3.1. Basic flow features

The schlieren images in figure 3 illustrate an exemplary case of the CWsBLI, with the key features of the flow structure schematically depicted in figure 3 $(c)$ . The upstream and downstream boundaries of the compression waves were determined based on the variation in grey value resulting from changes in the density gradient, together with the inviscid solutions. The observed flow structure can be interpreted as follows. A separation bubble and a separated shear layer are induced by the APG, producing a separation shock. The compression waves interact with the separation shock and impinge on the separated shear layer, generating a series of expansion waves. Time-averaged schlieren images show that the expansion fans generated by the CWsBLI exhibit greater dispersion.

Figure 3.

$(a)$ An instantaneous schlieren image of the CWsBLI for $\varphi =12^{\circ }$ and $d=27.8{\text{ mm}}$ . $(b)$ An average intensity map from $1.0\times {10^5}$ schlieren images for the same $\varphi$ and $d$ . $(c)$ A schematic illustration (not to scale) of key flow features of the CWsBLI. Here, $i$ is the upstream boundary of the compression waves, $\textit{ii}$ is the downstream boundary of the compression waves, $\textit{iii}$ is the separation shock induced by the CWsBLI, $\textit{iv}$ is expansion fans induced by the compression waves, $\textit{v}$ is the separated shear layer, and $\textit{vi}$ is expansion fans induced by the model.

Figure 4.

Time-averaged schlieren images of CWsBLIs compared to ISWBLIs at the same $\varphi$ : $(a)$ $\varphi =10^{\circ }$ , $(b)$ $\varphi =12^{\circ }$ . The white dotted line represents either the incident shock or the boundaries of the compression waves. The origin is defined as the intersection point between the incident shock or the downstream boundary of the compression waves and the wall, and $L_{\textit{single}}$ denotes the interaction length of the ISWBLI.

Prior to defining the interaction length of CWsBLI, we describe the interaction characteristics in terms of the separation scale. Figure 4 shows the flow structures resulting from CWsBLI in various configurations, in contrast to the incident SWBLI (ISWBLI). All schlieren images are aligned in the flow direction according to the downstream boundary of the compression waves. By varying $\varphi$ , comparing the characteristics of CWsBLI reveals that when $d$ is relatively small, the induced flow separation aligns closely with that of ISWBLI. As $d$ increases, the separation scale decreases progressively.

The pressure distributions within the interaction region are shown in figure 5. In this figure, the mean location of the shock foot, which was identified according to the peak of the pressure spectrum, was set as the origin. Sensor positions were normalised by $L_{\textit{single}}$ . For ISWBLIs with different $\varphi$ , the typical two-step pressure rise can be observed: an initial separation-induced pressure rise followed by a reattachment-induced pressure rise. In CWsBLI cases, the initial pressure rise exhibits a trend similar to that of a typical SWBLI, marked by a sharp pressure increase. In the subsequent region, instead of a pressure plateau, the pressure continues to increase due to the persistent influence of compression waves when $d \leq L_{\textit{single}}$ . The slope of this pressure rise closely corresponds to the second pressure rise in a typical SWBLI. When $d \gt L_{\textit{single}}$ , the pressure distribution becomes comparable to that of an attached SWBLI. It is important to note that as a result of the influence of expansion waves downstream of the model, the maximum pressure ratios are significantly lower than the theoretical inviscid values ( $p_{3,\textit{in}v}/p_1$ ). Furthermore, the larger the value of $\varphi$ , the more pronounced this deviation becomes. The primary reason is that to minimise blockage, models with large $\varphi$ are designed with a relatively compact configuration, thus reducing the distance between the expansion fan and the interaction region. Consequently, the effect of the pressure drop becomes more substantial.

Figure 5.

Pressure distributions along the bottom-wall centreline: $(a)$ for $\varphi =10^{\circ }$ , $p_{3,\textit{in}v}/p_1 = 3.18$ in ISWBLI, and $p_{3,\textit{in}v}/p_1 = 3.17$ in CWsBLI, respectively; $(b)$ for $\varphi =12^{\circ }$ , $p_{3,\textit{in}v}/p_1 = 3.90$ in ISWBLI, and $p_{3,\textit{in}v}/p_1 = 3.88$ in CWsBLI, respectively.

Schlieren images show that the separation shock appears wider after time-averaged processing due to shock foot dispersion into multiple compression waves and the spanwise integration effect of the schlieren system. Therefore, maximum luminance points within the separation shock region were extracted at various heights without being affected by the expansion fan induced by the model. A straight line obtained through linear fitting served as an approximation of the separation shock. Figure 6 $(a)$ illustrates the variation in angles of the separation shock that correspond to different $d$ relative to the ISWBLI with $d=0$ mm. The angle of the separation shock in the ISWBLI reaches a maximum value. For CWsBLIs, the shock angle decreases as $d$ increases until it approaches $L_{\textit{single}}$ . When $d \gt L_{\textit{single}}$ , the angle gradually increases again. The uncertainty associated with the separation shock angle is estimated to be approximately 0.5 %. While this level of uncertainty is comparable to the difference in shock angles observed between the cases $d = 72.0\text{ mm}$ and $d = 75.1\text{ mm}$ , where the interaction lengths are both close to $L_{\textit{single}}$ , the flow behaviour in the immediate vicinity of this critical point remains indeterminate. Nevertheless, the trends derived from data sufficiently distant from the critical point are considered reliable.

Figure 6.

$(a)$ Angles of the separation shocks and $(b)$ development of the separated shear layers for CWsBLIs with $\varphi =12^{\circ }$ and varying $d$ , compared to those of the ISWBLI.

Schlieren images also show that the development of the separated shear layer varies with $d$ . The minimum luminance points in the shear layer were identified, and a quadratic curve fitting method determined the trend line of shear layer development (see figure 6 $b$ ). In this figure, the starting point of the shear layer is defined as the origin, and all positions are normalised with respect to $L_{\textit{single}}$ . For the ISWBLI with $d=0$ mm, the shear layer angle reaches its maximum value, indicating the largest flow deflection and shock angles. For CWsBLIs, the downward-deflected flow induced by compression waves affects the shear layer. Broader coverage range reduces both the shear layer angle and the shock angle. When $d$ approaches $L_{\textit{single}}$ , the shear layer developments for $d=64.9$ , 72.0 and 75.1 mm become similar, leading to nearly identical separation shock angles. When $d$ exceeds $L_{\textit{single}}$ , variations in the shear layer angle become negligible, while the angle of the separation shock increases due to changes in the upstream Mach number caused by partial compression waves.

Figure 7 $(a)$ presents maps of the frequency-premultiplied power spectral density ( $fG(f)$ ) of the pressure fluctuations with $\varphi =12^{\circ }$ . The spectra are normalised by the variance of the local wall pressure below $20.0\text{ kHz}$ . In all scenarios, the peak energy should be located at the convection frequency $U_{1}/\delta _{1}=89.51\text{ kHz}$ ; however, these peaks fall near $20\text{ kHz}$ , which is caused by the filtering of the acquisition system. The presence of a weak low-frequency component below 500 Hz can be observed at the shock foot. Subsequently, high-frequency fluctuations emerge, indicating the incipient instability of the separated shear layer. For ISWBLI and CWsBLI with a relatively large separation scale, the subsequent shedding of the shear layer is found to shift towards lower frequencies as it develops along the recirculation region.

Figure 7.

$(a)$ Contours of the frequency-premultiplied pressure fluctuation power spectral density within the interaction region and $(b)$ distributions of the dominant frequency $f_{ch}$ in the shear layer evolution of the CWsBLI with $\varphi =12^{\circ }$ and increasing $d$ , in comparison to the ISWBLI.

Figure 8.

$(a)$ Evolution of the dominant frequency of low-frequency oscillations at the separation shock foot, and $(b)$ evolution of the dominant frequency of shear layer unsteadiness for the CWsBLI with $\varphi =12^{\circ }$ and increasing $d$ , in comparison to the ISWBLI.

Figure 7 $(b)$ presents the streamwise evolution of the characteristic frequency $f_{ch}$ under varying values of $d$ . This $f_{ch}$ is determined directly from the spectral peaks recorded at each sensor location, with estimated uncertainty approximately $5\,\%$ . Across all cases, $f_{ch}$ exhibits a rapid increase in the low frequencies observed at the separation shock foot. It attains a local maximum at the inception of the shear layer, and subsequently decreases monotonically throughout the recirculation region until vortex shedding is initiated. As $d$ increases, the characteristic frequency at the vortex shedding location gradually rises, and this process can be categorised into two different types. For $d \leq L_{\textit{single}}$ , the separation scale remains relatively unchanged, which means that the shear layer length does not vary significantly. However, with the expanding influence of compression waves, the upstream boundary gradually moves closer to the inception of the shear layer development. Consequently, the vortex structures forming within the shear layer are disrupted by compression waves and break up prematurely. As a result, these vortices do not develop as fully as they do in the case of SWBLI or when $d$ is small, which ultimately leads to an increase in dominant frequency. When $d \gt L_{\textit{single}}$ , as described in figure 4, the reduction in the separation scale results in a shorter shear layer and smaller vortex structures, which in turn causes an increase in the characteristic frequency at the vortex shedding location. This trend is contrary to what has been previously observed in studies in which the separation scale increased (Li et al. Reference Li, Li, Ye and Zhang2024).

Figure 8 $(a)$ depicts the evolution of the low-frequency components associated with the separation shock oscillation and the characteristic frequency during the development of the shear layer with respect to $d$ . The Strouhal numbers ( $St_{L_{\textit{sep}}}=\textit{fL}_{\textit{sep}}/U_1$ ) corresponding to these frequencies, computed based on their respective separation scales, fall within the low-frequency range. However, $St_{L_{\textit{sep}}}$ associated with the shear layer transitions from medium to high frequency, indicating a decrease in flow separation and a transition towards turbulence-dominated behaviour. It is evident that the characteristic frequency changes as $d$ approaches $L_{\textit{single}}$ . When $d \leq L_{\textit{single}}$ , the frequency of the low-frequency oscillation gradually increases. One possible explanation is that under conditions of larger-scale separation, internal mechanisms contribute to the instability, while compression waves interact with the shear layer, thereby influencing the shedding rate. This can be observed in the continuously increasing shedding frequency shown in figures 7 $(b)$ and 8 $(b)$ . As $d$ increases, the impact of compression waves on the shear layer becomes more significant. When $d \gt L_{\textit{single}}$ , the separation scale decreases, and the rate of the entrainment-recharge process accelerates. However, the dominant frequency of the low-frequency oscillation drops rapidly. The possible reason for this is that the decrease in the separation scale leads to a change in the dominant mechanism of the unsteadiness, from ‘downstream mechanism’ to ‘upstream mechanism’. For example, low-frequency disturbances caused by the thin and pointed leading edge of the curved surface can be the main cause of the low-frequency oscillation of the separation shock. Figure 8 $(b)$ summarises the evolution of the characteristic frequency within the shear layer with respect to $d$ . When the separation scale is relatively large ( $d$ is small), the characteristic frequency remains in the medium to high frequency range, with $St_{L_{\textit{sep}}} = 0.2{-}1.0$ . When the separation scale is small, the characteristic frequency shifts to the high-frequency band, showing turbulent characteristics.

3.2. Length of interaction

A clear and reasonable definition of the interaction length is essential for developing the scaling law. The interaction length, as a critical parameter, quantifies the extent of the disturbed boundary layer and can estimate the separation scale under specific conditions. For ISWBLIs, it refers to the distance between the mean location of the separation shock (or reflected shock) and the extrapolated wall impact point of the incident shock; for compression corners, it denotes the distance to the corner. This study defines the interaction length of CWsBLI ( $ L_{\textit{com}}$ ) as the distance between the mean location of the separation shock (or reflected shock) and the extrapolated wall impact point of the downstream boundary of the compression waves, as illustrated in figure 9.

Figure 9.

Control volume for the CWsBLI.

The flow model depicting the interaction between the compression waves and the turbulent boundary layer is presented in figure 9. Under the inviscid assumption, the compression waves reflected by the wall can be equivalently replaced by an oblique shock wave (indicated by the dotted blue line). When viscous effects are considered, the boundary layer is subjected to an APG due to compression waves. As a result, the reflected shock is translated upstream by $ L_{\textit{com}}$ to ensure mass and momentum conservation. This length is defined as the interaction length. Based on this, a control volume enclosed by five surfaces (labelled $a$ to $e$ ) is defined, and mass conservation equations are derived for both inviscid and viscous cases:

(3.1) \begin{align}&\qquad\qquad\quad \rho _{1} U_{1} H_{cv1}+\rho _{2} V_{2} L_{cv}-\rho _{3} U_{3} H_{cv3}=0, \end{align}

(3.2) \begin{align}& \rho _{1} U_{1}\!\left (H_{cv1}-\delta _{1}^{*}\right )+\rho _{2} V_{2}\!\left (L_{cv}-L_{\textit{com}}\right )-\rho _{3} U_{3}\left (H_{cv3}-\delta _{3}^{*}\right )=0. \end{align}

By substituting (3.2) into (3.1), the interaction length can be derived as

(3.3) \begin{equation} L_{\textit{com}} = \frac {\rho _{3}U_{3}\delta _{3}^{*} - \rho _{1}U_{1}\delta _{1}^{*}}{\rho _{2}V_{2}} = \frac {\rho _{3}U_{3}\delta _{3}^{*} - \rho _{1}U_{1}\delta _{1}^{*}}{\rho _{2}U_{2}\sin (\varphi )}. \end{equation}

The dimensionless characterisation of the interaction length was obtained by utilising the geometric shape factor $G_3$ and the boundary layer displacement thickness $\delta _{1}^{*}$ :

(3.4) \begin{equation} L_{\textit{com}}^* = \frac {L_{\textit{com}}}{\delta _{1}^{*}}G_{3} = \frac {\rho _{3}U_{3}\delta _{3}^{*}}{\rho _{1}U_{1}\delta _{1}^{*}} - 1 = \frac {\dot {m}_{out}^{*}}{\dot {m}_{in}^{*}} - 1, \end{equation}

where

(3.5) \begin{equation} G_3 = \sin (\varphi ) \frac {\rho _2 U_2}{\rho _1 U_1}=\sin (\varphi ) \frac {M_{2}}{M_{1}} \left ({\frac {1+0.5(\gamma -1)M_{1}^2}{1+0.5(\gamma -1)M_{2}^2}}\right )^{\frac {\gamma +1}{2(\gamma -1)}}, \end{equation}

and the right-hand side of (3.4) will be replaced by the interaction intensity. The scaling of the interaction length for both CWsBLI and SWBLI is identical. The only distinction lies in the methodology used to determine $G_3$ : in CWsBLI, $G_3$ is calculated based on the Prandtl–Meyer function and isentropic relations, while in SWBLI, $G_3$ is established using the oblique shock wave relation. Equation (3.4) relates the interaction length to geometric parameters, incoming flow conditions, and interaction intensity. Among these factors, the interaction intensity serves as a critical determinant of the length scale for both SWBLI and CWsBLI, traditionally quantified by the pressure rise ( $S_{e}^*$ proposed by Souverein et al. Reference Souverein, Bakker and Dupont2013) or the pressure ratio ( $S_{r}^*$ proposed by Mahalingesh et al. Reference Mahalingesh, Piponniau and Dupont2024).

Figure 10.

$(a)$ Variation of the interaction length with different $\varphi$ and $d$ , and $(b)$ scaling for CWsBLI based on the method proposed by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024).

Based on the definition of the interaction length for CWsBLIs and the scaling law proposed by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024), figure 10 presents the variations of $L_{\textit{com}}$ under different $\varphi$ and $d$ , together with the corresponding scaling for $L_{\textit{com}}$ . It can be seen that when $d$ is smaller than $L_{\textit{single}}$ , the interaction length essentially remains consistent with $L_{\textit{single}}$ . Possible reasons for the deviations may include the influence of the downstream expansion fan of the model on the interaction, which decreases the pressure rise during reattachment. This inference aligns well with the conclusion drawn by Fan et al. (Reference Fan, Wang, Lin, Yang and Tang2023). It is crucial to note that in scenarios where $\varphi$ remains constant and the coverage range $d$ exceeds $L_{\textit{single}}$ , the pressure rise across the interaction region remains largely unaffected by further increases in $d$ , irrespective of whether the results are obtained from inviscid solutions or experimental measurements. This variety in trends was greater than the uncertainty associated with the measurement techniques. This poses significant challenges in establishing the separation criterion for CWsBLIs, since an identical pressure rise or pressure ratio corresponds to the same interaction state. As shown in figure 10 $(b)$ , based on the pressure ratio method, the scaling did not collapse the dataset of the CWsBLIs. Such data points fell on a straight vertical line under the same $\varphi$ , with shorter lengths for longer coverage ranges. A likely reason is that the interaction intensity scaling does not account for the effect of the coverage range of the compression waves. Hence this study highlights the need to consider changes in the coverage range of compression waves when evaluating the interaction intensity.

Figure 11.

Classification of the CWsBLI and its equivalent ISWBLI case.

3.3. Scaling analysis based on equivalent interaction intensity

The underlying reason for this misjudgement lies in the fact that the primary factor influencing the separation of the boundary layer is the pressure gradient. Our subsequent development of the scaling law starts from this consideration. Rather, if we were to adopt current methods, the influence of spatial scale would inevitably be mitigated. According to the aforementioned observations, under the same $\varphi$ , the interaction length of the corresponding ISWBLI serves as a critical demarcation point. When $d$ is either greater than or less than this value, the resulting length scales of the CWsBLI differ. Based on this characteristic, this study categorises CWsBLI into two types: type I, where $d \leq L_{\textit{single}}$ , and type II, where $d \gt L_{\textit{single}}$ , as illustrated in figure 11. The ISWBLI induced by an incident shock with flow deflection angle $\varphi$ is taken as the reference flow. An equivalent transformation approach is employed to convert CWsBLIs with varying coverage ranges into ISWBLIs exhibiting uniform interaction intensity. The key hypothesis is that two interactions producing the same interaction length must be driven by the same average APG, resulting in the same pressure rise in the interaction region. In CWsBLIs, the flow passing through the compression waves is assumed to be an isentropic process. Based on this classification and the associated hypotheses, subsequent discussions will focus on the scaling of these two types.

3.3.2. Type II: $d \gt L_{\textit{single}}$

For type II ( $d \gt L_{\textit{single}}$ ), the pressure rise within the interaction region originates from the upstream boundary of the compression waves, which distinguishes it from the previous type. In this context, the CWsBLI is substituted with an equivalent ISWBLI. The equivalent ISWBLI has the same Reynolds number, upstream Mach number, interaction length and interaction intensity as the original CWsBLI. Subsequently, the scaling can be performed in accordance with the methods proposed by Souverein et al. (Reference Souverein, Bakker and Dupont2013) or Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024). During this process, two key considerations must be taken into account.

First, it is the pressure gradient rather than the pressure rise that primarily governs the flow separation and the mass-flow deficit ratio across the interaction. Assuming that the pressure rise induced by the wave system remains constant, the increase in the coverage range $d$ directly contributes to a reduction in the average pressure gradient, which in turn reduces the size of the separation zone. Consequently, the average pressure gradient induced by compression waves, ${\Delta p_{\textit{com},\textit{in}v}}/d$ , which is also assumed to be equivalent to that in the separation region (region II in figure 11), serves as a critical parameter for the equivalent conversion. Therefore, the ISWBLI to be considered equivalent has the same pressure gradient as the CWsBLI:

(3.6) \begin{equation} \frac {{\Delta } p_{\textit{com},\textit{in}v}}{{\Delta } x}=\frac {{\Delta } p_{\textit{equ},\textit{in}v}}{{\Delta } x}. \end{equation}

Under condition $d \gt L_{\textit{single}}$ , the pressure gradient imposed on the boundary layer by the compression waves is given by $(p_{3,\textit{com},\textit{in}v} - p_{1})/d$ . Consider an equivalent ISWBLI scenario with the same pressure gradient and interaction length $L_{\textit{com}}$ . The pressure rise for the equivalent ISWBLI is

(3.7) \begin{equation} {\Delta p_{\textit{equ},\textit{in}v} = (p_{3,\textit{com},\textit{in}v} - p_{1})\frac {L_{\textit{com}}}{d}}. \end{equation}

This study uses the pressure ratio method proposed by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024) to characterise the intensity of CWsBLI. The equivalent pressure rise can be converted into the equivalent pressure ratio as

(3.8) \begin{equation} {\left ( \frac {p_{3,\textit{in}v}}{p_{1}} \right )_{\textit{equ}} = \frac {\Delta p_{\textit{equ},\textit{in}v} + p_{1}}{p_{1}} = \left ( \frac {p_{3,\textit{com},\textit{in}v}}{p_{1}} - 1 \right )\frac {L_{\textit{com}}}{d} + 1}. \end{equation}

Second, it is essential to ensure that two equivalent interactions exhibit identical interaction lengths. Given the Mach number $M_1$ and the interaction intensity determined previously, the flow deflection angle $\varphi _{\textit{equ}}$ of the equivalent ISWBLI can be established, which is crucial for determining $G_3$ in subsequent calculations. This consideration highlights an important issue. According to existing scaling laws for SWBLI, the interaction length can be determined solely based on $M_1$ and $\varphi _{\textit{equ}}$ . However, for CWsBLIs, this theoretically computed interaction length often deviates from the experimentally observed value $L_{\textit{com}}$ . To address this discrepancy, a correction is necessary to the initial separation pressure $p_{\textit{sep}}$ of the boundary layer, grounded in physical evidence.

The CWsBLI can be decomposed into an infinite series of discrete interactions that occur between each individual compression wave and the boundary layer. Within the interaction region, the pressure progressively increases in the streamwise direction, while the boundary layer’s ability to resist flow separation gradually decreases. This decline in separation resistance is attributed to the cumulative impact of disturbances generated by each wave interaction. When the pressure rise and the corresponding reduction in the boundary layer’s resistance exceed a critical threshold, flow separation is triggered. As shown in figure 11, the CWsBLI can be divided into two discrete parts: one partially covered by compression waves without flow separation (region I) and the other corresponding to the separated region (region II), which is covered by the remaining compression waves. The compression waves in region I initially interact with the boundary layer. Although this interaction does not lead to separation, the velocity deficits introduced into the boundary layer can influence the subsequent interaction. In region II, the compression waves interact with a perturbed boundary layer that differs from the corresponding boundary layer in the equivalent ISWBLI. Consequently, the interaction intensity $p_{\textit{sep}}$ responsible for the initial separation of the boundary layer must also be adjusted. Here, ${L_{\textit{com}}}/{d}$ also serves as a correction factor to determine this pressure ratio:

(3.9) \begin{equation} \left ( \frac {p_{\textit{sep}}}{p_{1}} \right )_{\textit{equ}} = 1 + \frac {\gamma\kern-1pt M_{1}^{2}}{2k}\left(\frac {L_{\textit{com}}}{d}\right)^{m}. \end{equation}

Considering the scenario of type I ( $d \le L_{\textit{single}}$ ), CWsBLIs with various $d$ can be equivalent to ISWBLI with the same flow deflection angle. The influence of $d$ can be deemed negligible. Therefore, introducing the correction factor ${L_{\textit{com}}}/{d}$ is unnecessary. To account for this behaviour, a piecewise function $\xi$ is defined as follows: $\xi = 1$ when $d \le L_{\textit{single}}$ , and $\xi ={L_{\textit{com}}}/{d}$ when $d \gt L_{\textit{single}}$ . By dividing (3.8) by (3.9), the dimensionless intensity of CWsBLI, denoted as $S_{\textit{com}}^*$ , is obtained as

(3.10) \begin{align} S_{\textit{com}}^* &= \frac {( {p_{3,\textit{com},\textit{in}v}}/{p_1} - 1)\xi + 1}{1 + (\gamma\kern-1pt M_{1}^2/2k)\xi ^m}, \quad k=\begin{cases}3.0, & \textrm{if}\,{\textit{Re}}_{\theta } \le 1 \times 10^4, \\ 2.5, & \textrm{if}\,{\textit{Re}}_{\theta } \gt 1 \times 10^4,\end{cases} \nonumber\\ \xi &=\begin{cases}1, & \textrm{if}\,d \le L_{\textit{single}}, \\ {L_{\textit{com}}}/{d}, & \textrm{if}\,d \gt L_{\textit{single}}.\end{cases} \end{align}

In addition to characterising the interaction intensity, scaling the interaction length is another essential step. In the work by Souverein et al. (Reference Souverein, Bakker and Dupont2013), a simplified model for the SWBLI was developed, leading to the derivation of the correction factor $G_3$ , which quantifies the effects of geometric parameters and the Mach number of the incoming flow. For the equivalent ISWBLI, the flow deflection angle $\varphi _{\textit{equ}}$ can be determined based on the equivalent pressure ratio $(p_{3,\textit{in}v} / p_1)_{\textit{equ}}$ using the oblique shock relations, given a specified incoming Mach number $M_1$ :

(3.11) \begin{equation} \left ( \frac {p_{3,\textit{in}v}}{p_{1}} \right )_{\textit{equ}} = f( M_{1}, \varphi _{\textit{equ}}) \Rightarrow \varphi _{\textit{equ}}. \end{equation}

The geometric correction factor $G_3$ for the equivalent ISWBLI is given by $ G_{3} = {\sin (\varphi _{\textit{equ}})\sin (\beta )}/{\sin (\beta - \varphi _{\textit{equ}})}$ . The appropriate scaling, using $L_{\textit{com}}^*$ as the non-dimensional interaction length of the CWsBLI, is

(3.12) \begin{equation} L_{\textit{com}}^* = \frac {L_{\textit{com}}}{\delta _{1}^{*}} \frac {\sin (\varphi _{\textit{equ}})\sin (\beta )}{\sin (\beta - \varphi _{\textit{equ}})} = \frac {\dot {m}_{out}^{*}}{\dot {m}_{in}^{*}} - 1. \end{equation}

According to the relationship proposed by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024), a first-order (linear) polynomial fit was made in the current analysis, which is expressed as

(3.13) \begin{equation} L_{\textit{com}}^{*}(\xi ) = a\, S_{\textit{com}}^*(\xi ) + b, \end{equation}

where $a = 2.6$ and $b = -1.4$ if the interaction was attached ( $S_{\textit{com}}^* \le 1$ ), and $a = 12.6$ and $b = -11.5$ if the interaction was separated ( $S_{\textit{com}}^* \gt 1$ ). In the definition of equivalent interaction intensity, the parameter $m$ is introduced to adjust the pressure required for the separation of the boundary layer. This parameter is derived by fitting the experimental data presented in figure 10 $(a)$ . Under the constraints imposed by (3.13), solve the optimisation problem of minimising the root mean square error, resulting in $m=0.65$ . It should be noted that the value of $m$ can be influenced by flow conditions such as the Mach number and the Reynolds number. In this study, these parameters are calibrated under specific experimental conditions. Future work should investigate how these parameters vary under different flow conditions in order to enhance the general applicability of the scaling law. For out-of-sample scenarios, when $ d \leq L_{\textit{single}}$ , $ \xi = 1$ , and consequently CWsBLI is equivalent to ISWBLI, with $ L_{\textit{com}} = L_{\textit{single}}$ . If $ d \gt L_{\textit{single}}$ , for a given $d$ , then the unknown in (3.13) is only $ L_{\textit{com}}$ and can be found.

Figure 12 illustrates the applicability of the scaling laws when comparing the current CWsBLI data with the SWBLI data from Souverein et al. (Reference Souverein, Bakker and Dupont2013) and Touré & Schülein (Reference Touré and Schülein2020). It should be noted that the experimental dataset from Souverein et al. (Reference Souverein, Bakker and Dupont2013) and Touré & Schülein (Reference Touré and Schülein2020) uses different values of $k$ , as indicated by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024). Although data from various types of interaction were included, the same normalised method, $S_{\textit{com}}^*$ , was applied consistently. It was clear that this new scaling law worked in fact for CWsBLI as well. For separated turbulent SWBLIs ( $S_{r}^*\gt 1$ ), the interaction intensity exhibits an identical expression form, with a coefficient of determination of 0.99 for both methods, as shown in table 3. Compared with the method proposed by Mahalingesh et al. (Reference Mahalingesh, Piponniau and Dupont2024), the approach presented in this study demonstrates a superior fitting performance in separated CWsBLI ( $R^2=0.94$ versus $R^2=0.88$ ). This improvement is primarily attributed to the incorporation of the effect of the coverage range on the interaction intensity within the model. In the case of SWBLI, the spatial length $ L_{\textit{single}}$ required for the flow to achieve a pressure rise within the SWBLI zone corresponds to the distance required for the loss of momentum to reach zero. As a result, the influence of $ L_{\textit{single}}$ becomes negligible, leading to a simplified expression based solely on the pressure rise or the pressure ratio. In contrast, for CWsBLI, the spatial length required for the pressure rise is affected by the coverage range of the compression waves, resulting in a distinct formulation of interaction intensity. By introducing the concept of equivalent interaction intensity, this study more accurately accounts for the impact of the coverage range of compression waves on the interaction length.

Table 3.

Coefficient of determination of first-order polynomial fit ${L}^{*}=aS^{*}_r+b$ .

Figure 12.

Non-dimensional interaction length and interaction intensity scaling in both SWTBLIs and CWsBLIs.

3.4. Discussion

As analysed in existing research, the interaction intensity exhibits only a weak dependence on the Reynolds number through the coefficient $k$ . In the case of SWBLIs, the interaction length is directly influenced by the Mach number and the flow deflection angle. In contrast, for CWsBLIs, in addition to these two parameters, the interaction intensity $S_{\textit{com}}^{*}$ is also closely related to the spatial extent of the compression waves – or more fundamentally, to the pressure gradient associated with the pressure distribution. This relationship enables the graphical determination of the $M{-}\varphi{-}d$ combination, which reveals a distinct tendency of the flow to separate.

Figure 13 illustrates the variation of $S_{\textit{com}}^{*}$ with $d$ for different Mach numbers and flow deflection angles. The total pressure ratio in the CWsBLI configuration arises from two distinct components: the flow passing through compression waves and a separation shock. It should be emphasised that the initial pressure ratio employed in the calculation of $S_{\textit{com}}^{*}$ is determined based on the assumption of an isentropic process. However, under non-ideal conditions, shock waves may develop during the compression process. To account for this phenomenon, this study adopts an extreme case approach, applying the oblique shock relation to approximate the pressure ratio achieved during compression. The figure also presents a comparison of the evaluation of $S_{\textit{com}}^{*}$ based on the isentropic assumption and the oblique shock relation. Furthermore, it should be emphasised that the parameter $m$ in the aforementioned equations is determined based on specific Mach numbers and flow deflection angles. Additional data across a range of operating conditions would be required to validate their broader applicability. Hence the current analysis remains qualitative.

Figure 13.

Dependence of $S_{\textit{com}}^{*}$ on the Mach number and the flow deflection: $(a)$ $M_{1}=2.0$ , $(b)$ $M_{1}=2.5$ , $(c)$ $M_{1}=3.0$ , $(d)$ $M_{1}=3.5$ , $(e)$ $M_{1}=4.0$ . The coverage range $d$ has been normalised based on the interaction length of the ISWBLI under corresponding Mach numbers and flow deflection angles. For the condition $M_{1} = 2.0$ , a Mach reflection is formed at $\varphi = 14^{\circ }$ ; therefore the corresponding results are not included. Similarly, under the conditions $M_{1} = 3.5$ and $M_{1} = 4.0$ with $\varphi = 10^{\circ }$ , the results are also omitted. The primary reason is that the interaction intensity of the SWBLI under these conditions is relatively low. Even in the presence of SWBLIs, the value of $S_r^*$ generally remains below 1.2.

It can be observed that when the Mach number and the flow deflection angle are small, the results obtained using the isentropic assumption and the oblique shock relation are in close agreement. As both the Mach number and the flow deflection angle increase, the discrepancy between the two methods gradually widens. For instance, when $M_{1}=4$ , $\varphi =14^{\circ }$ and $d/L_{\textit{single}}\approx 1.6$ , the deviation in $S_{\textit{com}}^{*}$ becomes 6.8 %. This discrepancy arises primarily from the calculation of the pressure rise. At a constant flow deflection angle, the pressure rise achieved through an isentropic process is greater than that resulting from an oblique shock, especially at higher Mach numbers. Consequently, under the same $d/L_{\textit{single}}$ and higher Mach numbers, if shock waves occur during the compression process, then employing the isentropic relation for calculations may lead to an overestimation of the actual pressure ratio, as well as the interaction intensity $S_{\textit{com}}^{*}$ , thus producing an overestimated interaction length. However, this deviation remains relatively minor at lower Mach numbers, typically in the range Mach 2–2.5.

Although there is a certain deviation between the ideal isentropic process and the actual scenario involving shock waves, the overall trends exhibited by both remain consistent. It can be observed that at a constant flow deflection angle and Mach number, the interaction intensity decreases as the coverage range increases (see (3.8)). For a fixed flow deflection angle and normalised coverage range, an increase in Mach number tends to delay the onset of flow separation, confirming the well-known property. Conversely, an increase in the flow deflection angle promotes flow separation.

The scaling law proposed in this study shows good performance under current experimental conditions; however, additional data obtained under diverse conditions are required to validate its broader applicability.

First, the proposed method exhibits potential advantages in addressing complex SWBLI scenarios, such as dual SWBLIs. For example, previous studies by Li et al. (Reference Li, Zhang, Tan, Jin and Li2022, Reference Li, Zhang, Tan, Sun, Yu, Jin and Zhou2023) have indicated that the separation scale in such interactions is significantly influenced by the spacing between the shock waves. By converting dual SWBLIs into equivalent single shock wave configurations, the method offers a simplified yet effective scaling approach for analysing these intricate flow phenomena. Future work should aim to further validate the applicability of this method across a broader range of wave system configurations.

Second, this study employed a curved surface design based on tracing the path of a streamline within the expansion waves passing over a sharp corner. However, the aerodynamic configurations of the actual high-speed inlets may differ due to various factors. For example, considering the constraints of the wall pressure distribution, various aerodynamic profiles can emerge. Therefore, a broader range of curved surface configurations should be taken into account.

In addition, the experiment was conducted at Mach number 2.3 with a turbulent boundary layer. For inlets operating over a wide Mach number range, the upstream flow conditions of the cowl will vary. The interaction characteristics associated with laminar and transitional CWsBLI may differ from those observed in turbulent flows, particularly under conditions involving low Reynolds numbers or high Mach numbers. The applicability of the scaling law under varying Mach numbers, Reynolds numbers, boundary layer conditions and wall conditions requires further validation.

Finally, this study did not account for the influence of three-dimensional effects on flow separation. In practical applications, these factors can significantly affect the dynamics of the boundary layer and the extent of flow separation. Therefore, future research should integrate three-dimensional numerical simulations with experimental investigations to examine how these parameters modify the scaling laws.