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On the flow around a vibrating cantilever pair with different phase angles
European Journal of Mechanics B/Fluids 34 (2012) 146–157
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On the flow around a vibrating cantilever pair with different phase angles Minsuk Choi a , Sang-Youp Lee b , Yong-Hwan Kim c,∗ a
Department of Mechanical Engineering, Myongji University, Yongin 449-728, Republic of Korea
b
Biomedical Research Institute, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea
c
New Growth Business Department, Growth and Investment Division, POSCO, Seoul 135-777, Republic of Korea
article
info
Article history: Received 24 June 2011 Received in revised form 27 December 2011 Accepted 31 January 2012 Available online 8 February 2012 Keywords: Vibrating cantilever Counter-rotating vortex Piezoelectric fan Unsteady simulation
abstract The unsteady flow fields generated by cantilevers were simulated using a commercial flow solver. The motion of cantilevers was described realistically with a user-define-function in the flow solver, and it was matched well with the experimentally measured one. For validation, the flow induced by a single vibrating cantilever was compared with the experimental data qualitatively, and the numerical results clearly showed that the numerical method can generate realistic flow phenomena. In the pair-wise configuration, the effect of phase angle difference between two cantilevers was analyzed by using the unsteady flow and the time-averaged velocity field. The flow was significantly affected by the phase angle difference, thus it was symmetric to the center in some cases but asymmetric in others. It was found that the interaction of counter-rotating vortices generated by each cantilever changes significantly with the phase angle difference, and this phase difference is the main factor that determines overall flow features. From the viewpoint of cooling effectiveness, the cantilever pair vibrating in counter-phase is a much more effective way to generate the airflow than the other tested cases. © 2012 Elsevier Masson SAS. All rights reserved.
1. Introduction Rotary type fans are generally used as air-cooling components in conventional electric devices such as computers and projectors. However, many mechanical and electric components require small and effective cooling devices because they are decreasing in size thanks to the rapid development of manufacturing techniques. Thus, recently the designers have been either reducing the size of fans or trying to develop an entirely new type of cooling devices. Although conventional fans work effectively and efficiently in various applications, many problems are involved in decreasing the size of fans for small components. It is very difficult to reduce the size of essential components in a fan because they include the rotor, bearing, motor and others. Moreover, the fan performance drops significantly with small radial size, because most of the load is on the outer parts in a rotary fan. To obtain a sufficient airflow with a smaller radius, the rotational speed of the fan has to be increased. Some researchers are exploring the possibility of replacing the conventional fan with an oscillating flat plate which is suitable for cooling small electric components. A fast vibrating speed is required for generating sufficient airflow for cooling small electric components, and it can be easily acquired using a piezoelectric
∗
Corresponding author. Tel.: +82 2 3457 5322; fax: +82 2 3459 1988. E-mail addresses: [email protected] (M. Choi), [email protected] (S.-Y. Lee), [email protected] (Y.-H. Kim). 0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.01.021
device. When the AC power is applied to a piezoelectric material on a thin plate, the electrical potential difference across the material changes periodically and causes the plate to vibrate; consequently, the plate works as a piezoelectric fan. It is easy to make a piezoelectric fan smaller in size and change the vibrating amplitude with a different voltage. Due to these advantages, the flow around the vibrating plate has attracted much attention. Toda [1,2] tried to resolve the airflow generated by a piezoelectric fan and investigated its effectiveness in cooling. His experiment revealed that the fan can be used as a cooling device to keep the surface temperature of a transistor below a critical value. Ihara and Watanabe’s [3] experiments on visualization of the flow field around vibrating flat plates arranged both singly and in pairs have found that the distance between two plates has large effects on the flow field generated by them in 180° out-of-phase. Tsutsui et al. [4] and Takato et al. [5] have also investigated the flow around a vibrating cantilever and the cooling effectiveness using LDV (Laser Doppler Velocimetry) measurements. However, their experimental results were insufficient to clearly resolve the motion of vortices generated by an oscillating plate due to the low spatial resolution of LDV. Recently, many researchers have been trying to build a small cantilever fan and show the effectiveness of the oscillating plate as a cooling device using analytical, computational and experimental methods [6–9]. In the flow field measurements around a vibrating cantilever, Kim et al. [10,11] have made a large contribution to identification of the cyclic generation of counter-rotating vortices
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Nomenclature c CCW CW f0 h0 l p T VT
ϕ ω
Effective length of cantilever Counter-clockwise Clockwise Vibrating frequency Maximum tip deflection amplitude Actual length of cantilever Static pressure Period Maximum tip speed Phase angle Vorticity
induced by the cantilever in a quantitative manner with highresolution PIV (Particle Image Velocimetry) measurements and wavelet transform. Choi et al. [12] have succeeded in numerically obtaining a 2D flow field formed by the vibrating cantilever and reported that static pressure difference across the cantilever tip plays an important role in the formation and development of counter-rotating vortices. 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 [3] have performed numerical simulations using the discrete vortex method and experimented on the smoke visualization to investigate the flow field around two plates oscillating in-phase and in counter-phase. However, the data from the simulation and experiment is insufficient to resolve the flow field in detail. This work is the continuation of the previous works [10–12] to investigate the vortex-generation mechanism induced by vibrating cantilevers in pairs using numerical simulations. With proper validations, numerical simulations are more appropriate to investigate the flow change depending on small variations in each parameter. This paper is focused on the analysis of the interaction of two vibrating cantilevers with different phase angles. 2. Validations for single vibrating cantilever The numerical results for a single cantilever were validated in the previous paper of Choi et al. [12]. The trajectories of vortex cores and its size were compared with the experimental data of Kim et al. [10] to confirm the accuracy of the computed flow fields quantitatively. These comparisons showed that the computation can capture the overall features of the vortex movement and the vortex generation process observed in the experiment. This section, therefore, compares the computational data and experimental results qualitatively. 2.1. Geometry of a vibrating cantilever Experimental velocity data measured by PIV technique were used for the validation of numerical results with a vibrating cantilever. The details of the experimental setup and the measurement can be found in Kim et al. [10,11]. This section begins with a brief description of the shape of the cantilever. The piezoelectric fan is a thin metal plate fixed to the apparatus in a cantilevered manner with its bottom end clamped and top end free, the bottom part of which is coated with a piezoelectric material. The length (l) of the cantilever is 31 mm and its thickness is 0.13 mm, but its moving part, referred to the effective length (c), is 25.4 mm. The vibrating frequency (f0 ) in this study is 180 Hz and this value matches
Fig. 1. Definition of the phase angle of the deflected cantilever tip.
with the fundamental natural frequency of the plate. The tip deflection is fixed at 0.054 of h0 /c, where h0 is the maximum tip deflection amplitude, and the corresponding maximum tip speed (VT ) is 1.54 m/s. Fig. 1 shows the relationship of the phase angles to the deflection of the cantilever tip. 2.2. Numerical methods and boundary conditions Two dimensional flow fields around a vibrating cantilever were calculated at the mid-span of the experimental plate using a commercial 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. Since the mesh near the cantilever was significantly affected and severely deformed by its movement, the size of the computational domain was set to be much larger than the cantilever, allowing the flow to develop with least effects of the boundaries. The inlet and outlet boundaries were located 110h0 and 260h0 from the tip of the cantilever, where the effect of the vortices generated by the cantilever 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. In order to measure the effects of the cantilever only, no-slip condition was applied on the wall of the cantilever, while the slip (symmetric) condition was used on the walls and stiffener. The computational domain was divided into two sub-domains to implement the movement of the cantilever, as shown in Fig. 2(a). The mesh in the stationary part does not move during simulations, while the nodes in the deformed part change their positions according to the movement of the cantilever. Fig. 2(a) also shows the initial mesh composed of triangles around the cantilever. One of the most challenging parts in this numerical simulation is to describe the movement of the cantilever as close to the real one as possible. In the previous study, Kim et al. [10] have measured the shapes of deflected cantilever at eight instants, finding that the deflection trajectories at the tip center were presented by the sine function. By combining the shape at the maximum deflection and the sine function, the motion of the cantilever was reconstructed and implemented using UDF (User Define Function) in Fluent 6.3. Another problem arose from the deformed mesh around the vibrating cantilever. The dynamic mesh was implemented by using the spring-based smoothing method
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(a) Initial mesh.
(b) Deformed mesh. Fig. 2. Computational mesh and deformation.
(a) Tip deflection trajectories.
(b) Shapes of deflected cantilever. Fig. 3. Movement of cantilever plate vibrating at 180 Hz.
combined with the local re-meshing method. In the spring-based smoothing method, all edges between two nodes are treated as interconnected springs [13]. By applying the smoothing method only, a highly skewed mesh was observed around the tip and in the region surrounding the cantilever, severely affecting the formation and propagation of the vortex generated by the cantilever. The local re-meshing occurred in a mesh cell when its skewness was higher than a critical value, improving the mesh quality during an unsteady calculation, as confirmed in Fig. 2(b) after 40 cycles. 2.3. Comparison between experimental and numerical results Before comparing the numerical results with the experimental data, it is necessary to check whether the modeled cantilever motion is realistic or not. In the experiment, the cantilever deflections were measured along the center of the plate for several phase angles using a laser vibrometer in situ. As mentioned earlier, in the simulation, the cantilever motion was implemented by combining the measured maximum deflection and the sine function. The cantilever deflections in both cases were compared with each other. Fig. 3(a) shows the tip deflection trajectories of the cantilever for one vibration cycle. Clearly, the motion of the tip in both cases is very nearly sinusoidal and two trajectories agree well with each other. The rms difference between the model and the real cantilever is less than 3.3% of the maximum tip deflection. As shown in Fig. 3(b), the flexible solid bars representing the modeled cantilever are well-matched to the experimentally measured vibration shape. These comparisons ensure that the
cantilever model in the simulation moves with a flexible shape like the real one in the experiment. In the unsteady flow simulations, a static pressure or velocity history is generally used to determine whether a computational result converges or not. One numerical sensor was placed at a distance of 5h0 downstream of the cantilever tip, as shown in Fig. 2(a), measuring the unsteady velocity magnitude 500 times a period. The unsteady flow simulation continued for 40 periods to obtain fully-converged results. The velocity magnitude history at the numerical probe shows the cyclic motions with the same fluctuating amplitude after 25 periods as shown in Fig. 4, revealing that a computational result is fully-converged and the flow around the cantilever is also fully-developed. The history has two fluctuations of the velocity per every period, because two counter-rotating vortices are generated by the cantilever movement and they pass by the sensor alternately. The top of Fig. 5 shows the visualized flows around the vibrating cantilever by smoke particles. The four particle images show the development of a pair of counter-rotating vortices generated by the vibrating cantilever during a single cycle. As the vortex pair moves downstream, each vortex increases in size but decreases in intensity, finally becoming undistinguishable due to diffusion. The quantitative flow fields measured by PIV technique and shown in the middle of Fig. 5 identify the precise location of each vortex. The velocity fields shown at the bottom of Fig. 5 were obtained from the computational results during a cycle after the unsteady flow field has had enough time to develop. The flow fields obtained by the experiment and the computation are well-matched to each other
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of the maximum tip speed of the cantilever in both experimental and computational results. However, the vortex intensity in the computation is smaller than that in the experiment. Referring to the experimental results of Kim et al. [10], a pair of counterrotating vortices has the maximum strength at the center plane and weakens near the end walls, because the vortices generated by the cantilever interact with the end walls. Therefore, we speculate that the weak vortices in the computation might have been caused by the lack of 3D effects, which cannot be resolved with 2D simulations. Fig. 4. Velocity magnitude history at vertex located at 5h0 downstream of cantilever tip.
3. Computational results for two vibrating cantilevers
at every instant. The maximum velocity is observed between the counter-rotating vortices, and its value reaches almost four times
Ihara and Watanabe [3] have found that the phase difference between two cantilevers has a great influence on the induced flow
Fig. 5. Flow field around vibrating cantilever during a cycle: Smoke visualization (top), PIV measurement (middle) and computational results (bottom). Source: Reproduced with data of Kim et al. [10] and Choi et al. [12].
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(a) Initial mesh.
(b) Deformed mesh (in-phase). Fig. 6. Computational mesh for cantilever pair.
Fig. 7. Velocity magnitude histories of sensor (R) at 1ϕ = 180° for grid-dependency check. Table 1 Mesh size in computational domain. Grid No.
1 2 3 4
Cell number Stationary part
Deformed part
Total
8,364 9,665 14,706 19,362
25,838 43,314 59,510 80,040
34,202 52,979 74,216 99,402
field. Therefore, on the basis of the validation of the computational results in the single vibrating cantilever, the same numerical procedure was applied to the pair-wise configuration of two cantilevers with eight phase angle differences: 0° (in-phase), 45°, 90°, 135°, 180° (counter-phase), 225°, 270° and 315° out-of-phases. The results are also compared with the flow field around the single cantilever. The size of the entire computational domains is the same as the case with the single cantilever, but two cantilevers are located 8h0 away from each other in the deformed region as shown in Fig. 6. Three numerical sensors were installed at an axial distance of 5h0 downstream of the cantilever tip to check the performance of cantilever pair with phase angle differences and to determine whether the unsteady flow is fully-developed or not. Three sensors record the unsteady velocity magnitude 500 times a period. In the course of checking the grid-dependence of the numerical results, four grids with different resolutions listed in Table 1 were tested. The velocity magnitude histories of the four grids show the cyclic motions after 25 periods as shown in Fig. 7, which means that the flow around two cantilevers has been fully-developed. In the last two periods, Grids 3 and 4 have nearly the same histories although the velocity variations are affected by the number of cells. Therefore, Grid 3 was selected for the simulation of the flow around two cantilevers.
Fig. 8 shows the velocity magnitude histories at three numerical probes for four phase angle differences during 40 periods. Surprisingly, the flow velocity in the cantilever pair decreased in magnitude due to the interaction between two cantilevers, in contrast to the flow in the single cantilever shown in Fig. 4. In particular, the velocity magnitude at 1ϕ = 0° is much smaller than the other cases, while the velocity magnitude at 1ϕ = 180° is similar to the case of the single cantilever. The right sensor at 1ϕ = 90° has the largest value among the three sensors, while the left sensor at 1ϕ = 270° has the largest value. This is why the flow field at 1ϕ = 270° is the mirror image of 1ϕ = 90° with respect to the center line (x/h0 = 0). Another important thing, as can be seen in Fig. 8, is that the symmetry of the flow field generated by the cantilever pair is depending on the difference in phase angle. In other words, the velocities measured at the left and right sensors have the same variations at 1ϕ = 0° and 180°, which reveal that the flow field is symmetric according to the center line. In other cases, the velocity measured by one sensor is much larger than the other. This means that a significantly asymmetric flow field has been developed around the cantilever pair. To investigate the formation of counter-rotating vortices in detail, a time sequence of velocity vectors and vorticity distributions were calculated and the results are shown in Figs. 9, 12 and 14, where the phase angle is based on the location of the right cantilever. In addition, the static pressure difference across the tip of each cantilever was measured at the location of 0.5h0 inboard of the cantilever tip, and the results are shown in Figs. 10, 11 and 13. Ihara and Watanabe [3] have speculated that the static pressure difference along the cantilever, which is induced by the difference of the velocity of the plate along the surface, generates the flow in a downstream direction. Recently, Choi et al. [12] have found using numerical simulations that the static pressure difference plays an important role in generating the counterrotating vortices and its value varies periodically during a cycle.
M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157
(a) 1ϕ = 0°.
(b) 1ϕ = 90°.
(c) 1ϕ = 180°.
(d) 1ϕ = 270°.
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Fig. 8. Velocity magnitude histories.
Fig. 9. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 0°.
The static pressure difference (1p) is calculated by subtracting the static pressure on the right side from the static pressure on the left side, so that the clock-wise vortex is generated with a positive pressure difference and the counter-clock-wise vortex is generated
with a negative pressure difference. Of course, 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.
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Fig. 10. Static pressure difference across tip of each cantilever at 1ϕ = 0°.
Fig. 11. Static pressure difference across tip of each cantilever at 1ϕ = 90°.
At the first stage, the flow generated by two cantilevers vibrating in-phase is introduced for four different instants. During a period, four vortices are always observed as shown in Fig. 9. The intensity of the vortices generated by the cantilever pair is weaker as shown in Fig. 9(a) than the vortex by the single cantilever shown in Fig. 5(a), because the magnitude of static pressure difference in the cantilever pair is smaller than that in the single cantilever as shown in Fig. 10. The counter-clockwise vortex from the right cantilever (CCW2) is ahead of the vortex from the left cantilever (CCW1) due to the flow in the downstream direction between two cantilevers observed in the time-averaged velocity fields, even though two vortices are formed at the same time. At this moment, CCW1 and CCW2 are interacting with the clockwise vortices, CW1 and CW2, respectively, generating a strong flow in the downstream direction between them. In addition, CCW2 is also interacting with CW1, making a weak reversed flow in the upstream direction which keeps them from propagating to the downstream. However, the interaction between CCW1 and CW2 is negligible. Two counterclockwise vortices grow large at the same axial position of theirs during the development till ϕ = 45°. When each cantilever moves a little more to the right, CCW1 and CCW2 separate from the tip and new clockwise vortices are initiated when the pressure difference is zero as confirmed in Fig. 10. At ϕ = 90°, the clockwise vortices are already formed and start interacting with CCW1 and CCW2 respectively. At this instant, CCW1 and CCW2 propagate downstream rapidly since the interaction between two cantilevers is negligible. Once the clockwise vortices grow large at ϕ = 135°, the interaction between new CW1 and CCW2 starts and generates a weak reversed flow in the upstream direction again, keeping CCW2 from moving forward. Due to the interaction between vortices from each cantilever, two saddle points indicated as a white circle with arrows are formed in the vector field. In the case of 1ϕ = 0° as shown in Fig. 9, two saddle points are aligned near the center line and the flow between them always moves upstream. The flow field from ϕ = 180° to ϕ = 360° is a mirror image of the flow field mentioned above with respect to the center line, so they are not shown here.
At 1ϕ = 90°, the flow feature is different from the case in phase. In order to check the initial intensity of each vortex, it is necessary to investigate the static pressure difference across the tip of each cantilever as shown in Fig. 11. The static pressure difference at the right cantilever varies with the similar phase to the single cantilever. In comparison to the single cantilever, the fluctuating amplitude at the right cantilever is similar above 1p = 0 but the amplitude is large below 1p = 0, implying that the counterclockwise vortex is stronger than the clockwise vortex in the right cantilever. The pressure difference in the left cantilever varies with a similar fluctuating amplitude but in different phase, compared with the single cantilever. The interaction between vortices from each cantilever grows stronger at 1ϕ = 90° as shown in Fig. 12(a). Four vortices interact with one another and form a saddle point among them. In particular, the clockwise vortex from the left cantilever (CW1) and the counter-clockwise vortex from the right cantilever (CCW2) come closer to each other and interact more significantly than the inphase case, generating a strong flow to the upstream direction. Due to the strong reversed flow, CW1 cannot propagate in the downstream direction after separating from the tip of the left cantilever. CCW2 develops to a large vortex at the same position and separates from the right cantilever as shown in Fig. 12(b) and (c). When the right cantilever starts moving to the left just after ϕ = 90°, CW1 weakens at the same position significantly and the reversed flow also becomes less. Consequently, CCW2 moves rapidly to the downstream direction as shown in Fig. 12(d). The interaction between vortices from each cantilever grows considerably, but the interaction between counter-rotating vortices from the same cantilever remains strong. At ϕ = 135°, the interaction in front of the right cantilever is much stronger than the left cantilever and the flow induced by the right cantilever moves in the right-up direction. At ϕ = 180°, CCW1 stays at the same position during development even after being separated from the tip as shown in Fig. 12(e), because the new CW1 is so weak that there is no strong axial flow between the counter-rotating vortices from the left cantilever. It is worth noting that there is no saddle point in the flow field, which means that the interaction between two cantilevers is small. In Fig. 12(f), the counter-rotating vortices from each cantilever generate a strong flow in the right-up direction, consequently making the flow lean to the right. Because of the flow moving to the right, CW1 stretches to the right as shown in Fig. 12(g) and the counterrotating vortices from the left cantilever (CW1 and CCW1) align on the same horizontal position as shown in Fig. 12(h). By the way, from ϕ = 270° to 0°, two cantilevers recede away from each other so that the pressure between them decreases significantly, consequently causing the pressure difference of the right cantilever to have a larger value than the single cantilever as shown in Fig. 11. As a result, the strong counter-rotating vortex, CCW2, is formed. If two cantilevers move in counter-phase with 180° out-ofphase, the fluctuating amplitude of the static pressure difference increases significantly in comparison to the other cases as shown in Fig. 13. When the cantilevers are getting close to each other (ϕ = 90° → 270°), they push the air to the center and the static pressure between two cantilevers increases, resulting in a large static pressure difference across the tip. When the cantilevers are receding away from each other (ϕ = 270° → 360°, 0° → 90°), the air is entrained from the center to each side and the static pressure between two cantilevers decreases, also resulting in a large static pressure difference across the tip. The absolute static pressure difference is much larger in receding of the cantilever pair than in getting close. This means that the size of vortices formed at the outer parts of two cantilevers is smaller than at the inner part between them. It is evident as shown in Fig. 14 that the large and strong vortices from each cantilever are symmetric to the center line at
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Fig. 12. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 90°.
1ϕ = 180°. The counter-rotating vortices from one cantilever are interacting violently with the vortices from the other cantilever, making a saddle point among four vortices. The reversed flow observed between CW1 and CCW2 is stronger than other cases. During the development, CW1 and CCW2 stay at the same position and make the reversed flow even stronger as shown in Fig. 14(b),
while CCW1 and CW2 propagate to the downstream. CW1 and CCW2 cannot move to the downstream even after separating from the tip due to the reversed flow between them as shown in Fig. 14(c). Once CCW1 and CW2 are formed and grow strong while CW1 and CCW2 become weak, the interaction between the vortices from the same cantilever is stronger than that between the
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M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157 Table 2 Axial velocity at y/h0 = 5. Phase angle difference (1ϕ ) 0° 45° 90° 135° 180°
Fig. 13. Static pressure difference across tip of each cantilever at 1ϕ = 180°.
vortices from different cantilevers, giving a chance to two vortices (CW1 and CCW2) to move forward as shown in Fig. 14(d). At this moment, another saddle point is formed. As CW1 and CCW2 become less strong quickly by moving to the downstream, two saddle points are getting close as shown in Fig. 14(e) and finally disappear as in Fig. 14(f). At ϕ = 225°, the air between two cantilevers flows to the downstream with a large speed. Just after initiating CW1 and CCW2, the counter-rotating vortices from each cantilever generate a strong flow in the downstream direction as shown in Fig. 14(g). With CW1 and CCW2 growing large, the interaction between these two vortices is growing strong enough to generate a reversed flow between two cantilevers again as shown in Fig. 14(h). Fig. 15 shows trajectories of the center of counter-rotating vortices from each cantilever during a cycle at three phase angle differences, 1ϕ = 0°, 90° and 180°. Here, the center refers to the location of the maximum absolute vorticity. As mentioned above, the trajectories of the counter-rotating vortices at 1ϕ = 0° and 180° are nearly symmetric to the center line, while those at 1ϕ = 90° are severely leaned to the right. From the viewpoint of the cantilever, the vortex movement in the cantilever pair is totally different from that of the single cantilever. As shown in Fig. 15(a), the computed trajectories of the counter-rotating vortices with the single cantilever are nearly symmetric with respect to the center of the cantilever. Due to the interaction between CW1 and CCW2 in all cases with the cantilever pair, however, the counter-rotating vortices generated by each cantilever move toward the inside of two cantilevers. In particular, the trajectory of two vortices outside of the two cantilevers, namely CCW1 and CW2, are inclined severely to the inside at 1ϕ = 0° and 180°. A time-averaged flow field is also important because the cantilever pair should pump sufficient air continuously for cooling a small device. A total of 32 image sets were captured during a cycle and the corresponding velocity fields were averaged to obtain Figs. 16 and 17. To begin with the axial flow, Fig. 16 shows the axial velocity distribution along the x-direction at three different y-positions, y/h0 = 1, 5, and 10. The maximum axial velocity appears in front of the cantilever tip at y/h0 = 1 in all cases. In the case of the single cantilever, the averaged axial velocity has a symmetric distribution to the cantilever center and its maximum velocity is about 3.8 m/s at y/h0 = 1. As the vortices move toward the downstream of the cantilever, the maximum axial velocity decreases but the axial velocity along x-direction increases gradually with the diffusion. With two vibrating cantilevers in phase as shown in Fig. 16(b), the axial velocity at y/h0 = 1 is locally symmetric to each cantilever but the maximum axial velocity is much smaller than the single cantilever, because of the interaction between CW1 and CCW2 as shown in Fig. 9. As the phase difference increases, the axial velocity distribution at y/h0 = 1 becomes asymmetric to the center but it is still symmetric locally to the cantilever center at 1ϕ = 90°. The axial velocity in front of the right
Axial velocity (m/s) Max value
Averaged value
1.158 1.652 2.104 2.264 2.019
0.583 0.624 0.744 0.865 0.908
cantilever is larger than the left one because the overall flow field is inclined to the right. In counter-phase as shown in Fig. 16(d), the axial velocity distribution at y/h0 = 1 is symmetric to the center and also to each cantilever, and its maximum value is larger than other cases. In addition, there is a region with a negative axial velocity between two cantilevers at y/h0 = 1. It is worth noting that the vibrating cantilever pair in counter-phase can obtain more sufficient air for cooling a small device than other cases at the further downstream, y/h0 = 10. To quantify the performance of the cantilever pair with different phase angles, the maximum and the averaged values of the timeaveraged axial velocity were extracted. The latter was obtained at the section of y/h0 = 5 by averaging the axial velocity from −10 to 10 of x/h0 , because the axial velocity is negligible beyond the boundaries. The results are listed in Table 2. The case with 1ϕ = 135° is the best in the maximum and the case with 1ϕ = 180° is the best in the averaged value. The mass flow rate is proportional to the averaged value, and the cantilever vibrating in counter-phase is more effective than other cases. The time-averaged velocity magnitude distribution during a cycle offers considerably more insight into the flow field as shown in Fig. 17. The flow field at only five phase angle differences are shown here, because other cases with 1ϕ = 225°, 270° and 315° are mirror symmetric to the cases with 1ϕ = 135°, 90° and 45° respectively. The flow field is symmetric with respect to the center line at 1ϕ = 0° but the flow field is very weak in comparison to the other cases. With the phase angle difference increasing, the flow starts leaning to the right and the velocity magnitude increases in front of the right cantilever. At 1ϕ = 90°, the flow asymmetry reaches its maximum value, and the difference of the velocity magnitude around the cantilever pair also reaches the maximum value. With further increase in phase angle difference, the velocity magnitude around the left cantilever increases and the value around the right cantilever decreases, making the flow field become less asymmetric. At 1ϕ = 180°, the flow field is completely symmetric to the center line again and the velocity magnitude in front of each cantilever is much larger than that of the case with 1ϕ = 0°. Here it is worth noting that the flow stream with a considerable velocity magnitude (∼1 m/s) is formed between two cantilevers, while the flow stream along the cantilever surface is negligible in the single cantilever configuration as shown in Fig. 16(e). As mentioned above, the static pressure difference across the tip of each cantilever is affected by the phase angle difference, and it makes the flow field around the cantilever pair have a different pattern. 4. Conclusions Using an advanced numerical method, the interaction of counter-rotating vortices from the cantilever pair was investigated in several phase angle differences. The computed flow field was mainly analyzed by using velocity, vorticity and static pressure. This approach enabled us evaluate the qualitative and quantitative effect of the phase angle difference between two cantilevers. The numerical results led us to make the following conclusions.
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Fig. 14. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 180°.
1. The numerical results show that the counter-rotating vortices from each cantilever interact with each other considerably depending on the phase angle difference. The interaction grows larger from 1ϕ = 0° to 1ϕ = 180°, and it has its maximum intensity in the cantilever pair vibrating in counter-phase. Considering the vortex center and the time-averaged flow, the overall
flow generated by the cantilever pair is symmetric to the center line only in phase and counter-phase, while it is leaned to one side in other cases. 2. Due to the interaction of the counter-rotating vortices, one or two saddle points are generated in the flow field, consequently
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(a) Single cantilever.
(b) 1ϕ = 0°.
(c) 1ϕ = 90°.
(d) 1ϕ = 180°. Fig. 15. Trajectories of the center of counter-rotating vortices.
(a) Single cantilever.
(b) 1ϕ = 0°.
(c) 1ϕ = 90°.
(d) 1ϕ = 180°. Fig. 16. Axial velocity distribution downstream of cantilevers.
forming the flow region with a negative axial velocity between the vortices and the cantilevers. 3. The cantilever pair vibrating in counter-phase is more effective in generating the airflow than other cases because of the strong interaction of vortices induced from each cantilever. Two cantilevers in phase have the worst performance among the test
cases and their performance is even inferior to that of the single cantilever. Acknowledgment This work was supported by 2011 Research Fund of Myongji University.
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Fig. 17. Time-averaged velocity magnitude.
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. [3] A. Ihara, H. Watanabe, On the flow around flexible plates, oscillating with large amplitude, J. Fluids Struct. 8 (1994) 601–619. [4] T. Tsutsui, M. Akiyama, H. Sugiyama, K. Shimanaka, Heat transfer enhancement around a rectangular cylinder set in near wake generated by elastically vibrating flat plate, Trans. JSME Ser. B 62 (1996) 3667–3674. [5] K. Takato, T. Tsutsui, M. Akiyama, H. Sugiyama, Numerical analysis of flow around vibrating elastic plate, Trans. JSME Ser. B 64 (1998) 3720–3728. [6] P. Burmann, A. Raman, S.V. Garimella, Dynamics and topology optimization of piezoelectric fans, IEEE Trans. Compon. Packag. Technol. 25 (2002) 592–600.
[7] T. Acikalin, A. Raman, S.V. Garimella, Two-dimensional streaming flow induced by resonating thin beams, J. Acoust. Soc. Am. 114 (2003) 1785–1795. [8] 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. [9] M. Kimber, S.V. Garimella, Measurement and prediction of the coolding characteristics of a generalized vibrating piezoelectric fan, Int. J. Heat Mass Transfer 52 (2009) 4470–4478. [10] 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. [11] 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. [12] M. Choi, C. Cierpka, Y.-H. Kim, Vortex formation by a vibrating cantilever, J. Fluids Struct. (in revision). [13] Fluent User Guide (ver. 6.3), Fluent Inc., 2006 (subsidiary of ANSYS).
Contents lists available at SciVerse ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
On the flow around a vibrating cantilever pair with different phase angles Minsuk Choi a , Sang-Youp Lee b , Yong-Hwan Kim c,∗ a
Department of Mechanical Engineering, Myongji University, Yongin 449-728, Republic of Korea
b
Biomedical Research Institute, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea
c
New Growth Business Department, Growth and Investment Division, POSCO, Seoul 135-777, Republic of Korea
article
info
Article history: Received 24 June 2011 Received in revised form 27 December 2011 Accepted 31 January 2012 Available online 8 February 2012 Keywords: Vibrating cantilever Counter-rotating vortex Piezoelectric fan Unsteady simulation
abstract The unsteady flow fields generated by cantilevers were simulated using a commercial flow solver. The motion of cantilevers was described realistically with a user-define-function in the flow solver, and it was matched well with the experimentally measured one. For validation, the flow induced by a single vibrating cantilever was compared with the experimental data qualitatively, and the numerical results clearly showed that the numerical method can generate realistic flow phenomena. In the pair-wise configuration, the effect of phase angle difference between two cantilevers was analyzed by using the unsteady flow and the time-averaged velocity field. The flow was significantly affected by the phase angle difference, thus it was symmetric to the center in some cases but asymmetric in others. It was found that the interaction of counter-rotating vortices generated by each cantilever changes significantly with the phase angle difference, and this phase difference is the main factor that determines overall flow features. From the viewpoint of cooling effectiveness, the cantilever pair vibrating in counter-phase is a much more effective way to generate the airflow than the other tested cases. © 2012 Elsevier Masson SAS. All rights reserved.
1. Introduction Rotary type fans are generally used as air-cooling components in conventional electric devices such as computers and projectors. However, many mechanical and electric components require small and effective cooling devices because they are decreasing in size thanks to the rapid development of manufacturing techniques. Thus, recently the designers have been either reducing the size of fans or trying to develop an entirely new type of cooling devices. Although conventional fans work effectively and efficiently in various applications, many problems are involved in decreasing the size of fans for small components. It is very difficult to reduce the size of essential components in a fan because they include the rotor, bearing, motor and others. Moreover, the fan performance drops significantly with small radial size, because most of the load is on the outer parts in a rotary fan. To obtain a sufficient airflow with a smaller radius, the rotational speed of the fan has to be increased. Some researchers are exploring the possibility of replacing the conventional fan with an oscillating flat plate which is suitable for cooling small electric components. A fast vibrating speed is required for generating sufficient airflow for cooling small electric components, and it can be easily acquired using a piezoelectric
∗
Corresponding author. Tel.: +82 2 3457 5322; fax: +82 2 3459 1988. E-mail addresses: [email protected] (M. Choi), [email protected] (S.-Y. Lee), [email protected] (Y.-H. Kim). 0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.01.021
device. When the AC power is applied to a piezoelectric material on a thin plate, the electrical potential difference across the material changes periodically and causes the plate to vibrate; consequently, the plate works as a piezoelectric fan. It is easy to make a piezoelectric fan smaller in size and change the vibrating amplitude with a different voltage. Due to these advantages, the flow around the vibrating plate has attracted much attention. Toda [1,2] tried to resolve the airflow generated by a piezoelectric fan and investigated its effectiveness in cooling. His experiment revealed that the fan can be used as a cooling device to keep the surface temperature of a transistor below a critical value. Ihara and Watanabe’s [3] experiments on visualization of the flow field around vibrating flat plates arranged both singly and in pairs have found that the distance between two plates has large effects on the flow field generated by them in 180° out-of-phase. Tsutsui et al. [4] and Takato et al. [5] have also investigated the flow around a vibrating cantilever and the cooling effectiveness using LDV (Laser Doppler Velocimetry) measurements. However, their experimental results were insufficient to clearly resolve the motion of vortices generated by an oscillating plate due to the low spatial resolution of LDV. Recently, many researchers have been trying to build a small cantilever fan and show the effectiveness of the oscillating plate as a cooling device using analytical, computational and experimental methods [6–9]. In the flow field measurements around a vibrating cantilever, Kim et al. [10,11] have made a large contribution to identification of the cyclic generation of counter-rotating vortices
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Nomenclature c CCW CW f0 h0 l p T VT
ϕ ω
Effective length of cantilever Counter-clockwise Clockwise Vibrating frequency Maximum tip deflection amplitude Actual length of cantilever Static pressure Period Maximum tip speed Phase angle Vorticity
induced by the cantilever in a quantitative manner with highresolution PIV (Particle Image Velocimetry) measurements and wavelet transform. Choi et al. [12] have succeeded in numerically obtaining a 2D flow field formed by the vibrating cantilever and reported that static pressure difference across the cantilever tip plays an important role in the formation and development of counter-rotating vortices. 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 [3] have performed numerical simulations using the discrete vortex method and experimented on the smoke visualization to investigate the flow field around two plates oscillating in-phase and in counter-phase. However, the data from the simulation and experiment is insufficient to resolve the flow field in detail. This work is the continuation of the previous works [10–12] to investigate the vortex-generation mechanism induced by vibrating cantilevers in pairs using numerical simulations. With proper validations, numerical simulations are more appropriate to investigate the flow change depending on small variations in each parameter. This paper is focused on the analysis of the interaction of two vibrating cantilevers with different phase angles. 2. Validations for single vibrating cantilever The numerical results for a single cantilever were validated in the previous paper of Choi et al. [12]. The trajectories of vortex cores and its size were compared with the experimental data of Kim et al. [10] to confirm the accuracy of the computed flow fields quantitatively. These comparisons showed that the computation can capture the overall features of the vortex movement and the vortex generation process observed in the experiment. This section, therefore, compares the computational data and experimental results qualitatively. 2.1. Geometry of a vibrating cantilever Experimental velocity data measured by PIV technique were used for the validation of numerical results with a vibrating cantilever. The details of the experimental setup and the measurement can be found in Kim et al. [10,11]. This section begins with a brief description of the shape of the cantilever. The piezoelectric fan is a thin metal plate fixed to the apparatus in a cantilevered manner with its bottom end clamped and top end free, the bottom part of which is coated with a piezoelectric material. The length (l) of the cantilever is 31 mm and its thickness is 0.13 mm, but its moving part, referred to the effective length (c), is 25.4 mm. The vibrating frequency (f0 ) in this study is 180 Hz and this value matches
Fig. 1. Definition of the phase angle of the deflected cantilever tip.
with the fundamental natural frequency of the plate. The tip deflection is fixed at 0.054 of h0 /c, where h0 is the maximum tip deflection amplitude, and the corresponding maximum tip speed (VT ) is 1.54 m/s. Fig. 1 shows the relationship of the phase angles to the deflection of the cantilever tip. 2.2. Numerical methods and boundary conditions Two dimensional flow fields around a vibrating cantilever were calculated at the mid-span of the experimental plate using a commercial 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. Since the mesh near the cantilever was significantly affected and severely deformed by its movement, the size of the computational domain was set to be much larger than the cantilever, allowing the flow to develop with least effects of the boundaries. The inlet and outlet boundaries were located 110h0 and 260h0 from the tip of the cantilever, where the effect of the vortices generated by the cantilever 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. In order to measure the effects of the cantilever only, no-slip condition was applied on the wall of the cantilever, while the slip (symmetric) condition was used on the walls and stiffener. The computational domain was divided into two sub-domains to implement the movement of the cantilever, as shown in Fig. 2(a). The mesh in the stationary part does not move during simulations, while the nodes in the deformed part change their positions according to the movement of the cantilever. Fig. 2(a) also shows the initial mesh composed of triangles around the cantilever. One of the most challenging parts in this numerical simulation is to describe the movement of the cantilever as close to the real one as possible. In the previous study, Kim et al. [10] have measured the shapes of deflected cantilever at eight instants, finding that the deflection trajectories at the tip center were presented by the sine function. By combining the shape at the maximum deflection and the sine function, the motion of the cantilever was reconstructed and implemented using UDF (User Define Function) in Fluent 6.3. Another problem arose from the deformed mesh around the vibrating cantilever. The dynamic mesh was implemented by using the spring-based smoothing method
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(a) Initial mesh.
(b) Deformed mesh. Fig. 2. Computational mesh and deformation.
(a) Tip deflection trajectories.
(b) Shapes of deflected cantilever. Fig. 3. Movement of cantilever plate vibrating at 180 Hz.
combined with the local re-meshing method. In the spring-based smoothing method, all edges between two nodes are treated as interconnected springs [13]. By applying the smoothing method only, a highly skewed mesh was observed around the tip and in the region surrounding the cantilever, severely affecting the formation and propagation of the vortex generated by the cantilever. The local re-meshing occurred in a mesh cell when its skewness was higher than a critical value, improving the mesh quality during an unsteady calculation, as confirmed in Fig. 2(b) after 40 cycles. 2.3. Comparison between experimental and numerical results Before comparing the numerical results with the experimental data, it is necessary to check whether the modeled cantilever motion is realistic or not. In the experiment, the cantilever deflections were measured along the center of the plate for several phase angles using a laser vibrometer in situ. As mentioned earlier, in the simulation, the cantilever motion was implemented by combining the measured maximum deflection and the sine function. The cantilever deflections in both cases were compared with each other. Fig. 3(a) shows the tip deflection trajectories of the cantilever for one vibration cycle. Clearly, the motion of the tip in both cases is very nearly sinusoidal and two trajectories agree well with each other. The rms difference between the model and the real cantilever is less than 3.3% of the maximum tip deflection. As shown in Fig. 3(b), the flexible solid bars representing the modeled cantilever are well-matched to the experimentally measured vibration shape. These comparisons ensure that the
cantilever model in the simulation moves with a flexible shape like the real one in the experiment. In the unsteady flow simulations, a static pressure or velocity history is generally used to determine whether a computational result converges or not. One numerical sensor was placed at a distance of 5h0 downstream of the cantilever tip, as shown in Fig. 2(a), measuring the unsteady velocity magnitude 500 times a period. The unsteady flow simulation continued for 40 periods to obtain fully-converged results. The velocity magnitude history at the numerical probe shows the cyclic motions with the same fluctuating amplitude after 25 periods as shown in Fig. 4, revealing that a computational result is fully-converged and the flow around the cantilever is also fully-developed. The history has two fluctuations of the velocity per every period, because two counter-rotating vortices are generated by the cantilever movement and they pass by the sensor alternately. The top of Fig. 5 shows the visualized flows around the vibrating cantilever by smoke particles. The four particle images show the development of a pair of counter-rotating vortices generated by the vibrating cantilever during a single cycle. As the vortex pair moves downstream, each vortex increases in size but decreases in intensity, finally becoming undistinguishable due to diffusion. The quantitative flow fields measured by PIV technique and shown in the middle of Fig. 5 identify the precise location of each vortex. The velocity fields shown at the bottom of Fig. 5 were obtained from the computational results during a cycle after the unsteady flow field has had enough time to develop. The flow fields obtained by the experiment and the computation are well-matched to each other
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of the maximum tip speed of the cantilever in both experimental and computational results. However, the vortex intensity in the computation is smaller than that in the experiment. Referring to the experimental results of Kim et al. [10], a pair of counterrotating vortices has the maximum strength at the center plane and weakens near the end walls, because the vortices generated by the cantilever interact with the end walls. Therefore, we speculate that the weak vortices in the computation might have been caused by the lack of 3D effects, which cannot be resolved with 2D simulations. Fig. 4. Velocity magnitude history at vertex located at 5h0 downstream of cantilever tip.
3. Computational results for two vibrating cantilevers
at every instant. The maximum velocity is observed between the counter-rotating vortices, and its value reaches almost four times
Ihara and Watanabe [3] have found that the phase difference between two cantilevers has a great influence on the induced flow
Fig. 5. Flow field around vibrating cantilever during a cycle: Smoke visualization (top), PIV measurement (middle) and computational results (bottom). Source: Reproduced with data of Kim et al. [10] and Choi et al. [12].
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(a) Initial mesh.
(b) Deformed mesh (in-phase). Fig. 6. Computational mesh for cantilever pair.
Fig. 7. Velocity magnitude histories of sensor (R) at 1ϕ = 180° for grid-dependency check. Table 1 Mesh size in computational domain. Grid No.
1 2 3 4
Cell number Stationary part
Deformed part
Total
8,364 9,665 14,706 19,362
25,838 43,314 59,510 80,040
34,202 52,979 74,216 99,402
field. Therefore, on the basis of the validation of the computational results in the single vibrating cantilever, the same numerical procedure was applied to the pair-wise configuration of two cantilevers with eight phase angle differences: 0° (in-phase), 45°, 90°, 135°, 180° (counter-phase), 225°, 270° and 315° out-of-phases. The results are also compared with the flow field around the single cantilever. The size of the entire computational domains is the same as the case with the single cantilever, but two cantilevers are located 8h0 away from each other in the deformed region as shown in Fig. 6. Three numerical sensors were installed at an axial distance of 5h0 downstream of the cantilever tip to check the performance of cantilever pair with phase angle differences and to determine whether the unsteady flow is fully-developed or not. Three sensors record the unsteady velocity magnitude 500 times a period. In the course of checking the grid-dependence of the numerical results, four grids with different resolutions listed in Table 1 were tested. The velocity magnitude histories of the four grids show the cyclic motions after 25 periods as shown in Fig. 7, which means that the flow around two cantilevers has been fully-developed. In the last two periods, Grids 3 and 4 have nearly the same histories although the velocity variations are affected by the number of cells. Therefore, Grid 3 was selected for the simulation of the flow around two cantilevers.
Fig. 8 shows the velocity magnitude histories at three numerical probes for four phase angle differences during 40 periods. Surprisingly, the flow velocity in the cantilever pair decreased in magnitude due to the interaction between two cantilevers, in contrast to the flow in the single cantilever shown in Fig. 4. In particular, the velocity magnitude at 1ϕ = 0° is much smaller than the other cases, while the velocity magnitude at 1ϕ = 180° is similar to the case of the single cantilever. The right sensor at 1ϕ = 90° has the largest value among the three sensors, while the left sensor at 1ϕ = 270° has the largest value. This is why the flow field at 1ϕ = 270° is the mirror image of 1ϕ = 90° with respect to the center line (x/h0 = 0). Another important thing, as can be seen in Fig. 8, is that the symmetry of the flow field generated by the cantilever pair is depending on the difference in phase angle. In other words, the velocities measured at the left and right sensors have the same variations at 1ϕ = 0° and 180°, which reveal that the flow field is symmetric according to the center line. In other cases, the velocity measured by one sensor is much larger than the other. This means that a significantly asymmetric flow field has been developed around the cantilever pair. To investigate the formation of counter-rotating vortices in detail, a time sequence of velocity vectors and vorticity distributions were calculated and the results are shown in Figs. 9, 12 and 14, where the phase angle is based on the location of the right cantilever. In addition, the static pressure difference across the tip of each cantilever was measured at the location of 0.5h0 inboard of the cantilever tip, and the results are shown in Figs. 10, 11 and 13. Ihara and Watanabe [3] have speculated that the static pressure difference along the cantilever, which is induced by the difference of the velocity of the plate along the surface, generates the flow in a downstream direction. Recently, Choi et al. [12] have found using numerical simulations that the static pressure difference plays an important role in generating the counterrotating vortices and its value varies periodically during a cycle.
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(a) 1ϕ = 0°.
(b) 1ϕ = 90°.
(c) 1ϕ = 180°.
(d) 1ϕ = 270°.
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Fig. 8. Velocity magnitude histories.
Fig. 9. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 0°.
The static pressure difference (1p) is calculated by subtracting the static pressure on the right side from the static pressure on the left side, so that the clock-wise vortex is generated with a positive pressure difference and the counter-clock-wise vortex is generated
with a negative pressure difference. Of course, 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.
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Fig. 10. Static pressure difference across tip of each cantilever at 1ϕ = 0°.
Fig. 11. Static pressure difference across tip of each cantilever at 1ϕ = 90°.
At the first stage, the flow generated by two cantilevers vibrating in-phase is introduced for four different instants. During a period, four vortices are always observed as shown in Fig. 9. The intensity of the vortices generated by the cantilever pair is weaker as shown in Fig. 9(a) than the vortex by the single cantilever shown in Fig. 5(a), because the magnitude of static pressure difference in the cantilever pair is smaller than that in the single cantilever as shown in Fig. 10. The counter-clockwise vortex from the right cantilever (CCW2) is ahead of the vortex from the left cantilever (CCW1) due to the flow in the downstream direction between two cantilevers observed in the time-averaged velocity fields, even though two vortices are formed at the same time. At this moment, CCW1 and CCW2 are interacting with the clockwise vortices, CW1 and CW2, respectively, generating a strong flow in the downstream direction between them. In addition, CCW2 is also interacting with CW1, making a weak reversed flow in the upstream direction which keeps them from propagating to the downstream. However, the interaction between CCW1 and CW2 is negligible. Two counterclockwise vortices grow large at the same axial position of theirs during the development till ϕ = 45°. When each cantilever moves a little more to the right, CCW1 and CCW2 separate from the tip and new clockwise vortices are initiated when the pressure difference is zero as confirmed in Fig. 10. At ϕ = 90°, the clockwise vortices are already formed and start interacting with CCW1 and CCW2 respectively. At this instant, CCW1 and CCW2 propagate downstream rapidly since the interaction between two cantilevers is negligible. Once the clockwise vortices grow large at ϕ = 135°, the interaction between new CW1 and CCW2 starts and generates a weak reversed flow in the upstream direction again, keeping CCW2 from moving forward. Due to the interaction between vortices from each cantilever, two saddle points indicated as a white circle with arrows are formed in the vector field. In the case of 1ϕ = 0° as shown in Fig. 9, two saddle points are aligned near the center line and the flow between them always moves upstream. The flow field from ϕ = 180° to ϕ = 360° is a mirror image of the flow field mentioned above with respect to the center line, so they are not shown here.
At 1ϕ = 90°, the flow feature is different from the case in phase. In order to check the initial intensity of each vortex, it is necessary to investigate the static pressure difference across the tip of each cantilever as shown in Fig. 11. The static pressure difference at the right cantilever varies with the similar phase to the single cantilever. In comparison to the single cantilever, the fluctuating amplitude at the right cantilever is similar above 1p = 0 but the amplitude is large below 1p = 0, implying that the counterclockwise vortex is stronger than the clockwise vortex in the right cantilever. The pressure difference in the left cantilever varies with a similar fluctuating amplitude but in different phase, compared with the single cantilever. The interaction between vortices from each cantilever grows stronger at 1ϕ = 90° as shown in Fig. 12(a). Four vortices interact with one another and form a saddle point among them. In particular, the clockwise vortex from the left cantilever (CW1) and the counter-clockwise vortex from the right cantilever (CCW2) come closer to each other and interact more significantly than the inphase case, generating a strong flow to the upstream direction. Due to the strong reversed flow, CW1 cannot propagate in the downstream direction after separating from the tip of the left cantilever. CCW2 develops to a large vortex at the same position and separates from the right cantilever as shown in Fig. 12(b) and (c). When the right cantilever starts moving to the left just after ϕ = 90°, CW1 weakens at the same position significantly and the reversed flow also becomes less. Consequently, CCW2 moves rapidly to the downstream direction as shown in Fig. 12(d). The interaction between vortices from each cantilever grows considerably, but the interaction between counter-rotating vortices from the same cantilever remains strong. At ϕ = 135°, the interaction in front of the right cantilever is much stronger than the left cantilever and the flow induced by the right cantilever moves in the right-up direction. At ϕ = 180°, CCW1 stays at the same position during development even after being separated from the tip as shown in Fig. 12(e), because the new CW1 is so weak that there is no strong axial flow between the counter-rotating vortices from the left cantilever. It is worth noting that there is no saddle point in the flow field, which means that the interaction between two cantilevers is small. In Fig. 12(f), the counter-rotating vortices from each cantilever generate a strong flow in the right-up direction, consequently making the flow lean to the right. Because of the flow moving to the right, CW1 stretches to the right as shown in Fig. 12(g) and the counterrotating vortices from the left cantilever (CW1 and CCW1) align on the same horizontal position as shown in Fig. 12(h). By the way, from ϕ = 270° to 0°, two cantilevers recede away from each other so that the pressure between them decreases significantly, consequently causing the pressure difference of the right cantilever to have a larger value than the single cantilever as shown in Fig. 11. As a result, the strong counter-rotating vortex, CCW2, is formed. If two cantilevers move in counter-phase with 180° out-ofphase, the fluctuating amplitude of the static pressure difference increases significantly in comparison to the other cases as shown in Fig. 13. When the cantilevers are getting close to each other (ϕ = 90° → 270°), they push the air to the center and the static pressure between two cantilevers increases, resulting in a large static pressure difference across the tip. When the cantilevers are receding away from each other (ϕ = 270° → 360°, 0° → 90°), the air is entrained from the center to each side and the static pressure between two cantilevers decreases, also resulting in a large static pressure difference across the tip. The absolute static pressure difference is much larger in receding of the cantilever pair than in getting close. This means that the size of vortices formed at the outer parts of two cantilevers is smaller than at the inner part between them. It is evident as shown in Fig. 14 that the large and strong vortices from each cantilever are symmetric to the center line at
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Fig. 12. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 90°.
1ϕ = 180°. The counter-rotating vortices from one cantilever are interacting violently with the vortices from the other cantilever, making a saddle point among four vortices. The reversed flow observed between CW1 and CCW2 is stronger than other cases. During the development, CW1 and CCW2 stay at the same position and make the reversed flow even stronger as shown in Fig. 14(b),
while CCW1 and CW2 propagate to the downstream. CW1 and CCW2 cannot move to the downstream even after separating from the tip due to the reversed flow between them as shown in Fig. 14(c). Once CCW1 and CW2 are formed and grow strong while CW1 and CCW2 become weak, the interaction between the vortices from the same cantilever is stronger than that between the
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M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157 Table 2 Axial velocity at y/h0 = 5. Phase angle difference (1ϕ ) 0° 45° 90° 135° 180°
Fig. 13. Static pressure difference across tip of each cantilever at 1ϕ = 180°.
vortices from different cantilevers, giving a chance to two vortices (CW1 and CCW2) to move forward as shown in Fig. 14(d). At this moment, another saddle point is formed. As CW1 and CCW2 become less strong quickly by moving to the downstream, two saddle points are getting close as shown in Fig. 14(e) and finally disappear as in Fig. 14(f). At ϕ = 225°, the air between two cantilevers flows to the downstream with a large speed. Just after initiating CW1 and CCW2, the counter-rotating vortices from each cantilever generate a strong flow in the downstream direction as shown in Fig. 14(g). With CW1 and CCW2 growing large, the interaction between these two vortices is growing strong enough to generate a reversed flow between two cantilevers again as shown in Fig. 14(h). Fig. 15 shows trajectories of the center of counter-rotating vortices from each cantilever during a cycle at three phase angle differences, 1ϕ = 0°, 90° and 180°. Here, the center refers to the location of the maximum absolute vorticity. As mentioned above, the trajectories of the counter-rotating vortices at 1ϕ = 0° and 180° are nearly symmetric to the center line, while those at 1ϕ = 90° are severely leaned to the right. From the viewpoint of the cantilever, the vortex movement in the cantilever pair is totally different from that of the single cantilever. As shown in Fig. 15(a), the computed trajectories of the counter-rotating vortices with the single cantilever are nearly symmetric with respect to the center of the cantilever. Due to the interaction between CW1 and CCW2 in all cases with the cantilever pair, however, the counter-rotating vortices generated by each cantilever move toward the inside of two cantilevers. In particular, the trajectory of two vortices outside of the two cantilevers, namely CCW1 and CW2, are inclined severely to the inside at 1ϕ = 0° and 180°. A time-averaged flow field is also important because the cantilever pair should pump sufficient air continuously for cooling a small device. A total of 32 image sets were captured during a cycle and the corresponding velocity fields were averaged to obtain Figs. 16 and 17. To begin with the axial flow, Fig. 16 shows the axial velocity distribution along the x-direction at three different y-positions, y/h0 = 1, 5, and 10. The maximum axial velocity appears in front of the cantilever tip at y/h0 = 1 in all cases. In the case of the single cantilever, the averaged axial velocity has a symmetric distribution to the cantilever center and its maximum velocity is about 3.8 m/s at y/h0 = 1. As the vortices move toward the downstream of the cantilever, the maximum axial velocity decreases but the axial velocity along x-direction increases gradually with the diffusion. With two vibrating cantilevers in phase as shown in Fig. 16(b), the axial velocity at y/h0 = 1 is locally symmetric to each cantilever but the maximum axial velocity is much smaller than the single cantilever, because of the interaction between CW1 and CCW2 as shown in Fig. 9. As the phase difference increases, the axial velocity distribution at y/h0 = 1 becomes asymmetric to the center but it is still symmetric locally to the cantilever center at 1ϕ = 90°. The axial velocity in front of the right
Axial velocity (m/s) Max value
Averaged value
1.158 1.652 2.104 2.264 2.019
0.583 0.624 0.744 0.865 0.908
cantilever is larger than the left one because the overall flow field is inclined to the right. In counter-phase as shown in Fig. 16(d), the axial velocity distribution at y/h0 = 1 is symmetric to the center and also to each cantilever, and its maximum value is larger than other cases. In addition, there is a region with a negative axial velocity between two cantilevers at y/h0 = 1. It is worth noting that the vibrating cantilever pair in counter-phase can obtain more sufficient air for cooling a small device than other cases at the further downstream, y/h0 = 10. To quantify the performance of the cantilever pair with different phase angles, the maximum and the averaged values of the timeaveraged axial velocity were extracted. The latter was obtained at the section of y/h0 = 5 by averaging the axial velocity from −10 to 10 of x/h0 , because the axial velocity is negligible beyond the boundaries. The results are listed in Table 2. The case with 1ϕ = 135° is the best in the maximum and the case with 1ϕ = 180° is the best in the averaged value. The mass flow rate is proportional to the averaged value, and the cantilever vibrating in counter-phase is more effective than other cases. The time-averaged velocity magnitude distribution during a cycle offers considerably more insight into the flow field as shown in Fig. 17. The flow field at only five phase angle differences are shown here, because other cases with 1ϕ = 225°, 270° and 315° are mirror symmetric to the cases with 1ϕ = 135°, 90° and 45° respectively. The flow field is symmetric with respect to the center line at 1ϕ = 0° but the flow field is very weak in comparison to the other cases. With the phase angle difference increasing, the flow starts leaning to the right and the velocity magnitude increases in front of the right cantilever. At 1ϕ = 90°, the flow asymmetry reaches its maximum value, and the difference of the velocity magnitude around the cantilever pair also reaches the maximum value. With further increase in phase angle difference, the velocity magnitude around the left cantilever increases and the value around the right cantilever decreases, making the flow field become less asymmetric. At 1ϕ = 180°, the flow field is completely symmetric to the center line again and the velocity magnitude in front of each cantilever is much larger than that of the case with 1ϕ = 0°. Here it is worth noting that the flow stream with a considerable velocity magnitude (∼1 m/s) is formed between two cantilevers, while the flow stream along the cantilever surface is negligible in the single cantilever configuration as shown in Fig. 16(e). As mentioned above, the static pressure difference across the tip of each cantilever is affected by the phase angle difference, and it makes the flow field around the cantilever pair have a different pattern. 4. Conclusions Using an advanced numerical method, the interaction of counter-rotating vortices from the cantilever pair was investigated in several phase angle differences. The computed flow field was mainly analyzed by using velocity, vorticity and static pressure. This approach enabled us evaluate the qualitative and quantitative effect of the phase angle difference between two cantilevers. The numerical results led us to make the following conclusions.
M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157
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Fig. 14. Instantaneous velocity fields (vectors) and vorticity distribution (contours) at 1ϕ = 180°.
1. The numerical results show that the counter-rotating vortices from each cantilever interact with each other considerably depending on the phase angle difference. The interaction grows larger from 1ϕ = 0° to 1ϕ = 180°, and it has its maximum intensity in the cantilever pair vibrating in counter-phase. Considering the vortex center and the time-averaged flow, the overall
flow generated by the cantilever pair is symmetric to the center line only in phase and counter-phase, while it is leaned to one side in other cases. 2. Due to the interaction of the counter-rotating vortices, one or two saddle points are generated in the flow field, consequently
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M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157
(a) Single cantilever.
(b) 1ϕ = 0°.
(c) 1ϕ = 90°.
(d) 1ϕ = 180°. Fig. 15. Trajectories of the center of counter-rotating vortices.
(a) Single cantilever.
(b) 1ϕ = 0°.
(c) 1ϕ = 90°.
(d) 1ϕ = 180°. Fig. 16. Axial velocity distribution downstream of cantilevers.
forming the flow region with a negative axial velocity between the vortices and the cantilevers. 3. The cantilever pair vibrating in counter-phase is more effective in generating the airflow than other cases because of the strong interaction of vortices induced from each cantilever. Two cantilevers in phase have the worst performance among the test
cases and their performance is even inferior to that of the single cantilever. Acknowledgment This work was supported by 2011 Research Fund of Myongji University.
M. Choi et al. / European Journal of Mechanics B/Fluids 34 (2012) 146–157
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Fig. 17. Time-averaged velocity magnitude.
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