# Analytic Solution of Optimal Aspect Ratio of Bionic Transverse V-Groove for Drag Reduction Based on Vorticity Kinetics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Influence of Boundary Vortex Stability on Drag-Reduction Performance

## 3. Theoretical Solution of AR for Maintaining the Stability of Boundary Vortices

#### 3.1. Induced Velocity Induced by Image Vortices

#### 3.2. Migration Velocity Decomposed by Total Velocity

#### 3.3. Solution of Aspect Ratio Based on Equalization between Induced Velocity and Migration Velocity

$A\left(\frac{a}{2},0i\right)$ | ${A}^{\prime}\left(\frac{a}{2}+\Delta x\mathrm{cos}2\theta ,\Delta xi\mathrm{sin}2\theta \right)$ |

$B\left(-\frac{a}{2},0i\right)$ | ${B}^{\prime}\left(-\frac{a}{2}+\Delta x\mathrm{cos}2\theta ,-\Delta xi\mathrm{sin}2\theta \right)$ |

$C\left(0,bi\right)$ | ${C}^{\prime}\left(\mathsf{\Delta}x,bi\right)$ |

$A\left(\frac{a}{2},0i\right)$ | ${A}^{\prime}\left(\frac{a}{2}-\Delta x\mathrm{cos}2\theta ,-\Delta xi\mathrm{sin}2\theta \right)$ |

$B\left(-\frac{a}{2},0i\right)$ | ${B}^{\prime}\left(-\frac{a}{2}-\Delta x\mathrm{cos}2\theta ,\Delta xi\mathrm{sin}2\theta \right)$ |

$C\left(0,bi\right)$ | ${C}^{\prime}\left(-\mathsf{\Delta}x,bi\right)$ |

_{1}and K

_{2}are two empirical parameters equal to 0.2 and 0.1, respectively. Then, Equation (28) is derived, describing the maximum offset distance to maintain boundary vortex sloshing.

## 4. Numerical Verification

#### 4.1. Numerical Methodology

#### 4.2. Stability of Boundary Vortices and Drag-Reduction Rate of Transverse Grooves with Different ARs

## 5. Conclusions

- (1)
- The velocity potential of the groove sidewalls to the boundary vortex is described by an image vortex model, thus establishing the relationship between the AR and the induced velocity. Secondly, the velocity profile of the migration flow is obtained by decomposing the total velocity inside the groove, by which the relationship between the AR and the migration velocity is established. Finally, the analytical solution of the optimal AR ($A{R}_{opt}=2.15$) is obtained based on the kinetic conditions (i.e., the induced velocity is equal to the migration velocity) of the boundary vortex stability and the value of the critical ARs ($A{R}_{min}=0.75$ and $A{R}_{max}=6.15$) for which the boundary vortex can slosh inside the groove is obtained. Without considering the adverse pressure gradient and external disturbance, the motion forms of the boundary vortex inside the groove can be divided into three forms with the variation in the AR.
- (2)
- The theoretical model for solving the optimal AR ($A{R}_{opt}$) and critical ARs ($A{R}_{min}$ and $A{R}_{max}$) is validated by investigating the motion of the boundary vortices and the drag-reduction rate of the groove for ARs of 0.5, 1, 2, 5, and 8 with large eddy simulations. For AR = 2, the boundary vortex is stable inside the groove, corresponding to the maximum drag-reduction rate. When the AR is closer to 2, i.e., AR = 1 and AR = 5 (corresponds to the interval $A{R}_{min}<AR<A{R}_{opt}$ and $AR<A{R}_{opt}<A{R}_{max}$), the boundary vortices slosh periodically inside the groove and the magnitude of the vertical velocity fluctuations is similar in both cases. This periodic motion of the boundary vortex in the groove is classified as the “vortex sloshing” phenomenon. When the AR is far from 2, i.e., AR = 0.5 and AR = 8 (corresponds to the interval $AR<A{R}_{min}$ and $AR>A{R}_{max}$), the boundary vortices are shed from the shear layer at the leeward side and migrate downstream with the mainstream, which is classified as the “vortex shedding” phenomenon and corresponds to the minimum drag-reduction rate.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the slip surface over transverse grooves. (

**a**) Stable boundary vortex; (

**b**) unstable boundary vortex.

**Figure 2.**Influence of the stability of boundary vortex on drag-reduction performance. (

**a**) Stable boundary vortex; (

**b**) boundary vortex moves upstream; (

**c**) boundary vortex moves downstream.

**Figure 6.**Influence of geometric parameters on velocity distribution within the groove. (

**a**) Variation in velocity profiles induced by image vortices with AR at the groove centerline when $b$ is a constant value and h = 0.2 mm; (

**b**) variation in the virtual distance ${R}_{vd}$ representing the total induced velocity with b and AR at the center of the boundary vortex.

**Figure 10.**Kinematic analyses of boundary vortices when AR < 2.15. (

**a**) Coordinates of boundary vortex and image vortices; (

**b**) moving path of boundary vortex.

**Figure 11.**Kinematic analyses of boundary vortices when AR > 2.15. (

**a**) Coordinates of boundary vortex and image vortices; (

**b**) moving path of boundary vortex.

**Figure 17.**The 3D isometric of the instantaneous coherent structures with indicator ${\lambda}_{ci}$. (

**a**) Identification of the random perturbation vortices in the computational domain with ${\lambda}_{ci}>80$, and (

**b**) identification of boundary vortices inside the groove with ${\lambda}_{ci}>540$.

**Figure 18.**The vorticity contours of boundary vortices within grooves with (

**a**) AR = 0.5, (

**b**) AR = 8 at $Re=5.44\times {10}^{4}$.

**Figure 19.**The vorticity contours of boundary vortices within grooves with (

**a**) AR = 1, (

**b**) AR = 5 at $Re=5.44\times {10}^{4}$.

**Figure 20.**The vorticity contours of boundary vortices within grooves with AR = 2 at $Re=5.44\times {10}^{4}$.

**Figure 21.**Time histories of the dimensionless vertical velocities at intersection point of groove centerline and horizontal line; (

**a**) $Re=5.44\times {10}^{4}$ and (

**b**) $Re=9.8\times {10}^{4}$.

**Figure 22.**Variance of dimensionless vertical velocity fluctuations at the intersection point of groove centerline and horizontal line.

**Figure 23.**Drag-reduction rate varies with AR. (

**a**) Total drag-reduction rate; (

**b**) reduction rate of viscous drag; (

**c**) increased rate of pressure drag.

Dimensionless Parameters | Nodes | ||
---|---|---|---|

${L}_{x}^{+}$ | 2490 + 311 + 311 | 300 + 1000 + 60 | |

${L}_{y}^{+}$ | 466 | 80 | |

${L}_{z}^{+}$ | 311 | 80 | |

$\Delta {x}^{+}$ | Groove | 0.3 | 1000 |

Other | <10 | 300 + 60 | |

$\Delta {y}^{+}$ | 0.02~10 | 80 | |

$\Delta {z}^{+}$ | 3.9 | 80 |

$\mathit{\Delta}{\mathit{x}}^{+}\left(\mathbf{Groove}\right)$ | $\mathit{\Delta}{\mathit{z}}^{+}$ | Drag (N) | Streamline |
---|---|---|---|

0.3 | 3.9 | 0.00314 | |

5.2 | 0.00313 | ||

1.2 | 3.9 | 0.00314 | |

5.2 | 0.00315 |

Variables | Value |
---|---|

${N}_{1}$ (coarse) | 2,404,420 |

${N}_{2}$ (medium) | 4,123,710 |

${N}_{3}$ (fine) | 8,703,310 |

${r}_{12}$ | 1.19 |

${r}_{23}$ | 1.28 |

p | $\approx $5.77 |

${g}_{p}^{12}$ | 0.0860% |

${g}_{p}^{23}$ | 0.0159% |

$GC{I}^{12}$ | 0.0795% |

$GC{I}^{23}$ | 0.026% |

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**MDPI and ACS Style**

Li, Z.; He, L.; Zuo, Y.; Meng, B.
Analytic Solution of Optimal Aspect Ratio of Bionic Transverse V-Groove for Drag Reduction Based on Vorticity Kinetics. *Aerospace* **2022**, *9*, 749.
https://doi.org/10.3390/aerospace9120749

**AMA Style**

Li Z, He L, Zuo Y, Meng B.
Analytic Solution of Optimal Aspect Ratio of Bionic Transverse V-Groove for Drag Reduction Based on Vorticity Kinetics. *Aerospace*. 2022; 9(12):749.
https://doi.org/10.3390/aerospace9120749

**Chicago/Turabian Style**

Li, Zhiping, Long He, Yueren Zuo, and Bo Meng.
2022. "Analytic Solution of Optimal Aspect Ratio of Bionic Transverse V-Groove for Drag Reduction Based on Vorticity Kinetics" *Aerospace* 9, no. 12: 749.
https://doi.org/10.3390/aerospace9120749