#### Abstract

A flap can be used to control wing camber and as a high-lift device. A convex corner is a simplified model of the upper surface of a flap. At transonic speeds, shock-induced boundary layer separation (SIBLS) occurs at greater freestream Mach numbers and deflection angles. This results in energy losses and a reduction in aerodynamic performance. This study installs ramp-type vortex generators (VGs) upstream of a convex corner, and the effect of the height of the VG on SIBLS is determined. As the height of the VG increases, the magnitude of the mean surface pressure upstream of the corner increases and downstream expansion decreases, which results in a reduction in lift. A reduction in peak surface pressure fluctuations, the separation length, and the frequency of shock oscillation is also determined. For flow control and lift enhancement, micro-VGs are more effective.

#### 1. Introduction

Flaps are located at the trailing edges of a fixed wing to control the wing camber and are used to increase lift during takeoff and landing. In the cruise, variable-geometry wing camber control has benefits for transport aircraft that uses deflected flaps [1]. However, the effect of compressibility must be considered. Chung [2] used a convex corner model to simulate the upper surface of a deflected flap. There is significant expansion near the corner’s apex and downstream compression. As the freestream Mach number, , and the deflection angle, , increase, there is a transition from subsonic to transonic flow and SIBLS occurs. A similarity parameter, (), is used to characterize a compressible convex corner flow (peak Mach number, , peak pressure fluctuations, and shock oscillation) [3]. If , SIBLS occurs and induces low-frequency, high-amplitude shock oscillations [4].

Flow separation reduces the aerodynamic efficiency. Passive or active device has been used to control separated flow [5–8]. A VG is a passive device [9]. An array of VGs is normally attached to a surface at an angle of incidence, , to introduce streamwise vorticity into the flow near the surface, and the boundary layer is energized to resist adverse pressure gradients and SIBLS. The ratio of the height of a VG, , to the incoming boundary layer thickness, , () is a dominant parameter. Increasing generates stronger vortices [10]. Lee et al. [11] showed that VGs for which prevent SIBLS, but the device drag increases. Micro-VGs () incur a smaller drag penalty [12–14].

Ramp-type VGs are robust structurally and are eminently suited to applications in high-speed flows [15–18]. A ramp-type VG forms a counter-rotating vortex pair and is advected by the mean flow. Energetic air is transferred from the primary vortex pair, and the velocity close to the surface depends on the liftoff of the vortices [19]. The gap between ramp-type VGs creates secondary counter-rotating vortices and reduces the interaction between counter-rotating vortex pairs [20]. For an oblique shock reflection with microramp VGs, the momentum flux that is added to the separation bubble increases linearly with . VGs stabilize the interaction (a reduction in shock unsteadiness) more effectively as increases. A distance of 5.7 *δ* is required for full boundary layer mixing [21].

A deflected flap controls wing camber for an increase in lift, and a convex corner models its upper surface. In the transonic flow regime, SIBLS and peak pressure fluctuations occur for [2]. Shock wave control technologies are required. This study uses ramp-type VGs positioned upstream of a convex corner ( and 15°) for and 0.89 to minimize the adversarial effects of SIBLS. The value of is 0.2, 0.5, 1.0, and 1.5. The mean and fluctuating pressure distributions in the streamwise direction are determined. The locations for flow separation and reattachment with and without the presence of VGs are visualized using oil flow visualization. The effect of VGs on SIBLS and shock oscillation is determined. Before discussing the results, details of the experiment setup are outlined next.

#### 2. Experimental Technique

##### 2.1. Transonic Wind Tunnel

The experiments were conducted in a blowdown transonic wind tunnel at the Aerospace Science and Technology Research Center of the National Cheng Kung University. The facility comprises two compressors, two air dryers (dew point in normal operation ≈ -40°C), a cooling water system, three storage tanks (total m^{3}), and a tunnel. The constant area test section is and is 1500 mm in length. The test section was assembled using perforated top/bottom walls with 6% porosity (to alleviate reflected shock and expansion waves) and solid side walls (to reduce background noise). The stagnation pressure, , is kPa, and the stagnation temperature is 28°C–32°C. is 0.83 and . This determines the respective unit Reynolds numbers: 2.33 and /m.

##### 2.2. Test Models

The test model comprised a flat plate and an instrumentation plate with/without an array of ramp-type VGs, as shown in Figure 1. The flat plate is 450 mm long and 150 mm wide. The interchangeable instrumentation plate is 170 mm in length and 150 mm in width. A convex corner with of 13° and 15° is located at 500 mm from the leading edge of the flat plate. The value of at 25 mm upstream of the convex corner is approximately 7 mm [2], and the respective Reynolds numbers based on are 1.63 and for and 0.89. The test without the presence of VGs is the baseline case.

Lin [12] showed that a VG is effective at least 20 *h* upstream of the location of flow separation. This study uses an array of ramp-type VGs that are positioned 3 mm upstream of the convex corner. Seventeen pressure taps are machined perpendicular to the surface of the test model along the centerline: 5 pressure taps ahead of the VG and 12 pressure taps downstream of the convex corner. The spacing between the pressure taps is 6 mm. The configuration of the VGs is shown in Figure 2, and the parameters of the VGs are shown in Table 1. The length, , is 1 *δ*, and the width, , is 0.5 *δ*. The spacing, , between the VGs is 3 *δ*. This gives an array with seven VGs [22]. The value of is 15° [23].

##### 2.3. Instrumentation and Data Acquisition System

Kulite pressure transducers (XCS-093-25A, B screen) with a natural frequency of 200 kHz were used to measure the mean and fluctuating surface pressures. The nominal outer diameter of the Kulite sensors is 2.36 mm, and the diameter of the pressure-sensitive element is 0.97 mm. External amplifiers (Ectron Model E713) with a roll-off frequency of approximately 140 kHz were used to improve the signal-to-noise ratio. A National Instrument (CI SCXI) was used to trigger all input channels and to record data. Ten Kulite sensors were installed in the streamwise direction on a flat plate. The experimental uncertainty for the mean surface pressure coefficient, , and the surface fluctuating pressure coefficient, , is determined. Using 10 Kulite sensors, the standard deviation for is 1.24% and the mean value for is 0.97%.

Oil flow visualization technique is used to determine the points at which separation and reattachment occur and the separation length, . A compound mixture of titanium dioxide, oil, and oleic acid was applied to the surface of the instrumentation plate. An accumulation of titanium dioxide signifies the separation location. The end of the deflected streak line denotes the flow reattachment which exhibits a corrugated pattern if there are VGs, as shown in Figure 3. The mean positions for separation and reattachment are calculated.

#### 3. Results and Discussion

##### 3.1. Mean Surface Pressure Distribution

The distribution of for is shown in Figure 4. For the baseline case, there is a near incipient separation for and shock oscillation for [24]. The horizontal axis is the normalized streamwise location, (), and the origin is the corner’s apex. There is no data for to 0 because this is the position of the VGs. The sonic condition () is also shown for reference. For the baseline case, the distribution exhibits slight upstream expansion, significant expansion near the corner’s apex, and downstream compression. For VGs for which and , the VGs have little effect on the distribution. The minimum , , is located farther downstream than that for the baseline case because the VGs induce expansion waves [25]. An increase in affects the distribution. For VGs for which , there is an increase in the value of upstream of the corner as increases. This shows an increase in device drag, particularly for and 1.5. An increase in the value of downstream of the corner represents less expansion or a decrease in the value of as increases. The distribution for is similar to that for . For the baseline case, there is a distinctive kink at which indicates SIBLS [2]. If there are VGs, the kink moves downstream. This represents a delay in flow separation.

The distribution for is shown in Figure 5. For the baseline case, compression occurs farther downstream than that for . Bouhadji and Braza [26] showed that a hyperbolic character for a high subsonic flow results in a sweep for a perturbation action propagating downstream. This causes greater convection for . For and 15°, the compression for occurs farther downstream, so there is a greater convection effect, but not for . has a similar effect on the distribution as that for .

For the baseline case, Figure 6 shows the effect of on and . The value of (0.267-0.303) decreases (or increases) as increases. For VGs for which and 0.5, the value of (0.266-0.299) and shows a minor deviation from the baseline case. An increase in the value of results in an increase in and a decrease in . For VGs for which and 1.5, there is less flow expansion near the convex corner, , so the lift force is reduced if a deflected control surface is used for a fixed wing.

##### 3.2. Surface Pressure Fluctuations

The distribution for is shown in Figure 7. For the baseline case, there is a minor variation in the value of upstream and downstream of the corner, but not near the corner’s apex. The respective peak values for , for and 15°, are 6.9% and 6.2% at . VGs reduce the amplitude of . For , there is a reduction in the values of to 1.6% and 1.0% for and 15°. An increase in results in a more decrease in . For (more extensive separated flow), VGs have less effect on than that for (near incipient separation), so the effectiveness of VGs depends on the status of the separated flow (incipient separation or strong shock oscillation).

The distribution for is shown in Figure 8. The respective values of are 8.6% and 10.3% at for and 15°. For VGs for which , there is a reduction in the value of at for and for . The location of for is larger than that for is due to a hyperbolic character for a high subsonic flow [26]. This is consistent with the downstream pressure recovery process for a convex corner flow, as shown in Figure 5. An increase in results in a decrease in , particularly for .

The relationship between the value of (6.2%-10.3%) and for the baseline case is shown in Figure 9. The presence of VGs results in a decrease in the value of (4.6%-10.2%), particularly as increases because stronger induced vortices affect shock oscillation.

##### 3.3. Shock Oscillation

SIBLS results in intense pressure fluctuations [27–30]. Dolling and Brusniak [31] studied the intermittent pressure signals to determine the unsteadiness of shock motion (shock oscillation). A conditional analysis technique (two-threshold method, THM) was used to determine the shock zero-crossing frequency, . The upper and lower thresholds are and . The time between consecutive passages of shock over a pressure sensor is determined. The pressure signal is then converted into a boxcar of amplitude unity, as shown in Figure 10.

Figure 11 shows the relationship between and . For the baseline case, the value of ranges from 1320 Hz to 640 Hz. There is a decrease as increases. Variation in is related to the length of a separation bubble [4]. The presence of VGs reduces the shock oscillation where .

For SIBLS, the unsteady shock motion is related to successive contractions and expansions of a separation bubble. The normalized separation length, (), can be used as a length scale to characterize shock unsteadiness [4]. The variation of with is shown in Figure 12. The value of (6.00-8.35) increases as increases. The presence of VGs results in a decrease in the value of (3.35-6.19). The decrease is more significant as increases. This result agrees with the results of Verma and Manisankar [32].

The variation in the Strouhal number, (), with is shown in Figure 13, where is the peak velocity. For the baseline case, the value of ranges from 0.09 to 0.12. When the VGs are positioned upstream of the convex corner, there is a decrease in the value of , , and . For , the value of (0.007-0.02) decreases as increases because there are stronger induced vortices.

#### 4. Conclusions

This study determines the effect of the height of ramp-type VGs on a transonic convex corner flow. VGs for which have a minor effect on the distribution. The presence of VGs induces streamwise vorticity that propagates downstream. This results in an extension in the low-pressure region and a slower downstream pressure recovery process, particularly for . An increase in (1.0 and 1.5) results in a mean surface pressure upstream of the convex corner that is greater in magnitude and less expansion near the corner. This indicates that there is greater device drag and a reduction in the effectiveness of the VGs. The values for (4.6%-10.2%) and (3.35-6.19) decrease as increases because the presence of VGs prevents SIBLS. The analysis in terms of (929 Hz-126 Hz) and (0.007-0.059) shows less shock oscillation, particularly for VGs for which .

#### Nomenclature

: | Spacing between vortex generators |

: | Shock zero-crossing frequency, Hz |

: | Height of vortex generator |

: | Normalized height of vortex generator, h/δ |

: | Mean separation length |

: | Normalized mean separation length, L/δ |

: | Length of vortex generator |

: | Freestream Mach number |

: | Peak Mach number |

: | Stagnation pressure |

: | Local mean surface pressure |

: | The minimum pressure coefficient |

SIBLS: | Shock-induced boundary layer separation |

: | Strouhal number, |

THM: | Two-threshold method |

: | Coordinate along the centerline of model surface |

: | Normalized streamwise distance, x/δ |

: | Peak velocity |

VG: | Vortex generator |

: | Width of vortex generator |

: | Angle of incidence of vortex generator |

: | Similarity parameter, |

: | Incoming boundary-layer thickness |

: | Convex corner angle, degree |

: | Standard deviation of surface pressure |

: | Fluctuating pressure coefficient |

: | Peak fluctuating pressure coefficient. |

#### Data Availability

Data is available upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The authors acknowledge the financial support from the Ministry of Science and Technology, Taiwan (MOST 110-2923-E-006-006-MY3).