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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 V_{PN} 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. [

_{SW} and L_{SW}, respectively. Their reduced forms G_{SW}/s and L_{SW}/s are fixed to 2 and 4.5, respectively. A sole kinetic parameter, the Reynolds number Re, is defined by ρV_{PN} s/μ, where ρ, V_{PN} and μ denote the density of fluid, the mean velocity at a primary-nozzle exit and the viscosity of fluid, respectively. In the regular-oscillation experiment, the connecting tube with a length L has a circular cross section with an inside diameter d. Their reduced forms d/s and L/s vary from 1.2 to 1.4 and from 100 to 300, respectively. On the other hand, in the single-port-control experiment, a flow-increment rate coefficient K (see later for its definition) varies from 1 × 10^{−4} to 3.5 × 10^{−4} according to the results in the regular oscillation experiment.

(a) Basic nozzle dimensions | |||
---|---|---|---|

Primary-nozzle-throat spacing | s | (m) | 0.01 |

Control-port spacing | b | (m) | 0.01 |

Gap between side walls | G_{SW} | (m) | 0.02 |

Streamwise length of side walls | L_{SW} | (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 | G_{SW}/s | 2 | |

Reduced side-wall length | L_{SW}/s | 4.5 | |

(b) Kinetic parameter | |||

Flow velocity at primary-nozzle exit | V_{PN} | (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 | dV_{T}/dt | (m/s^{2}) | 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 Q_{T} 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 Q_{T} and hot-wire anemometer signal V_{T}. _{T} is detected by a measuring bar (No. 6) attached to a float on the anterior water surface of the U-tube. We record the value of the measuring bar by a camcorder (No. 5). These signals are recorded and analysed by a personal computer (No. 9). _{T} based on the flow velocity V_{T} is measured by

the hot-wire anemometer, together with the actual Q_{T} by the hot-wire anemometer measured by a camcorder. We compensate Q_{T} on the basis of this result, then determine Q_{T} 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. [_{PN} at a primary-nozzle exit, fluid density ρ and fluid viscosity μ. The three geometric parameters are the spacing s of a primary-nozzle throat, the length L of a connecting tube and the inner diameter d of a connecting tube. We regard s, V_{PN} and ρ as characteristic scales. Then, according to the dimensional analysis, we get

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 _{T} in the connecting tube obtained by the hot-wire anemometer (No. 13 in _{T} is closely periodic, as well as Δp. Again, the periodicity is not rigorous due to high-frequency random fluctuations superimposed. While the wave form of V_{T} seems to be not sinusoidal but non-isosceles, it is rather different from the wave form of Δp in

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. [_{T}, as shown in

where V_{T} 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 ≤ Re_{CT} < 2900, then

where γ = 9.8 × 10^{−4} Re_{CT} − 1.852. And, if 2900 ≤ Re_{CT} < 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 V_{T} in _{T} does not include randomly-fluctuating components.

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 J_{P} of mass flux, in addition to the time integral J_{M} of momentum flux for comparison. As mentioned in Section 1, J_{M} is proposed by Funaki et al. [_{P} is the accumulated mass, which could be essentially regarded as the accumulated flow work by pressure to the fluid inside the re-circulation regions; strictly speaking, it is the product of the accumulated volume (or the quotient of the accumulated mass divided by ρ) and the pressure difference between a re-circulation region (Funaki et al. [_{P} as the accumulated flow work. Specifically speaking, the integrals J_{P} and J_{M} are defined as follows.

where V_{CP} 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), _{0} and t_{SW}. At t = τ_{SW}, Δp jumps from zero to a positive value. This jump of Δp corresponds to the jet’s switching onto a side wall from the opposite side wall. On the other hand, V_{T} is still negative even at t = τ_{SW}, that is, the fluid flows in the connecting tube from the jet-switching side to the opposite un-jet-switching side. Thereafter, Δp monotonically decreases with time t toward zero. On the other hand, V_{T} monotonically increases with t toward a certain positive value, crossing zero at t = t_{0}. Then, V_{T} becomes positive at t > t_{0}. In other words, the flow in the connecting tube is reversed at t = t_{0} and afterwards continues to accelerate. Finally, at t = t_{SW}, Δp jumps from zero to a certain negative value, corresponding to the jet’s switching from the side wall on the jet-switching side onto the opposite side wall. And, V_{T} begins to decelerate toward the next reverse of the connecting-tube flow from a certain positive value. In summary, there exists no reversed flow during the supposed 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 _{UNV} (=0.01) attains

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) κ_{UNV} 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 _{UNV} (=0.012) assuming a triangular-wave pressure difference with C_{Δp}_{AMP} = 0.11 shows good agreement with the experiment. In order to confirm the effectiveness of the empirical formula Equation (13), we have calculated the comparison between all the experiments and the corresponding predictions based on_{Δp}_{AMP} and κ_{UNV} are approximated to be constant without any dependences upon the three factors like L/s, d/s and Re in the prediction.

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 dV_{T}/dt ought to be constant, but vary with time t. However, as seen in Figures 7(b)-(d), V_{T} could increase approximately with a constant acceleration from the reversed time t_{0} to the jet-switching time t_{SW}. 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. _{T}’s like _{0} to t_{SW}, because K depends upon t in a strict sense. We can confirm the dependence of K upon L/s, d/s and Re. The actual range of K in the regular oscillation varies from 1 × 10^{−}^{4} to 5 × 10^{−4}. So, we next conduct the single-port-control experiments in a range of K from 1 × 10^{−4} to 3.5 × 10^{−4}, keeping a constant acceleration of the inflow as closely as possible. To be exact, although K in the regular oscillation is not the same as that in the single-port-control experiments, the order of K is the same.

_{T} and V_{EX} and a pressure p_{TE} where V_{EX} and p_{TE} denote the flow velocity near one side wall at the FFJN’s exit and the pressure at the connecting-tube end adjacent to the chamber and the control port. ^{−4} and Re = 8800, and ^{−4} and Re = 8800. In each figure, figures (a), (b) and (c) represent V_{T}, V_{EX} and p_{TE}, respectively.

At first, we see _{T} starts to increase from zero with time t at t = t_{0} (=0.6 s). We should note that the instant at t = 0 merely represents the time when each measurement starts. The increasing manner is not strictly linear, but almost linear with a constant acceleration during the duration t = 0.6 - 1.1 s. At t = t_{SW}, the jet from the primary nozzle switches from the beforehand-jet-attached side to the opposite afterward-jet-attached side. In order to determine t_{SW}, this switching is preliminarily observed by flow visualisation using smoke together with simultaneous measurements of V_{T}, V_{EX}, p_{TE} and so on. Actually, corresponding to this jet switch at t = t_{SW}, V_{EX} in _{TE} in

same time, respectively. Then, we can determine K or dV_{T}/dt from _{T}/dt is time-mean which is obtained by such three data as t_{0}, t_{SW} and the V_{T} at t = t_{SW}. Of course, the above features can be seen in

^{−4} and at various values of d/s. Each plot represents the ensemble mean over five trials in the single-port-control experiments. And, a solid line represents proposed empirical formula Equation (13) for

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 V_{PN} 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 _{P} to determine f. Second, to discuss the physics of J_{P} further, we have conducted another experiment in single- port control, instead of the experiment in regular jet oscillation using the ordinary nozzle

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 J_{P} 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 J_{P} 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)

C_{Δp}_{AMP}: coefficient of pressure-difference amplitude º Δp_{AMP}/(1/2 ρV_{PN}^{2})

d: (inner) diameter of a connecting tube (m)

f: frequency (Hz)

G_{SW}: gap between side walls (m)

J_{M}: time integral of momentum flow per unit span (kg/s)

J_{P}: time integral of mass flow per unit span (kg/m)

K: flow-increment-rate coefficient

L: length of a connecting tube (m)

L_{SW}: streamwise length of a side wall (m)

p: pressure (Pa)

Δp: pressure difference between two connecting-tube ends (Pa)

Δp_{AMP}: (half) amplitude of Δp (Pa)

Q: (volumetric) flow rate (m^{3}/s)

Re: Reynolds number, º ρV_{PN} s/μ

Re_{CT}: connecting-tube Reynolds number, º ρV_{T} d/μ

S: span (m)

s: spacing of a primary-nozzle throat (m)

St: Strouhal number, º f s/V_{PN}

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/m^{3})

τ_{SW}: time at former jet’s switching (s)