Experimental Identification of a New Secondary Wave Pattern in Transonic Cascades with Porous Walls

Abstract

Turbomachinery shock wave patterns occur as a natural result of operating at off-design points and are accountable for some of the loss in performance. In some cases, shock wave–boundary layer (SW-BLIs) interactions may even lead to map restrictions. The current paper refers to experimental findings on a transonic linear cascade specifically designed to mitigate shock waves using porous walls on the blades. Schlieren visualization reveals two phenomena: Firstly, the shock waves were dissipated in all bladed passages, as predicted by the CFD studies. Secondly, a lower-pressure wave pattern was observed upstream of the blades. It is this phenomenon that the paper reports and attempts to describe. Attempts to replicate this pattern using Reynolds-averaged Navier–Stokes (RANS) calculations indicate that the numerical method may be too dissipative to accurately capture it. The experimental campaign demonstrated a 4% increase in flow rate, accompanied by minimal variations in pressure and temperature, highlighting the potential of this approach for enhancing turbomachinery performance.

1. Introduction

Compressor cascades are essential tools for simulating and studying the behavior of airflow through gas compressor systems. These cascades, which consist of a linear array of blades, replicate the configuration of compressor blades in real-world turbomachinery. Their primary purpose is to investigate aerodynamic characteristics such as pressure distribution, flow separation, and boundary layer behavior under controlled conditions. This is particularly valuable for assessing and improving the performance of blades in high-efficiency, high-performance applications like jet engines and industrial turbines. In compressor studies, cascades enable the examination of complex interactions between the blades and the airflow, with a focus on phenomena like shock–boundary layer interactions (SBLIs). SBLI is a critical aspect of compressor performance, as it can lead to flow separation, increasing drag and potentially causing flow instability, especially at high Mach numbers or adverse pressure gradients. The resulting flow degradation due to SBLIs can significantly impact compressor efficiency. Understanding and managing these interactions is essential for improving overall performance and maintaining stability in high-speed compressible flows [1,2].

Shock wave–boundary layer interaction (SBLI) is a critical aerodynamic phenomenon that plays a vital role in the performance and stability of various high-speed systems, such as transonic wings, helicopter rotor blades, and turbine cascades [3,4]. SBLIs occur when a shock wave interacts with the surface boundary layer, causing complex flow dynamics, such as high pressure, flow separation, and turbulence. This leads to increased drag, reduced lift, and unsteadiness, negatively affecting aerodynamic efficiency and structural stability [5]. To mitigate the adverse effects of SBLIs, a variety of flow control strategies have been studied, broadly categorized into active and passive methods. Active flow control techniques involve the use of external energy to manipulate the flow field and are known for their effectiveness. However, these methods often come with added complexity, higher power consumption, and potential reliability challenges [6]. On the other hand, passive control methods, such as vortex generators, micro-ramp devices, and surface modifications, offer a simpler and more robust approach. These passive techniques do not require external energy input, making them attractive for practical applications due to their simplicity, cost-effectiveness, and reliability [7]. One prominent passive control strategy involves the use of perforated plates with shallow cavities strategically positioned beneath the shock wave. These surface modifications alter the shock wave topology, reduce the severity of shock wave–boundary layer interaction, and minimize the extent of boundary layer separation [8]. By adjusting the cavity’s design parameters, such as its length and the configuration of the perforated plate, the flow characteristics can be significantly influenced. For example, shorter cavities tend to generate large λ-foot shock structures, which are indicative of stronger interaction regions. In contrast, longer cavities often result in the formation of oblique shock waves, which are less intense and lead to more stable flow patterns. Additionally, employing full-chord perforations can create a gradual compression effect, which smooths out the shock structures, further enhancing flow stability and reducing drag [9].

Numerical simulations using computational fluid dynamics (CFD) tools play a crucial role in studying SBLIs and evaluating control methods. These simulations often use turbulence models like Spalart–Allmaras [10] and specific transpiration models to accurately represent surface ventilation effects. Perforated plates have emerged as a practical and effective passive control method to mitigate the adverse effects of SBLIs in high-speed aerodynamic applications [11,12]. These plates, often integrated into passive ventilation systems, allow air to bleed through a strategically designed network of holes, which modifies the boundary layer characteristics and reduces shock-induced flow disturbances. The effectiveness of perforated plates in controlling SBLIs depends on various design parameters, including hole size, spacing, and overall porosity [13]. Generally, smaller holes and higher porosity enhance the bleeding effect, which stabilizes the boundary layer and reduces the likelihood of flow separation. In contrast, larger holes or lower porosity may result in less effective control of SBLIs due to reduced airflow management [14,15,16]. Experimental studies have demonstrated that the use of perforated plates can significantly alter shock wave structures, transforming strong, normal shocks into more favorable oblique shocks or λ-shock patterns. This transformation not only reduces SBLI intensity but also mitigates adverse effects such as increased drag, flow separation, and large-scale flow unsteadiness [17].

Porous bleed systems are another widely studied passive method for SBLI control. These systems allow part of the boundary layer flow to bleed through a perforated surface, reducing the momentum deficit near the wall and stabilizing the boundary layer [18]. Recent studies show that porous bleed systems are effective in controlling both supersonic and subsonic boundary layer flows. However, their effectiveness varies depending on the flow regime, requiring distinct characterizations for supersonic and subsonic conditions due to different flow topologies inside the bleed holes [19]. In supersonic conditions, porous bleed systems have been validated through both experimental and numerical studies, demonstrating that three-dimensional effects significantly influence boundary layer profiles and corner flows, ultimately impacting overall aerodynamic performance [20]. These three-dimensional interactions can lead to variations in pressure distribution and flow separation, which need to be carefully managed to optimize the effectiveness of the porous bleed system. In subsonic flows, the use of porous bleeding introduces a trade-off by increasing near-wall momentum while simultaneously reducing momentum in the outer boundary layer [21]. Despite significant advancements, challenges persist in accurately modeling real-world interactions involving porous bleed systems, especially concerning design parameters such as hole diameter, spacing, and porosity levels. Discrepancies between numerical predictions and experimental results, particularly in complex three-dimensional scenarios, emphasize the need for refined modeling approaches [22,23].

Schlieren visualization is a crucial tool for studying the effectiveness of perforated plates in managing shock wave–boundary layer interactions (SBLIs) [24]. This non-intrusive, optical technique leverages variations in the air’s refractive index to visualize shock waves and other flow features associated with SBLIs. By using schlieren visualization, this enables researchers to directly observe how perforated plates influence the shock interaction region. For example, it visualizes the formation of oblique shock structures and the reduction in shock strength when perforated plates are applied [25]. These visual insights are vital for validating numerical models, ensuring their accuracy in predicting real-world behavior, and optimizing the design of perforated plates for specific aerodynamic applications [26]. Furthermore, schlieren imaging provides valuable information about the transient behavior of SBLIs, helping researchers understand the dynamic response of the flow to passive control strategies.

Using perforated plates as a passive control method in turbomachinery presents several compelling advantages compared to other techniques. Their ability to transform strong shock waves into weaker patterns effectively reduces overall pressure losses, enhancing system efficiency [27,28]. Perforated plates also provide excellent hydraulic performance by minimizing cavitation risks, making them suitable for high-performance applications [29]. Furthermore, their geometric versatility allows for tailored designs that can be easily integrated into existing systems, unlike more rigid methods such as vortex generators [30]. This adaptability, coupled with the potential for net efficiency improvements, positions perforated plates as a superior choice for passive flow control in turbomachinery.

Perforated plates are recognized as an effective passive method for controlling SBLIs, which poses a significant challenge in high-speed aerodynamic applications. The strategic design and implementation of these plates, coupled with advanced diagnostic tools like schlieren visualization, provide valuable insights that can enhance aerodynamic performance. Therefore, this paper focuses on the application of perforated plates on the blades of a vaned diffuser, a critical component of a microgas turbine engine. This study investigates this setup to improve understanding of SBLI control in microgas turbines, contributing to the overall optimization of high-speed systems.

2. Materials and Methods

2.1. Facility Description

The blades examined in this study are integral components of the diffuser vanes typically used in mixed-flow centrifugal compressors, which are crucial for microgas turbine engine applications. These perforated vanes were fabricated using advanced additive manufacturing techniques, specifically through Laser Powder Bed Fusion (L-PBF), also known as Selective Laser Melting (SLM). This cutting-edge manufacturing process provides precise control over the material’s microstructure, enabling the production of complex geometries with intricate details that would be challenging to achieve with conventional manufacturing methods [31].

Figure 1 provides a visual comparison between the vanes with porous walls (for passive flow control) and a reference vane with a plain wall (no flow control). The specific geometric details of the porous vanes are listed in

Table 1. Geometrical details of aerodynamic profiles.

	Main Blade	Splitter
Length [mm]	73	48
Maximum thickness [mm]	7	7
Height [mm]	10	10
Porosity ratio—suction side area [%]	20.5	48
Porosity ratio—pressure side area [%]	30	23
Number of orifices—pressure side	104	88
Number of orifices—suction side	56	88
Hole diameter [mm]	0.75	0.75

. The difference between patterns is significant because, in the classical setup, between the rows of holes, there is a fully solid wall. This solid wall serves as an anchor for smaller shock waves. The design and construction of the facility, including its linear cascade functionality, are comprehensively detailed in [28]. Importantly, the geometry of the manufactured blade closely aligns with the design requirements, demonstrating an average dimensional deviation of only ±0.04 mm from the original 3D CAD model. These small deviations indicate that the flow characteristics predicted by the CFD model are likely to be representative of the actual flow conditions observed in the experimental setup.

The experimental model is integrated into the facility via a 3D-printed PLA adapter designed with a flange that enables the transition from a circular to a rectangular cross-section. This adapter is equipped with both a static pressure sensor and a total pressure sensor for accurate measurements and monitoring.

Figure 2 illustrates the linear cascade setup used for the experiments. The setup consists of five channels, where the channels are described as the space between the main vanes. In the case of the first one, it is the space between the rig’s curved wall (outer wall) and the first main vane.

Figure 3 presents the Piping and Instrumentation Diagram (P&ID) of the facility, which outlines the critical components and equipment required to operate the linear cascade. The P&ID includes the following elements: 1—air source (screw compressor); 2—high-pressure vessel; 3—pressure regulator; 4—flowmeter; 5—static pressure sensor; 6—total pressure sensor; 7—schlieren system, which consists of a laser-pumped white light source (LS-WL1), two f/6 parabolic mirrors with 60″ focal length, a knife edge for light cut-off, and a Phantom VEO 710L high-speed camera; 8—linear cascade experimental model, along with a data acquisition system and a remote control panel.

This comprehensive setup ensures precise control and measurement of flow conditions within the linear cascade facility. The schlieren optics system, in conjunction with the high-speed camera, enables direct visualization of the flow and the detection of shock phenomena.

Light travels from a point-like source with a diameter equal to 1 mm to the first parabolic mirror. It is then collimated between the twin parabolic mirrors. The second mirror reflects the light back, projecting the image of the light source over the test area. The resulting light source image contains all the information from the test area, with the result being a path-integrated representation of the entire region. The schlieren system used to investigate the current phenomenon achieves the necessary sensitivity for visualizing shock waves through two methods. First, it employs beam deflection using the knife-edge technique, with a knife-edge cut-off to enhance contrast. Second, it uses a gradual color filter, which reveals the phenomenon through varying color gradients.

The configuration and positioning schematics of the schlieren system in relation to the experimental model can be observed in Figure 3, while the physical setup is presented in Figure 4. Specific data about the schlieren equipment are presented in

Table 2. Schlieren system equipment and recording parameters.

Equipment Type	Name and Relevant Features	Manufacturer
Two twin parabolic mirrors	Al-plated parabolic mirrors Effective focal length (EFL) = 1524 mm	Edmund Optics (Edmund Optics, Barrington, NJ, USA)
White light source	$\mathrm{LSW}1,\lambda$ = 440–750 nm, 550 mW	Lightsource.tech (Lightsource Tech, Göttingen, Germany)
High-speed camera	Phantom Veo 710L CMOS Recording settings: Spatial resolution (pixels): 1280 × 82,012-bit depth Exposure (color filter): 2 μs Exposure (knife edge): 0.81 μs Sample rate: 400 fps	Phantom Ametek (Vision Research Inc., Charlottetown, PE, Canada)
Knife edge	Cutter blade	-
Color filter	Gradual color distribution, photographic film	Custom Made Exposed Fujifilm (Tokyo, Japan) Superia Extra 400 Photographic, 35 mm fine grain film

. The off-axis beam angles were maintained to be $\alpha <10°$, in order to minimize optical distortions [32].

This facility’s design and instrumentation provide a robust platform for the study and analysis of flow dynamics in advanced turbine applications, allowing for the optimization of diffuser vane designs in microgas turbine engines.

2.2. Numerical Simulation Setup

The numerical model of the linear cascade was developed using the pressure-based solver, Ansys CFX (version 2022 R2), configured with a high-order Total Variation Diminishing (TVD) scheme to effectively capture shock phenomena and complex flow features characteristic of compressible cascade flows [33]. An ideal gas assumption was applied, with density treated as a function of both pressure and temperature, and the energy equation was included to account for the thermal effects associated with shock formation and propagation. Turbulence was modeled using the Shear Stress Transport (SST) k-omega model, selected for its balance between accuracy and computational efficiency, particularly in regions with adverse pressure gradients and separation [34].

To accurately capture shock structures, a comprehensive grid independence study was conducted for the entire cascade, evaluating seven different grid resolutions ranging from coarse to highly refined meshes. Each grid employed unstructured tetrahedral cells with low skewness to ensure solution precision. The coarser grids (30.5, 40.7, and 46.3 million cells) focused on refinement around the airfoil surfaces and the wake region to achieve a y+ value close to 1, critical for near-wall accuracy. Grid variations were introduced by adjusting layer counts in the radial and axial directions and refining the overall mesh geometry to improve flow representation near high-gradient regions. For finer grids (54.5, 64.98, 76.5, and 90.7 million cells), additional refinement in the blade test area increased mesh density, allowing for a more detailed capture of shock wave structures and boundary layer development. The number of layers in the inflation region was also increased to improve boundary layer definition. A gradual transition was applied from the denser regions near the blade to the far field to maintain smooth cell size transitions, minimizing interpolation errors between adjacent cells.

As shown in Figure 5, total pressure drop served as the primary metric for evaluating grid convergence, and the results demonstrated that beyond a resolution of 64.5 million cells, further refinements had minimal impact on solution accuracy, with changes in the total pressure drop dropping below 0.5%. The selected 65 million-cell grid resolution (marked with the red box) balances computational efficiency with accuracy, ensuring that a broad range of flow structures, including detailed shock wave and boundary layer interactions, are well captured in the shock wave region.

The cascade’s three-dimensional grid was generated using Ansys Meshing, resulting in an unstructured grid with approximately 65 million elements. The average orthogonal quality of the grid is 0.77, with a skewness ratio of 0.22. The overall mesh size is 1.4 × 10⁻² m; in the proximity of the test area, one refinement zone was applied with a size of 5 × 10⁻⁴ m. To accurately capture detailed pressure gradients and the interactions between shocks and boundary layers (SBLIs), mesh refinements were applied along the vane surfaces and within the flow channels (see Figure 6 and Figure 7). The inflation feature of the blade surface is defined as having 15 layers; the first layer has a height of 1 × 10⁻⁶ m and a growth rate of 1.3. Inflation is also defined for the lateral walls of the cascade with 7 layers, a transition ratio of 0.15, and a growth rate of 1.3. The y+ distribution on the blade surface has a maximum of 1.5, which ensures adequate resolution of near-wall turbulence effects. The perforated plate area exhibits y+ values up to 25 due to heightened shear forces in this region, balancing computational efficiency with acceptable near-wall resolution. The mesh was further validated to ensure numerical stability and accuracy, particularly in regions sensitive to SBLIs and other high-gradient flow features (Figure 8).

Figure 9 illustrates the computational domain, including boundary conditions and overall dimensions. The numerical simulations were designed to closely replicate the experimental campaign conditions. At the inlet, a total pressure of 187,000 Pa and a total temperature of 294 K were specified. The outlet boundary condition was set to an atmospheric pressure of 101,150 Pa, matching the conditions observed during the experimental campaign. This setup corresponds to a working point with an approximate gauge static pressure of 80,000 Pa.

The sidewalls, top, and bottom boundaries of the computational domain, as well as the vane surfaces, were modeled as adiabatic walls with no surface roughness, ensuring no heat transfer across these boundaries. The simulation was executed as a steady-state problem, with convergence criteria set to residuals below 10⁻³ for mass, momentum, and energy, ensuring reliable and accurate results. Additionally, residuals were carefully monitored to ensure they dropped below a predefined threshold of 10⁻⁵ for mass, momentum, and energy. To further validate the reliability and stability of the simulation results, the mass flow rates at the inlet and outlet boundaries were cross-checked, confirming consistent and accurate flow behavior throughout the system.

For the experimental campaign, measurement uncertainties associated with the pressure transducer used at the entrance of the linear cascade were assessed. The transducer’s accuracy is specified as 0.8% of the measured pressure, according to the manufacturer’s manual [35]. To quantify the uncertainty, the root-sum-square method was applied, allowing for the combination of the transducer’s inherent accuracy with other relevant sources of variability. This analysis yielded an estimated uncertainty of approximately 640 Pa, reflecting the instrument’s accuracy over the range of pressures measured during the experiment.

3. Results

3.1. Experimental Results

This study employs knife-edge cut-off schlieren and color schlieren techniques, both relying on detecting refractive index variations caused by density gradients in the observed phenomenon. The primary goal was to qualitatively identify and localize shock structures, focusing on their presence, distribution, and characteristics. Challenges in the experimental setup, such as light distribution through quartz, resulted in darker regions (“fringes”) in certain images, visible due to refractive variations in the quartz material. The knife-edge intensifies these fringes, creating high-sensitivity areas in darker regions. These fringes, artifacts of refractive index variations rather than structural features, only appear with the knife-edge cut-off. Exposure times of 0.81 μs for knife-edge schlieren and 2.01 μs for color schlieren were used, balancing image brightness and quality. While a non-stationary flow can cause image blurring, further reducing exposure would darken images, impacting clarity. Extensive parameter adjustments were made to optimize imaging for the qualitative analysis of shock structures, despite challenges in image quality due to system constraints.

The linear cascade facility, as previously described, was tested under three distinct flow regimes, with inlet pressures set at 70,000, 80,000, and 90,000 Pa gauge. During these tests, the schlieren system was precisely aligned to capture detailed flow behavior and shock structures. The experimental campaign included two configurations: one with baseline vanes (no flow control) and another with vanes incorporating passive control mechanisms using perforated plates with 0.75 mm holes. The aim was to evaluate the impact of this passive flow control on the characteristics of both the flow and shock waves. The schlieren images obtained from the passive control tests are shown in Figure 10, Figure 11 and Figure 12.

In the passive control configuration depicted in Figure 10b, a notable change in the shock structure was observed. The distinct normal shock waves seen in the baseline case (Figure 10a) were replaced by a series of smaller, less intense compression waves, highlighted within the white rectangle. These shock trains, although present, are significantly less pronounced compared to the strong normal shock waves seen in the baseline configuration. The schlieren visualization reveals flow detachment at the leading edge of the main blade and downstream of the control region on the secondary blades. This visualization clearly captures areas of disrupted flow, where separation points form and propagate, particularly in regions influenced by the control mechanisms. While the perforated plates effectively reduced shock intensity in other regions, this localized detachment compromised flow stability and generated higher drag, especially in the first three channels. This case illustrates that, although the passive control strategy provides notable benefits in shock attenuation, it introduces specific challenges in flow management that require additional adjustments to maintain overall aerodynamic efficiency.

At an inlet static pressure of 80,000 Pa gauge (Figure 11), significant alterations in the normal shock structure occur, resulting in multiple small compression waves, particularly near the leading edge of the main blade. In this region, the absence of an additional wall to fully confine the channel causes the leading edge of the main vane to behave more like a standalone airfoil than as part of a confined cascade, resulting in different shock formations compared to a fully bounded channel. Towards the outlet of the channel, the interaction with the perforated plate transforms the normal shock into a series of oblique shock waves. Furthermore, oblique compression waves begin to form in Channel 3, specifically in the area between the splitter and the main blade, as indicated by the white rectangle. This modification in shock structure shows that the passive control strategy not only reduces the intensity of normal shocks but also alters the overall flow pattern within the cascade, leading to complex shock interactions and flow phenomena.

At the highest tested inlet static pressure of 0.9 bar gauge (Figure 12), the normal shock waves in the baseline case are positioned further downstream compared to the lower pressure conditions. In this scenario, the first main blade experiences a significant transformation of the normal shock wave, converting it into a lambda-type shock structure, marked in white. This behavior aligns with findings from previous studies [9], where similar shock transformations were observed in single airfoils under comparable flow conditions.

Additionally, near the leading edges of the second and third main blades, the pattern of smaller compression waves reappears. These waves, which are much less intense than the primary shocks, indicate a recurring flow behavior influenced by the inlet pressure and the passive control strategy. The formation of these compression waves underscores the complex interaction between the flow and the blade geometry, especially under higher pressure conditions, where the flow dynamics become increasingly intricate.

In all cases studied, significant reductions in shock wave intensity were observed in channels where the shock waves were initially strong. This reduction occurs when the fluid interacts with the porous surface, as part of the fluid enters the cavity beneath the porous wall, effectively dissipating the energy of the shock wave. This interaction between the fluid and the porous surface leads to shock attenuation, resulting in a series of weaker shock waves aligned “in a row”. These weaker shock waves are more distinctly visible in the filtered images. This phenomenon highlights the effectiveness of the passive control strategy in modifying shock characteristics and improving flow conditions within the linear cascade.

3.2. Aerodynamic Results

Figure 13 compares the static pressure distributions between the no-control and passive control scenarios, highlighting the differences introduced by the porous blades. Both the static pressure field (Figure 13) and the density gradient (Figure 14) show a reduction in shock wave intensity and a modification in shock structure, particularly noticeable in the first three channels of the cascade. This reduction is characterized by a less severe shock structure and smoother pressure variations. In Channels 4 and 5, the intensity of the leading-edge shock waves also decreases, indicating that the porous blade design effectively mitigates shock strength (Figure 14).

The implementation of perforated plates with underlying cavities does not result in a significant reduction in total pressure loss. While there is an enhancement in shock wave attenuation, this benefit is offset by increased viscous losses due to the roughness introduced by the perforated plates. This added roughness leads to higher frictional losses, counteracting the advantages gained from the reduced shock wave intensity.

Overall, the total pressure loss in the cascade with porous blades is measured at 39,492 Pa, which is only slightly different from the baseline configuration. This indicates that although the porous blades improve shock wave characteristics, their impact on overall pressure loss is minimal, maintaining a pressure loss level close to that of the reference setup.

Figure 15 provides a detailed view of Channel 3, highlighting the velocity vectors and illustrating the role of the cavity in flow behavior. The cavity enables flow recirculation from downstream to upstream of the shock wave. This recirculation induces a blowing effect, creating a fluidic ramp within the main flow and altering the shock structure into a more oblique, less dissipative configuration.

However, in the current passive control strategy, the intensity of the blowing effect is self-regulated by the strength of the shock wave. A particularly strong shock wave can lead to intense blowing, which could cause boundary layer detachment. This phenomenon is observed with the splitter in Channel 2, where the strong shock wave results in significant flow separation. This highlights a limitation of the passive control approach in managing flow dynamics under certain conditions.

Figure 16 shows the position of the reference lines located upstream and downstream of the test area, positioned at a distance of 0.013 m from the leading and trailing edges of the main blade. The corresponding pressure and velocity distributions along these reference lines are displayed in Figure 17 and Figure 18, providing insights into flow behavior and variations across the test section. In front of the blades, the pressure exhibits a sinusoidal variation, with five distinct peaks corresponding to the influence of shock wave formation near the leading edges of the blades. This oscillatory behavior highlights the periodic interaction of airflow with the blade surfaces, particularly emphasizing the lower pressure conditions where shock waves develop. In the first channel, where the pressure is lower behind the blades, the intensity of the shock waves is higher, coupled with increased flow velocity compared to the last channel. In contrast, the last channel shows a higher pressure and lower velocity, resulting in reduced shock wave intensity.

While the porous wall effectively reduces shock wave intensity, it also presents drawbacks. The main disadvantage is the occurrence of boundary layer separations caused by cross-flow interactions between the secondary flow within the cavity of the porous wall and the main flow, which typically moves in a nearly perpendicular direction. These separations contribute to additional pressure losses that affect the flow. Although the pressure loss observed in this study is not substantial, it is important to note that these losses can vary based on the specific operating conditions.

As previously noted, the numerical simulations were conducted for a static pressure of 80,000 Pa, while the computational analysis yielded a static pressure of 78,640 Pa. Comparing the experimental data with the simulation results revealed a 1.7% discrepancy between the measured inlet static pressure and the computational outcome. Differences in flow rates were also observed: For the reference case, the experimental mass flow rate was 0.342 kg/s, representing a 13.47% variation from the numerical value. In the case of the porous wall, the measured mass flow rate was 0.358 kg/s, with a 9.97% difference compared to the numerical result.

$$\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}[\%]={\displaystyle \frac{{\mathrm{P}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}-{\mathrm{P}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}}{{\mathrm{P}}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{d}}}}\times 100$$

(1)

Further differences between the CFD analysis and experimental observations are evident in the density gradient distribution. Although the numerical model successfully captures the overall position and intensity of the shock waves at a macro level, certain finer flow structures were not accurately represented in the simulation. These discrepancies highlight areas for potential refinement in the numerical modeling approach to improve the accuracy of simulations in capturing complex flow dynamics.

3.3. Analysis of Aerodynamic Phenomena Observed During the Experimental Campaign

The scenario examined in this study, both numerically and experimentally, involves an inlet static pressure of 80,000 Pa. This setting represents a borderline case, positioned between the operational limits of a traditional cascade and those of the current porous wall implementation. Although not the most aerodynamically favorable pressure, this scenario provides a valuable opportunity to investigate the system’s behavior under challenging conditions.

One significant advantage of this scenario is the ability to observe progressively weaker shock waves across four of the five visualized channels in the cascade. This setup allows for a gradual assessment of the porous wall’s effectiveness, illustrating its impact on shock wave strength and flow behavior. The observations range from minor boundary layer separations due to interactions with secondary flow within the porous cavity to more significant separations, particularly evident in the first two channels studied.

Under nearly identical flow conditions, the porous cascade exhibited a 4% increase in flow rate compared to the traditional setup. Figure 19 depicts the evolution of the experimental sequences, showing the comparable duration and stability of measured thermodynamic parameters, including mass flow rate, temperature, and pressure. The pressure differences between the cases are minimal, while temperature variations are approximately 1 K, corresponding to a change in the speed of sound of less than 0.6 m/s (0.17%). These small variations slightly favor the porous material configuration. Figure 19 also illustrates how the flow rate evolves with varying inlet pressures. It is consistently observed that the flow rate through the porous cascade remains higher, even outside the equilibrium zone (80,000 Pa gauge).

A black mask was applied to the schlieren images shown in Figure 10, Figure 11, Figure 12 and Figure 20, covering the blades and background to emphasize the phenomenon and reduce the impact of any light scattering outside the region of interest.

While flow rates vary slightly among the cascade channels—being higher in the upper channel and progressively lower in subsequent channels—the shock waves observed on traditional blades are dissipated more effectively with the porous wall. This dissipation is associated with the formation of smaller waves oriented at 60° relative to the initial shock direction, which interact minimally with the flow and channel walls. The distribution of flow rates across individual channels also appears to be affected, with a noticeable increase through the first channel.

It is notable that in areas where perpendicular shock waves previously formed on the blade profile, the presence of porous material now generates a train of smaller waves that propagate upstream. These smaller waves are relatively weak and diminutive, making their tracking over time challenging due to the spatial and temporal resolution required, which exceeds the capabilities of the current experimental setup. Despite this limitation, the distinct pattern and regularity with which these waves appear near the porous wall suggest a marked deviation from behavior observed with isolated profiles. Previous studies have shown that a high-intensity shock wave typically redistributes into multiple lower-intensity waves [36,37]. However, in the present case, an (likely unsteady) interference phenomenon among these smaller waves leads to the formation of observed wave trains, as depicted in Figure 20.

Given that the perforation pattern is of a “checkerboard” type, with each row of holes interleaved with a row of smaller holes, the absence of a continuous surface between adjacent rows of perforations may facilitate the visibility of this phenomenon. Without this spacing, the small waves formed on the surfaces between the holes might have either obscured the visualization or suppressed the phenomenon entirely.

Additionally, the apparent upstream propagation of shock waves in the supersonic flow could be advantageous for feedback-type adjustments of specific parameters, potentially offering more dynamic control over flow characteristics. This phenomenon is also observable at higher inlet pressures; however, the secondary flow from the porous wall cavity induces boundary layer separations that are too disruptive to overall flow dynamics. To explore these high-pressure regimes further, modifications such as filling the cavity with bulk porous material or installing partition walls may be necessary.

Figure 21 provides two detailed views of shock waves for intermediate blades. In the passive control scenario, small waves (indicated by the red rectangle) are observed that weakly interfere and dissipate quickly, leading to a noticeable change in the flow distribution across the cascade. Instead of a gradual reduction towards the lower channels, a periodic pattern emerges, repeating every two channels. In addition to revealing the secondary wave patterns associated with the porous material, macroscopic shock waves are observed in regions where none were initially present. These macroscopic waves exhibit a periodic behavior across the network, appearing every other channel. This periodicity suggests a global property of the network that cannot be explained solely by the observed 4% increase in flow rate.

A similar pattern is observed with the main blades, as shown in Figure 22. In the traditional cascade setup, the intensity of the shock waves on the main blade (indicated by the red rectangle) diminishes as the flow rate decreases in the corresponding channels. However, in the porous cascade, shock wave trains are notably more pronounced in the even-numbered channels, while the shock waves in Channels 1 and 3 appear significantly weaker. Variations in the direction, intensity, shape, and location of the shock waves along the blade are evident. This behavior cannot be explained solely by the increased flow rate through the cascade, suggesting complex interactions between the porous surfaces, shock wave dynamics, and channel flow distribution that warrant further investigation.

Analytical modeling of the aero-acoustic phenomena in the porous plate was performed to quantitatively support the experimental data. The first thought was to model the porous cavity as a Helmholtz resonator with multiple orifices, [+1, +2]:

$${\mathrm{f}}_{\mathrm{t}}={\displaystyle \frac{\mathrm{c}}{2\mathsf{\pi }}}\sqrt{{\displaystyle \frac{{\mathrm{A}}_{h\mathrm{o}\mathrm{l}\mathrm{e}.1}}{{\mathrm{V}}_{\mathrm{c}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}}\left({\mathrm{L}}_{\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{k}.1}+{\mathrm{a}}_{\mathrm{e}\mathrm{n}\mathrm{d}}·{\mathrm{d}}_{h\mathrm{o}\mathrm{l}\mathrm{e}.1}\right)}}}$$

(2)

$${\mathrm{f}}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}={\displaystyle \frac{\mathrm{c}}{2\mathsf{\pi }}}\sqrt{{\displaystyle \frac{\sum {\mathrm{A}}_{h\mathrm{o}\mathrm{l}\mathrm{e}.\mathrm{i}}}{{\mathrm{V}}_{\mathrm{c}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}}\sum \left({\mathrm{L}}_{\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{k}.\mathrm{i}}+{\mathrm{a}}_{\mathrm{e}\mathrm{n}\mathrm{d}}·{\mathrm{d}}_{h\mathrm{o}\mathrm{l}\mathrm{e}.\mathrm{i}}\right)}}}$$

(3)

$${\mathrm{f}}_0=\sqrt{{{\mathrm{f}}_{\mathrm{t}}}^2+{{\mathrm{f}}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}}^2}$$

(4)

which, for the current geometry, (d = 0.75 mm, V_cavity = 240.2 mm³, L_neck = 1 mm, 97 holes) would yield a wavelength of 1.048 mm. In this case, Langfeldt’s model for a multiple-hole Helmholtz resonator was used, with the end correction factor of a_{_end} = 0.85, corresponding to flanged holes [+3, +4]. Using optical estimation, the wavelength presented here lies between 1–1.1 mm.

Although a good visual agreement can be observed, a more thorough investigation method, more suited for aero-acoustics, is definitely required for definitive confirmation.

The wave’s front angle observed coincides with the complementary angle of the bladed array. This makes sense intuitively; however, more studies are also required here to make sure that this is a true correlation and not just a coincidence.

4. Conclusions

This study investigated the impact of a porous wall design on the global flow rate and internal flow structure within a linear cascade, using both numerical and experimental methods at an inlet static pressure of 80,000 Pa gauge. The results revealed that the porous wall effectively dissipated shock waves, leading to a substantial reduction in shock intensity throughout the channels. In Channels 4 and 5, supersonic shocks on the secondary blade were nearly eliminated, and shocks near the leading edge of the main blade were significantly weakened, demonstrating the porous wall’s capability to reduce shock strength consistently.

The porous wall also influenced boundary layer behavior by introducing secondary flow in the porous cavity, which interacted with the main flow and led to varying levels of boundary layer separation. Although minor in some areas, these separations were more pronounced in the first two channels, highlighting the limitations of the porous wall under higher flow rates and pressures and its potential impact on flow stability. Additionally, the design led to a 4% increase in overall flow rate compared to the traditional cascade configuration, with only minimal differences in pressure and temperature (~1K) between the two setups, indicating a minor influence on thermodynamic properties.

Instead of forming perpendicular shock waves on blade profiles, the porous wall generated a train of smaller waves that propagated upstream. This distinct pattern, not seen with isolated blade profiles, suggested that the porous material redistributed high-intensity waves into periodic, lower-intensity formations, underscoring the complex interactions induced by the porous wall. While the porous wall provided clear advantages in shock attenuation and flow rate increase, it also introduced challenges, particularly in high-pressure environments, where flow separation became problematic. Potential improvements could involve filling the cavity with bulk porous material or adding partition walls to better manage flow dynamics.

In summary, the porous wall implementation offered significant insights into its effects on flow behavior and shock dynamics within a linear cascade. Although it successfully reduced shock wave intensity and increased the flow rate, its influence on boundary layer separation and periodic wave formations indicates that further refinement is necessary. Future research should aim to optimize the porous wall design, especially for high-pressure conditions, to maximize its aerodynamic benefits.