Effects of Perforated Plates on Shock Structure Alteration for NACA0012 Cascade Configurations

Abstract

To alleviate the shock boundary layer interaction adverse effects, various active or passive flow control strategies have been investigated in the literature. This research sheds light on the behavior of perforated plates as passive flow control techniques applied to NACA0012 airfoils in cascade configurations. Two identical perforated plates with shallow cavities underneath are accommodated on the upper and lower surfaces of each airfoil in the cascade arrangement. Six different cascade arrangements, including a baseline configuration with no control applied, are additively manufactured, with different perforated plate orifice sizes in the range of 0.5–1.2 mm. A high-speed wind tunnel with Schlieren optical diagnosis and wall static pressure taps is used to investigate the changes in the shock waves pattern triggered by the perforated plates. Steady 3D density-based numerical simulations in Ansys FLUENT are conducted for further analysis and validation. In the cascade configuration, the perforated plates alter the shock structure, and the strong normal shock wave is replaced by a weaker X-type shock structure. Eventually, a 1% penalty in overall total pressure loss is induced by the perforated plates because of the negative loss balance between the reduced shock losses and the enhanced viscous losses. Further studies on perforated plate geometrical features are needed to improve this outcome in a cascade arrangement.

1. Introduction

When the shock wave induces an interaction on the boundary layer, this is known as shock boundary layer interaction (SBLI), and this occurs in nearly all high-speed flows beyond transonic speeds. This phenomenon might arise in external aerodynamics (airfoils or other control surfaces), but it is also very common for internal flows (turbomachinery blade rows or intakes). Irrespective of the flow nature, the SBLI is responsible for non-negligible losses that eventually affect the overall performance of the machine where they develop. Normal shock waves alone are contributing with the wave drag to the overall drag budget, whereas their interaction with the boundary layer usually leads to separation and enhanced viscous drag. Additionally, the SBLI introduces high unsteadiness in the flow field, with strong detrimental effects on the performance of either single airfoils, or cascade configurations.

In recent decades, special attention has been dedicated in the scientific community to shock wave effects and interactions with the boundary layer [1], together with different control strategies to alleviate the repercussions of the SBLI [2]. Since the shock strength and boundary layer characteristics might be expected to have the primary impact on the SBLI, both active (plasma jets [3], air-jet vortex generators [4], micro jets [5], suction [6], or bleed [7]) and passive control methods (slots [8], bumps [9], micro vortex generators [10], perforated plates [11]) have been developed, focusing either on changing the shock structure into a less dissipative one (perforated plates, bumps) or on altering the boundary layer before the interaction region (vortex generators) [12].

The use of perforated plates with a cavity underneath as passive control methods proved to be effective in changing the shock structure from a strong normal shock to a ‘lambda’-type shock structure [13], or eventually to weak compression waves [14] with reduced wave-induced drag. The perforated plates control approach was applied to both external (single airfoils [15,16,17], helicopter blades [18]) and internal flows (nozzles [19], intakes [20,21], vaned diffusers [22]). The perforated plates, as thoroughly investigated in the EUROSHOCK I [23] and II [24] projects, are usually installed in the normal shock anchoring position so that the cavity beneath favors a flow recirculation from the high-pressure region downstream of the shock structure region to the low-pressure region upstream. The onset of this transpiration flow creates a fluid ramp upstream of the normal shock, with the possibility of altering its structure into an oblique one. The magnitude of the transpiration mass flow rate [25] depends not only on the high pressure downstream of the normal shock wave but also on the cavity and perforated plates’ geometrical parameters (e.g., cavity depth, hole diameter, plate thickness, length of perforated area). Extensive parametric studies were performed by Raghunathan [26] to assess the main parameters influencing the effects of the perforated plates on the flow, such as shock strength, porosity, location of porous region, type of porosity distribution, type of porous surface, or cavity depth. This study revealed no appreciable improvement in drag reduction with the change in cavity depth. However, the forward-facing holes inclined 60° to the surface normal, and together with rounded or chamfered corners, they proved to perform better than normal to surface holes with sharp corners.

The manufacturing of perforated plates with shallow cavities underneath is challenging as very small holes, with diameters below 1 mm [27], are usually required. Larger holes, or poor manufacturing, might lead to significant disturbances in the flow, leading to enhanced viscous losses and unsteadiness in the flow. Precision drilling is usually carried out using laser or electron beam technologies. The successful employment of perforated plates is thus conditioned by the loss balance between reduced shock losses (transition to less dissipative shock structures) and increased viscous losses (increased roughness induced by the perforated plate) [28]. Moreover, in the presence of strong transpiration flows, the boundary layer may be significantly degraded in the control region due to fluid injection [24]. Thus, the gain in terms of wave drag reduction might be canceled, or even surpassed, by viscous drag increase.

Typically, the experimental infrastructure for studying the effects of perforated plates on different flow surfaces relies on high-speed wind tunnels with optical access for either Schlieren visualization [14] or LDV (Laser Doppler Velocimetry) [11] measurements. Whereas the Schlieren technique is mainly used for qualitative investigations of the changes in the shock structure when passive flow control is applied, the LDV can provide quantitative information regarding the transpiration flow on the perforated plate, together with boundary layer measurements. The research apparatus is usually complemented by high-speed wall static pressure ports along the flow path, together with pressure ports in the shallow cavity [13].

Usually, the experimental apparatus yields limited data, and complex numerical modeling [29,30] is needed to validate and extend the results. Several attempts to numerically model the flow through perforated plates are documented in the literature [31,32,33]. Dedicated boundary conditions for transpiration flows were developed and included in RANS in-house flow solvers and successfully validated against experimental data for pressure distributions, velocity profiles, Mach contours, or boundary layer properties. An immersed boundary approach coupled to a high-speed turbulent flow solver was used by Roy et al. [34] to study the SBLI on a flat plate in the presence of porous media at Mach 1.3. A parametric study revealed an enhancement of viscous losses once the medium porosity was increased.

The aim of this paper is to experimentally and numerically assess the behavior of perforated plates with shallow cavities underneath as a passive control strategy applied to NACA0012 cascade configurations. One baseline batch of airfoils with no control applied, and five different batches with perforated plates applied (holes diameter ranging from 0.5 mm to 1.2 mm), are additively manufactured and tested in a high-speed wind tunnel with optical access for Schlieren visualization. The experimental results, in terms of wall static pressure and instantaneous Schlieren images, are compared against the numerical data obtained with Ansys FLUENT. Density gradient and Mach number contour plots, together with static pressure distributions on the airfoils and boundary layer velocity profiles, are thoroughly analyzed to meet the research objective of the current paper, i.e., to characterize the change in the shock structure due to the use of perforates plates. As the scientific literature on the topic at hand is scarce, this paper’s innovative character relies on the implementation of perforated plates in a new working environment with cascade effects.

2. Wind Tunnel Facility

A high-speed ‘Eiffel’-type open wind tunnel manufactured by Gunt (HM 172 [35]) is employed for the entire experimental campaign. The facility can be operated continuously at different flow regimes thanks to a vacuum pump downstream that draws air from the environment at ambient conditions. The tunnel is delivered with three interchangeable upper walls to achieve different axial flow contours and so different flow Mach numbers up to 1.8. The side walls and the bottom wall are flat by design. In the current work, the flat upper contour is employed, leading to a rectangular flow channel of 25 mm width, 100 mm height, and 900 mm length.

The facility, with all its components, is depicted in Figure 1. The ‘Eiffel-type’ air inlet, together with the inlet honeycomb (1), deliver straight uniform flow to the test section via a rectangular wind tunnel duct. A Schlieren optics system (2) is aligned on the test section (3), which is optically accessible from the environment by means of two plexiglass windows. The control panel (4) contains the fan throttling button (5), together with the emergency shut down button and a manometer, as an analog user interface. The switch cabinet (6) connects the entire facility to the power supply line. The static pressure on the bottom wall of the wind tunnel is recorded by 18 static ports of 0.5 mm in diameter. The static taps are placed 50 mm apart over an overall flow channel length of 850 mm (Figure 2). The distance between pressure tap #1 and pressure tap #18 (i.e., the above mentioned 850 mm) subsequently defines the numerical domain. The anchoring point of the airfoils in the test section is placed 550 mm downstream of pressure tap #1 and directly corresponds to pressure tap #12. The pressure signal is captured by a digital scanning valve (7) and transmitted to a PC via an USB connection, handled by dedicated software. The measurement range of the Sensortechnics HCX001D6V differential pressure gauges inside the scanning valve is [0, −1] bar with an accuracy of 10 mbar.

The Z-type Schlieren system [36] depicted in Figure 3 allows for the direct observation of the high-speed high-frequency phenomena in the wind tunnel test section by means of a Phantom VEO710L high-speed camera operated at 24 k FPS at 512 × 512 pixels resolution with 1 µs exposure time. Two f/8 11 cm diameter parabolic mirrors capture the incoming beam from a point light source, convert it into a parallel beam that passes through the test section, and subsequently converge it by an adjustable knife diaphragm for light cutoff. To keep the optical distortions small, the beam angles between the camera and the light source with respect to the main optical axis are set to 10° [37].

To accommodate the three airfoils cascade configuration rather than single airfoils or single bluff bodies, the front and back optical access windows were both replaced with new monoblock machined plexiglass parts. The front window with three holes for cascade setup is presented in Figure 4, together with the six sealing gaskets that were additively manufactured by Bambu Lab X1-Carbon 3D Printer (manufactured by Shenzhen Tuozhu Technology Co., Ltd., Shenzhen, China) [38] using PLA.

To ensure the correct null incidence and the airfoils alignment in the cascade, a PLA jig (Figure 5) was additively manufactured by the same Bambu Lab X1-Carbon 3D Printer [38]. The high-quality 3D printer allowed for tight tolerances to minimize the alignment error of the airfoils in the cascade setup (Figure 6). Moreover, the jig facilitates fast and accurate mounting and dismounting of the entire cascade whenever the geometry of the airfoils is altered between various tests during the experimental campaign. The sealed test section after the incidence jig was extracted, and this is shown in Figure 7.

Based on the existing literature manufacturing recommendations for perforated plates with cavities underneath included in single airfoils [14], PLA additive manufacturing by the Bambu Lab X1-Carbon 3D Printer [38] was used for all the airfoils. This allows for rapid prototyping with precise (tight tolerances), accurate (smooth aerodynamic surfaces), and low-cost manufactured airfoils, especially in the perforated area. For the current experimental campaign, six different batches of three airfoils each were manufactured, and their main geometrical characteristics are reported in

Table 1. Main characteristics of the NACA 0012 airfoils as part of the cascade setup.

Name	Material	Hole Diameter	No. of Holes
A1	PLA	-	-
A2	PLA	0.5 mm	19 × 24
A3	PLA	0.65 mm	15 × 18
A4	PLA	0.8 mm	12 × 15
A5	PLA	1 mm	10 × 12
A6	PLA	1.2 mm	8 × 10

. All the airfoils are NACA 0012 with a 75 mm chord length. One batch of airfoils will be referred to as the baseline (A1), with no control applied (Figure 8). The other five batches (A2–A6) have different hole diameters for the perforated plates, in the range 0.5–1.2 mm, as shown in Figure 9. For the airfoils employing the passive control strategy, the perforated plate with a cavity underneath is applied on both the suction and the pressure side at the same chordwise location (Figure 10 and Figure 11) as the airfoils are operating in the cascade configuration. The cavity length is 20 mm measured along the camber mean line of the airfoils. The perforated plates’ orifices are placed two diameters apart both chordwise and spanwise. All the airfoils are fixed in the plexiglass windows using 3 mm nominal diameter screws and threaded insert nuts in the airfoils, as shown in Figure 10 and Figure 11.

3. Results

The wind tunnel described in the previous section, along with the Z-type Schlieren system, was used to experimentally investigate the NACA 0012 cascade configurations. Six different configurations were installed and tested in the wind tunnel, as detailed in Table 1. The wind tunnel’s regime remained the same across all tests, thanks to its capability to set and monitor the vacuum pump’s operating parameters. Based on the inlet total pressure and port #1 static pressure, the computed inlet Mach number was 0.45 for all cases, also subsequently validated by numerical simulations. The high-speed camera was operated at 24,000 frames per second with a resolution of 512 × 512 pixels and an exposure time of 1 μs.

Instantaneous Schlieren images for airfoils A1–A6 in cascade configuration are presented in Figure 12. From top to bottom, the airfoils in the cascade will be referred to as 1, 2, and 3, and they are also indicated in Figure 12a. The optical knife slices the light in a way that high-pressure and high-density areas appear dark (e.g., the airfoil’s stagnation point and the shock waves), while low-pressure and low-density areas appear light (e.g., the local acceleration near the leading edge). The white or dark spots interfering with the visualization, particularly near the airfoils, result from internal stress on the plexiglass windows caused by tightened screws. The vertical ‘interference-like’ pattern is also attributed to the quality and inherent manufacturing imperfections of the plexiglass.

In the baseline case, where no control is applied on the airfoils (Figure 12a), normal shocks form not only in the main flow channels but also in the lateral ones (formed with the wind tunnel walls). The shocks in the main channels exhibit a normal structure without any oblique characteristics, indicating that they are strong normal shocks with a significant associated total pressure loss. Despite the presence of these normal shocks, the Schlieren visualization highlights tiny flow detachment regions downstream of the normal shock on both the upper and lower surfaces of airfoil 2 in the cascade (see the orange rectangle in Figure 12a).

In the case of the 0.5 mm perforated plate shown in Figure 12b, the shock structure is noticeably altered by the passive control strategy on both the upper and lower surface of airfoil 2. The control mechanism transforms the normal shock into an oblique ‘lambda-type’ structure, where two oblique shocks from the lower and upper neighboring surfaces intersect to form an ‘X’-shaped pattern (see the orange rectangles in Figure 12b). In the upper lateral channels adjacent to the wind tunnel wall, a single oblique shock is found. Although the shock pattern shifts to an oblique configuration in the inner channels, normal shocks still develop at the end of the control region (see the white rectangles in Figure 12b). Additionally, significant flow detachment is observed starting from the beginning of the control region on the lower surface of airfoil 1 and the upper surface of airfoil 3, possibly due to an overly strong blowing effect in the cavity, increased surface roughness, or an intrinsic change in the flow angle. Compared to the baseline case in Figure 12a, the middle airfoil exhibits earlier flow detachment due to enhanced roughness introduced by the perforated plate too. However, the flow detachment is considerably weaker than the one experienced by the lower surface of airfoil 1 and the upper surface of airfoil 3, as mentioned before.

In the case of the 0.65 mm perforated plate shown in Figure 12c, a similar alteration of the shock structure is observed, as presented in Figure 12b. However, multiple compression lines intersect to form a complex ‘X’-shaped pattern. The earlier flow detachment on airfoil 2, and especially on the airfoil 1 lower surface and the airfoil 3 upper surface, is also present. A shock wave is also present at the end of the control region of airfoil 2 for both the upper and lower surfaces (see the orange rectangle in Figure 12c). This shock creates more favorable conditions for the flow detachment on the lower surface of airfoil 1 and the upper surface of airfoil 3.

As the hole size increases to 0.8 mm (Figure 12d), 1 mm (Figure 12e), and eventually to 1.2 mm (Figure 12f), more pronounced ‘X’-type oblique shock structures emerge in the inner channels. These structures are again accompanied by a downstream shock at the end of the control region.

The experimental static pressure measured on the bottom wall of the wind tunnel is reported in Figure 13 and Figure 14 for all tests A1–A6. For each test, 20 s of continuous data were recorded and then time-averaged. Based on the T-student distribution and the number of data points acquired for each static pressure port, the confidence interval for each experimental point was estimated using 1.725 as the distribution parameter [39]. The axial position of the cascade is marked with a black thick line on the bottom axis. The anchoring point of each airfoil in the cascade, which is actually the quarter-chord point of the airfoil, axially coincides with static pressure tap #12. Thus, pressure tap #11 is located 31.25 mm upstream the cascade leading edge, while pressure tap #13 is located 6.25 mm upstream the cascade trailing edge.

For the first eleven static pressure ports, the static pressure readings and inherently the decreasing slope statistically coincide given the computed confidence interval. The perforated plate hole diameter impacts the minimum static pressure value and the pressure recovery profile towards the wind tunnel exit. Thus, in the low-pressure region, marked in red, minimum static pressure is achieved for the 1 mm configuration, while the largest static pressure is obtained for the 0.8 mm case. Whereas the minimum for the no control, 0.65 mm, and 1.2 mm cases is located very close to one another, the 0.5 mm case leads to a slightly lower minimum pressure.

To complement the experimental results, numerical simulations were performed in Ansys FLUENT for two specific cases: the baseline case with no control applied and the passive control case with 0.5 mm perforated plate arrangement. The computational domain, which perfectly matches the wind tunnel geometry and cascade configuration, is presented in Figure 15 (a cascade with applied flow control is used as an example here). The side walls, along with the top and bottom walls and the surface of the airfoils, are modeled as adiabatic no-slip walls. For the inlet and outlet surfaces, specific boundary conditions are applied according to the experimental measurements in the wind tunnel. The atmospheric pressure during the experimental campaign was set to 101,313 Pa, while the inlet honeycomb total pressure loss was previously estimated to be around 1% [14]. Thus, an inlet total pressure of 100,300 Pa was imposed on the inlet patch. The overall setup of the numerical simulation is summarized in

Table 2. Three-dimensional wind tunnel simulation setup.

Simulation	RANS (Steady) Density-Based
Turbulence model	k-omega SST
Inlet	Total pressure: 100,300 Pa Total temperature: 297.4 K
Working fluid	Air (ideal gas)
Reference pressure	0 atm
Outlet	Static pressure: 72,319 Pa
Walls	Smooth adiabatic no-slip wall

. The use of this setup to such wind tunnel cases is appropriate, as documented in [14].

The computational domain for both scenarios was discretized using unstructured grids in Ansys Meshing. For the sake of the example, only the 35 million cells grid for the 0.5 mm control case is shown in Figure 16. The baseline case grid, with less complexity as the no-control region, was discretized following the same meshing strategy and using 30 million cells.

Local refinement was applied in the cascade region and in the perforated plate control region to capture the large shock gradients and the recirculation flow inside the cavity (Figure 16b). The overall mesh size is $5·{10}^{-3} \mathrm{m}$, while the reference size for the two refinement zones is $1.5·{10}^{-3} \mathrm{m}$ in the cascade region and $4·{10}^{-4} \mathrm{m}$ in the cavity region. Inflation at the airfoil surfaces (Figure 16e,f) was also set to accurately capture boundary layer gradients (13 layers, growth rate 1.35, transition ratio 0.11). The inflation was also applied to the wind tunnel lateral walls, with nine layers, a growth rate of 1.5, and a first-layer height of $1.25·{10}^{-6} \mathrm{m}$. The presence of the perforated plate with a cavity underneath posed a significant meshing challenge (Figure 16c), requiring a tradeoff between the number of cells and accuracy. In the current grid, each hole is resolved with 30 cells on its diameter (Figure 16d).

To verify that the mesh is compatible with the turbulence model and flow regime, the y+ distribution is plotted in Figure 17 for both cases. In the baseline ‘no-control’ case, y+ values are generally below 1, except for a few cells near the lateral walls. For the ‘control’ case, y+ values are somewhat higher, particularly in the holes where shear effects are more pronounced.

Figure 18 shows a grid independence study performed for the 0.5 mm passive control case. Five cases were included in the study with grids in the range of 15 to nearly 45 million cells. For the coarsest grid, the cavity refinement zone reference size was set to $9·{10}^{-4} \mathrm{m}$ and then gradually reduced down to $1·{10}^{-4} \mathrm{m}$ for the finest one. Beyond 35 million cells, additional refinement proved to have almost no impact on the performance parameter defined as the non-dimensional total pressure loss ratio between the inlet and the outlet of the computational domain.

The density gradient, Mach number, and static pressure distributions in the proximity of the cascade are plotted in Figure 19, Figure 20 and Figure 21. Examining the density gradient in Figure 19a, the baseline case reveals the development of normal shocks in both the inner and outer channels, with the latter being confined by the wind tunnel’s lateral walls. Due to lateral walls effects, the stagnation points for airfoils 1 and 3 are slightly shifted. As a result, airfoil 1 experiences a negative incidence, making its lower surface more susceptible to shock–boundary layer interactions. Thus, the shock wave in conjunction with the negative incidence leads to flow separation downstream of the normal shock, as confirmed by the Mach number distribution in Figure 20a too. The same applies to the upper surface of airfoil 3 in positive incidence operating conditions. In contrast, the upper and lower surfaces of airfoil 2, which operates at nearly zero incidence, seem unaffected by the shock wave.

In the case of the 0.5 mm perforated plate, the density gradient in Figure 19b shows a significant change in the shock structure introduced by the passive control. In the lateral channels, a ‘lambda’-type shock is followed by a normal shock, causing flow separation on the upper surface of airfoil 1 and on the lower surface of airfoil 3, together with boundary layer thickening at the lateral walls. Inside the inner channels, the perforated plate induces an ‘X-type’ shock structure with a lower intensity than the original normal shock. The null incidence flow remains valid for airfoil 2, considering the position of the stagnation point. Moreover, this airfoil did not experience flow separation once the passive control was implemented, and just a small thickening of the boundary layer was observed. However, for airfoils 1 and 3, flow separation still occurs on the lower surface of airfoil 1 and the upper surface of airfoil 3. Compared to the baseline case, where separation was caused by shock–boundary layer interaction and incidence (either negative or positive), the flow detachment here might also be attributed to the increased roughness introduced by the perforated plates, with less contribution from the shock, which is weaker now.

A reference axial line corresponding to the wind tunnel bottom wall (Figure 22a) is used to extract the numerical static pressure distributions for both scenarios (no control and passive control 0.5 mm). The numerical data are compared against the experimental measurements in Figure 23. The black line indicates the actual axial position of the cascade in the wind tunnel. The decreasing slope of the first eleven static pressure readings is correctly captured by CFD for both cases. However, the CFD figures lie slightly outside the computed confidence intervals due to limited accuracy in estimating the total pressure loss on the inlet honeycomb, which dictates the inlet total pressure imposed in the simulations. The maximum difference between the CFD and experimental results for the first eleven ports is around 2.5% for port number 9.

The sharp decrease in static pressure, corresponding to port number 12, is correctly predicted by CFD, while the axial position this phenomenon takes place in shows a very good match between CFD and the experiment. The minimum numerical static pressure achieved with the 0.5 mm passive control is lower than the one in the no-control case, as it is also the case judging the experimental results from port number 12. As explained before, based on Figure 20a, the shock structure in the lower end channel (confined between the wind tunnel flat wall and the lower surface of airfoil 3) is altered from a strong normal shock to an oblique configuration. Despite the fact that perforated plates locally increase the viscous losses, the reduction in shock loss dominates the energy balance in this case, allowing for a higher acceleration and thus a lower minimum static pressure for the passive control case, as also confirmed by the experiments.

For the last six static pressure ports, the CFD passive control case pressure recovery matches the experiment within the confidence interval. Conversely, the CFD slope for the no-control case reduces a bit around port number 13, which slightly departs the numerical and experimental results.

A reference axial line is also defined for the mid lower channel (Figure 22b) to extract key quantities such as static pressure, Mach number, and density gradient for both cases.

In Figure 24, the static pressure along the lower mid-channel reference line shows that the two cases are quite similar until the flow reaches the local acceleration region. The minimum static pressure in the no-control case is lower than that achieved with the control mechanism. The pressure recovery in the no-control case is also slightly more abrupt compared to the passive control case, indicating that the perforated plate reduces the shock intensity. The Mach number results in Figure 25 support the pressure distribution observations, showing a higher maximum Mach number in the no-control case and a very sharp decrease to subsonic values. Finally, the density gradient plot in Figure 26 clearly demonstrates the reduction in shock-induced total pressure loss achieved by the perforated plate passive control strategy.

The non-dimensional streamwise velocity profiles on the upper surface of airfoil 2 are reported in Figure 27 for nine rakes, for which the relative chordwise position is depicted in Figure 27j. Three rakes are placed upstream to the cavity, three rakes are positioned in the cavity region, while the last three are located downstream to the cavity. Spanwise, all the rakes are placed in the same plane that corresponds to the middle row of the orifices. The reference inlet velocity used for non-dimensionalizing the plots is 154.7 m/s. For the first three rakes in Figure 27a–c, no significant difference is observed in the velocity profile slope. However, slightly lower velocities are achieved in the passive control case due to an upstream influence from the perforated plate. For Figure 27a,b, the non-dimensional velocity increases from 1.5 to 2 due to airfoil curvature, and, more precisely, due to the convergence of the cascade channel. For rake 4 in Figure 27d, the perforated plate blowing effect, together with the additional viscous losses induced by the orifices, lead to a thickened and less ‘filled’ velocity profile for the passive control case compared to the baseline one. For rakes 5 and 6, the velocity profile slopes are, again, very close, as the normal shock wave in the baseline case interacts with the boundary layer, thickening it. Larger velocities are achieved in the passive control case, as the normal shock in the baseline one is responsible for significant total pressure loss and, hence, velocity loss. This is supported by the position of the normal shock wave with respect to rakes 5 and 6 according to the airfoil static pressure distribution shown in Figure 28. For the last three rakes, in Figure 27g–i, the velocity profile for the passive control case is less ‘filled’, which is an indication of momentum deficit close to the surface of the perforated plate.

The static pressure distribution on the second airfoil of the cascade is plotted in Figure 28 for both the no-control and 0.5 mm passive control cases. For the control case, by ‘holes’, one refers to the distribution along the mid row of orifices, while by ‘no holes’, one understands the distribution in between the midspan holes’ rows. The slope of the initial pressure decrease is the same for both cases prior to the perforated plate control region. Additionally, the slopes for the upper and lower surfaces of the airfoil coincide as it is a symmetric airfoil which operates in null incidence conditions. The no-control case decreases in pressure and almost reaches 0.4 bar, followed by a normal shock that leads to a 0.55 bar value with quite a sharp gradient. The perforated plate alters the pressure profile into a plateau with a little bump where the control region begins. This corresponds to a weaker shock wave, whose position is shifted towards the beginning of the cavity compared to the original position of the normal shock wave in the no-control case.

Figure 29 presents a qualitative comparison between the numerical and experimental results for the baseline no-control case and the 0.5 mm perforated plate passive control case. In the baseline case (Figure 29a,c), the position of the normal shock waves in the inner channels is slightly downstream in the experimental results compared to the numerical predictions. This discrepancy might be justified by a cumulative contribution from inlet honeycomb total pressure loss estimation (which sets the inlet total pressure boundary condition for CFD), manufacturing imperfections (mainly roughness), and imprecise alignment of the cascade (even if a dedicated jig is used, the flows near critical conditions are very sensitive to minor area, and so flow path, changes). Additionally, for the baseline case, both the numerical and experimental results capture the shift in the stagnation point positions. However, the downstream flow detachment on the lower surface of airfoil 1 and upper surface of airfoil 3 is more pronounced in the numerical results, whereas the experiment shows only minor flow separation.

The alteration of the shock structure due to the perforated plates is captured by both CFD and the experiment in Figure 29b,d. In CFD, the behavior is more uniform and symmetrical, with ‘X-type’ oblique shock waves in the inner channels. In the lateral channels, an oblique shock is followed by a normal one, which is also observed in the top channel of the experiment. Overall, the experiment highlights the changes in shock structure for the inner channels. However, due to the experimental uncertainties mentioned earlier, the flow field in the experiment is not perfectly following the numerical data. Additionally, the two inner channels are not perfectly symmetrical in terms of the observed phenomena.

Comparative mass-averaged numerical results between the baseline case and the 0.5 mm passive control case are presented in

Table 3. Mass-averaged comparative CFD results.

	No Control	Passive Control 0.5 mm
Inlet mass flow [kg/s]	0.410677	0.411173
Outlet mass flow [kg/s]	−0.410333	−0.411065
Static pressure inlet [bar]	0.868955	0.868282
Static pressure outlet [bar]	0.72319	0.72319
Inlet Mach number	0.45	0.45
Inlet velocity [m/s]	154.7	154.7
Total pressure inlet [bar]	1.003	1.003
Total pressure outlet [bar]	0.903356	0.890994
Total pressure loss %	9.93%	11.1%

. For both cases, the inlet-to-outlet mass flow rate imbalance is on the order of ${10}^{-3} \mathrm{kg}/\mathrm{s}$, which is below 0.25% of the wind tunnel mass flow rate. Despite having the same inlet total pressure, the passive control case results in slightly higher overall dissipation (approximately 1%), even though the perforated plates reduce shock intensity. The additional viscous losses introduced by the perforated plates are slightly larger than the reduction in total pressure loss achieved by weakening the shocks. To optimize this outcome, further studies are needed to refine the perforated plate geometry, including factors such as hole inclination, hole pattern, cavity depth, and other relevant design parameters of the perforated plate with cavity underneath.

4. Conclusions

In the current work, the passive flow control technique employing perforated plates with shallow cavities underneath was applied to airfoils in a cascade configuration. As documented in the literature survey section, significant research efforts were dedicated to perforated plates applied to either single airfoils or bare walls, with scarce studies on the behavior of perforated plates in cascade arrangements. The use of flow control techniques in such configurations is highly relevant for high-speed turbomachinery applications, where shock-induced losses have a non-negligible impact on the overall loss budget. The reduction in shock-induced losses coupled with flow separation mitigation might extend the operational envelope for compressors. Additionally, perforated plates might be included inside turbine vanes to provide starting conditions.

In this study, one baseline batch, with no control applied, and five batches of airfoils with perforated plates with orifice diameters in the range of 0.5–1.2 mm were additively manufactured and tested in a high-speed wind tunnel. The Schlieren optical diagnosis revealed a change in shock structure, from a strong normal shock to a weaker ‘X-type’ oblique shock pattern, whose complexity depends on the orifice size.

A steady 3D density-based CFD study was conducted in Ansys FLUENT on the passive control configuration featuring 0.5 mm holes using boundary conditions derived directly from wind tunnel measurements of inlet and outlet pressure. The numerical static pressure distribution along the bottom wall acceptably matched the experimental results, with deviations of 2.5% near the inlet section. Both experimental and numerical Schlieren visualizations confirmed the changes in the shock structure induced by the perforated plate, from a normal shock to an ‘X-type’ oblique shock pattern. However, according to mass-averaged CFD results, an overall 1% increase in total pressure loss was noted in the passive control scenario. This is motivated by the increased viscous losses resulting from the perforated plates that were marginally greater than the reduction in total pressure loss gained from diminishing the shocks’ intensity.

The non-dimensional wall velocity profiles showed boundary layer thickening in the perforated plate area similar to the post-shock boundary layer thickening phenomena in the no-control case. However, in the passive control case, the velocity deficit was smaller, leading to a more energized flow in proximity to the airfoil. The mechanisms responsible for boundary layer thickening were thus different. In the baseline, no-control case, the normal shock wave strongly interacted with the boundary layer. Conversely, in the passive control case, the weaker shock system together with the combined blowing effect from the cavity and the increased roughness of the perforated plate affected the boundary layer.

The static pressure distributions showed a clear alteration of the shock structure when passive control was applied. The strong pressure gradient corresponding to the normal shock in the baseline case was converted into a pressure plateau with a weaker reminiscent gradient at the beginning and at the end of the control cavity.

To make the effect of the perforated plates in cascade arrangements positive, further research is necessary to optimize the geometry of the perforated plate considering aspects such as hole inclination, hole pattern, cavity depth, and other pertinent design parameters of the perforated plate system together with the cavity underneath. Future research also intends to transfer current knowledge to relevant turbomachinery flow channels, either linear cascades for reduced complexity or full annular configurations.