# Parameter Influence on Porous Bleed Performance for Shock-Wave/Boundary-Layer Interaction Control

Julian Giehler, Pierre Grenson, Reynald Bur

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## ► To cite this version:

Julian Giehler, Pierre Grenson, Reynald Bur. Parameter Influence on Porous Bleed Performance for Shock-Wave/Boundary-Layer Interaction Control. Towards Effective Flow Control and Mitigation of Shock Effects in Aeronautical Applications, 201, Springer, pp.127 - 147, 2025, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 978-3-031-86605-0. 10.1007/978-3-031-86605-0_7. hal-05158524

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https://hal.science/hal-05158524v1

Submitted on 11 Jul 2025

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# Parameter Influence on Porous
Bleed Performance for
Shock-Wave/Boundary-Layer
Interaction Control

Julian Giehler, Pierre Grenson, and Reynald Bur

**Abstract** This chapter examines the influence of hole diameter, porosity, thickness-to-diameter ratio, and stagger angle on the performance of porous bleed control in mitigating the negative effects of shock-wave/boundary-layer interactions. A detailed numerical study focuses on the control of an irregular shock reflection, or Mach reflection, where a separation bubble below the shock foot is present in the uncontrolled case. Implementing bleed control modifies the flow field significantly, with variations in bleed rates upstream and downstream of the shock because of the external flow characteristics. The findings indicate that smaller hole diameters enhance bleed efficiency and control effectiveness, while porosity levels and thickness-to-diameter ratios exhibit complex trends, with a medium thickness-to-diameter ratio and a stagger angle of 45° emerging as optimal configurations for effective shock-wave/boundary-layer control.

**Keywords** Supersonic flow · Shock wave—boundary layer interaction · Flow control · Numerical simulations

## 1 Introduction

The deceleration of the airflow to subsonic conditions forces the presence of shock-wave/boundary-layer interactions in supersonic air intakes [1]. Thus, the boundary-layer separation may occur, accompanied by performance losses. Porous bleed systems are a common technique to control shock-wave/boundary-layer interactions in supersonic air intakes [2, 3]. These systems remove the low-momentum portion of the boundary layer through perforated plates, leading to a non-separating boundary layer due to control of its size and characteristics.

While the working principle is simple, the control effect of bleed systems is still challenging to forecast. Typical bleed systems consist of hundreds of holes, resulting

J. Giehler · P. Grenson (✉) · R. Bur

DAAA, ONERA, Institut Polytechnique de Paris, Meudon, France
e-mail: pierre.grenson@onera.fr
in the impracticality of three-dimensional simulations of the entire system in the pre-design phase because of the excessive numerical costs. Thus, bleed models, which can be applied as a boundary condition in RANS solvers, are desired. However, state-of-the-art bleed models are not able to capture important geometrical parameters of the system [4].

Our previous numerical study revealed parameter influences on bleed efficiency and control effectiveness when a bleed is applied to control a supersonic boundary layer [5]. The bleed efficiency describes the capability of removing the highest possible bleed rate for a given pressure drop from the external wall to the bleed plenum. On the contrary, the control effectiveness is unlinked to the efficiency and describes the ability to achieve the maximum effect on the boundary layer, i.e., the increase of the momentum in the near-wall region, for a given bleed rate. In the following, a parametric study on the control of a shock-wave/boundary-layer interaction using porous bleed is presented, showing the influences of four geometrical parameters: hole diameter, porosity level, thickness-to-diameter ratio, and stagger angle.

# 2 Methodology

## 2.1 Governing Fluid Equations and Flow Solver

The flow is modeled by the compressible Reynolds-averaged Navier-Stokes equations, which describe the behavior of an ideal gas in motion, where **u** denotes the velocity vector [$u, v, w$]:

$$ \text{Continuity:} \qquad \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \qquad (1) $$

$$ \text{Momentum:} \qquad \frac{\partial(\rho u)}{\partial t} + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) = -\nabla p + \nabla \cdot \tau_v + \nabla \cdot \tau_t \qquad (2) $$

$$ \text{Total energy:} \qquad \frac{\partial \rho E}{\partial t} + \nabla \cdot [(\rho E + p)\mathbf{u}] = \nabla \cdot [(\tau_v + \tau_t) \cdot \mathbf{u}] - \nabla \cdot \varphi - \nabla \cdot \varphi_t \qquad (3) $$

The total energy $E$ combines internal energy $e$ and kinetic energy $\frac{\|\mathbf{u}\|^2}{2}$. For a calorically perfect gas, internal energy $e$ is given by $c_v T$, where $c_v = 717.63 \text{ J kg}^{-1} \text{ K}^{-1}$ is the specific heat at constant volume, and $T$ is the temperature. For Newtonian fluids, the stress tensor $\tau_v$ is determined by the velocity gradient $\nabla \mathbf{u}$ and dynamic viscosity $\mu$:

$$ \tau_v = \mu \left[ \nabla \mathbf{u} + \nabla \mathbf{u}^T - \frac{2}{3} (\nabla \cdot \mathbf{u}) \mathbf{I} \right] \qquad (4) $$

Dynamic viscosity is calculated through Sutherland's law [6] from the static temperature obtained via the ideal gas law $T = p/(R\rho)$, where $R = 287.058 \text{ J kg}^{-1} \text{ K}^{-1}$.
The heat flux vector $\varphi$ follows Fourier's law as $\varphi = -\lambda_f \nabla T$, where $\lambda_f$ represents the thermal conductivity coefficient.

Reynolds tensor $\tau_t$ and turbulent heat flux $\varphi_t$ are determined using the Spalart-Allmaras turbulence model with quadratic constitutive relation [7], based on the Boussinesq hypothesis with the turbulent Prandtl number $Pr_t = 0.9$ and modeled turbulent viscosity coefficient $\mu_t$. The turbulence model sensitivity was verified in our previous study [5].

To numerically solve the compressible Navier-Stokes equations, the ONERA-Safran finite-volume solver elsA [8] is employed, utilizing a second-order-accurate Roe upwind scheme with the minmod limiter and Harten entropic correction for spatial derivatives, alongside a backward-Euler implicit local time-stepping scheme.

## 2.2 Geometry and Mesh

Three-dimensional simulations of the S8Ch wind tunnel at the ONERA in Meudon, where experiments were performed [9], were conducted. A convergent-divergent nozzle upstream of the working section generates a M = 1.62 flow. A shock generator with a wedge angle of $\alpha = 10.5^\circ$ is placed in the nozzle exit to provoke an oblique shock wave. The wedge angle was selected to obtain a so-called Mach reflection—an irregular shock reflection at the wall. Thus, the presence of a normal shock wave in the vicinity of the wall is guaranteed, enabling the investigation of the porous bleed under both supersonic and subsonic flow conditions. The numerical domain is illustrated in Fig. 1.

The structured mesh is generated using the in-house pre-processing tool and mesh-generator Cassiopee [10]. A fully parameterized mesh allows the variation of the bleed parameters. Therefore, each hole is modeled out of five blocks using a butterfly mesh. The four other blocks are part of a C-grid, including the wall boundary layer on the plate and plenum sides (see Fig. 2). For a proper resolution of the boundary layer, the minimum wall-normal cell size is $y^+ \approx 1(0.2 \times 10^{-3} \text{ mm})$. Inside the holes, the wall is equally meshed. The cell-to-cell growth ratio is 1.1, the maximum value to

<figure><img src="image_4.png" /><figcaption>Fig. 1 Computational domain with shock generator [4]</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 2** View on mesh around the holes [5]</figcaption></figure>

accurately predict the local mass flow rate, thanks to a preliminary mesh sensitivity study [5]. More details on the reference simulations can be found in [5, 11].

## 2.3 Evaluation of Bleed Efficiency and Control Effectiveness

The efficiency of porous bleed systems is typically quantified by the sonic flow coefficient. In this study, the scaling of Slater [12] is used to compute the surface sonic flow coefficient

$$Q_{sonic,w} = \frac{\dot{m}_{bl}}{\dot{m}_{sonic,w}}, \qquad (5)$$

where the surface sonic mass flow rate $\dot{m}_{sonic,w}$ normalizes the bleed mass flow rate $\dot{m}_{bl}$. The surface sonic mass flow rate can be computed using the static flow quantities at the wall as follows:

$$\dot{m}_{sonic,w} = p_w A_{bl} \left( \frac{\gamma}{RT_w} \right)^{1/2} \left( \frac{\gamma + 1}{2} \right)^{-\frac{\gamma+1}{2(\gamma-1)}} \qquad (6)$$

With the aim to analyze the local mass flow rate for each hole, the local extraction of the hole mass flow rate $\dot{m}_{bl,i}$ and the local wall pressure is required. In the current study, the mass flow rate is extracted locally in each hole instead of using the bleed mass flow rate extracted at the plenum exit. Therefore the flow momentum in the plate normal direction is integrated
$$ \dot{m}_{bl,i} = \int_S \rho v dS \qquad (7) $$

at the hole center corresponding to half the plate thickness, as illustrated in Fig. 3. The (global) bleed mass flow rate can be easily computed by summing all hole mass flow rates:

$$ \dot{m}_{bl} = \sum_{i=1}^{N} \dot{m}_{bl,i} \qquad (8) $$

The static wall pressure and temperature cannot be extracted at the hole position since its value is disturbed by the local velocity caused by the flow into the holes. Therefore, a circular patch around the hole is defined with its size $A_o$ fitting the porosity $\phi$ concerning the hole area $A_h$:

$$ \phi = \frac{A_{bl,i}}{A_o} \qquad (9) $$

The external wall quantities are then extracted by averaging their values above the patch size (see Fig. 3). Cells that partially lie within the circular patch are weighted using the areal fraction inside the circle. A similar method is applied to the plenum pressure, where the patch is equal in size but placed three hole diameters below the wall instead of using the wall pressure, which can be highly affected by the under-expanded jet at the hole exit. Higher patch distances lead to local information losses as the plenum pressure becomes more uniform with further distance from the holes.

The bleed effectiveness describes the ability of the bleed system to control the flow. In this investigation, the aim of the porous bleed is to increase the momentum in the wall vicinity. Therefore, characteristic quantities of the boundary layer are required, which are challenging to extract even in this relatively simple case of a flat plate because shock waves and expansion fans induced by the suction holes profoundly modify the boundary layer downstream of the region. As a result, we selected the wall shear stress to evaluate the effectiveness of the bleed system. The fuller and/or thinner the boundary layer, the higher the wall shear stress. Thus, comparing the wall shear stress upstream and downstream of the bleed regions enables the evaluation of the effectiveness. Therefore, the wall shear stress along the span is extracted on a line 10 mm upstream of the leading edge of the first hole and 10 mm downstream of the rear edge of the last hole. Using these quantities, we define the rise in the wall shear stress as

$$ \epsilon_{\tau} = \frac{\tau_{w,d} - \tau_{w,u}}{\tau_{w,u}}, \qquad (10) $$

quantifying the relative increase of the wall shear stress along the bleed region. However, it is essential to note that this value is case-sensitive and dependent on the inflow conditions, which are kept constant in this study.
<figure><img src="image_6.png" /><figcaption>**Fig. 3** Positions for the local extractions [5]</figcaption></figure>

Moreover, the bleed mass flow rate must be constant relative to the inflow to evaluate the effectiveness. Hence, the displacement mass flow rate

$$ \dot{m}_{\delta_{1,c}} = \int \delta_{1,c} \rho_{\infty} u_{\infty} dz, \qquad (11) $$

which describes the theoretical missing mass flow rate required to obtain an inviscid flow, is computed. The ratio of bleed mass flow rate and displacement mass flow rate is kept constant ($\dot{m}_{bl}/\dot{m}_{\delta_{1,c}} = \text{const.}$).

## 3 Flow Topology in the Uncontrolled Case

Before controlling a shock-wave/boundary-layer interaction, the base flow without any control is presented for a wedge angle of the shock generator of $\alpha = 10.5^{\circ}$. Figure 4 shows the flow field in the area. Three different shock waves are apparent in the outer inviscid flow: the oblique shock provoked by the shock generator, the reflected shock, and a normal shock—the so-called Mach stem—close to the wall. All three shocks meet in the so-called triple point. Downstream of the Mach stem, the flow is subsonic and has a higher entropy because of the higher total pressure losses caused by the normal shock. In contrast, the flow remains supersonic in the
<figure><img src="image_7.png" /><figcaption>Fig. 4 Mach number field for the uncontrolled case with SBLI</figcaption></figure>

upper part. As a result, a slip line is present between the supersonic and subsonic regions, separating the areas of different entropy.

Closer to the wall, a typical Lambda shock foot is observed. Again, an oblique shock is apparent, induced by the boundary-layer growth resulting from the adverse pressure gradient propagating upstream in the subsonic part of the boundary layer. Further downstream, we observe another normal shock. Again, the two shock waves meet with the Mach stem in a triple point.

A closer look at the flow field in the vicinity of the wall (see zoom-in view in Fig. 4) reveals the presence of a separation bubble. This separation bubble is limited in size because of an expansion fan generated by the rear edge of the shock generator, which accelerates the flow and leads to its reattachment. The aim of the porous bleed system is the mitigation of the separation bubble.
# 4 Geometry Influence on the Control

The flow field for a controlled case is presented in Fig. 5 for a bleed configuration with a hole diameter of $D = 2$ mm, a length-to-diameter ratio of $L/D = 1$, a porosity level of $\phi = 22.67\%$, and a stagger angle of $\beta = 60^\circ$. Figure 5a shows the flow on the hole-cutting plane. The wedge angle is equal to the uncontrolled case with $\alpha = 10.5^\circ$.

A different flow field is observed compared to the uncontrolled case. Again, the oblique shock is irregularly reflected, resulting in a Mach stem, as shown in Fig. 5b. However, no lambda shock is apparent, resulting from mitigating the flow separation. However, close to the shock, the Mach number in the vicinity of the wall is low, indicating the presence of a small local flow separation. Moreover, the suction through the perforated plate causes additional compressible effects. At the beginning of the perforated plate, an expansion fan is caused, and the so-called trailing shock is apparent at the end of the plate.

Following the streamlines in Fig. 5a, a slight thinning of the supersonic boundary layer is observed upstream of the shock, resulting in the expansion fan at the beginning of the plate. After passing the shocks, the streamlines are bent towards the wall, which indicates a high mass removal downstream of the shock where the flow is subsonic. No shock-induced flow separation is notable as the low-momentum flow is sucked into the plenum. At the end of the plate, a flow deflection in the wall-parallel direction is observed, resulting in the trailing shock.

<figure><img src="image_8.png" /><figcaption>**Fig. 5** Mach number field for an SBLI control; gray patches illustrate the out-of-plane holes ($TR = 0.7$)</figcaption></figure>
The zoom-in view on the left-hand side shows the flow field inside a bleed hole upstream of the shock where the external flow is supersonic. The suction rate in this region is relatively low, the holes are unchoked, and the flow inside the hole becomes supersonic only in a small area as illustrated by the sonic line (dotted). Moreover, a sizeable separated region is observed. At the front of the hole, an expansion fan is induced by the suction, while further downstream, the so-called barrier shock is present, which redirects the flow either into the hole or in the wall-parallel direction.

A completely different flow field is apparent on the zoom-in view on the right, presenting the flow inside the hole in the subsonic region downstream of the shock. Here, the external flow is subsonic. Thus, no barrier shock is evident, and the expansion fan is limited to a small area close to the leading edge, where the flow is accelerated to supersonic conditions again. The flow inside the hole is choked, as visualized by the sonic line. The separated region inside the holes in the subsonic region is significantly lower. Moreover, the streamlines directed toward the plate indicate a substantial thinning of the boundary layer. Altogether, the flow inside the holes is found to behave like in the case of supersonic boundary-layer bleeding upstream of the shock, and equally to subsonic boundary-layer bleeding downstream of the shock [11].

The flow on the hole-cutting plane between the holes is detailed in Fig. 5b, where the pattern of the shock reflection, as well as the expansion fan and the trailing shock induced by the porous bleed, are illustrated. In the supersonic region upstream of the shock, small out-of-plane effects caused by the hole located outside the plane are evident in the form of irregularities in the Mach number inside the boundary layer. Downstream of the shock, where the external flow is subsonic, no out-of-plane effects are apparent. Close to the shock, the Mach number in the vicinity of the wall is low, which indicates the presence of a small local flow separation.

CFD simulations have been performed for different hole diameters, porosity levels, thickness-to-diameter ratios, and stagger angles over a range of plenum exit throat ratios. Here, the effect of the parameters on the control of the SBLI is investigated.

## 4.1 Hole Diameter

The flow fields for the different hole diameters are shown in Fig. 6. On the left, the contours are extracted on the hole-cutting plane so that the flow inside the holes is visible. The flow on the offset plane between the two rows is shown on the right side. From top to bottom, the hole diameter is increased. The gray patches illustrate the position of the second row of bleed holes.

A view of Fig. 6a reveals significant differences between the flow fields. For the smallest diameter $D = 0.25$ mm, the flow field upstream of the shock is very homogeneous. There is no apparent penetration of the barrier shocks in the boundary layer. The shock is located at $x = 83$ mm (dotted vertical lines), and no lambda foot is found. This is a consequence of the strong thinning of the upstream boundary layer. Thus, the interaction of the shock with the boundary layer is effectively mitigated with
<figure><img src="image_9.png" /><figcaption>**Fig. 6** Mach number contours for different hole diameters; gray patches illustrate the second row hole positions; dotted line highlights shock location for $D = 0.25$ mm</figcaption></figure>

no apparent flow separation. Downstream of the shock, a strong transpiration flow is evident with the high-velocity jets in the plenum. As a result, no flow separation is apparent. At the end of the plate, the trailing shock, which leads to a weak thickening of the boundary layer, is visible.

With larger hole diameters, a slight upstream displacement of the shock foot is apparent. Moreover, for hole diameters $D \ge 1.\,\text{mm}$, the flow field is less homogeneous, and penetration of the barrier shocks through the boundary layer is found. This also affects the shape of the shock wave, which is strongly bent in the vicinity of the holes. For the largest diameter of $D = 4\,\text{mm}$, a strong interaction of the holes with the boundary layer can be observed. Every hole creates an expansion fan and a barrier shock that penetrates the boundary layer and the incoming oblique shock wave. The shock wave is curved, and the shock foot is approximately 3 mm further upstream than for the smallest holes. Also, downstream of the shock, differences can be noted: The boundary layer is thicker, and the trailing shock is less prominent. These findings indicate a lower bleed efficiency.
In Fig. 6b, the effect of the hole diameter is more significant. For small hole diameters $D \le 1$ mm, the flow field looks identical to the left-hand side. This means that the flow field is very homogeneous in the spanwise direction. For larger holes $D \ge 2$ mm, the impact of the barrier shock and the expansion waves is evident. Moreover, the Mach number in the vicinity of the wall below the shock foot is very low. Here, a lambda shock foot is observable, which indicates the presence of local flow separation.

Figure 7 shows the curves of the external wall pressure and the plenum pressure, which are scaled by the stagnation pressure $p_t$, and the surface sonic flow coefficient along the bleed region. The shock position is located between $\hat{x}/L_w = 75\%$ and $80\%$, as seen by the pressure rise in Fig. 7a. Upstream of the shock, The larger the hole, the higher the wall pressure. Moreover, the shock is more smeared for larger hole diameters. Hence, better control is achieved for small holes. Downstream of the shock, the differences are smaller. Only at the end of the plate, a larger pressure rise for tiny holes is found, leading to the assumption of a stronger trailing shock as the pressure information can spread upstream inside the boundary layer.

The surface sonic flow coefficient (Fig. 7c) behaves similarly to the wall pressure. Upstream of the shock, the flow coefficient is the highest for the smallest holes. Passing the shock foot, the sonic flow coefficient sharply increases as the pressure

<figure><img src="image_10.png" /><figcaption>Fig. 7 Comparison of the flow quantities for different hole diameters</figcaption></figure>
difference from the external wall to the plenum increases. Downstream of the shock, a lower efficiency is found for the largest diameter of $D = 4$ mm. The differences between the other hole diameters are small.

Also, the pressure inside the cavity slightly varies downstream of the incident shock, as shown in Fig. 7b. Since the bleed rate is higher in this area, the transpiration flow is stronger, leading to secondary flows inside the cavity plenum. Also, losses induced by the under-expanded jet are increased. However, the pressure difference inside the plenum is low compared to the change in the external wall pressure. Thus, the assumption of constant pressure inside the cavity, as used in the bleed modeling, is valid.

The surface sonic flow coefficient as a function of the pressure ratio is visualized in Fig. 8. Additionally, the bleed models of Slater [12] and Grzelak et al. [13] are shown. The model of Slater [12] is validated for supersonic external flows while the model of Grzelak et al. [13] is derived from a numerical study without external flow. In our previous study [4], we found a low prediction error of the first model upstream of the shock, while the second one predicts the bleed effect downstream of the shock better.

The trend for the sonic flow coefficient is challenging to grasp in one view since an extensive range of working regimes is covered by one throat ratio. For low throat ratios, the plenum pressure can exceed the external wall pressure upstream of the shock. Thus, the air is not removed but added to the boundary layer (blowing). Consequently, the momentum inside the boundary layer is decreasing instead of increasing as desired.

<figure><img src="image_11.png" /><figcaption>Fig. 8 Variation of the surface sonic flow coefficient for different hole diameters for the SBLI</figcaption></figure>
Downstream of the shock, the pressure ratio is higher regardless of the throat ratio. Thus, the sonic flow coefficient sharply increases. Since the flow momentum in the vicinity of the wall is low, the size of the separated region decreases, resulting in a larger vena contracta area and hence a higher flow coefficient than for the supersonic bleeding. The bleed operates here in subsonic flow conditions, and the trend is similar to that of the model of Grzelak et al. [13].

A diameter dependence on the flow coefficient is apparent, similar as for supersonic and subsonic flows. The higher the hole diameter, the lower the sonic flow coefficient. However, this trend is smaller for the holes downstream of the shock, where the hole flow is choked, and the external flow is subsonic.

Interestingly, by observing the mean plate values, large holes are found to be slightly more efficient for low throat ratios. The sonic flow coefficient is equal for all cases, but the pressure ratio is higher. The reason seems to be the low “blowing efficiency” for large holes.

A more detailed view of the control of the shock-induced flow separation is shown in Fig. 9, which displays the streamwise wall shear stress below the shock foot. The absolute values for the coordinates are given to grasp the size of the separated region. For the largest holes of $D = 4$ mm, a large region of reversed flow is apparent between the rows starting from $x = 78$ mm. This region is induced by the adverse pressure gradient of the incident shock and ends with the end of the plate ($x = 90$ mm), where the flow reattaches due to the strong bleed rate, but also because of the expansion fan caused by the rear edge of the shock generator.

For smaller holes, the upstream influence of the incident shock inside the boundary layer is smaller, as indicated by the smaller extent of the region of reversed flow. With regard to Fig. 6b, it can be stated that the smaller hole sizes lead to a higher flow momentum inside the boundary layer, resulting in a shorter interaction length. The size of the lambda shock foot is smaller, and the separation bubble below decreases in size. Directly downstream of the bleed holes, there is no flow separation apparent, as the momentum in the wall vicinity is too high to separate. Thus, the flow separation is limited to the spanwise area between the holes, which is a function of the porosity level and the hole diameter. Hence, smaller diameters have a better control effect as the solid area between two holes is reduced.

The boundary-layer profiles downstream of the bleed region are illustrated in Fig. 10. The observed effects are similar to the investigations for the boundary-layer bleeding but even stronger. The use of tiny holes guarantees a higher momentum of the flow in the wall vicinity downstream of the impinging shock.

The application of large holes to control SBLIs results in strong variations of the boundary-layer profiles along the span. As observed in Giehler et al. [5], large hole diameters lead to a less homogeneous flow for both supersonic and subsonic conditions. The appearance of the adverse pressure gradient caused by the shock wave enforces this effect. Fluctuations in the sonic height result in variations of the interaction length along the span. Thus, the flow momentum between the holes is further decreased and negative effects sum up. Consequently, the variations along the span increase.
<figure><img src="image_12.png" /><figcaption>**Fig. 9** Streamwise wall shear stress below the shock foot for different hole diameters; dashed lines mark $\tau_{w,x} = 0$ contour line</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 10** Boundary layer 10 mm downstream of the bleed region for $D = 0.25$ mm, 1.00 mm and 4.00 mm (left to right); gray patch illustrates the envelope of the boundary-layer profiles along the spanwise direction; red area details the compressible displacement thickness and green area the difference to the inflow</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11** Rise in the wall shear stress as a function of the bleed rate $\dot{m}_{bl}$ and the pressure ratio $p_{pl}/p_t$ for varying hole diameters; dotted line in (b) highlights isentropic pressure ratio for $M = 1.62$</figcaption></figure>

The rise of the wall shear stress as a function of the bleed rate and pressure ratio $p_{pl}/p_t$ is shown in Fig. 11. The stagnation pressure $p_t$ is used to scale the plenum pressure since the wall pressure $p_w$ varies along the bleed because of the incident shock. A view at Fig. 11a reveals significant differences in the boundary-layer bleeding: for low bleed rates, a degradation of the wall shear stress is apparent. This is a result of a lower Mach number downstream of the impinging shock, but also of the presence of blowing for lower bleed rates as highlighted in Fig. 8. As long as blowing occurs, no differences between the different hole diameters are found. With reaching a plenum pressure below the external wall pressure, the bleed operates in full-suction mode, resulting in differences in the effectiveness between the different hole diameters.

Figure 11b confirms that a pressure lower than the isentropic static pressure upstream of the shock is necessary to obtain a higher wall shear stress. If the plenum pressure is higher, the blowing occurs upstream of the shock, which eliminates diameter influences. The lower the plenum pressure, the higher the effectiveness. However, tiny holes are advantageous, which are linked to the lower extent of the reversed flow region below the shock foot.

## 4.2 Porosity Level

The influence of the porosity level on the SBLI is shown in Fig. 12. Again, the sonic flow coefficients and the models of Slater [12] and Grzelak et al. [13] are illustrated. Generally, it must be noted that comparing different porosity levels is challenging as the global bleed mass flux strongly differs. The higher the bleed mass flux, the better
<figure><img src="image_15.png" /><figcaption>Fig. 12 Variation of the surface sonic flow coefficient for different porosity levels for the SBLI</figcaption></figure>

the control of the SBLI because of the higher momentum close to the wall. Figure 12 still allows the comparison of the efficiency of different porosity levels.

For low throat ratios, blowing takes place upstream of the shock for all porosity levels with higher intensity for low-porosity plates. The reason is the weaker control of the shock. As a result, the location of the shock foot moves upstream, and the plenum pressure and the blowing rate increase.

Downstream of the shock, the differences between the different porosity levels are minor. All points of the last holes show a similar trend. For unchoked conditions, high porosity levels seem to be slightly more efficient, whereas the contrary is the case for choked holes. Also, the findings of Grzelak et al. [13] cannot be confirmed as they found a higher efficiency for larger porosity levels. However, their study was limited to small porosity levels and no tangential flow.

Since the porosity level is strongly linked to the achievable bleed rate, this parameter has an essential impact on the effectiveness of the bleed. The higher the bleed rate, the higher the effect. Figure 13 demonstrates the higher rise in the wall shear stress for higher porosity levels. As already shown for the hole diameter, a certain amount of bleed rate is required to obtain an equal wall shear stress downstream of the bleed compared to upstream. However, the maximum bleed rate for the lowest porosity level of $\phi = 5.67\%$ is not high enough to reach this point.

Nevertheless, a difference between the porosity levels in terms of effectiveness is observed in Fig. 13a: the lower the porosity level, the lower the required bleed rate to obtain the same wall shear stress downstream of the SBLI compared to upstream.
<figure><img src="image_16.png" /><figcaption>**Fig. 13** Rise in the wall shear stress as a function of the bleed rate $\dot{m}_{bl}$ and the pressure ratio $p_{pl}/p_t$ for varying hole diameters; dotted line in (b) highlights isentropic pressure ratio for $M = 1.62$</figcaption></figure>

## 4.3 Thickness-to-Diameter Ratio

As already observed by Giehler et al. [5], the thickness-to-diameter ratio has no significant effect on the boundary layer. Therefore, only the sonic flow coefficient is regarded in Fig. 14. Like for the other parameters, blowing occurs for small throat ratios, which is unaffected by the thickness-to-diameter ratio. With increasing throat diameter, effects on the sonic flow coefficient become apparent. Upstream of the shock, where the flow is supersonic, large ratios are advantageous if the holes are unchoked. This is in line with the results from Giehler et al. [5]. For low-pressure ratios, this effect inverts, and lower thicknesses are found to be more efficient.

In contrast, downstream of the shock, where the pressure ratio is always lower because of the jump in the external wall pressure, high thickness-to-diameter ratios are observed to negatively affect the sonic flow coefficient. In this area, the tangential flow momentum is very low, resulting in a small size of the separated region inside the holes. Thus, the flow is fully attached inside the holes, and friction increases with further hole length, i.e., plate thickness. Grzelak et al. [13] also found a degradation of the bleeding performance for $T/D \ge 1.5$. However, no separated region was induced by external flow in their study, and the observed hole diameters and porosity levels were smaller.

The difference between subsonic and supersonic conditions complicates the definition of an optimal geometry for the control of SBLIs. However, thickness-to-diameter ratios $T/D > 4$ result in high friction losses and should be avoided. Since the highest mass fluxes are achieved downstream of the shock, a higher sonic flow coefficient in this area is advantageous. Thus, the thickness-to-diameter ratio should not be higher than $T/D = 2$.
<figure><img src="image_17.png" /><figcaption>**Fig. 14** Variation of the surface sonic flow coefficient for different thickness-to-diameter ratios for the SBLI</figcaption></figure>

## 4.4 Stagger Angle

The last investigated geometrical parameter is the stagger angle, which is unconsidered in all bleed models. Its effect on the sonic flow coefficient is shown in Fig. 15. The findings are similar to those from Giehler et al. [5]: for subsonic conditions, no difference in the sonic flow coefficient is apparent. For supersonic conditions, the stagger angles $\beta = 30^{\circ}$ and $45^{\circ}$ are preferable to use since the pressure losses are lower. Overall, the effects are in a negligible range.

More interesting is the effect on the SBLI control. Therefore, the streamwise wall shear stress is extracted and illustrated in Fig. 16 below the shock foot. Thus, the size of the separation bubble between the holes can be estimated. From top to bottom, the stagger angle is increased, leading to higher spanwise but lower streamwise distances between the holes.

For the stagger angles $\beta = 30^{\circ}$ and $45^{\circ}$, the region of reversed flow is very small. The high density of holes in the spanwise direction prevents the occurrence of one large separated area, as the flow reattaches directly downstream of the bleed holes because of the high transpiration flow, leading to an energizing of the boundary layer in this region. On the contrary, a large area of reversed flow is found for large stagger angles, as the spanwise distance between the bleed holes is larger, and the near-wall momentum is low. Thus, the effectiveness of the bleed is expected to be smaller.

The rise in the wall shear stress is illustrated in Fig. 17. Surprisingly, no significant differences are observed if the rise in the wall shear stress is shown as a function of the bleed rate, as seen in Fig. 17a. However, a higher effectiveness is achieved for
<figure><img src="image_18.png" /><figcaption>Fig. 15 Variation of the surface sonic flow coefficient for different stagger angles for the SBLI</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 16 Streamwise wall shear stress below the shock foot for different stagger angles; dashed lines mark $τ_{w,x} = 0$ contour line</figcaption></figure>

$β = 45°$ if the pressure ratio $p_{pl}/p_t$ is kept constant. Thus, it can be stated that a 45° staggering has the biggest control effect on an SBLI.
<figure><img src="image_20.png" /><figcaption>**Fig. 17** Rise in the wall shear stress as a function of the bleed rate $\dot{m}_{bl}$ and the pressure ratio $p_{pl}/p_t$ for varying stagger angles; dotted line in (b) highlights isentropic pressure ratio for $M = 1.62$</figcaption></figure>

**5 Conclusion**

This chapter studies the influence of the hole diameter, the porosity, the thickness-to-diameter ratio, and the stagger angle on the performance of a porous bleed to control a shock-wave/boundary-layer interaction. A comprehensive numerical database is generated, and the bleed efficiency and the control effectiveness are evaluated. The selected flow case is an irregular shock reflection at the wall—a so-called Mach reflection. In the uncontrolled case, a separation bubble is observed below the shock foot.

The bleed control significantly changes the flow field. The first observed effect is linked to the different bleed rates upstream and downstream of the shock. The pressure rise due to the shock leads to a higher wall pressure downstream of the shock, which results in higher suction rates. Additionally, the flow downstream of the shock is subsonic, leading to an increased bleed efficiency. Hence, the tangential flow velocity decreases along the shock, while the transpiration velocity increases. The consequence is a different flow angle downstream of the shock compared to the flow upstream. Thus, a deflection angle along the shock is generated, reducing the shock intensity. The simulations show a significant reduction in the height of the Mach stem, which implies a “more regular” shock reflection.

Overall, smaller hole diameters are found to enhance bleed efficiency and control effectiveness. The smaller the holes, the smaller the size of the interaction length. The porosity level shows opposing trends for bleed efficiency and control effectiveness. High porosity levels are beneficial in the control effectiveness as more mass can be removed, even though the sonic flow coefficient is slightly lower. The thickness-to-diameter ratio is only linked to the bleed efficiency as it influences the flow field inside the holes. Large ratios are advantageous upstream of the shock, where the bleed rates
are low, and small ratios are better in the subsonic region. Thus, a medium value may be the optimum. Finally, the stagger angle shows only small influences. An angle of 45° is the optimum since the separated area below the shock foot is the lowest.

**Competing Interests** The authors have no conflicts of interest to declare that are relevant to the content of this chapter.

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