the vortex generation process if they are too close, while there would be no interaction between them if they are too far from each other. In the present study, two vibrating cantilevers with different distances between them were tested in phase or in counter-phase in order to find an optimal distance for the highest performance. The distance between the cantilevers was changed from 4h0 to 20h0 .
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Fig. 2. Flow field around vibrating cantilever: smoke visualization (left), PIV measurement (middle) and computational results (right). Source: Reproduced with data of Kim et al. [17] and Choi et al. [20].
3.2. Numerical methods and boundary conditions Two dimensional flow fields around a vibrating cantilever pair were simulated using a commercially available flow solver, Fluent 6.3. The 2D incompressible Reynolds-averaged Navier–Stokes equations, including continuity and momentum equations, were solved with SIMPLE algorithm. A second-order upwind scheme was used for the convection and diffusion terms and an implicit time marching was used to get unsteady flow fields. The flow field around the cantilever was assumed to be turbulent and the turbulent viscosity was obtained by the standard k–ε model. The movement of the cantilever in the computation was implemented using a UDF (user-defined-function), which describes the measured shapes of the deflected cantilever at eight instants in the experiment of Kim et al. [17]. The modeled cantilever was matched well to the experimentally measured vibration shape as shown in the previous study of Choi et al. [20]. The computational domain was divided into two sub-domains, stationary and deformed parts, owing to the motion of the cantilevers. The dynamic mesh
in the deformed part was implemented by using the spring-based smoothing method combined with the local re-meshing method. The mesh in the stationary part does not change during computations. The size of the computational domain was set to be much larger than the cantilever to allow the flow to develop with the lowest possible effects from the boundaries. The inlet and outlet boundaries were taken to be at a distance of 110h0 and 260h0 from the tip of the cantilever, where the effect of the vortices generated by the cantilevers could not be felt, and only the standard atmospheric static pressure was specified on the boundaries. There are two side walls at 100h0 away from the cantilever, and these walls were also present for the experiment. In order to measure the effects of the cantilever only, the no-slip condition was applied on the surface of the cantilever, while the slip (symmetric) condition was used on the side-walls and stiffener. The accuracy of these numerical methods and boundary conditions has already been validated in the previous study and the interested reader is referred to Choi et al. [20].
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Fig. 3. Schematic diagram of the computational geometry.
3.3. Vortex detection method
4.1. Cantilever pair vibrating in counter-phase
To extract the trajectories of the vortices, i.e., the locations of the vortex cores, an automatic detection algorithm, based on the Mexican hat wavelet, was applied to the absolute vorticity field. The same method was already applied to the experimental data of Kim et al. [18] and the computational results of Choi et al. [20] for tracking the vortices. For the current data processing, the numerical velocity fields were interpolated on an equidistant grid with a spacing of 1x = 1y = 0.2h0 . Based on this representation for each instant, the vorticity was calculated using second-order central differences. The λ2 -criterion was then used to filter high vorticity regions in the shear layer developed along the cantilever. Furthermore, the position of the two cantilevers was extracted to remove data points that lie too close to the cantilever surface. The algorithm now correlates the vorticity field with wavelets of different scales. This paper is not intended to discuss all the details of the method, but the interested reader is referred to Cierpka et al. [22] for a deeper treatment of the method. The main benefit of the wavelet method is that the procedure can be calibrated using the analytical Lamb–Oseen vortex model. In the first step, the possible size of structures is identified and the wavelet scales are adapted by the algorithm automatically, which provides the best spatial resolution. In the next step, maxima for the correlations are identified. The position of these maxima in the correlation plane between the wavelet and vorticity fields indicates the center of a vortex and the wavelet scale provides a direct measure of the vortex core size. Additional information such as the center velocity or the vorticity can now be extracted.
It is important for two vibrating cantilevers to generate sufficient air flow continuously for cooling small devices. A total of 32 data sets were captured for the computational results during a cycle and the corresponding flow fields were averaged to obtain a time-averaged flow field, which is suited to quantify the cooling effectiveness. Fig. 4 shows the time-averaged axial velocity distribution along the x-direction at three different ypositions, y/h0 = 1, 5 and 10. In all cases, the maximum velocity appears at y/h0 = 1 just downstream of the cantilever tip. As the vortices move downstream of the cantilever, the maximum axial velocity decreases and the width of the vortex-affected area increases gradually due to the diffusion. For the single cantilever, the averaged axial velocity is symmetric to the cantilever center and its maximum value is 3.8 m/s at y/h0 = 1. With two vibrating cantilevers in counter-phase, the axial velocity has a symmetric pattern to the center (x/h0 = 0) at all three y-positions and it is locally symmetric to each cantilever at y/h0 = 1. When two cantilevers are too close within 4h0 , the maximum axial velocity at y/h0 = 1 is much smaller than that observed using a single cantilever, because of the strong interaction between the vortices from each cantilever. However, the maximum velocity and total flow rate further downstream of the cantilever, y/h0 = 5 and 10, are much larger than those in the case of a single cantilever. When the distance between the cantilevers increases above 6h0 , the maximum axial velocity at y/h0 = 1 is larger than or similar to the single cantilever. In addition, a region with a negative axial velocity is clearly observed between two cantilevers at y/h0 = 1 in the cases of d = 6h0 and 8h0 . The axial velocity distribution at y/h0 = 10 has one peak if the distance is smaller than 8h0 but it has two peaks if the distance is larger than 10h0 . As the distance increases above 12h0 , the axial velocity distribution at the three different y-positions clearly shows that the interaction of the vortices from each cantilever is very weak and it looks like that the cantilevers work independently of each other. To quantify the effect of the distance between the cantilevers on the performance of them, the maximum axial velocity and the flow rate were obtained from the time-averaged axial velocity at y/h0 = 10. Each variable was normalized by the value of the single cantilever and the results are shown in Fig. 5. In particular, the latter was calculated by integrating the time-averaged axial
4. Computational results Checking the grid-independence of the numerical results was already conducted for a cantilever pair and five grids with different resolutions were tested in the previous study of Choi et al. [21]. The same grid resolution, which had obtained the grid-independent solution in the previous study, was applied to the present study. The computational grid was composed of triangular cells in the deformed part around the cantilevers and quadrilateral cells in the stationary part far from the cantilevers. Total cell number was about 74,000.
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(a) Single cantilever.
(b) d = 4h0 .
(c) d = 6h0 .
(d) d = 8h0 .
(e) d = 10h0 .
(f) d = 12h0 .
(g) d = 14h0 .
(h) d = 16h0 . Fig. 4. Time-averaged axial velocity distribution downstream of the cantilevers in counter-phase.
velocity from −20 to 20 of x/h0 . The maximum axial velocity is observed at a distance of d = 4h0 and, as the distance between two cantilevers increases, the maximum axial velocity decreases gradually to become 1.1 times the value of the single cantilever. As the distance increases, the mass flow rate increases to have its maximum value at d = 14h0 and decreases slowly thereafter. In the case of d < 8h0 , as shown in Fig. 4, the strong interaction between the counter-rotating vortices from each cantilever generates a smooth axial velocity distribution with a peak at the center (x/h0 = 0). Although the maximum axial
velocity is larger than in the other cases, the mass flow rate is smaller because of the large mixing loss caused by the vortex interaction. In the case of d > 8h0 , however, the interaction of the vortices is weaker and the axial velocity distribution has two peaks symmetrical to the center. As the distance increases further, the maximum axial velocity would converge to the value of the single cantilever and the mass flow rate doubles in comparison to the single cantilever. Here, there are some important points to determine which distance is optimal for the cantilever pair vibrating in counter-phase. It is evident that a larger value in the
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(i) d = 18h0 .
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(j) d = 20h0 . Fig. 4. (continued)
Fig. 5. Maximum time-averaged axial velocity and normalized mass flow rate in counter-phase at y/h0 = 10.
axial velocity and mass flow rate is better for cooling. The location of the maximum axial velocity is another important factor and it is chosen to be optimal at the center. Therefore, the authors propose the distance between the cantilevers of 6h0 and 8h0 best suited for the cooling of small electric devices, which is also equal to twice the size of the fully-grown vortex generated by the single cantilever. To investigate the interaction of the counter-rotating vortices from each cantilever in detail, velocity vectors and vorticity distribution were calculated and the results are shown in Fig. 6, where the phase angle is based on the movement of the right cantilever. When two cantilevers vibrate in counter-phase, the large and strong vortices from each cantilever are symmetric to the center line. In the case of ϕ = 0° and d = 4h0 , the vortices from the left cantilever (CW1 and CCW1) push the flow horizontally to the right while the vortices from the right cantilever (CW2 and CCW2) push the flow to the left. CW1 and CCW2 interact violently with each other and generate a strong reversed flow between them. CCW1 and CW2 also generate a flow in the downstream direction but its intensity is much smaller than the reversed flow. Consequently, a saddle point is clearly visible between the four vortices. At ϕ = 180° and d = 4h0 , the interaction between the vortices from the same cantilever is stronger than that between the vortices from the different cantilevers, generating a flow in the left-up or right-up direction. Because CW1 and CCW2 at this instant are too weak to generate the reversed flow, the resulting flow moves downstream. At ϕ = 0° and d = 8h0 , the reversed flow still exists between CW1 and CCW2 but its intensity decreases significantly in comparison to the case of ϕ = 0° and d = 4h0 . The interaction between CW1 and CCW1 or between CW2 and CCW2 pushes the flow in the downstream direction. At ϕ = 180° and d = 8h0 , CW1 and CCW2 are still strong enough to interact with CCW1 and CW2 respectively, generating a flow in the downstream direction, while the interaction between CW1 and CCW2 weakens because of the increased distance between two cantilevers. The distance between two cantilevers at d = 12h0 is larger than the
size of the fully-grown vortex, so that the interaction between CW1 and CCW2 is very weak even at ϕ = 0°. The vortices from each cantilever show similar patterns to the single cantilever, and it seems that the cantilevers generate the counter-rotating vortices independently of each other. The interaction between the vortices from the same cantilever is always stronger than that from different cantilevers. In the three cases shown in Fig. 6, the reversed flow between two cantilevers always exists at ϕ = 0° and causes the inner vortices, CW1 and CCW2, to lag behind the outer vortices, CCW1 and CW2. It is evident that the vortices from the different cantilevers interact violently with each other and cause large momentum loss if two cantilevers are too close, whereas the cantilever works independently of each other if the distance is large enough. Two numerical pressure sensors were installed on each side of the cantilever at 0.5h0 away from the tip and they measured the static pressure on both sides at the 16 time instants. The static pressure difference (1p) is calculated by subtracting the static pressure on the right side from the static pressure on the left side and the results are shown in Fig. 7. The static pressure difference on the left cantilever is symmetric of that on the right cantilever with respect to the zero line, so it is not shown here. In the previous study of Choi et al. [20], it was found that the static pressure difference is a major driving force in vortex formation and its value changes periodically during a cycle. Due to the method for calculating the static pressure difference, the clockwise vortex is generated with a positive pressure difference and the counter-clock-wise vortex is generated with a negative pressure difference. In addition, when the static pressure difference crosses the zero line, an already full-grown vortex separates from the tip and a new vortex with the opposite rotational direction is initiated. The absolute value of the maximum and minimum static pressure differences in all cases is larger than those for the single cantilever, implying that the counter-rotating vortices generated by the cantilever pair in counter-phase are stronger than the vortices generated by the single cantilever. At d = 4h0 , in particular, the vortex formed at the inner part of the cantilevers (CCW2 in Fig. 6(a)) is much stronger than at the outer part (CW2 in Fig. 6(b)), which is also observed in the static pressure difference. The pressure difference variation on the right cantilever converges to the value on the single cantilever as the distance between two cantilevers increases, but the fluctuating amplitude of the static pressure difference is still larger than the single cantilever even at d = 20h0 . This is the main reason why the maximum axial velocity induced by the cantilever pair at y/h0 = 1 is larger than that by the single cantilever. Fig. 8 shows the trajectories of the counter-rotating vortex pair obtained by the vortex detection method described above. The vortex center is blanked for clarity if the normalized vorticity, i.e. ωh0 /VT , is smaller than 3. For the single cantilever, the
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Fig. 6. Instantaneous velocity fields (vectors) and vorticity distribution (contours) for the cantilever pair vibrating in counter-phase.
trajectories of two counter-rotating vortices are symmetric to the cantilever. Two vortices are alternately generated by the single cantilever and the induced flow by the vortices moves in the downstream direction. When two cantilevers are sufficiently close, all trajectories are symmetric to the center but asymmetric to each cantilever. At d = 4h0 , two inner vortices cannot propagate to the downstream region because of the reversed flow between them, while two outer vortices are forced to move toward the inside of two cantilevers due to continuity. As the distance between two cantilevers increases, the reversed flow becomes weaker and the interaction between two outer vortices is negligible. This is why the trajectories of four vortices at d = 8h0 are more straight than those at d = 4h0 . At d = 12h0 , the trajectories are also symmetric to each cantilever and look similar to those for the single cantilever.
4.2. Cantilever pair vibrating in phase In the previous study of Choi et al. [21], the performance of the cantilever pair vibrating in phase was lower than the cantilever pair vibrating in counter-phase. Fig. 9 shows the time-averaged axial velocity distribution along the x-direction. At d = 4h0 , the maximum axial velocity at three different y-positions is much smaller than that of the single cantilever. Moreover, a strong reversed flow appears near the center line between two cantilevers, owing to the strong interaction between the counter-rotating vortices from each cantilever. As the distance increases, the maximum velocity at all three y-positions increases and the reversed flow region disappears. In the cases of d > 12h0 , the time-averaged velocity distribution splits into two distinctive distributions, indicating that
M. Choi et al. / European Journal of Mechanics B/Fluids 43 (2014) 154–165
Fig. 7. Static pressure difference across the tip of the right cantilever in counterphase.
the interaction between two cantilevers is very weak. It should be noted that the maximum axial velocity at y/h0 = 1 induced by the cantilever pair vibrating in phase is always smaller than the single cantilever. The maximum axial velocity and the flow rate at y/h0 = 10 were again calculated from the time-averaged flow field and the results are shown in Fig. 10. The maximum axial velocity has the smallest value at d = 4h0 and monotonically increases with the increase of the distance. The mass flow rate induced by the vibrating cantilevers in phase has the same trend as the maximum axial velocity. However, the mass flow rate is always smaller than twice the value of the single cantilever. Moreover, the maximum axial velocity is also smaller than the value for the single cantilever case. As the distance increases further, the maximum axial velocity would converge to the value of the single cantilever and the mass flow rate would double in comparison to the single cantilever, like the cantilever pair in counter-phase. When two cantilevers vibrate in phase, the flow pattern generated by the cantilevers is similar and independent of the distance
(a) Single cantilever.
(c) d = 8h0 .
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between them as shown in Fig. 11. The counter-rotating vortices from the same cantilever generate the flow in the left-up and the right-up directions at ϕ = 0° and ϕ = 180° respectively and these interactions are favorable in the cantilever performance. The reversed flow in the right-down and the left-down directions is also induced by the vortices from the different cantilevers (CW1 and CCW2) and it has a negative effect on the cantilever performance. If two cantilevers are too close to each other, as shown in Fig. 11(a) and (b), the negative effect of the vortex interaction increases and degrades the cantilever performance. In addition, the violent vortex interaction weakens the vortices after separating from the cantilever. As the distance between the cantilevers increases, the interaction between the vortices from the different cantilevers becomes weaker, while the interaction between the vortices from the same cantilever remains similar or becomes stronger. If d > 12h0 , the flow fields indicate that both cantilevers work independently of each other. The static pressure difference across the tip of the right cantilever shows the similar pattern to the single cantilever as shown in Fig. 12, but the absolute value of the maximum and minimum static pressure difference is always smaller than that in the single cantilever. This indicates that the counter-rotating vortices generated by the cantilever pair in phase are weaker than those by the single cantilever. Although the fluctuating amplitude of the pressure difference increases with the increase of the distance, the fluctuating amplitude even at d = 20h0 is still smaller than the single cantilever. If the distance increases further, the pressure difference at each instant converges to the value of the single cantilever. When two cantilevers are vibrating in phase, as shown in Fig. 13, the trajectories of the vortices are symmetric to each cantilever as well as to the center. However, the four vortices are initially weak due to small pressure differences across the cantilever tip and the induced flow is also weak in comparison to the single cantilever. The reversed flow generated by the interaction between two inner vortices also has a negative effect on the movement of the vortices. In particular, the vortices at d = 4h0 hardly move in the downstream direction and become weaker very quickly
(b) d = 4h0 .
(d) d = 12h0 .
Fig. 8. Trajectories of the center of counter-rotating vortices generated by the cantilever pair vibrating in counter-phase.
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(a) Single cantilever.
(b) d = 4h0 .
(c) d = 6h0 .
(d) d = 8h0 .
(e) d = 10h0 .
(f) d = 12h0 .
(g) d = 14h0 .
(h) d = 16h0 .
(i) d = 18h0 .
(j) d = 20h0 . Fig. 9. Time-averaged axial velocity distribution downstream of the cantilevers in phase.
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near the cantilever tip. As the distance between two cantilevers increases, the vortices propagate further downstream, because larger pressure difference induces stronger vortices. The vortex centers at d = 12h0 follow similar trajectories to the single cantilever up to y/h0 = 4. 4.3. Comparison with other studies
Fig. 10. Maximum time-averaged axial velocity and normalized mass flow rate in phase at y/h0 = 10.
Three research papers in the reference lists were considering a cantilever pair. Ihara and Watanabe [4] investigated numerically the flow field induced by two cantilevers with different distances between them. They simulated the flow field using the discrete vortex method combined with the singularity method and compared their numerical results with the experimental visualization
Fig. 11. Instantaneous velocity fields (vectors) and vorticity distributions (contours) for the cantilever pair vibrating in phase.
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Fig. 12. Static pressure difference across the tip of the right cantilever in phase.
qualitatively. In their numerical results where the distance is below 4h0 , the averaged axial velocity induced by the cantilever pair was the same as for the single cantilever for in-phase vibration but it was smaller than for the single cantilever for counter-phase vibration. In the present study, the cantilever pair in counter-phase shows better performance than the cantilevers vibrating in phase for the flow generation at d = 4h0 . The main reason for the difference is the existence of the reversed flow between the two cantilevers. In the time-averaged flow field of Ihara and Watanabe [4], no reversed flow was obtained between the two cantilevers vibrating in phase using the discrete vortex method. However, the reversed flow can be seen between two cantilevers vibrating in phase at d = 4h0 in Fig. 9(b) of the present study. Kimber and Garimella [13] measured the temperature distribution using an infrared camera on a heated surface, which was installed normal to a cantilever pair, and calculated the heat transfer coefficients in order to analyze the cooling effectiveness of the cantilevers. In their experiments, the distance between the two cantilevers vibrating in phase was changed from 0.5h0 to 4h0 . The
(a) Single cantilever.
(c) d = 8h0 .
cantilever pair is more effective than the single cantilever when the distance is below 1.5h0 , but less effective in cooling the heated surface when the distance is between 2h0 and 4h0 with respect to the area-averaged Nusselt number at the center of the cantilever width. Due to the existence of the front wall and the 3D effects, the flow pattern induced by the cantilever pair in their study is slightly different from the flow obtained in the present study and the direct comparison is difficult. However, it is possible to conjecture the performance of the cantilever pair used in the present study when the distance is below 4h0 . In Fig. 10, the maximum time-averaged axial velocity and normalized mass flow rate decrease monotonically as the distance between two cantilevers vibrating in phase decreases from 20h0 to 4h0 , implying that the distance reduction causes the cantilever pair to be less effective in cooling. In particular, the predicted cooling effect of the cantilever pair based on the maximum time-averaged axial velocity and normalized mass flow rate is smaller than that of the single cantilever at d = 4h0 , which is coincident with the result of Kimber and Garimella [13]. Therefore, the authors conjecture that the maximum time-averaged axial velocity and normalized mass flow rate might decrease continuously till d = 2h0 but increase around d = 1.5h0 to form a local peak when two cantilevers vibrate in phase. Kimber et al. [14] found the air damping has a large effect on the vibrating amplitude of an array of cantilevers when they are close to each other in the face-to-face configuration. The vibrating amplitude significantly increases with the cantilevers vibrating in phase, while it significantly decreases with the cantilevers vibrating in counter-phase. This implies that the air damping has a positive effect on the cantilever pair in phase but it has a negative effect on the cantilever pair in counter-phase. However, the present study was intended to investigate the effect of the distance between the cantilevers on the flow generation. Therefore, all simulations were performed for a specific amplitude which was the same as the single cantilever, rather than for a specific applied power. In a viewpoint of the applied power on the cantilevers, it is possible that the optimal distance between the vibrating cantilevers would change. Another important finding of Kimber
(b) d = 4h0 .
(d) d = 12h0 . Fig. 13. Trajectories of the center of counter-rotating vortices generated by the cantilever pair vibrating in phase.
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et al. [14] was that the effect of the air damping is negligible and the two cantilevers behave independently when the distance is above 12h0 , which is the same value as found in the present study. 5. Conclusions In the present study, the performance of a cantilever pair vibrating in counter-phase or in phase was investigated numerically for several distances between the two cantilevers. The performance was mainly evaluated by the time-averaged axial velocity and the mass flow rate downstream of the cantilevers. The following conclusions could be made from the numerical results. 1. The cantilever pair vibrating in counter-phase is superior to the single cantilever in generating the flow, unless the distance between the two cantilevers is too close. However, the cantilever pair vibrating in phase is inferior to the single cantilever in the maximum velocity regardless of the distance. Therefore, it is required to operate two cantilevers in counterphase with a proper distance between them. 2. When two cantilevers are vibrating in counter-phase, it is possible to find the optimal distance between them, at which the counter-rotating vortices from each cantilever interact favorably without interruption between each other. The optimal distance between two cantilevers is equal to twice the size of the fully-grown vortex generated by a single cantilever. The value for the optimal distance found in the present study is 6h0 ∼ 8h0 . 3. When the distance is larger than three times the vortex size, i.e. 12h0 , the interaction between the vortices from each cantilever is negligible and the two cantilevers seem to work independently from each other. References [1] M. Toda, Theory of air flow generation by a resonant type PVF2 bimorph cantilever vibrator, Ferroelectrics 22 (1979) 911–918. [2] M. Toda, Voltage-induced large amplitude bending device—PVF2 bimorph—its properties and applications, Ferroelectrics 32 (1981) 127–133.
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