# Validating a Reduced-Order Model for Synthetic Jet Actuators Using CFD and Experimental Data

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Reduced-Order Model (ROM)

#### 2.1. Brief Description of the ROM

#### 2.2. Estimating ROM Parameters $K$ and $\beta $

## 3. Validation Methodologies

#### 3.1. Numerical Validation Using Transient Computational Fluid Dynamics (CFD) Modelling

#### 3.1.1. Complementary 3D Simulations

#### 3.2. Experimental Validation

^{3}and ${A}_{d}=44$ cm

^{2}, corresponding to a 75 mm circular diaphragm representing a Visaton FR8 (4 Ohm, 10 W) loudspeaker [16].

#### 3.3. Comparing Synthetic Jet Actuator Performance for 2D and 3D Geometries

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$A$ | Cross-sectional area, m^{2} |

$a$ | Orifice aspect ratio ($a=b/h$) |

$Bl$ | Electromagnetic force factor (${F}_{d}=Bl\cdot i$) |

$b$ | Spanwise length of slot orifice, m |

$C$ | Linear damping coefficient, N/(m*s) or capacitance, F |

$c$ | Speed of sound, m/s |

${d}_{h}$ | Hydraulic diameter, m |

$e$ | Actuator voltage, V |

${F}^{+}$ | Dimensionless frequency ($fL/{U}_{\infty}$) |

${F}_{d}$ | Actuator driving force, N |

$f$ | Actuation frequency, Hz |

$h$ | Orifice slot width, m |

$i$ | Actuator current, A |

$j$ | Imaginary unit ($j=\sqrt{-1})$ |

$K$ | Nozzle non-linear damping coefficient or stiffness, N/m |

$L$ | Length, m or inductance, H |

${L}_{0}$ | Synthetic jet stroke length, m |

$M$ | Mass, kg |

${p}_{c}$ | Cavity acoustic pressure, Pa |

$R$ | Resistance, Ohm |

$Re$ | Reynolds number ($Re={U}_{0}{d}_{h}/\nu $) |

$s$ | Laplace variable ($s=j2\pi f$), s^{−1} |

$t$ | Time, s |

$U$ | Velocity, m/s |

${U}_{0}$ | Characteristic velocity (${U}_{0}={L}_{0}f$), m/s |

$V$ | Volume, m^{3} |

$x$ | Displacement, m |

$Z$ | Impedance, Ω |

Subscripts | |

0 | Characteristic synthetic jet scale |

c | Actuator cavity |

d | Actuator diaphragm |

e | Electrical (referring to the actuator driver) |

emf | Electromotive force |

H | Helmholtz resonance |

n | Actuator nozzle or orifice |

p | Parallel |

s | Series |

$\infty $ | Free stream |

Greek symbols | |

$\alpha $ | Electromechanical coupling coefficient, N/V |

$\beta $ | Nozzle added mass coefficient |

$\rho $ | Density, kg/m^{3} |

$\nu $ | Kinematic viscosity, m^{2}/s |

Abbreviations | |

CFD | Computational fluid dynamics |

ROM | Reduced-order model |

SJA | Synthetic jet actuator |

## References

- Smith, B.L.; Glezer, A. The formation and evolution of synthetic jets. Phys. Fluids
**1998**, 10, 2281–2297. [Google Scholar] [CrossRef] - Glezer, A.; Amitay, M. Synthetic jets. Annu. Rev. Fluid Mech.
**2002**, 34, 503–529. [Google Scholar] [CrossRef] - Holman, R.; Utturkar, Y.; Mittal, R.; Smith, B.L.; Cattafesta, L. Formation criterion for synthetic jets. AIAA J.
**2005**, 43, 2110–2116. [Google Scholar] [CrossRef] - McGuinn, A.; Farrelly, R.; Persoons, T.; Murray, D.B. Flow regime characterisation of an impinging axisymmetric synthetic jet. Exp. Therm. Fluid Sci.
**2013**, 47, 241–251. [Google Scholar] [CrossRef][Green Version] - You, D.; Moin, P. Active control of flow separation over an airfoil using synthetic jets. J. Fluids Struct.
**2008**, 24, 1349–1357. [Google Scholar] [CrossRef] - Dandois, J.; Garnier, E.; Sagaut, P. Numerical simulation of active separation control by a synthetic jet. J. Fluid Mech.
**2007**, 574, 25–58. [Google Scholar] [CrossRef] - Persoons, T.; McGuinn, A.; Murray, D.B. A general correlation for the stagnation point Nusselt number of an axisymmetric impinging synthetic jet. Int. J. Heat Mass Transf.
**2011**, 54, 3900–3908. [Google Scholar] [CrossRef][Green Version] - Fanning, E.; Persoons, T.; Murray, D.B. Heat transfer and flow characteristics of a pair of adjacent impinging synthetic jets. Int. J. Heat Fluid Flow
**2015**, 54, 153–166. [Google Scholar] [CrossRef] - Smith, B.L.; Glezer, A. Jet vectoring using synthetic jets. J. Fluid Mech.
**2002**, 458, 1–34. [Google Scholar] [CrossRef][Green Version] - Smith, B.L.; Glezer, A. Vectoring of adjacent synthetic jets. AIAA J.
**2005**, 43, 2117–2124. [Google Scholar] [CrossRef] - Persoons, T.; O’Donovan, T.S. A pressure-based estimate of synthetic jet velocity. Phys. Fluids
**2007**, 19. [Google Scholar] [CrossRef][Green Version] - Shuster, J.M.; Smith, D.R. Experimental study of the formation and scaling of a round synthetic jet. Phys. Fluids
**2007**, 19. [Google Scholar] [CrossRef] - Smith, B.L.; Swift, G.W. A comparison between synthetic jets and continuous jets. Exp. Fluids
**2003**, 34, 467–472. [Google Scholar] [CrossRef] - Crittenden, T.M.; Glezer, A. A high-speed, compressible synthetic jet. Phys. Fluids
**2006**, 18. [Google Scholar] [CrossRef] - Kordík, J.; Trávníček, Z. Novel nozzle shapes for synthetic jet actuators intended to enhance jet momentum flux. Actuators
**2018**, 7. [Google Scholar] [CrossRef] - Persoons, T. General reduced-order model to design and operate synthetic jet actuators. AIAA J.
**2012**, 50, 916–927. [Google Scholar] [CrossRef] - McCormick, D. Boundary layer separation control with directed synthetic jets. In Proceedings of the 38th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 2000. [Google Scholar]
- Gallas, Q.; Holman, R.; Nishida, T.; Carroll, B.; Sheplak, M.; Cattafesta, L. Lumped element modeling of piezoelectric-driven synthetic jet actuators. AIAA J.
**2003**, 41, 240–247. [Google Scholar] [CrossRef] - Kordík, J.; Trávníček, Z.; Safarik, P. Experiments on resonance frequencies of synthetic jet actuators. J. Flow Vis. Image Proc.
**2010**, 17, 203–214. [Google Scholar] [CrossRef] - Chiatto, M.; Capuano, F.; Coppola, G.; Luca, L.D. LEM characterization of synthetic jet actuators driven by piezoelectric element: A review. Sensors
**2017**, 17, 1216. [Google Scholar] [CrossRef] [PubMed] - Kordík, J.; Trávníček, Z. Optimal diameter of nozzles of synthetic jet actuators based on electrodynamic transducers. Exp. Therm. Fluid Sci.
**2017**, 86, 281–294. [Google Scholar] [CrossRef] - Beranek, L.L. Acoustics; McGraw-Hill: New York, NY, USA, 1954; pp. 116–128. [Google Scholar]
- Burnett, D.S.; Soroka, W.W. Tables of rectangular piston radiation impedance functions, with application to sound transmission loss through deep apertures. J. Acoust. Soc. Am.
**1972**, 51, 1618–1623. [Google Scholar] [CrossRef] - Kooijman, G.; Ouweltjes, O. Finite difference time domain electroacoustic model for synthetic jet actuators including nonlinear flow resistance. J. Acoust. Soc. Am.
**2009**, 125, 1911–1918. [Google Scholar] [CrossRef] [PubMed] - Alimohammad, S.; Fanning, E.; Persoons, T.; Murray, D.B. Characterization of flow vectoring phenomenon in adjacent synthetic jets using CFD and PIV. Comput. Fluids
**2016**, 140, 232–246. [Google Scholar] [CrossRef] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng.
**2008**, 130. [Google Scholar] [CrossRef] - Persoons, T.; Saenen, T.; Oevelen, T.V.; Baelmans, M. Effect of flow pulsation on the heat transfer performance of a minichannel heat sink. J. Heat Transf.
**2012**, 134. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic diagram and equivalent electrical networks of a reduced-order model of a synthetic jet actuator (SJA) with (

**b**) piezoelectric and (

**c**) electromagnetic driver: from driving voltage $e$ to diaphragm deflection ${x}_{d}$, cavity pressure ${p}_{c}$ and nozzle velocity ${U}_{n}$ [16].

**Figure 2.**2D computational domain and mesh (the diaphragm deformation is exaggerated and only indicative. The full solution domain is not shown for the sake of clarity).

**Figure 3.**(

**a**) Schematic diagram of an instrumented SJA with (

**b**) rectangular orifice (not to scale), instrumented with (i) a hot-wire anemometer probe for nozzle velocity ${U}_{n}$, (ii) microphone for cavity pressure ${p}_{c}$ and (iii) laser displacement sensor for ${x}_{d}$ [16].

**Figure 4.**ROM validation in terms of the fluidic model ${U}_{n}/{p}_{c}$ (Equation (11) with $K=1.552$ and $\beta =0.615$) (lines) versus experimental data (markers) for a synthetic jet actuator with rectangular orifice of aspect ratio 30:1. Markers represent cavity pressure magnitudes (⚪) 100 Pa, (☐) 200 Pa, (◇) 500 Pa, (△) 1000 Pa. Cases A and B (see Table 2) are indicated with red and blue markers, respectively.

**Figure 5.**(

**a**) Diaphragm deflection ${x}_{d}\left(\omega t\right)$, (

**b**) cavity pressure ${p}_{c}\left(\omega t\right)$ and (

**c**) nozzle velocity ${U}_{n}\left(\omega t\right)$ as a function of phase angle $\omega t$ for case A (see Table 2), comparing experimental results (─), 2D CFD results (⚪), and 3D CFD results (☐).

**Figure 6.**(

**a**) Diaphragm deflection ${x}_{d}\left(\omega t\right)$, (

**b**) cavity pressure ${p}_{c}\left(\omega t\right)$ and (

**c**) nozzle velocity ${U}_{n}\left(\omega t\right)$ as a function of phase angle $\omega t$ for case B (see Table 2), comparing experimental results (─), 2D CFD results (⚪), and 3D CFD results (☐).

**Figure 7.**ROM validation in terms of the fluidic model ${U}_{n}/{p}_{c}$ given by Equation (11) (horizontal axis) versus (⚪) 2D CFD results, (☐) 3D CFD results and (◇) experimental results, for the same cases included in Figure 4 and Table 2. Cases A and B are indicated with red and blue markers, respectively.

**Table 1.**List of experimental synthetic jet investigations with details of the type of orifice and actuator used.

Study | Orifice | Fluid | Actuator Type | $\text{}{\mathit{L}}_{0}/{\mathit{d}}_{\mathit{h}}\text{}$ | $\text{}\mathit{R}\mathit{e}\text{}$ | ${\mathit{U}}_{0},\text{}\mathbf{m}/\mathbf{s}$ | $\mathit{f}$, HZ |
---|---|---|---|---|---|---|---|

Smith and Glezer [10] | Slot ($h=0.5$ mm, $a=147$) | Air | Piezoelectric disk | 14.6 | 596 | 8.5 | 577 |

Shuster and Smith [12] | Circular (${d}_{h}=25.4$ mm) | Water | Oscillating piston | 1.0 3.0 | 1000–10,000 | 0.04–0.39 | 1.6–5.2 |

Smith and Swift [13] | Slot ($h=5.1$ mm, $a=47$) | Air | Loudspeakers | 6.9–41 | 1360–28,790 | 2.0–41.8 | 29–102 |

Crittenden and Glezer [14] | Circular (${d}_{h}=$ 1.6 mm–4.8 mm) | Air | Oscillating piston | >76 | 989–35,830 | 9.0–108 | 25–200 |

Kordík and Trávníček [15] | Circular (${d}_{h}=10$ mm) | Air | Loudspeaker | 6.3–15.8 | 2400–7100 | 3.5–10.3 | 55–65 |

Current study | Slot ($h=1.5$ mm, $a=30$) | Air | Loudspeaker | 0.5–40 | 80–1836 | 0.2–9.5 | 41–164 |

**Table 2.**List of cases included in this investigation. Cases A and B are indicated respectively as red and blue markers in subsequent figures. Values in parenthesis are reduced-order model (ROM) predictions at the operating conditions.

Case | Frequency $\frac{\mathit{\omega}}{{\mathit{\omega}}_{\mathit{H}}}\left(=\frac{\mathit{f}}{{\mathit{f}}_{\mathit{H}}}\right)\text{}$ | Pressure Amplitude ${\mathit{p}}_{\mathit{c}}$, Pa | Velocity-to -Pressure Ratio $\left(\frac{\mathit{\rho}\mathit{c}{\mathit{U}}_{\mathit{n}}}{{\mathit{p}}_{\mathit{c}}}\right)\sqrt{\frac{{\mathit{A}}_{\mathit{n}}{\mathit{L}}_{\mathit{n}}^{\prime}}{{\mathit{V}}_{\mathit{c}}}}$, in dB | ||
---|---|---|---|---|---|

Experimental (${\mathit{f}}_{\mathit{H}}^{\left(\mathit{e}\mathit{x}\mathit{p}\right)}$ = 307 HZ) | 2D Computational Fluid Dynamics (CFD) (${\mathit{f}}_{\mathit{H}}^{\left(2\mathit{D}\right)}$ = 509 HZ) | 3D CFD (${\mathit{f}}_{\mathit{H}}^{\left(3\mathit{D}\right)}$ = 332 HZ) | |||

A | 0.13 | 106 | 13.1 (13.0) | 15.0 (15.1) | 11.8 (13.0) |

B | 0.53 | 202 | 4.8 (5.1) | 6.5 (9.1) | 6.1 (5.5) |

C | 0.13 | 208 | 10.9 (10.7) | 12.0 (12.3) | - |

D | 0.27 | 509 | 9.6 (9.1) | 13.5 (12.5) | - |

E | 0.27 | 1027 | 6.8 (6.3) | 8.7 (8.5) | - |

F | 0.27 | 202 | 3.5 (3.7) | 6.5 (6.2) | - |

G | 0.53 | 504 | 4.6 (4.2) | 5.5 (6.8) | - |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Persoons, T.; Cressall, R.; Alimohammadi, S.
Validating a Reduced-Order Model for Synthetic Jet Actuators Using CFD and Experimental Data. *Actuators* **2018**, *7*, 67.
https://doi.org/10.3390/act7040067

**AMA Style**

Persoons T, Cressall R, Alimohammadi S.
Validating a Reduced-Order Model for Synthetic Jet Actuators Using CFD and Experimental Data. *Actuators*. 2018; 7(4):67.
https://doi.org/10.3390/act7040067

**Chicago/Turabian Style**

Persoons, Tim, Rick Cressall, and Sajad Alimohammadi.
2018. "Validating a Reduced-Order Model for Synthetic Jet Actuators Using CFD and Experimental Data" *Actuators* 7, no. 4: 67.
https://doi.org/10.3390/act7040067