Vortex dynamics and mechanisms of heat transfer enhancement in synthetic jet impingement

International Journal of Thermal Sciences 112 (2017) 153e164

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Vortex dynamics and mechanisms of heat transfer enhancement in synthetic jet impingement Luis Silva-Llanca a, *, Alfonso Ortega b nica, Benavente 980, La Serena, Chile Universidad de La Serena, Departamento de Ingeniería Meca Villanova University, Laboratory for Advanced Thermal and Fluid Systems, Mechanical Engineering Department, 800 East Lancaster Avenue, Villanova, PA 19085, USA a

b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 February 2016 Received in revised form 8 September 2016 Accepted 16 September 2016

When confined flow is oscillated from and into a quiescent volume, a periodic coherent flow that resembles that of a conventional jet can be generated. Such a jet, termed a synthetic jet, has been investigated for thermal management by causing it to impinge onto a heated surface. Because of its fluctuating nature, the impinging jet thus formed is dominated by vortices that are advected towards the surface. This surface-vortex interaction is key to understanding the fundamental mechanisms of convective heat transfer by the impinging synthetic jet, which motivates this work along with the search for the improvement of the system thermal efficiency. A canonical geometry was developed to investigate the flow and heat transfer of a purely oscillatory jet that is not influenced by the manner by which it is produced. The unsteady two-dimensional Navier-Stokes equations and the convection-diffusion equation were solved using a finite volume approach in order to capture the complex time dependent flow field. The Q-criterion (Hunt et al., 1988), which defines vortices as connected fluid regions with positive second invariant of the velocity gradient tensor was utilized to identify vortices without ambiguity. A definition of the jet characteristic velocity was developed rigorously based on the vortex dynamics produced by the jet. It is equivalent to the common definition accepted in the literature, which has been successfully used to match the dynamics of synthetic and steady jets, but which was developed using heuristic reasoning. When the primary vortex advects in a direction parallel to the target surface it gives rise to a secondary vortex with opposite net vorticity. This secondary vortex is largely responsible for enhancement of the heat transfer within the wall jet region. Under certain conditions, vortex coalescence occurs, leading to degradation in the heat transfer enhancement due to the reduction in the number of secondary vortices interacting with the heated surface. By understanding, quantifying and predicting the mechanisms that drive the phenomenon of vortex merging, optimum conditions of operation are demonstrated, ultimately leading to higher efficiencies by maximizing the heat transfer at similar pumping costs. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Convective heat transfer Impinging synthetic jets Vortex dynamics Vortex merging

1. Introduction Synthetic jets are generated by the ejection and injection of fluid from or to an exiting orifice (e.g. a nozzle), resulting in the spawning of a counter-rotating vortex pair with each cycle, and consequently forming a downward moving train of vortex pairs, as seen in Fig. 1. Even though they induce zero net mass flow per cycle, positive net momentum is produced due to the difference in the dynamics of the fluid between the first part of the cycle (forward

* Corresponding author. E-mail address: [email protected] (L. Silva-Llanca). http://dx.doi.org/10.1016/j.ijthermalsci.2016.09.021 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.

stroke) compared to the second part (back stroke). This flow can be used as the basis for a cooling method by impinging the unsteady vortical flow onto a heated surface. The formation and evolution of a slit synthetic jet was investigated by Smith and Glezer [1]. They showed that at the end of the forward stroke, a pair of vortices is generated at the jet exit. Impelled by its own positive momentum, this vortex pair advects away rapidly enough to be only weakly affected by the suction part of the cycle. Analytical and experimental comparisons between synthetic and steady jets have been presented [2,3]. Smith and Swift [2] matched the Reynolds numbers within the range 695 < Re < 2200, finding correspondent flow characteristics, where

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Fig. 1. Schematic representation of the vortex dynamics during synthetic jet impingement.

the normalized time mean velocity data were similar for both cases. They reported that the synthetic jet width grows linearly with respect to the downstream direction, just as in a steady jet, although it spreads more rapidly. Agrawal and Verma [3] used similarity analysis to compare synthetic and conventional jets, finding similar streamwise variation of velocity and spread rate; their conclusions were corroborated with empirical data. The origin of the characteristic velocity used to define Re such that the steady and unsteady jets can be compared was not established in these initial studies. The present paper shows that the characteristic velocity arises from the analysis of the selfinduced velocity of the co-rotating vortex pair. Several studies have been performed to understand the fluid flow and/or the heat transfer generated by an impinging synthetic jet [4e17]. Optimum jet operating conditions have been found for the heat transfer at certain jet-to-surface distances [4,9,11,13,14,18,19]. It has been claimed that when the wall was located within the so-called intermediate-flow regime, a more effective mixing with colder ambient air augments local heat transfer rates. When compared to conventional steady impinging jets, synthetic jets can offer higher heat transfer rates except in cases when, for example, vortex merging is present [11,12]. A notable phenomenon sometimes observed with the advection of vortices is the merging or coalescence of consecutive co-rotating vortices, as shown schematically in Fig. 1. Merging of contiguous vortex pairs has been previously reported by experimental observations [20]. The power spectrum of the instantaneous velocity near the jet exit detected the presence of a frequency equal to one half the driving frequency (sub-harmonic), which was related to vortex coalescence. Silva and Ortega [11] correlated the onset of vortex merging to a decrease in the overall heat transfer. We note here that many if not most previous experimental investigations include actuator resonant artifacts either by intent or by mistake [4,6,7,9,13,15,17e19]. Optimum heat transfer has been found at resonance frequencies [6,7,9,15,17,19], which is an intrinsic

characteristic of the actuator and the jet issuing cavity, rather than pure flow physics. In general, the experiments have been carried such that the excitation voltage is kept constant and the frequency is varied; procedure that also modifies the jet Reynolds number by maximizing it at resonance conditions. Unless a study were mainly focused on the actuator performance, this simultaneous adjustment of the Reynolds number and the frequency may obscure any conclusion regarding their individual effect over the impinging flow and the heat transfer fundamentals. Silva-Llanca et al. [12] tackled this issue in their experiment by modifying the excitation voltage when the frequency varied, so that the flow could be produced at constant Reynolds numbers. Disregarding this matter renders the data set available in the literature hard to compare, as the results are strongly related to the device utilized in each experiment. Numerical experiments facilitate the idealization of the convective phenomena and are less prone to artifacts. Several computational studies of the heat transfer phenomenon have been presented [10e13,21]. Utturkar et al. [13] achieved good agreement between numerical and experimental data assuming laminar flow throughout. This assumption was found to be accurate due to the high effective viscosity that dampens the small vorticity scale near the jet orifice. Wang et al. [21] simulated the vibration of the diaphragm used to produce the pulsating flow as a sine wave velocity function at the inlet boundary. Large eddy simulation was the technique chosen, showing acceptable agreement with velocity and temperature distributions that were experimentally obtained. Silva-Llanca et al. [12] compared idealized numerical data from a laminar synthetic jet to large-scale experiments, up to six times the size of the previously published canonical geometry [11]. Excellent agreement was found under a given range of Reynolds numbers, frequencies and jet to wall separation. Some disagreement in the data was attributed to phenomena that were not accounted for in the numerical simulations such as: comparable conduction heat loses to convective heat transfer, transition to turbulence, and plausible three dimensional effects. The synthetic jet, as a coherent flow, is dominated by its vortical nature, wherein the train of vortices plays an essential role in the free jet, stagnation, and wall jet zones. Locating and sizing vortices unambiguously, even when they can be identified as coherent, rotating structures in the flow, is challenging. Different Galilean invariant (frame of reference independent) definitions of a vortex have been previously proposed [22e25]. Hunt et al. [24] defined vortices as regions with positive second invariant of the velocity gradient, Q, with the additional condition that the pressure be lower than ambient. Dallmann [23], Vollmers et al. [25] and Chong et al. [22] defined vortices based on the eigenvalues of the velocity gradient Vv to classify the local streamline pattern around any point in a flow in a reference frame moving with the velocity of that point. They proposed that a vortex core is a region with complex eigenvalues of Vv, which implies that the local streamline pattern is closed or spiral in a reference frame moving with the point. 2. Motivation and goals It has been previously shown that impinging synthetic jets remove heat more efficiently than steady jets, and it has been hypothesized that the vortical nature of the flow is responsible for this enhancement in the heat transfer. However, fundamental exploration of the vortex-surface interactions that are claimed to be responsible for heat transfer enhancement has not been performed, and certainly not in a pure flow that is absent of actuator artifacts. The impinging synthetic jet flows previously studied by the authors have been created either experimentally or computationally such that they are independent to the effect of the actuator geometry

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and actuation physics. The complex time-dependent flows might be linked to the method of creation, particularly in the near field close to the nozzle exit. The goals of the present work are to: (1.) Use a canonical geometry free of artifacts to investigate the fundamental mechanisms of vortex-surface and vortex-to-vortex interactions, and their role in heat transfer enhancement and detriment. (2.) Find conditions of operation to optimize the heat transfer, based on a comprehensive understanding of the physics.

3.1. Non-dimensional parameters The stroke length L0 is the quantity that characterizes the amount of fluid ejected during the forward stroke (0 < t/P < 0.5), which can also be utilized to define the jet characteristic velocity U0 as [1]: P

Z2 L0 ≡

3. Physical situation, mathematical model and experimental validation

155

vin ðtÞ dt

(5)

0

U0 ≡L0 f

(6)

A purely oscillatory jet was created to minimize, or eliminate, actuation effects on the oscillatory flow. A canonical geometry was designed to produce such a flow, depicted in Fig. 2. The fluid was considered to flow with no shearing between two, adiabatic parallel plates separated by a distance w so as to eliminate any contributions of a viscous boundary layer on the formation of the jet after leaving the nozzle. The inlet flow was imposed as an oscillating uniform flow. The plates had zero thickness. The following assumptions were adopted for the numerical simulation: (i) Laminar, unsteady, two-dimensional flow, (ii) negligible viscous dissipation, and (iii) temperature independent thermophysical properties. The governing conservation equations for mass, momentum and thermal energy are given as follows: Continuity:

The global Reynolds number Re was defined in terms of the characteristic velocity U0 and the jet width w. Such an approach has been used successfully to compare continuous and synthetic jets [2].

V,v ¼ 0

Alternatively, the Strouhal number St can be used as nondimensional frequency, however the use of St prevents a clear separation between velocity and frequency effects which has been one of the failings of previous investigations. As such it is preferable to use Womersley number and Reynolds number as the independent parameters of the problem. The pulsating flow impinges over a uniform heat flux wall located at a distance H from the channel exit thereby causing convective heat transfer from the wall to the jet. The heat transfer coefficient h and the Nusselt number Nu are defined as:

(1)

Equation of motion:

Dv ¼ Vp þ nV2 v Dt

(2)

Conservation of energy:

DT ¼ aV2 T Dt

(3)

The oscillating uniform velocity at the channel inlet obeys a sinusoidal function as:

vin ðtÞ ¼ vmax sinð2pftÞ

Re ¼

U0 w

The non-dimensional frequency was defined in terms of the Womersley number, which relates the pulsatile flow frequency to viscous effects as:

U¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pfw2

hðx; tÞ ¼

q00 Tw ðx; tÞ  T∞

(9)

hðx; tÞ w k

(10)

Nuðx; tÞ ¼

Nuavg

(8)

n

(4)

Where vmax is the inlet velocity amplitude and f is the jet frequency of period P ¼ 1/f.

(7)

n

1 ¼ 4P

ðnþ4ÞP Z

Nuðx; tÞ dt

(11)

Nuavg dx

(12)

nP

Nuavg ¼

Fig. 2. Schematic of the physical domain that represents the canonical geometry.

1 10w

Z5w 5w

Where q00 is the heat flux per unit area; Tw and T∞ are the wall temperature and ambient temperature, respectively. The time averaged Nusselt number Nuavg was calculated using the four cycles that follow after the problem reached steady state at cycle n, where the value of n can vary depending on the jet parameters; e.g., finding the steady state at the largest frequency can take up to three times more cycles than in the lower frequency cases. The time and space averaged Nusselt number Nuavg was estimated within the spatial range 5x/w5. The parameters that characterize the jet were varied as follows: Re ¼ {406;508}, U ¼ {12.7;17.9;22} and H/ w ¼ {5;7.5;10}.

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3.2. Numerical approach and experimental validation The transient incompressible laminar Navier-Stokes equations were approximated via a finite volume formulation using the commercial software ANSYS Fluent™. The maximum inlet velocity was 25 m/s, leading to a peak Mach number on the order of 0.1, thus the incompressible flow assumption can be considered as valid; moreover, in the scaled-up experiments the maximum inlet velocity ranged between 4 and 6 m/s depending on the jet width. The velocity and pressure fields were coupled using the SIMPLE algorithm due to its computational efficiency, especially in unsteady flows with small time steps. For the momentum equations, First Order Upwind was chosen as the interpolation scheme, whereas the QUICK method was used in the energy equation. The grid was formulated as structured near the areas of interest to ensure accuracy, whereas an unstructured formulation was utilized sufficiently far from the impingement zone so that fewer nodes were used without affecting the final outcome. This hybrid grid formulation improved the code numerical efficiency. After comparing different mesh densities, grid independence was met for a number of nodes on the order of 1.5  105. The time step was likewise contrasted as function of the jet period P ¼ 1/f as: Dt ¼ P/n, with n ¼ 50,100,200. Time step independence was met for n ¼ 100. The numerical data presented in this work are the same as those used by Silva and Ortega [11], thus further details about the numerical procedure can be found in that previous paper. A detailed depiction of the heated surface used in the experiments is shown in Fig. 3. Under certain conditions the synthetic jet may produce heat transfer coefficients sufficiently low to make the convective heat transfer comparable to conduction; balsa wood was utilized as the substrate to decrease heat losses due to its low thermal conductivity. Phenolic wood was added underneath to

improve the surface mechanical properties, especially to prevent deflection. Fig. 3 shows a 50:1 near surface cross section view that details the way the heater and the thermocouples were attached to the substrate. A 0.0762 mm (0.003 inch) thick layer of double sided adhesive was added to support the heater which was covered with a second layer of the same adhesive in which the 40 gage K-type thermocouples were embedded. The outer 0.0254 mm (0.001 inch) thick stainless steel sheet was layered on top to smoothen the surface and to avoid inducing turbulence in the vicinity of the thermocouples. The 40 gage alumel and chromel wires were terminated in the back of the surface, then extended to the measuring equipment with 20 gage wires for data acquisition. The synthetic jet rig and its main components are shown schematically in Fig. 4. A sinusoidal analog signal was generated from the laptop computer using the software Labview™, which was electrically amplified before its arrival into the speaker, consequently oscillating the speaker diaphragm and actuating the jet. An ice bath was used as an ice-point reference in order to decrease the temperature reading uncertainty. The sinusoidal voltage signal generated from the computer V ¼ Asin(2pft) was calibrated such that its amplitude A and frequency f were correlated to the jet Reynolds number using a hotwire anemometer. This sensor was placed exactly at the jet exit with its probe holder in the upstream direction, so the forward flow (first half of the jet cycle) remained undistorted. Those data are sufficient to calculate U0 and Re, thus the backstroke data distorted by the probe holder were neglected. For a fixed jet width w, and based on its definition, Re is only a function of U0 which only depends on vmax. From a mathematical perspective this implies that in order to study pure frequency effects on the problem, the velocity amplitude vmax must remain

Fig. 3. Heated surface geometry used in the experiments (dimensions in mm).

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157

opposite circulation (intensity) G1. If the vortices are assumed to be irrotational, G1 can be approximated as [26]:

G1 z2pa1 vj vj ≡

2 P

Z

P 2

(13)

vmax sinð2pftÞ dt

(14)

0

The vortex radius a, is defined in terms of its equivalent area Aeq as a≡Aeq/p. The downward component of the velocity induced by the other three vortices upon Q1 can be written in terms of the vortex intensities and length scales as follows:

vG ¼



G1 2pð2a1 Þ

þ









G0 a0 d a1 G a0 b2 þ d b2 þ a1 b2 þ 0 þ  2p b2 b2 b2 2p 1 þ 4 ab12 ðd þ a0 Þ (15)

Equation (15) can be simplified to vG ¼ G1/4pa under the following assumptions: G1 > G0, d < a1 < a 0≪ b2. By combining Eqns. (13)e(15), the vortex translational velocity can be rewritten as: Fig. 4. Synthetic jet experiment schematic (not to scale).

unchanged as f is modified in Eq. (4). It was empirically found that vmax was a function of both A and f, thus every time f was varied, A needed to be adjusted accordingly to maintain a constant Reynolds number. From the point of view of the fundamental physics of the jet, modifying f without varying A may mislead the interpretations of the data, since Re is being changed as well. It is important to remark that this is an exclusively experimental issue, since in Eq. (4) it is simple to parametrically adjust in the numerical approach. After each temperature measurement, the surface was laterally displaced using a linear motion system attached to a computer controlled stepper motor. The time interval between steps was sufficient so that the surface reached steady state before the data acquisition. A detailed description of the experimental procedure may be found in Silva-Llanca et al. [12]. Fig. 5 presents the time averaged Nusselt number distribution obtained from the numerical code (solid lines) and the experiments (symbols) in four cases, where good to excellent agreement was found between the two approaches. Fig. 6 shows the time and space averaged Nusselt number for the experiments and the simulations throughout the data set that best compared between the two approaches in Ref. [12]. The non-dimensional frequency U introduced a mostly diminishing effect upon the heat transfer, except at the lower spacing H and the higher Re. This effect reinforces one the motivations of this work, which is to adequately explain this previously observed phenomenon [11,12] by elucidating the fundamental physics during a synthetic jet impingement.

4. A characteristic velocity definition based on vortex dynamics A schematic of the idealized vortex dynamics between two consecutive counter-rotating vortex pairs, sometimes referred to as “dipoles”, is depicted in Fig. 7. The corresponding vortices in each pair have intensities (circulations) of equal magnitude but opposite orientation: G1 ¼ G1 and G0 ¼ G0. It is assumed that the azimuthal velocity of the vortex Q1 and Q1 equals the mean jet exit velocity vj during the first half of the period P, in other words, the ejected fluid completely rolls into two axisymmetric vortices of

vG ¼

vj 1 ¼ 2 P

Z

P 2

vmax sinð2pftÞ dt

(16)

0

The translational vortex velocity, as given by Eq. (16), provides an unambiguous definition of the jet characteristic velocity. This approach matches the definition originally proposed by Smith and Glezer [1] that is commonly utilized in the synthetic jet literature, usually referred to as U0.

5. Vortex identification A vortex can be intuitively distinguished as a flow structure, although its proper identification lacks uniqueness. The location of local pressure minima might be one way to detect a coherent structure (CS) in a given domain; however, this approach is only accurate in steady laminar flows. Moreover, it can be proven that a pressure minimum is neither necessary nor sufficient to establish the presence of a CS. Vortices can be distinguished by the use of closed streamlines, however this technique depends on the frame of reference selected, therefore leading to possible ambiguities. Furthermore, a particle within the closed region might not complete an entire circuit during the vortex survival time. Particularly for the synthetic jet phenomenon, the vorticity magnitude u has been the preferred method for locating CS [9,10,13,14,19,27]. This approach may be successful in free shear flows, for example, in regions sufficiently far from a stationary boundary. The disadvantage of this technique is that u cannot differentiate between swirling motion and pure shearing. For instance, the vorticity magnitude does not allow the identification of vortex cores in a shear flow, especially when the magnitude of the shear is comparable to the vorticity magnitude within the vortex. Therefore, for a synthetic jet impinging over a wall, this criterion tends to be ambiguous as u is large in the vicinity of the boundary layer. In addition, the geometry of the vortices depends on the vorticity threshold applied, leading to arbitrariness as a result of the lack of a convention to define the range of u where one can affirm that a vortex exists. Several Galilean invariant techniques for vortex identification have been developed. Among the most widely used is the Q-criterion [24] which defines vortices as connected fluid regions with a positive second invariant of Vv, where Q is defined as:

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Fig. 5. Time averaged Nusselt number distribution; comparison between numerical (solid line) and experimental (symbols) data.

Fig. 6. Heat transfer comparison between numerical and experimental data.

 1 1 Q ≡ u2i;i  ui;j uj;i ¼ ðjjujj2  jjSjj2 Þ 2 2

(17)

From the right hand side of Eq. (17) it can be inferred that this technique distinguishes zones where the vorticity magnitude prevails over the strain-rate (S) magnitude. Dallmann [23], Vollmers et al. [25] and Chong et al. [22] defined vortices based on the eigenvalues of the velocity gradient Vv to classify the local streamline pattern around any point in a flow in a reference frame moving with the velocity of that point. They proposed that a vortex core is a region with complex eigenvalues of Vv, which leads to:

D¼

3  3  1 1 Q R >0 þ 3 2

(18)

where. R ¼ det(ui,j) Jeong and Hussain [28] presented a method based on dynamic considerations. By taking the symmetric part of the gradient of the Navier-Stokes equation, the following equation was derived:

DSi;j 1  nSij;kk þ uik ukj þ Sik Skj ¼  p;ij Dt r

(19)

In this case only the contribution of the term S2 þ U2 was considered, which is a symmetric tensor. Therefore, in order for pressure minima to occur, the second eigenvalue of S2 þ U2, l2, must be less than zero within the vortex core. The three vortex identification criteria are compared in Fig. 8. Even though these methods differ in their formulation and mathr ematical approach, the contour lines overlapped identically. Kola [29] summarized these three criteria, and pointed out their limitations for specific cases and/or conditions. In this work, simplifications such as incompressible flow, constant properties and twodimensionality, led to negligible differentiation between the three criteria. The Q-criterion was chosen to identify coherent structures, since it is the least computationally expensive of the three. Fig. 8 highlights the level of ambiguity that could arise in the analysis when vorticity is used as a marker to identify vortices near the wall.

6. Behavior and role of primary and secondary vortices The arrival of the primary vortex Q1 to the vicinity of the wall induces the formation of a secondary vortex Qs with opposite net circulation, as illustrated schematically in Fig. 9. This phenomenon is due to the nature of vorticity production at a stationary boundary. Lighthill [30] first elucidated the appearance of a flux of vorticity at a no-slip wall into the flow domain. The x-component of the NavierStokes equation at the static wall (y ¼ 0) in terms of vorticity is given by:

n

vu 1 vp ¼ vy r vx

(20)

Equation (20) reveals that a positive pressure gradient leads to the generation of a negative vorticity flux at the wall. The circulation of fluid around a given closed curve C enclosing the vortex equivalent area Aeq (determined by the iso-contours Q ¼ 0), is defined as [26]:

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159

Fig. 9. Schematic of the vortex dynamics observed instantaneously in the vicinity of the wall.

G≡

I C

Fig. 7. Schematic of the idealized vortex dynamics produced during the generation of a synthetic jet.

Z v,ds ¼ A

eq

V  v ,dA |fflfflffl{zfflfflffl}

(21)

¼u

One notable feature of the Q-method can be appreciated from Fig. 8. As previously mentioned, some areas of non-negligible vorticity may become ambiguous in the region near the wall (blue zone), and the presence of these secondary vortices may not be as evident as when the Q-method is utilized to identify them. The instantaneous fluid flow and Nusselt number Nu(t) distribution at the wall, as well as the time averaged Nusselt number Nuavg, are shown in Fig. 10. The gray zones represent the areas where the identification criterion distinguishes the presence of a vortex (Q > 0). The overlap between the solid blue line and the symbols for Nuavg show the excellent level of agreement between the idealized computations and the experiments [12]. The creation of primary vortices Q1 at the jet exit and their downward translation towards the heated wall are clearly identifiable, as well as the subsequent formation of the secondary vortices Qs. In the vicinity of the stagnation zone, Nu peaks as Q1 approaches the wall due to the impingement of colder fluid induced by the counterclockwise

Fig. 8. Instantaneous fluid flow and vortex identification criteria at two different instants for H/w ¼ 10, U ¼ 12.7 (a)e(b) and H/w ¼ 5, U ¼ 22 (c)e(d). The vorticity field is normalized as u* ¼ u w/U0.

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rotation of Q1. When Q1 turns and starts its spanwise translation, the heat transfer enhancement depends almost entirely on the passage of Qs. This effect is especially noticeable in Fig. 10 (c), where Nu(t) presents a secondary local maximum for 7  x/w  9.5, which coincides with the region where Qs is “sweeping” the heated surface. The appearance of secondary vortices near the solid wall has been reported for conventional impinging jets [31e33]. Chung et al. [31] and Rohlfs et al. [32] studied the problem through direct numerical simulations, where they also observed that these secondary vortices enhanced the instantaneous wall Nusselt number. Vortex merging can be observed. In Fig. 10 (e) only one large vortex can be identified (Q1 þ Q0), which resulted from the coalescence of the two vortices observed in Fig. 10 (d). Similarly, in the region x/w > 6, the two vortices that appear in Fig. 10 (e) have coalesced in Fig. 10 (f). Silva and Ortega [11], showed that the effect of pairing on Nu was directly proportional to H/w and U (f). They illustrated that this phenomenon affected velocity and temperature locally, causing detrimental effects on time and space averaged heat transfer. The trajectories of the primary vortex cores c1, for four different cases, are shown in Fig. 11, where the symbols are colored by the vortex normalized intensity G* ¼ G/G1. The instantaneous position of the vortices is located as the small region where Q is maximized. As Q1 advects downstream, and since its self-induced velocity ðvG Þ1 is larger than that of the preceding vortex ðvG Þ0 , the vortex

separation b decreases, thereby causing the consecutive co-rotating vortices (Q1 and Q0) to begin interacting. For H/w  10, this interaction occurs well before the vortices have reached the wall. The trajectory of Q1 shifted towards the centerline because of the counterclockwise rotation of G0 that generates velocity upon G1, with a dominant negative component, in the x direction. This situation is advantageous for heat removal, since Q1 advects closer to the stagnation zone, increasing the magnitude of the velocity field in the vicinity of x/w ¼ 0. Furthermore, when the vortex approaches the wall nearer to x/w ¼ 0, it causes Qs to develop sooner, allowing it to sweep a larger area of the heated surface, and thereby leading to higher rates of heat transfer. This vortex interaction is consistent with observations by previous authors [4,9,11,13,14,18,19], where the time and space averaged Nu is optimized at intermediate jet-tosurface distances 12  H/w  18. Once the flow field reaches its periodicity or quasi-steady state, the vortices follow equal paths per cycle, except for those cases where merging is present, since the flow field now replicates every two cycles. Particularly in Fig. 11, the trajectory for the case H/ w ¼ 10 and U ¼ 22 (which exhibits merging) corresponds to the trailing vortex, shown as Q1 in Fig. 10 (d). The evolution of the primary vortex intensity is shown in Fig. 12, where t ≡ t vG/H is the time normalized by the characteristic time it takes for a vortex to reach the wall, in other words, the time a vortex travels a distance H at its self-induced velocity vG. The intensity peaks approximately half way through the first cycle (t z 0.25P). As

Fig. 10. Instantaneous fluid flow and Nusselt number distribution, gray zones indicate areas with Q > 0. (a)e(c) H/w ¼ 5, U ¼ 12.7; (d)e(f) H/w ¼ 10, U ¼ 22.

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161

Fig. 11. Primary vortex trajectories for Re ¼ 508, (a) U ¼ 12.7 and (b) U ¼ 22 (H/ w ¼ 5 , and 10 △).

the vortex undergoes pairing, sudden increase can be observed in the data as discontinuities in G. If the intent is to increase U at constant Re, the stroke length needs to be decreased, meaning that a vortex with higher f will rotate at a faster rate. This implies that increasing f, does not necessarily augment the intensity of the vortices generated. As can be seen in Fig. 12, f is inversely proportional to the intensity, where the ratio between peak G for U ¼ 12.7 (gray) and U ¼ 22 (white) was approximately two. The trajectories of the secondary vortices are illustrated in Fig. 13, for four different cases. Their instantaneous locations are found in the same manner as for the primary vortices. It can be seen that their cores remained within one jet width from the wall throughout. The evolution of the secondary vortex intensity in the four cases is shown in Fig. 14. Since the near wall advection of primary vortices is responsible for the generation of the secondary vortices, the evolution of G was consistent with the observations from Fig. 12, where G was inversely proportional to U. This suggests that increasing the frequency might reduce the heat transfer due to vortices with lower near-wall velocity magnitude; however, the actual benefit of incrementing the frequency lies in the capacity of the jet to generate more vortices, since this augments the number of secondary vortices sweeping the heated wall at a given time. In other words, the magnitude of the heated area being swept by the secondary vortices caused greater impact over the heat transfer, than the velocity that these vortices could induce in the vicinity of the wall. Vortex pairing plays a critical role in decreasing this covered area, as it reduces the number of primary vortices by a factor of two, resulting in fewer secondary vortices and lower time averaged rates of heat transfer. The presence of merging could be corroborated in the velocity field via the Fast Fourier Transform (FFT), where in the instantaneous velocity evolution at any given location, such as the

Fig. 13. Secondary vortex trajectories for Re ¼ 508, (a) U ¼ 12.7 and (b) U ¼ 22 (H/ w ¼ 5 , and 10 △).

Fig. 14. Secondary vortex intensity (circulation) G vs non-dimensional time t for Re ¼ 508, H/w ¼ 5 ,, H/w ¼ 10 △ (U ¼ 12.7 gray - U ¼ 22 white).

vicinity of the stagnation zone, a significant effect should appear at a frequency exactly one half the forcing frequency, since a new phenomenon (pairing) is emerging every two cycles. Silva and Ortega [11] proved the latter via FFT and phase plot analyses, where merging was quantified and correlated to a decrease in the heat transfer. 7. Physical mechanisms for merging in synthetic jets Vortex merging significantly influences the fluid dynamics and heat transfer phenomena, most significantly causing a decrease in the overall Nusselt number [11,12]. In order to elucidate the mechanisms that can lead two consecutive co-rotating vortex pairs to coalesce, the process is divided into three distinctive stages: (a) the creation of the vortex during the forward stroke, (b) the beginning of its interaction with the vortex remnant from the previous cycle, and (c) the completion of merging. A depiction of these three stages is shown in Fig. 15, where the velocity field was set in a frame of reference relative to the primary vortex Q1. Recall that both vorticity and Q are frame of reference independent. 7.1. First stage of merging (a)

Fig. 12. Primary vortex intensity (circulation) G vs non-dimensional time t for Re¼508, H/w¼5 ,, H/w¼10 △ (U¼12.7 gray - U¼22 white).

The first stage begins with the initiation of the forward stroke and lasts until the vortex Q1 begins interacting with the remnant vortex Q0 (from the previous cycle). This occurs when the ratio of the vortex radius squared to its distance with Q0, a2/b, reaches a certain critical value. This means that the vortex Q1 has grown sufficiently to induce an effect on Q0. The self-induced velocity of

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Q0 is much smaller than that of Q1, therefore Q1 approaches Q0 at a rate db/dt that scales more strongly with ðuG Þ1 . Thus, ðuG Þ1 is used in the scaling instead of the relative velocity between the vortices ððuG Þ1  ðuG Þ0 Þ since ðuG Þ1 [ðuG Þ0 . The time that it takes for Q1 to reach the zone of significant interaction with ðuG Þ0 is defined in terms of the vortex translational velocity ðuG Þ1 and the initial separation ðbi Þa as ðbi Þa =ðuG Þ1 . This characteristic time is used to define a non-dimensional time for stage (a) such that:

ta ≡t

ðuG Þ1 ðbi Þa

(22)

Fig. 16 shows the evolution of the ratio between vortex radius a and vortex distance b, normalized by the jet width w as (a2)/(bw). At the end of stage (a), different physics begin to drive the phenomenon and the assumptions that led to the non-dimensional groups proposed no longer hold. This manifests in the data departure observed in Fig. 16, which happened at the critical ratio ða2 =bwÞ crit z0:15 and critical time taz2.8, as indicated by the dashed lines. During the first cycle of a new vortex, the majority of its displacement takes place during the forward stroke. In other words, the initial trajectory of Q1 is mostly correlated to its ejection from the channel. Therefore, it is assumed that the initial distance between consecutive vortex pairs ðbi Þa scales with the radius of Q1. Furthermore, during the forward stroke it is expected that the area of the vortex will scale with the fluid displacement as Aeq~L0w/2, hence:

a¼

rffiffiffiffiffiffiffi Aeq

p

rffiffiffiffiffiffiffiffiffi L0 w ; 2p

z

a a ¼ qffiffiffiffiffiffiffi

(23)

L0 w 2p

The normalized vortex radius a* is shown as a function of time in Fig. 17, where the data are displayed for the range at which stage (a) occurs (0  ta  2.8). It can be inferred that the assumptions leading to Eq. (23) are valid and therefore it can predict the vortex radius growth during stage (a). Consequently, the vortex separation at which stage (a) ends, and stage (b) begins, can be estimated as:

2 t 32 Za 6:674 ðbi Þb ¼ vmax sinð2pfta Þ dt 5 ; w 0

ta ¼ 2:8

ðbi Þa ðvG Þ0

(24)

Fig. 16. Evolution of the ratio between the vortex radius squared and the co-rotating vortex separation a2/b during the first stage of merging.

7.2. Second stage of merging (b) The normalized vortex separation as a function of nondimensional time is illustrated in Fig. 18. The presence of a clockwise rotating vortex (blue), adjacent to Q1 (sometimes referred to as a “ghost vortex” [34]) can be observed from Fig. 15 (b). This vortex plays a key role during this stage as it induces a fluid flow that stretches Q1 by advecting its vorticity, and induces a downward velocity onto Q0. In an idealized situation, a stretched vortex behaves much like a vortex sheet, which generates a fluid flow parallel to it, whose velocity magnitude and direction are related to the sheet intensity [26]. The downward effect of the ghost vortex and the upward velocity induced by the stretched Q1 combined, maintain Q0 approximately at a constant distance from Q1. It is proposed that the duration of this stage has a dependence of the form t ¼ fn(G,w,n). A dimensional analysis suggests that a possible combination of non-dimensional groups is:

P1 ≡

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðt  ta Þ ; w

P2 ≡

G1 n

(25)

Where ta is the time at which stage (a) ends ta z2:8ðbi Þa =ðvG Þ1 . P1 can be interpreted as the non-dimensional distance between Q1 and the ghost vortex. This characteristic length scales with the size of the jet exit, thereby justifying the use of w instead of b. P2 is the

Fig. 15. Vorticity and velocity fields in a reference frame moving with the primary vortex during the first (a), second (b) and third (c) stage of merging. The vorticity field is normalized as u* ¼ u w/U0, whereas the velocity (vector) field is normalized using the characteristic velocity U0.

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163

8. Heat transfer optimality by avoiding merging As previously mentioned, the theoretical time for a vortex to reach the wall may be defined in terms of its translational velocity vG and the jet-to-surface distance H as H/vG. This quantity may be used to normalize the time when stage (c) starts as:

t ≡

non-dimensional vortex intensity, sometimes defined as the vortex Reynolds number. The non-dimensional time for stage (b) tb emerges as:

G1 nw2

(26)

From Fig. 18 it can be seen that t b z9  104 nw2 =G21 . This time signifies the end of stage 2 as indicated by the divergence of the data beyond this time in Fig. 18.

7.3. Third stage of merging (c) The final stage occurs when the areas of Q1 and Q0 intersect. This stage can be clearly seen by comparing Fig. 15 (b) and (c). During this stage the time is normalized with the vortex intensity G1 and the vortex separation b. In a frame of reference moving with Q1, Q0 is advected towards Q1 at a velocity induced by G1. Thus, the theoretical time for the vortices to fully merge must be proportional to G1 and inversely proportional to their separation b. The non-dimensional time for stage (c) is defined as:

tc ≡½t  ðta þ tb Þ

G1 ðbi Þ2b

(28)

It can be seen in Fig. 15 (c) that, even though the Q-criterion indicates that the vortices merged, their cores are still separated by a finite (non-negligible) distance. This explains the fact that b remained greater than zero for all cases. The evolution of b, as seen in Fig. 19, suggests that the cases studied can be separated into two groups:

Fig. 17. Normalized vortex radius vs time for ta  2.8.

t b ≡ðP1 P2 Þ2 ¼ ðt  t a Þ

time to reach third stage ta þ tb ¼ time for vortex to travel H H=vG

(27)

Fig. 19 depicts how the vortex separation b evolves as a function of tc. By predicting the interaction between consecutive vortices and its evolution, the onset of vortex merging may be delayed in favor of the heat transfer, which is discussed in the next section.

Fig. 18. Normalized vortex separation vs non-dimensional time ta during second stage of merging.

1. Q1 and Q0 separated (b increased). 2. Q1 and Q0 approached each other (b decreased) for tc<1, then maintained their separation until tcz3. Table 1 details t* for all the cases shown in Fig. 19. The two clusters of data observed in Fig. 19 can now be differentiated by their values of t*. Based on what was demonstrated in section 6, the earlier the vortices reach (c) before approaching the heated wall, the stronger merging effects will be. In other words, smaller values of t* will lead to weaker heat transfer from the wall. This behavior is consistent with previous observations [11,12], where the cases with t*<0.43 showed lower heat transfer rates. The expression for the normalized time t* can be re-written in terms of the jet non-dimensional parameters as:

 1 "  1  2 # H Re 2 Re t ¼ þ 570 2:8 w U U2 

(29)

As previously mentioned, for given Re and H/w, increasing the frequency also increases the heat transfer until coalescence affects the impinging flow. In practice, Eq. (29) can be used to optimize the heat transfer by setting the jet parameters to be close to t*z0.5, then solving for U. The relationship between the jet parameters and coalescence presented in this section was numerically and empirically observed in Silva and Ortega [11] and Silva-Llanca et al. [12]. This suggested approach maximizes the heat removal at similar pumping power, resulting in a more energy efficient practice.

Fig. 19. Normalized vortex separation vs non-dimensional time tc during third stage of merging.

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Table 1 Normalized time at which the second stage of merging ends with respect to the theoretical time a vortex displaces a distance H. *

H/w

Re

U

t

5 7.5 7.5 10 10 10 10 10 10

508 406 508 406 406 406 508 508 508

22 22 22 12.7 17.9 22 12.7 17.9 22

0.788 0.525 0.565 0.501 0.426 0.424 0.534 0.423 0.394

[5] [6] [7] [8] [9]

[10]

[11]

9. Conclusions The most relevant aspects of this work can be summarized as follows: (i) A new formulation for a jet characteristic velocity was introduced in Eq. (16). It was shown to be consistent with what has been used in the literature to match synthetic and steady jets. (ii) The three vortex identification criteria utilized, resulted in identical results for this flow due to simplifications inherent to the canonical geometry used. (iii) After the vortex pair arrives at the vicinity of the heated wall, it generates a secondary vortex with opposite circulation. This was identified as the main mechanism for heat transfer enhancement in the wall jet region. (iv) It has been observed in the literature that vortex coalescence reduces the heat transfer capacity of the jet. We show that the dominant reason for this is that merging decreases the number of secondary vortices sweeping the heated wall. (v) Three distinctive phases were identified in the vortex coalescence process. The time necessary for its completion was found to be directly proportional to the Reynolds number and inversely proportional to the frequency. (vi) The ratio between the time for merging and the time for a vortex to reach the wall was introduced, which may be utilized to find optimum conditions for operation of the synthetic jet and to improve its energy efficiency.

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19] [20] [21]

[22] [23] [24]

[25]

Acknowledgment [26]

This work was supported by the Villanova University James R. Birle endowment to the senior investigator. Some aspects of the investigation were also supported by CONICYT-Chile (82140056) under project CONICYT PAI/Concurso Nacional Apoyo al Retorno de Investigadores/as desde el Extranjero, Convocatoria 2014 Folio 82140056. References [1] Smith BL, Glezer A. The formation and evolution of synthetic jets. Phys Fluids 1998;10(9):2281e97. [2] Smith BL, Swift GW. A comparison between synthetic jets and continuous jets. Exp Fluids 2003;34(4):467e72. [3] Agrawal A, Verma G. Similarity analysis of planar and axisymmetric turbulent synthetic jets. Int J Heat Mass Transf 2008;51(25):6194e8. [4] Arik M. An investigation into feasibility of impingement heat transfer and

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