Analysis of external flow characteristics in two-dimensional synthetic jets under excitation produced by cavity resonance

3.1. Flow behavior of synthetic jet in transverse direction

Using flow control of the synthetic jet actuator needs to understand its parameters. For an isolated two-dimensional SJA, whose characteristic in motionless flow major depend on three kinds of parameters [21, 22]: (1) the actuator geometric parameters, including the exit width $w$, exit depth $h$, cavity width $w_c$ and cavity depth $H$; (2) the implementing parameters, including the membrane oscillation $\delta$ (or piston amplitude) and excitation frequency $f_e$; (3) the working fluid parameters, including the fluid density $\rho$ and kinematic viscosity $\nu$. These parameters of governing equation for behavior of a synthetic jet can be expressed as follows:

where $U_0$ is a time-averaged ejected velocity. According to the Buckingham $\pi$ theory, then $w$, $\rho$ and $f_e$ are selected as repeatable fundamental dimensions. Moreover, the six dimensionless $\pi$ parameters can be formed in the following equation:

Above six independent $\pi$ parameters via the Buckingham π theory can be obtained as ${\pi }_1=\delta /w\text{,}$${\pi }_2=h/w\text{,}$${\pi }_3=H/w\text{,}$${\pi }_4=w_c/w\text{,}$${\pi }_5=U_{0}w/\nu \text{,}$ and ${\pi }_6=U_{0}T/w$. These parameters of ${\pi }_1$-${\pi }_4$ are fixed values with respect to design of a synthetic jet. Both ${\pi }_5$ and ${\pi }_6$ belong to operating parameters of a synthetic jet actuator. The quantity ${\pi }_5$ is a dimensionless parameter of the Reynolds number. Additionally, the quantity ${\pi }_6$ can be expressed as dimensionless stroke length. Besides, Eq. (3) is still simplified as Eq. (4):

The stroke length $L_0$ indicates the synthetic jet travel distance from exit orifice while the fluid is ejected out the cavity. $L_0=U_{0}T$, where $T$ is the entire period of the oscillating membrane and $U_0$ is the synthetic jet average orifice velocity, defined as in the Eq. (5):

Fig. 2Flow visualization for formation of a synthetic jet using excitation of cavity resonance with 15 V sine wave (Total ejected period plus sucked period T= 0.00625 second)

a)$t/T=$0

b)$t/T=$ 0.1

c)$t/T=\mathrm{}$0.2

d)$t/T=\mathrm{}$0.3

e)$t/T=\mathrm{}$0.4

f)$t/T=\mathrm{}$0.5

g)$t/T=\mathrm{}$0.6

h)$t/T=\mathrm{}$0.7

i)$t/T=\mathrm{}$0.8

j)$t/T=\mathrm{}$0.9

k)$t/T=\mathrm{}$1

Flow visualization for formation of the synthetic jet at the cavity resonance has been obtained that its dimensionless stroke length is less than that of the resonance of diaphragm and the response of its exit velocity at the SJA orifice is strong better than other excitation frequencies except the resonance of diaphragm.

For the resonance frequency in cavity, it takes place at 160 Hz, the flow characteristic of external SJA needs to discuss further in this investigation. The characteristics of resonance with the SJA cavity has been done in some literatures [16-18, 21-29], but these results at excitation of cavity resonance aren’t still described clear about the external flow characteristics of the SJA.

In order to obtain flow visualization outside the cavity at excitation of cavity resonance, using a high-speed camera of MotionPro X4 is to capture images with sampling rate setting at 2000 pictures per second. The oscillating cycle of membrane for the excitation mechanism in the periodic time is 0.00625 second. These images in shown Fig. 2 reveal formation of the synthetic jet at cavity resonance with 15 V sine wave from flow visualization. While the flow ejected out the cavity, the dimensionless stroke length at cavity resonance is calculated as $L_0/w=$1.5. Evidently, the flow stops traveling along downstream of center from exit orifice during the flow sucked into the cavity. In contrast, the flow begins to travel into the cavity. To this point, it makes a difference between the resonance of diaphragm and the cavity resonance. From the results in Fig. 2, it needs to obtain also flow field outside the synthetic jet actuator with the hot-wire measurements.

In Fig. 3, the synthetic jet is excited with a sine wave of 160 Hz at 15 V, wherein the mean velocities ($\stackrel{-}{u}$) of synthetic jets were measured along different transverses perpendicular to the exit slot are measured. Here $H$ refers to the 8 mm width of the exit slot.

Fig. 3(a) shows the three-dimensional relation among the average velocity, transverse distance ($y$-axis) and downstream distance ($x/H$) for the description of the behaviors of synthetic jets. It appears that the optimal mean velocity of the synthetic jet occurs at $y=$ 0. In other words, the velocity at the center of the orifice exit is the highest than any of transverses. To gain a more comprehensive understanding of vortex formation, we compared the transverse mean velocities of the synthetic jet from $y=$ –40 mm to 40 mm (–5$\le y/H\le$ 5). From the results in Fig. 3(b), it is clear that the mean velocity of $y/H$ (in the range of -0.5 to 0.5), whereas the mean velocities of $x/H$ ranges (in the range of 0.5 to 1) are all beyond 6 m/s. Beyond this point, the transverse mean velocities continue decreasing as the synthetic jet moves away from the orifice. The results in Fig. 3(b) indicate that the mean velocities reached approximately 1 m/s in every transverse from $y=$ –20 mm to $y=$20 mm when the synthetic jet moves downstream as far as $x/H=$ 5. According to the above description, plane regions of 0$\le x/H\le$ 5 and –2.5$\le y/H\le \mathrm{}$2.5 could be used for flow control.

Fig. 3a) Three-dimensional relationships among mean velocity, y-axis, and x/H; b) mean velocity versus y-axis of a synthetic jet

a)

b)

3.2. Flow behavior of synthetic jet in downstream direction

In the following, our work focuses on the relationship between the streamwise mean velocities and traveling distance for several transverses at $y/H=$ 0, 0.5, 1, 1.5, 2, 2.5 and 3, as shown in Fig. 4. For the transverse $y/H=$ 0, the downstream mean velocity reached 8 m/s at the exit of the orifice ($x=$ 0) and then decreased suddenly to 7.16 m/s at $x=$ 4 mm (or $x/H=$ 0.5). One explanation would be that the flow was ejected from the orifice by the oscillation of the membrane before a vortex was produced.

The mean velocity begins increasing after the flow passes this point ($x/H=$ 0.5), which is an indication that the pair of vortices have developed. When the flow passes $x/H=$ 1, its mean velocity reaches another peak (i.e., the mean center velocity reaches 7.5 m/s), whereupon it drops rapidly until the flow is pulled into the cavity and disappears in the air. This result can also be explained as follows. When the pair of vortices become sufficiently large, the velocity begins decreasing. It is interesting to note that the mean velocity is inversely proportional to the distance from the orifice exit except at $x/H=$ 0 and $x/H=$ 0.5 in the case of $y/H=$ 0. The same trend in Fig. 4 means that the mean velocities are also inversely proportional to distance from the orifice exit in the case of transverses of $y/H=$ 0.5, $y/H=$ 1, $y/H=$ 1.5, $y/H=$ 2, $y/H=$ 2.5, and $y/H=$ 3. Fig. 5 presents the results obtained by taking the logarithm of the data in Fig. 4. The slopes obtained by curve fitting fall between –7/10 and –3/5 for the synthetic jet under excitation of cavity resonance using a sine wave 160 Hz at 15 V.

Fig. 4Mean velocity versus downstream distance from synthetic jet

Fig. 5Logarithm of mean velocity and downstream distance from synthetic jet

3.3. Characteristic frequencies of flow field

This section concentrates on the study of characteristic frequency from near to far field associated with a synthetic jet under excitation from cavity resonance. The original data of velocity at the center of the orifice exit ($x/H=$0 and $y/H=$ 0) were obtained by fast Fourier transform, as shown in Fig. 6. The excitation frequency is 160 Hz; however, there are two characteristic frequencies at 80 Hz and 160 Hz in the flow field. When the synthetic jets locate at different transverses ($y/H=$ 0.5, 1, 1.5 and 2) for the same downstream location ($x/H=$ 0.5), the variation of characteristic frequencies can be observed in Fig. 7.

Initially, data in the exit center of the orifice ($x/H=$ 0 and $y/H=$ 0) was obtained using cross hot-wire measurement, which manifest as frequencies at 80 Hz and 160 Hz. The generation of the characteristic frequency at 80 Hz can be attributed to the collision of the upper and lower vortices of the synthetic jet. The other characteristic frequency at 160 Hz is an excitation frequency. It should be noted that the density of the power spectrum at the characteristic frequency at 160 Hz is higher than at other frequencies.

The cross hot-wire probe was moved to $x/H=$ 0.5 for four different transverses, as shown in Fig. 7. The sequence of development of the vortex in the synthetic jet can be clearly observed in Figs. 7(a) to 7(d). When the vortex began forming at $x/H=$ 0.5 and $y/H=$ 0 at the downstream center of the orifice exit, only one a characteristic frequency was observed at 80 Hz, at which point the other characteristic frequency at 160 Hz disappeared, as shown in Fig. 7(a). As the vortex of the synthetic jet gradually enlarged, the hot-wire probe was moved to another location at $x/H=$ 0.5 and $y/H=$ 0.5. At this point, the characteristic frequency at 80 Hz suddenly increased, as shown in Fig. 7(b), whereupon the velocity power spectra of this characteristic frequency decreased gradually with an increase of $y/H$, as shown in Figs. 7(c) and 7(d). The hot-wire probe continued to move downstream until $x/H=$ 8, at which point only the characteristic frequency at 80 Hz remained within the flow field, and it gradually dissipated into the flow field.

Fig. 6Relationship between velocity power spectra and characteristic frequency at center of orifice exit (x/H= 0 and y/H= 0)

Fig. 7Relationship between velocity power spectra and characteristic frequency at different downstream locations from a synthetic jet