2.1. Numerical method and flow conditions

The 3-D compressible Navier–Stokes equations are solved in the conservative form, and are presented in the Cartesian coordinate system as

(2.1) \begin{align} \frac {\partial \rho }{\partial t} + \frac {\partial \rho u_{j}}{\partial x_{j}} = 0, \\[-30pt]\nonumber\end{align}

(2.2) \begin{align} \frac {\partial \rho u_{i}}{\partial t} +\frac {\partial \rho u_{i} u_{j}}{\partial x_{j}} = - \frac {\partial p}{\partial x_{i}} + \frac {1}{Re}\frac {\partial \tau _{ij}}{\partial x_{j}} ,\\[-30pt]\nonumber\end{align}

(2.3) \begin{align} \frac {\partial \rho E}{\partial t} + \frac {\partial (\rho E + p) u_{j}}{\partial x_{j}} = \frac {1}{(\gamma -1)Re Pr M_{\infty }^2}\frac {\partial }{\partial x_{j}}\left (\mu \frac {\partial T}{\partial x_{j}}\right ) + \frac {1}{Re}\frac {\partial \tau _{ij}u_{i}}{\partial x_{j}}, \end{align}

showing the non-dimensional form of the mass conservation equation, three momentum conservation equations and the energy conservation equation, respectively. The indices $i$ and $j$ run from 1 to 3. In the equations, $\rho = \rho ^*/\rho _\infty ^*$ is the non-dimensional density, $u_1=u=u^*/U_\infty ^*$ , $u_2=v=v^*/U_\infty ^*$ and $u_3=w=w^*/U_\infty ^*$ are the non-dimensional velocity components respectively in the $x$ , $y$ and $z$ directions scaled with free-stream velocity $U_\infty ^*$ ; $E = e + 1/2 \rho (u^2+v^2+w^2)$ is the total energy per unit mass, with $e$ as specific internal energy. The corresponding conservative variables are $\rho$ , $\rho u$ , $\rho v$ , $\rho w$ and $\rho E$ . The terms $p$ , $T$ are the non-dimensional pressure and temperature, respectively, while $\tau _{ij} = \mu [ \partial u_{i}/\partial x_j + \partial u_{j}/\partial x_i -2/3 (\partial u_{k}/\partial x_k) \delta _{ij} ]$ is the viscous stress tensor, where $\mu$ is the non-dimensional dynamic viscosity given by Sutherland’s law, with a Sutherland temperature of $T_S^*=110.4\,{\textrm {K}}$ , and $\delta _{ij}$ is the Kronecker delta function. The various physical variables are normalised using the corresponding free-stream values. However, pressure is normalised using the free-stream dynamic pressure term, $\rho _\infty ^{*} U_\infty ^{*2}$ , i.e. $p = p^*/\rho _\infty ^{*} U_\infty ^{*2}$ , while the unit total energy $E$ is normalised by $ U_\infty ^{*2}$ . The dimensional quantities are denoted by a superscript $ ^{*}$ , which is dropped for non-dimensional quantities unless mentioned otherwise. Also, the subscript ` $_{\infty }$ ’ represents the free-stream conditions at the inflow. Here $x=x^*/\delta _{\textit{inlet}}^{*}$ , $y=y^*/\delta _{\textit{inlet}}^{*}$ and $z=z^*/\delta _{\textit{inlet}}^{*}$ are the non-dimensional coordinates scaled with the displacement thickness $\delta _{\textit{inlet}}^{*}=0.075$ (mm) at the inflow. The characteristic fluid dynamic time scale is $\delta _{\textit{inlet}}^{*}/U_\infty ^*$ .

The OpenSBLI solver (Lusher, Jammy & Sandham Reference Lusher, Jammy and Sandham2021), which is an open-source finite-difference-based solver, is used on structured Cartesian coordinate systems for the shock-reflection set-up. A local Lax--Friedrichs flux splitting approach is used for the inviscid fluxes in characteristic space. Different variations of flux reconstruction schemes, i.e. weighted essentially non-oscillatory (WENO) and targeted essentially non-oscillatory (TENO), are available to compute the inviscid fluxes. As noted in earlier literature, the TENO scheme is less dissipative than the WENO schemes and, hence, an adaptive version of sixtth-order TENO is used to perform the present simulations (Lusher et al. Reference Lusher, Jammy and Sandham2021). The viscous fluxes are computed using fourth-order central differences, while a third-order Runge–Kutta scheme is used for time integration.

Figure 1. Two-dimensional schematic of the numerical set-up, where the computational domain is demarcated with a red dashed line.

A 2-D schematic of the computational set-up is shown in figure 1. The computational domain, marked with a red dashed line, has extents $0 \leqslant x \leqslant 375$ , $0 \leqslant y \leqslant 140$ , $0 \leqslant z \leqslant 27.32$ , and the number of points ( $N_x$ , $N_y$ , $N_z$ ) = (2050, 325, 200). The origin is located at the beginning of the computational domain. The grids are stretched in the wall-normal ( $y$ ) direction using a tangent hyperbolic stretching function, while the grids are uniform in both streamwise ( $x$ ) and spanwise ( $z$ ) directions. All the distances are scaled with the displacement thickness $\delta _{\textit{inlet}}^{*}=0.075$ (mm) at the inflow plane, which is initialised using a similarity solution for a Mach 1.5 flow with a unit Reynolds number of $10^{7}$ (m⁻¹). Hence, the simulation Reynolds number based on this $\delta _{\textit{inlet}}^{*}$ is $Re=750$ .

The reference conditions are the same as Sansica et al. (Reference Sansica, Sandham and Hu2016), and table 1 summarises the aerodynamic parameters. At the wall, no-slip and isothermal boundary conditions (where the wall temperature is set to the laminar adiabatic wall temperature, i.e. $T_{\textit{wall}} = T^*_{\textit{wall}}/T^*_{\infty } \approx 1.381$ ) are used. Here, the reference free-stream temperature is $T^*_{\infty } = 202.17$  K. An extrapolation method is used at the inflow (for pressure) and outflow, while the span is periodic. The top boundary has shock jump conditions for a wedge angle of 2.5 $^{\circ }$ at $x=20$ , resulting in a pressure rise of $p_{3}/p_{1}= 1.28$ , where $p_{3}$ indicates the pressure state after the reflected shock. The Reynolds number at the location of inviscid shock impingement from the leading edge of the flat plate is $Re_{\tilde {x}_{imp}} = 1.95 \times 10^{5}$ . These are further depicted in the schematic of the domain in figure 1.

Table 1. Aerodynamic flow conditions.

Disturbances are applied, upstream of the separation bubble, as a body-forcing term in the continuity equation, and a sample oblique wave representation with a particular frequency and spanwise wavenumber is given as

(2.4) \begin{equation} {\rho '}(x,y,z,t) = {\textrm {Real}} \left[A_0 \exp \left[- (x-x_c)^2 - (y-y_c)^2\right]{\exp} \big[i(\pm \beta z - \omega t)\big]\right], \end{equation}

where $A_0$ represents the amplitude of the forcing, while ( $x_c$ , $y_c$ ) = (20, 4) are the coordinates where the forcing is centred, which is roughly located at the edge of the shear layer. The forcing takes a maximum value at the central location and then tapers off in both $x$ and $y$ directions due to the first exponential term in (2.4). The last exponential term introduces variation in the spanwise and temporal dimensions, representing an oblique wave that travels at different angles with respect to the $z$ direction depending upon the + or − sign. The values of the spanwise wavenumber and circular frequency ( $\beta$ and $\omega$ , respectively) are obtained from the linear stability theory (Sansica et al. Reference Sansica, Sandham and Hu2016).

Various combinations of the simple deterministic forcing represented by (2.4) are used to trigger flow transition, the simplest of which is a pair of oblique waves with single circular frequency as used in Sansica et al. (Reference Sansica, Sandham and Hu2016). As the modifications represent a key point in this study, an entire section (see § 4) has been devoted to a comprehensive and detailed treatment of them. At this point in the text it is important to emphasise that, when two frequencies are forced, the combination is designed to be periodic over one cycle of the difference frequency (i.e. $T=2\pi /\Delta \omega$ ). This periodicity should be evident in the response flow field under these forcings, and once this was ensured, the wall pressure data was collected over one cycle of this difference frequency to evaluate the frequency spectrum.

2.2. High-order analysis

High-order spectra, corresponding to the Fourier transform of high-order correlation functions, are the preferred tools to study nonlinear interactions since they allow the analysis of the quadratic couplings present in the governing Navier–Stokes equations on a scale-by-scale basis.

One relevant high-order spectrum is the bispectrum (Tynan et al. Reference Tynan, Moyer, Burin and Holland2001). It is formally defined as the Fourier transform of the triple correlation, given by

(2.5) \begin{equation} {\textit{Bis}}_{\textit{FGH}}\!\left (\boldsymbol{x_F},\boldsymbol{x_G},\boldsymbol{x_H},f_1,f_2\right )= \langle \hat {F}\!\left (\boldsymbol{x_F},f_1\right )\, \hat {G}\!\left (\boldsymbol{x_G},f_2\right )\, \hat {H}^*\!\left (\boldsymbol{x_H},f_1+f_2\right ) \rangle , \end{equation}

where $\langle \rangle$ denotes the averaging operation over time segments and possibly the homogeneous direction. Here $\hat {F}$ , $\hat {G}$ and $\hat {H}$ are the temporal Fourier transforms at the locations $\boldsymbol{x_{F}}, \boldsymbol{x_{G}}$ and $\boldsymbol{x_{H}}$ , and the superscript $^{*}$ indicates the complex conjugate. The bispectrum reveals the energy content associated with the cross-interaction between $\hat {F}$ and $\hat {G}$ ( $\hat {F} \times \hat {G}$ ) and a third signal $\hat {H}$ at the frequency $f_1+f_2$ . This tool has been used extensively in the work of Mauriello (Reference Mauriello2024), where a broadband stochastic forcing was used to stimulate the boundary layer transition in the case of a transition SBLI at Mach 1.7. It has been proven to be very powerful in highlighting the triadic interactions that occur between the oblique modes, i.e. the coherent structures responsible for the transition to the turbulent state of the boundary layer, and the structures of a 2-D nature that emerge at the separation point. In the present work, the modal transition has been fostered and a deterministic forcing has been applied (see § 4), plus the periodicity of the present simulations (one-period simulation) imposes that, for the lowest frequency $f_{min} = 1/T$ , only a single segment, encompassing fully the period, can be considered. It therefore excludes the possibility of averaging over segments leading to a meaningless value of the normalised form of the bispectrum, i.e. the bicoherence ( $Bic = 1$ ). With this in mind, the bispectral analysis presented above is reformulated in terms of spanwise wavenumbers taking advantage of the time/space duality found for both the oblique mode and the low-frequency unsteadiness (Mauriello Reference Mauriello2024). This version of the bispectrum is given by

(2.6) \begin{align} &{\textit{Bis}}_{\textit{FGH}}\big(({x_F}, {y_F}),({x_G}, {y_G}),({x_H}, {y_H}),k_{z_{1}},k_{z_{2}}\big)\nonumber\\ &\quad=\langle \hat {F}\big(({x_F}, {y_F}),k_{z_{1}},t\big)\hat {G}\big(({x_G}, {y_G}),k_{z_{2}},t\big)\hat {H}^*\big(({x_H}, {y_H}),k_{z_{1}}+k_{z_{2}},t\big)\rangle, \end{align}

where $\langle \rangle$ denotes a time average over one period. In this way, it is possible to detect the wavenumbers responsible for nonlinear interactions among the fixed locations $({x_F}, {y_F}),({x_G}, {y_G}),({x_H}, {y_H})$ . By time averaging in the wavenumber space, information about the temporal behaviour is lost, but can be partially recovered by introducing a time delay $\tau$ . The time delay can be introduced for the two time series $F$ and $G$ , consequently $\tau _{1}$ and $\tau _{2}$ identify the time lag occurring with respect to the third time series $H(f)$ . The formula can be written as

(2.7) \begin{align}&{\textit{Bis}}_{\textit{FGH}}\big(({x_F}, {y_F}),({x_G}, {y_G}),({x_H}, {y_H}),k_{z_{1}},k_{z_{2}}, \tau _{1}, \tau _{2} \big)\nonumber\\&\quad=\langle \hat {F}\big(({x_F}, {y_F}),k_{z_{1}}, t + \tau _{1}\big)\hat {G}\big(({x_G}, {y_G}),k_{z_{2}}, t + \tau _{2}\big)\,\hat {H}^*\big(({x_H}, {y_H}),k_{z_{1}}+k_{z_{2}}, t \big)\rangle,\end{align}

In addition to the standard bispectrum maps defined in (2.7), optimal bispectrum maps can be extracted. These maps are optimal in the sense that, for all possible time-delay pairs, the optimal time delay $\tau _{opt}$ is such that it maximises the bispectral energy content. For the sake of simplicity, the Fourier transforms $\hat {F}$ , $\hat {G}$ and $\hat {H}$ will all be referred to by the letter $G$ in the remainder of the text, and will be distinguished by subscript numbers running from 1 to 3.

5.1. Frequency, wavenumber and location

Figure 10. Streamwise distribution of the power spectra for each normalised wavenumber at selected frequencies: the left column indicates low frequency $(f_{2}-f_{1})$ and the right column indicates high frequency $(f_{2}+f_{1})$ . The white vertical lines indicate the position of the separation point (solid pattern) and the reattachment point (dashed pattern). (a,b) Crossing waves, (c,d) parallel beating waves.

More information about the interactions can be obtained from a spectral analysis of the resulting flow. Figure 10 displays the power spectrum of wall pressure fluctuations in the spanwise wavenumber domain, at fixed frequencies and for each combination of oblique waves. The wavenumber is presented as a multiple of the imposed wavenumber $\beta$ in the form $k = k_{z}/\beta$ . This approach enables a direct comparison of the results with the theoretical ones presented in table 6. The white vertical lines indicate the position of the separation point (solid line) and the reattachment point (dashed line). For simplicity, we use frequency $f$ instead of circular frequency $\omega$ . Consequently, the left column plots the flow organisation in the low-frequency dynamics $(f_{2}-f_{1})$ , while the right column shows the space arrangement for $(f_{2}+f_{1})$ . Furthermore, the first row illustrates the results for the crossing family (a,b), while the second row shows the parallel beating family (c,d).

In the high-frequency dynamics (right column), the crossing combination of waves gives rise to 2-D waves $k= 0$ , originating around the reattachment point and extending downstream. However, a similar downstream contribution at $k= 0$ is absent for the parallel beating family at high frequency $(f_{2}+f_{1})$ (see panel d). This region is populated by 3-D structures whose value of $k$ is equal to −2.

A detailed examination of the slow dynamics (left column) reveals that only the parallel combination (see panel c) leads to 2-D structures, originating before the separation point. Conversely, in the case of crossing waves, the spectral content for 2-D structures does not emerge at this same location, thereby confirming their absence in the slow dynamics at the separation point. These observations are consistent with the findings of the previous section, i.e. that the appearance of low-frequency unsteadiness only occurs for wavenumber combinations that are two dimensional.

5.2. Nonlinear analysis in terms of wavenumber

The previous subsection showed that oblique mode families interact in such a way as to produce 2-D and 3-D flow features at specific locations in both slow and fast dynamics. Exploiting the periodicity of the present simulation (one period of the difference mode $\Delta \omega$ ), the extended version of the higher-order spectral analysis, as presented in § 2.2, is applied in order to detect any possible triadic interactions. In addition, a further simplification has been introduced. In the study of Mauriello et al. (Reference Mauriello, Larchevêque and Dupont2022) it was observed from bicorrelation maps that the quadratic interactions between oblique modes are maximal for null time delay between the modes (see figure 8 of their paper, high bispectral content is observed along the diagonal shown in the bispectrum map). In short, for a time $\tau _{1} = \tau _{2} = \tau$ , the nonlinear interaction between oblique modes is at its maximum energy activity. Consequently, the same time delay $\tau _{1} = \tau _{2}$ is set for the source sensors used to extract possible quadratic couplings between oblique modes with respect to the target sensor. Using these assumptions, (2.7) is reduced to

(5.4) \begin{align} &{\textit{Bis}}_{\textit{FGH}}\big(({x_F}, {y_F}),({x_G}, {y_G}),({x_H}, {y_H}),k_{z_{1}},k_{z_{2}}, \tau \big)\nonumber\\ &\quad=\big\langle \hat {F}\big(({x_F}, {y_F}),k_{z_{1}}, t + \tau \big)\hat {G}\big(({x_G}, {y_G}),k_{z_{2}}, t + \tau \big)\hat {H}^*\big(({x_H}, {y_H}),k_{z_{1}}+k_{z_{2}}, t \big) \big\rangle. \end{align}

Optimal bispectral maps are presented in this section. The optimality results from the time delay $\tau _{opt}$ , which maximises ${\textit{Bis}}_{F,G,H}(k_{z_{1}},k_{z_{2}},\tau )$ . For the sake of simplicity, we drop the subscript ‘ $_{z}$ ’ in the wavenumber $k$ , and the three signals will all be denoted by the sole letter G. Subscripts from 1 to 3 are used to distinguish the signals. The first two signals $G_1$ and $G_2$ have been chosen as source signals and are located between the forcing location and the separation point at $x = 45$ . At this location the spectral decomposition of the wall pressure fluctuations has the same power content in each of the cases (see figure 8) and the flow field in this region is described solely by the dynamics of the oblique modes. This implies full knowledge of the power contribution of the source sensors $G_{1}$ and $G_{2}$ , which is the same for both families. Consequently, it is natural that the target sensor $G_{3}$ (located either at the separation or at the reattachment points) will have a power contribution that depends only on the power due to the quadratic couplings, which varies according to the case under consideration. An alternative approach would be to use the bicoherence to quantify the level of nonlinear coupling. However, in the latter case, the normalisation used to define the bicoherence yields a measure of the strength of the quadratic coupling regardless of the level of quadratic power involved, thus highlighting a set of quadratic couplings that have no dynamical impact due to negligible energy content. In contrast, the norm of the bispectrum directly reveals the energy content associated with the nonlinear couplings. The location of the three sensors is the same for all further analyses, unless clearly stated.

Figure 11. Modal forcing: maps of the norm of the optimal wavenumber bispectrum, with pairs resulting in a bicoherence value higher than 0.25 encircled in black. Left column: the target sensor $G_{3}$ is located at the separation point; right column: the target sensor $G_{3}$ is located at the reattachment point. (a,b) Crossing waves; (c,d) parallel beating waves. All maps show quadratic interactions with the two source sensors $G_{1} = G_{2}$ located at $x = 45$ .

Figure 11 shows maps of the norm of the optimal bispectrum for each oblique mode combination. In all maps, the two source sensors $G_{1}$ and $G_{2}$ are located at $x= 45$ for the reason previously explained, whereas the destination sensor $G_{3}$ is located either at the separation point (left column) or at the reattachment point (right column). This approach allows the 2-D and/or 3-D nature of the flow features responsible for the nonlinear coupling between the upstream region and the separation and reattachment locations to be highlighted. Note that the spanwise wavenumbers are presented with subscripts $_{1}$ and $_{2}$ to indicate that the quadratic couplings result from all possible wavenumbers detected by sensor $G_{1}$ (i.e. $k_{1}$ ) and $G_{2}$ (i.e. $k_{2}$ ). Also note that $(k_{1}, k_{2})$ pairs resulting in squared bicoherence values higher than $0.25$ are encircled by a black thick line in order to demarcate couplings effectively resulting in a high level of relative quadratic power.

When considering the crossing waves combination at the separation point (see figure 11 a), the set of wavenumbers resulting from quadratic nonlinearities of the oblique modes are dominated by couplings involving at least one oblique mode, i.e. $k_{1,2} = \pm 1$ , all with a similar amount of quadratic power (orange circles). In contrast, for the parallel wave case, the bispectral map is dominated by the combination $(k_{1},k_{2}) = (- 1,1)$ , for which the level of bispectral content is higher (see figure 11 c). This means that quadratic couplings at the separation point lead to 2-D flow features with $k_{3} = k_{1} + k_{2} = 0$ . However, it is evident that 2-D quadratic combinations also appear for the subsequent $k_{2} = - k_{1}$ couplings visible along the diagonal, despite the decrease in the bispectral power content. Nevertheless, at this stage of the analysis, it is still not possible to infer whether the quadratic interactions arise from the interaction of the oblique modes after they have passed through the shock interaction system and, thus, have the possibility of flowing back through the separated region, or whether they are the beginning of pure triadic interactions that are about to develop and will continue to develop along the shear layer. Another important difference between crossing waves and parallel beating waves, when observed at the separation point, is the resulting nonlinear contribution introduced to the mean field $(k_{1},k_{2}) = (0,0)$ that occurs in the case of the combination of parallel waves. This nonlinear modulation of the mean field is also present for this particular combination when the target sensor is positioned at the reattachment point (see figure 11 d). However, the corresponding level of the bicoherence is low, indicating that the power issued from this coupling only contributed to a small amount of the total power at $k_{3} = 0$ .

On the other hand, strong quadratic couplings for all integer multiples of the fundamental wavenumber $k_{1,2} = 1$ appear for the crossing wave family at the reattachment point (see figure 11 b). The oblique modes at the reattachment region interact nonlinearly, initiating the cascade process towards higher wavenumbers (smaller scales) typical of the turbulent kinetic energy cascade. For small $k_{1,2}$ , the same organisation of the quadratic power is observed as for the separation point. The combination of $k_{1,2}=1$ is largest, indicating a direct quadratic interaction between oblique modes resulting in a 3-D organisation of the flow.

When restricted to the reattachment point, the parallel waves show a cascade process towards higher $k_{1,2}$ that is at its early stages, as only a few cascading combinations of the fundamental harmonic are visible (see figure 11 d). These results support the previous finding that the crossing arrangement is more prone to turbulence breakdown than the parallel wave arrangement.

Figure 12. Norm of the optimal wavenumber bispectrum for both crossing ( $\times$ ) and parallel beating ( $\big/\big/$ ) waves extracted for the resulting 2-D ( $k_3=0$ ) and 3-D ( $k_3= 2$ ) flow characteristics. The vertical black lines indicate the separation (solid) and reattachment (dashed) points.

Limiting attention to combinations of 2-D ( $ (k_1,k_2 )= (- 1,+1 )$ ) and 3-D ( $ (k_1,k_2 )= (+1,+1 )$ ) wavenumbers from the previous maps, information on the streamwise evolution of the norm of the optimal bispectrum is extracted and presented in figure 12. Note that in this figure, the target sensor $G_{3}$ is no longer limited to the two locations of separation and reattachment, but extracts information for each point in the $x$ direction. For the clarity of the figure, the crossing waves are indicated in the legend by the symbol $\times$ , while the parallel beating family is indicated by $\big/\big/$ . The vertical black lines indicate the separation (solid line) and reattachment (dashed line) locations for each wave family. A high value of the optimal bispectral content is observed in the separated flow region for the oblique mode coupling resulting in 2-D spanwise organisation of the flow in the case of parallel beating waves, confirming that most of the quadratic couplings result in $k_{3} = 0$ (see green line) for such an arrangement. In the same region, nonlinear couplings between the oblique modes resulting in 3-D (parallel beating waves) and 2-D and 3-D (crossing waves) flow characteristics begin to develop within the separation bubble. After the shock interaction, quadratic couplings saturate, with significantly higher plateau levels in the case of the crossing waves, especially if 3-D interactions ( $k_{3} = 2$ ) are considered. Those associated with the occurrence of the turbulent energy cascade dominate, confirming the occurrence of the turbulent energy cascade, which is observed to be less pronounced in the case of parallel beating waves for $k_{3} = 2$ .

Figure 13. Norm of the time-filtered wavenumber bispectrum. The filter is applied to the target sensor $G_{3}$ , which spans all $x$ locations. It retains either the frequencies associated with the low-frequency dynamics $(f_{2}\,{-}\,f_{1})$ (indicated in the legend with $G_{3} (f_{2} - f_{1})$ ) or those associated with the high-frequency dynamics $(f_{2} + f_{1})$ (indicated in the legend with $G_{3} (f_{2} + f_{1})$ ). Symbols are used when the lines overlap perfectly. The green diamond symbols indicate the presence of self-quadratic couplings, i.e. $f_{1}+f_{1} = 2 f_{1}$ and $f_{2}+f_{2} = 2 f_{2}$ . The vertical black lines indicate the separation (solid) and reattachment (dashed) locations.

The norm of the optimal bispectrum does not directly indicate the frequency range in which the quadratic couplings occur. In this context, the time-filtered optimal bispectral maps are computed. They are obtained by bandpass filtering in time the target signal $G_{3}(k_{1}+k_{2},t - \tau )$ . The filter retains either the frequencies associated with the low-frequency dynamics $(f_{2} - f_{1})$ or those associated with the high-frequency dynamics $(f_{2} + f_{1})$ . This enables the whole frequency spectrum to distinguish whether the predominant contribution comes from quadratic interactions occurring at low or high frequencies. In each plot of figure 13, the separation (solid line) and reattachment (dashed line) locations are shown to ease visualisation of the separated region. In addition, the colour code uses black for the unfiltered target signal $G_{3}$ , red for low-pass filtering and blue for all signals that retain only the high-frequency range. The total frequency spectrum for the 2-D periodicity in the case of the crossing arrangement (see panel a) is fully dominated by quadratic couplings occurring at high frequency $(f_{2} + f_{1})$ , and the contribution of nonlinear couplings at low frequency is marginal. The opposite situation is observed for the same 2-D organisation of the flow in the case of parallel beating waves (see panel c). Most of the nonlinear interaction is detected in the low-frequency range (see the red diamonds overlapping the black line). The high value of the optimal bispectral content through the separated region again supports the observation that nonlinear coupling due to oblique modes in the specific case of parallel arrangement is responsible for the appearance of 2-D flow features at the separation point. These flow features are in turn sustaining the low-frequency motion of the head shock. In short, in the wavenumber space, the trace of the low-frequency unsteadiness is two dimensional. Indeed, for the crossing waves with periodicity $k_{3}=0$ , only quadratic couplings acting at high frequency are observed, and in § 4 we have observed that in such a case the interaction is steady. The 3-D periodicity requires careful analysis, as the quadratic couplings also take into account beatings of a different nature. Although most of the contribution to the total bispectral content comes from the low-frequency (crossing waves) and high-frequency (parallel beating waves) range, a complete overlap of the plots is not observed. The full bispectral power is only recovered when the self-quadratic coupling of each single oblique mode is taken into account, i.e. $f_{1}+f_{1} = 2 f_{1}$ and $f_{2}+f_{2} = 2 f_{2}$ in the frequency range that best maximises the total quadratic power. This is shown in panels (b) and (d) with the green diamond symbols.

Figure 14. Time-delay map extracted from the real part of the wavenumber bispectrum. The first row shows the crossing waves and the second row shows the parallel beating waves. The vertical black lines indicate the separation (solid) and reattachment (dashed) points.

5.3. The direction of quadratic couplings

Information on the directionality of the quadratic motion can be extracted by mapping the norm of the bispectrum into the time-delay space domain, as shown in figure 14. The contours represent the norm of the bispectral power, and information on the time periodicity and direction of the motion is available from its pattern and the slope associated with it. Note that the contours are saturated such that low amplitude activity can be highlighted. The inverse of the ratio of $\tau$ to $x$ directly gives the value of a propagation velocity associated with the quadratic coupling under consideration, normalised with the external velocity, i.e. $U_{B}/U_{\infty }$ . In each map, the location of the separation and reattachment points is indicated by black vertical lines, solid for the former and dashed for the latter. Propagation velocities deduced from the map in various regions of the flow thus delineated are listed in table 7. Note that the streamwise evolution has been divided into three regions: from the forcing location ( $x_{\textit{forcing}}= 20$ ) to the separation point, within the recirculating region and downstream of the reattachment point.

Downstream motion is observed for crossing waves, regardless of the 2-D (see figure 14 a) or 3-D (see figure 14 b) nature of the flow structures. However, the fundamental periodicity is different. A short period corresponding to $T= 1/(f_{1} + f_{2})$ is observed for $k_{3}=0$ . This result is consistent with the previous observations, for which most of the quadratic activity concerns the high-frequency range of the total spectrum (see figure 13 a).

Indeed, the upstream region of the flow is dominated by slow periodic dynamics corresponding to $T= 1/(f_{2} - f_{1})$ , in agreement with the results presented in the previous section (see figure 13 b). But, as seen in this plot, there are also quadratic couplings of lower amplitude associated with self-interactions of oblique modes towards frequencies $2f_{1}$ and $2f_{2}$ . In the time-delay domain of figure 14, the sum of these two waves of similar frequencies is visualised, through beating, as a wave at frequency $(f_{1}+f_{2})$ being modulated in amplitude by a wave at frequency $(f_{1}-f_{2})$ . It hence results in spots of high-frequency ripples with a width equal to $T/2$ as seen, for instance, in the first half of the separated region or downstream of the reattachment point. In the Fourier space, however, the only contribution to the $(f_{2}-f_{1})$ range comes from the quadratic interaction between modes 1 and 2.

Although barely visible, the same observations apply to the parallel beating waves in the case of $k_{3}=2$ organisation of the flow (see figure 14 d). Both crossing and parallel beating wave cases for $k_{3}=0$ are free of this self-quadratic interaction, since the self-coupling of a single mode towards zero wavenumber corresponds to zero frequency. Consequently, there is no secondary modulation in time in figures 14(a) and 14(c).

In all these cases, the quadratic power couplings move from upstream to downstream, with the exception of the 2-D parallel beating waves. In this case, figure 14(c) clearly shows that there is an upstream motion of period $T= 1/(f_{2} - f_{1})$ within the separated region. The corresponding value of the propagation velocity is $U_{B}/U_{\infty } = - 0.17$ . Such a value is in agreement with the values suggested by Larchevêque (Reference Larchevêque2016), Bonne et al. (Reference Bonne, Brion, Garnier, Bur, Molton, Sipp and Jacquin2019), and Threadgill et al. (Reference Threadgill, Little and Wernz2021). Moreover it falls in the range of values observed in Mauriello et al. (Reference Mauriello, Larchevêque and Dupont2022), who studied transitional SBLI in a similar flow configuration and observed an upstream motion from the reattachment point towards the separation point. The value they found is $U_{B}/U_{\infty } = - 0.09$ for the features sustaining the low-frequency dynamics and $U_{B}/U_{\infty } = - 0.25$ for the frequencies falling in the medium range. In our case, after $x = 197$ , the direction of the motion changes to downstream, with a speed of $ U_{B} = 0.12 U_{\infty }$ .

Table 7. Value of the propagation velocity of the bispectral content normalised by the external velocity, i.e. $U_{B}/U_{\infty }$ , for each region of the flow: from the forcing location to the separation point, within the recirculating region and downstream of the reattachment point.

5.4. Phenomenology of the nonlinear interactions

The basic phenomenology can now be proposed as follows. As the oblique modes convect downstream in the separated shear layer, they grow in amplitude, developing a high amplitude towards the end of the separation bubble, achieving a maximum quadratic power transfer. After this power input, structures having zero spanwise wavenumber and period $T=1/(f_{2}-f_{1})$ split into two parts just upstream of the reattachment point. A part is convected downstream with the reattaching flow, and is possibly reinforced by further quadratic interactions between the oblique modes still persisting in that region. The other part enters the lower region of the separated zone and is subject to an upstream propagation up towards (and beyond) the separation point. This upstream propagation occurs without further quadratic power supply because of the low oblique mode amplitude in that region. The separation point and the reattaching boundary layer therefore undergo motion similar in amplitude but opposite in sign. In this context, the work of Dupont et al. (Reference Dupont, Haddad and Debiève2006) noted that the rear part of the interaction for an oblique reflected shock geometry also exhibits some degree of unsteadiness that is in quasi-linear dependence with the reflected shock motion, reinforcing the idea that the separation bubble is in a breathing motion. In addition, the work of Touber & Sandham (Reference Touber and Sandham2009) on a turbulent shock-induced separation bubble interaction had already observed a jump in the velocity phase associated with the wall pressure perturbations. However, in their work, this jump is observed to occur at one-third of the length of the separation zone. In the present work, the shift in the velocity direction occurs near the point of minimum $C_{f}$ at $x=200$ (see figure 7), after which the skin friction begins to increase. The same observation applies to the other time-delay maps, although instead of a net change in direction, an increase in bispectral activity is observed.

We observed in § 5.2 that, at the separation point, both crossing waves and parallel beating waves experienced quadratic couplings towards a wavenumber equal to zero (see figures 11 a and 11 c). It is now possible to add that the nonlinearities observed at the separation point in the case of parallel beating waves result from quadratic interactions of the oblique modes that, after passing through the shock interaction, succeed in entering the separation bubble via the reattachment point and travelling up to the separation point. These triadic interactions result in 2-D structures. In contrast, the 2-D structures that populate the separation point in the case of crossing waves are the result of quadratic interactions in the incoming boundary layer. They begin to develop between the oblique modes and among families of them.

Irrespective of the 2-D or 3-D spanwise structure of the flow and the specific arrangement of the oblique modes in the two families, an upstream motion is observed for $x \leqslant 20$ . This point corresponds to the location of the forcing $x_{\textit{forcing}}$ , and since the perturbations are introduced as density perturbations (leading to pressure perturbations), they are free to move in all directions, including upstream within the subsonic layer of the compressible boundary layer.

5.5. The frequency-space organisation of the quadratic coupling

The time delay $\tau$ introduced in (5.4) can be exploited to expand the wavenumber bispectrum ${\textit{Bis}}_{1,2,3}(k_{z_{1}},k_{z_{2}},\tau )$ into a Fourier series in time . This results in the frequency-transformed wavenumber bispectrum $\widehat {\textit{Bis}}_{1,2,3}(k_{z_{1}},k_{z_{2}},f_n)$ for discrete frequencies $f_n={n}/{T}$ such that

(5.5) \begin{equation} {\textit{Bis}}_{1,2,3}(k_{z_{1}},k_{z_{2}},\tau )=\sum _{n=-\frac {T}{2\Delta \tau }}^{+\frac {T}{2\Delta \tau }} \widehat {\textit{Bis}}_{1,2,3}(k_{z_{1}},k_{z_{2}},f_n)\,e^{2i\pi \frac {n\tau }{T}} .\end{equation}

The space transform provides information about the nature of the structures, while the time transform allows the amount of nonlinear content associated with each frequency and each streamwise location to be determined. Overall, the frequency-transformed wavenumber bispectrum directly highlights the relevant frequency content of the nonlinear coupling involved between selected wavenumbers. For simplicity, the subscript $_{z}$ is dropped in the following discussion.

Figure 15. Streamwise-frequency distribution of the norm of the frequency-transformed wavenumber bispectrum for selected wavenumber pairs. (a,b) Crossing waves; (c,d) parallel beating waves. The white vertical lines indicate the separation (solid) and reattachment (dashed) points.

Figure 15 shows the norm of the space--frequency-transformed bispectrum for each combination of oblique waves. The streamwise organisation of the quadratic interactions is plotted for each frequency. Only results for selected wavenumbers $k_{1} + k_{2} = k_{3} = 0$ (left column) and $k_{1} + k_{2} = k_{3} = 2$ (right column) are presented. Note that, for $k_{3}=0$ , the wavenumber bispectrum is real, resulting in a time-Fourier series that is even, i.e. symmetric with respect to the origin $f = 0$ . On the other hand, for $k_{3}=2$ , the bispectrum is inherently complex and the positive and negative frequency regions are no longer symmetric, even in the Hermitian sense. If we take into account the asymmetries inherent in the different arrangements of the waves, the present bispectra are therefore able to reflect these spanwise asymmetries.

In the case of the crossing wave family, figure 15(a) shows that from the reattachment point onwards, the quadratic coupling, associated with the transition mechanism, takes place. The cascade process starts from the forcing frequencies and fills the spectrum with exact nonlinear harmonics of the fundamental frequency, becoming increasingly difficult to visualise in the plot because of their sharp frequency distribution and the use of the logarithmic scale. Evidence of quadratic interactions is clearly detected in the low-frequency range for $k_{3} = 2$ (see figure 15 b), and confirm the results presented in table 6. Specifically, this confirms that 2-D flow features $k_{3} = 0$ lack the low-frequency dynamics.

Figure 15(c) shows that high values of bispectral content are observed for parallel beating waves when $k_{3} = 0$ . Nevertheless, at the separation point as well as upstream and downstream of it, the range of frequency associated with quadratic couplings spreads up to $f \sim 0.005$ . On the contrary, this combination of waves, when quadratically interacting to give $k_{3} = 2$ , does not support the low-frequency dynamics, and only the frequency corresponding to the sum of the forcing contribution emerges (see figure 15 d). Note that this frequency is negative because the wavenumber $k_{3}=2$ under consideration is positive. For the parallel family, $k_3=2$ can be obtained from oblique modes only by considering an additive quadratic coupling, i.e. the one involving $k_{1}=+\beta$ and $k_{2}=+\beta$ . It translates, through the ansatz (2.4), into an ( $- \omega _1$ , $- \omega _2$ ) coupling into the frequency space. As a consequence of the Hermitian symmetry of the space--time transform, the quadratic interaction resulting in the positive ( $\omega _1+\omega _2$ ) frequency is found for the negative wavenumber $k_{3}=- 2$ .

6.1. A beating crossing combination

The spectral analysis of the pressure fluctuations at the wall showed that an approximately steady interaction (lacking 2-D low-frequency content) is observed in the case of the crossing waves family, while an unsteady interaction was found for the parallel beating arrangement. A third combination of waves that combines both parallel waves and crossing waves is considered next. The aim is to explore whether this new arrangement of waves still retains the properties of parallel waves to stimulate unsteadiness as well as the ability of crossing waves to facilitate the breakdown to turbulence. The arrangement is called beating crossing waves and it is expressed through the notation of (4.2) as

(6.1) \begin{align} \mbox{Beating crossing waves:} \quad \rho '(x,z,t) & = \rho _{1}^{\prime+}(x,z,t) + \rho _{1}^{\prime-}(x,z,t) + \rho _{2}^{\prime+}(x,z,t) \nonumber\\&\quad + \rho _{2}^{\prime -}(x,z,t). \end{align}

Figure 16. Panel (a) shows the streamwise evolution of the skin friction coefficient $C_{f}$ for all combinations of waves. The black dashed horizontal lines indicates $C_{f} = 0$ . Panel (b) plots the spectral decomposition of the wall pressure fluctuation for the sole beating crossing waves case. The white vertical lines indicate the separation (solid) and reattachment (dashed) points.

Figure 16(a) plots the streamwise evolution of the skin friction coefficient, comparing the different cases. The beating crossing waves have the shortest separation zone with $L_{\textit{int}}/\delta ^{*}_{\textit{inlet}} = 44$ ( $L_{\textit{sep}}/\delta ^{*}_{\textit{inlet}} = 73.4$ ) and result in a transitioning boundary layer downstream of the interaction. This result is consistent with the observation that this family possesses a double crossing wave combination and, as observed from the previous results, the breakdown to turbulence is facilitated.

Since double parallel beating waves are also included, low-frequency unsteadiness is expected to develop. Figure 16(b) shows the spectral decomposition of the wall pressure fluctuations for the beating crossing waves. The same normalisation and structure of the spectra shown in figure 8 is followed here. Downstream of the reattachment point and starting from the forcing frequencies, the energy content cascades over all their harmonics and fills the spectrum up to high frequencies consistent with the results extracted from the $C_{f}$ plot that indicates transition to turbulence. When looking at the separation point, beating crossing waves show high energy activity that results in an unsteady interaction. As observed, the parallel combination induces slow motion at $St_{L_{\textit{int}}} = 0.037$ . For the beating crossing family, the head shock moves at $St_{L_{\textit{int}}} = 0.028$ . This difference is due to the change of $L_{\textit{int}}$ . Hence, the inclusion of the parallel combination of the two wave families proves once again to be responsible for the slow 2-D motion of the head shock, but this time combined with transition to turbulence. Note that all the quadratic metrics used in § 5 to analyse the crossing and parallel cases have been applied to the beating crossing combination. The results, not presented here for the sake of conciseness, confirm that this more complex family combines the individual quadratic features of the two simpler cases.

6.2. A streaky crossing combination

Low-speed velocity streaks are often present in laminar boundary layers, for example, when free-stream turbulence or roughness modifies the laminar base flow. They are steady in time ( $\omega = 0$ ), but have a non-zero spanwise wavenumber (Schmid & Henningson Reference Schmid and Henningson2001). Hence, it is of interest to take advantage of them to modify the simple crossing case to achieve a 2-D spanwise organisation of the flow at the separation point without altering the incoming frequency content and verify whether the slow bubble breathing motion occurs. The targeted nonlinear coupling involving the streaks and the oblique modes will inherently result in the same frequency as the corresponding nonlinear coupling between the sole oblique modes, but it must result in $k_{z} = 0$ in the separated flow region. Consequently, the spanwise dimension of the streaks must be carefully chosen. For this purpose, half of the spanwise dimension of the 3-D unstable waves was chosen ( $\beta _{\textit{streak}} = 2\beta _{OM}$ ). The mathematical representation of the streaks is therefore given by $\rho '_{\textit{streak}}(x,z, 0) = {\textrm {Real}} [A_{0}\,\hat {\rho } (y) e^{i(2 \beta z)}]$ . By adding the streak to the crossing waves family, the new combination we are considering is called streaky crossing waves, and it is mathematically described by

(6.2) \begin{align} \begin{array}{l} \mbox{Streaky crossing waves:}\quad \rho '(x,z,t) = \rho _{1}^{\prime +}(x,z,t) + \rho _{2}^{\prime -}(x,z,t) + \rho^{\prime}_{\textit{streak}}(x,z,0). \end{array} \end{align}

By following the same mathematical approach presented in § 5, for the streaky crossing family, we find that the only nonlinear combination involving the oblique modes with frequency $(f_{2}-f_{1})$ is given by the same relation of (5.2). Nevertheless, if the contribution of the streak is included, many possible nonlinear combinations of the triad of modes can result in the $(f_{2}-f_{1})$ frequency. The simplest ones are two consecutive quadratic coupling or a single cubic coupling. The five simplest possible nonlinear couplings towards the low-frequency range therefore read

(6.3) \begin{equation} f_{\textit{LF}} = f_{2} - f_{1}\longleftrightarrow \left \{\begin{array}{r@{\,}c@{\,}ll} \beta + \beta & = & 2\beta & {\textrm{(quadratic coupling)}},\\[3pt] \left (\beta +\beta \right ) + 2\beta & = & 4\beta & {\textrm {(consecutive quadratic couplings)}},\\[3pt] \left (\beta +\beta \right ) - 2\beta & = & 0 & {\textrm {(consecutive quadratic couplings)}},\\[3pt] \beta + \beta + 2\beta & = & 4\beta & {\textrm {(cubic coupling)}},\\[3pt] \beta + \beta - 2\beta & = & 0 & {\textrm {(cubic coupling)}}. \\ \end{array}\right . \end{equation}

The first line corresponds to interactions between oblique modes only, whereas the four subsequent lines are related to couplings involving both the oblique modes and streaks. Thus, the streaky family potentially results at low frequency in both 2-D and 3-D flow features, once either quadratic or cubic couplings involving streaks are taken into account.

Figure 17(a) shows the spectral content of the pressure fluctuations field extracted at the wall for the streaky crossing case. In contrast with the simple crossing arrangement, a low-frequency activity of mostly constant level is found in the separated flow region. It results in a low-frequency power at the separation point being 1.5 orders of magnitude larger for the streaky case compared with the original crossing case. Although such a value is more than four orders of magnitude lower than the one found for the parallel arrangement, it is a clear indication that streaks nonlinearly promote flow structures at low frequency that are sustained in the separated flow region.

Figure 17. Streaky wave family: power spectra of the wall pressure fluctuations (a) and wavenumber bispectrum (b) for sensor $G_3$ located at the separation point, denoted by the white solid vertical lines on plot (a).

Recalling the results from Mauriello (Reference Mauriello2024) and from § 5, it is tempting to postulate that such a sustainability is related to two-dimensionality. Candidate nonlinear couplings having such a property correspond to lines 3 and 5 of (6.3). The former, being quadratic in nature, can be tested using the bicoherence defined in (5.4). Its imprint on the $k_1-k_2$ bispectral map at separation plotted in figure 17(b) would include the primary interaction between oblique modes at $ (k_{1},k_{2} )= (1,1 )$ , already present in the simple crossing case, as well as the secondary interaction between the structures resulting from the primary interaction and the streaks at $ (2,-2 )$ .

No such extra quadratic coupling is found when comparing figures 11(a) and 17(b) and, in fact, the norms of the bispectra at separation are similar for the pure crossing and streaky crossing cases for all wavenumber pairs under consideration. This demonstrates that, at least in the separated region of the flow, streaks do not contribute to quadratic coupling. Moreover, bicoherence levels are generally lower for the streaky crossing case. This, coupled with similar norms for the bispectra, is a clear indication that the extra power found in the separated region for the streaky case in figure 17(a) does not originate from any additional quadratic coupling.

The cubic coupling hypothesis appears therefore as the most probable explanation. Note that it could be formally tested by computing the wavenumber trispectrum. However its definition involves four distinct time series, three time delays and only limited obvious hypotheses to reduce the dimension of the input space they span. This makes its computation quite cumbersome, even if restricted to a small number of wavenumber triads. As a consequence, a formal demonstration of the actual occurrence of cubic couplings involving streaks at the separation point has not been carried out.

Table 8 integrates table 6 with all possible combinations at both low frequency $(\omega _{2} - \omega _{1})$ and high frequency $(\omega _{2} + \omega _{1})$ for the new families of oblique mode, i.e. the beating crossing waves and streaky crossing waves. The couplings provided in table 8 supports the results presented in §§ 6.1 and 6.2. Low-frequency spectral content emerges at the separation point when oblique modes are arranged in parallel or include structures that result in a spanwise wavenumber ( $k_z$ ) of zero, similar to what we have observed with the streak. These 2-D structures are responsible for the phenomenon of the low-frequency unsteadiness.

Table 8. Summary of the couplings for all modal forcing combinations. The subscript ‘ $_{\textit{LF}}$ ’ and ‘ $_{HF}$ ’ in $k$ indicate the low-frequency dynamics $(\omega _{2}-\omega _{1})$ and the high-frequency dynamics $(\omega _{2}+\omega _{1})$ , respectively.

6.3. Effect of the frequency

At this point it is important to make a few more comments. First, both parallel beating waves and simple crossing wave configurations had the same injected energy level (see table 3). Despite that, they exhibited very differing characteristics in terms of their influence on the unsteady dynamics and transition process. Secondly, regardless of the specific configuration, the system was forced with two high frequencies, such that their difference fell in the low range of the frequency spectrum, i.e. $\Delta St_{L_{\textit{int}}} \simeq 0.04$ . Although the forcing is set in the high-frequency range, thanks to quadratic coupling we observed low frequencies emerging in the spectrum. Nevertheless, it would be interesting to observe the possible consequences for both the low-frequency dynamics and the transition process when the resulting, quadratically induced, Strouhal number falls within a higher range of the frequency spectrum. To this end, we decided to investigate the arrangement of parallel beating waves since they resulted in a strong response at the quadratic couplings in the low-frequency dynamics. The forcing frequency was chosen so that the frequency separating the motion of the two parallel wave families is ten times larger than for the original one, with $\Delta St_{L_{\textit{int}}} \simeq 0.4$ . This (similar) value was observed in the work of Mauriello (Reference Mauriello2024), in which the interaction between an incoming laminar boundary layer and an impinging reflected shock system at $M=1.7$ was studied, and the nonlinear coupling between the multiple boundary layer modes and the flow features at the separation point were examined. In that work, it was found that the resulting nonlinearities occurring between oblique modes of different frequencies were progressively damped when the frequency difference exceeded $\Delta St_{L_{\textit{int}}} = 0.35$ .

Figure 18. Power spectrum of the wall pressure fluctuations for the parallel beating waves family with forcing frequency difference $\Delta S_{L_{\textit{int}}} \simeq 0.4$ . The white vertical lines indicate the separation (solid) and reattachment (dashed) points.

Figure 18 shows the power spectrum of the wall pressure fluctuations for the parallel beating waves with a medium forcing frequency difference of $\Delta St_{L_{\textit{int}}} \simeq 0.4$ . For simplicity, in the following discussion we refer to this case as $MF$ forcing, whereas the original parallel beating case with the low forcing frequency difference of $\Delta St_{L_{\textit{int}}} \simeq 0.04$ , whose corresponding power spectrum is plotted in figure 8(b), will be referred to as $LF$ forcing.

When comparing figures 8(b) and 18, it appears that the region downstream of the reattachment point experiences a different transition state, with a slightly more developed energy cascade in the case of $MF$ forcing. Such a difference is consistent with the slightly reduced size of the separated region in that case. One can postulate that the alteration of the early transition is due to the nonlinear process that broadens the frequency ranges $\omega ^n$ associated with the $n$ th nonlinear harmonics of the oblique modes. The broadening is progressively achieved through a succession of forward and backward interactions involving the two kinds of oblique modes: $(\omega ^n)\times (\pm \omega _{1})\rightarrow (\omega ^n\pm \omega _{1})$ followed by $(\omega ^n\pm \omega _{1})\times (\mp \omega _{2})\rightarrow (\omega ^n\pm \omega _{1}\mp \omega _{2})$ . The frequency extent of the process therefore scales with the $(\omega _{2}-\omega _{1})$ difference, making the $MF$ -forcing case result more rapidly in the overlapping of the broadened frequency ranges, thus leading to a fuller spectrum at earlier stations.

Besides the transition mechanism, the larger frequency difference also affects the low-frequency dynamics. Note that the lack of energy content for frequencies below $f=0.0035$ for the $MF$ -forcing case is a direct consequence of the medium frequency-difference forcing that results in a periodicity in time being inversely proportional to $\Delta St_{L_{\textit{int}}}$ . The power contents of the lowest frequency ranges, being equal to $\Delta \omega /(2\pi )$ for both cases, is rather similar alongside the last third of the separated region ( $180\lt x\lt 220)$ and downstream of the reattachment. In the first two thirds of the bubble, however, a significantly lower power is found for the $MF$ -forcing case compared with the $LF$ -forcing case.

Figure 19. Wavenumber bispectra of the parallel beating waves family with forcing frequency difference $\Delta S_{L_{\textit{int}}}\simeq 0.4$ for the 2-D ( $k_{z_{3}}=0$ ) coupling: time-delay map of the real part (a) and norm of the time-Fourier transform (b) for the streamwise coordinate. The vertical lines indicate the separation (solid) and reattachment (dashed) points. The horizontal dashed lines in panel (a) denote the time boundary of a single period since data have been duplicated through time periodicity for better visualisation.

The power content of the low-frequency range in the separated region having been associated in § 5, for the parallel case, with 2-D quadratic coupling, the various bispectral metrics described in that section have also been applied to the $MF$ -forcing case. Overall, these analyses yielded similar results to those of the $LF$ -forcing case, but two noticeable differences were found in the separated region. The first is related to the upstream propagation velocity of the 2-D bispectral content, which is found to be about 2.5 times larger than for the $LF$ -forcing case, as deduced from the time-delay map of the norm of the bispectrum for the $k_{z_{3}}=0$ coupling, plotted in figure 19(a). It is worth noting that a similar velocity ratio was found in Mauriello et al. (Reference Mauriello, Larchevêque and Dupont2022) between the lower and upper frequency ranges of the low-frequency, upstream propagating structures resulting from quadratic interactions between multiples oblique modes.

Beyond a velocity change, the quadratically induced structures of the $MF$ -forcing case are also subject to an extra damping when moving upstream in the separated region. The norm of the time-Fourier-transformed wavenumber bispectrum associated with the 2-D ( $k_{3}=0$ ) coupling experiences a 1.5 order of magnitude drop between $x=180$ and $x=155$ , as seen in figure 19(b). In the first half of the bubble this yields a difference of more than one order of magnitude between the bispectral powers at the lowest frequency found in the $LF$ -forcing and $MF$ -forcing cases. This fully explains the similar differences seen on the power spectrum in figures 8(b) and 18. It therefore seems that the downstream part of the separated region acts as a low-pass filter (in the wavenumber space) with respect to the 2-D structures originating in the quadratic coupling of oblique modes. This is reminiscent of the low-pass filter nature of the shock/bubble system suggested by Touber & Sandham (Reference Touber and Sandham2011) for a turbulent interaction, even if the upstream part of the separated region was considered in this work. Moreover, this is compatible with the mechanism suggested by Bugeat et al. (Reference Bugeat, Robinet, Chassaing and Sagaut2022).