2.1. Governing equations and calculation schemes

The numerical simulations are based on the three-dimensional Navier–Stokes equations, non-dimensionalised using free-stream variables, as expressed in (2.1)

(2.1) \begin{equation} \frac {\partial Q}{\partial t}+\frac {\partial f_{c}}{\partial x}+\frac {\partial g_{c}}{\partial y}+\frac {\partial h_{c}}{\partial z}-\frac {1}{Re_{\infty }}\left (\frac {\partial f_{v}}{\partial x}+\frac {\partial g_{v}}{\partial y}+\frac {\partial h_{v}}{\partial z}\right )=0, \end{equation}

among them, the convective and viscous fluxes in the $x$ -direction can be expressed as (2.2)

(2.2) \begin{equation} f_{c}=\left ( \begin{array}{c} \rho u \\ \rho u u+p \\ \rho u v \\ \rho u w \\ (\rho e +p)u \\ \end{array}\right )\! , \quad f_{v}=\left ( \begin{array}{c} 0 \\ \tau _{11} \\ \tau _{12} \\ \tau _{13} \\ u \tau _{11}+v \tau _{12}+w \tau _{13}-q_{1} \\ \end{array}\right )\! , \end{equation}

where

(2.3) \begin{align} \tau _{\textit{ij}} &= \mu \left (\frac {\partial u_{i}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{i}}\right )-\frac {2}{3}\mu \frac {\partial u_{k}}{\partial x_{k}}\delta _{\textit{ij}}, \quad q_{i}=-\frac {\mu }{(\gamma -1)Re_{\infty }M\!a_{\infty }^{2}Pr}\frac {\partial T}{\partial x_{i}}, \nonumber\\[10pt] e & = \frac{T}{\gamma (\gamma-1) Ma_\infty^2 } + \frac{u^2 + v^2 + w^2}{2}, \quad p=\frac {\rho T}{\gamma Ma_\infty ^2}, \quad \mu =\frac {1+T_s/T_\infty }{T+T_s/T_\infty }T^{3/2}. \end{align}

Variables in (2.1)–(2.3) with the subscript ‘ $\infty$ ’ denote free-stream values, while those without subscripts, such as density $\rho$ , streamwise velocity $u$ , wall-normal velocity $v$ , spanwise velocity $w$ , pressure $p$ and temperature $T$ are dimensionless quantities obtained by normalisation with free-stream parameters. Moreover, $\gamma$ is the specific heat capacity ratio, $\delta_{\textit{ij}}$ represents the Kronecker $\delta$ , and $Ma$ , $Re$ , $Pr$ denote the Mach number, Reynolds number and Prandtl number, respectively.

This study employs the open-source finite-difference DNS solver, OpenCFD-SC2.0, developed by Li et al. (Reference Li, Fu and Ma2008, Reference Li, Fu, Ma and Liang2010), which is tailored for structured three-dimensional curvilinear grids. Known for its ease of deployment and clear program structure, the solver has demonstrated high computational accuracy and robustness across various benchmarks, including OSWTBLI (Tong et al. Reference Tong, Lai, Duan, Dong, Yuan and Li2022b , Reference Tong, Yuan, Lai, Duan, Sun and Dongc ), compression corners (Zhu et al. Reference Zhu, Yu, Tong and Li2016; Tong et al. Reference Tong, Yu, Tang and Li2017) and supersonic plate turbulent boundary layers (Liang & Li Reference Liang and Li2013; Tong et al. Reference Tong, Dong, Lai, Yuan and Li2022a ). For numerical discretisation, convective fluxes are split using the local Lax–Friedrichs method and reconstructed with the high-order TENO6-SA scheme (Zhang et al. Reference Zhang, Jiang, Qin, Qu and Yan2024), which incorporates the piecewise spectral optimisation method to achieve a broader resolved bandwidth with equivalent accuracy and grid resolution. This provides potential advantages for simulating multi-scale turbulence. Viscous fluxes are discretised using the sixth-order central difference scheme, while time integration is performed with the classic third-order Runge–Kutta method.

Figure 1.

Schematic of computational domain and boundary conditions for SWTBLI simulations. The iso-surface is the second invariant of velocity gradient $Q = 10$ and coloured by the wall-normal height.

2.2. Simulation conditions and configurations

The computational domain for simulating the OSWTBLI problem, as illustrated in figure 1, has sizes of $[L_x \times L_y \times L_z] = [100 \times 10 \times 4]$ mm. The front boundary at $x = 0$ mm is defined as a laminar boundary layer inlet with $M\!a_\infty = 2.25$ , derived from the Blasius solution at a distance of 127 mm from the leading edge of the plate. Detailed free-stream parameters are listed in table 1. The rear boundary is a supersonic outlet, while the side boundaries are periodic. The bottom boundary is an isothermal wall with $T_w=1.9T_\infty$ , closely approximating the temperature of an adiabatic wall under the current flow conditions. The subscript $w$ denotes the variable at the wall. To generate a canonical turbulent boundary layer (TBL), this study adopts the transition-inducing method based on blowing and suction, as referenced in Pirozzoli, Grasso & Gatski (Reference Pirozzoli, Grasso and Gatski2004). A synthetic jet band is implemented at $x=x_a \sim x_b = 7.5 \sim 20.2$ mm from the leading edge, with wall-normal velocity disturbances determined by (2.4)

(2.4) \begin{align} v(x,y=0,z,t) &= A f(x) g(z) h(t), \quad x_a\leqslant x\leqslant x_b \nonumber\\ f(x) &= \frac {4}{\sqrt {27}}\sin (\theta )[1-\cos {(\theta )}], \quad \theta =2\pi \frac {x-x_{a}}{x_{b}-x_{a}} \nonumber\\ g(z) &= \sum _{l=1}^{l_{\textit{max}}}\frac {0.8^{l-1}}{g_{0}}\sin \left(2\pi l\frac {z}{L_z}+\phi _{l}\right), \quad g_0 = \sum _{l=1}^{l_{\textit{max}}}0.8^{l-1} \nonumber\\ h(t) &= \sum _{m=1}^{m_{\textit{max}}}\frac {0.8^{m-1}}{h_{0}}\sin (m \beta t+\phi _{m}), \quad h_0=\sum _{m=1}^{m_{\textit{max}}}0.8^{m-1}. \end{align}

Here, $l_{ {max}}$ and $m_{ {max}}$ are set to 10 and 5, respectively, while $\phi _{l}$ and $\phi _{m}$ are random numbers in the range $[0, 2\pi ]$ . The amplitude $A$ is assigned a larger value of 0.1 to accelerate transition. Under these settings, canonical turbulent structures fully develop at $x\gt 55$  mm after approximately 100 dimensionless time units. Consequently, this study selects $x=x_{\textit{ref}}=70$ mm as the nominal shock impingement point and reference station. For simplicity, the dimensionless coordinates $ [x_0 \times y_0 \times z_0 ]$ are defined as $ [ (x-x_{\textit{ref}} ) \times y \times z ]/\textrm {mm}$ . Other flow variables are non-dimensionalised using the boundary layer thickness $\delta _0$ , friction velocity $u_\tau$ , wall kinematic viscosity $\nu _w$ and wall density $\rho _w$ at $x_{\textit{ref}}$ , facilitating their representation in either outer or inner scales. To examine variations in SWTBLI with different interaction intensities, two incident shock angles are considered. For the weak interaction, the incident shock angle is set to $33.2^\circ$ , which corresponds to a classic OSWTBLI simulation that has been extensively analysed. For the strong interaction, the incident shock angle is increased to $37^\circ$ . As shown in figure 1, the incident shock and the reflected separation shock divide the upper boundary into three regions. The boundary conditions for these regions significantly influence the shock position and intensity. The regions upstream of the incident shock and downstream of the reflected shock are set as penetrating boundary conditions, consistent with the standard turbulent boundary layer simulations without an incident shock. In the intermediate region, post-shock conditions are precisely specified based on the oblique shock relations for the two incident angles, allowing the flow to adaptively generate the correct shock pattern.

Table 1.

Free-stream conditions: Mach number, unit Reynolds number, temperature, Prandtl number, specific heat capacity ratio and specific gas constant.

All simulations employed a consistent grid resolution of $[3500 \times 300 \times 350]$ , with finer grids allocated to the interaction zone for improved accuracy. Specifically, in the streamwise direction, 900 progressively refined grids cover the turbulent boundary layer generation region ( $x=0 {-} 55$ mm), 2500 uniform grids are utilised in the interaction region ( $x=55 {-} 85$ mm) and 100 progressively coarser grids are applied in the downstream buffer zone ( $x=85 {-} 100$ mm). The transitions between these regions are uniformly smoothed to avoid discontinuities. Additionally, wall-normal grids are arranged based on (2.5), with more points concentrated near the wall to capture boundary layer structures

(2.5) \begin{equation} y_j=L_y\left \{ 1- \frac {\tanh \left [ \kappa (1-j/N_y) \right ]}{\tanh \kappa } \right \}, \quad j=0,1,\cdots \!,N_y, \quad \kappa =3. \end{equation}

Spanwise grids are uniformly distributed. Under these configurations, the grid scales within the interaction region reach $[\Delta x^+ \times \Delta y_w^+ \times \Delta z^+] = [4.65 \times 0.39 \times 4.43]$ , which are comparable to those in recent DNS studies (Bernardini et al. Reference Bernardini, Della Posta, Salvadore and Martelli2023; Yu et al. Reference Yu, Dong, Liu, Tang, Yuan and Xu2023). The superscript $+$ denotes a variable normalised in wall units.

2.3. Validation

To verify the accuracy of the simulation results, figure 2 presents the van Driest transformed velocity ( $u_{V\!D}$ ) at different streamwise stations, measured after turbulence has fully developed and in the absence of the incident oblique shock, as defined in (2.6)

(2.6) \begin{equation} u_{V\!D}^+=\frac {1}{u_\tau }\int _0^{\overline {u}}\sqrt {\frac {\overline {\rho }}{\overline {\rho }_w}}{\textrm{d}}\overline {u}. \end{equation}

In this study, $-$ and $\sim$ represent the Reynolds average and Favre (density-weighted) average, respectively. The corresponding fluctuation variables are denoted as $'$ and $''$ , respectively. Unless otherwise specified, all mean results in this study are averaged over both time and the spanwise direction.

Figure 2.

Comparison of the van Driest transformed mean velocity $u_{V\!D}^+$ profiles at different streamwise stations and similar simulations (Fang et al. Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011b ).

The calculated profiles in figure 2 clearly conform to the classical law of the wall in the viscous sublayer ( $y^+ \lt 5$ ) and logarithmic region ( $ 30 \lt y^+ \lt 100$ ) and exhibit excellent agreement with recent DNS simulations under similar flow conditions (Fang et al. Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011b ), confirming that the flow has transitioned into fully developed turbulence for $x_0\geqslant -15$ . In addition, the density-scaled Reynolds stress components, $({\overline {\rho }}/{\overline {\rho }_w})\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , $({\overline {\rho }}/{\overline {\rho }_w})\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , $({\overline {\rho }}/{\overline {\rho }_w}) \widetilde {w^{\prime\prime}w^{\prime\prime}}$ and $({\overline {\rho }}/{\overline {\rho }_w}) \!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ at the reference station are compared with the incompressible DNS data by Wu & Moin (Reference Wu and Moin2010), and the results are shown in figure 3(a). These results also demonstrate satisfactory consistency, verifying the correctness of the simulation results.

Figure 3.

( $a$ ) Density-scaled Reynolds stresses: $({\overline {\rho }}/{\overline {\rho }_w})\widetilde {u^{\prime \prime}u^{\prime \prime}}$ (), $({\overline {\rho }}/{\overline {\rho }_w})\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ (), $({\overline {\rho }}/{\overline {\rho }_w})\widetilde {w^{\prime \prime}w^{\prime \prime}}$ () and $({\overline {\rho }}/{\overline {\rho }_w})\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ (). Lines are present results at $x_{\textit{ref}}$ , symbols are incompressible DNS results from Wu & Moin (Reference Wu and Moin2010). ( $b$ ) Spanwise two-point correlations: $C_{uu}$ (), $C_{vv}$ (), $C_{ww}$ (). The results at $y^+ = 10$ and 100 are denoted as thin solid and thick dashed lines, respectively.

The spanwise two-point correlations of velocity fluctuations at various heights at $x_{\textit{ref}}$ are presented in (2.7). The two-point correlation for velocity fluctuation $u_{i}^{\prime }$ is defined as follows:

(2.7) \begin{equation} C_{u_{i} u_{i}}(x, y, r_z)=\frac {\overline {u_{i}^{\prime }(x, y, z) u_{i}^{\prime }(x, y, z+r_z)}}{\overline {u_{i}^{\prime }(x, y, z) u_{i}^{\prime }(x, y, z)}}, \quad r_z \in \left [0,L_z/2\right ]. \end{equation}

Figure 3(b) illustrates that all two-point correlations decrease as the spanwise interval increases, with slower decay at higher wall-normal heights. Nevertheless, as the spanwise interval $r_z$ approaches $L_z/2$ , all correlations approach zero, confirming that the spanwise domain length designed in this study is sufficient and appropriate.

3.1. Time-averaged flow field and separation scale comparison

For a free-stream Mach number of approximately 2.25, an oblique shock with a $33.2^\circ$ incident angle generates a mild and intermittent separation bubble. This computational set-up is a classical benchmark for studying OSWTBLI, first analysed via DNS by Pirozzoli & Grasso (Reference Pirozzoli and Grasso2006) and later reproduced in numerous numerical studies (Fang et al. Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020; Tong et al. Reference Tong, Lai, Duan, Dong, Yuan and Li2022b , Reference Tong, Yuan, Lai, Duan, Sun and Dongc ). To further explore the impact of phenomena such as larger-scale separation bubbles and increased unsteadiness associated with stronger incident shocks on the turbulence amplification mechanisms in SWTBLI, this study introduces a stronger oblique shock with an incident angle of $\beta =37$ $^\circ$ . Figure 4 presents the mean wall skin-friction coefficient ( $C_{\!f}$ ) and the mean wall pressure coefficient ( $C_{\!p}$ ) for the two incident angles, as defined in (3.1)

(3.1) \begin{equation} C_{f}=\frac {2 \left (\mu \partial \overline {u}/ \partial y\right )_w }{\rho _\infty u_\infty ^2}, \quad C_{\!p}=\frac {\overline {p}_w }{p_\infty }. \end{equation}

Figure 4.

Streamwise distributions of mean wall skin-friction coefficient ( $C_{\!f}$ ) and the mean wall pressure coefficient ( $C_{\!p}$ ) under weak ( $a$ ) and strong ( $b$ ) interactions.

Although variations in computational conditions, such as the boundary layer thickness at the nominal impingement location, Reynolds number (Laguarda et al. Reference Laguarda, Hickel, Schrijer and van Oudheusden2024) and the wall-to-recovery temperature ratio (Bernardini et al. Reference Bernardini, Asproulias, Larsson, Pirozzoli and Grasso2016), may lead to differences in $C_{\!f}$ , the overall shape of the $C_{\!f}$ curve for weak interaction shown in figure 4(a) remains similar to the results from Vyas, Yoder & Gaitonde (Reference Vyas, Yoder and Gaitonde2019) and Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020). It exhibits a distinctive ‘W’-shaped profile in the separation zone, a trend likely influenced by the behaviour observed at lower Reynolds numbers, as reported in Laguarda et al. (Reference Laguarda, Hickel, Schrijer and van Oudheusden2024).

Comparing the $C_{\!f}$ curves for strong and weak interactions reveals distinct differences. In the weak interaction case, the separation zone is smaller, and the maximum negative $C_{\!f}$ occurs downstream of the separation point (S). In contrast, the strong interaction exhibits an enlarged separation zone and a deformed ‘W’-shaped $C_{\!f}$ . While the first half of the ‘W’ remains similar to that in the weak interaction, the second half is significantly extended. Additionally, a $C_{\!f}$ plateau emerges in the region $x_0 \approx -7.5\,\textrm{to}\,{-}3.7$ , with the maximum negative $C_{\!f}$ appearing further downstream, closer to the reattachment point (R). Some characteristics of $C_{\!f}$ in strong interaction are consistent with the $M\!a_{\infty }=2.0$ , $\beta =40$ $^\circ$ case at low Reynolds numbers in Laguarda et al. (Reference Laguarda, Hickel, Schrijer and van Oudheusden2024) and the $M\!a_\infty =2.3$ , $\beta =36.9$ $^\circ$ case in Gross, Little & Fasel (Reference Gross, Little and Fasel2022). Moreover, in the strong interaction, $C_{\!p}$ exhibits a plateau near the region where the $C_{\!f}$ plateau occurs, suggesting the presence of large-scale separation bubbles and the formation of reattachment compression waves. Conversely, under weak interaction, $C_{\!p}$ increases smoothly in the streamwise direction near the reference station.

To provide a clearer comparison of the differences in separation bubbles between strong and weak interactions, figure 5 presents the numerical schlieren visualised using the mean density gradient magnitude in the interaction regions, along with the sonic line and the mean streamline near the separation zone.

Figure 5.

Contours of mean density gradient magnitude fields under weak ( $a$ ) and strong ( $b$ ) interactions together with sonic line (line).

The numerical schlieren in figure 5 clearly indicate that the primary flow structures in both cases share common features, including the incident shock, reflected separation shock, expansion waves behind the incident shock foot and the detached free shear layer originating from the separation shock foot. However, notable differences arise due to variations in the separation bubble characteristics. The $C_{\!f}$ and $C_{\!p}$ curves in figure 4 imply that a complete separation bubble forms under strong interaction, with a streamwise length of approximately 12 mm and a wall-normal height of around 1.4 mm. In contrast, the separation bubble in the weak interaction case is significantly less distinct. The local zoomed-in view reveals that its streamwise length is approximately 1.8 mm, while its wall-normal height is much smaller, measuring around 0.036 mm.

Regardless of the separation bubble size, the deceleration and upward deflection of the mean flow at the initial separation stage remain generally consistent. Consequently, the compression waves tend to converge around $y_0 = 1$ , forming the separation shock in both cases. In addition, the incident shock foot is typically positioned above the top of the separation bubble, while the separation shock foot is located upstream of the separation point. A larger separation bubble, as displayed in the strong interaction case, significantly increases the height of the incident shock foot and extends the distance between the foot of the separation and the incident shocks. This explains the phenomenon observed in figure 5, where the separation shock in strong interaction forms further upstream of the shock intersection point, whereas in weak interaction, the separation shock is generated closer to the intersection point.

Furthermore, the larger separation zone observed in the strong interaction indicates the presence of a more extensive subsonic region. Although the overall shape of the sonic line is similar at both incident angles, its position in strong interaction is significantly elevated, further confirming the existence of a larger subsonic region. A comparison between the schlieren contour and the sonic line reveals that the position of the detached free shear layer, as indicated by the density gradient magnitude, aligns well with the sonic line in both interactions. However, notable differences emerge downstream of the interaction region. Under weak interaction, the height of the free shear layer steadily increases, while under strong interaction, its height gradually decreases. The impact of these differences in the mean flow field on turbulence amplification will be discussed comprehensively in subsequent sections.

3.2. Amplification process of turbulence kinetic energy and Reynolds stresses

The previous subsection highlighted significant differences in the scale and shape of the separation bubble under strong and weak interactions, as observed through numerical schlieren, $C_{\!f}$ , and $C_{\!p}$ plots. These variations are expected to influence the TKE amplification mechanism, necessitating further exploration. To this end, figure 6 presents the mean TKE contours for different incident angles, accompanied by the zero-velocity streamline and dividing streamline to clearly indicate the separation bubble boundaries. The definition of TKE is provided in (3.2)

(3.2) \begin{equation} \textrm {TKE}=K=\frac {1}{2}\big( \widetilde {u^{\prime \prime}u^{\prime \prime}}+\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}+\widetilde {w^{\prime \prime}w^{\prime \prime}} \big ). \end{equation}

Figure 6.

Contours of mean TKE fields under weak ( $a$ ) and strong ( $b$ ) interactions together with zero-velocity streamline (line) and dividing streamline (line).

As shown in figure 6, the TKE peak reaches its maximum near the separation shock foot in both cases, with weaker amplification observed downstream of the incident shock foot. According to Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020), downstream TKE amplification is predominantly influenced by shear effects, whereas upstream amplification, though also affected by shear, is primarily governed by stronger flow deceleration. Differences in separation bubble structures lead to distinct TKE distribution patterns. Figure 6 indicates that the amplified TKE is consistently located above the dividing streamline. In the strong interaction, the local TKE peak appears to be elevated due to the increased wall-normal height of the separation bubble, resulting in a rise-and-fall pattern in the wall-normal direction as the flow evolves. Furthermore, the strong interaction exhibits a ‘secondary amplification’ of TKE with reduced intensity in the downstream recovery region, which is not observed in the weak interaction.

To better understand the TKE amplification process and identify its dominant factors, figure 7 presents the mean results of three normal stress components ( $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ ) and the shear stress component $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ for both interaction cases.

Figure 7.

Contours of mean Reynolds stresses $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ (a,b), $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ (c,d), $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ (e,f) and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ (g,h) under weak (a,c,e,g) and strong (b,d,f,h) interactions together with zero-velocity streamline (line) and dividing streamline (line).

The results in figure 7(a,b) highlight that $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , the dominant contributor to TKE in most regions of this SWTBLI problem, exhibits a similar evolution trend to that of TKE, with significant amplification occurring downstream of the separation shock. However, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ follows a different pattern. After being amplified near the separation shock foot, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ does not weaken significantly near the incident shock foot but instead dissipates gradually as the flow evolves. The two wall-parallel components, $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ and $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , exhibit similar streamwise evolution trends. While $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ also amplifies near the separation shock, its amplification is less intense than that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , and it weakens downstream of the incident shock. The shear stress component, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ , behaves differently from the normal stress components. Its local peak between the separation shock foot and the incident shock foot is closer to the wall. This behaviour aligns with the results of Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020) and may be related to the distinct amplification mechanisms of shear and normal stresses. For weak interaction, the intensity of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ remains comparable upstream and downstream of the incident shock foot. In contrast, under strong interaction, the detached free shear layer downstream of the incident shock foot experiences a significant increase in $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ , indicating a ‘secondary amplification.’ Similarly, all normal stresses under the strong interaction also exhibit ‘secondary amplification’ downstream of the incident shock. This phenomenon is absent in the weak interaction but resembles the initial amplification near the separation shock.

To provide a clearer comparison of stress amplification peak intensities and heights, figure 8 presents the streamwise evolution of local peak values, peak heights and amplification factors for various stress components. The amplification factor is normalised using the stress peaks of canonical turbulent boundary layers prior to turbulence amplification, as defined in (3.3)

(3.3) \begin{equation} \varphi _\alpha (x_0)=\frac {\varLambda _\alpha (x_0)}{\varLambda _\alpha (x_s)}, \end{equation}

where, $\alpha$ represents the TKE or any Reynolds stress component, $\varLambda$ denotes the local peak values at a given streamwise station $x_0$ and the undisturbed station, $x_s$ , is set to $x_0 = -5$ and $-12.5$ for weak and strong interactions, respectively.

The streamwise evolution of local peaks reflects the proportional contribution of each stress component, while the amplification factor provides a direct comparison of the extent of amplification relative with free-stream conditions. Notably, the stress peak plots exclude the influence of shocks to facilitate a more accurate analysis of turbulence development trends and amplification mechanisms.

Figure 8.

Streamwise evolutions of peak values (solid line) (a,b), peak heights (dashed line) (a,b) and amplification factors (solid line) (c,d) for TKE (black) and Reynolds stresses $\widetilde {u^{\prime \prime}u^{\prime \prime}}/2$ (), $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}/2$ (), $\widetilde {w^{\prime \prime}w^{\prime \prime}}/2$ (), $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ () under weak (a,c) and strong (b,d) interactions.

The streamwise evolution of peaks and amplification factors of Reynolds stresses and TKE, as depicted in figure 8, indicates that, despite significant differences in shock intensity and separation bubble structures between strong and weak interactions, the maximum peak values and amplification factors for each stress component during the initial amplification remain comparable. This suggests that shock intensity does not significantly enhance turbulence amplification, implying that the direct effect of shocks is not the primary mechanism driving turbulence amplification in SWTBLI. Moreover, regardless of interaction strength, the peaks of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and TKE have similar streamwise evolution trends, even though their specific values differ, reaffirming the dominant role of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ in the TKE amplification process. Furthermore, as shown in the enlarged view, both strong and weak interactions exhibit the same initial amplification order for normal stresses: $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ and $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ amplification beginning even upstream of the separation shock foot, as reported by Yu et al. (Reference Yu, Zhao, Tang, Yuan and Xu2022). For shear stress $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ , initial amplification begins alongside $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ but at a slower rate. However, as $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ amplification progresses, the $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ amplification rate significantly accelerates, eventually matching the growth of other components.

The differences between strong and weak interactions are also evident. In the weak interaction case, both $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibit significant amplification near the separation shock foot, followed by rapid decay upon crossing the incident shock foot, forming distinct amplification peaks that collectively shape the TKE peak. Meanwhile, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ undergoes notable amplification downstream of the separation shock foot but decays more gradually, resulting in a flatter peak. The shear stress $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ amplifies after the separation shock foot, with its decay upstream and subsequent increase downstream of the incident shock foot forming distinct double peaks in the amplification factor plot. Under strong interaction, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ rapidly amplifies near the separation shock foot. However, as discussed in $\S$ 3.1, the distance between the separation shock foot and the incident shock foot is larger in the strong interaction. Within this region, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ exhibits an approximately uniform decay rate, forming a ‘slope platform’ in the amplification factor plot. Near the incident shock foot, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ experiences a rapid stepped decay, followed by a weaker ‘secondary amplification.’ For $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , the second-largest normal stress component contributing to TKE, no sharp amplification peak is observed near $x_0=-9.6$ as seen in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . However, it still exhibits a ‘slope platform’ upstream of the incident shock foot, a stepped decay near the incident shock foot and a ‘secondary amplification’ downstream. In contrast, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , the smallest contributor to TKE, exhibits relatively minor variations in its local peak values along the streamwise direction. However, its amplification factor follows a pattern similar to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , except for the absence of a stepped decay process. The shear stress $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ also amplifies and forms a ‘slope platform’ downstream of the separation shock foot. Unlike the normal stresses, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ displays a more pronounced ‘secondary amplification’ downstream of the incident shock foot, with its second peak significantly exceeding the first.

The streamwise evolution of peak heights also exhibits distinctions between strong and weak interactions. As illustrated in figure 8(a,b), the peak heights of stress components are significantly higher under strong interaction than weak interaction after amplification. However, within the same interaction, the streamwise evolution of peak heights across different stress components remains similar, with their peak positions generally aligning with the corresponding free shear layer. In the turbulent amplification region upstream of the incident shock, peak heights of all stress components gradually increase. Regardless of interaction strength, at the same streamwise station, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ consistently exhibits the highest peak position, while $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ maintain similar peak positions. This behaviour may be attributed to the earliest amplification of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . The TKE, as the sum of the three normal stresses, reaches a peak height slightly lower than that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . In the downstream recovery region, the streamwise evolution of peak heights differs notably between strong and weak interactions. For weak interaction, peak heights continue to increase, whereas in strong interaction, a significant decrease occurs upstream of the reattachment point before $x_0 \approx 0$ , followed by a slight downward trend further downstream.

In summary, turbulence amplification differs significantly between strong and weak interactions. In weak interaction, all normal stresses exhibit a single peak amplification, except for $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ , which displays a double peak. In contrast, strong interaction features both a ‘slope platform’ and ‘secondary amplification’ for all stress components, with the boundaries of the ‘slope platform’ closely corresponding to the $C_{\!f}$ and $C_{\!p}$ plateaus. The unique amplification mechanisms observed in strong interaction, along with differences in the initial amplification sequence of stress components, will be further investigated in subsequent sections.

4.1. Budgets of turbulent kinetic energy and Reynolds stresses

The TKE budgets provide a comprehensive view of energy production, transfer and dissipation, providing deeper insight into turbulence mechanisms (Yu et al. Reference Yu, Zhao, Tang, Yuan and Xu2022). Additionally, analysing TKE budgets offers guidance for improving Reynolds-averaged Navier–Stokes models (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011a ). Using the continuity and momentum equations, an unaveraged transport equation for Reynolds stresses can be derived, as presented in (4.1), with a detailed derivation available in Clark (Reference Clark2020)

(4.1) \begin{align} \frac {\partial \rho u_{i}^{\prime \prime } u_{j}^{\prime \prime }}{\partial t} & = -\frac {\partial \rho \widetilde {u}_k u_{i}^{\prime \prime } u_{j}^{\prime \prime }}{\partial x_k} - \frac {\partial \rho u_k^{\prime \prime } u_{i}^{\prime \prime } u_{j}^{\prime \prime }}{\partial x_k} - \left (\frac {\partial p^{\prime } u_j^{\prime \prime }}{\partial x_i} + \frac {\partial p^{\prime } u_i^{\prime \prime }}{\partial x_j} \right )+ \left ( \frac {\partial u_i^{\prime \prime }\tau _{jk}^{\prime }}{\partial x_k} +\frac {\partial u_j^{\prime \prime }\tau _{ik}^{\prime }}{\partial x_k} \right ) \nonumber \\ & \quad -\, \rho u_k^{\prime \prime } \left ( u_j^{\prime \prime }\frac {\partial \widetilde {u}_i }{\partial x_k}+ u_i^{\prime \prime }\frac {\partial \widetilde {u}_j }{\partial x_k} \right ) +p^{\prime }\left ( \frac {\partial u_j^{\prime \prime }}{\partial x_i} + \frac {\partial u_i^{\prime \prime }}{\partial x_j} \right ) + \left (u_i^{\prime \prime } \frac {\partial \overline {\tau }_{jk}}{\partial x_k} + u_j^{\prime \prime } \frac {\partial \overline {\tau }_{ik}}{\partial x_k}\right ) \nonumber \\ & \quad -\, \left ( u_i^{\prime \prime } \frac {\partial \overline {p} }{\partial x_j} + u_j^{\prime \prime } \frac {\partial \overline {p} }{\partial x_i} \right ) - \left ( \tau _{ik}^{\prime } \frac {\partial u_j^{\prime \prime }}{\partial x_k} + \tau _{jk}^{\prime } \frac {\partial u_i^{\prime \prime }}{\partial x_k} \right ). \end{align}

When $i=j$ and all terms are Reynolds averaged, the TKE transport equation is obtained, as shown in (4.2)

(4.2) \begin{align} \frac {\partial \overline {\rho } K}{\partial t} &= \underbrace {-\frac {\partial \overline {\rho } \widetilde {u}_k K}{\partial x_k}}_{C_K} \underbrace {- \frac {\partial }{\partial x_k}\left ( \frac {1}{2} \overline {\rho u_i^{\prime \prime } u_{i}^{\prime \prime } u_{k}^{\prime \prime }} \right )}_{DT_K} \underbrace {- \frac {\partial \overline {p^{\prime } u_i^{\prime \prime }} }{\partial x_i}}_{Dp_K} + \underbrace {\frac {\partial \overline {u_i^{\prime \prime }\tau _{ik}^{\prime }} }{\partial x_k}}_{DV_K} \nonumber \\ & \quad \underbrace {- \overline {\rho u_i^{\prime \prime } u_k^{\prime \prime }} \frac {\partial \widetilde {u}_i }{\partial x_k}}_{P_K} + \underbrace {\overline {p^{\prime } \frac {\partial u_i^{\prime \prime }}{\partial x_i}} }_{\varPi _K} + \underbrace {\overline {u_i^{\prime \prime }} \left (\frac {\partial \overline {\tau }_{ik}}{\partial x_k} - \frac {\partial \overline {p} }{\partial x_i}\right )}_{M_K} \underbrace {- \overline {\tau _{ik}^{\prime } \frac {\partial u_i^{\prime \prime }}{\partial x_k}} }_{\varepsilon _K }. \end{align}

Among these terms, $C_K$ represents the convective term, $DT_K$ the turbulent diffusion term, $Dp_K$ the pressure diffusion term, $DV_K$ the viscous diffusion term, $P_K$ the production term, $\varPi _K$ the pressure-strain term, $M_K$ the mass flux term and $\varepsilon _K$ the dissipation term.

Figure 9.

Comparison of TKE budget profiles. Lines are present results at $x_{\textit{ref}}$ , symbols are DNS results from Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020).

Figure 10.

Streamwise evolution of TKE budget at different stations under the weak interaction. For the line legend, see figure 9.

The TKE budgets at the reference point are presented in figure 9, exhibiting high consistency with the results in Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020). In the near-wall region, the balance of TKE is primarily between viscous diffusion and dissipation, while in the logarithmic region, TKE is predominantly produced by $P_K$ , transported to the near-wall region by diffusion terms and dissipated by $\varepsilon _K$ . These results confirm that a canonical turbulent boundary layer is successfully established through blowing and suction, validating the accuracy of this study. A minor net gain in the total sum of budget terms near the $P_K$ peak is observed, likely due to flow non-equilibrium. This phenomenon, also reported by Vyas et al. (Reference Vyas, Yoder and Gaitonde2019), was attributed to the delayed energy redistribution following $P_K$ amplification, leading to increased anisotropy and a net gain in the budgets.

4.2. Streamwise evolution of budget terms in weak SWTBLI

To systematically analyse the energy transfer and evolution of TKE budget terms in the SWTBLI, the TKE budgets at different streamwise stations under weak interaction are first presented in figure 10. Note that the horizontal axis represents both the streamwise position $x_0$ and the dimensionless budget values added to corresponding $x_0$ .

The streamwise evolution of the TKE budget in figure 10 for weak interaction reveals that in the undisturbed upstream region at $x_0=-4$ , all budget terms conform to the behaviour of canonical turbulent boundary layers. At $x_0=-3.5$ , upstream of the separation shock foot, a negative peak in the convective term $C_K$ emerges, balanced by an increase in the production term $P_K$ . As previously discussed in the streamwise evolution of the TKE peak, figure 10 also highlights the initial TKE amplification occurring upstream of the separation shock foot. The $C_K$ term represents energy transported by the mean flow (Yu et al. Reference Yu, Zhao, Tang, Yuan and Xu2022), while $P_K$ serves as the primary TKE production term. This indicates that, even without significant turbulence amplification due to shocks, upstream-propagating disturbances below the sonic line lead to slight flow deceleration upstream of the separation point. The energy from mean flow deceleration is absorbed by $P_K$ as the initial energy source for TKE amplification, consistent with the conclusions of Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020). At $x_0=-3$ , near the separation shock foot, intensified mean flow deceleration increases the negative $C_K$ peak, which shifts farther from the wall. Simultaneously, $P_K$ amplifies significantly, indicating stronger turbulence production that corresponds to rapid TKE growth in figure 8(c). To balance the increased $P_K$ , energy is transferred both toward the wall and outward through the turbulent diffusion term $DT_K$ . Within the interaction region at $x_0=-2$ , while the overall budget balance remains unchanged, wall-normal budget profiles deviate significantly from those of canonical turbulence. Both the viscous diffusion term $DV_K$ and dissipation term $\varepsilon _K$ are amplified near the wall at approximately $y^+ \lt 4$ and remain balanced. The $P_K$ peak reaches its maximum across the flow field and moves farther from the wall, matching the TKE peak height evolution in figure 8(a). Downstream of the incident shock foot at $x_0=-1\,\rm {to}\,0$ , budget term profiles remain qualitatively similar to those at $x_0=-2$ , but the intensity of all terms, particularly $P_K$ , decreases significantly. The appearance of positive $C_K$ near the $P_K$ peak suggests that post-shock expansion acceleration weakens $P_K$ . At a further downstream station, $x_0=1$ , all budget terms remain largely unchanged, and $P_K$ exhibits an inner peak at $y^+ \approx 7$ , signalling the re-establishment of the turbulent boundary layer.

Summarising the streamwise evolution of the TKE budget, a clear correlation is observed between TKE amplification and the production term $P_K$ . The initial turbulence amplification appears to be driven by energy extracted from the mean flow by the $P_K$ term, with this process occurring upstream of the separation shock.

The TKE budgets alone cannot explain the differences in the streamwise amplification order of various normal and shear stresses or the disparities in the peak evolution trends of each stress component. To further investigate the interrelations between these stresses and their streamwise evolution, the transport equations for $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ are derived by specifying the subscript $ij$ as 11, 22, 33 and 12 in (4.1) and applying Reynolds averaging. The resulting budget plots for these four stress components in weak interactions are presented in figures 11–14.

The budget plots of the normal stress components depicted in figures 11–13 reveal that the dominant component $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ follows a streamwise evolution pattern similar to TKE across all streamwise stations, except for the pressure-strain term. The pressure-strain term $\varPi _K$ , which exists only in compressible flows, governs energy exchange between kinetic and internal energy. Given that the SWTBLI in this study includes extensive subsonic regions and exhibits significant compressibility effects only near shocks, $\varPi _K$ is nearly zero in the TKE budgets shown in figure 10. Consequently, the pressure-strain terms $\varPi _{11}$ , $\varPi _{22}$ and $\varPi _{33}$ primarily describe inter-component energy transfer (Yu et al. Reference Yu, Zhao, Tang, Yuan and Xu2022).

Figure 11.

Streamwise evolution of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ budget at different stations under the weak interaction. For the line legend, see figure 9.

Figure 11 illustrates that the $\varPi _{11}$ term in the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ budgets is predominantly negative in the outer peak for approximately $y^+ \gt 7$ , with its peak position nearly coinciding with $P_{11}$ . The intensity of $\varPi _{11}$ appears to be influenced by the strength of $P_{11}$ . Notably, a significant production term is observed only in the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component, indicating that TKE production originates from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and is subsequently redistributed to other components via the pressure-strain term. At $x_0=-3$ , upstream of the separation shock foot, $\varPi _{11}$ activity is negligible in the near-wall region for the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component. However, within the interaction region at $x_0=-2$ and the downstream recovery region, positive $\varPi _{11}$ values emerge near the wall, signifying a significant change in energy redistribution, where energy transfer from other components to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ becomes more pronounced in the near-wall region.

Figure 12.

Streamwise evolution of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ budget at different stations under the weak interaction. For the line legend, see figure 9.

As illustrated in figure 12, the wall-parallel component $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibits behaviour similar to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ in the viscous diffusion term $DV$ and dissipation term $\varepsilon$ , where $DV_{33}$ and $\varepsilon _{33}$ achieve an approximate energy balance near the wall. However, in the outer peak where $y^+ \gt 7$ , $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ lacks energy production from the $P_{33}$ term and primarily relies on energy transfers from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . In contrast to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , in the undisturbed region at $x_0=-4$ , the $\varPi _{33}$ term for $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ shows noticeable positive values in the near-wall region. After entering the interaction region at $x_0=-2$ , downstream of the separation shock foot, the $\varPi _{33}$ term increases further in the inner region at $y^+ \lt 4$ , forming a double peak. As the flow develops downstream, energy in all budget terms decreases, but the double peak of $\varPi _{33}$ remains visible at $x_0=1$ . These observations imply that in the undisturbed region, the $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ component primarily transfers energy to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ in the inner region. However, downstream of the separation shock foot, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ simultaneously transfers energy to both $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ with greater intensity. In the outer region for approximately $y^+ \gt 9$ , the main energy source for $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ is $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component, and these distinct energy sources contribute to the double-peak pattern observed in $\varPi _{33}$ for $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ .

Figure 13.

Streamwise evolution of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ budget at different stations under the weak interaction. For the line legend, see figure 9.

The wall-normal $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ component behaves differently compared with the wall-parallel components, as highlighted in figure 13. Two distinct energy transfer processes in the outer and inner peaks are evident throughout the flow evolution. In the outer peak for approximately $y^+\gt 10$ , energy transferred from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ generates $\varPi _{22}$ , with some energy dissipated by the dissipation term $\varepsilon _{22}$ and the rest transported to the inner region via the pressure diffusion $Dp_{22}$ and turbulent diffusion $DT_{22}$ terms. In the inner region, energy is primarily balanced by positive contributions from $Dp_{22}$ and negative contributions from $\varPi _{22}$ , where the negative $\varPi _{22}$ indicates energy transferred from $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ . This energy transfer was also observed by Gungor et al. (Reference Gungor, Maciel and Gungor2022) in DNS studies of APG turbulent boundary layers and may be attributed to the ‘splatting effect’ caused by the wall obstructing normal velocity (Moin & Kim Reference Moin and Kim1982; Mansour, Kim & Moin Reference Mansour, Kim and Moin1988), which leads to energy redistribution to other components. The $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ budget terms also amplify considerably after the separation shock foot and gradually decay downstream. At $x_0=-3\,\textrm{to}\, {-}2$ , near the separation shock foot, a negative $P_{22}$ is observed at the peak position of $\varPi _{22}$ , while a positive $P_{22}$ peak appears downstream at $x_0 = 0$ . These changes are likely due to flow deflection caused by the detached shear layer and separation bubble, where the flow accelerates away from the wall at the leading edge of the separation bubble and decelerates towards the wall at the trailing edge. Similar to how streamwise deceleration generates $P_k$ and acceleration decreases $P_k$ in TKE budgets, these wall-normal velocity changes lead to different $P_{22}$ signs for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ upstream and downstream of the separation. The negative $P_{22}$ upstream may inhibit the amplification of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , while the positive $P_{22}$ downstream appears to mitigate its decay.

Summarising the discussion on normal stresses, the dominant component $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ exhibits a streamwise budget evolution nearly identical to that of the total TKE. The wall-parallel components $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ behave similarly in terms of viscous diffusion ( $DV$ ) and dissipation ( $\varepsilon$ ), while the wall-normal component $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ follows a completely distinct pattern. In regions where $y^+ \gt 7$ , energy generation for $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ is driven by the production term $P_{11}$ , with partial energy transfer to $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ subsequently occurring through the pressure-strain term $\varPi$ . This explains the earliest amplification of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ along the streamwise direction. However, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ amplifies downstream of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , likely influenced by the negative $P_{22}$ near the separation shock foot. Near the wall, the energy transfer direction differs due to the ‘splatting effect.’ In the undisturbed region, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ transfers more energy to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , but downstream of the separation shock foot, the intensified inter-component energy transfer results in the development of double peaks in $\varPi _{33}$ for $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , with energy being transferred nearly equally to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ .

Figure 14.

Streamwise evolution of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ budget at different stations under the weak interaction: ( $a$ ) $x_0=-4\, \textrm{to}\,{-}2.5$ , ( $b$ ) $x_0=-2 \, \rm {to}\, 1$ . For the line legend, see figure 9.

The budgets of the shear stress $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ in figure 14 reveal two distinct balance modes in the inner and outer peaks divided by $y^+ \approx 5$ throughout the flow evolution process. In the outer peak, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is produced by $P_{12}$ and balanced by negative $\varPi _{12}$ . In contrast, the inner region is dominated by a balance between $Dp_{12}$ and $\varPi _{12}$ , as these two terms collectively form the pressure term ( $\overline {v^{\prime \prime}\partial p'/\partial x}+\overline {u^{\prime \prime}\partial p'/\partial y}$ ), which is zero at the wall. In the undisturbed region at $x_0=-4$ , $Dp_{12}$ is positive and $\varPi _{12}$ is negative in the inner region. Since $\varPi _{12}$ balances different terms in the viscous sublayer and logarithmic region, this results in double peaks along the wall-normal direction. A significant change occurs upstream of the separation shock foot at $x_0=-3.5$ , where $Dp_{12}$ shows its first negative peak at $y^+ \approx 5$ , accompanied by a corresponding positive peak in $\varPi _{12}$ . Near the separation shock foot at $x_0=-3$ , the signs of $Dp_{12}$ and $\varPi _{12}$ in the inner region reverse compared with the undisturbed region. A similar reversal occurs at $x_0=-2.5$ in the interaction zone, returning to canonical turbulent conditions at $x_0=-2$ . During the flow through the separation compression waves, all budget terms experience significant amplification, reaching their maximum values near $x_0=-2$ . In the subsequent downstream evolution, these budget terms gradually dissipate, while their balance relationships remain unchanged. Notably, the peak position of $P_{12}$ gradually increases downstream, consistent with the observations in figure 8(a). Although the $P_{12}$ peak at $x_0 = -1$ has decayed compared with that at $x_0 = -2$ , a slight increase is observed at $x_0 = 0$ , possibly due to enhanced shear layer instability caused by the incident shock, which will be discussed in detail in $\S$ 4.4. This corresponds to the double-peak pattern of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ in the streamwise evolution of weak interaction, as depicted in figure 8(c).

In conclusion, through budget analysis of TKE and Reynolds stresses under weak interaction, this study briefly identifies the primary energy sources and inter-component energy transfer relationships during turbulent amplification in SWTBLI. It is found that the dominant component $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and shear stress $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ are amplified by the production term $P$ , while $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ derive energy from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ through the pressure-strain term $\varPi$ . The differences in energy sources may explain the delayed initial amplification of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ compared with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ . Furthermore, the double-peak pattern of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ and its production term in weak interaction, along with the variation in initial amplification rates, will be further explored in $\S$ 4.4.

4.3. Streamwise evolution of budget terms in strong SWTBLI

To explore the underlying physical mechanisms responsible for the ‘slope platform’ region and the ‘secondary amplification’ observed in strong interactions, the budgets of various Reynolds stresses are calculated at different streamwise stations. The results for TKE and individual stress components are presented in figures 15 and 17–20.

Figure 15.

Streamwise evolution of TKE budget at different stations under the strong interaction: ( $a$ ) $x_0=-12\ \textrm{to}\, -2$ , ( $b$ ) $x_0=-2\,\rm {to}\,8$ . For the line legend, see figure 9.

The streamwise evolution of the TKE budget for strong interaction, depicted in figure 15, reveals that the overall trends for each budget term largely align with those observed under weak interaction. The production term $P_K$ , responsible for turbulence generation, still reaches its maximum downstream of the separation shock foot near $x_0=-10$ . At $x_0=-8$ , near the leading edge of the separation bubble, this $P_K$ peak decays by approximately half, but the decay rate slows considerably at $x_0=-6\,\textrm{to}\, -4$ upstream of the incident shock foot. Throughout this process, the $P_K$ peak height steadily moves away from the wall, corresponding to the first amplification of TKE observed in figure 8(d). The slow decay region for $P_K$ , absent in weak interaction, arises from the larger separation zone in strong interaction, which extends the distance between the incident and separation shocks. This elongated separation allows the flow more space to develop after the initial amplification near the separation shock foot. In addition, the absence of other energy amplification sources in this region, such as shocks, leads to slow dissipation behaviour similar to that of the recovery region. As the flow expands and accelerates downstream of the incident shock foot at $x_0 = -2$ , the $P_K$ peak intensity rapidly decays, similar to weak interaction. However, the larger wall-normal scale of the separation bubble in strong interaction causes a significant decrease in the $P_K$ peak height as the bubble shrinks after crossing its upper vertex, a phenomenon nearly absent in weak interaction. In the downstream region at $x_0 = 0\,\rm{to}\, 4$ , although the relationships among all budget terms remain consistent, a slight overall increase in their intensities is observed. This trend, also absent in weak interaction, suggests a potential new energy source, similar to a shock, located at the trailing edge of the reattachment point. This energy increase seems to result from flow compression induced by deflection at the trailing edge of the separation bubble in strong interaction. Supporting this hypothesis, figure 16 presents instantaneous snapshots of the density gradient magnitude for strong and weak interactions, which demonstrate that, although reattachment shocks are not discernible in the averaged results in figure 5, intermittent compression waves occur near the reattachment point. These downstream-propagating shocklets are stronger and more widespread in strong interaction and interact with the detached shear layer, enhancing flow compression and shear instability, ultimately causing the ‘secondary amplification’ of turbulence. This behaviour supports the findings of Kang & Lee (Reference Kang and Lee2024) for strong interaction at compression corners. After this compression, all budget terms are amplified until $x_0 = 6$ . Downstream of this station, the budget pattern resembles that of the recovery region in weak interaction, with all budget terms slowly decaying except for a gradual increase in the inner peak of $P_K$ .

Figure 16.

Contours of instantaneous density gradient magnitude fields under weak ( $a$ ) and strong ( $b$ ) interactions.

The budget analysis for strong interaction reveals that $P_K$ , as the primary production term of TKE, maintains a strong correlation with the streamwise evolution of TKE. Compared with weak interaction, strong interaction introduces two unique features: energy dissipation due to stable flow evolution downstream of the separation shock, and energy amplification caused by flow compression near the reattachment point. These correspond to the ‘slope platform’ region and the ‘secondary amplification’ process observed in TKE and Reynolds stresses, respectively. To better understand the energy transfer and amplification mechanisms within these regions, which exist only in strong interaction, a detailed budget analysis of each stress component is also performed.

Figure 17.

Streamwise evolution of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ budget at different stations under the strong interaction: ( $a$ ) $x_0=-12\,\textrm{to}\,{-}2$ , ( $b$ ) $x_0=-2\,\rm {to}\, 8$ . For the line legend, see figure 9.

Figure 18.

Streamwise evolution of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ budget at different stations under the strong interaction: ( $a$ ) $x_0=-12\,\textrm{to}\,{-}2$ , ( $b$ ) $x_0=-2\,\textrm{to}\,8$ . For the line legend, see figure 9.

Figure 19.

Streamwise evolution of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ budget at different stations under the strong interaction: ( $a$ ) $x_0=-12\,\textrm{to}\,{-}2$ , ( $b$ ) $x_0=-2\,\textrm{to}\,8$ . For the line legend, see figure 9.

Figures 17–20 show that the streamwise evolution of the stress components under strong interaction generally follows a pattern similar to that of weak interaction. However, distinct behaviours emerge in the ‘slope platform’ region between the separation and incident shocks. For the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component, its maximum $P_{11}$ occurs downstream of the separation shock foot at $x_0\approx -10$ . Here, the negative terms balancing $P_{11}$ , mainly including the convective term $C_{11}$ , the turbulent diffusion term $DT_{11}$ and the pressure-strain term $\varPi _{11}$ , reach their maximum negative values near $y^+ \approx 20 {-} 40$ , with contributions stronger than those of the dissipation term $\varepsilon _{11}$ . As the flow evolves downstream to $x_0=-8$ , the $P_{11}$ peak rapidly decays, reducing its intensity by about half, similar to TKE. However, at this station, $P_{11}$ is primarily balanced by $\varPi _{11}$ , leading to a smaller relative decrease in $\varPi _{11}$ . Further downstream at $x_0 = -6\,\textrm{to}\,{-}4$ where upstream of the incident shock foot, budget terms such as $P_{11}$ and $\varPi _{11}$ gradually dissipate. As previously analysed, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ transfers energy to $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ through the pressure-strain term $\varPi$ to amplify turbulence. Near the amplification peak at $x_0=-10\,\textrm{to}\,{-}8$ , changes in the balance relationship of $P_{11}$ contribute to a slower streamwise decay of $\varPi _{11}$ . As confirmed in figures 18 and 19, the $\varPi$ terms for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibit slight decreasing trends. This explains why $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ displays a distinct peak near $x_0=-9.6$ in the streamwise evolution plots of the strong interaction in figures 8(b) and 8(d), while the $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ plots are smoother. Further downstream of the amplification peaks, the stable balance between $P_{11}$ and $\varPi _{11}$ in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ leads to the formation of the ‘slope platform’ region for all three normal stress components. Another notable distinction between strong and weak interactions lies in the recovery region downstream of the incident shock foot at $x_0=-2\,\rm{to}\,8$ . Here, the emergence of reattachment compression waves significantly amplifies all budget terms, resulting in a ‘secondary amplification’ of $P_{11}$ in the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component and $\varPi$ in the $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ components. Since the amplified energy primarily originates from $P_{11}$ in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , and the balance between $P_{11}$ and $\varPi _{11}$ remains stable in the outer region by $y^+ \gt 100$ , the amplification regions and rates of all three normal stress components appear relatively similar, as illustrated in figure 8(d).

Figure 20.

Streamwise evolution of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ budget at different stations under the strong interaction: ( $a$ ) $x_0=-12\,\textrm{to}\,{-}2$ , ( $b$ ) $x_0=-2\,\textrm{to}\,8$ . For the line legend, see figure 9.

As depicted in figure 19, the dominant energy production term for the $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ component is $P_{12}$ , identical to that of the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component. Consequently, the streamwise evolution of $P_{12}$ in $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is also similar to that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and TKE under strong interaction. Downstream of the separation shock foot at $x_0=-11$ , $P_{12}$ experiences significant amplification, peaking at $x_0=-10$ and then decaying to approximately half its strength by $x_0=-8$ , followed by a slow decay region between $x_0=-6\,\textrm{to}\,{-}4$ . Similar to weak interaction, the weakest $P_{12}$ peak appears near the incident shock foot at $x_0=-4\,\textrm{to}\,{-}2$ . However, at downstream stations of $x_0=-2\,\textrm{to}\,2$ , despite the gradual decay of $Dp_{12}$ and $\varPi _{12}$ in the near-wall region, $P_{12}$ begins to recover, potentially driven by enhanced shear instability induced by the incident shock. In contrast to weak interaction, further downstream at $x_0=4\,\textrm{to}\,8$ in strong interaction, intermittent reattachment compression waves amplify all major contributing terms, including $Dp_{12}$ and $\varPi _{12}$ , in the near-wall region. Simultaneously, the peak intensity of $P_{12}$ undergoes an additional enhancement. Summarising this process, the streamwise evolution of $P_{12}$ in $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ upstream of the incident shock foot follows a pattern nearly identical to that in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . During the initial amplification stage, the amplification factors of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ exhibit comparable trends, as illustrated in figure 8(d). However, downstream of the incident shock foot, $P_{12}$ in $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ experiences dual amplification, driven by both the incident shock and the reattachment compression waves. As a result, the downstream amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ under strong interaction is more pronounced than that of the normal stress components. This observation aligns with the higher amplification rate of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ observed during the ‘secondary amplification’ process in the streamwise peak evolution plots of Reynolds stresses in figure 8(d).

In summary, this subsection reaffirms that the dominant contribution budgets and energy redistribution mechanisms established under weak interaction also hold for strong interaction, as demonstrated through budget term analysis. The streamwise evolution of these dominant terms shows a clear correlation with their corresponding stress components, exhibiting the unique ‘slope platform’ region upstream of the incident shock and the ‘secondary amplification’ process downstream. This study concludes that during the initial amplification in strong interaction, the primary production term $P_{11}$ of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ decays rapidly after peaking near the separation shock foot, followed by a slower and more stable decay downstream. In contrast, $\varPi _{11}$ , which transfers energy to the $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ components, shows consistent dissipation in this region. The uniform decay of dominant budget terms responsible for stress amplification forms the ‘slope platform’ region, with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ displaying a sharp peak at its leading edge. Since TKE is dominated by $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , it exhibits a similar behaviour. Additionally, the dominant production term for $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is also $P_{12}$ , leading to an initial amplification process resembling that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . The ‘secondary amplification’ is likely caused by intermittent travelling reattachment compression waves, which amplify all normal stress components at comparable rates due to their shared energy source. However, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ undergoes dual amplification by the incident and reattachment shocks, resulting in a greater intensity and a higher amplification rate during the ‘secondary amplification’ process.

4.4. Budget decomposition and dominant amplification mechanisms for Reynolds stresses

As discussed in the previous subsection, the production and transfer of energy are primarily governed by the $P$ and $\varPi$ terms. To analyse the dominant flow states and inter-component energy transfer mechanisms during stress amplification under varying strengths of SWTBLI, this subsection further decomposes the $P$ and $\varPi$ terms for different stress components. As expressed in (4.3), each normal stress component consists of two $P$ sub-terms and one $\varPi$ sub-term, while the shear stress component $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is decomposed into four $P$ sub-terms and two $\varPi$ sub-terms

Figures 21–25 display the mean results of $P$ , $\varPi$ and their sub-terms for $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ under both strong and weak interactions. For brevity, the $P_{33}$ term and its sub-terms are omitted, as they are nearly zero throughout the entire flow field.

Figure 21.

Contours of mean production terms $P_{11}$ (a,b), their sub-terms $P_{11}^x$ (c,d), $P_{11}^y$ (e,f) and pressure-strain term $\varPi _{11}$ (g,h) for $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component under weak (a,c,e,g) and strong (b,d,f,h) interactions.

The $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ component is the dominant contributor to TKE in this SWTBLI problem, and its $P_{11}$ term nearly represents turbulence production. As a result, the mean contours of $P_{11}$ in figure 21(a,b) and TKE in figure 7(a,b) show strong consistency for both strong and weak interactions. The decomposition of $P_{11}$ further supports the conclusion from Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020), stating that the $P_{11}^x$ term ( $-2\overline {\rho } \widetilde {u^{\prime \prime}u^{\prime \prime}} \partial \widetilde {u} / \partial x$ ), related to flow deceleration, dominates turbulence amplification upstream of the incident shock. Meanwhile, the $P_{11}^{y}$ term ( $-2\overline {\rho } \!\widetilde{\ u^{\prime\prime}v^{\prime\prime}} \partial \widetilde {u} / \partial y$ ), associated with shear, is comparatively weaker but begins to amplify near the separation shock foot, peaking along the shear layer and extending into the downstream recovery region. Although ‘secondary amplification’ regions are observed in the $P_{11}$ contours for both strong and weak interactions, their underlying mechanisms differ significantly. In the weak interaction case, expansion waves downstream of the incident shock accelerate the flow, causing $\partial \widetilde {u} /\partial x$ to become positive and generating negative $P_{11}^x$ . Additionally, an energy loss occurs in $P_{11}^y$ at the incident shock foot, primarily due to the weaker local intensity of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ in $P_{11}^y$ relative to nearby regions, as illustrated in the averaged results of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ and $P_{12}$ in figures 7(g) and 24(a). A more detailed analysis of this $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ deficit will be provided in later discussions. Both $P_{11}^x$ and $P_{11}^y$ decrease near the incident shock foot, resulting in a local minimum of $P_{11}$ at this station. However, as the flow develops downstream, the effects of flow acceleration from expansion waves and the $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ deficit diminish, enabling $P_{11}$ to temporarily recover to its upstream $P_{11}^y$ intensity before gradually dissipating. This recovery process explains the ‘secondary amplification’ observed in weak interaction. In strong interaction, while a similar process occurs, the larger distance between the separation and incident shocks leads to further dissipation of $P_{11}^y$ before it reaches the incident shock foot. Consequently, $P_{11}$ downstream of the incident shock cannot recover to a higher intensity. Figures 21(f) and 24(b) reveal that, despite the rapid recovery of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ after the incident shock, the $P_{11}^y$ in strong interaction only reaches the lower energy levels upstream of the incident shock foot. The ‘secondary amplification’ of $P_{11}^y$ occurs further downstream, and as discussed in $\S$ 4.3, is likely driven by reattachment compression waves that enhance flow compression and shear instability.

Figure 22.

Contours of mean production terms $P_{22}$ (a,b), their sub-terms $P_{22}^x$ (c,d), $P_{22}^y$ (e,f) and pressure-strain term $\varPi _{22}$ (g,h) for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ component under weak (a,c,e,g) and strong (b,d,f,h) interactions.

As illustrated in figure 22, $P_{22}$ is primarily contributed by $P_{22}^y$ ( $-2\overline {\rho } \!\widetilde{\ v^{\prime\prime}v^{\prime\prime}} \partial \widetilde {v} /\partial y$ ), and its trends for strong and weak interactions are not significantly different. Upstream of the incident shock, $P_{22}^y$ is negative due to the wall-normal acceleration of $\widetilde {v}$ , while downstream, it becomes positive as wall-normal velocity decelerates. The primary energy source for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ remains the transfer from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ via the $\varPi$ term. In the weak interaction, the $\varPi _{22}$ distribution in figure 22(g) consistently exhibits negative values near the wall, which intensify downstream of the separation shock. Although the same energy transfer mechanism occurs in strong interaction as discussed in $\S$ 4.3, the $\varPi _{22}$ contour in figure 22(h) appears less distinct due to the larger spatial scale of the strong interaction region. However, upon examining the enlarged view, similar negative $\varPi _{22}$ patterns can be observed, which are related to the ‘splatting effect.’ Therefore, while strong interaction significantly affects the shape of the separation bubble and subsequently affects the turbulence amplification process, the overall scale of energy transfer and the corresponding dominant budget terms remain similar. Particularly near the separation shock foot, the behaviours under strong and weak interactions are nearly identical, with comparable initial amplification factors as shown in figure 8(c,d). This suggests that the primary difference in turbulence amplification between strong and weak interactions arises not from the gradient changes with the incident shock angle, but from differences in the size and shape of the separation zone, which influence the flow evolution and ultimately the amplification path.

Figure 23.

Contours of mean pressure-strain term $\varPi _{33}$ for $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ component under weak ( $a$ ) and strong ( $b$ ) interactions.

In this SWTBLI problem, $P_{33}$ is negligible due to the absence of spanwise shocks. Nonetheless, figure 23 reveals significant amplification of $\varPi _{33}$ downstream of the separation shock, especially near the wall and within the detached shear layer.

Figure 24.

Contours of mean production terms $P_{12}$ (a,b) and their sub-terms $P_{12}^{x1}$ (c,d), $P_{12}^{y1}$ (e,f), $P_{12}^{x2}$ (g,h), $P_{12}^{y2}$ (i,j) for $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ component under weak (a,c,e,g,i) and strong (b,d,f,h,j) interactions.

Figure 25.

Contours of mean pressure-strain terms $\varPi _{12}$ (a,b) and their sub-terms $\varPi _{12}^x$ (c,d), $\varPi _{12}^y$ (e,f) for $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ component under weak (a,c,e) and strong (b,d,f) interactions.

Figure 24 presents the decomposition of the $P_{12}$ term, revealing that the deficit in $P_{12}$ at the incident shock foot results from the combined effects of $P_{12}^{x1}$ ( $ \overline {\rho }\widetilde {u^{\prime \prime}u^{\prime \prime}} \partial \widetilde {v} /\partial x$ ), which negatively contributes at the shock foot, and $P_{12}^{y2}$ ( $\overline {\rho } \!\widetilde{\ v^{\prime\prime}v^{\prime\prime}} \partial \widetilde {u} /\partial y$ ), which remains positive in the shear layer but gradually weakens downstream after amplification. The other two sub-terms have negligible contributions. From the averaged normal velocity contour in figure 26(a,b), it is evident that after crossing the separation shock, the wall-normal velocity $\widetilde {v}$ steadily increases along the streamwise direction, reaching a maximum before the incident shock foot, and then rapidly decelerating with a steeper gradient. This deceleration near the incident shock foot explains the negative contribution of $P_{12}^{x1}$ . In contrast, $\partial \widetilde {u} /\partial y$ stays positive along the shear layer with relatively small variations, and the peak of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ also remains stable in the interaction zone, ensuring a more uniform amplification of $P_{12}^{y2}$ . During the initial turbulence amplification stage, $P_{12}^{y2}$ serves as the dominant production sub-term for $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ . The APG in the interaction zone alters the upstream velocity profile, slowing the streamwise velocity and weakening its wall-normal gradient near the wall. However, the peak position of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is higher than that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ in the canonical turbulent boundary layer. As shown in figure 26(c,d), $\partial \widetilde {u} /\partial y$ not only remains positive at the height of the $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ peak but also increases along the streamwise direction due to velocity profile changes induced by the APG. Therefore, the initial amplification position of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is close to that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . However, since $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ does not amplify at this station, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ initially grows at a slower rate. Once $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ begins to amplify, $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ exhibits amplification rates comparable to those of the other normal stresses. In addition, as demonstrated in figure 25, the distinct behaviour of $\varPi _{12}$ between the near-wall region and the shear layer arises from the combined effects of $\varPi _{12}^{x}$ ( $-\overline {p' \partial v^{\prime \prime}/\partial x}$ ) and $\varPi _{12}^{y}$ ( $-\overline {p' \partial u^{\prime \prime}/\partial y}$ ). Enlarged contours for strong interaction reveal that $\varPi _{12}^{y}$ dominates near the wall, while $\varPi _{12}^{x}$ prevails in the outer region. The sign change of $\varPi _{12}$ along the wall near the separation shock foot, as revealed in $\S$ 4.2, is also observed in figure 25(e,f). According to the balance relationship discussed in $\S$ 4.2, the pressure diffusion term at the wall should be dominated by $Dp_{12}^y$ ( $\partial \overline {p'u^{\prime \prime}}/\partial y$ ), consistent with findings by Chan, Schlatter & Chin (Reference Chan, Schlatter and Chin2021). To further analyse energy transfer changes at the wall, figure 27 presents the $\overline {p'u^{\prime \prime}}$ contours under strong and weak interactions. It is evident from figure 27 that $\overline {p'u^{\prime \prime}}$ is positive near the wall upstream of the shocks but exhibits significant negative values in regions with large gradients, such as shocks and slip lines. Therefore, this sign change in $\overline {p'u^{\prime \prime}}$ and its wall-normal gradient near the wall is likely due to the incomplete detachment of the shear layer originating from the separation shock foot. Additionally, streamwise velocity reversal and destruction of canonical wall-bounded turbulent structures induced by flow separation near the separation shock foot may further alter the wall-normal energy transfer described by $Dp_{12}^y$ term, ultimately resulting in a positive–negative–positive pattern of $Dp_{12}^y$ along the streamwise direction near the wall, with its balance term $\varPi _{12}^{y}$ exhibiting an opposite negative–positive–negative pattern.

Figure 26.

Contours of mean wall-normal velocity (a,b) and wall-normal gradient of mean streamwise velocity (c,d) under weak (a,c) and strong (b,d) interactions. The enlarged view is redrawn at different level range (0–11) with the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ ( dash-dotted line) and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ ( dashed line) peak positions.

Figure 27.

Contours of mean pressure–velocity fluctuation correlation $\overline {p'u^{\prime \prime}}$ under weak ( $a$ ) and strong ( $b$ ) interactions.

This section has identified and compared the key factors responsible for producing and amplifying different stress components, as well as the mechanisms of inter-component energy transfer under both strong and weak interactions. The findings reveal that $P_{11}^x$ and $P_{11}^y$ in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ primarily drive TKE amplification, with excess energy redistributed to other normal stress components via $\varPi$ . In the $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ component, $P_{22}^y$ dominates $P_{22}$ , contributing negatively upstream of the incident shock and positively downstream. Meanwhile, $P_{33}$ is negligible. For the $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ component, $P_{12}$ is primarily determined by the negative $P_{12}^{x1}$ and positive $P_{12}^{y2}$ , resulting in a local valley in $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ stress near the incident shock foot. The initial turbulence amplification near the separation shock foot exhibits little difference between strong and weak interactions. The redistribution mechanism allows $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ to amplify first, followed by $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , while $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ amplifies last, possibly due to negative $P_{22}^y$ . The amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ occurs simultaneously due to velocity profile changes induced by the APG, though $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ exhibits a slower initial amplification rate, influenced by $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ . Additionally, inter-component energy transfer in the inner layer under $y^+ =10$ , as reflected by the $\varPi$ terms, is almost unaffected by the strength of the incident shock. Instead, the differences between strong and weak interactions primarily stem from the distinct flow evolution processes caused by changes in the scale of the separation zone. In strong interaction, the larger distance between the incident and separation shocks enhances flow dissipation, resulting in the formation of the ‘oblique platform’ and differences in the amplification mechanism for $P_{11}^y$ . Furthermore, the increased height of the separation bubble in the wall-normal direction under strong interaction leads to greater flow deflection downstream of the incident shock foot. This deflection promotes a gradual reattachment process downstream, influencing the evolution of the peak height of dominant production terms and Reynolds stresses in the recovery region, resulting in different trends between strong and weak interactions in figure 8. In addition, the enhanced flow deflection near the reattachment point also likely generates compression waves, contributing to the ‘secondary amplification’ process observed in strong interaction.

5.1. Spectra of Reynolds stresses

The weighted spanwise spectra of various Reynolds stresses at different streamwise stations during the initial turbulence amplification stage for both strong and weak interactions are calculated using (5.1) and presented in figures 28–31. For clarity, all spectra are normalised by the absolute values of their respective local peaks.

Figure 28.

Streamwise evolution of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ weighted spanwise spectra at different stations under weak ( $a$ ) and strong ( $b$ ) interactions. The contour colours, from light to dark, represent 10 %, 30 %, 50 %, 70 % and 90 % of the local peak value.

Figure 29.

Streamwise evolution of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ weighted spanwise spectra at different stations under weak ( $a$ ) and strong ( $b$ ) interactions. For the legend, see caption of figure 28.

Figure 30.

Streamwise evolution of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ weighted spanwise spectra at different stations under weak ( $a$ ) and strong ( $b$ ) interactions. For the legend, see caption of figure 28.

Figure 31.

Streamwise evolution of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ weighted spanwise spectra at different stations under weak ( $a$ ) and strong ( $b$ ) interactions. For the legend, see caption of figure 28.

To further analyse the scale evolution of different stress components during turbulence amplification, figure 32 summarises the streamwise evolution of energy, height and scale at the spectral peaks in figures 28–31.

Figure 32.

Streamwise evolutions of energy (a,b), height (c,d) and scale (e,f) at spectral peaks for Reynolds stresses $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ (), $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ (), $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ () and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ () under weak (a,c,e) and strong (b,d,f) interactions. The symbols represent the spectra depicted in figures 28–31.

Upstream of the separation shock, figure 28 reveals that both strong and weak interactions retain the spanwise spectral characteristics of canonical turbulent boundary layers. Specifically, the spectral peak of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ at $\lambda _z^+ \approx 100$ corresponds to the characteristic streaks of wall-bounded turbulence. As the flow crosses the separation shock, turbulence amplification begins, with each stress component exhibiting distinct changes in amplification magnitude, spanwise scale and peak height. For the weak interaction shown in figure 32(a,c,e), the energy, height and scale at the spectral peak of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ increase monotonically along the streamwise direction. In contrast, consistent with the stress peak height evolutions discussed in $\S$ 3.2, the spectral peak height of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ decreases significantly after the separation point, accompanied by a reduction in its spectral peak scale. For $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ , another wall-parallel component the same as $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , the spectral peak scale decreases notably upstream of the separation point, while its peak height also slightly decreases. Downstream of the separation point, $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ closely follows the behaviour of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , with energy, height and scale increasing uniformly, and its plots are closest to those of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ in figure 32(c,e). As concluded in $\S$ 4.4, the amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is closely associated with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is also an important part of the production term in $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . Consequently, as illustrated in figure 32(a,e), although $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ exhibits smaller energy and scale amplification compared with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , its overall trend remains similar to $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , showing monotonic scale amplification.

In strong interaction, the spectral peak energy, scale and height evolve similarly to those in weak interaction, with noticeable decreases in the spectral peak height of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and the scale of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ near the separation point. This streamwise evolution appears to equalise the height and scale of all stress components during turbulence amplification, potentially related to inter-component energy transfer mechanisms. A more detailed analysis of this phenomenon, incorporating the pressure-strain spectrum, will be presented in $\S$ 5.3. Compared with the weak interaction, the strong interaction spectra exhibit larger scales and higher peak heights during turbulence amplification, consistent with the peak value and height evolution trends of stresses in figure 8(b). Upstream of the separation shock, the larger spanwise scales in strong interaction may result from the more pronounced unsteady effects of shocks. Unlike the turbulent amplification evolutions shown in figure 8(d) of $\S$ 3.2, the spectral peak energy does not exhibit a typical ‘slope platform’ region. This is particularly evident for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , where the spectral peak energy remains nearly constant or even increases in this area. Examining the $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ spectrum in figure 28 within the ‘slope platform’ region reveals that the width of energy above 50 %, 70 % or 90 % of the peak value narrows as the flow evolves downstream. This suggests that, although the peak energy remains relatively unchanged, the overall energy continues to decay, consistent with the turbulence amplification trends discussed in $\S$ 3.2. Furthermore, this narrowing of energy width implies that in the ‘slope platform’ region, where the flow remains undisturbed by shocks, the scales become more concentrated as the flow evolves, promoting the formation of large-scale structures.

Figure 33.

Instantaneous streamwise velocity flow fields near the peak position of TKE under weak ( $a$ ) and strong ( $b$ ) interactions.

Figure 33 presents instantaneous snapshots of streamwise velocity near the peak of TKE. Figure 33(a) illustrates that, under weak interaction, velocity streaks of canonical wall-bounded turbulence are observed upstream of the separation shock. The scale of velocity streaks increases noticeably downstream of the separation shock and becomes fully amplified in the downstream recovery zone. In contrast, figure 33(b) reveals a slightly different scale evolution of streaks under strong interaction. While canonical velocity streaks also amplify at the separation shock foot, small-scale structures emerge within the ‘slope platform’ region, eventually developing into large-scale streaks at the tail of this region. This evolution of velocity streaks confirms the conclusions drawn from the spanwise spectrum analysis. The presence of large-scale streaks upstream of the incident shock may arise from the shock-free, elongated ‘slope platform’ region, which provides the necessary space for the formation and concentration of large-scale structures.

5.2. Spectra of production terms

Building on the Reynolds stress spectra analysis, this study highlights the differences in the evolution of scale structures between strong and weak interactions. Extending this analysis, the spanwise spectra of the dominant production terms are examined in this subsection. The results indicate that, consistent with the findings in $\S$ 5.1, the scale of production terms in strong interaction remains larger than in weak interaction. However, the spectral peak evolution trends exhibit no significant differences between the two cases. In both interactions, the scales of the main production terms and their dominant sub-terms exhibit a consistent increasing trend in the streamwise direction, and the energy and height at spectral peaks evolve similarly to the time-averaged results presented in figures 21–24 of $\S$ 4.4. For simplicity, only results that deviate from the time-averaged peak value evolution are provided. Figure 34 illustrates the streamwise evolution of energy, height and scale at the spectral peaks of the principal $P_{11}$ term and its two components, $P_{11}^x$ and $P_{11}^y$ , for both strong and weak interactions.

Figure 34.

Streamwise evolutions of energy (a,b), height (c,d) and scale (e,f) at spectral peaks for production terms $P_{11}$ (black solid line), $P_{11}^x$ ( dashed line) and $P_{11}^y$ ( dash-dotted line) under weak (a,c,e) and strong (b,d,f) interactions.

Figure 35.

Weighted spanwise spectra of production terms $P_{11}$ ( $a$ ), $P_{11}^x$ ( $b$ ) and $P_{11}^y$ ( $c$ ) at $x_0=-2.5$ under weak interactions. For the legend, see caption of figure 28.

A comprehensive analysis of figure 34 indicates that both strong and weak interactions share similar evolution trends. During turbulence amplification, the scale and height of $P_{11}$ terms in figure 34(c–f) steadily increase along the streamwise direction. However, the distinct production mechanisms of the sub-terms $P_{11}^x$ and $P_{11}^y$ lead to differences in their evolution patterns for height and scale. Downstream of the separation shock, $P_{11}$ evolves closer to $P_{11}^y$ in weak interaction and closer to $P_{11}^x$ in strong interaction. This divergence may stem from the stronger shock in the strong interaction, which intensifies the deceleration effect and enhances the contribution of $P_{11}^x$ . Despite this, spectral peak energy plots in figure 34(a, b) reveal that $P_{11}$ peaks appear downstream of the separation shock for both interactions, with $P_{11}^y$ dominating this spectral peak. This observation appears inconsistent with $\S$ 4.4 and Fang et al. (Reference Fang, Zheltovodov, Yao, Moulinec and Emerson2020), which suggest that deceleration primarily drives initial turbulence amplification in low-Reynolds-number simulations. A closer inspection of the flow near the spectral peak of $P$ at $x_0=-2.5$ in weak interaction provides insight into this discrepancy. As shown in figure 32(a), the spectral peak energy of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ at this station is approximately six times that of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ . Although the leading-edge separation shock induces significant deceleration, the streamwise gradient of velocity $\widetilde {u}$ remains more than an order of magnitude smaller than its wall-normal gradient near the boundary layer, numerically explaining the dominance of $P_{11}^y$ in spectral peak energy. Further examination of $P$ spectra at the peak station in figure 35 reveals that, while $P_{11}^y$ exhibits a higher spectral peak energy, $P_{11}^x$ covers a broader range, making $P_{11}^x$ more prominent in time-averaged results at the current Reynolds number. Consequently, the total $P_{11}$ spectrum integrates features of both sub-terms, with a peak position resembling $P_{11}^y$ and a broader distribution similar to $P_{11}^x$ . The sharp peak of $P_{11}^y$ and the broader range of $P_{11}^x$ stem partially from differences in the spectra of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ , suggesting that turbulence amplification driven by shear effects is more efficient within specific scale ranges, while amplification driven by deceleration has a broader influence across multiple scales. In the self-sustaining cycles of wall-bounded turbulence, mechanisms such as the Orr mechanism, which regenerates streamwise vortices from meandering streaks (Jiménez Reference Jiménez2013, Reference Jiménez2015; Hwang & Bengana Reference Hwang and Bengana2016), and the ‘lift-up’ mechanism, which forms streaks through linear advection of streamwise vortices (Farrell & Ioannou Reference Farrell and Ioannou2012; de Giovanetti et al. Reference de Giovanetti, Sung and Hwang2017; Kawata & Tsukahara Reference Kawata and Tsukahara2021), both rely on the contribution of mean shear. This close relationship between velocity streaks and shear effects leads this study to conclude that shear-driven amplification, represented by $P_{11}^y$ , primarily influences streamwise-elongated streaky structures, while deceleration-driven amplification, represented by $P_{11}^x$ , amplifies multi-scale structures within wall-bounded turbulence.

5.3. Spectra of pressure-strain term

The pressure-strain term governs the energy transfer between normal stresses and serves as the amplification source for some stress components. Therefore, unlike the previously discussed Reynolds stresses and production terms, the spectra of the pressure-strain term can exhibit opposing signs at different heights within the same streamwise station, leading to the emergence of double peaks. Figure 36 presents the evolution of energy, height and scale at the spectral peaks of each pressure-strain term under strong and weak interactions during the initial amplification stage. Solid lines indicate the outer peak, while dashed lines represent the inner peak. To ensure consistency in comparison, the peak energy values are presented as absolute magnitudes, regardless of their original sign.

Figure 36.

Streamwise evolutions of energy (a,b), height (c,d) and scale (e,f) at spectral peaks for pressure-strain terms $\varPi _{11}$ (), $\varPi _{22}$ () and $\varPi _{33}$ () under weak (a,c,e) and strong (b,d,f) interactions. Solid lines represent the outer peak, while dashed lines denote the inner peak.

According to the analysis in $\S$ 4.1, the energy of the pressure-strain terms for normal stresses originates from the $P_{11}$ of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . Figure 36 reveals that the evolution of the outer peaks of the pressure-strain terms follows a similar pattern to the $P$ terms, with both strong and weak interactions exhibiting gradual increases in height and scale during the initial amplification stage. Although the inner peak height remains nearly identical between strong and weak interactions, the inner peak scale increases steadily under strong interaction, in contrast to weak interaction, where the inner peak scale remains nearly constant along the streamwise direction. This discrepancy may result from the extended plateau in strong interaction allowing for more sufficient development of inner-layer structures, the larger separation zone leading to an increase in inner-layer scale or the influence of larger-scale structures in the outer region under strong interaction leaving ‘footprints’ on the inner region. Nonetheless, the pressure-strain spectra during the initial amplification near the separation shock foot still behave similarly for both interactions. Given that the peak evolution trends of the mean Reynolds stresses in $\S$ 3.2 align completely during the initial amplification stages of strong and weak interactions, figure 37 presents the spanwise pressure-strain spectra of the normal stress components at several representative stations under weak interaction for clarity and conciseness.

Figure 37.

Streamwise evolution of $\varPi _{11}$ ( $a$ ), $\varPi _{22}$ ( $b$ ) and $\varPi _{33}$ ( $c$ ) weighted spanwise spectra at different stations under weak interaction. The contour colours, from light to dark, represent 10 %, 30 %, 50 %, 70 % and 90 % of the local peak value, with for positive and for negative peaks.

Comparing the pressure-strain spectra of various stress components in figure 37 reveals that at $x_0 = -2.5$ for weak interaction, an inner peak gradually emerges in $\varPi _{33}$ , with its height and scale closely matching those of $\varPi _{22}$ at $x_0 = -2.5\,\textrm{to}\,{-}2$ . This indicates that energy is primarily transferred from $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ in the inner region at the initial amplification process, consistent with the conclusions drawn from the stress component budgets in $\S$ 4.2. For the outer peaks, the peak heights, scales and overall shapes of $\varPi _{33}$ and $\varPi _{11}$ are relatively consistent, but $\varPi _{22}$ shows a noticeable difference in scale compared with $\varPi _{11}$ . This suggests that during the initial turbulence amplification stage, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ primarily transfers energy to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ through the pressure-strain term, which explains the earlier amplification of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ relative to $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ from a scale perspective.

Based on the spectral contours in figure 37 and the preceding analysis, it can be concluded that energy transfer is characterised by consistency in height and scale. Moreover, the streamwise evolution of the pressure-strain spectra, along with the spanwise spectra of production terms and Reynolds stresses in figures 32, 34 and 36, demonstrates a delay in scale amplification relative to energy amplification, consistent with the results in Yu et al. (Reference Yu, Zhao, Tang, Yuan and Xu2022). Therefore, employing the evolution of stress spectra under weak interaction in figure 32 for illustration, the inter-component energy transfer mechanism during the initial turbulence amplification of SWTBLI can be revealed from a scale perspective. Specifically, although the energy of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ is initially amplified due to the continuous deceleration effect of the separation shock or compression waves at $x_0=-3.5$ , its spectral height and scale remain consistent with those of canonical turbulence due to the delayed amplification of these characteristics. Until approximately $x_0=-3$ , to achieve energy balance, $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ transfers its excessively amplified energy to other components through the pressure-strain term, leading to subsequent amplifications in spectral height and scale. Downstream of this station, the wall-normal and spanwise stresses are influenced by the energy transfer from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , resulting in changes in their spectral peak heights. The peak height of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ decreases to approach that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , accompanied by a reduction in scale. Since the characteristic scale of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ in canonical turbulence is larger than $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , this reduction aligns their scales. Meanwhile, $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ also experiences a similar decrease in spectral peak height and scale. However, the scale reduction in $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ increases the scale disparity between $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ , since their characteristic scales are naturally closer in canonical turbulence. Therefore, during the initial turbulence amplification near the separation shock foot, the spectral peak height and scale of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ become more aligned with those of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , while $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ diverges further in characteristic scale from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . This suggests that energy from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ is primarily transferred to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ first from a scale perspective, resulting in the earlier amplification of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ compared with $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ .

To summarise the discussion in this section, spanwise spectrum analysis reveals that under strong interaction, the streamwise stress spectrum becomes progressively concentrated along the ‘slope platform’ region, corresponding to the gradual formation of uniform large-scale streaks. In addition, deceleration effects influence structures across a broader range of scales, whereas shear effects primarily act on specific scales, amplifying particular structures such as velocity streaks more effectively. Furthermore, it is found that the characteristic of energy transfer necessitates the consistency of height and scale. As energy is redistributed from $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ , the peak heights of $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ and $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ gradually approach that of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ . However, this height decrease leads to a corresponding reduction in scale, increasing the scale disparity between $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ . This process facilitates the transfer of more energy to $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ through the pressure-strain term, explaining its upstream amplification relative to $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ from a scale perspective.