This research deals with the oscillation mechanism of a flip-flop jet nozzle with a connecting tube, based on the measurements of pressures and velocities in the connecting tube and inside the nozzle. The measurements are carried out varying: 1) the inside diameter d of the connecting tube; 2) the length L of the connecting tube and 3) the jet velocity VPN from a primary-nozzle exit. We assume that the jet switches when a time integral reaches a certain value. At first, as the time integral, we introduce the accumulated flow work of pressure, namely, the time integral of mass flux through a connecting tube into the jet-reattaching wall from the opposite jet-un-reattaching wall. Under the assumption, the trace of pressure difference between both the ends of the connecting tube is simply modeled on the basis of measurements, and the flow velocity in the connecting tube is computed as incompressible flow. Second, in order to discuss the physics of the accumulated flow work further, we conduct another experiment in single-port control where the inflow from the control port on the jet-reattaching wall is forcibly controlled and the other control port on the opposite jet-un-reattaching wall is sealed, instead of the experiment in regular jet’s oscillation using the ordinary nozzle with two control ports in connection. As a result, it is found that the accumulated flow work is adequate to determine the dominant jet- oscillation frequency. In the experiment in single-port control, the accumulated flow work of the inflow until the jet’s switching well agrees with that in regular jet’s oscillation using the ordinary nozzle.
The flip-flop jet nozzle (hereinafter, referred to as FFJN) is regarded as one kind of fluidic oscillator, which is oscillating devices among the fluidics. The fluidics, or the elements in fluid logic, is applications of the Coanda effect where a jet reattaches to a solid side wall, and has been researched since the 1960s [
In such a context, there have been many researches on the FFJN in both fundamental and practical approaches [
Among them, in order to reveal the flow inside the FFJN, we have carried out the measurements of unsteady flow-velocity distributions by an ultrasound-velocity-profile (UVP) monitor which gives us instantaneous information with higher accuracy in comparison to the conventional particle-image velocimetry [
Our purpose is to elucidate the oscillation mechanism of the FFJN. In the present study, we focus upon a dominant jet-oscillation frequency of the FFJN, based on the measurements of pressures and velocities in the connecting tube and inside the FFJN, and attempt to find out the universal number which determines the jet-oscillation frequency. The measurements are carried out, varying: 1) the inside diameter d of the connecting tube; 2) the length L of the connecting tube and 3) the jet velocity VPN from a primary-nozzle exit. We assume that the jet switches when a time integral reaches a certain value. At first, as this time integral which can be the universal number for the jet’s switching, we introduce the accumulated flow work of pressure, namely, the time integral of mass flux through a connecting tube into the jet-reattaching wall from the opposite jet-un-reattaching wall. Under the assumption, the trace of pressure difference between both the ends of the connecting tube is simply modeled on the basis of measurements. Such modeling is the same as Funaki et al. [
More specifically, in our previous study Funaki et al. [
To predict the dominant frequency is very useful and strongly needed in many practical aspects, as we have not yet established any general prediction methods as mentioned above. One of the main factors preventing the establishment is the spatial-and- temporal complexity of the flow inside the FFJN: for example, quasi-steady approaches are not suitable even for very-low dominant frequencies and the momentum-theory approaches are difficult in setting the control volume. Therefore, the present approach could be effective for a breakthrough, in addition to our previous study Funaki et al. [
(a) Basic nozzle dimensions | |||
---|---|---|---|
Primary-nozzle-throat spacing | s | (m) | 0.01 |
Control-port spacing | b | (m) | 0.01 |
Gap between side walls | GSW | (m) | 0.02 |
Streamwise length of side walls | LSW | (m) | 0.045 |
Span | S | (m) | 0.05 |
Aspect ratio of primary-nozzle throat | A, º S/s | 5 | |
Reduced control-port spacing | b/s | 1 | |
Reduced side-wall gap | GSW/s | 2 | |
Reduced side-wall length | LSW/s | 4.5 | |
(b) Kinetic parameter | |||
Flow velocity at primary-nozzle exit | VPN | (m/s) | 11.3 - 34.7 |
Reynolds Number | Re | 7500 - 23,000 | |
(c) Basic connecting-tube dimensions: parameters for regular oscillation | |||
Connecting-tube length | L | (m) | 1.0, 1.5, 2.0, 2.5, 3.0 |
Connecting-tube diameter | d | (m) | 0.012, 0.013, 0.014 |
Reduced connecting-tube length | L/s | 100, 150, 200, 250, 300 | |
Reduced connecting-tube diameter | d/s | 1.2, 1.3, 1.4 | |
(d) Parameter for single-port control | |||
Flow-increment rate | dVT/dt | (m/s2) | 1.8 - 30.7 |
Flow-increment-rate coefficient | K | 1, 1.5, 2, 2.5, 3, 3.5 (×10−4) |
oscillation experiment. The control port on the opposite side is sealed by a plug, being flush with a side wall. As well as the regular-oscillation experiment, the working fluid is air, which is provided by an air compressor (No. 10 in the figure) into a primary nozzle (No. 20) of the FFJN, through an air dryer (No. 11), a pressure regulator (No. 3), a flow meter and a long straight duct. Volumetric flow rate into the FFJN measured by the flow meter is compensated using both the temperature and the pressure detected by a thermocouple and a pressure transducer which are placed adjacent to the flow meter. The inflow from the control port is driven by a blower through a tube (No. 1), which is regulated by a flow-control value (No. 4). In the single-port-control experiment, the jet from the primary nozzle is reattached to the side wall with the control port in advance. Then, we force the jet to switch by the inflow, from the side wall to the opposite side wall without the control port. Pressures and velocities at several points are simultaneously measured by two pressure transducers (Nos. 15 & 16) and two hot-wire anemometers (Nos. 17 & 18), respectively. These signals are recorded and analysed by a personal computer (No. 7).
In the single-port-control experiment, we quantitatively characterise the magnitude of the inflow using volumetric flow rate QT from the control port through the tube, which is detected by a hot-wire anemometer at the tube end adjacent to a chamber and the control port. So, prior to the single-port-control experiment, we need the calibration between QT and hot-wire anemometer signal VT.
the hot-wire anemometer, together with the actual QT by the hot-wire anemometer measured by a camcorder. We compensate QT on the basis of this result, then determine QT in the single-port-control experiment.
The frequency f of jet’s oscillation is important not only from an academic viewpoint but also from a industrial viewpoint. According to Raman et al. [
where
All the symbols in
with experimental constants such as C = 0.068, α = −0.72, β = 1.37 and γ = 0.22. The experimental constants are determined using the least-squares method based on all the experimental results in
The empirical formula Equation (3) is practically useful not only for the present FFJN in the present test ranges of the governing parameters, but restricted due to the lack of theoretical background. Then, we consider more generally focusing upon the jet-oscillation frequency.
measurements, we get
Now, we summarise all the experiments concerning the pressure difference Δp from a quantitative point of view. Concerning the fluctuating period or the fluctuating frequency of Δp, we have already proposed Equation (3). Then, we next consider the fluctuating amplitude of Δp. To conclude, the pressure-difference-amplitude coefficient
At this stage, we attempt to purify these wave forms by a simple model which is the same as Funaki et al. [
where VT denotes the flow velocity averaged over a cross section of the connecting tube to be exact. λ is the resistance coefficient of pipe flow by Spriggs [
If 1900 ≤ ReCT < 2900, then
where γ = 9.8 × 10−4 ReCT − 1.852. And, if 2900 ≤ ReCT < 1,000,000, then
We numerically solve Equation (4) by the fourth-order Runge-Kutta method. To confirm numerical accuracy, we compare several computations with different time steps. As a result, we can see that the wave form of the computed VT in
Now, we consider the physical background of the present approach. We assume that the jet switches, when the accumulation of the inflow into a jet-switching side wall from the connecting tube through a control port, and/or of the outflow from the opposite un-jet-switching side wall into the other control port and the connecting tube, reaches a certain value. As the accumulation, we examine the time integral JP of mass flux, in addition to the time integral JM of momentum flux for comparison. As mentioned in Section 1, JM is proposed by Funaki et al. [
where VCP denotes the flow velocity at the control port. In Equation (8) and Equation (9), a decaying factor w is given by Equation (10).
where κ denotes a damping constant. We should note that these integrals are amounts per unit span. To specify the integral interval in Equation (8) and Equation (9),
For convenience, all the integrals are usually normalised as follows.
in the present study, we have confirmed that L/s is the most influential upon St among the three (also see
At first, we see
Second, we see
To conclude, concerning the influences of the other two governing parameters d/s and Re in addition to the influence of L/s, we summarise all the results in the experimental ranges such as L/s = 100 - 300, d/s = 1.2 - 1.4 and Re = 7000 - 20,000 as follows. 1) κUNV is almost constant (≈0.012) being independent of both d/s and Re, 2)
on the basis of
As a result, the predicted St based on the empirical formula Equation (13) for
As will be revealed in the latter half of the present study, the inflow from one control port on the jet-reattaching wall is crucial for jet switching, while the outflow into the other control port on the opposite jet-un-reattaching wall is not crucial. At the present stage, although we do not have exact information to discuss the details of the jet switching mechanism, it seems acceptable that to weaken/destabilize the re-circulation region on the jet-reattaching wall could be a trigger of the jet’s switching. In this context, the jet switching is possibly controlled by a certain accumulated amount from the control port, such as
In the previous subsection, we have introduced
At first, we need to characterise the inflow from a quantitative point of view. Then, we assume a constant acceleration of the inflow or the connecting-tube flow, and define the flow-increment-rate coefficient K as follows.
K means a normalised acceleration of fluid in the tube. From a theoretical point of view, K or the acceleration dVT/dt ought to be constant, but vary with time t. However, as seen in Figures 7(b)-(d), VT could increase approximately with a constant acceleration from the reversed time t0 to the jet-switching time tSW. In other words, we could suppose the connecting-tube flow and the inflow continue to accelerate linearly until the instant when the jet switches. Under this situation, K becomes an appropriate parameter.
Next, we estimate the range of K in the actual regular oscillation of the ordinary FFJN with two control ports.
At first, we see
same time, respectively. Then, we can determine K or dVT/dt from
In order to reveal the oscillation mechanism of a flip-flop jet nozzle (FFJN) with a connecting tube, we have carried out the measurements of pressures and velocities in the connecting tube and inside the FFJN specially focusing on the jet-oscillation frequency f, varying: 1) the diameter d of the connecting tube; 2) the length L of the connecting tube and 3) the jet velocity VPN from a primary-nozzle exit. Obtained results are as follows. We have proposed an empirical formula to determine f, and confirmed its validity. Then, to consider f more generally, we assume that the jet switches when a time integral reaches a certain value. At first, as the time integral, we have introduced the accumulated flow work
with two control ports in connection. As the result, we have confirmed good agreement between the single-port control and the regular jet oscillation. This agreement suggests that JP from the connecting tube to the FFJN inside can be a key parameter to explain the jet’s switching, in addition to the validity of JP in practical aspects to estimate the dominant jet frequency of the FFJN.
Inoue, T., Nagahata, F. and Hirata, K. (2016) On Switching of a Flip-Flop Jet Nozzle with Double Ports by Single-Port Control. Journal of Flow Control, Measurement & Visualization, 4, 143- 161. http://dx.doi.org/10.4236/jfcmv.2016.44013
A: aspect ratio of a primary-nozzle throat, º S/s
b: breadth of a control port (m)
CΔpAMP: coefficient of pressure-difference amplitude º ΔpAMP/(1/2 ρVPN2)
d: (inner) diameter of a connecting tube (m)
f: frequency (Hz)
GSW: gap between side walls (m)
JM: time integral of momentum flow per unit span (kg/s)
JP: time integral of mass flow per unit span (kg/m)
K: flow-increment-rate coefficient
L: length of a connecting tube (m)
LSW: streamwise length of a side wall (m)
p: pressure (Pa)
Δp: pressure difference between two connecting-tube ends (Pa)
ΔpAMP: (half) amplitude of Δp (Pa)
Q: (volumetric) flow rate (m3/s)
Re: Reynolds number, º ρVPN s/μ
ReCT: connecting-tube Reynolds number, º ρVT d/μ
S: span (m)
s: spacing of a primary-nozzle throat (m)
St: Strouhal number, º f s/VPN
t: time (s)
V: flow velocity (m/s)
w: decaying factor
λ: friction coefficient of pipe
κ: damping constant
μ: viscosity of fluid (Pa∙s)
ρ: density of fluid (kg/m3)
τSW: time at former jet’s switching (s)