## 2.1 Acoustic propulsion measurements

In order to address the propulsion properties of synthetic jets, we investigated the resulting propulsion of jets produced through a 4” diameter loudspeaker (20 W max nominal power, 4 Ω impedance). The loudspeaker was driven by a sinusoidal voltage signal produced by a function generator with fine control on frequency and amplitude. Similarly to the work of Ingard and Labate (1950), the thrust of the jet was measured by the force exerted on a screen placed in the jet stream (Fig. 1).

In Fig. 2 we show the results of the thrust (*T*) as a function of the frequency of the driving voltage (*F*) for four different voltage amplitudes (*V*). The four curves present similar asymmetric and very regular thrust distribution with a maximum around *F* = 70 Hz. Throughout the remaining of this manuscript, we discuss many relevant parameters that influence the thrust yield of this kind of system, but now we would like discuss with regard to basic aspects of jet propulsion.

Considering a conventional jet propulsion system operating in ideal condition, we can model the thrust (*T*), power (*P*) and air flow by the equations 1, 2 and 3 respectively:

$$P=\frac{1}{2}\frac{dm}{dt}{v}^{2}$$

2

$$\frac{dm}{dt}= \rho Av$$

3

where\(m\) is the mass, \(v\)is the velocity of the flow at the nozzle, \(A\) is the transversal area of the nozzle and \(\rho\) is the density of the medium. Combining the equations one can get

$${P}^{2}=\frac{{T}^{3}}{4\rho A}$$

4

from where we apprehend the general correlation \({P}^{2}\propto {T}^{3}\) of jet propulsion systems.

In our setup we use a loudspeaker to produce synthetic jets by means of electric power. Typically, loudspeakers yield about 1% of the electric power in sound pressure. It is reasonable to assume that the produced sound pressure is proportional to the electric power input to the loudspeaker and that the power of the system is given by , where is the effective impedance of the speaker coil. Furthermore, by inferring that our set up can be modeled as an ideal jet propulsion system and as a regular electric system, we can expect a correlation between driving voltage and thrust as . In the inset of Fig. 2 we observe that the maximum thrust value of each curve to the ¾ power as a function of the driving voltage results in a noteworthy linear behavior. The dashed line is a guide to the eye where the slope was adjusted to the experimental data. As discussed, the constant of proportionality depends on several parameters related to the intricate fluid dynamics of the experiment, as the physics of synthetic jets is distinct from traditional jet propulsion systems, and the intricacies determining the yield of the loudspeaker, that may also account for the systematic shift of the data towards lower values of thrust. For this reason, we have to be cautious when considering the inset as a result *per se*. Still, the correlation is remarkable and an indication, along with the striking regular behavior of the four curves, that the measurements are sound.

## 2.3 Theoretical and numerical considerations

Since the initial publications by Rogers (1858) and Helmholtz (1858), several studies and models on vortex rings have been carried out, and their evolution in parallel with mathematical and computational tools over this period is remarkable. A classic view of the phenomenon is presented by Lamb (1993), who models the Stokes stream function of the vortex ring in cylindrical coordinates as being

$$\psi \left(r,x\right)=\frac{\kappa }{2\pi }(r+R)\left[K\left(E\left(\lambda \right)\right)-E\left(\lambda \right)\right]$$

5

where \(r\) and \(R\) are respectively the flow radius and the ring radius of the vortex ring, \(\kappa\) is the vortex filament of strength, or “circulation”, in its constant half-plane, \(\lambda\) is the ratio

$$\lambda =\frac{R-r}{R+r}$$

6

and \(K\left(\lambda \right)\) and \(E\left(\lambda \right)\) are the complete elliptic integral of the first kind over \(\lambda\) and the complete elliptic integral of the second kind, respectively. The vorticity is assumed to be a combination of the filament strength and the coordinates of the vortex filament, with expression

$$\omega \left(r,x\right)=\kappa \delta (r-{r}^{\text{'}})\delta (x-{x}^{\text{'}})$$

7

with \(({r}^{\text{'}},{x}^{\text{'}})\) as the cylindrical coordinates of the vortex filament. \(\delta \left(x\right)\) is the Dirac delta function. This model of an ideal vortex ring establishes the boundary situation in which the rotating flux ring has zero radius, and the ring is a circular line.

If we consider that the vortex ring has a thickness, the vortex ring can be approximated by a disk with a radius much smaller than the radius of the ring. So, we have \(r/R\ll 1\), which leads us to \(1-{\lambda }^{2}\approx 4r/R\), and in the limit \({r}_{1}\to 0,\) \(\lambda \to 1\). Under these conditions, the elliptical integrals have as a solution

$$K\left(\lambda \right)=\frac{1}{2}\text{ln}\left(\frac{16}{1-{\lambda }^{2}}\right)$$

8

and

$$E\left(\lambda \right)=1$$

9

We can also consider the approximation of a uniform vorticity, formally \(\omega \left(r,x\right)={\omega }_{0}\), which leads us to a Stokes stream function given by

$$\psi \left(r,x\right)=\frac{\kappa {\omega }_{0}}{2\pi }R\iint \text{ln}\left(\frac{8R}{{r}_{1}}-2\right)dr\text{'}dx\text{'}$$

10

whose most important results for our study are the circulation

$${\Gamma }=\pi {\omega }_{0}{a}^{2}$$

11

with \(a\)being the radius of the disk, the hydrodynamic impulse

$$I=\rho \pi {\Gamma }{R}^{2}$$

12

and the kinetic energy

$$E=\frac{1}{2}\rho {{\Gamma }}^{2}R\text{ln}\left(\frac{8R}{a}-\frac{7}{4}\right).$$

13

This thin-core vortex ring model allows us to estimate the translational speed for the ring:

$$U=\frac{E}{2I}+\frac{3}{8\pi }\frac{{\Gamma }}{R}$$

14

For the simulations, we solve the Navier-Stokes equations properly simplified for the problem, considering the fluid (air) as compressible, adiabatic and with its non-zero viscosity. Integration over time was done using *FreeFem + +* software (Hecht 2012), which was also used to generate the mesh, initial conditions and resulting data.

The simulations were carried out considering the symmetries of the problem and the recommendations proposed by Guyon and Mulleners (2021). The generated scene represents a 2D section that contains the main axis, corresponding to the advance movement of the ring. The initial conditions are: 1 atm and constant density throughout the environment, with zero speeds. The only region different from these conditions is the speaker output, represented by a wall that has its velocity in the x direction changed by a time interval equivalent to one pulse to mimic the ring generation. After that, the speed on the wall becomes zero again. In Fig. 4, the details of the edges of the calculation scenario are presented, together with the grid mesh.

We consider an incompressible, isothermal Newtonian flow (density \(\rho\) =const, viscosity *µ* = const, no gravity), with a velocity field \(\overrightarrow{V}= ({u}_{r} ,u\theta ,uz )\) in its cylindrical form:

$$\frac{1}{r}\frac{\partial \left(r{u}_{r}\right)}{\partial r}+\frac{1}{r}\frac{\partial \left({u}_{\theta }\right)}{\partial \theta }+\frac{\partial {u}_{z}}{\partial z}=0$$

15

$$\rho \left(\frac{\partial {u}_{r}}{\partial t}+{u}_{r}\frac{\partial {u}_{r}}{\partial r}+\frac{{u}_{\theta }}{r}\frac{\partial {u}_{r}}{\partial \theta }-\frac{{u}_{\theta }^{2}}{r}+{u}_{z}\frac{\partial {u}_{r}}{\partial z}\right)=-\frac{\partial P}{\partial r}$$

(16)

$$\rho \left(\frac{\partial {u}_{\theta }}{\partial t}+{u}_{r}\frac{\partial {u}_{\theta }}{\partial r}+\frac{{u}_{\theta }}{r}\frac{\partial {u}_{\theta }}{\partial \theta }-\frac{{u}_{r}{u}_{\theta }}{r}+{u}_{z}\frac{\partial {u}_{\theta }}{\partial z}\right)=-\frac{1}{r}\frac{\partial P}{\partial \theta }$$

(17)

$$\rho \left(\frac{\partial {u}_{z}}{\partial t}+{u}_{r}\frac{\partial {u}_{z}}{\partial r}+\frac{{u}_{\theta }}{r}\frac{\partial {u}_{z}}{\partial \theta }+{u}_{z}\frac{\partial {u}_{z}}{\partial z}\right)=-\frac{\partial P}{\partial z}$$

18

and

$$p=\rho \stackrel{-}{R}\theta$$

19

where \(\stackrel{-}{R}\) is the ideal gas constant and \(\theta\) is the temperature.

Here, we present the most relevant results of the simulations. Figure 5 show the evolution of gas pressure and velocity at the same instants, namely t = 4, 6 and 8 ms. The instant \({t}_{0}=0\) is the moment when the pressure pulse at the loudspeaker output returns to zero. According to the simulations the ring is formed immediately after \({t}_{0}\), presenting the nominal conditions expected from this phenomenon.

As the simulation progresses in time, two regions of low pressure, one ahead and the other behind, accompany the ring. This is a natural consequence of the phenomenon's propagation, but it can be intuitively explained as the edges of the high-speed region of the interior of the ring.

By introducing a normalized kinetic energy quantity, given by

$$E\text{'}=\frac{1}{2}{{\Gamma }}^{2}R\text{ln}\left(\frac{8R}{a}-\frac{7}{4}\right)$$

20

one can estimate the point of best (or maximum) transport of energy. Figure 6 presents the evolution of ring radius, hydrodynamic impulse and normalized kinetic energy in the vortex ring. It is possible to distinguish an instant at t = 0.94, which corresponds to the maximum kinetic energy. For subsequent times, the kinetic energy will quickly disappear, and the vortex ring is disaggregated.

In the experiments the vortex rings are formed sequentially by the sinusoidal voltage input and each following ring is slightly affected by the distribution of velocity and pressure left by the previous one. The low-pressure region ahead of the new ring binds to its counterpart in the preceding ring. As a final result, a "tube" is formed that conducts the air flow and preserves the life of the rings for a longer time.

On the other hand, for higher frequencies, a proper residence time for the air is not satisfied and the close presence of a former ring can be disruptive to the stability of the next coming ring, as we show in the experimental results in the next sessions. However, in suitable conditions, we can use this "vortex ring driven tube" model to estimate the final numbers, which connect the simulations with the experiments.

Assuming the pulse diameter (region 1 of Fig. 4) as 23 mm and using the frequency of 100 Hz as a time base, we can denormalize the simulation results. As a result, a maximum pressure of 10 Pa has been obtained in the circulating inner region of the ring (Fig. 5a). Moreover, as can be seen in Fig. 5b, a vortex ring structure with a central axial velocity of 13 m/s embedded in stagnant surroundings is compatible with a 7.2 m/s propagation velocity of the vortex ring measured by Schlieren instrumentation. Under the hypothesis that the tube maintains its internal diameter constant throughout the experiment, this brings us to a force of 10 mN applied to the bulkhead/sensor.

From the observations, it is possible to conclude that vortex rings do not propagate indefinitely. In fact, there are three main reasons that end up with a vortex ring: dissipation, self-induced instabilities, and space intervals between rings. The dissipation occurs due to exchanges of energy in infinitesimal packets between the ring and the environment. This energy loss occurs at the ring surface and is mainly linked to the viscosity of the medium. Its effect is to reduce circulation which, along the process, ends up accelerating the generation of self-induced instabilities and the destruction of the ring. Self-induced instabilities were thoroughly investigated by Windall and Sullivan (1973), both formally and experimentally. These perturbations are characterized by resonant longitudinal oscillations that tend to increase in amplitude with the passage of time. The main result of these researchers is that the order of instability follows \(ln(a/R)\) and tends to dissipate the ring as \(a/R\) approaches unity.

If the space between two rings is small enough for there to be significant interference between the circulation of both, we will observe the self-destruction of the two rings. This is easy to understand, since, as the circulation is the same for all rings ("forward" on the inside and backwards on the outside), the surface velocities of the fluid will have opposite directions at the point where the rings touch. This last form of dissipation gives us the ideal frequency for the experiments. If the frequency is too high, the rings get closer and destroy each other. If it is too low, the contribution to the formation of the "tube" with the airflow will be less effective.

## 2.4 Relevant parameters for acoustic jet propulsion

In previous sessions we presented the behavior of the thrust generated by acoustic jets as a function of the amplitude and the frequency of the applied voltage. We also established, by Schlieren photography and simulations, the nature of the jets as being constituted of a sequence of vortex rings, thus being consistent with the synthetic jet literature. In this session, we present experimental results for the behavior of the system’s propulsion as a function of two other relevant parameters highlighted in Fig. 1, the nozzle exit diameter (D) and tapering angle (α).

In Fig. 7 we present the measurements of the thrust as a function of the frequency of the driving voltage (using \(V\) = 1.86 V) for 9 different nozzle exit diameters. In all curves, the tapering angle was the same (α = 60°). For the sake of visualization, the experimental points are shown for three representative curves, namely for D = 15,5 mm (smallest nozzle exit diameter of the set), D = 24.5 mm (maximum yield) and D = 35 mm (largest diameter). All other curves are represented by lines, as the behavior of the thrust varies monotonically and consistently from one curve to another. All curves present the same general trend observed in Fig. 2, i.e., an asymmetric curve with a peak of maximum thrust around \(F\) = 100 Hz. It is worth mentioning that in the thrust measurements in between 160 and 200 Hz, a moderate instability was observed, in the sense that the position of the screen used to measure the force swayed slightly, turning the measurement in that region rather less precise, as can be sensed by the oscillation of the data points in this range of the graph. This effect was more or less observed in all measurements and, as mentioned in the last session, simulations indicate that this is related to unsuitable residence time and coupling with the thrust measurement screen. Interestingly the instability of the screen diminished for frequencies higher than 200 Hz. However, one has to take into account that the thrust is also strongly mitigated in that range of frequencies.

In the inset, we depict the maximum thrust of each curve as a function of the corresponding nozzle exit diameter, showing the consistent variation of the behavior of the curves and that the maximum thrust happened for D = 24.5 mm (red closed circle).

The influence of the nozzle tapering angle (α) on the thrust can be examined in Fig. 8. In (a) we observe the behavior of the thrust as a function of the driving frequency for a fixed nozzle exit diameter D = 27,5 mm but for 4 different values of the tapering angle, namely \(\alpha =\) 60°, 85°, 97° and 180°. The last value means that, instead of a cone shaped nozzle, the loudspeaker has been covered with a flat screen with a round orifice in the center. Interestingly, the tapering angle seem not to have significant impact on the thrust yield of the propulsion system as all curves essentially overlap. An important remark is that the thrust measurement instability for \(F>\) 160 Hz turned out to be severe for \(\alpha =\) 180° (the flat screen), precluding measurements beyond 150 Hz value. In Fig. 8b, we performed the same study depicted in the inset of Fig. 7, i.e., the maximum thrust as a function of the nozzle exit diameter, but for 3 different tapering angles, namely α = 60°, 85° and 180°. Surprisingly, the largest value of trust has been obtained for the flat screen, and for considerably larger values of exit diameters. Arguably, nozzle engineering for propulsion systems seldom relies on flat obstructions to the flow, and this reveals how this study points to interesting new physics of jet propulsion.