# Numerical Characterisation of Active Drag and Lift Control for a Circular Cylinder in Cross-Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

## 3. Results and Discussion

^{2}= 0.908 in Figure 8b. The higher the actuation frequency, the more the bottom separation point moves downstream, towards the rear of the cylinder (Figure 8b).

^{2}= 0.931). As ${\theta}_{\mathsf{\Delta}sep}$ increases, the drag force decreases (Figure 10b). This contrasts the known relationship between the separation point and drag for aerofoils, where delaying separation results in a drag reduction [10,15,16,17,18].

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

${C}_{D}$ | Coefficient of drag | |

${C}_{L}$ | Coefficient of lift | |

${C}_{p}$ | Coefficient of pressure | |

${C}_{{p}_{b}}$ | Coefficient of pressure at rear of cylinder | |

$D$ | Cylinder diameter | m |

${d}_{o}$ | Synthetic jet orifice width | m |

${f}_{a}$ | Synthetic jet actuation frequency | Hz |

${F}^{+}$ | Dimensionless actuation frequency | |

I | Turbulence intensity | |

k | Turbulent kinetic energy | m^{2}/s^{2} |

L | Characteristic length scale | m |

P | Pressure | Pa |

Re | Reynolds number of the freestream flow | |

$R{e}_{{\overline{U}}_{o}}$ | Reynolds number of the synthetic jet | |

$Sr$ | Strouhal number of vortex shedding | |

$S{r}_{o}$ | Strouhal number of the synthetic jet | |

$S{t}_{o}$ | Stokes number of the synthetic jet | |

${U}_{dia}$ | Pseudo diaphragm velocity | m/s |

${\overline{U}}_{o}$ | Averaged velocity at the orifice | m/s |

${U}_{p}$ | Peak velocity at the orifice | m/s |

${U}_{\infty}$ | Freestream velocity | m/s |

${u}_{o}$ | Velocity at centreline of jet orifice | m/s |

x, y | Spatial coordinates | m |

Greek symbols and abbreviations | ||

$\theta $ | Circumferential coordinate | (rad) |

v | Kinematic viscosity | m^{2}/s |

τ | Wall shear stress | Pa |

$\omega $ | Angular velocity | rad/s |

CFL | Courant-Friedrichs-Lewy number | |

RMS | Root-mean-squared | |

SST | Shear stress transport | |

URANS | Unsteady Reynolds-averaged Navier-Stokes |

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**Figure 2.**Schematic diagram of the circular cylinder with internal synthetic jet cavity and pseudo diaphragm, represented by an inlet velocity boundary condition.

**Figure 3.**Computational grid for (

**a**) the full domain and (

**b**) a close-up view of the vicinity of the cylinder and synthetic jet actuator cavity and orifice.

**Figure 4.**Vorticity plots for formation criteria set out by Holman et al. [37] for a synthetic jet through two-dimensional simulation in quiescent flow at a fixed jet Stokes number $S{t}_{o}$ = 15.8 and Reynolds number $R{e}_{{\overline{U}}_{o}}$ = 479 ($R{e}_{{\overline{U}}_{o}}/S{t}_{o}^{2}$ = 1.92), during the (

**a**) blowing phase and (

**b**) suction phase.

**Figure 5.**Modification in drag coefficient ${C}_{D}$ as a function of the synthetic jet Stokes number $S{t}_{o}$ for (- -o- -) Catalano et al. [25] and (‒o‒) the present study.

**Figure 6.**Modification of (- -Δ- -) RMS lift coefficient ${C}_{L}^{\prime}$ and (‒o‒) drag coefficient ${C}_{D}$ as a function of the synthetic jet Stokes number $S{t}_{o}$.

**Figure 7.**Schematic diagram of the near-field locations of interest on the cylinder surface: (i) stagnation point at the front of the cylinder ($\theta $ = 0), (ii) top and bottom separation points ${\theta}_{se{p}_{1}}$ and ${\theta}_{se{p}_{2}}$ and (iii) the angle between both separation points ${\theta}_{\mathsf{\Delta}sep}$.

**Figure 8.**Comparison of the modification of the separation points as a function of actuation frequency: (

**a**) Top separation point location ${\theta}_{se{p}_{1}}$; (

**b**) Bottom separation point location ${\theta}_{se{p}_{2}}$.

**Figure 9.**Correlation between the separation point locations and the drag force, for the range of actuation frequencies: (

**a**) Top of the cylinder ${\theta}_{se{p}_{1}}$; (

**b**) Bottom of the cylinder ${\theta}_{se{p}_{2}}$.

**Figure 10.**Comparison of the magnitude of the separation angle ${\theta}_{\mathsf{\Delta}sep}$ (i.e., angle between top and bottom separation points) and the drag force, for the range of actuation frequencies: (

**a**) Change in ${\theta}_{\mathsf{\Delta}sep}$ with $S{t}_{o}$; (

**b**) Change in drag force ${F}_{x}$ with ${\theta}_{\mathsf{\Delta}sep}$.

**Figure 11.**Change in drag force ${F}_{x}$ and minimum pressure over the rear of the cylinder as a function of $S{t}_{o}$: (

**a**) Change in pressure and drag with $S{t}_{o}$; (

**b**) Change in drag with pressure.

**Figure 12.**Effect of time and spatially averaged turbulence intensity at the rear of the cylinder on ${C}_{D}$ as a function of $S{t}_{o}$ : (

**a**) Change in turbulence intensity and ${C}_{D}$ with $S{t}_{o}$; (

**b**) Change in ${C}_{D}$ with turbulence intensity.

**Figure 13.**Alteration of pressure gradient across the jet orifice $\frac{dP}{d\theta}$ and the corresponding near-wake turbulence intensity $I$ as a function of $S{t}_{o}$ : (

**a**) Change in turbulence intensity and $\frac{dP}{d\theta}$ with $S{t}_{o}$; (

**b**) Change in turbulence intensity with $\frac{dP}{d\theta}$.

**Figure 14.**Change in the surface pressure coefficient ${C}_{p}$ distribution: (

**a**) $S{t}_{o}$ = 2.43, $I=9.4\%$ and ${C}_{D}$ = 1.62; (

**b**) $S{t}_{o}$ = 4.86, $I=10.2\%$ and ${C}_{D}$ = 1.53.

**Figure 15.**Time and spatially averaged turbulence intensity $I$ in the lee of the cylinder, for a range of synthetic jet actuation frequencies, with corresponding Stokes number values of (‒‒o‒‒) $S{t}_{o}=0$, (‒‒x‒‒) $2.43$, (‒‒Δ‒‒) $3.44$, (‒‒□‒‒) $4.21$, (- - o - -) $4.68$, (- - x - -) $5.43$, (- - Δ - -) $5.95$, (- - □ - -) $6.43$.

**Table 1.**Effect of mesh density on characteristic dimensionless numbers averaged over the quasi-steady portion of flow time after 20 seconds of simulated flow time, for a cylinder in cross-flow at 𝑅𝑒 = 3900. Percentage values in brackets represent deviations for all results to values reported by Catalano et al. [25] and Beaudan and Moin [35].

Mesh Size | Drag Coefficient ${\mathit{C}}_{\mathit{D}}$ | RMS Lift Coefficient ${\mathit{C}}_{\mathit{L}}\prime $ | Vortex Shedding $\mathit{S}\mathit{r}$ | Calculation Time (Approx.) |
---|---|---|---|---|

66,000 | 1.63 (error = −6%) | 1.22 (error = −14%) | 0.36 (error = +37%) | 24 hours |

113,000 | 1.66 (error = −5%) | 1.28 (error = −10%) | 0.32 (error = +22%) | 48 hours |

253,000 | 1.67 (error = −4%) | 1.33 (error = −6%) | 0.29 (error = +10%) | 96 hours |

512,000 | 1.69 (error = −3%) | 1.38 (error = −3%) | 0.27 (error = +3%) | 192 hours |

Literature [25,35] | 1.74 | 1.42 | 0.26 |

Drag Coefficient ${\mathit{C}}_{\mathit{D}}$ | RMS Lift Coefficient ${\mathit{C}}_{\mathit{L}}$’ | Vortex Shedding $\mathit{S}\mathit{r}$ | Rear Surface Pressure Coefficient ${\mathit{C}}_{{\mathit{P}}_{\mathit{b}}}$ | Flow Separation Locations ${\mathit{\theta}}_{\mathit{s}\mathit{e}{\mathit{p}}_{1,2}}$ | |
---|---|---|---|---|---|

Other published results [25,35] | 1.74 | 1.42 | 0.26 | −2.16 | 72° and 288° [25] 75° and 285° [35] |

Present study | 1.67 | 1.33 | 0.29 | −2.07 | 77° and 283° |

Deviation | (−4%) | (−6%) | (+10%) | (+4%) | ($\pm $3–6%) |

**Table 3.**Operating parameters of the synthetic jet actuator in the present study, in terms of the Reynolds number $R{e}_{{\overline{U}}_{o}}$, actuation frequency ${f}_{a}$, Stokes number $S{t}_{o}$ and Strouhal number $S{r}_{o}$.

$\mathit{R}{\mathit{e}}_{{\overline{\mathit{U}}}_{\mathit{o}}}$ | ${\mathit{f}}_{\mathit{a}}$ | $\mathit{S}{\mathit{t}}_{\mathit{o}}$ | $\mathit{R}{\mathit{e}}_{{\overline{\mathit{U}}}_{\mathit{o}}}/\mathit{S}{\mathit{t}}_{\mathit{o}}^{2}$ | $\mathit{S}{\mathit{r}}_{\mathit{o}}$ |
---|---|---|---|---|

12 | 0.9 Hz | 2.43 | 2.03 | 0.49 |

12 | 1.9 Hz | 3.44 | 1.01 | 0.99 |

12 | 2.8 Hz | 4.21 | 0.68 | 1.48 |

12 | 3.8 Hz | 4.86 | 0.51 | 1.97 |

12 | 4.7 Hz | 5.43 | 0.41 | 2.46 |

12 | 5.7 Hz | 5.95 | 0.34 | 2.96 |

12 | 6.6 Hz | 6.43 | 0.29 | 3.45 |

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**MDPI and ACS Style**

McDonald, P.; Persoons, T.
Numerical Characterisation of Active Drag and Lift Control for a Circular Cylinder in Cross-Flow. *Appl. Sci.* **2017**, *7*, 1166.
https://doi.org/10.3390/app7111166

**AMA Style**

McDonald P, Persoons T.
Numerical Characterisation of Active Drag and Lift Control for a Circular Cylinder in Cross-Flow. *Applied Sciences*. 2017; 7(11):1166.
https://doi.org/10.3390/app7111166

**Chicago/Turabian Style**

McDonald, Philip, and Tim Persoons.
2017. "Numerical Characterisation of Active Drag and Lift Control for a Circular Cylinder in Cross-Flow" *Applied Sciences* 7, no. 11: 1166.
https://doi.org/10.3390/app7111166