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

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## 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

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**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) | - |

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**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