# Unsteady Simulation of Transonic Buffet of a Supercritical Airfoil with Shock Control Bump

^{*}

## Abstract

**:**

_{B}/c = 0.55 demonstrated a good buffet control effect. The lift-to-drag ratio of the buffet case was increased by 5.9%, and the root mean square of the lift coefficient fluctuation was decreased by 67.6%. Detailed time-averaged flow quantities and instantaneous flow fields were analyzed to demonstrate the flow phenomenon of the shock control bumps. The results demonstrate that an appropriate “λ” shockwave pattern caused by the bump is important for the flow control effect.

## 1. Introduction

## 2. Numerical Method and Validation

#### 2.1. Numerical Method

^{6}. Consequently, an equilibrium stress-balanced equation was solved to model the energy-containing eddies in the inner turbulent boundary layer [44,45]. The wall-modeled equation was solved on an additional one-dimensional grid with 50 grid points to obtain the velocity profile of the inner layer of the boundary layer. The shear stress provided by the wall-modeled equation was applied as the boundary condition of the Navier–Stokes equation. The velocity of the upper boundary of the wall-modeled equation was interpolated from the third layer of the LES grid. The numerical method is not the focus of this paper. Consequently, detailed numerical schemes and formulas are not presented here. Readers can refer to our previous work on high-Reynolds separated flows [44,45].

#### 2.2. Computational Grid

^{−4}c. The increasing ratio in the normal direction was less than 1.1. The first grid layer can be located in the logarithmic layer of the turbulent boundary layer because the wall-modeled equation can provide the shear stress boundary condition of the wall. When the SCB was installed on the airfoil surface, the wall surface was deformed to simulate the effect of the bump while keeping the grid topology and grid number the same.

#### 2.3. Validation of the Baseline OAT15A Airfoil

^{6}. [14] The nondimensional time step was ΔtU

_{∞}/c = 1.37 × 10

^{−4}. Approximately 180 time units (tU

_{∞}/c) were computed for each case, and the last 100 time units were applied for flow field statistics. It cost approximately 2.0 × 10

^{5}core hours for each flow case on an Intel cluster with a 2.0 GHz CPU.

^{+}and Δy

^{+}of the first grid layer were collected to demonstrate the grid resolution of the present computation. Figure 2a shows the Δx

^{+}and Δy

^{+}of the LES grid collected by an angle of attack α = 2.5°. Δy

^{+}was approximately 50 on the upper surface and 80 on the lower surface, while Δx

^{+}was approximately twice the Δy

^{+}. The first grid point in the wall normal direction was located at the logarithmic layer of a turbulent boundary layer, which satisfies the requirement for a WMLES. The Δy

^{+}of the first grid layer of the wall-modeled equation (Δy

^{+}

_{wm}) was always less than 1.0, as shown in Figure 2b. This is sufficient to describe the viscous sublayer of the turbulent boundary layer. The spanwise correlation [43] of the pressure fluctuation was computed to test the spanwise domain. As shown in Figure 3, the correlations of both x/c = 0.4 and 0.8 damp fast in the spanwise direction, which demonstrates that the present spanwise domain size (0.075 c) is adequate.

^{6}, which is higher than the present computation. Although the flow condition was slightly different from the present computation, the time-averaged lift coefficients of the present computation were located between the two series of experimental data.

## 3. Numerical Study of Shock Control Bump

#### 3.1. Shape of the Shock Control Bump

_{B}) is superposed on the baseline airfoil. h/c is the relative height of the bump, x

_{B}/c is the central location and l

_{B}is the bump length. Figure 10 shows the geometries of bumps with different parameters. Four bumps are computed in this paper, as shown in Figure 10a. The lengths of the bumps are 0.2c. The first three bumps are all located at x

_{B}/c = 0.55, which is the shockwave location of the baseline OAT15A airfoil. The relative heights h/c of the first three bumps are 0.004, 0.008 and 0.012. The fourth bump is located slightly upstream x

_{B}/c = 0.50 and has the same height as Bump 2. Figure 10b shows the geometries of the airfoil superposed with the bumps.

#### 3.2. Time-Averaged Characteristics of the Airfoil with Bumps

_{∞}/c = 1.37 × 10

^{−4}. The flow field of the baseline airfoil is used as the initial condition of the bump cases. More than 130 time units (tU

_{∞}/c) are computed for each case. The last 80 time units are used for time averaging to obtain the aerodynamic coefficients. Six cases were computed, which are α = 3.5° for all four bumps and α = 2.5° for Bump 2 and Bump 4.

_{f}of the baseline airfoil is plotted in the figure. The wall friction coefficient of the upper surface is strongly affected by the bumps. The friction first declines when approaching the bump and then increases on the bump, which coincides with the tendency of the flow acceleration on the bump (Figure 12).

#### 3.3. Fluctuation Characteristics of the Airfoil with Bumps

_{∞}are plotted along the vertical direction in each figure. It is clear that the high-speed region of the baseline airfoil (Figure 19a) oscillates periodically. When Bump 1 is installed on the upper surface (Figure 19b), the oscillation is weakened, but periodical oscillation still exists. The high-speed regions of Bumps 2, 3 and 4 are almost constant over time. However, strong flow acceleration regions appear at the bump locations of Bumps 3 and 4. Moreover, an instantaneous reverse flow (u < 0.0) can be seen before Bumps 3 and 4 at 0.4 < x/c < 0.5, which means that an inappropriate set of SCBs may induce an unexpectedly low-speed region before the bump. Although Bump 2 has a good control effect for the shock oscillation, an instantaneous reverse flow (u < 0.0) region also exists after the bump, which coincides with the mean flow contour shown in Figure 12b.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Δx

^{+}and Δy

^{+}of the first grid layer and the wall-modeled grid (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 2.5°).

**Figure 5.**Comparison of the computed lift coefficient with experimental data [48].

**Figure 6.**Comparison of pressure coefficients at different angles of attack (Ma = 0.73, Re = 3.0 × 10

^{6}).

**Figure 8.**Comparison of streamwise velocity fluctuation (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°). (

**a**) Experimental data (taken from [49]), (

**b**) present computation.

**Figure 13.**Time-averaged pressure coefficients for the airfoil with bumps (Ma = 0.73, Re = 3.0 × 10

^{6}).

**Figure 14.**Wall friction coefficient of the baseline airfoil and airfoil with bumps (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 15.**Velocity profile and Reynolds stresses of the airfoil with bumps (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 16.**RMS of the pressure fluctuation on the upper surface (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 17.**RMS contours of the pressure fluctuations for Bump 2 and Bump 4 (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 18.**Power spectral density of the pressure fluctuations at three locations (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 19.**Instantaneous streamwise velocity of the first LES grid layer near the wall (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 20.**Instantaneous streamwise velocity contours of the airfoil (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

**Figure 21.**Instantaneous Kutta wave propagating from the trailing edge (Ma = 0.73, Re = 3.0 × 10

^{6}, α = 3.5°).

Configuration | Angle of Attack | Lift Coefficient | RMS of Lift Coefficient | Drag Coefficient | RMS of Drag Coefficient | Lift-to-Drag Ratio | Pitching Moment |
---|---|---|---|---|---|---|---|

Baseline airfoil | 2.5° | 0.919 | 0.006884 | 0.03009 | 0.000724 | 30.54 | −0.1335 |

3.5° | 0.968 | 0.037572 | 0.04823 | 0.003737 | 20.07 | −0.1262 | |

Bump 1 (h/c = 0.004, x _{B}/c = 0.55) | 3.5° | 0.965 (−0.3%) | 0.027828 (−25.9%) | 0.04634 (−3.9%) | 0.003107 (−16.8%) | 20.82 (+3.7%) | −0.1252 (+0.8%) |

Bump 2 (h/c = 0.008, x _{B}/c = 0.55) | 2.5° | 0.866 (−5.8%) | 0.004527 (−34.2%) | 0.03174 (+5.5%) | 0.000662 (−8.6%) | 27.28 (−10.6%) | −0.1260 (+5.6%) |

3.5° | 0.953 (−1.5%) | 0.012188 (−67.6%) | 0.04484 (−7.0%) | 0.001569 (−58.0%) | 21.25 (+5.9%) | −0.1229 (+2.6%) | |

Bump 3 (h/c = 0.012, x _{B}/c = 0.55) | 3.5° | 0.917 (−5.3%) | 0.008297 (−77.9%) | 0.04642 (−3.8%) | 0.001343 (−64.1%) | 19.75 (−1.6%) | −0.1184 (+6.2%) |

Bump 4 (h/c = 0.008, x _{B}/c = 0.50) | 2.5° | 0.870 (−5.3%) | 0.003852 (−44.0%) | 0.03395 (+12.8%) | 0.000549 (−24.2%) | 25.63 (−16.1%) | −0.1271 (+4.8%) |

3.5° | 0.925 (−4.4%) | 0.016171 (−56.9%) | 0.04559 (−5.5%) | 0.002168 (−42.0%) | 20.29 (+1.1%) | −0.1193 (+5.5%) |

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**MDPI and ACS Style**

Zhang, Y.; Yang, P.; Li, R.; Chen, H.
Unsteady Simulation of Transonic Buffet of a Supercritical Airfoil with Shock Control Bump. *Aerospace* **2021**, *8*, 203.
https://doi.org/10.3390/aerospace8080203

**AMA Style**

Zhang Y, Yang P, Li R, Chen H.
Unsteady Simulation of Transonic Buffet of a Supercritical Airfoil with Shock Control Bump. *Aerospace*. 2021; 8(8):203.
https://doi.org/10.3390/aerospace8080203

**Chicago/Turabian Style**

Zhang, Yufei, Pu Yang, Runze Li, and Haixin Chen.
2021. "Unsteady Simulation of Transonic Buffet of a Supercritical Airfoil with Shock Control Bump" *Aerospace* 8, no. 8: 203.
https://doi.org/10.3390/aerospace8080203