On Switching of a Flip-Flop Jet Nozzle with Double Ports by Single-Port Control

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.

• Flip-Flop Jet Nozzle
• Flowmeter
• Fluidics
• Mixing
• Flow Control

1. Introduction

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 [1] - [5] . The FFJN retains useful features as well as other flow-induced-vibration devices: namely, 1) low production cost and high reliability due to non-mechanically-moving parts; 2) usability due to a linear frequency response in proportion to flow rate; and 3) robustness against fluid density, temperature, pressure and composition. Owing to the above features, the FFJN is applicable for such products as flow meters, fuel injectors, micro mixers and various control devices to disturb the shear layer or to enhance heat transfer, not only for single- phase flows but also for multi-phase flows. Thus, the FFJN is often called a “fluidic oscillator” or “oscillatory-jet-type flowmeter” in different applications.

In such a context, there have been many researches on the FFJN in both fundamental and practical approaches [6] - [22] . Besides, various simplified FFJN’s without control ports, chambers, feedback loop (or connecting tube) and so on, have been reported [23] - [30] .

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 [19] . Because the flow inside the FFJN is usually turbulent with random perturbations in addition to a dominant jet’s oscillation, we conducted conditional sampling by the phase-averaging technique using the flow velocity in the connecting tube as a reference signal. However, the oscillation mechanism of the FFJN has not been fully understood yet. In general, it is still difficult to predict the jet-oscillation frequency, which depends upon such various parameters as connecting-tube length, connecting-tube volume, flow rate, nozzle’s geometries and so on according to Raman et al. [12] , even in the present state.

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. [31] . Using this model, the flow velocity in the connecting tube is computed by the Runge-Kutta method 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 one control port on the jet-reattaching wall is forcibly controlled by a blower-and-value system and the other control port on the opposite jet-un-reattaching wall is sealed by a plug, as opposed to the experiment in regular oscillation using the ordinary FFJN with two control ports in connection.

More specifically, in our previous study Funaki et al. [31] , the accumulated flow work was investigated, namely the time integral of momentum flux. However, there is a possibility of predicting the jet-oscillation frequency more directly. So, we attempt to examine two kinds of integrals; namely, the time integral of momentum flux, and the time integral of mass flux.

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. [31] .

2. Experimental Method

2.1. Model: FFJN

Figure 1 shows the present model of a FFJN, together with its main dimensions. The FFJN consists of a primary nozzle, two side walls, two control ports, two chambers and a connecting tube. The two chambers with the control ports on the side walls are linked to each other by the connecting tube, to cause regular jet oscillation. The basic dimensions of the FFJN are determined according to Viets [8] . In addition to the experiment in regular jet’s oscillation using the ordinary FFJN with two control ports in connection, we conduct another experiment in single-port-control where we seal one control port by a plug and forcibly feed fluid from the other un-sealed control port.

Figure 1

Model: a flip-flop jet nozzle (FFJN)

Table 1 summarises the present experimental parameters. The chief geometric and kinetic parameters are as follows. In a characteristic length scale the spacing s of the primary-nozzle throat and the nozzle span S are fixed to 0.01 m and 0.05 m, respectively. So, the corresponding aspect ratio A (º S/s) is equal to 5. The control port with a breadth b is on each side wall, where b is fixed to the same as s in the present study. The gap between the side walls and the streamwise length of the side walls are denoted by G_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.

Table 1

Experimental parameters

(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)

2.2. Experimental Apparatus for Regular Oscillation

Figure 2 shows the schematic diagram of the present experimental apparatus in a regular-oscillation experiment. The main part of an ordinary FFJN with a primary nozzle (No. 6 in Figure 2), two control ports, two chambers (Nos. 11 & 12) and the connecting tube (No. 5) consists of acrylic-resin plates and a PVC tube. The working fluid is air, which is provided by an air compressor (No. 1) into the primary nozzle (No. 6) of the FFJN, through an air dryer (No. 2), a pressure regulator (No. 3), a flow meter (No. 4) and a long straight duct. The rectification duct in the upstream of the primary nozzle is straight and enough long (200 s = 2 m) to suppress the pulsation included in primary-nozzle jet. 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. Pressures and velocities at several points are simultaneously measured by four pressure transducers KYOWA PGM-G (Nos. 7 - 10), and two hot-wire anemometers KANOMAX 7000 with I-type probes (Nos. 13 & 14), respectively. More specifically, two (Nos. 7 & 8) of the four pressure transducers are near the connecting-tube ends, one (No. 9) of the four is in the upstream of the primary nozzle, and the other (No. 10) of the four is on the side- wall inside the FFJN. One of the two hot-wire anemometers (No. 13) is for the measurement of the flow velocity inside the connecting-tube, and the other one (No. 14) is for the measurement of the flow velocity just outside the FFJN exit. The hot-wire anemometers are calibrated using a Pitot tube outside the experimental apparatus for each measurement whose duration is less than 2 hours. These signals are recorded and analyzed by a spectrum analyzer (No. 17) and a personal computer (No. 18).

2.3. Experimental Apparatus for Single-Port Control

Figure 3 shows the schematic diagram of the present apparatus in the single-port experiment. The FFJN has a single control port instead of dual control ports, as in the regular

Figure 2

Experimental apparatus for regular oscillation in ordinary FFJN operation

Figure 3

Experimental apparatus for single-port control

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. Figure 4 shows the schematic diagram of the apparatus for the calibration. The working fluid is air, which is provided by a blower (No. 1 in the figure) through a tube with a diameter d into a U-tube with wider cross section than the tube. Pressure and velocity at the tube end are simultaneously measured by a pressure transducer (No. 13) and a hot-wire anemometer (No. 14), respectively. The actual value of volumetric flow rate Q_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). Figure 5 shows the result of the calibration, where the time history of the flow rate Q_T based on the flow velocity V_T is measured by

Figure 4

Experimental apparatus for the calibration of volumetric flow rate Q T

Figure 5

Flow rate Q T based on V T measured by a hot-wire anemometer, together with that by a camcorder

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.

3. Results and Discussion

3.1. Regular Oscillation: Jet-Oscillation Frequency and Connecting-Tube Flow

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. [12] , f depends upon such various parameters as connecting-tube length, connecting-tube volume, flow rate, nozzle’s geometries and so on. However, it is still difficult to predict f even in the present stage. So, we first propose an empirical formula to determine f. In the present study, we get the experimental value f by the Fourier analysis on the flow-velocity fluctuation detected at the FFJN exit (No. 14 in Figure 2). As governing parameters for f, we suppose three geometric ones in addition to mean velocity V_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 denotes an arbitrary function. In this Equation (1), we consider a normalised f, namely, the Strouhal number St, instead of f. The definition of St is given by

All the symbols in Figure 6 represent the experimental results of St plotted against Re at L/s = 100 - 300, d/s = 1.2 - 1.4 and Re = 7000 - 20,000. While there exist minor random scatterings, these are not due to lower stability of the present periodic flow phenomenon but due to unknown factors included in the actual experimental apparatus. From the results, we can see the following three tendencies. That is, we can see that 1) St monotonically decreases with increasing L/s, that 2) St monotonically increases with increasing d/s and that 3) St monotonically increases with increasing Re. Then, we assume the following power function as.

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 Figure 6. We should note that L/s is the most influential upon St among the three non-dimensional governing parameters in the present test ranges. The curves in Figure 6 show this empirical formula Equation (3). We can see that the empirical formula almost agrees with the experiment.

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. Figure 7(a) shows a typical example of the present experiments; that is, the time history of the pressure difference Δp between both the connecting tube ends at L/s = 100, d/s = 1.2 and Re = 13,000. The wave form in Figure 7(a) indicates some features which are commonly seen in all the present experiments; namely, 1) close periodicity with high-frequency random fluctuations and 2) the wave form characterised by the two similar non-isosceles triangles with positive and negative signs during each jet-oscillation period. The second feature is commonly observed, whenever the wave form is almost periodic. In fact, Fourier-transform analyses on the present results always are characterised by one remarkable and stable spectrum peak repre- senting a dominant periodicity, together with some secondary peaks representing higher harmonics due to the periodic but non-sinusoidal wave form. By means of simultaneous

Figure 6

Strouhal number St against Reynolds number Re, in regular FFJN oscillation. Symbols and lines denote experiments and an empirical formula, respectively

Figure 7

Time histories of pressure difference Δp between both connecting-tube ends and velocity V T in a connecting tube, in regular FFJN oscillation at L/s = 100, d/s = 1.2 and Re = 13,000

measurements, we get Figure 7(b), which is the corresponding time history of the flow velocity V_T in the connecting tube obtained by the hot-wire anemometer (No. 13 in Figure 2). We can see that V_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 Figure 7(a).

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, which is the normalised half value of the difference between the ensemble mean of the maximum Δp and the ensemble mean of the minimum Δp, almost keeps a constant value of about 0.11 through all the present measurements, being independent of L/s, d/s and Re.

At this stage, we attempt to purify these wave forms by a simple model which is the same as Funaki et al. [31] . This is because, the experimental raw data including random fluctuations like Figure 7(a) and Figure 7(b) are not suitable for further delicate discussion on the switching mechanism of the jet. At first, Δp is simply modeled as a right-angled triangular wave, as shown in Figure 7(c). Then, using the modeled Δp, we compute V_T, as shown in Figure 7(d). The computational procedure is as follows. As the compressibility is negligible through all the present experiments, the governing equations of motion can be described by

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 [32] and JSME [33] , and are defined as follows. If, then

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 Figure 7(d) is almost the same as the experiment at one in Figure 7(b), not only qualitatively but also quantitatively. Of course, the periodicity in Figure 7(d) is rigorous, as the computed V_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. [31] . J_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. [19] ) and the connecting-tube end (or the chamber) on the same side. So, we hereinafter refer to J_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), Figure 7(c) and Figure 7(d) show the definitions of t₀ 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 = t₀. Then, V_T becomes positive at t > t₀. In other words, the flow in the connecting tube is reversed at t = t₀ 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.

Figure 8 shows the normalised integral (in Figure 8(a)), together with (in Figure 8(b)) plotted against the time constant κ, in a range of L/s = 100 - 300 at d/s = 1.3 and Re = 14,000. While we suppose three governing parameters like L/s, d/s and Re

Figure 8

Normalised integrals and against time constant κ, at L/s = 100 - 300, d/s = 1.3 and Re = 14,000.

in the present study, we have confirmed that L/s is the most influential upon St among the three (also see Figure 6). So, we first examine the influence of L/s prior to d/s and Re.

At first, we see Figure 8(a). tends to decrease with increasing κ at each L/s. As the decreasing manner of at each L/s is not the same but depends upon L/s, the curve with a certain L/s tends to cross the other curves with different L/s’s. Fortunately, we can find all the crosses in a narrow range of κ ≈ 0.01. In other words, we can expect that is the universal number to determine the jet-oscillation period (or St): that is, the jet switches when with κ = κ_UNV (=0.01) attains (=0.2).

Second, we see Figure 8(b). tends to decrease with increasing κ at each L/s, and the decreasing manner of depends upon L/s, as well as. The curves with each L/s tend to cross one another in a range of κ ≈ 0.006. As the range is wider than that for, all the curves in Figure 8(b) seems to focus less clearly than Figure 8(a). Thus, is considered to be more adequate than, for the universal number to determine the jet-oscillation frequency. We have successfully found out the superiority of to. As the reason for this finding, we should notice the present range of St, which is much wider than that in Funaki et al. [31] .

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) is almost independent of d/s. On the other hand, 3) we cannot ignore the influence of Re upon, which tends to monotonically decrease toward a constant value with increasing Re. This tendency is empirically given by

on the basis of for L/s = 100 - 300 like Figure 8(a) at various values of d/s and Re. Three constants in Equation (13) are derived using the least-squares method on the basis of all the results. This formula seems consistent, as the needed accumulated flow work for jet switching increases with decreasing Re.

As a result, the predicted St based on the empirical formula Equation (13) for together with κ = κ_UNV (=0.012) assuming a triangular-wave pressure difference with C_ΔpAMP = 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. As a result, the standard deviation of St is less than 0.0025 for all the present results. It should be remarked that both C_ΔpAMP 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 or. It seems difficult to discuss the superiority or only from a theoretical point of view because the boundary of the re-circulation region is not exactly impermeable but complicated, rather soft, unsteady, turbulent and so on. However, as is one-order moment flux higher than, more directly tends to affect the volume of the re-circulation region than.

3.2. Single-Port Control

In the previous subsection, we have introduced as the universal number for jet switching. However, there exists one remaining question as to whether either or both of the inflows from the connecting tube into the jet-switching side wall and the outflow from the un-jet-switching side wall into the connecting tube are dominant. Another question is whether is effective even in a quasi-steady situation at very low jet-oscillation frequency. So, in this subsection, in order to discuss the physics, or the switching mechanism of the jet further, we conduct the single-port-control experiments. By some preliminary experiments, we have confirmed that the jet’s switching is sensitive to the inflow, but not to the outflow. So, we will exclusively examine the effectiveness of concerning the inflow, below.

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 t₀ 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. Figure 9 shows K plotted against Re, in the regular oscillation at L/s = 100 - 300 and Re = 5000 - 25,000. Figure 9(a) and Figure 9(b) denote the results at d/s = 1.2 and 1.4, respectively. More specifically, Figure 9 is based on the computed V_T’s like Figure 7(d). We should note that K is the time-mean value from t₀ 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.

Figure 10 and Figure 11 show typical examples in the single-port-control experiment, namely, the time histories of two velocities V_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. Figure 10 denotes the results at K = 1 × 10⁻⁴ and Re = 8800, and Figure 11 denotes those K = 3.5 × 10⁻⁴ and Re = 8800. In each figure, figures (a), (b) and (c) represent V_T, V_EX and p_TE, respectively.

At first, we see Figure 10. In Figure 10(a), V_T starts to increase from zero with time t at t = t₀ (=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 Figure 10(b) and p_TE in Figure 10(c) step down and up toward constant values at the

Figure 9

Flow-increment-rate coefficient K against Re, in regular FFJN oscillation

Figure 10

Time histories of velocities and pressure in single-port control at K = 1.0 × 10 −4 and Re = 8800

Figure 11

Time histories of velocities and pressure in single-port control at K = 3.5 × 10 −4 and Re = 8800

same time, respectively. Then, we can determine K or dV_T/dt from Figure 10. To be exact, dV_T/dt is time-mean which is obtained by such three data as t₀, t_SW and the V_T at t = t_SW. Of course, the above features can be seen in Figure 11 as well as Figure 10.

Figure 12 summarises all the results in the single-port-control: measured’s in the single-port-control experiments are plotted against Re in a range of Re = 8000 - 20,000. Figure 12(a) denotes at d/s = 1.3 and various values of K. And, Figure 12(b) denotes at K = 3 × 10⁻⁴ 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 based on the regular oscillation using the ordinary FFJN with two control ports in connection. We can see good agreement of the single-port control with the empirical formula for the regular oscillaton. This agreement suggests that or the accumulated flow work of not the outflow but the inflow from the connecting tube to the FFJN inside could be a key parameter for jet switching to explain the oscillation mechanism of the FFJN, in addition to the validity of in practical aspects to estimate the jet frequency of the FFJN.

4. Conclusion

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 of pressure through the connecting tube into the jet- reattaching wall from the opposite jet-un-reattaching wall. Under this assumption, we have conducted computations, and shown the effectiveness of J_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

Figure 12

Time histories of velocities and pressure in single-port control at K = 1 × 10 −4 and Re = 20,000

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.

Nomenclature

A: aspect ratio of a primary-nozzle throat, º S/s

b: breadth of a control port (m)

C_ΔpAMP: coefficient of pressure-difference amplitude º Δp_AMP/(1/2 ρV_PN²)

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 (m³/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/m³)

τ_SW: time at former jet’s switching (s)

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