# Global Surface Pressure Pattern for a Compressible Elliptical Cavity Flow Using Pressure-Sensitive Paint

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{W}= 8.7 × 10

^{4}) showed that the shear layer grows almost linearly downstream of the separation point. There are symmetrical cellular structures inside the cavity volume if the minor axis is aligned with the streamwise direction. For shallow elliptical cavities, there is a nominally two-dimensional feature, but this is not the case for intermediate-depth elliptical cavities. For a yawed elliptical cavity (freestream velocity = 27 m/s; W = 72.5 mm; W/L = 0.5; Re

_{W}= 9.1 × 10

^{4}) [16], there is a highly asymmetric flow regime. A nominally two-dimensional flow regime is observed for large yaw angles, and the effect of yaw angle on the flow regimes for the shallowest and deepest cavities is minor. A strong resemblance is found between the flow regimes associated with elliptical and rectangular cavities for similar yaw angle and depth configurations. In the study of Khadivi and Savory [17] (freestream velocity = 18.3 m/s; W = 72.5 mm; W/L = 0.5; Re

_{W}, = 8.7 × 10

^{4}), particle image velocimetry and computational fluid dynamics simulations were conducted. The 3D structure of the flow was pronounced in asymmetric regimes with large yaw angles in which there was a formation of a trailing vortex.

## 2. Experimental Setup

#### 2.1. Transonic Wind Tunnel

^{3}with a maximum pressure of 50 bars. The dew point of high-pressure air flowing through the air dryers is maintained at −40 °C under normal operating conditions. Airflow from the supply tanks to the stagnation chamber is controlled by a sleeve rotating valve which continuously throttles the flow to maintain a specific stagnation pressure, p

_{o}. A flow-conditioning module, comprising acoustic baffles, three screens and a honeycomb inside a stilling chamber, reduces noise and turbulence. A nozzle and two downstream choke flaps monitor the value of M. The operating Mach number ranges from 0.2 to 1.4, and the simulated Reynolds number is up to 20 million per meter. The square test section (600 × 600 mm

^{2}and 1500 mm in length) has perforated top/bottom walls and solid sidewalls. Downstream of the test section, a divergent subsonic diffuser decelerates the flow exiting to the atmosphere. The test conditions were recorded using a National Instruments system (Austin, TX, USA), including PXIe-8840 RT, PXI-7846, PXI-6511, and PXI-6513 NI-SCXI.

#### 2.2. Models and Test Conditions

^{2}) and an interchangeable instrumentation plate (170 × 150 mm

^{2}) with an elliptical cavity, as shown in Figure 1. Two side fences were also installed to prevent crossflow. Experimental conditions are shown in Table 1. The boundary layer thickness for a naturally developed turbulent boundary layer, δ, was approximately 7 mm at a value of M of 0.83 [19]. The Reynolds number based on the boundary layer thickness, Re

_{δ}, was 1.69 × 10

^{5}. The stagnation pressure was 172 ± 1 kPa, and the stagnation temperature was 28–32 °C. The values of ε for elliptical cavities were 0 (a cylindrical cavity), 0.66 and 0.87. The geometrical configurations of the cavities are listed in Table 2. The value of L (= 43 mm) is fixed, and W varies with the values of 43.0, 32.3 and 21.5 mm. The value of L/H ranges from 4.43 to 21.50 as the depth, H, varies from 2.0 to 9.7 mm.

#### 2.3. Pressure Measurements

_{p}(= (p

_{w}− p

_{∞})/q), was 0.43%, where p

_{w}and p

_{∞}are the mean and freestream pressures, respectively. The dynamic pressure is denoted as q.

_{ref}and P

_{ref}are the respective reference intensities for the emission and the applied pressure. B represents pressure sensitivity, and A is a constant.

_{2}mesoporous particles measuring 2 μm in diameter were added to the mixture to increase the oxygen permeability in the polymer binder and the PSP sensors. The respective absorption and emission spectra were measured at 411–467 nm and 597 nm. The PSP was illuminated using two light-emitting diode (LED) light sources (Revox SLG-55; Kanagawa, Japan). These light sources have a maximum brightness of 2100 lumens and a wavelength range of 300–800 nm. Two 550 nm low-pass filters were placed on the LEDs to filter incident light within the range of the luminescent signal wavelength. The emitted signal was captured using a 16-bit scientific Complementary Metal–Oxide Semiconductor camera (CMOS) (PCO. edge 3.1; Regensburg, Germany) with a 605 ± 15 nm high-pass optical filter to eliminate excitation light. The instrument has a resolution of 2048 × 1536 pixels and a pixel ≈0.3 mm. The sampling rate was set as 20 frames per second. The irreversible photo-degradation process proceeded at a rate of approximately 1% per hour.

_{ref}and an applied pressure that varied from 0.9 to 1.18 bar. Uncertainty due to spatial non-uniformities in the emitted intensity, paint thickness and luminophore concentration was reduced using a ratio between images at P and P

_{ref}(or wind-on and wind-off images from wind tunnel tests). The value of B was 0.64%/kPa, and the temperature sensitivity for PSP was −1.12%/°C.

## 3. Results and Discussion

#### 3.1. Global Pressure Pattern

^{*}= x/L) is in the streamwise direction. For ε = 0 (a cylindrical cavity, Figure 3a), there is a uniform pressure distribution until x

^{*}≈ 0.7, and an adverse pressure gradient occurs upstream of the rear face. This corresponds to a typical pressure distribution for an open-type cavity flow [5,6]. For ε = 0.66 (L/W = 1.33, Figure 3b) and 0.87 (L/W = 2.0, Figure 3c), there is a slight favorable pressure gradient until x

^{*}≈ 0.7–0.8, and the region in which C

_{p}has a positive value is narrower. An arc-shaped pressure pattern near the rear face becomes more significant as ε increases because of the effect of the curvature of the rear wall. An increase in the value of C

_{p}signifies greater inflow and outflow near the rear face as ε or L/W increases, which is similar to rectangular cavities [10].

_{p}downstream of the front edge in comparison with that for L/H = 4.43. For ε = 0.66 (Figure 4b), the region in which there is an adverse pressure gradient is expanded upstream, and the surface pressure pattern resembles a transitional–open cavity flow. This is also true for ε = 0.87 (Figure 4c). There is an open-type cylindrical cavity flow for L/H = 4.43−7.17 [14]. In this context, the boundary between different flow types varies for cylindrical and elliptical cavities.

_{p}is positive moves upstream. The shear layer separates at the front edge and impinges on the cavity floor. This phenomenon becomes more significant as ε increases, as shown in Figure 5b (ε = 0.66) and Figure 5c (ε = 0.87). The surface pressure pattern is for a transitional–closed or closed cavity so that there is early transition from an open to a closed cavity flow as ε increases. For L/H = 21.50, Figure 6a shows a typical closed-type cylindrical cavity. The region in which C

_{p}is positive moves further upstream, and an arc-shaped pressure pattern is shown in Figure 6b,c for ε = 0.66 and 0.87. Hering and Savory [15] demonstrated that there is an increase in entrainment over the sidewall for rectangular cavities with a greater value for L/W. For elliptical cavities, there is early shear layer impingement on the cavity floor as ε increases.

#### 3.2. Mean Surface Distribution

^{*}= 0; ε = 0; L/H = 4.43–21.50), as measured using Kulite sensors and PSP. The two measurement techniques give comparable results. A greater deviation is observed near the rear edge. There is significant flow expansion near the front and rear edges for L/H = 14.33 and 21.50, and the respective peak pressure values, C

_{p,peak}, at x

^{*}= 0.92 are 0.29 and 0.31 as the value of L/H increases. There is a plateau (x

^{*}= 0.37−0.46) in the C

_{p}distribution for L/H = 21.50, which corresponds to a transitional–closed cavity. For L/H = 4.43 and 6.14, the static pressure distribution is uniform on the cavity floor, representing a typical open cavity flow, and rear-edge expansion is less significant than for L/H = 14.33 and 21.50.

^{*}= 0.36−0.78 for L/H = 21.50, so an increase in ε results in a lower value for a closed-type cavity. The value of C

_{p,peak}at x

^{*}= 0.92 is 0.28, which is slightly less than that for ε = 0. However, there is greater flow expansion near the rear edge because of an increase in the entrainment over the sidewall [15]. For L/H = 14.33, there is less expansion near the rear edge, and the value of C

_{p,peak}at x

^{*}≈ 0.91 is 0.32. The flow resembles that for a transitional–closed cavity. For L/H = 6.14 and 4.43, the effect of ε is less significant. There is a decrease in the value of C

_{p,peak}(= 0.19 and 0.14 for ε = 0.66 and 0.87, respectively) as ε increases. This is also true for L/H = 4.43. The surface pressure distribution for ε = 0.87 (L/W = 2.0), as shown in Figure 9, resembles that for ε = 0.66.

_{p}, corresponding to the flow mass exchange. For ε = 0, there is a slight variation in the value of ΔC

_{p}for an open cavity (L/H ≤ 6.14). An increase in the value of L/H increases the value of ΔC

_{p}, following a decrease for a transitional–closed cavity (L/H = 21.50). Flow expansion increases if there is an increase in the value of ε due to an increase in entrainment over the sidewall, particularly for L/H = 14.33 (a transitional cavity). This determines the effect of ε on flow development near the rear edge.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | constant for PSP calibration curve |

B | pressure sensitivity for PSP |

C_{p} | pressure coefficient, (p–p_{∞})/q |

C_{p} | pressure coefficient |

C_{p,peak} | peak pressure coefficient |

H | cavity depth |

I | intensity of the emission |

I_{ref} | reference intensity for emission |

L | the length of the major axis |

M | freestream Mach number |

p_{w} | mean surface pressure |

p_{∞} | freestream pressure |

q | dynamic pressure |

W | the length of the minor axis |

x | coordinate in the streamwsie direction |

y | coordinate in the spanwise direction |

x^{*} | x/δ |

y^{*} | y/δ |

δ | boundary layer thickness |

ε | eccentricity |

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**Figure 7.**Mean surface pressure distribution for ε = 0 (red solid line: PSP data; black points: Kulite data).

**Figure 8.**Mean surface pressure distribution for ε = 0.66 (red solid line: PSP data; black points: Kulite data).

**Figure 9.**Mean surface pressure distribution for ε = 0.87 (red solid line: PSP data; black points: Kulite data).

M | p_{o}, Pa | Re_{δ} | T_{o} |
---|---|---|---|

0.83 | 1.72 × 10^{5} | 1.69 × 10^{5} | 28–32 °C |

L/H | L, mm | W, mm | ε |
---|---|---|---|

4.43 | 43 | ||

6.14 | 43 | 43.0, 32.3, 21.5 | 0, 0.66, 0.87 |

14.33 | 43 | ||

21.50 | 43 |

Luminophore | Polymer | Solvent | Particle | Spray Area |
---|---|---|---|---|

Ru(dpp) | RTV-118 | Toluene | SiO_{2} | 37.5 cm^{2} |

2.5 mg | 250 mg | 5 ml | 125 mg |

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**MDPI and ACS Style**

Huang, Y.-X.; Chung, K.-M.
Global Surface Pressure Pattern for a Compressible Elliptical Cavity Flow Using Pressure-Sensitive Paint. *Aerospace* **2024**, *11*, 159.
https://doi.org/10.3390/aerospace11020159

**AMA Style**

Huang Y-X, Chung K-M.
Global Surface Pressure Pattern for a Compressible Elliptical Cavity Flow Using Pressure-Sensitive Paint. *Aerospace*. 2024; 11(2):159.
https://doi.org/10.3390/aerospace11020159

**Chicago/Turabian Style**

Huang, Yi-Xuan, and Kung-Ming Chung.
2024. "Global Surface Pressure Pattern for a Compressible Elliptical Cavity Flow Using Pressure-Sensitive Paint" *Aerospace* 11, no. 2: 159.
https://doi.org/10.3390/aerospace11020159