Transition in synthetic jets

Sensors and Actuators A 187 (2012) 105–117

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Transition in synthetic jets Václav Tesaˇr ∗ , Jozef Kordík Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v.v.i., Prague, Czech Republic

a r t i c l e

i n f o

Article history: Received 19 November 2011 Received in revised form 8 August 2012 Accepted 10 August 2012 Available online 1 September 2012 Keywords: Turbulence Synthetic jet Transition Velocity spectra

a b s t r a c t This paper is a sequel to several recent papers that analysed experimental data on axisymmetric synthetic jets using similarity transformation of governing equations involving a one-equation model of turbulence. Unusual feature there is variation of parameters (“constants”) in transformed equations along the synthetic jet axis. Interestingly, this variation is non-monotonous: there is an abrupt change at a certain distance from the nozzle. Velocity spectra of one synthetic jet have earlier shown this change to be associated with disappearance of sharp peaks in the spectra, characteristic for presence of organised vortices. The change thus indicates transition into fully chaotic turbulence. In the present paper, the study of the transition is extended to a whole range of synthetic jets, from Stokes number Sk = 11.5–108.8 (corresponding to frequencies from ∼10 Hz to ∼100 Hz). In a search for a rule governing the transition, the vortices in the organised part of the synthetic jets were modelled as spherical objects. It is found that the transition occurs at the axial distance from the nozzle equal to ∼10 diameters of these spherical model objects. © 2012 Elsevier B.V. All rights reserved.

1. Introduction: synthetic jets Pulses of fluid flow from a nozzle alternating with reverse-flow suctions lead to an interesting rectification phenomenon. The asymmetry between the far reaching outflow and only near-field character of the suction (Fig. 1) generates a time-mean flow that propagates from the nozzle in a manner resembling a steady turbulent submerged jet. The rectification, a consequence of the non-linearity of governing equations, has been known at audible frequencies as the “acoustic streaming” [1]. Under the name “loudspeaker wind” it became a popular observation in early radio sets of 1930s. The effect found application in simple devices where it replaced mechanical suction valve and delivery valves. The resultant simplicity brings very low cost but also poor efficiency, which made the phenomenon uninteresting for applications other than those – such as, e.g. fish-tank aerators – in which low device price was at premium. Probably the first discussion in scientific literature was the Dauphinne’s description in 1957 of a blower for air circulation in a thermostat [2]. Later, Walkden and Kell’s [3] pump with two nozzles driven in anti-parallel achieved a somewhat higher efficiency which – together with robustness and (when made from refractory material) resistance to high temperatures – made possible use in pumping difficult to handle liquids like, e.g. molten salts. Tippetts and Swithenbank [41] developed the idea further

∗ Corresponding author. E-mail address: [email protected] (V. Tesaˇr). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.08.021

for pumping hazardous liquids in nuclear fuel re-processing [4]. Currently it finds use in microfluidic pumps where is welcome the absence of mechanical suction and delivery valves – which are difficult to manufacture, cause malfunctions, and their wear limits the lifetime. Present first author, in his investigations of these jet-type rectifiers [5], discovered in early 1980s an interesting law [6–8] that governs the entrainment of outer fluid into the zero-time-mean jets. Apart from the frequency of the flow pulsation, an important factor characterising the zero-time-mean flowrate jets is the “extrusion length” s [m]. It is the length of an imagined fluid column pushed through the nozzle in each of the outflow phases [4] keeping a constant cross section area equal to that of the nozzle exit and having volume equal to the amount of fluid actually displaced. If the outflow pulses are short, resulting in s small relative to the nozzle exit diameter d, the generated flow contains a system of pure vortex rings. At large s/d the generated vortical objects trail behind (as suggested in Fig. 1) a “tail” wake. It was the character of the system of vortex rings produced at small s/d that led Glezer and Amitay [9] to use it, under newly coined name “synthetic jet”, as a laboratory model of the vortical structures in steady jets. Synthetic jets removed the problem of phase jitter irregularity that made earlier studies of natural jets structure very difficult. In the wealth of publications that appeared inspired by [9], synthetic jets became popular in many engineering uses. The main applications are actuators for active control of fluid flow separation and/or transition on aeroplane wings and turbomachinery blades, e.g. [10,11], where the most important advantage is the absence of the supply

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2. Quasi-similarity and transition Nomenclature cZ d f ff l kT rv rH S Sf Sk s s0.5 V W X1 X1crit X1∗ ∈ ε εax ı0.5



coefficient of turbulence dissipation rate [–] nozzle exit diameter [m] oscillation frequency (variable) [Hz] forcing (driving) frequency (constant of an experiment) [Hz] length scale of momentum transporting vortices [m] Tollmien factor [–] radius of the toroid model body [m] radius of the spherical vortex model [m] spectral density of measured velocity vector magnitude [m3 /s2 ] spectral density at forcing frequency [m3 /s2 ] Stokes number [–] equivalent “extruded column” length [m] parameter of spreading of the diameter ı0.5 [–] volume [m3 ] velocity [m/s] axial distance from the nozzle exit [m] axial distance to transition [m] axial distance from the virtual origin [m] turbulence dissipation rate [m2 /s3 ] similarity-transformed fluctuation energy [–] value of ε on the jet axis [–] convention jet diameter [m] viscosity (kinematic) [m2 /s]

pipes which otherwise (leading from a central source to each one of all actuators) complicate the wing design. Synthetic jet actuators may be designed by numerical flowfield computations, e.g. [39], the results being, of course, valid only for the particular case. Obviously, it is highly desirable to have at hand a universal mathematical model valid for any synthetic jet flowfield, simple enough and yet reasonably accurate to be useful for engineering calculations.

Present authors published such a useful model in Refs. [12,13]. It was set up in a direct analogy to earlier similarity-based models of steady turbulent jets [23]. The underlying idea of the approach was Stokes’ (1856) similarity treatment of aerodynamic resistance of a drastically simplified pendulum [14], where the mutually similar distributions of various quantities (such as velocity profiles) made possible conversion into transformed co-ordinates in the terms of which the governing partial differential equations are converted into a set of ordinary differential equations, much easier to solve. This approach was applied to laminar jets by Schlichting in 1933 [15]. Attempts to set up an analogy valid for turbulent jets were actually proposed as early as 1920s but ended without real success. Tollmien’s solution [16] introduced a useful hypothesis about the turbulence length scale but used an inadequate algebraic model of turbulence, while the later Görtler’s solution [17] (actually for planar jet, but adaptable to the axisymmetric case of interest here) was based on the totally wrong Prandtl’s “das neue Modell” [18]. Only later models of turbulence with transport equations for the parameters used in evaluation of turbulent viscosity made possible successful similarity solutions of steady turbulent axisymmetric jets [19–21]. The solution with the one-equation model in [20,21] has led to practically the same results as those obtained, at a price of considerably more effort, with the two-equation turbulence model [21–23]. The conceptual advantage of the two-equation approach is no need of any a priori assumptions about the turbulence length scale l, a quantity essential for evaluation of turbulent viscosity. Nevertheless, in the central, paraxial region of the jet – which is of primary interest in applications – the two-equation model solution [22,23] results in the length scale practically the same as in the one-equation approach: the values are more or less constant over the jet cross section and linearly increase with the axial distance (the distance X1∗ measured from virtual origin, the position of which is found by extrapolation), in reasonable agreement with the Tollmien’s assumption [16] l = kT s0.5 X1∗

that forms the basis of the one-equation model for jet flows. The Tollmien’s factor kT defines the size of the most energetic vortices and s0.5 a coefficient of the linear growth of convention-defined diameter ı0.5 of the jet on the axial distance X1∗ . While the more laborious two-equation turbulence model solution leads, after the similarity transformation, to a system of 7 ordinary differential equations to be solved simultaneously, there are only 5 similarity-transformed simultaneous ordinary equations [20] to be solved with the one-equation model. A certain problem is these five equations contain an undetermined dimensionless parameter cZ /kT s0.5 , where cZ is the coefficient in the definition of dissipation rate of fluctuations (constant value cZ = 0.164 is usually taken). Integrations of the ordinary differential equations with different magnitudes of this parameter result in different shapes of the profiles of velocity and other variables. In [20], comparisons with known experimental results have led to the value for fully developed steady jets cZ = 9.108 kT · s0.5

Fig. 1. Schematic representation of the jet-type fluidic rectification effect. The vortical structure created in the outflow phase gains a momentum preventing it to return when, in the subsequent inflow phase, the direction of the flow in the nozzle is reversed.

(1)

(2)

Another problem encountered in the solutions is the absence of any obvious numerical value for one of the five boundary conditions needed on the jet axis (where the integration usually starts, progressing in the radial direction). The unknown quantity is εax , the similarity-transformed magnitude of the specific energy of unsteady motions. What is known is this quantity gradually approaches the value ε = 0 outside the jet. This typical problem of split boundary conditions was solved by repeated integration of

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Fig. 2. Demonstration of the typical sudden change in the character of the dependence of jet flow parameters on the axial distance. The variable presented here as a function of the axial distance is the length scale l of momentum-transporting vortices (related to the constant nozzle exit diameter d). Data were obtained at different forcing frequencies ff in two experimental series A and B differing in the nozzle size and shape.

the five equations with various magnitudes of the parameter Eq. (2) and looking for the value εax with which the solution reaches the correct final value ε = 0. For the value presented in Eq. (2) this final zero is obtained with εax = 0.07641

(3)

Of course, the similarity solution is not valid in the nozzle exit and also in the transition region immediately downstream from it. With turbulence absent in the core of a steady jet, there is in these locations εax = 0. As the axial distance X1 from the nozzle increases, the magnitude εax grows from the zero value to the fully developed steady jet value Eq. (3). There is a general belief in the literature that this increase of the magnitudes εax is fast – so that the jet becomes fully developed already at about X1∗ ∼10d. This belief in the fast development is seemingly supported by velocity profiles, which at this axial distance indeed usually reasonably correspond to the data for developed jet. However, present authors’ results obtained for quantities other than velocity do not support this belief. The transformed energy ε of fluctuation is seen in [12] not reaching the fully developed value of Eq. (3) at distances as large as X1∗ ∼60d (at which the jet ceases to be practically useful). When the similarity solution idea was applied to the synthetic jets in [12,13], a similar problem with the unknown parameter cZ /kT s0.5 was also encountered. Moreover, to make the idea applicable at all, several simplifying assumptions were necessary. One of them is the summary handling of the kinetic energy of all unsteady motions in the jet, not discriminating between the momentum transport by organised vortical motions and by the chaotic turbulence. The other simplification is an assumed isotropy of these unsteady motions. This is a rather bold step, considering the dominant spatial orientation of the organised vortex rings. Nevertheless, these simplifications still lead to good agreement with experimental data. However, no universal numerical values corresponding to Eqs. (2) and (3) could be found. A locally valid similarity, or quasi-similarity [12], had to be adopted, with the value of the model parameter cZ /kT s0.5 as well as the magnitude of εax dependent on the streamwise position X1∗ along the jet axis. Even more surprisingly, this variation of the parameter cZ /kT S0.5 was found non-monotonous. At a certain critical axial position, the character of the dependence on the axial distance X1 undergoes a

sudden, rather dramatic change. A typical example is presented in Fig. 2 (a re-plotted version of a similar diagram in [13]), showing dependence of the length scale l on axial position. Distributions of other quantities also show such two differently behaving regions. A hypothesis was formulated already in [12,13], attributing this change to the loss of the organised character of the vortices. In the present paper, these two regimes and the transition between them are investigated using an unusual approach: analysis of velocity spectra. Other authors of synthetic jet studies did measure and present such spectra but were unable to analyse them because doing so requires knowing some quantities not directly measurable and rather difficult to evaluate–like the local turbulence dissipation rate ∈ [J/kg s = m2 /s3 ], which plays the central role in the spectra theory due to Kolmogorov [32]. It was the advantage of the present analysis that experimental data evaluated by means of the quasi-similarity transformation solutions made these quantities available. 3. Experiments Laboratory measurements were performed in several experimental series, some of them with different nozzle sizes and shapes. Measurements with nozzles of 4 mm diameter with rounded entrance (their geometry is presented in [13]) and others with 6 mm diameter bevelled entrance were initially labelled A and B, e.g. as shown in Fig. 2. Since the results, when plotted in similarity co-ordinates, were practically mutually indistinguishable, later measurements concentrated on the rounded-entrance 4 mm nozzle. The crucial part of the used synthetic-jet generating actuator is shown in Fig. 3. Its essential component was a standard loudspeaker of 120 mm nominal membrane diameter, type ARN-165-01/4, supplied by the manufacturer TVM Valaˇsské Meziˇríˇcí. The membrane, because of its conical shape, was practically stiff, moving as a whole. The motion – when driven by voice coil – was made possible by the easily deformable rubber strip at the membrane circumference. The strip, shaped in the meridian cross section as a wide letter “U”, is seen in Fig. 3. The displacement cavity, the changes in volume of which moved the air through the nozzle, was set up by fixing in front of the membrane a rigid metal plate, with the nozzle in its centre. Aerodynamic properties of the nozzle were

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Fig. 3. The actuator used to generate the synthetic jets in the experiment. The main part is a standard loudspeaker, with the space in front of the membrane closed by a metal plate having a quadrant nozzle [24] in its centre. Position of the virtual origin varies with varying forcing frequency ff ; mostly its position is in front of the nozzle, so that X1∗ < X1 .

investigated earlier and are discussed in [25]. The fluid was air at laboratory conditions. Air temperature in different measurement runs was within the range from 21.0 ◦ C to 24.1 ◦ C, the value being practically constant during each run. Also the barometric pressure values, between 97.5 kPa and 98.5 kPa, did not practically vary in the course of individual measurement runs. Although the loudspeaker catalogue value of maximum handled power was 100 W, experience with mechanical failures in previous experiments (voice coil separated from the membrane due to overstressing caused by the high hydraulic resistance of the small nozzle, for which the loudspeaker was not designed) has led to driving the actuator in all these experiments at only 22 W electric power This power level was adjusted to the same constant value prior to each run. Since some of the measured air velocities, especially at large distances from the nozzle, were quite small and easily disturbed, the actuator and test space were surrounded at all sides, at about ∼1 m distance, by cardboard walls preventing disturbances caused by draughts in the laboratory room. To avoid influencing by thermal air currents, the room heating was switched off and the experimenter left the room for the duration of each measurement run. The velocities in the synthetic jet were measured by hot-wire anemometer CTA 54T30 (DANTEC Dynamics) with single-wire probe type 55P16. The overheat ratio was set to the value recommended by the manufacturer, corresponding to wire temperature 241 ◦ C. The low-pass filter was set to 10 kHz. The probe was calibrated – and then re-calibrated after each approximately 2 hours of operation – using a calibration nozzle with measured either pressure drop (at higher velocities) or volume flow rate by precise digital thermal mass flow meter Bronkhorst EL-FLOW F-201A-50kAAD-33-V. The calibration measurements were possible down to the velocity 0.39 m/s, below which the free convection from the heated wire made the results questionable. The calibration curve fitted to the data was a fifth-order polynomial. A complete table of uncertainty estimates for all variables was presented earlier in [13]. In another preliminary procedure, the actuator frequency dependence characteristic was measured, again at the constant 22 W electric driving. For these tests, velocity measuring hot-wire probe was positioned in the exit plane of the nozzle. Throughout the frequency range of interest, from 10 Hz to 100 Hz, it was established that the velocity magnitude (evaluated as averages of outflow halfperiod, since the probe could not distinguish negative velocity in the remaining inflow half-period) was practically constant. This is

not surprising as similar flat characteristics were also found e.g. by Gallas et al. [36], Hong [37] or Chaudhari et al. [38]. In the present case the invariance was promoted by the application of constant driving power. In the subsequent experiments, the velocities in the synthetic jet were actually measured at several radial positions, with computercontrolled traversing gear moving the probe along the radius (perpendicular to the nozzle axis). As documented in [27], the effect on the evaluated velocity spectrum of the off-axis radial position of the probe was only quantitative – a decrease of the spectral density magnitude. Since the aim of the spectral analysis here was detection of qualitative changes (the disappearance of the organised-motion peaks), the data used for discussing here were only those measured on the jet axis. They were obtained at the axial distances X1 between 15 and 90 nozzle exit diameters. The steps between individual axial positions were 5d = 20 mm. In some cases, the data at very short distances, especially near the low frequency end of the range, were discarded because of the occurrence of temporary negative flow direction – air motion towards the nozzle – which the probe was unable to interpret properly. Otherwise the velocity did not change its sign over most of the measurement region, apart from some exceptions – also discarded – very far from the nozzle, where the time-mean velocity component was so small that some eddy motions could result in temporarily negative velocities. There were typically 15 measurement locations along the axis to which the probe was first moved by traversing gear and then held there stationary until 262 144 instantaneous velocity samples at 15 kHz rate were taken and stored. This procedure was then repeated 8 times with the different driving frequencies ff [Hz]. The exact driving frequency values, as presented in Table 1, were computed by discrete-time spectral analysis. They reveal small deviations from the originally desired integer values – mainly

Table 1 List of actual forcing frequencies in the experimental investigations and the corresponding values of the Stokes number Sk. ff = 10.47 Hz ff = 20.02 Hz ff = 30.46 Hz ff = 39.98 Hz ff = 50.14 Hz ff = 60.03 Hz ff = 80.78 Hz ff = 101.07 Hz

Sk = 11.5 Sk = 22.0 Sk = 33.3 Sk = 43.6 Sk = 54.6 Sk = 65.0 Sk = 87.0 Sk = 108.8

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Fig. 4. An example of velocity spectra obtained at different driving frequencies ff on the synthetic jet axis at a relatively short distance X1 from the nozzle. Except at the highest driving frequencies, almost all spectra here have very strong organised components, characterised by the sharp isolated peaks.

caused by the hardly avoidable changes that took place in the course of the relatively long duration, time-consuming measurement runs. The driving (forcing) frequency ff itself does not fully specify the character of the alternating flowfields. Much more information is provided by Stokes number Sk, a dimensionless quantity which has analogous role in oscillatory flows as Reynolds number in steady flows. Sk =

ff d 2

v

Eq. (4), at his time the meanings of the variables characterising fluid viscosity were not completely clear and this may be the reason why some authors prefer to call this expression a Roshko number, after the much later (in 1952 [26]) but more clearly defined usage. 4. Analysis of spectra 4.1. Disappearance of organised motion

(4)

where  [m2 /s] is the molecular (laminar-flow) viscosity. The Stokes number values in Table 1 are those evaluated for the experiment B (with the smaller nozzle diameter d). Although Stokes – in his study of damping of clock pendulum in [14] – was undeniably the first scientist who used the dimensionless expression corresponding to

To obtain data for the spectral analysis, authors computed the frequency spectra from the stored 15 × 8 = 120 velocity data sets (at the 15 locations X1 along the jet axis and 8 driving frequencies). Each set consisted of the 262 144 instantaneous velocity magnitude samples. The algorithms for spectra computation was the standard discrete-time Fourier transformation, available in the

Fig. 5. Another example of the evaluated velocity spectra, this time demonstrating the effect of varied driving frequency on the character of spectra at a large distance X1 from the nozzle. As in previous Fig. 4, the frequency f on the horizontal co-ordinate is normalised with respect to the forcing frequency ff . At this particular axial position X1 /d = 70, there is only at the lowest Sk a peak indicating a remnant of the organised motion.

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Fig. 6. Three examples of typical velocity spectra at different distances along the jet axis at the same driving frequency. The subcritical region is dominated by the peaks of organised motion. The transition between the two regimes is associated with the disappearance of the peaks.

MATLAB library. Typical results obtained are shown in Fig. 4 for a constant axial location and eight values of the driving frequency – the latter presented in the non-dimensional form of the Stokes numbers, Eq. (4). Similar measured and evaluated spectra of synthetic jets – never so many as processed here – were already presented in literature, e.g. in [28–31]. However, none of these authors did go further than just presentation of spectra examples. There was no analysis, apart from [27] – the only reference in existing literature on synthetic jets in which the spectra were analysed. Data obtained from the spectra offer a way to evaluation of various parameters of interest, as was demonstrated for the case of single driving frequency in [27]. In the present paper the interest focuses on a single aspect, the transition into turbulence. This is revealed by the different character of the spectra in the subcritical (organised motion) and supercritical (chaotic turbulent) regimes. The change in the character may be seen in the examples presented in Figs. 4–6. In [12,13] the locations at which the dependence of the parameter cZ /kT s0.5 on the axial distance X1 from the nozzle show values computed by the quasi-similarity approach the sudden change, as in Fig. 2, were interpreted as the locations of the transition into turbulence. Significantly, at these same distances X1 disappear the dominant features in the spectrum – the high narrow peaks at the forcing frequency and its integer multiples, indicative of the organised motions. In Fig. 4, apart from the cases of very high driving frequencies (at which the distance X1 = 30d is at the transition conditions or, as in the case of ff = 100 Hz, it is already beyond it) these peaks clearly dominate. In the supercritical region, interpreted as the chaotic turbulence, they are absent.

At much lower levels below the organised-motion peaks, there is present even in the subcritical regime a continuous component in the spectrum characteristic for the chaotic turbulent noise. This noise is not important because its spectral density magnitudes are smaller than the organised motion amplitudes by more than two decimal orders of magnitude – note the use of logarithmic scale in Fig. 4. The main peak in Fig. 4 (reaching up to S = 8 m3 /s2 ) is by almost three decimal orders of magnitude above the chaotic turbulence component (which in “A” is at spectral density level S ∼ 10−2 m3 /s2 ). Besides the peak at the driving frequency, there are in the sub-critical regime also lower but still quite strong peaks at the harmonic multiples of ff , Fig. 4. Their presence documents a non-sinusoidal character of the velocity waves propagating past the measuring probe. This character is, of course, a consequence of the strong non-linearities governing the flowfield, because the electric input into the flow-generating actuator was purely harmonic. The frequency f on the horizontal co-ordinate is in Figs. 5 and 6 normalised with respect to the forcing frequency ff . This makes more apparent the changes of the spectra resulting from the change in the forcing frequency. At frequencies lower than ff – in the segment labelled A in Fig. 4 – the spectrum is dominated by large chaotic vortices. The spectral density distribution there seems to be practically constant. As in other cases of turbulent flows, this constancy is not real but due to aliasing (a motion past the probe of an actually smaller eddy with its axis misaligned relative to the direction of its motion is interpreted as a much larger vortex). On the other side, at large f, the energy of motion decreases with increasing frequency. This is the region dominated by the cascade processes of spectral transport towards the small scales as larger vortices deform smaller ones. Another example of typical spectra and the influence of the driving frequency ff is presented in Fig. 5, differing from Fig. 4 by relatively large streamwise distance X1 = 70d (i.e. at X1 = 280 mm) of the probe from the nozzle. The visibly lower spectral density values than in previous Fig. 4 indicate the considerable loss of the jet energy en route from the nozzle to this far downstream location. All spectra at this large downstream distance, with the single exception of the case of very low ff ∼ 10 Hz that shows a remnant of coherence, exhibit the continuous character typical for the conditions far downstream from the investigated transition. In this region of energetic cascade, the spectra of Fig. 5 may be divided into two segments, the inertial segment B obeying the Kolmogorov – 5/3 power law slope [32] and the dissipation segment with C steeper slope (and hence more rapid loss of energy). A comparison of the subcritical, transitional, and supercritical spectra at identical driving frequency is presented in Fig. 6. It shows in a particularly evident manner that the transition seen in Fig. 2 for ff = 30 Hz to occur roughly near to X1 ∼ 50d is the transition into turbulence since the spectra at increasing distances show disappearance of the peaks typical for the organised motion. Further downstream the flowfield has a fully turbulent character and is practically not distinguishable from last development stages of a steady jet (though Ref. [13] mentions some differences in numerical values of some parameters). A new phenomenon discussed in the present paper, one that could not be observed earlier, is the strong influence of change in driving frequency on the character of the spectra. From Fig. 4 it is obvious that a high driving frequency makes the synthetic jet much more dissipative – a fact of importance in applications. For example, in the use of synthetic jets for flow separation control there is a trend to operate the actuators at quite high frequencies, Symptomatic in this respect is the influential paper [40] by Glezer and collaborators, who warn against the danger of the actuation al low frequency, capable of amplifying unsteady components of global aerodynamic forces on the wing or turbine blade. They recommend a driving frequency higher – typically by a decimal order

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Fig. 7. Decrease of spectral density values with increasing axial distance X1 for a particular driving frequency ff = 20.02 Hz, plotted in semi-logarithmic co-ordinates and shown fitted by straight lines of exponential laws. There are three distinct regions D, R, and T each having a different decay rate – the slope of the fitted line.

of magnitude – than the shedding frequency of separating flow which is usually chosen by many researchers. The important fact of higher momentum loss in high-frequency synthetic jets has in this context seemingly eluded attention (perhaps because of the lack of an information as the only spectral analysis available, Ref. [27], is for only a single driving frequency). 4.2. Decay laws A particularly useful way how to identify location of the transition in synthetic jets is to plot the values of spectral density at a

particular driving frequency as a function of the streamwise distance and then fitting to this plot a trend line. Initially, in the subcritical flow, the plotted values are those of the tips of the driving-frequency peaks. Then, in the supercritical range after the disappearance of the peaks, the values are taken in the continuous part of the spectrum. As a by-product of identifying the transition, this identification procedure also provides interesting information about the laws of energy decay in the jet. As an example, the data for the streamwise dependence of spectral density at the driving frequency ff = 20.02 Hz are presented in the following Fig. 7. It is evident that the data points there may be

Fig. 8. The first step in the transition, from D into R, associated with increase in decay rate. The location of this first transition was identified by the intersections of the fitted lines of exponential decay.

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Fig. 9. Fitted exponential trend lines to the values of spectral density along the jet axis in the region near to the transition. There are two line families, R and T. Intersections of the line pairs from different families at the same driving frequency define the transition line. Strong dependence of the transition on the driving frequency is evident.

fitted in the used semi-logarithmic co-ordinates by straight lines, showing the decay is exponential. There are three fitted lines, each for a different segment which are labelled D, R, and T. In the regime D the line shows the relatively slower streamwise decrease of the fundamental frequency peaks. The decay is then more rapid in the subsequent segment R. The spectra there show only remnants of the organised character. Obviously, the energy of the peaks is here transferred rather fast to the non-organised motions. The streamwise extent of this re-distribution regime R is quite short. Further downstream, there is the third segment T of the fully developed turbulence. In the available experimental data sets the segment D could be identified with reasonable accuracy only at very low driving frequencies – which means, because of the constant nozzle exit velocity, flows with very large initial organised vortices. Two examples are in Fig. 8. On the other hand, it was easily possible to identify in all the data sets the segment R and particularly the position of its high-frequency end, in the intersection of the fitted exponential lines for the two segments R and T. It is this intersection point (Fig. 7) that is referred to here as the position of transition into turbulence. Procedure of its identification at a particular driving frequency ff consisted of evaluating the numerical parameters of the two exponential fits to the data points – in regimes R and T – and then solution of the equation which results from putting the two spectral density values Sf equal.

4.3. First transition, D–R The diagram in Fig. 8 shows that the decay rate in the regime D is relatively slow – it is the slower the larger are the vortical structures (i.e. the lower is the frequency). Apparently, the processes in this regime do not involve much viscous damping. The first transition into the subsequent regime R further away along the jet axis is seen

in Fig. 7 to take place when the available power of the structures (characterised by their spectral density) decreases below a certain critical level. In Fig. 8 this level – indicated there by the fitted thick horizontal line – is seen to be same in the two cases that differed in their driving frequency values. This constant value at the first transition D–R is one of the findings of this paper.

4.4. Decay in regimes R and T Data obtained for these regimes R and T were more plentiful. Interest concentrated on the intersections of the exponential lines fitted to the data points, as shown in the in the next Fig. 9 – they indicate the defined position of the transition into turbulence. It is apparent that the transition location X1 varies quite strongly with the driving frequency ff . Contrary to the transition D–R in Fig. 8, the line connecting the transition intersections (and called therefore “transition line” in Fig. 9) appears to be here not horizontal, but has a certain slope – a small positive one, nevertheless seemingly systematic (the rather small scatter of most data points here should be noted). Also apparent in Fig. 9 are the demonstrations of different aerodynamic mechanisms in the two regimes R and T. In the fully turbulent regime T the slopes of the lines are practically the same (deviations may be reasonable attributable to data scatter). On the other hand, in the subcritical regime R the decay is much more rapid and the slopes of the fitted lines depend considerably on the driving frequency ff . At higher ff the remains of the organised structures are smaller and their energy is more rapidly dissipated. It is useful to characterise the intensity of exponential decays by their halving distance. This is defined as the increment in streamwise distance necessary for the amplitude of the unsteady motions decreasing to one half of its upstream value. For the lines fitted in Fig. 9 the halving distances were evaluated and are plotted in

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Fig. 10. The halving distances indicate the intensity of the dissipative losses. They are the distances over which the amplitude of the unsteady motion decreases to one half of its upstream value. Here they were evaluated as the slopes of the fitted exponential lines in Fig. 9.

Fig. 10. They are not constant, decreasing (i.e. exhibiting more rapid loss of energy) with increasing driving frequency. The rapidity of the decay is noteworthy. In the turbulent regime T, the synthetic jet loses one half of its available power by the fluid moving by only ∼10 mm further downstream. However, in the regime R the decay is even faster, by a decimal order of magnitude. The transitional unsteady motions in the jet lose full one half of their power as they move downstream by a step as short as ∼1 mm! This is a surprisingly short travelled distance compared with the typical size of the moving vortical objects, which is evaluated below to be equal to ∼5 nozzle exit diameters, i.e. ∼20 mm.

Fig. 11. The main result of the investigations: the dependence of the transition position on the driving frequency. Turbulence becomes dominant nearer to the nozzle if the driving frequency is increased.

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Fig. 12. The size of the vortical structures generated in a synthetic-jet actuator characterised by the equivalent length s [m] of the fluid column imagined as being extruded through the nozzle in the expulsion part of the cycle.

5. Dependence of the transition on driving frequency The cardinal question discussed in the present paper is dependence of the critical distance X1crit on the driving frequency ff . The answer was found by analysing the intersection positions of the fitted exponentials in Fig. 9. The results are presented in Fig. 11. The distances from the nozzle vary there in a quite wide range, from X1crit ∼ 30d up to more than X1crit ∼ 60d. Note that in the latter case this means the synthetic jet generated in a 4 mm diameter nozzle is not yet fully turbulent at the distance as large as ∼250 mm. Such long distances are found at very low frequencies (and hence at large size of the vortical structures). It is possible to argue that this influence of the driving frequency ff on the transition as presented in Fig. 10 is the consequence of the faster oscillation being capable of decomposing the organised motions more rapidly because of the consequently higher acting shear stresses. Available evidence, however, also suggests the key effect of the frequency may be in the smaller size (and therefore easier destruction) of the structures generated at increased driving frequency while (as in the present case) the nozzle exit velocity is kept constant.

Fig. 13. Importance of the outer scale, characterised by the “extruded column length” s. For the large s/d values, which are typical for the present experiments, cf. Fig. 12, the generated ring vortices develop a “tail” trailing behind the vortex structure.

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Fig. 14. The downstream decay in the fully turbulent regime T may be fitted by a universal line correlating it with the initial size of the coherent structures (here characterised by the extrusion length s).

6. Scale of structures in synthetic jets 6.1. “Extrusion” length To study the effect of the size of the generated structures, it is necessary to choose a suitable quantity to characterise their dimensions. A quantity suitable for the present purpose is the length of the imagined column s [m] of fluid extruded from the displacement cavity through the nozzle during each outflow part of the oscillation period, as discussed in [6–8]. In the piston-type actuators – as was used, e.g. in the experiments described in [8], the magnitude s is evaluated quite accurately and easily from the piston stroke and the known diameters of the piston and the nozzle exit. Neglecting compressibility (or applying the correction discussed in [42]), the length s is evaluated as that of an imagined cylinder with cross section equalling the exit area of the nozzle and volume V which equals the volume of the fluid displaced out from the nozzle during the displacement part of the period. The use of spring-supported and deformable membranes for the displacement made evaluation of the actually displaced volume difficult and this led Glezer and Amitay [9] to introduce the definition of the equivalent quantity L0 , computed by integration of nozzle exit velocity over the expulsion part of the cycle – which in these cases need not be equal to one half of the period, cf. [34]. It was the value L0 that was actually computed in the present investigations. However, the constancy of the nozzle exit velocity together with the effectively rigid actuator membrane made the extrusion length simply inversely proportional to the driving frequency, as is demonstrated in Fig. 12, and in fact equal to s as defined in [8]. One aspect of the importance of the length s is shown in Fig. 13: it influences the character of the generated vortical structures in the synthetic jets. 6.2. Extrusion length similarity Presented in Fig. 14 is an interesting fact about the decay of energy of unsteady motions in the final, fully turbulent flow regime

T. The equal slope in the semi-logarithmic co-ordinates (Fig. 9), regardless of the driving frequency, requires only horizontal shifting dependent on the “extrusion length” s, as shown in Fig. 19 to obtain an interesting similarity law for turbulence in synthetic jets. 6.3. Simple vortical structure models Of course, the fluid does not leave the nozzle as a constant-crosssection column, since immediately after passing through the exit it starts forming the vortical structure. For a better comparison of the size of the structure relative to the nozzle, it might be reasonable to convert the volume V = d2 s/4 of the imaginary extruded column into the equal-volume toroidal shape. For such a torus, as shown in Fig. 15, there is:



2rv S = d d

(5)

where rv is the radius of the circle (Fig. 15) that would generate the torus by its imagined revolving about the jet axis. The other circle that characterises a torus is the loci of such minor radius circles. For simplicity, the diameter of this circle torus tube centres in this expression Eq. (5) is assumed to be equal to the nozzle exit diameter d. Despite the fact that similar tori are presented by many authors as the shapes of the vortices that constitute synthetic jets (their very name suggesting the flow being “synthesised” from such toroidal vortices), they may be actually formed only in the cases with very small s/d values, much smaller than those in Fig. 12. At s/d values typical for the experiments discussed in this paper, the radius rv (defined in Fig. 15), would be much larger than d/2, making the toroidal shape impossible. For simple equivalence of volumes assumed in Eq. (5), the limiting case rv = d/2 occurs at s/d = , which is ten times smaller than the lowest value plotted for the present experiments in Fig. 12.

V. Tesaˇr, J. Kordík / Sensors and Actuators A 187 (2012) 105–117

Fig. 15. A simple model intended to provide a better measure of the size of vortices generated downstream from the quadrant-shaped nozzle than the ratio s/d in Fig. 12. Numerical values show, however, that the structures in the present investigations could not have this toroidal shape.

Obviously, a reasonable model of the vortical structure formed by outflow from the nozzle must be different. A better choice is the Hill’s spherical vortex (cf. [35]) shown in Fig. 16. Its radius rH is determined, again for the simple equivalence of volumes, by

 3

3s rH = 16d d

(6)

and plotted as a function of the parameters of present investigation in Figs. 17 and 18. This simplistic model approach, of course, ignores the formation of the “tail”, shown e.g. in Figs. 1 and 13. Its consideration would make the resultant size of the structure model smaller than shown in Figs. 17 and 18 where the objects are much larger than expected to be the case. Even the smallest spherical object for which the data are presented in Fig. 12 has its diameter much larger than the diameter of the nozzle. In the low

Fig. 16. A better simple model useful for characterisation of the size of the vortical objects formed in the expulsion part of the period. The main simplification is the neglect of the “tail” trailed by real instability structures. The radius rH is plotted in Fig. 17. The size of the spherical model shown here, 4.66-times larger than the diameter d of the nozzle exit, is typical.

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Fig. 17. Size of the spherical model of the vortical structure formed downstream from the nozzle exit, plotted as a function of the equivalent extrusion length s, presented in Fig. 12. Both radius diameter rH and s are here non-dimensionalised by relating them to the nozzle exit diameter d.

frequency case Sk = 11.5, the spherical model is shown in Fig. 18 to be of eight times (!) larger diameter than the nozzle exit. No wonder such large entities in the flow can keep and carry considerable organised movement momentum quite far downstream.

7. An invariant of the transition The final target of scientific research in general should be identification of invariants of the investigated problem. In the present case, study of the transition in synthetic jets, the last Fig. 19 offers such an (approximate) invariance: the critical distances X1crit are almost independent of the driving frequency when nondimensionalised with respect to the diameter 2 rH of the spherical vortex model. The small deviations from the invariance seen in Fig. 19 may be plausibly explained by the formation of the “tail”, neglected in the computations leading to Fig. 19. The critical distance at which synthetic jets rather suddenly undergoes transition into developed turbulence thus seems to be a practically constant (near to 10) multiple of the spherical vortex size.

Fig. 18. Size of the geometric spherical model (admittedly simplistic, not considering the fluid volume in the “tail”) introduced by Eq. (6). The volume displaced through the nozzle decreases with increasing values of the Stokes number, Eq. (4).

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synthetic jet turbulence, with dependence on the extrusion length s – as opposed to the inner, Kolmogoroff similarity [27]. Acknowledgments Authors gratefully acknowledge the support by research plan AV0Z20760514 from the Grant Agency of the Academy of Sciences of the Czech Republic and also by grant Nr. P101/11/J019 received ˇ – the Czech Science Foundation. Author V.T. is grateful from GACR ˇ for the support by research grant TA 02020795 and J.K. to TACR ˇ grant GPP101/12/P556. wishes to extend his thanks to GACR References Fig. 19. The (nearly) invariant expression for the transition into turbulence in synthetic jets, based in the very simple geometric model according to Fig. 16: the flow becomes fully turbulent at the axial distance roughly equal to 10 diameters of the spherical vortex.

8. Conclusions Analysis of velocity spectra combined with the quasi-similarity solutions can provide interesting information about synthetic jets, as was demonstrated in the previous paper [27] and in [33] – which was limited to analysing the spectra of only one synthetic jet case, generated at a particular Stokes number Sk = 33.3 (driving frequency ff = 30.46 Hz). In the present paper, the analysis is extended to a range of cases covering a full decimal order of the driving frequency magnitudes. In particular, the attention is here concentrated on investigations of the transition into turbulence. This was already earlier hypothesised to take place as the observed rather abrupt change in the character, at a distance from the nozzle decreasing with increasing frequency of fluid displacement in the actuator. The investigation is based on the mathematical model of synthetic jets model with the model parameter Eq. (2) varying along the jet axis. The abrupt change in the parameter was identified, using the fitted decay laws from spectra, as the location at which disappear organised motions in the jet (identified by the presence of peaks in frequency spectrum). An important fact found now is the critical location, where the transition into the final fully turbulent regime takes place, decreases with increasing the driving frequency. This may be associated (certainly in the present case of constant time-mean nozzle exit velocity) with decreased initial size of the vortical structures generated by outflows from the nozzle. When these structures are modelled by a simple spherical vortex having its volume equal to the volume of the fluid leaving the exit during each cycle, the transition occurs at the streamwise distance from the nozzle equal to ∼10 diameters of the spherical vortex model object. Among other interesting results found are the following: (a) The existence was established of the two regimes D and R prior to the transition into turbulence. (b) Change from D to R takes place when the magnitude of spectral density at the forcing frequency peaks decreases to a particular value as shown in Fig. 8. (c) There is a transition regime R in which the loss of energy (as characterised by the power-halving axial distance) is exceptionally rapid – more rapid than in the fully turbulent regime T. (d) The energy decay rate is more rapid at a higher driving frequency ff (Fig. 10). (e) In the fully chaotic turbulent flow regime T the dissipation rate (Figs. 9 and 16) is not dependent upon the driving frequency ff . This leads to an interesting “outer” similarity law for the

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Biographies Prof. Ing. Václav Tesaˇr CSc. – received his Ing. degree in mechanical engineering and later CSc (an equivalent of PhD) from Czech Technical University in Prague, Czech Republic. From 1984 to 1998 was there the Head of the Department of Fluid Mechanics and Thermodynamics at the Faculty of Mechanical Engineering. He was visiting professor at Keio University in Japan and later visiting professor at Northern Illinois University, DeKalb, USA. Between 1999 and 2005 he was Professor at the Department of Chemical and Process Engineering at the University of Sheffield in United Kingdom. Since 2006 is Senior Research Scientist at the Institute of Thermomechanics, Academy of Sciences of the Czech Republic. His research interests are focused on shear flows – in particular jets and wall jets – and also their applications in fluidics. An author of 4 textbooks, more than 400 papers, and monograph “Pressure-Driven Microfluidics”, published in the U.S.A. In 2010 he was awarded in Britain the Moulton Medal. He is named as the inventor on more than 200 Patents, mainly on various fluidic devices. Ing. Jozef Kordik Ph.D. – received his Ing. degree in mechanical engineering from ˇ CVUT Czech Technical University in Prague, Czech Republic in 2007. From 2006 (initially on part-time basis) has been employed at the Institute of Thermomechanics, Academy of Sciences of the Czech Republic as a researcher working on experimental investigations – especially anemometric measurements – of unsteady flows. He is also active in the field of computer processing of complex experimental data. He has recently received his PhD degree having defended a thesis on the subject of synthetic and hybrid-synthetic jets.