Micro-Vortex Generators on Transonic Convex-Corner Flow

: A convex corner models the upper surface of a deﬂected ﬂap and shock-induced boundary layer separation occurs at transonic speeds. This study uses micro-vortex generators (MVGs) for ﬂow control. An array of MVGs (counter-rotating vane type, ramp type and co-rotating vane type) with a height of 20% of the thickness of the incoming boundary layer is installed upstream of a convex corner. The surface pressure distributions are similar regardless of the presence of MVGs. They show mild upstream expansion, a strong favorable pressure gradient near the corner’s apex and downstream compression. A corrugated surface oil ﬂow pattern is observed in the presence of MVGs and there is an onset of compression moving downstream. The counter-rotating vane type MVGs produce a greater reduction in peak pressure ﬂuctuations and the ramp type decreases the separation length. The presence of MVGs stabilizes the shock and shock oscillation is damped. η respective values for the CRV type for f s are 535 Hz and 70 Hz for M = 0.83 and 0.89.


Introduction
A deflected trailing-edge flap increases camber and lift at a constant angle of attack (or lift-to-drag ratio) for transport aircraft. There is also a higher buffet boundary (structural response due to pressure fluctuations) and a greater pitching moment [1]. In a strong adverse pressure gradient (or shock region) at transonic speeds, flap deflection leads to boundary layer separation when the critical Mach number (≈1.3) is achieved [2]. A convex corner can be used to model the upper surface of a deflected flap. The typical mean surface pressure pattern for a turbulent boundary layer over a convex corner was determined by Chung [3]. There is initial mild upstream expansion, a strong favorable pressure gradient near the corner's apex and downstream compression (λ shock structure at transonic speeds, as shown in Figure 1). The flow characteristics are dominated by the freestream Mach number, M, and angle of the convex corner, η. A similarity parameter β (= M 2 η/ 1 − M 2 ) was proposed for compressible flow around a convex corner, which involves transition from subsonic to transonic flow (β ≈ 8), shock-induced boundarylayer separation (SIBLS) (β > 13) and peak pressure fluctuations [4]. These peak pressure fluctuations are associated with low-frequency, large-scale shock oscillation in the range of several hundred hertz to several kilohertz [5].
SIBLS significantly reduces aerodynamic efficiency and increases energy loss [6][7][8]. Passive vortex generators (VGs) with a height, h, that is of the order of the thickness of the incoming boundary layer, δ, are used to control boundary layer separation [9][10][11][12]. The geometric factors for VGs also include length, spacing and positions, but the effect of h is the most significant [13]. The device is typically normal to the surface and is positioned at an angle of incidence, α, to the local flow, to produce streamwise vortices in the flow close to the surface [14]. Therefore, VGs that are located upstream of the flow separation energize the boundary layer or increase near-wall momentum, which is transferred from the outer flow (or wall-ward momentum transfer).  A convex corner is used to model the upper surface of a deflected flap for an increase in lift. In the transonic flow regime, SIBLS occurs for β > 13, which results in peak pressure fluctuations due to shock oscillation [23]. This affects aerodynamic characteristics for flight vehicles, such as wave drag and buffet boundary. Shock wave control technologies Sub-δ-scale or micro VGs (h/δ = h* < 1, MVGs) that have a rectangular, delta or trapezoidal shape ensure effective mixing over a region that is several times their own height [15][16][17][18] and device-induced streamwise vortices may last up to h* of 100 [13]. MVGs also induce less device (or parasitic) drag than larger scale VGs.
MVGs with counter-rotating vanes (CRV), co-rotating vanes (Vane) or ramps (Ramp) are most commonly used to control boundary-layer separation [14]. For CRV MVGs, vanes are grouped in pairs and produce vortices of alternating sign, as shown in Figure 2. There is a significant reduction in separation-induced fluctuations in shear stress on the upper surface for a high-lift airfoil [15]. Vane MVGs are the most common type of VG which energizes the low-momentum boundary layer flow near the surface. Large vortical structures and up-sweep flows are attenuated [19]. The amplitude of the forced shock wave oscillation is also reduced by delaying the upstream displacement of the leading shock [20]. In the presence of Ramp MVGs, a horseshoe vortex system and a vortex pair form and the velocity is greater close to the surface (unsteady wake trailing downstream), so energetic air is transferred from the primary vortex pair [21,22]. to the surface [14]. Therefore, VGs that are located upstream of the flow separation energize the boundary layer or increase near-wall momentum, which is transferred from the outer flow (or wall-ward momentum transfer). Sub-δ-scale or micro VGs (h/δ = h* < 1, MVGs) that have a rectangular, delta or trapezoidal shape ensure effective mixing over a region that is several times their own height [15][16][17][18] and device-induced streamwise vortices may last up to h* of 100 [13]. MVGs also induce less device (or parasitic) drag than larger scale VGs.
MVGs with counter-rotating vanes (CRV), co-rotating vanes (Vane) or ramps (Ramp) are most commonly used to control boundary-layer separation [14]. For CRV MVGs, vanes are grouped in pairs and produce vortices of alternating sign, as shown in Figure 2. There is a significant reduction in separation-induced fluctuations in shear stress on the upper surface for a high-lift airfoil [15]. Vane MVGs are the most common type of VG which energizes the low-momentum boundary layer flow near the surface. Large vortical structures and up-sweep flows are attenuated [19]. The amplitude of the forced shock wave oscillation is also reduced by delaying the upstream displacement of the leading shock [20]. In the presence of Ramp MVGs, a horseshoe vortex system and a vortex pair form and the velocity is greater close to the surface (unsteady wake trailing downstream), so energetic air is transferred from the primary vortex pair [21,22].  A convex corner is used to model the upper surface of a deflected flap for an increase in lift. In the transonic flow regime, SIBLS occurs for β > 13, which results in peak pressure fluctuations due to shock oscillation [23]. This affects aerodynamic characteristics for flight vehicles, such as wave drag and buffet boundary. Shock wave control technologies A convex corner is used to model the upper surface of a deflected flap for an increase in lift. In the transonic flow regime, SIBLS occurs for β > 13, which results in peak pressure fluctuations due to shock oscillation [23]. This affects aerodynamic characteristics for flight vehicles, such as wave drag and buffet boundary. Shock wave control technologies (MVGs, bleeding or suction) can be used to minimize the adversarial effects of SIBLS. This study uses MVGs (CRV, Ramp and Vane types) with a value of h* = 0.2 to alleviate flow separation. The effectiveness of MVGs is determined using surface mean/fluctuating pressure measurements and surface oil flow visualization. Before discussing the results, details of the experiment setup are outlined next.

Transonic Wind Tunnel
This experiment was conducted in the blowdown transonic wind tunnel, as shown in Figure 3, at the Aerospace Science and Technology Research Center in National Cheng Kung University (ASTRC/NCKU). The facility consists of compressors, air dryers (dew point ≈ −40 • C), a cooling water system, three air storage tanks (180 m 3 ) and a tunnel. The constant-area test section has a cross-sectional area of 600 mm × 600 mm and a length of 1500 mm. The test section was assembled with solid sidewalls for a reduction in background noise and perforated top/bottom walls with 6% porosity. Holes were inclined at 60 • to minimize the wall interference. The operating Mach number ranges from 0.2 to 1.4. The stagnation pressure, P o , is 172 ± 0.5 kPa and the stagnation temperature, T o , is 28-32 • C. The unit Reynolds number has a maximum value of 2 × 10 7 /m. The respective flow angularity and the centerline Mach number uniformity in the test section are 0.2 • and 0.005. study uses MVGs (CRV, Ramp and Vane types) with a value of h* = 0.2 to alleviate flow separation. The effectiveness of MVGs is determined using surface mean/fluctuating pressure measurements and surface oil flow visualization. Before discussing the results, details of the experiment setup are outlined next.

Transonic Wind Tunnel
This experiment was conducted in the blowdown transonic wind tunnel, as shown in Figure 3, at the Aerospace Science and Technology Research Center in National Cheng Kung University (ASTRC/NCKU). The facility consists of compressors, air dryers (dew point ≈ −40 °C), a cooling water system, three air storage tanks (180 m 3 ) and a tunnel. The constant-area test section has a cross-sectional area of 600 mm × 600 mm and a length of 1500 mm. The test section was assembled with solid sidewalls for a reduction in background noise and perforated top/bottom walls with 6% porosity. Holes were inclined at 60° to minimize the wall interference. The operating Mach number ranges from 0.2 to 1.4. The stagnation pressure, Po, is 172 ± 0.5 kPa and the stagnation temperature, To, is 28-32 °C. The unit Reynolds number has a maximum value of 2 × 10 7 /m. The respective flow angularity and the centerline Mach number uniformity in the test section are 0.2° and 0.005.

Test Model
The test model consists of a flat plate and an interchangeable instrumentation plate with/without an array of MVGs, as shown in Figure 4. The flat plate with a leading edge of 3° is 450 mm long and 150 mm wide for a naturally developed, turbulent boundary layer [24]. The interchangeable instrumentation plate is 170 mm long and 150 mm wide. A sharp convex corner with an angle, η, of 13° or 15° is located 500 mm from the leading edge of the flat plate. Two side fences of (13.5 × 4.5 × 0.5 cm) were also installed at both sides of the instrumentation plate to prevent crossflow. The freestream Mach number, M, is 0.83 and 0.89 ± 0.01. The value of δ at 25 mm upstream of the convex corner is approximately 7 mm [23,25]. The Reynolds number based on the boundary layer thickness, Reδ, is 1.63 and 1.69 × 10 5 . The baseline experiment, denoted as the baseline case, does not use MVGs.

Test Model
The test model consists of a flat plate and an interchangeable instrumentation plate with/without an array of MVGs, as shown in Figure 4. The flat plate with a leading edge of 3 • is 450 mm long and 150 mm wide for a naturally developed, turbulent boundary layer [24]. The interchangeable instrumentation plate is 170 mm long and 150 mm wide. A sharp convex corner with an angle, η, of 13 • or 15 • is located 500 mm from the leading edge of the flat plate. Two side fences of (13.5 × 4.5 × 0.5 cm) were also installed at both sides of the instrumentation plate to prevent crossflow. The freestream Mach number, M, is 0.83 and 0.89 ± 0.01. The value of δ at 25 mm upstream of the convex corner is approximately 7 mm [23,25]. The Reynolds number based on the boundary layer thickness, Re δ , is 1.63 and 1.69 × 10 5 . The baseline experiment, denoted as the baseline case, does not use MVGs.
of the corner. The spacing between the pressure taps was 6 mm. Three types of MVGs were used: CRV, Vane and Ramp. Sketches of the MVGs are shown in Figure 5 and the parameters for the MVGs are shown in Table 1. For all MVGs, the length, l, and height, h, are 1 δ and 0.2 δ (lower device drag), respectively. The width for the CRV and Vane types, wv, is 0.2 δ and for the Ramp type, wr, is 0.5 δ. The spacing, D, between MVGs is 3 δ, which corresponds to an array with seven MVGs. For the CRV type, the front, d, and rear, s, spacing are 1 δ and 0.5 δ, respectively. The value of α is 15°.   The rear face of an array of MVGs is located 3 mm upstream of the convex corner. Seventeen pressure taps were machined perpendicular to the surface of the test model along the centerline: 5 pressure taps ahead of the MVGs and 12 pressure taps downstream of the corner. The spacing between the pressure taps was 6 mm. Three types of MVGs were used: CRV, Vane and Ramp. Sketches of the MVGs are shown in Figure 5 and the parameters for the MVGs are shown in Table 1. For all MVGs, the length, l, and height, h, are 1 δ and 0.2 δ (lower device drag), respectively. The width for the CRV and Vane types, w v , is 0.2 δ and for the Ramp type, w r , is 0.5 δ. The spacing, D, between MVGs is 3 δ, which corresponds to an array with seven MVGs. For the CRV type, the front, d, and rear, s, spacing are 1 δ and 0.5 δ, respectively. The value of α is 15 • . The rear face of an array of MVGs is located 3 mm upstream of the convex corner. Seventeen pressure taps were machined perpendicular to the surface of the test model along the centerline: 5 pressure taps ahead of the MVGs and 12 pressure taps downstream of the corner. The spacing between the pressure taps was 6 mm. Three types of MVGs were used: CRV, Vane and Ramp. Sketches of the MVGs are shown in Figure 5 and the parameters for the MVGs are shown in Table 1. For all MVGs, the length, l, and height, h, are 1 δ and 0.2 δ (lower device drag), respectively. The width for the CRV and Vane types, wv, is 0.2 δ and for the Ramp type, wr, is 0.5 δ. The spacing, D, between MVGs is 3 δ, which corresponds to an array with seven MVGs. For the CRV type, the front, d, and rear, s, spacing are 1 δ and 0.5 δ, respectively. The value of α is 15°.

Parameters
Value

Instrumentation and Data Acquisition System
Flush-mounted Kulite pressure transducers (Model XCS-093-25A, B screen) with a natural frequency of 200 kHz were used to measure the mean, P w , and fluctuating, σ p , surface pressures. The nominal outer diameter of the Kulite sensor is 2.36 mm and the diameter of its pressure-sensitive element is 0.97 mm. Figure 6 shows that external amplifiers (Ectron Model 753A) with a roll-off frequency of approximately 140 kHz were used to improve the signal-to-noise ratio. A National Instruments (CI SCXI) was used to trigger all input channels and to record data. An experiment for a flat plat using 10 Kulite sensors was conducted to determine the experimental uncertainty. The uncertainty for the mean surface pressure coefficient, P w /P o, and the surface fluctuating pressure coefficient, σ p /P w , is respectively determined using the standard deviation of surface pressure and the mean value of surface pressure fluctuations for 10 Kulite sensors. The respective value for P w /P o and σ p /P w is 1.24% and 0.97%. Shock oscillation occurs in shock wave/boundary layer interactions. An unsteady shock moving forward and backward across a pressure sensor results in intermittent pressure signals, as shown in Figure 7a. A two-threshold method (THM, a conditional analysis technique) [26] was used to determine shock zero-crossing frequency, f s , where T 1 (the lower threshold) = P w + 3σ p and T 2 (the upper threshold) = P w + 6σ p , as shown in Figure 7a. P w > T 2 represents the flow downstream of the shock. The counter records the time between successive shock waves until P w < T 1 . The time between consecutive passages of the shock over the pressure transducer (shock motion) is used to calculate f s which determines the unsteady characteristics of the separation shock wave motion. Figure 7b shows the conversion of the pressure signal into a boxcar of amplitude unity and varying frequency.

Mean Surface Pressure Distributions
At M = 0.83, the mean surface pressure distributions along the centerline of the instrumentation plate for η = 13° and 15° are shown in Figure 8. For the baseline case, they respectively correspond to near incipient separation and intense shock oscillation [3]. The Oil-flow visualization technique was used to determine the surface flow pattern. A mixture of titanium dioxide, oil, oleic acid and kerosene was applied to the surface of instrumentation plate. If there is shock-induced boundary layer separation, titanium dioxide accumulates at the separation position. The end of the deflected streamlines is denoted as the reattachment position.

Mean Surface Pressure Distributions
At M = 0.83, the mean surface pressure distributions along the centerline of the instrumentation plate for η = 13 • and 15 • are shown in Figure 8. For the baseline case, they respectively correspond to near incipient separation and intense shock oscillation [3]. The vertical axis represents P w /P o and the horizontal axis corresponds to the normalized streamwise location, X* (= X/δ). The origin of the horizontal axis is at the corner's apex and there is no data for X* = −2 and 0 because of the presence of MVGs. The sonic condition (P w /P o = 0.5283) is also shown for reference. The P w /P o distributions with/without MVGs are similar in that they show upstream mild expansion, strong expansion near the corner's apex and downstream compression. The presence of MVGs has a minor effect on the amplitude of P w /P o upstream of the corner.  For η = 13 • , the minimum value for P w /P o , (P w /P o ) min , occurs at X* = 0.43, but not for the Vane MVGs (X* = 1.29). For the baseline case with η = 15 • , there is a distinctive kink at X* ≈ 2, which indicates SIBLS [23]. The pressure plateau moves downstream if MVGs are used. The location for (P w /P o ) min moves downstream because of trailing-edge expansion induced by MVGs [27]. P w /P o has a lower amplitude downstream of the corner.
The P w /P o distributions for M = 0.89 are shown in Figure 9. The general flow characteristics are the same as those for M = 0.83. For the baseline case, the low pressure region is longer (X* = 0-3) or the onset of downstream compression occurs later than that for M = 0.83. Bouhadji and Braza [28] studied transonic flow around an aerofoil in terms of the compressibility and viscous effects. For M ≈ 0.9, they found that convection effect results in a sweep for perturbation action much farther downstream because of the highly hyperbolic character of the flow. When MVGs are installed, the low pressure region is observed at X* = 0-6.4. Therefore, all three types of MVGs produce stronger convection and force the shock farther downstream.
For the baseline case, the values for (P w /P o ) min or the local peak Mach number, M peak , are reasonably well correlated with β [4]. An increase in β results in stronger expansion near the corner's apex. For β = 16.1 (M = 0.83 and η = 13 • ), Figure 10 shows that the value for (P w /P o ) min (or M peak ) is not significantly affected by the presence of MVGs. If β increases, the presence of MVGs results in a decrease in (P w /P o ) min or an increase in M peak . This is not the case for β

Surface Pressure Fluctuations
The distribution of surface pressure fluctuations on a lifting surface corresponds to fluctuating loads, particularly during SIBLS and shock oscillation. For the baseline case for M = 0.83, Figure 11 shows that the peak value for σp/Pw, (σp/Pw)max, is observed at X* = 2.14, which corresponds to SIBLS [23]. The presence of MVGs results in a decrease in the value of (σp/Pw) upstream and near the corner's apex (X* = −5.86-3.00) for η = 13°.
Verma and Manisankar [18] showed that there is a reduction in (σp/Pw)max, when MVGs are present. This is also true for the test configuration in this study. If Vane MVGs are used, the location of (σp/Pw)max moves downstream, which corresponds to (Pw/Po)min (or Mpeak) in Figure 8. For η = 15°, the effect of MVGs is less significant. For M = 0.89, the distributions of surface pressure fluctuations are shown in Figure 12. These are similar to

Surface Pressure Fluctuations
The distribution of surface pressure fluctuations on a lifting surface corresponds to fluctuating loads, particularly during SIBLS and shock oscillation. For the baseline case for M = 0.83, Figure 11 shows that the peak value for σp/Pw, (σp/Pw)max, is observed at X* = 2.14, which corresponds to SIBLS [23]. The presence of MVGs results in a decrease in the value of (σp/Pw) upstream and near the corner's apex (X* = −5.86-3.00) for η = 13°.
Verma and Manisankar [18] showed that there is a reduction in (σp/Pw)max, when MVGs are present. This is also true for the test configuration in this study. If Vane MVGs are used, the location of (σp/Pw)max moves downstream, which corresponds to (Pw/Po)min (or Mpeak) in Figure 8. For η = 15°, the effect of MVGs is less significant. For M = 0.89, the distributions of surface pressure fluctuations are shown in Figure 12. These are similar to

Surface Pressure Fluctuations
The distribution of surface pressure fluctuations on a lifting surface corresponds to fluctuating loads, particularly during SIBLS and shock oscillation. For the baseline case for M = 0.83, Figure 11 shows that the peak value for σ p /P w , (σ p /P w ) max , is observed at X* = 2.14, which corresponds to SIBLS [23]. The presence of MVGs results in a decrease in the value of (σ p /P w ) upstream and near the corner's apex (X* = −5.86-3.00) for η = 13 • . gressively in three-dimensional flows than in two-dimensional flows. In terms of (σp/Pw)max (Figure 13), CRV MVGs are the most effective at control for SIBLS. Ramp MVGs are less effective because the vortices are weaker and there is rapid lift-off from the surface [27]. Lee and Loth [30] determined that streamwise vortices are sustained at greater Mach numbers. The onset of flow separation occurs farther downstream for M = 0.89 but the MVGs are still effective to mitigate surface pressure fluctuations.  Verma and Manisankar [18] showed that there is a reduction in (σ p /P w ) max , when MVGs are present. This is also true for the test configuration in this study. If Vane MVGs are used, the location of (σ p /P w ) max moves downstream, which corresponds to (P w /P o ) min (or M peak ) in Figure 8. For η = 15 • , the effect of MVGs is less significant. For M = 0.89, the distributions of surface pressure fluctuations are shown in Figure 12. These are similar to those for M = 0.83, but the location of (σ p /P w ) max moves farther downstream (X* = 3.86), which corresponds to the P w /P o distributions that are shown in Figure 9. The variation in (σ p /P w ) max with/without MVGs is shown in Figure 13. For the baseline case, (σ p /P w ) max is mainly associated with SIBLS and shock oscillation. The value of (σ p /P w ) max decreases if MVGs are used. For CRV MVGs, the primary flow passes through the split of the counter vanes and interacts with the flow that develops from each half of the device. The counter-rotating vortices are larger and stronger or flow becomes highly three-dimensional. There is a similar mechanism for Ramp MVGs. For the baseline case, extensive SIBLS occurs for β ≈ 20 [4]. In Figure 13, the data for β = 18.5 do not follow the global trend because of the presence of MVGs on M peak and shock oscillation. This determines that the effectiveness of MVGs depends on the size of separation bubble.

Surface Oil Flow Visualization
Oil-flow visualization technique was used to determine the surface flow pattern. Examples for η = 13° and M = 0.83 with/without MVGs are shown in Figure 11. For the baseline case (Figure 14a), the surface flow pattern is primarily two-dimensional, in that there is little variation in the separation and reattachment locations in the spanwise direction. Separation and reattachment respectively occur at X* ≈ 2.14 and 8.14 (or the normalized separation length, L* = L/δ, of 6.00).
The presence of MVGs results in a corrugated pattern for separation and reattachment, which is most significant for CRV and Ramp types because there are counter-rotating vortices. There are sharper spikes than for Vane MVGs. This agrees with the results of the study by Lee et al. [31]. Figure 14 is used to determine the mean position for separation and reattachment.
The presence of MVGs delays flow separation, particularly for the CRV type. The respective mean separation and reattachment locations are at X* = 4.71 and 9.86 (or L* = 5.15), as shown in Figure 14b. For all MVGs, mean separation occurs slightly farther downstream. Mean reattachment for Ramp MVGs occurs farther upstream. A plot of L* versus Mpeak is shown in Figure 15. Ramp MVGs result in a greater reduction in L* and Vane MVGs has the least effect. Delery [29] showed that separation appears and develops more slowly and more progressively in three-dimensional flows than in two-dimensional flows. In terms of (σ p /P w ) max (Figure 13), CRV MVGs are the most effective at control for SIBLS. Ramp MVGs are less effective because the vortices are weaker and there is rapid lift-off from the surface [27]. Lee and Loth [30] determined that streamwise vortices are sustained at greater Mach numbers. The onset of flow separation occurs farther downstream for M = 0.89 but the MVGs are still effective to mitigate surface pressure fluctuations.

Surface Oil Flow Visualization
Oil-flow visualization technique was used to determine the surface flow pattern. Examples for η = 13 • and M = 0.83 with/without MVGs are shown in Figure 11. For the baseline case (Figure 14a), the surface flow pattern is primarily two-dimensional, in that there is little variation in the separation and reattachment locations in the spanwise direction. Separation and reattachment respectively occur at X* ≈ 2.14 and 8.14 (or the normalized separation length, L* = L/δ, of 6.00).
The presence of MVGs results in a corrugated pattern for separation and reattachment, which is most significant for CRV and Ramp types because there are counter-rotating vortices. There are sharper spikes than for Vane MVGs. This agrees with the results of the study by Lee et al. [31]. Figure 14 is used to determine the mean position for separation and reattachment.
The presence of MVGs delays flow separation, particularly for the CRV type. The respective mean separation and reattachment locations are at X* = 4.71 and 9.86 (or L* = 5.15), as shown in Figure 14b. For all MVGs, mean separation occurs slightly farther downstream. Mean reattachment for Ramp MVGs occurs farther upstream. A plot of L* versus M peak is shown in Figure 15

Self-Sustained Shock Oscillation
THM was used to determine fs with/without MVGs, as shown in Figure 16. For the baseline case, the value of fs (640-1320 Hz) decreases as Mpeak increases [4]. There is a significant reduction in the value of fs when MVGs are installed, particularly for the CRV type. This is similar to the variation in (σp/Pw)max. The data also shows that the decrease interactions. L is used as the characteristic length scale in this study. The Strouha St (=fsL/Up) is determined, as shown in Figure 17, where Up is the peak velocity. Th range from 0.12 to 0.09 for the baseline case, when there is an increase in Mpeak. T ence of MVGs stabilizes the shock, so the value of St is smaller, less than 0.06. T shock oscillation is damped, particularly for M = 0.89 (St < 0.02).

Self-Sustained Shock Oscillation
THM was used to determine f s with/without MVGs, as shown in Figure 16. For the baseline case, the value of f s (640-1320 Hz) decreases as M peak increases [4]. There is a significant reduction in the value of f s when MVGs are installed, particularly for the CRV type. This is similar to the variation in (σ p /P w ) max . The data also shows that the decrease depends on M rather than η. The respective values for the CRV type for f s are 535 Hz and 70 Hz for M = 0.83 and 0.89. depends on M rather than η. The respective values for the CRV type for fs are 535 70 Hz for M = 0.83 and 0.89. Gonsalez and Dolling [32] showed that the peak fs is nondimensionalized in of the Strouhal number with the intermittent region length as the characterist scale, whose value lies within the range of 0.01-0.03 for compression ramp and interactions. L is used as the characteristic length scale in this study. The Strouhal St (=fsL/Up) is determined, as shown in Figure 17, where Up is the peak velocity. Th range from 0.12 to 0.09 for the baseline case, when there is an increase in Mpeak. T ence of MVGs stabilizes the shock, so the value of St is smaller, less than 0.06. T shock oscillation is damped, particularly for M = 0.89 (St < 0.02).   Gonsalez and Dolling [32] showed that the peak f s is nondimensionalized in the form of the Strouhal number with the intermittent region length as the characteristic length scale, whose value lies within the range of 0.01-0.03 for compression ramp and blunt-fin interactions. L is used as the characteristic length scale in this study. The Strouhal number St (=f s L/U p ) is determined, as shown in Figure 17, where U p is the peak velocity. The values range from 0.12 to 0.09 for the baseline case, when there is an increase in M peak . The presence of MVGs stabilizes the shock, so the value of St is smaller, less than 0.06. Therefore, shock oscillation is damped, particularly for M = 0.89 (St < 0.02).

Conclusions
This study conducts experiments involving transonic convex-corner flow (M = 0.83 and 0.89) with/without the presence of MVGs (h* = 0.2). The mean/fluctuating surface pressure distributions, the local peak Mach number, the peak pressure fluctuations and the shock oscillation (f s and St) are determined. The presence of MVGs induces streamwise vortices, so there is a greater expansion wave near the corner's apex (the shock and compression move downstream) and flow separation is delayed. This is also true for the peak pressure fluctuations and the f s . Shock oscillation is reduced, particularly for M = 0.89.
The surface oil flow pattern for CRV MVGs is three-dimensional and the counterrotating vortices delay flow separation. For all MVGs, mean separation occurs farther downstream. Mean reattachment for Ramp MVGs occurs farther upstream. The Ramp MVGs produce the shortest separation length. The presence of MVGs is beneficial for deflected trailing-edge flaps as a high-lift device at transonic speeds.  Strouhal number, f s L/U p U p peak velocity X coordinate along the centerline of model surface X* normalized streamwise distance, X/δ β similarity parameter, M 2 η/ 1 − M 2 δ incoming boundary-layer thickness η convex-corner angle, degree σ p standard deviation of surface pressure σ p /P w fluctuating pressure coefficient