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Numerical investigation for effects of actuator parameters and excitation frequencies on synthetic jet fluidic characteristics
Sensors and Actuators A 219 (2014) 100–111
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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Numerical investigation for effects of actuator parameters and excitation frequencies on synthetic jet fluidic characteristics Lv Yuan-wei a , Zhang Jing-zhou a,b,∗ , Shan Yong a , Tan Xiao-ming a a College of Energy and Power Engineering, Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 3 April 2014 Received in revised form 14 August 2014 Accepted 14 August 2014 Available online 23 August 2014 Keywords: Synthetic jet actuator Excitation frequency Geometric parameters Helmholz frequency Numerical simulation
a b s t r a c t Numerical investigation is performed to further address the effects of geometric parameters and excitation frequency of the actuator on the synthetic jet fluidic characteristics by utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model. The vibrating diaphragm is modeled as a movable wall varying in sinusoidal mode. Computations are carried out by using FLUENT software with the coupled user definition function (UDF) describing the diaphragm movement. The results show that the geometric parameters of the actuator, such as the cavity depth and diameter, as well as orifice thickness and diameter, have important influences on the synthetic jet fluidic characteristics. The velocity output could be maximized if the geometric parameters of the synthetic jet actuator are designed to ensure that the cavity acoustic Helmholtz resonance frequency is coincided with the diaphragm excitation frequency. For a fixed actuator cavity, when the diaphragm excitation frequency is consistent with the Helmholtz resonance frequency of actuator cavity, the relative pressure insides the cavity is obviously great during the ejection stroke while low during the suction stroke. In the presented actuator parameters, the synthetic jet is enhanced as the decrease of cavity depth for the fixed orifice. The change of cavity diameter in the vicinity of corresponding cavity acoustic resonance diameter has relatively weaker influence on the synthetic jet. When the diaphragm is excited at high frequency, small orifice diameter will restrict the ejection and suction capacity of the synthetic jet actuator. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A synthetic jet is a quasi-steady jet of fluid generated from the periodic motion of a diaphragm enclosed in a cavity with openings on one or more walls. The fluid inside the cavity is expelled through the orifice as the diaphragm is forced to move upwards. The flow separates at the edge of the orifice, inducing a vortex ring that moves outwards under its own momentum. When the diaphragm moves downwards to entrain fluid into the cavity, the vortex ring is sufficiently distant from the orifice that it is virtually unaffected by the entrainment of the fluid into the cavity. In this fashion, a train of vortex rings moving away from the orifice occurs, whereupon the coherent structures then interact, coalesce, and break down in a transition toward a quasi-steady jet [1,2]. This
∗ Corresponding author at: College of Energy and Power Engineering, Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. Tel.: +86 13913887006. E-mail address: [email protected] (J.-z. Zhang). http://dx.doi.org/10.1016/j.sna.2014.08.009 0924-4247/© 2014 Elsevier B.V. All rights reserved.
operational principle allows for synthetic jets to have the unique property that they are formed from the working fluid in which they are deployed. In contrast to conventional continuous jets, synthetic jets are able to transfer linear momentum without a net mass injection across the flow boundary. On the other hand, conventional jets are formed by the addition of both mass and momentum at the orifice. Unlike synthetic jets, the momentum flux in conventional jets remains conserved. Due to the difference in formation between the two jets, there is no potential core in synthetic jets as opposed to conventional jets. Additionally, the periodic vortical structures introduced into the flow exhibit an ability of synthetic jet to influence the environment at a variety of length scales. This vortex shedding phenomenon is important for many applications, such as control of separated flow [3–5], jet vectoring [6–8], mixing enhancement [9–11], thermal management and heat transfer enhancement [12–17]. A lot of studies have covered the development of the piezoelectric actuator as well as the general behavior and performance of the synthetic jet produced. Results indicated that excitation frequency and cavity orifice were important factors affecting the size
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and formation of coherent structures in synthetic jet flow. Crook et al. [18] studied the development of a round jet using an analytical model and compared his results with hot-wire measurements. The study was focused on the maximum velocity from the jet as function of actuator geometry and operation parameters. It was found that the peak velocity was originated at operating frequency of about 1400 Hz. Chen et al. [19] measured velocity profiles at the centerline of a plane jet, and jet exit velocity profiles along and across the jet slot with a single component hot-wire. They used an actuator with a diaphragm oscillating at a frequency ranging from 500 to 1000 Hz. Guy et al. [20,21] studied a plane synthetic jet produced by piezo-diaphragm with hot-wire measurements. Similar to Chen et al. [19], they observed two resonance frequencies, in this case, 700 and 1160 Hz. Furthermore, it was suggested that there was an optimum combination of all geometric parameters at which the actuator would operate at its full capacity. To address the response of a cavity, Gallas et al. [22,23] presented a simplified lumped element model (LEM) of a piezoelectric-driven synthetic jet actuator. In this model, the individual components of a synthetic jet were modeled as elements of an equivalent electrical circuit using conjugate power variables (i.e. resistor, inductor, and capacitor). Based on the lumped element model, they deduced that the output velocity versus actuation frequency curve had two local maxima corresponding to the actuator cavity acoustic resonance (Helmholtz resonance, fH ) frequency and the diaphragm structural resonance frequency (fD ). Persoons and O’Donovan [24] presented an analytical model derived from simplified gas dynamics, for estimating the synthetic jet velocity and actuator deflection, based on a cavity pressure measurement. It was concluded that low-power operation was achieved by matching actuator and Helmholtz resonance frequencies. Lockerby et al. [25] made numerical-simulation studies to describe the methodology on how Helmholtz resonance affected the interaction of active and nominally inactive micro-jet actuators with a laminar boundary layer. It was previously shown that the conditions for Helmholtz resonance were identical to those for optimizing actuator performance for maximum mass flux. The velocity output could be increased significantly if the synthetic jet actuator could be designed so that the acoustic and structural resonances coincided in order to increase output velocity. This approach had been verified by some following works [26–29]. Pavlova and Amitay [26] experimentally investigated the efficiency and mechanism of cooling a constant heat flux surface by impinging synthetic jet; also, comparison with continuous jet was presented. In their measurements, high frequency (1200 Hz) jets were found to be more effective at smaller axial distances and the low frequency (420 Hz) jets were found to be more effective at larger axial distances. Zhang and Tan [27] investigated experimentally the flow and heat transfer characteristics of a synthetic jet driven by piezoelectric actuator by utilizing particle image velocimetry, hot-wire anemometry and infrared camera. Two resonance frequencies of 540 Hz and 1140 Hz were easily identified. It was evident that the actuator produces fairly high velocities even when operated offresonance and the synthetic jet could not be formed under some frequencies. Chaudhari et al. [28] made a detailed experimental investigation wherein the effect of excitation frequency on the synthetic jet flow was studied for cavities of different depths and for orifices of different diameters. The exit velocity averaged over an excitation cycle indicated a lower and an upper bound on the frequency for the formation of a jet, and showed resonance at two frequencies. The resonant frequencies had been identified as being close to the diaphragm and the Helmholtz frequencies. Jain et al. [29] performed a numerical simulation to investigate the effect of various cavity parameters and orifice/cavity shapes on the ensuing synthetic jet flow. The simulation results showed that synthetic jets are more affected by changes in the geometric parameters of the orifice than those of the cavity.
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The momentum carried by the fluid away from the actuator cavity is strongly dependent on its design, which makes the operating frequency and the geometric parameters of the synthetic jet actuator very crucial aspects in the synthetic-jet assembly. However, the detailed knowledge about the design aspects on the ensuing jet deserves further illustrated, especially on the relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency. In order to further address the effects of actuator parameters and excitation frequency on the synthetic jet fluidic characteristics, a numerical investigation utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model on the synthetic jet actuator fluid field was performed in this study. The synthetic jet fluidic characteristics were explored under the different design parameters of the synthetic jet actuator that include radius and thickness of the orifice, as well as radius and depth of the cavity. The relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency was analyzed. 2. Numerical simulation procedure 2.1. Physical model and computational domain The physical model of a synthetic jet considered in the present is illustrated schematically in Fig. 1. It is composed of a vibrating diaphragm located at the bottom of a cavity, on the opposite face of which is a round orifice. The diaphragm is oscillating up and down in the sinusoidal or cosinoidal mode. The main geometric parameters of the synthetic jet actuator include diameter (do ) and thickness (ho ) of the orifice, as well as diameter (dc ) and depth (hc ) of the cavity. The computational domain consists of three zones including the cavity, the orifice, and the surrounding region into which the jet exits. According to the study of Jain et al. [29], the surrounding size of 30 orifice diameters (in the lateral direction) and 20 orifice diameters (along the axis) is chosen. For this surrounding size, jet near field parameters are not affected by pressure outlet boundary condition applied at the end of the outer region. In the present, a group of baseline geometric parameters are outlined as: dc = 45 mm, hc = 7 mm, do = 3 mm, and ho = 2.5 mm. To address the effects of actuator parameter on the synthetic jet fluidic characteristics individually, one of the geometric parameters is varied in a certain range while the other geometric parameters are maintained as their baseline values. 2.2. Numerical methods The unsteady Reynolds-average Navier–Stokes (RANS) equations are chosen as governing equations for the synthetic jet problems. The flow is assumed to be compressible and turbulent. The compressibility effects come into picture because of the rapid change in pressure/density due to the movement of the diaphragm. The computation is carried out by using the commercial CFD software FLUENT coupled with the user definition function (UDF) describing the diaphragm movement. The segregated, unsteady, symmetric solver is chosen with first-order implicit time scheme that is unconditionally stable with respect to time step size. Pressure-implicit with splitting of operators (PISO) based on a higher degree of approximation between the iterative corrections for pressure and velocity is chosen. Second-order upwind spatial discretization is used for the momentum, turbulent kinetic energy, and turbulent dissipation rate. According to Yoo and Lee [30], Menter’s k–ω shear stress transport (SST) turbulence model is used to account for the turbulent nature of the synthetic jet flows.
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Fig. 1. Physical model of synthetic jet flow simulation.
Successful computational fluid dynamics (CFD) synthetic jet flow simulations that incorporate the oscillating diaphragms displacement performance depend heavily on accurate approximations of the displacement profiles, including the instantaneous deflections and shape modes. As already mentioned, the most crucial part of the numerical simulation of the synthetic jet is the modeling of the vibrating diaphragm. Several approximations have been employed in the past to achieve this. These include using a velocity inlet boundary condition with an assumed profile at the location of the orifice exit [31,32] or using a moving piston boundary condition [33]. The swept volume is maintained identical to that in the case of an actually vibrating diaphragm. In the present simulations, a wall boundary condition with the diaphragm vibrating sinusoidally or cosinoidally in time such that the geometry of the wall is parabolic at any instant has been employed to simulate the vibrating diaphragm [29]
Y (r, t) =
A 1− 2
r 2 rc
sin(2ft)
(1)
where Y is the diaphragm deformation relative to its neutral position, A is the peak-to-peak displacement at the center, r is the lateral distance from the diaphragm center, rc is the radius of the cavity, f is the actuation frequency of diaphragm, t is the time. Eq. (1) is expected to represent a vibrating diaphragm realistically. The instantaneous logarithmic diaphragm velocity is derived by differentiating Eq. (1).
v(r, t) = fA 1 −
r 2 rc
cos(2ft)
(2)
In Fig. 1, the boundary conditions are specified at the cavity end and the ambient surrounding into which the jet is expelled. The diaphragm and cavity surfaces are considered as impermeable walls. The ambient air is specified with a pressure outlet (relative pressure is assumed as zero). The diaphragm is defined separately from the rest of the walls so that a user-defined function can be used to describe its movement.
2.3. Grid and time independence studies Similar to Jain et al. [29], the computational domain is meshed with unstructured mesh (triangular elements) for the cavity (up to the orifice exit). The tri-pave unstructured scheme is chosen to allow for relative displacement among the nodes on the diaphragm. The relative displacement among the nodes on the diaphragm is necessary which is not possible with the structured quadrilateral scheme. The outer field is meshed with structured quadrilaterals. Combining the unstructured and structured grids significantly reduces the simulation time by restricting the number of grid points to be re-meshed after each time step and still allows usage of the dynamic mesh feature of the software, which is applied at the vibrating diaphragm and cavity. Size functions to control and specify the size of the mesh spacing intervals between nodes are attached to the orifice edge, orifice exit vertex, and the diaphragm upper vertex. This ensures fine meshing in the regions where large gradients are present. By the grid independence test, the grid number is chosen as about 40,000. Owing to nonlinearities it is necessary to control the calculated value change to assure proper convergence through the use of under-relaxation parameters. The default under-relaxation parameters are kept at 0.3, 1.0, and 0.7 for pressure, density, and momentum, respectively. The solution is then initialized at the diaphragm with initial value of zero for the gage pressure, axial velocity, and lateral velocity. The default of 120 maximum iterations per time step is kept. A time step of 1/(1000f) is chosen based on the actuation frequency to allow for 1000 time steps per cycle. In the residual monitors, the convergence criteria are set as 10−6 . More details on these solvers could be found in the ANSYS Fluent Software User’s Guide [34].
2.4. Validation of simulation procedure The validation of presented simulation procedure is made by comparison with the experimental study of Mane et al. [35].
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Fig. 2. Instantaneous velocity vectors of a synthetic jet in one cycle. (a) 1/4 T, (b) 1/2 T, (c) 3/4 T, (d) 1 T.
A typical geometric configuration and operational condition is chosen for the present propose. According to the experimental parameters, the computational model is set as: cavity diameter (dc ) of 63.5 mm and cavity depth (hc ) of 5.5 mm, orifice diameter (do ) of 2 mm and thickness (ho ) of 1.6 mm. The peak-to-peak center diaphragm displacement (A) for the Bimorph actuator is 0.396 mm and the excitation frequency is 50 Hz. Fig. 2 presents the instantaneous velocity vectors of a synthetic jet in one cycle form computation. Note that the first several cycle results are affected by numerical transient and are therefore discarded. Here, the final data is analyzed only when the monitored results show stabilization over at least 10 cycles. As the diaphragm oscillates, the ambient air is alternately drawn into and expelled out from the cavity orifice. When the diaphragm moves upwards (corresponding to 1/4 T, as seen in Fig. 2(a)), a shear layer is formed at the orifice edge and the vorticity in the shear layer rolls up ambient air to form a vortex ring. When the diaphragm moves downwards to entrain fluid into the cavity (corresponding to 1/2 T and 3/4 T, as seen in Fig. 2(b) and (c)), the vortex ring is sufficiently distant from the orifice that it is virtually unaffected by the entrainment of the fluid into the cavity. In subsequent images, the new vortex pair undergoes the transition and development,
shedding into the surrounding fluid, wrapping around the cores of the primary vortices and dissipating downstream (corresponding to 1 T, as seen in Fig. 2(d)). Because of periodic formation of new vortex pairs near the orifice exit, continuous momentum is replenished to the downstream to maintain the quasi-continuity of the synthetic jet. Fig. 3 presents the comparison between numerical axial velocity profile and experimental result [35] at maximum expulsion and downstream distance of 2 mm from the orifice. It is shown that the results are slightly different in magnitude but follow a similar trend. The velocity at the center of the orifice is over-predicted by the numerical simulation while the velocity magnitudes beyond r/dc of 0.6 are under-predicted. It is also noticed that the presented computational results are in good agreement with the numerical simulation by Mane et al. [35]. The reason for the differences the velocity magnitudes beyond r/dc of 0.6 was brought out by Mane et al. [35], which could be due to the hotwire inability of detecting flow direction. The higher maximum experimentally measured velocities observed in the outer portion of the jet most likely occurs during the ingestion cycle when the fluid flow is from the outer edges. The numerical results over-predict the maximum measured experimental velocity at the orifice center by only 15%.
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where is air density, c is sound speed in the air, ho is orifice thickness, ro is orifice radius, and Vc is cavity volume. fD =
Fig. 3. Numerical and experimental axial velocity profiles at 2 mm downstream of the orifice.
3. Resonance frequencies Performance of a synthetic jet actuator is expected to be sensitive to the excitation parameters related to the frequency and amplitude of vibration, as well as the parameters related to the cavity and orifice geometry. The depth and diameter of the cavity along with the orifice length and diameter are the geometric parameters, while frequency and amplitude of actuation are the control parameters relevant to cavity design. Based on the lumped element modeling (LEM) by Gallas et al. [22,23], two resonance frequencies, the actuator cavity acoustic resonance (Helmholtz resonance, fH ) frequency and the diaphragm structural resonance frequency (fD ) are given by
fH =
1 2
4ho 3ro2
+
8 32 ro
Vc c 2
−0.5 (3)
1 2
m 6rc4
rc6 (1 − ˛2 ) 16E3
−0.5
(4)
where rc is diaphragm radius, E is diaphragm elastic modulus, ˛ is the Poisson ratio, is diaphragm thickness, and m is diaphragm mass. From formula (4), it is obvious that the diaphragm structural resonance frequency depends on the material elasticity, mass, dimensions of diaphragm. In the present simulation study, the diaphragm structural resonance frequency by itself cannot be determined, due to the assumption of zero thickness and massless wall of the diaphragm. However, as the maximum amplitude of the diaphragm is generally achieved at the structural resonance frequency, the excitation frequency to which the maximum amplitude of diaphragm is correspondent maybe regarded as the structural resonance frequency. In the present simulations, the movement of the vibrating diaphragm is defined as formula (1) and the maximum peak-topeak displacements at the center of diaphragm for all the excitation frequencies are fixed at A = 0.2 mm. This approach is not reasonable for a specific diaphragm excited under different frequency, due to the coupling of the amplitude and excitation frequency. But it is true that every excitation frequency may be regarded as the resonance frequency of corresponding diaphragm configuration. That is, under a specific excitation frequency, a corresponding diaphragm could be excited to achieve maximum amplitude. 4. Results and analysis 4.1. Effect of diaphragm excitation frequency To explore the effects of diaphragm excitation frequency on the synthetic jet, three groups of the actuator geometric parameters are designed in the present, as shown in Table 1. In these actuators, the cavity radius (rc ) and the orifice radius (ro ) are maintained as 22.5 mm and 1.5 mm, respectively. To make the Helmholtz frequencies for these groups are all fixed at 635 Hz, the cavity depth
Fig. 4. Effect of excitation frequency on time-averaged velocity at orifice exit during ejection stroke.
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Table 1 Geometry parameters of synthetic jet actuators with fixed Helmholtz frequency. Actuator
Geometric parameters
No. 1 No. 2 No. 3
rc = 22.5 mm, ro = 1.5 mm, hc = 7 mm, ho = 2.5 mm rc = 22.5 mm, ro = 1.5 mm, hc = 4.1 mm, ho = 5 mm rc = 22.5 mm, ro = 1.5 mm, hc = 2.91 mm, ho = 7.5 mm
(hc ) and orifice thickness (ho ) are varied according to formula (3) respectively. The geometric features for these actuators are marked as: (a) large cavity depth and small orifice thickness for actuator No. 1, (b) moderate cavity depth and moderate orifice thickness for actuator No. 2, and (c) small cavity depth and large orifice thickness for actuator No. 3. Fig. 4 shows the effects of diaphragm excitation frequency on the area-averaged velocity at orifice exit during the ejection stroke. The time-averaged velocity at orifice exit during the ejection stroke is defined as
V0 = f
T/2
v0 (t)dt
(5)
0
Here v0 is the area-averaged instantaneous velocity at orifice exit during the ejection stroke, T is the period of oscillation. V0 can be used for representing the intensity of synthetic jet. It is seen from Fig. 5 that the effects of diaphragm excitation frequency on the time-averaged velocity at orifice exit during the ejection stroke are significantly different in magnitude but follow a similar trend. When the diaphragm excitation frequency coincides with the actuator cavity Helmholtz resonance frequency, the time-averaged velocity at orifice exit during the ejection stroke reaches to the maximum value corresponding to each synthetic jet actuator. Taking actuator No. 2 for example, as the diaphragm excitation frequency is increased from 100 Hz to 635 Hz, V0 is varied approximately form 1 m/s to 14 m/s in nearly linear mode. As the diaphragm excitation frequency increases after 635 Hz, V0 undergoes rapid decline from 14 m/s to 5 m/s under 1400 Hz. When the diaphragm excitation frequency increases further, the intensity of synthetic jet gets weaker slowly. In generally, when the diaphragm excitation frequency coincides with the actuator cavity Helmholtz resonance frequency, the ejection and suction capacities of the gas near the orifice are enhanced, and the ejection and suction processes are also coincided together, resulting in the maximum synthetic jet velocity output. This result is in good agreement with the previous investigations. Another notable phenomenon is that the maximum velocity output corresponding to the actuator No. 3 is the highest, while the maximum velocity output corresponding to the actuator No. 1 is the lowest, although they are all excited under the optimum frequencies coincided with the acoustic resonances. In relative to actuator No. 3, the peak velocity corresponding to the cavity Helmholtz resonance frequency is reduced approximately by 12.5% for actuator No. 2, while 30% for actuator No. 1. It is indicated that the cavity dimension and orifice dimension have important effects on the synthetic jet fluidic characteristics. Fig. 5 shows the instantaneous pressure contours at t = 0.25 T under different diaphragm excitation frequencies. At this instant, the diaphragm moves upwards from the neutral station to the top peak station. Fig. 6 shows the instantaneous pressure contours at t = 0.75 T under different diaphragm excitation frequencies. At this instant, the diaphragm moves downwards from the neutral station to the bottom peak station. In these figures, the pressure is the relative pressure to the ambient pressure. It is evident that the relative pressure insides the cavity corresponding to the diaphragm excitation frequency of 635 Hz is obviously greater than the other diaphragm excitation frequencies during the ejection stroke of the cycle, as seen in Fig. 5(b), building up stronger ejection capacity.
Fig. 5. Instantaneous pressure contours during ejection stroke of the cycle (t = 0.25 T). (a) f = 50 Hz, (b) f = 635 Hz, (c) f = 1400 Hz.
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During the suction stroke of the cycle, the relative pressure insides the cavity corresponding to the diaphragm excitation frequency of 635 Hz is obviously lower than the other diaphragm excitation frequencies, as seen in Fig. 6(b), resulting in stronger suction capacity. When the diaphragm is excited at 50 Hz, the perturbation of the vibrating diaphragm to the internal air insides the cavity is very weak, with very small pressure difference between the actuator cavity and the surrounding, as seen in Figs. 5(a) and 6(a), thus it is difficult to form the synthetic jet at the orifice exit. When the diaphragm is excited at 1400 Hz, the ejection and suction of the gas near the orifice are difficult to coincide together because of the high operating frequency. Owing to the inertial action of the fluid inside cavity, the perturbation of the vibrating diaphragm to the air near the orifice is discrepant seriously to that near the diaphragm under high operating frequency. It is evident that the ejection and suction of the gas near the orifice are fall behind the diaphragm movement. During the ejection stroke of the diaphragm, the internal flow near the orifice is remained at the suction status as seen in Fig. 5(c). While during the suction stroke of the diaphragm, the internal flow near the orifice is remained at the ejection status as seen in Fig. 6(c). Phase angle versus diaphragm excitation frequency curve and velocity at the orifice exit versus time curves are presented in Fig. 7. As the compressible effect is very weak in the presented situation, the instantaneous velocity at orifice exit during the suction stroke is nearly the same as that during the ejection stroke, as seen in Fig. 7(a), which is coincidence with the pulsating nature of the synthetic jet flow [1,2]. As regards to the phase angle between the diaphragm motion and the velocity at the orifice exit, it is found that the phase angle is increased rapidly from 50 Hz to 1000 Hz and then it is varied slowly. Here a cycle of operation of diaphragm motion is assumed to be 0◦ where 270◦ is when the diaphragm is farthest from the orifice, as seen in Fig. 7(b). When the diaphragm is excited at optimum frequency of 635 Hz, the phase angle between the diaphragm motion and the velocity at the orifice exit is approximately 90◦ , which means that when the flow peaks at the orifice exit, the diaphragm motion is still toward the orifice. Thus, the peak velocity is sustained for a longer period. When the diaphragm is excited at a higher frequency, the phase difference between the diaphragm and the flow is closed to 180◦ . For large excitation frequency, by the time the flow peaks, the diaphragm may start moving away from the orifice thus reducing the pressure inside the cavity, which results in a decrease in the velocity. 4.2. Effects of cavity dimension
Fig. 6. Instantaneous pressure contours during suction stroke of the cycle (t = 0.75 T). (a) f = 50 Hz, (b) f = 635 Hz, (c) f = 1400 Hz.
Performance of a synthetic jet actuator is expected to be sensitive to the parameters related to the cavity and orifice geometry. In the present, the effects of the cavity dimension are investigated individually for a fixed orifice with radius (ro ) of 1.5 mm and thickness (ho ) of 2.5 mm. Fig. 8 presents effects of cavity depth on the average velocity at orifice exit during the expulsion stroke. Here the cavity depth (hc ) ranges from 2 mm to 30 mm, while the radius of the cavity (rc ) is maintained as 22.5 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the cavity depths at which the actuator cavity Helmholtz resonance frequencies will coincide with the corresponding diaphragm excitation frequencies are 17.6 mm, 7 mm and 3.51 mm, respectively. For convenience, they are named as cavity acoustic resonance depths. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the cavity depth of about 8 mm, which is discrepant seriously from the cavity acoustic resonance depth of 17.6 mm deduced from formula (3). However, it must be aware of that the effect of cavity depth on the synthetic jet performance
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Fig. 7. Effect of excitation frequency on instantaneous velocity and phase angle for actuator No. 3. (a) Velocity at orifice exit versus time curves and (b) phase angle versus excitation frequency curve.
under this lower excitation frequency seems very slight. In this diaphragm excitation frequency, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. The average velocity at orifice exit during the ejection stroke is ranged between 9.4 m/s and 10.2 m/s when the cavity depth is varied from 2 mm to 18 mm. As regards to the discrepancy between the cavity depth corresponding to the highest synthetic jet intensity and the acoustic resonance depth, it is should noticed that the Helmholtz frequency presented by Gallas et al. [22,23] is deduced on the simplified lumped element model of a synthetic jet actuator. The nonlinear inertial action of the fluid inside cavity is not considered enough by this model, which maybe induces negligible influence on the resonance frequency of the actuator. For a larger
cavity depth, the inertial action of the fluid inside cavity on the ejection and suction of the gas near the orifice behaves more strongly, which maybe responsible for resulting in the significant deviation of the real cavity acoustic resonance frequency to the Helmholtz frequency obtained from formula (3). When the diaphragm is excited at 635 Hz, the calculated cavity depth corresponding to the highest synthetic jet intensity is closed to the cavity acoustic resonance depth of 7 mm. When the diaphragm is excited at 900 Hz, the highest synthetic jet intensity is occurred at the cavity depth of 3.5 mm obtained from computation, which is perfectly the same as cavity acoustic resonance depth of 3.51 mm. As the diaphragm excitation frequency increases, the corresponding cavity acoustic resonance depth decreases, making
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Fig. 8. Effect of cavity depth on time-averaged orifice velocity.
the inertial action of the fluid inside cavity week on the ejection and suction of the gas near the orifice. Therefore, the cavity depth corresponding to the highest synthetic jet intensity becomes closed to the acoustic resonance depth, and the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. When the diaphragm is excited under 900 Hz, the peak velocity reaches up 23 m/s, which is approximately 130% increased in relative to 400 Hz case. Under higher excitation frequency, it is also found that the average velocity at orifice exit during the ejection stroke changes significantly in the vicinity of cavity acoustic resonance depth. The velocity versus cavity depth curve is much sharper as observed for the 900 Hz case than 400 Hz case. Once the cavity depth offsets from the cavity acoustic resonance depth, the velocity output of the synthetic jet will decreases dramatically. Fig. 9 presents effects of cavity radius on the average velocity at orifice exit during the expulsion stroke. Here the cavity radius (rc ) ranges from 10 mm to 40 mm, while the depth of the cavity (hc ) is maintained as 7 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the cavity acoustic resonance radiuses coincided with the corresponding diaphragm excitation frequencies are 35.7 mm, 22.5 mm and 15.9 mm, respectively. Here the definition of cavity acoustic resonance radius is similar to the cavity acoustic resonance depth. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the cavity radius of about 37 mm, which is discrepant a little from the cavity acoustic resonance radius deduced from formula (3). When the diaphragm is excited at 635 Hz or 900 Hz, the discrepancy between the cavity radius corresponding to the highest synthetic jet intensity and the acoustic resonance radius is nearly zero. It is observed that the change of cavity radius in the vicinity of cavity acoustic resonance radius has weaker influence on the average velocity at orifice exit during the ejection stroke in relative to the case of varying cavity depth. The calculated cavity radiuses where the peak velocities are achieved fit well with those corresponding cavity acoustic resonance radiuses. As the diaphragm excitation frequency increases, the corresponding cavity acoustic resonance radius decreases, making the inertial action of the fluid
inside cavity weak on the ejection and suction of the gas near the orifice. Therefore, the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. It is also noted that the peak synthetic jet velocities are affected weakly by the excitation frequency in these cases. The peak velocities corresponding to 400 Hz, 635 Hz and 900 Hz are 12.8 m/s, 14.4 m/s and 15 m/s, respectively. The presented effects of cavity dimension on the synthetic jet performance are similar to the previous works [29,35] in the trends, but somewhat different in the results due to the different geometric parameters of actuator adopted for investigation. In generally, the actuator cavity with little depth or radius has little volume of the cavity, which is contributed for reducing the inertial action of the fluid inside cavity. So that the response of the ejection and suction near the orifice is easily to step with the diaphragm excitation frequency. The Helmholtz resonance frequency determined by formula (3) is more suitable in these cases, and the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. Under high excitation frequency, the effect of cavity parameters is shown more sensitive on the synthetic jet. These simulations also suggest that Helmtoltz resonance frequency determined by formula (3) is in general identical to those for optimizing actuator performance in many circumstances. When the diaphragm is operated under relative low excitation frequency with large cavity acoustic resonance depth, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. For this situation, one could choose a relative small cavity in spite of the cavity acoustic resonance depth. 4.3. Effects of orifice dimension In the present, the effects of the orifice dimension are investigated individually for a fixed cavity with radius (rc ) of 22.5 mm and depth (hc ) of 7 mm. Fig. 10 presents effects of orifice thickness on the average velocity at orifice exit during the expulsion stroke. Here the orifice thickness (ho ) ranges from 0.5 mm to 12 mm, while the radius of the orifice (ro ) is maintained as 1.5 mm. Three diaphragm excitation
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Fig. 9. Effect of cavity radius on time-averaged orifice velocity.
frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the orifice acoustic resonance thicknesses coincided with the corresponding diaphragm excitation frequencies are 7.8 mm, 2.5 mm and 0.77 mm, respectively. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the orifice thickness of about 8.5 mm, which is a little discrepant from the orifice acoustic resonance thickness of 7.8 mm deduced from formula (3). When the diaphragm is excited at 900 Hz, the highest synthetic jet intensity occurs at the orifice thickness of about 0.8 mm, which is the same as the orifice acoustic resonance thickness. It is interesting to find that the synthetic jet peak velocity achieved under low excitation frequency is higher than that under high excitation frequency in this situation. The peak velocities
corresponding to 400 Hz, 635 Hz and 900 Hz are 15.2 m/s, 15.2 m/s and 13.3 m/s, respectively. Under lower diaphragm excitation frequency, the orifice acoustic resonance thickness corresponding to the highest synthetic jet intensity is bigger. The synthetic jet is affected weakly by the orifice thickness in the neighboring of orifice acoustic resonance thickness. For the relative larger orifice resonance acoustic thickness, although the inertial action of the fluid inside the orifice is enhanced, the gas with higher momentum gathers near the orifice exit and does not penetrates deeply into the cavity to be dissipated. When the diaphragm is excited at relative higher frequency, the synthetic jet is affected seriously by the orifice thickness in the neighboring of orifice acoustic resonance thickness. The velocity versus orifice thickness curve is sharper as observed for the 900 Hz case than the 400 Hz case, due
Fig. 10. Effect of orifice thickness on time-averaged orifice velocity.
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Fig. 11. Effect of orifice radius on time-averaged orifice velocity.
to worse coordination of the flow suction and ejection processes inside the cavity and the orifice. Fig. 11 presents effects of orifice radius on the average velocity at orifice exit during the expulsion stroke. Here the orifice radius (ro ) ranges from 0.2 mm to 3 mm, while the thickness of the orifice (ho ) is maintained as 2.5 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the orifice acoustic resonance radiuses coincided with the corresponding diaphragm excitation frequencies are 0.89 mm, 1.5 mm and 2.27 mm, respectively. It is seen that the orifice radius corresponding to the highest synthetic jet intensity increases as the diaphragm excitation frequency is increased. The calculated orifice radiuses where the peak velocities are achieved fit well with those corresponding orifice acoustic resonance radiuses determined from formula (3). When the diaphragm is excited at 900 Hz, the perturbation frequency of the vibrating diaphragm to the internal air insides the cavity is very high, small orifice radius restricts the ejection and suction capacity of the synthetic jet actuator. In this case, the optimum orifice radius should be larger. As the orifice radius is increased further, the velocity output is decreased due to the larger orifice exit area although being of high mass flux. Therefore, it is indicated that the orifice radius is an important parameter affecting the synthetic jet. The presented effects of orifice dimension on the synthetic jet performance are similar to the previous works [29,35] in the trends, but somewhat different in the results due to the different geometric parameters of actuator adopted for investigation. In generally, the response of the ejection and suction near the orifice is easily to step with the diaphragm excitation frequency for the actuator orifice with small thickness or radius. The Helmholtz resonance frequency determined by formula (3) is suitable in these cases. 5. Conclusions A numerical investigation utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model on the synthetic jet actuator fluid field is performed in this study. The synthetic jet fluidic characteristics are explored under the different design parameters of the synthetic jet actuator that include radius and thickness of
the orifice, as well as radius and depth of the cavity. The relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency is analyzed. The following conclusions are made from this study: (1) When the diaphragm excitation frequency is closed to the actuator cavity Helmholtz resonance frequency, the relative pressure insides the cavity is obviously greater during the ejection stroke of the cycle and lower during the suction stroke of the cycle, resulting in the maximum synthetic jet velocity output. When the diaphragm excitation frequency is lower than the actuator cavity Helmholtz resonance frequency, the perturbation of the vibrating diaphragm to the internal air insides the cavity is very weak. While for the diaphragm excited at higher frequency, the ejection and suction of the gas near the orifice are fall significantly behind the diaphragm movement. (2) For the fixed orifice parameters, the actuator cavity depth has more significant influences on the average velocity at orifice exit during the ejection stroke than the cavity diameter. As the cavity depth decreases, the corresponding Helmholtz resonance frequency increases. Under higher excitation frequency, the average velocity at orifice exit during the ejection stroke changes significantly in the vicinity of acoustic resonance depth. Once the cavity depth offsets from the cavity acoustic resonance depth, the velocity output of the synthetic jet will decreases dramatically. The change of cavity diameter in the vicinity of acoustic resonance diameter has relatively weaker influence on the synthetic jet. (3) For the fixed cavity parameters, the synthetic jet peak velocity under low excitation frequency is higher than that under high excitation frequency. Under lower diaphragm excitation frequency, the synthetic jet is affected weakly by the orifice thickness in the neighboring of orifice acoustic resonance thickness. When the diaphragm is excited at high frequency, the perturbation frequency of the vibrating diaphragm to the internal air insides the cavity is very high, small orifice diameter will restrict the ejection and suction capacity of the synthetic jet actuator. (4) The simulations suggest that Helmtoltz resonance frequency presented by Gallas et al. [22,23] is in general identical to those
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for optimizing actuator performance in many circumstances. When the diaphragm is operated under relative low excitation frequency with large cavity acoustic resonance depth, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. For this situation, one could choose a relative small cavity in spite of the cavity acoustic resonance depth. Acknowledgement The authors gratefully acknowledge the financial support for this project from the National Natural Science Foundation of China (Grant No.: 51306088). References [1] B.L. Smith, A. Glezer, The formation and evolution of synthetic jets, Phys. Fluids 10 (1998) 2281–2297. [2] A. Glezer, M. Amitay, Synthetic jets, Annu. Rev. Fluid Mech. 34 (2002) 503–529. [3] C. Lee, G. Hong, Q.P. Ha, A piezoelectrically actuated micro synthetic jet for active flow control, Sens. Actuators A 108 (2003) 168–174. [4] G. Hong, Effectiveness of micro synthetic jet actuator enhanced by flow instability in controlling laminar separation caused by adverse pressure gradient, Sens. Actuators A 135 (2006) 607–615. [5] S.Y. Zhang, S. Zhong, Experimental investigation of flow separation control using an array of synthetic jets, AIAA J. 49 (2011) 2637–2649. [6] B.L. Smith, A. Glezer, Jet vectoring using synthetic jets, J. Fluid Mech. 458 (2002) 1–34. [7] Z.B. Luo, Z.X. Xia, Y.G. Xie, Jet vectoring control using a novel synthetic jet actuator, Chin. J. Aeronaut. 20 (2007) 193–201. [8] B.C. Mather, F. Mohsen, B. Jean-Christophe, Aerodynamic flow vectoring of a wake using asymmetric synthetic jet actuation, Exp. Fluids 53 (2012) 1797–1813. [9] H. Wang, S. Menon, Fuel–air mixing enhancement by synthetic microjets, AIAA J. 39 (2001) 2308–2319. [10] Y.M. Liu, B.G. Wang, S.Y. Liu, Investigation of phase excitation on mixing control on coaxial jets, J. Therm. Sci. 18 (2009) 364–369. [11] Q.F. Xia, S. Zhong, Enhancement of laminar flow mixing using a pair of staggered lateral synthetic jets, Sens. Actuators A 207 (2014) 75–83. [12] R. Mahalingam, N. Rumigny, A. Glezer, Thermal management with synthetic jet ejectors, IEEE Trans. Compon. Packing Technol. 27 (2004) 439–444. [13] M. Chaudhari, B. Puranik, A. Agrawal, Effect of orifice shape in synthetic jet based impingement cooling, Exp. Therm. Fluid Sci. 34 (2010) 246–256. [14] M. Chaudhari, B. Puranik, A. Agrawal, Heat transfer characteristics of synthetic jet impingement cooling, Int. J. Heat Mass Transfer 53 (2010) 1057–1069. [15] S. Gao, J.Z. Zhang, X.M. Tan, Experimental study on heat transfer characteristics of synthetic jet driven by piston actuator, Sci. China Technol. Sci. 55 (2012) 1732–1738. [16] J.Z. Zhang, S. Gao, X.M. Tan, Convective heat transfer on a flat plate subjected to normally synthetic jet and horizontally forced flow, Int. J. Heat Mass Transfer 57 (2013) 321–330. [17] X.M. Tan, J.Z. Zhang, Flow and heat transfer characteristics under synthetic jets impingement driven by piezoelectric actuator, Exp. Therm. Fluid Sci. 48 (2013) 134–146. [18] A. Crook, A.M. Sadri, N.J. Wood, The development and implementation of synthetic jets for the control of separated flow, AIAA Paper 99-3176, 1999. [19] F.J. Chen, C. Yao, G.B. Beeler, et al., Development of synthetic jet actuators for active flow control at NASA Langley, AIAA Paper 2000-2405, 2000.
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[20] Y. Guy, J. McLaughlin, A. Morrow, Velocity measurements in a synthetic jet, AIAA Paper 2001-0118, 2001. [21] Y. Guy, J. McLaughlin, A. Morrow, Effect of geometric parameters on the velocity output of a synthetic jet actuator, AIAA Paper 2002-0126, 2002. [22] Q. Gallas, R. Holman, T. Nishida, et al., Lumped element modelling of piezoelectric-driven synthetic jet actuators, AIAA J. 41 (2003) 240–247. [23] Q. Gallas, G. Wang, M. Papila, et al., Optimisation of synthetic jet actuators, AIAA paper 2003-0635, 2003. [24] T. Persoons, T.S. O’Donovan, A pressure-based estimate of synthetic jet velocity, Phys. Fluids 19 (2007), 128104-1-4. [25] D.A. Lockerby, P.W. Carpenter, C. Davies, Is Helmholtz resonance a problem for micro-jet actuators? Flow Turbul. Combust. 78 (2007) 205–222. [26] A. Pavlova, M. Amitay, Electronic cooling with synthetic jet impingement, J. Heat Transfer 128 (2006) 897–907. [27] J.Z. Zhang, X.M. Tan, Experimental study on flow and heat transfer characteristics of synthetic jet driven by piezoelectric actuator, Sci. China Technol. Sci. 50 (2007) 221–229. [28] M. Chaudhari, G. Verma, B. Puranik, et al., Frequency response of a synthetic jet cavity, Exp. Therm. Fluid Sci. 33 (2009) 439–448. [29] M. Jain, B. Puranik, A. Agrawal, A numerical investigation of effects of cavity and orifice parameters on the characteristics of a synthetic jet flow, Sens. Actuators A 165 (2011) 351–366. [30] I. Yoo, S. Lee, Reynolds-averaged Navier–Stokes computations of synthetic jet flow using deforming meshes, AIAA J. 50 (2012) 1943–1955. [31] L.D. Kral, F.D. John, A.B. Cain, et al., Numerical simulation of synthetic jet actuators, AIAA Paper 97-1824, 1997. [32] D.B. Goldstein, Two-dimensional synthetic jet simulation, AIAA Paper 20000406, 2000. [33] S.R. Fugal, B.L. Smith, R.E. Spall, Displacement amplitude scaling of a two dimensional synthetic jet, Phys. Fluids 17 (2005), 045103-1-10. [34] ANSYS Fluent v12.0 Software User’s Guide, 2009. [35] P. Mane, K. Mossi, A. Rostami, et al., Piezoelectric actuators as synthetic jet: cavity dimension effects, J. Intell. Mater. Syst. Struct. 18 (2007) 1175–1190.
Biographies Lv Yuan-wei obtained his B.E. degree in 2012 from Nanjing University of Aeronautics and Astronautics (NUAA), PR China. He is currently pursuing the M.S. degree at the College of Energy and Power Engineering, NUAA. His research is focused on the active flow control, in particular synthetic jets. Zhang Jing-zhou is currently a Professor at Nanjing University of Aeronautics and Astronautics, PR China and a visiting Professor at Collaborative Innovation Center of Advanced Aero-Engine, PR China. He received B.S. degree in 1986 from Tsinghua University, PR China, and M.S. degree in 1989 from Southeast University, PR China, and Ph.D. degree in 1992 from Nanjing University of Aeronautics and Astronautics, PR China, respectively. His research interests include heat transfer enhancement, active cooling of aero-engine hot components, infrared radiation and mixer-ejector technology, etc. By now, He has published approximately 250 scientific papers in the above research regions and three books on the Heat transfer and Mixer-Ejector issues. Shan Yong is an assistant professor in the College of Energy and Power Engineering and Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics (NUAA) since 2009. He received B.E. degree in 2000, M.S. degree in 2003 and Ph.D. degree in 2006, all in NUAA. His research interests include gas dynamics, infrared radiation, and mixer-ejector technology. Tan Xiao-ming is an assistant professor in the College of Energy and Power Engineering and Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics (NUAA) since 2008. She received B.E. degree in 1998 and Ph.D. degree in 2006 in NUAA. Her research interests include heat transfer enhancement and active cooling of aero-engine hot components.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Numerical investigation for effects of actuator parameters and excitation frequencies on synthetic jet fluidic characteristics Lv Yuan-wei a , Zhang Jing-zhou a,b,∗ , Shan Yong a , Tan Xiao-ming a a College of Energy and Power Engineering, Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China b Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 3 April 2014 Received in revised form 14 August 2014 Accepted 14 August 2014 Available online 23 August 2014 Keywords: Synthetic jet actuator Excitation frequency Geometric parameters Helmholz frequency Numerical simulation
a b s t r a c t Numerical investigation is performed to further address the effects of geometric parameters and excitation frequency of the actuator on the synthetic jet fluidic characteristics by utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model. The vibrating diaphragm is modeled as a movable wall varying in sinusoidal mode. Computations are carried out by using FLUENT software with the coupled user definition function (UDF) describing the diaphragm movement. The results show that the geometric parameters of the actuator, such as the cavity depth and diameter, as well as orifice thickness and diameter, have important influences on the synthetic jet fluidic characteristics. The velocity output could be maximized if the geometric parameters of the synthetic jet actuator are designed to ensure that the cavity acoustic Helmholtz resonance frequency is coincided with the diaphragm excitation frequency. For a fixed actuator cavity, when the diaphragm excitation frequency is consistent with the Helmholtz resonance frequency of actuator cavity, the relative pressure insides the cavity is obviously great during the ejection stroke while low during the suction stroke. In the presented actuator parameters, the synthetic jet is enhanced as the decrease of cavity depth for the fixed orifice. The change of cavity diameter in the vicinity of corresponding cavity acoustic resonance diameter has relatively weaker influence on the synthetic jet. When the diaphragm is excited at high frequency, small orifice diameter will restrict the ejection and suction capacity of the synthetic jet actuator. © 2014 Elsevier B.V. All rights reserved.
1. Introduction A synthetic jet is a quasi-steady jet of fluid generated from the periodic motion of a diaphragm enclosed in a cavity with openings on one or more walls. The fluid inside the cavity is expelled through the orifice as the diaphragm is forced to move upwards. The flow separates at the edge of the orifice, inducing a vortex ring that moves outwards under its own momentum. When the diaphragm moves downwards to entrain fluid into the cavity, the vortex ring is sufficiently distant from the orifice that it is virtually unaffected by the entrainment of the fluid into the cavity. In this fashion, a train of vortex rings moving away from the orifice occurs, whereupon the coherent structures then interact, coalesce, and break down in a transition toward a quasi-steady jet [1,2]. This
∗ Corresponding author at: College of Energy and Power Engineering, Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. Tel.: +86 13913887006. E-mail address: [email protected] (J.-z. Zhang). http://dx.doi.org/10.1016/j.sna.2014.08.009 0924-4247/© 2014 Elsevier B.V. All rights reserved.
operational principle allows for synthetic jets to have the unique property that they are formed from the working fluid in which they are deployed. In contrast to conventional continuous jets, synthetic jets are able to transfer linear momentum without a net mass injection across the flow boundary. On the other hand, conventional jets are formed by the addition of both mass and momentum at the orifice. Unlike synthetic jets, the momentum flux in conventional jets remains conserved. Due to the difference in formation between the two jets, there is no potential core in synthetic jets as opposed to conventional jets. Additionally, the periodic vortical structures introduced into the flow exhibit an ability of synthetic jet to influence the environment at a variety of length scales. This vortex shedding phenomenon is important for many applications, such as control of separated flow [3–5], jet vectoring [6–8], mixing enhancement [9–11], thermal management and heat transfer enhancement [12–17]. A lot of studies have covered the development of the piezoelectric actuator as well as the general behavior and performance of the synthetic jet produced. Results indicated that excitation frequency and cavity orifice were important factors affecting the size
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and formation of coherent structures in synthetic jet flow. Crook et al. [18] studied the development of a round jet using an analytical model and compared his results with hot-wire measurements. The study was focused on the maximum velocity from the jet as function of actuator geometry and operation parameters. It was found that the peak velocity was originated at operating frequency of about 1400 Hz. Chen et al. [19] measured velocity profiles at the centerline of a plane jet, and jet exit velocity profiles along and across the jet slot with a single component hot-wire. They used an actuator with a diaphragm oscillating at a frequency ranging from 500 to 1000 Hz. Guy et al. [20,21] studied a plane synthetic jet produced by piezo-diaphragm with hot-wire measurements. Similar to Chen et al. [19], they observed two resonance frequencies, in this case, 700 and 1160 Hz. Furthermore, it was suggested that there was an optimum combination of all geometric parameters at which the actuator would operate at its full capacity. To address the response of a cavity, Gallas et al. [22,23] presented a simplified lumped element model (LEM) of a piezoelectric-driven synthetic jet actuator. In this model, the individual components of a synthetic jet were modeled as elements of an equivalent electrical circuit using conjugate power variables (i.e. resistor, inductor, and capacitor). Based on the lumped element model, they deduced that the output velocity versus actuation frequency curve had two local maxima corresponding to the actuator cavity acoustic resonance (Helmholtz resonance, fH ) frequency and the diaphragm structural resonance frequency (fD ). Persoons and O’Donovan [24] presented an analytical model derived from simplified gas dynamics, for estimating the synthetic jet velocity and actuator deflection, based on a cavity pressure measurement. It was concluded that low-power operation was achieved by matching actuator and Helmholtz resonance frequencies. Lockerby et al. [25] made numerical-simulation studies to describe the methodology on how Helmholtz resonance affected the interaction of active and nominally inactive micro-jet actuators with a laminar boundary layer. It was previously shown that the conditions for Helmholtz resonance were identical to those for optimizing actuator performance for maximum mass flux. The velocity output could be increased significantly if the synthetic jet actuator could be designed so that the acoustic and structural resonances coincided in order to increase output velocity. This approach had been verified by some following works [26–29]. Pavlova and Amitay [26] experimentally investigated the efficiency and mechanism of cooling a constant heat flux surface by impinging synthetic jet; also, comparison with continuous jet was presented. In their measurements, high frequency (1200 Hz) jets were found to be more effective at smaller axial distances and the low frequency (420 Hz) jets were found to be more effective at larger axial distances. Zhang and Tan [27] investigated experimentally the flow and heat transfer characteristics of a synthetic jet driven by piezoelectric actuator by utilizing particle image velocimetry, hot-wire anemometry and infrared camera. Two resonance frequencies of 540 Hz and 1140 Hz were easily identified. It was evident that the actuator produces fairly high velocities even when operated offresonance and the synthetic jet could not be formed under some frequencies. Chaudhari et al. [28] made a detailed experimental investigation wherein the effect of excitation frequency on the synthetic jet flow was studied for cavities of different depths and for orifices of different diameters. The exit velocity averaged over an excitation cycle indicated a lower and an upper bound on the frequency for the formation of a jet, and showed resonance at two frequencies. The resonant frequencies had been identified as being close to the diaphragm and the Helmholtz frequencies. Jain et al. [29] performed a numerical simulation to investigate the effect of various cavity parameters and orifice/cavity shapes on the ensuing synthetic jet flow. The simulation results showed that synthetic jets are more affected by changes in the geometric parameters of the orifice than those of the cavity.
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The momentum carried by the fluid away from the actuator cavity is strongly dependent on its design, which makes the operating frequency and the geometric parameters of the synthetic jet actuator very crucial aspects in the synthetic-jet assembly. However, the detailed knowledge about the design aspects on the ensuing jet deserves further illustrated, especially on the relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency. In order to further address the effects of actuator parameters and excitation frequency on the synthetic jet fluidic characteristics, a numerical investigation utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model on the synthetic jet actuator fluid field was performed in this study. The synthetic jet fluidic characteristics were explored under the different design parameters of the synthetic jet actuator that include radius and thickness of the orifice, as well as radius and depth of the cavity. The relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency was analyzed. 2. Numerical simulation procedure 2.1. Physical model and computational domain The physical model of a synthetic jet considered in the present is illustrated schematically in Fig. 1. It is composed of a vibrating diaphragm located at the bottom of a cavity, on the opposite face of which is a round orifice. The diaphragm is oscillating up and down in the sinusoidal or cosinoidal mode. The main geometric parameters of the synthetic jet actuator include diameter (do ) and thickness (ho ) of the orifice, as well as diameter (dc ) and depth (hc ) of the cavity. The computational domain consists of three zones including the cavity, the orifice, and the surrounding region into which the jet exits. According to the study of Jain et al. [29], the surrounding size of 30 orifice diameters (in the lateral direction) and 20 orifice diameters (along the axis) is chosen. For this surrounding size, jet near field parameters are not affected by pressure outlet boundary condition applied at the end of the outer region. In the present, a group of baseline geometric parameters are outlined as: dc = 45 mm, hc = 7 mm, do = 3 mm, and ho = 2.5 mm. To address the effects of actuator parameter on the synthetic jet fluidic characteristics individually, one of the geometric parameters is varied in a certain range while the other geometric parameters are maintained as their baseline values. 2.2. Numerical methods The unsteady Reynolds-average Navier–Stokes (RANS) equations are chosen as governing equations for the synthetic jet problems. The flow is assumed to be compressible and turbulent. The compressibility effects come into picture because of the rapid change in pressure/density due to the movement of the diaphragm. The computation is carried out by using the commercial CFD software FLUENT coupled with the user definition function (UDF) describing the diaphragm movement. The segregated, unsteady, symmetric solver is chosen with first-order implicit time scheme that is unconditionally stable with respect to time step size. Pressure-implicit with splitting of operators (PISO) based on a higher degree of approximation between the iterative corrections for pressure and velocity is chosen. Second-order upwind spatial discretization is used for the momentum, turbulent kinetic energy, and turbulent dissipation rate. According to Yoo and Lee [30], Menter’s k–ω shear stress transport (SST) turbulence model is used to account for the turbulent nature of the synthetic jet flows.
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Fig. 1. Physical model of synthetic jet flow simulation.
Successful computational fluid dynamics (CFD) synthetic jet flow simulations that incorporate the oscillating diaphragms displacement performance depend heavily on accurate approximations of the displacement profiles, including the instantaneous deflections and shape modes. As already mentioned, the most crucial part of the numerical simulation of the synthetic jet is the modeling of the vibrating diaphragm. Several approximations have been employed in the past to achieve this. These include using a velocity inlet boundary condition with an assumed profile at the location of the orifice exit [31,32] or using a moving piston boundary condition [33]. The swept volume is maintained identical to that in the case of an actually vibrating diaphragm. In the present simulations, a wall boundary condition with the diaphragm vibrating sinusoidally or cosinoidally in time such that the geometry of the wall is parabolic at any instant has been employed to simulate the vibrating diaphragm [29]
Y (r, t) =
A 1− 2
r 2 rc
sin(2ft)
(1)
where Y is the diaphragm deformation relative to its neutral position, A is the peak-to-peak displacement at the center, r is the lateral distance from the diaphragm center, rc is the radius of the cavity, f is the actuation frequency of diaphragm, t is the time. Eq. (1) is expected to represent a vibrating diaphragm realistically. The instantaneous logarithmic diaphragm velocity is derived by differentiating Eq. (1).
v(r, t) = fA 1 −
r 2 rc
cos(2ft)
(2)
In Fig. 1, the boundary conditions are specified at the cavity end and the ambient surrounding into which the jet is expelled. The diaphragm and cavity surfaces are considered as impermeable walls. The ambient air is specified with a pressure outlet (relative pressure is assumed as zero). The diaphragm is defined separately from the rest of the walls so that a user-defined function can be used to describe its movement.
2.3. Grid and time independence studies Similar to Jain et al. [29], the computational domain is meshed with unstructured mesh (triangular elements) for the cavity (up to the orifice exit). The tri-pave unstructured scheme is chosen to allow for relative displacement among the nodes on the diaphragm. The relative displacement among the nodes on the diaphragm is necessary which is not possible with the structured quadrilateral scheme. The outer field is meshed with structured quadrilaterals. Combining the unstructured and structured grids significantly reduces the simulation time by restricting the number of grid points to be re-meshed after each time step and still allows usage of the dynamic mesh feature of the software, which is applied at the vibrating diaphragm and cavity. Size functions to control and specify the size of the mesh spacing intervals between nodes are attached to the orifice edge, orifice exit vertex, and the diaphragm upper vertex. This ensures fine meshing in the regions where large gradients are present. By the grid independence test, the grid number is chosen as about 40,000. Owing to nonlinearities it is necessary to control the calculated value change to assure proper convergence through the use of under-relaxation parameters. The default under-relaxation parameters are kept at 0.3, 1.0, and 0.7 for pressure, density, and momentum, respectively. The solution is then initialized at the diaphragm with initial value of zero for the gage pressure, axial velocity, and lateral velocity. The default of 120 maximum iterations per time step is kept. A time step of 1/(1000f) is chosen based on the actuation frequency to allow for 1000 time steps per cycle. In the residual monitors, the convergence criteria are set as 10−6 . More details on these solvers could be found in the ANSYS Fluent Software User’s Guide [34].
2.4. Validation of simulation procedure The validation of presented simulation procedure is made by comparison with the experimental study of Mane et al. [35].
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Fig. 2. Instantaneous velocity vectors of a synthetic jet in one cycle. (a) 1/4 T, (b) 1/2 T, (c) 3/4 T, (d) 1 T.
A typical geometric configuration and operational condition is chosen for the present propose. According to the experimental parameters, the computational model is set as: cavity diameter (dc ) of 63.5 mm and cavity depth (hc ) of 5.5 mm, orifice diameter (do ) of 2 mm and thickness (ho ) of 1.6 mm. The peak-to-peak center diaphragm displacement (A) for the Bimorph actuator is 0.396 mm and the excitation frequency is 50 Hz. Fig. 2 presents the instantaneous velocity vectors of a synthetic jet in one cycle form computation. Note that the first several cycle results are affected by numerical transient and are therefore discarded. Here, the final data is analyzed only when the monitored results show stabilization over at least 10 cycles. As the diaphragm oscillates, the ambient air is alternately drawn into and expelled out from the cavity orifice. When the diaphragm moves upwards (corresponding to 1/4 T, as seen in Fig. 2(a)), a shear layer is formed at the orifice edge and the vorticity in the shear layer rolls up ambient air to form a vortex ring. When the diaphragm moves downwards to entrain fluid into the cavity (corresponding to 1/2 T and 3/4 T, as seen in Fig. 2(b) and (c)), the vortex ring is sufficiently distant from the orifice that it is virtually unaffected by the entrainment of the fluid into the cavity. In subsequent images, the new vortex pair undergoes the transition and development,
shedding into the surrounding fluid, wrapping around the cores of the primary vortices and dissipating downstream (corresponding to 1 T, as seen in Fig. 2(d)). Because of periodic formation of new vortex pairs near the orifice exit, continuous momentum is replenished to the downstream to maintain the quasi-continuity of the synthetic jet. Fig. 3 presents the comparison between numerical axial velocity profile and experimental result [35] at maximum expulsion and downstream distance of 2 mm from the orifice. It is shown that the results are slightly different in magnitude but follow a similar trend. The velocity at the center of the orifice is over-predicted by the numerical simulation while the velocity magnitudes beyond r/dc of 0.6 are under-predicted. It is also noticed that the presented computational results are in good agreement with the numerical simulation by Mane et al. [35]. The reason for the differences the velocity magnitudes beyond r/dc of 0.6 was brought out by Mane et al. [35], which could be due to the hotwire inability of detecting flow direction. The higher maximum experimentally measured velocities observed in the outer portion of the jet most likely occurs during the ingestion cycle when the fluid flow is from the outer edges. The numerical results over-predict the maximum measured experimental velocity at the orifice center by only 15%.
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where is air density, c is sound speed in the air, ho is orifice thickness, ro is orifice radius, and Vc is cavity volume. fD =
Fig. 3. Numerical and experimental axial velocity profiles at 2 mm downstream of the orifice.
3. Resonance frequencies Performance of a synthetic jet actuator is expected to be sensitive to the excitation parameters related to the frequency and amplitude of vibration, as well as the parameters related to the cavity and orifice geometry. The depth and diameter of the cavity along with the orifice length and diameter are the geometric parameters, while frequency and amplitude of actuation are the control parameters relevant to cavity design. Based on the lumped element modeling (LEM) by Gallas et al. [22,23], two resonance frequencies, the actuator cavity acoustic resonance (Helmholtz resonance, fH ) frequency and the diaphragm structural resonance frequency (fD ) are given by
fH =
1 2
4ho 3ro2
+
8 32 ro
Vc c 2
−0.5 (3)
1 2
m 6rc4
rc6 (1 − ˛2 ) 16E3
−0.5
(4)
where rc is diaphragm radius, E is diaphragm elastic modulus, ˛ is the Poisson ratio, is diaphragm thickness, and m is diaphragm mass. From formula (4), it is obvious that the diaphragm structural resonance frequency depends on the material elasticity, mass, dimensions of diaphragm. In the present simulation study, the diaphragm structural resonance frequency by itself cannot be determined, due to the assumption of zero thickness and massless wall of the diaphragm. However, as the maximum amplitude of the diaphragm is generally achieved at the structural resonance frequency, the excitation frequency to which the maximum amplitude of diaphragm is correspondent maybe regarded as the structural resonance frequency. In the present simulations, the movement of the vibrating diaphragm is defined as formula (1) and the maximum peak-topeak displacements at the center of diaphragm for all the excitation frequencies are fixed at A = 0.2 mm. This approach is not reasonable for a specific diaphragm excited under different frequency, due to the coupling of the amplitude and excitation frequency. But it is true that every excitation frequency may be regarded as the resonance frequency of corresponding diaphragm configuration. That is, under a specific excitation frequency, a corresponding diaphragm could be excited to achieve maximum amplitude. 4. Results and analysis 4.1. Effect of diaphragm excitation frequency To explore the effects of diaphragm excitation frequency on the synthetic jet, three groups of the actuator geometric parameters are designed in the present, as shown in Table 1. In these actuators, the cavity radius (rc ) and the orifice radius (ro ) are maintained as 22.5 mm and 1.5 mm, respectively. To make the Helmholtz frequencies for these groups are all fixed at 635 Hz, the cavity depth
Fig. 4. Effect of excitation frequency on time-averaged velocity at orifice exit during ejection stroke.
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Table 1 Geometry parameters of synthetic jet actuators with fixed Helmholtz frequency. Actuator
Geometric parameters
No. 1 No. 2 No. 3
rc = 22.5 mm, ro = 1.5 mm, hc = 7 mm, ho = 2.5 mm rc = 22.5 mm, ro = 1.5 mm, hc = 4.1 mm, ho = 5 mm rc = 22.5 mm, ro = 1.5 mm, hc = 2.91 mm, ho = 7.5 mm
(hc ) and orifice thickness (ho ) are varied according to formula (3) respectively. The geometric features for these actuators are marked as: (a) large cavity depth and small orifice thickness for actuator No. 1, (b) moderate cavity depth and moderate orifice thickness for actuator No. 2, and (c) small cavity depth and large orifice thickness for actuator No. 3. Fig. 4 shows the effects of diaphragm excitation frequency on the area-averaged velocity at orifice exit during the ejection stroke. The time-averaged velocity at orifice exit during the ejection stroke is defined as
V0 = f
T/2
v0 (t)dt
(5)
0
Here v0 is the area-averaged instantaneous velocity at orifice exit during the ejection stroke, T is the period of oscillation. V0 can be used for representing the intensity of synthetic jet. It is seen from Fig. 5 that the effects of diaphragm excitation frequency on the time-averaged velocity at orifice exit during the ejection stroke are significantly different in magnitude but follow a similar trend. When the diaphragm excitation frequency coincides with the actuator cavity Helmholtz resonance frequency, the time-averaged velocity at orifice exit during the ejection stroke reaches to the maximum value corresponding to each synthetic jet actuator. Taking actuator No. 2 for example, as the diaphragm excitation frequency is increased from 100 Hz to 635 Hz, V0 is varied approximately form 1 m/s to 14 m/s in nearly linear mode. As the diaphragm excitation frequency increases after 635 Hz, V0 undergoes rapid decline from 14 m/s to 5 m/s under 1400 Hz. When the diaphragm excitation frequency increases further, the intensity of synthetic jet gets weaker slowly. In generally, when the diaphragm excitation frequency coincides with the actuator cavity Helmholtz resonance frequency, the ejection and suction capacities of the gas near the orifice are enhanced, and the ejection and suction processes are also coincided together, resulting in the maximum synthetic jet velocity output. This result is in good agreement with the previous investigations. Another notable phenomenon is that the maximum velocity output corresponding to the actuator No. 3 is the highest, while the maximum velocity output corresponding to the actuator No. 1 is the lowest, although they are all excited under the optimum frequencies coincided with the acoustic resonances. In relative to actuator No. 3, the peak velocity corresponding to the cavity Helmholtz resonance frequency is reduced approximately by 12.5% for actuator No. 2, while 30% for actuator No. 1. It is indicated that the cavity dimension and orifice dimension have important effects on the synthetic jet fluidic characteristics. Fig. 5 shows the instantaneous pressure contours at t = 0.25 T under different diaphragm excitation frequencies. At this instant, the diaphragm moves upwards from the neutral station to the top peak station. Fig. 6 shows the instantaneous pressure contours at t = 0.75 T under different diaphragm excitation frequencies. At this instant, the diaphragm moves downwards from the neutral station to the bottom peak station. In these figures, the pressure is the relative pressure to the ambient pressure. It is evident that the relative pressure insides the cavity corresponding to the diaphragm excitation frequency of 635 Hz is obviously greater than the other diaphragm excitation frequencies during the ejection stroke of the cycle, as seen in Fig. 5(b), building up stronger ejection capacity.
Fig. 5. Instantaneous pressure contours during ejection stroke of the cycle (t = 0.25 T). (a) f = 50 Hz, (b) f = 635 Hz, (c) f = 1400 Hz.
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During the suction stroke of the cycle, the relative pressure insides the cavity corresponding to the diaphragm excitation frequency of 635 Hz is obviously lower than the other diaphragm excitation frequencies, as seen in Fig. 6(b), resulting in stronger suction capacity. When the diaphragm is excited at 50 Hz, the perturbation of the vibrating diaphragm to the internal air insides the cavity is very weak, with very small pressure difference between the actuator cavity and the surrounding, as seen in Figs. 5(a) and 6(a), thus it is difficult to form the synthetic jet at the orifice exit. When the diaphragm is excited at 1400 Hz, the ejection and suction of the gas near the orifice are difficult to coincide together because of the high operating frequency. Owing to the inertial action of the fluid inside cavity, the perturbation of the vibrating diaphragm to the air near the orifice is discrepant seriously to that near the diaphragm under high operating frequency. It is evident that the ejection and suction of the gas near the orifice are fall behind the diaphragm movement. During the ejection stroke of the diaphragm, the internal flow near the orifice is remained at the suction status as seen in Fig. 5(c). While during the suction stroke of the diaphragm, the internal flow near the orifice is remained at the ejection status as seen in Fig. 6(c). Phase angle versus diaphragm excitation frequency curve and velocity at the orifice exit versus time curves are presented in Fig. 7. As the compressible effect is very weak in the presented situation, the instantaneous velocity at orifice exit during the suction stroke is nearly the same as that during the ejection stroke, as seen in Fig. 7(a), which is coincidence with the pulsating nature of the synthetic jet flow [1,2]. As regards to the phase angle between the diaphragm motion and the velocity at the orifice exit, it is found that the phase angle is increased rapidly from 50 Hz to 1000 Hz and then it is varied slowly. Here a cycle of operation of diaphragm motion is assumed to be 0◦ where 270◦ is when the diaphragm is farthest from the orifice, as seen in Fig. 7(b). When the diaphragm is excited at optimum frequency of 635 Hz, the phase angle between the diaphragm motion and the velocity at the orifice exit is approximately 90◦ , which means that when the flow peaks at the orifice exit, the diaphragm motion is still toward the orifice. Thus, the peak velocity is sustained for a longer period. When the diaphragm is excited at a higher frequency, the phase difference between the diaphragm and the flow is closed to 180◦ . For large excitation frequency, by the time the flow peaks, the diaphragm may start moving away from the orifice thus reducing the pressure inside the cavity, which results in a decrease in the velocity. 4.2. Effects of cavity dimension
Fig. 6. Instantaneous pressure contours during suction stroke of the cycle (t = 0.75 T). (a) f = 50 Hz, (b) f = 635 Hz, (c) f = 1400 Hz.
Performance of a synthetic jet actuator is expected to be sensitive to the parameters related to the cavity and orifice geometry. In the present, the effects of the cavity dimension are investigated individually for a fixed orifice with radius (ro ) of 1.5 mm and thickness (ho ) of 2.5 mm. Fig. 8 presents effects of cavity depth on the average velocity at orifice exit during the expulsion stroke. Here the cavity depth (hc ) ranges from 2 mm to 30 mm, while the radius of the cavity (rc ) is maintained as 22.5 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the cavity depths at which the actuator cavity Helmholtz resonance frequencies will coincide with the corresponding diaphragm excitation frequencies are 17.6 mm, 7 mm and 3.51 mm, respectively. For convenience, they are named as cavity acoustic resonance depths. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the cavity depth of about 8 mm, which is discrepant seriously from the cavity acoustic resonance depth of 17.6 mm deduced from formula (3). However, it must be aware of that the effect of cavity depth on the synthetic jet performance
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Fig. 7. Effect of excitation frequency on instantaneous velocity and phase angle for actuator No. 3. (a) Velocity at orifice exit versus time curves and (b) phase angle versus excitation frequency curve.
under this lower excitation frequency seems very slight. In this diaphragm excitation frequency, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. The average velocity at orifice exit during the ejection stroke is ranged between 9.4 m/s and 10.2 m/s when the cavity depth is varied from 2 mm to 18 mm. As regards to the discrepancy between the cavity depth corresponding to the highest synthetic jet intensity and the acoustic resonance depth, it is should noticed that the Helmholtz frequency presented by Gallas et al. [22,23] is deduced on the simplified lumped element model of a synthetic jet actuator. The nonlinear inertial action of the fluid inside cavity is not considered enough by this model, which maybe induces negligible influence on the resonance frequency of the actuator. For a larger
cavity depth, the inertial action of the fluid inside cavity on the ejection and suction of the gas near the orifice behaves more strongly, which maybe responsible for resulting in the significant deviation of the real cavity acoustic resonance frequency to the Helmholtz frequency obtained from formula (3). When the diaphragm is excited at 635 Hz, the calculated cavity depth corresponding to the highest synthetic jet intensity is closed to the cavity acoustic resonance depth of 7 mm. When the diaphragm is excited at 900 Hz, the highest synthetic jet intensity is occurred at the cavity depth of 3.5 mm obtained from computation, which is perfectly the same as cavity acoustic resonance depth of 3.51 mm. As the diaphragm excitation frequency increases, the corresponding cavity acoustic resonance depth decreases, making
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Fig. 8. Effect of cavity depth on time-averaged orifice velocity.
the inertial action of the fluid inside cavity week on the ejection and suction of the gas near the orifice. Therefore, the cavity depth corresponding to the highest synthetic jet intensity becomes closed to the acoustic resonance depth, and the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. When the diaphragm is excited under 900 Hz, the peak velocity reaches up 23 m/s, which is approximately 130% increased in relative to 400 Hz case. Under higher excitation frequency, it is also found that the average velocity at orifice exit during the ejection stroke changes significantly in the vicinity of cavity acoustic resonance depth. The velocity versus cavity depth curve is much sharper as observed for the 900 Hz case than 400 Hz case. Once the cavity depth offsets from the cavity acoustic resonance depth, the velocity output of the synthetic jet will decreases dramatically. Fig. 9 presents effects of cavity radius on the average velocity at orifice exit during the expulsion stroke. Here the cavity radius (rc ) ranges from 10 mm to 40 mm, while the depth of the cavity (hc ) is maintained as 7 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the cavity acoustic resonance radiuses coincided with the corresponding diaphragm excitation frequencies are 35.7 mm, 22.5 mm and 15.9 mm, respectively. Here the definition of cavity acoustic resonance radius is similar to the cavity acoustic resonance depth. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the cavity radius of about 37 mm, which is discrepant a little from the cavity acoustic resonance radius deduced from formula (3). When the diaphragm is excited at 635 Hz or 900 Hz, the discrepancy between the cavity radius corresponding to the highest synthetic jet intensity and the acoustic resonance radius is nearly zero. It is observed that the change of cavity radius in the vicinity of cavity acoustic resonance radius has weaker influence on the average velocity at orifice exit during the ejection stroke in relative to the case of varying cavity depth. The calculated cavity radiuses where the peak velocities are achieved fit well with those corresponding cavity acoustic resonance radiuses. As the diaphragm excitation frequency increases, the corresponding cavity acoustic resonance radius decreases, making the inertial action of the fluid
inside cavity weak on the ejection and suction of the gas near the orifice. Therefore, the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. It is also noted that the peak synthetic jet velocities are affected weakly by the excitation frequency in these cases. The peak velocities corresponding to 400 Hz, 635 Hz and 900 Hz are 12.8 m/s, 14.4 m/s and 15 m/s, respectively. The presented effects of cavity dimension on the synthetic jet performance are similar to the previous works [29,35] in the trends, but somewhat different in the results due to the different geometric parameters of actuator adopted for investigation. In generally, the actuator cavity with little depth or radius has little volume of the cavity, which is contributed for reducing the inertial action of the fluid inside cavity. So that the response of the ejection and suction near the orifice is easily to step with the diaphragm excitation frequency. The Helmholtz resonance frequency determined by formula (3) is more suitable in these cases, and the maximum synthetic jet velocity output could be augmented when the diaphragm excitation frequency coincides with the cavity Helmholtz resonance frequency. Under high excitation frequency, the effect of cavity parameters is shown more sensitive on the synthetic jet. These simulations also suggest that Helmtoltz resonance frequency determined by formula (3) is in general identical to those for optimizing actuator performance in many circumstances. When the diaphragm is operated under relative low excitation frequency with large cavity acoustic resonance depth, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. For this situation, one could choose a relative small cavity in spite of the cavity acoustic resonance depth. 4.3. Effects of orifice dimension In the present, the effects of the orifice dimension are investigated individually for a fixed cavity with radius (rc ) of 22.5 mm and depth (hc ) of 7 mm. Fig. 10 presents effects of orifice thickness on the average velocity at orifice exit during the expulsion stroke. Here the orifice thickness (ho ) ranges from 0.5 mm to 12 mm, while the radius of the orifice (ro ) is maintained as 1.5 mm. Three diaphragm excitation
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Fig. 9. Effect of cavity radius on time-averaged orifice velocity.
frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the orifice acoustic resonance thicknesses coincided with the corresponding diaphragm excitation frequencies are 7.8 mm, 2.5 mm and 0.77 mm, respectively. When the diaphragm is excited at 400 Hz, the highest synthetic jet intensity occurs at the orifice thickness of about 8.5 mm, which is a little discrepant from the orifice acoustic resonance thickness of 7.8 mm deduced from formula (3). When the diaphragm is excited at 900 Hz, the highest synthetic jet intensity occurs at the orifice thickness of about 0.8 mm, which is the same as the orifice acoustic resonance thickness. It is interesting to find that the synthetic jet peak velocity achieved under low excitation frequency is higher than that under high excitation frequency in this situation. The peak velocities
corresponding to 400 Hz, 635 Hz and 900 Hz are 15.2 m/s, 15.2 m/s and 13.3 m/s, respectively. Under lower diaphragm excitation frequency, the orifice acoustic resonance thickness corresponding to the highest synthetic jet intensity is bigger. The synthetic jet is affected weakly by the orifice thickness in the neighboring of orifice acoustic resonance thickness. For the relative larger orifice resonance acoustic thickness, although the inertial action of the fluid inside the orifice is enhanced, the gas with higher momentum gathers near the orifice exit and does not penetrates deeply into the cavity to be dissipated. When the diaphragm is excited at relative higher frequency, the synthetic jet is affected seriously by the orifice thickness in the neighboring of orifice acoustic resonance thickness. The velocity versus orifice thickness curve is sharper as observed for the 900 Hz case than the 400 Hz case, due
Fig. 10. Effect of orifice thickness on time-averaged orifice velocity.
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Fig. 11. Effect of orifice radius on time-averaged orifice velocity.
to worse coordination of the flow suction and ejection processes inside the cavity and the orifice. Fig. 11 presents effects of orifice radius on the average velocity at orifice exit during the expulsion stroke. Here the orifice radius (ro ) ranges from 0.2 mm to 3 mm, while the thickness of the orifice (ho ) is maintained as 2.5 mm. Three diaphragm excitation frequencies are specified as 400 Hz, 635 Hz and 900 Hz. According to formula (3), the orifice acoustic resonance radiuses coincided with the corresponding diaphragm excitation frequencies are 0.89 mm, 1.5 mm and 2.27 mm, respectively. It is seen that the orifice radius corresponding to the highest synthetic jet intensity increases as the diaphragm excitation frequency is increased. The calculated orifice radiuses where the peak velocities are achieved fit well with those corresponding orifice acoustic resonance radiuses determined from formula (3). When the diaphragm is excited at 900 Hz, the perturbation frequency of the vibrating diaphragm to the internal air insides the cavity is very high, small orifice radius restricts the ejection and suction capacity of the synthetic jet actuator. In this case, the optimum orifice radius should be larger. As the orifice radius is increased further, the velocity output is decreased due to the larger orifice exit area although being of high mass flux. Therefore, it is indicated that the orifice radius is an important parameter affecting the synthetic jet. The presented effects of orifice dimension on the synthetic jet performance are similar to the previous works [29,35] in the trends, but somewhat different in the results due to the different geometric parameters of actuator adopted for investigation. In generally, the response of the ejection and suction near the orifice is easily to step with the diaphragm excitation frequency for the actuator orifice with small thickness or radius. The Helmholtz resonance frequency determined by formula (3) is suitable in these cases. 5. Conclusions A numerical investigation utilizing a two-dimensional unsteady Reynolds-averaged Navier–Stokes model on the synthetic jet actuator fluid field is performed in this study. The synthetic jet fluidic characteristics are explored under the different design parameters of the synthetic jet actuator that include radius and thickness of
the orifice, as well as radius and depth of the cavity. The relationship between the cavity Helmholtz resonance frequency and the diaphragm excitation frequency is analyzed. The following conclusions are made from this study: (1) When the diaphragm excitation frequency is closed to the actuator cavity Helmholtz resonance frequency, the relative pressure insides the cavity is obviously greater during the ejection stroke of the cycle and lower during the suction stroke of the cycle, resulting in the maximum synthetic jet velocity output. When the diaphragm excitation frequency is lower than the actuator cavity Helmholtz resonance frequency, the perturbation of the vibrating diaphragm to the internal air insides the cavity is very weak. While for the diaphragm excited at higher frequency, the ejection and suction of the gas near the orifice are fall significantly behind the diaphragm movement. (2) For the fixed orifice parameters, the actuator cavity depth has more significant influences on the average velocity at orifice exit during the ejection stroke than the cavity diameter. As the cavity depth decreases, the corresponding Helmholtz resonance frequency increases. Under higher excitation frequency, the average velocity at orifice exit during the ejection stroke changes significantly in the vicinity of acoustic resonance depth. Once the cavity depth offsets from the cavity acoustic resonance depth, the velocity output of the synthetic jet will decreases dramatically. The change of cavity diameter in the vicinity of acoustic resonance diameter has relatively weaker influence on the synthetic jet. (3) For the fixed cavity parameters, the synthetic jet peak velocity under low excitation frequency is higher than that under high excitation frequency. Under lower diaphragm excitation frequency, the synthetic jet is affected weakly by the orifice thickness in the neighboring of orifice acoustic resonance thickness. When the diaphragm is excited at high frequency, the perturbation frequency of the vibrating diaphragm to the internal air insides the cavity is very high, small orifice diameter will restrict the ejection and suction capacity of the synthetic jet actuator. (4) The simulations suggest that Helmtoltz resonance frequency presented by Gallas et al. [22,23] is in general identical to those
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for optimizing actuator performance in many circumstances. When the diaphragm is operated under relative low excitation frequency with large cavity acoustic resonance depth, the synthetic jet is affected hardly by the cavity depth in the neighboring of cavity acoustic resonance depth. For this situation, one could choose a relative small cavity in spite of the cavity acoustic resonance depth. Acknowledgement The authors gratefully acknowledge the financial support for this project from the National Natural Science Foundation of China (Grant No.: 51306088). References [1] B.L. Smith, A. Glezer, The formation and evolution of synthetic jets, Phys. Fluids 10 (1998) 2281–2297. [2] A. Glezer, M. Amitay, Synthetic jets, Annu. Rev. Fluid Mech. 34 (2002) 503–529. [3] C. Lee, G. Hong, Q.P. Ha, A piezoelectrically actuated micro synthetic jet for active flow control, Sens. Actuators A 108 (2003) 168–174. [4] G. Hong, Effectiveness of micro synthetic jet actuator enhanced by flow instability in controlling laminar separation caused by adverse pressure gradient, Sens. Actuators A 135 (2006) 607–615. [5] S.Y. Zhang, S. Zhong, Experimental investigation of flow separation control using an array of synthetic jets, AIAA J. 49 (2011) 2637–2649. [6] B.L. Smith, A. Glezer, Jet vectoring using synthetic jets, J. Fluid Mech. 458 (2002) 1–34. [7] Z.B. Luo, Z.X. Xia, Y.G. Xie, Jet vectoring control using a novel synthetic jet actuator, Chin. J. Aeronaut. 20 (2007) 193–201. [8] B.C. Mather, F. Mohsen, B. Jean-Christophe, Aerodynamic flow vectoring of a wake using asymmetric synthetic jet actuation, Exp. Fluids 53 (2012) 1797–1813. [9] H. Wang, S. Menon, Fuel–air mixing enhancement by synthetic microjets, AIAA J. 39 (2001) 2308–2319. [10] Y.M. Liu, B.G. Wang, S.Y. Liu, Investigation of phase excitation on mixing control on coaxial jets, J. Therm. Sci. 18 (2009) 364–369. [11] Q.F. Xia, S. Zhong, Enhancement of laminar flow mixing using a pair of staggered lateral synthetic jets, Sens. Actuators A 207 (2014) 75–83. [12] R. Mahalingam, N. Rumigny, A. Glezer, Thermal management with synthetic jet ejectors, IEEE Trans. Compon. Packing Technol. 27 (2004) 439–444. [13] M. Chaudhari, B. Puranik, A. Agrawal, Effect of orifice shape in synthetic jet based impingement cooling, Exp. Therm. Fluid Sci. 34 (2010) 246–256. [14] M. Chaudhari, B. Puranik, A. Agrawal, Heat transfer characteristics of synthetic jet impingement cooling, Int. J. Heat Mass Transfer 53 (2010) 1057–1069. [15] S. Gao, J.Z. Zhang, X.M. Tan, Experimental study on heat transfer characteristics of synthetic jet driven by piston actuator, Sci. China Technol. Sci. 55 (2012) 1732–1738. [16] J.Z. Zhang, S. Gao, X.M. Tan, Convective heat transfer on a flat plate subjected to normally synthetic jet and horizontally forced flow, Int. J. Heat Mass Transfer 57 (2013) 321–330. [17] X.M. Tan, J.Z. Zhang, Flow and heat transfer characteristics under synthetic jets impingement driven by piezoelectric actuator, Exp. Therm. Fluid Sci. 48 (2013) 134–146. [18] A. Crook, A.M. Sadri, N.J. Wood, The development and implementation of synthetic jets for the control of separated flow, AIAA Paper 99-3176, 1999. [19] F.J. Chen, C. Yao, G.B. Beeler, et al., Development of synthetic jet actuators for active flow control at NASA Langley, AIAA Paper 2000-2405, 2000.
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Biographies Lv Yuan-wei obtained his B.E. degree in 2012 from Nanjing University of Aeronautics and Astronautics (NUAA), PR China. He is currently pursuing the M.S. degree at the College of Energy and Power Engineering, NUAA. His research is focused on the active flow control, in particular synthetic jets. Zhang Jing-zhou is currently a Professor at Nanjing University of Aeronautics and Astronautics, PR China and a visiting Professor at Collaborative Innovation Center of Advanced Aero-Engine, PR China. He received B.S. degree in 1986 from Tsinghua University, PR China, and M.S. degree in 1989 from Southeast University, PR China, and Ph.D. degree in 1992 from Nanjing University of Aeronautics and Astronautics, PR China, respectively. His research interests include heat transfer enhancement, active cooling of aero-engine hot components, infrared radiation and mixer-ejector technology, etc. By now, He has published approximately 250 scientific papers in the above research regions and three books on the Heat transfer and Mixer-Ejector issues. Shan Yong is an assistant professor in the College of Energy and Power Engineering and Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics (NUAA) since 2009. He received B.E. degree in 2000, M.S. degree in 2003 and Ph.D. degree in 2006, all in NUAA. His research interests include gas dynamics, infrared radiation, and mixer-ejector technology. Tan Xiao-ming is an assistant professor in the College of Energy and Power Engineering and Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing University of Aeronautics and Astronautics (NUAA) since 2008. She received B.E. degree in 1998 and Ph.D. degree in 2006 in NUAA. Her research interests include heat transfer enhancement and active cooling of aero-engine hot components.