3.1 Pitch of lower wedge

The first excitation mechanism corresponds to the pitching of the lower wedge across the steady-state DSD. For the baseline conditions ($M_{\infty }=3$ and $\unicode[STIX]{x1D717}_{1}=25^{\circ }$), the steady-state DSD extends from the von Neumann condition $\unicode[STIX]{x1D717}_{2}^{n}=14.14^{\circ }$ until detachment at $\unicode[STIX]{x1D717}_{2}^{d}=17.43^{\circ }$; see figure 1(c). Both RI and MI are considered as the starting shock topology by setting the initial lower wedge deflection to $\unicode[STIX]{x1D717}_{2,0}=12^{\circ }$ for the former and $\unicode[STIX]{x1D717}_{2,0}=19^{\circ }$ for the latter. After reaching a converged steady-state solution, the lower wedge deflection is changed at a linear rate to enforce transition; see § 2.4. Rotational velocities corresponding to $M_{t}/M_{\infty }=0.1$, 0.01, 0.001 and 0.0001 are considered. A summary of relevant parameters can be found in table 1.

Figure 5. Lower flow deflection $\unicode[STIX]{x1D717}_{2}$ at transition as a function of the rotational velocity of the lower wedge: (▾, ▿) numerical data for $\text{MI}\rightarrow \text{RI}$ transition, (▴) numerical data for $\text{RI}\rightarrow \text{MI}$ transition, (●) $\text{RI}\rightarrow \text{MI}$ transition predictions based on shock polar theory evaluated in the (moving) $C_{2}$ frame of reference, and ($\unicode[STIX]{x1D717}_{2}^{n}$) steady-state von Neumann and ($\unicode[STIX]{x1D717}_{2}^{d}$) detachment conditions with corresponding DSD shaded in grey. Numerical data was obtained for $2g/w=0.84$. The flow deflection for the fastest $\text{MI}\rightarrow \text{RI}$ transition is labelled with an empty triangle (▿) because the Mach stem was still present when the lower flow deflection at the interaction location was $\unicode[STIX]{x1D717}_{2}=0^{\circ }$.

The post-processing method explained in § 2.3 proved to be robust and accurate at tracking the evolution of the quantities of interest over the integration time. For every rotational velocity considered, $\unicode[STIX]{x1D717}_{2}$ at transition was recorded when the Mach stem height became larger than zero for $\text{RI}\rightarrow \text{MI}$ transition, or became equal to zero for $\text{MI}\rightarrow \text{RI}$. Results are included in figure 5 as up-pointing and down-pointing triangles, respectively. As expected, unsteady effects become important for large rotational velocities, meaning that $\unicode[STIX]{x1D717}_{2}$ at transition differs significantly from predictions based on steady flow assumptions. Under these conditions, the MI can penetrate far into the RI domain. As the magnitude of the rotational velocity decreases, however, unsteady effects progressively vanish and the value of $\unicode[STIX]{x1D717}_{2}$ at transition approaches a constant. For $\text{RI}\rightarrow \text{MI}$ transition, see up-pointing triangles in figure 5, this value is clearly the steady-state theoretical detachment boundary $\unicode[STIX]{x1D717}_{2}^{d}$ associated with the baseline conditions. However, $\unicode[STIX]{x1D717}_{2}$ at transition does not approach the theoretical von Neumann deflection $\unicode[STIX]{x1D717}_{2}^{n}$ in the $\text{MI}\rightarrow \text{RI}$ transition case, as shown by the down-pointing triangles in the same figure. Instead, the data point corresponding to the slowest case falls approximately $0.4^{\circ }$ inside the DSD. We believe this could be a geometry effect related to the selected value of $2g/w$, which imposes a limitation on the minimum $\unicode[STIX]{x1D717}_{2}$ for which an MI is stable. As mentioned, a necessary stability requirement is that the mass flow through the Mach stem should also pass through the virtual throat formed by both sliplines ($s_{1}$ and $s_{2}$ in figure 1b) at sonic conditions. If for a particular wedge arrangement this is not possible, the system response is either a constantly increasing Mach stem until unstart, or constantly decreasing until transition to RI. The impelling cause that drives towards one or the other still remains an open question; in our computations the latter occurs. In order to further explore the influence of the geometry parameter $2g/w$, two additional $\text{MI}\rightarrow \text{RI}$ transitions at $M_{tip}/M_{\infty }=0.0001$ with $2g/w$ ratios of 1.05 and 0.63 were simulated (that is $0.84\pm 25\,\%$; see cases P09 and P10 in table 1). The results show a shift in the measured $\unicode[STIX]{x1D717}_{2}$ at transition from $14.51^{\circ }$ to $14.60^{\circ }$ and $14.45^{\circ }$, respectively. In line with this finding, relevant geometry effects have been also reported in the recent work of Grossman & Bruce (Reference Grossman and Bruce2018) on SWBLI at $M_{\infty }=2$.

Table 1. Summary of relevant parameters for the pitch analysis: $\unicode[STIX]{x1D717}_{2,0}$ corresponds to the wedge deflection in the initial steady-state; $2g/w$ is the ratio of vertical trailing edge distance to wedge hypotenuse (see figure 3); $M_{tip}/M_{\infty }$ relates the wedge tip Mach number to the free stream value; $\unicode[STIX]{x1D717}_{2}^{t}$ and $\unicode[STIX]{x1D719}_{2}^{t}$ the measured lower flow deflection and $C_{2}$ incidence at transition; $(\text{d}h_{ms}/\text{d}t)/u_{\infty }$ the Mach stem characteristic growth rate; and $\unicode[STIX]{x1D717}_{2,c}^{d}$ is the corrected detachment condition given by the shock polar analysis in the (moving) $C_{2}$ frame of reference. All angles are expressed in degrees.

We define the characteristic velocity scale associated with the $\text{RI}\rightarrow \text{MI}$ transition process as the maximum Mach stem growth rate, $(\text{d}h_{ms}/\text{d}t)/u_{\infty }$, occurring when the MI emerges from the interaction location. In a similar fashion as for the transitional $\unicode[STIX]{x1D717}_{2}$, the Mach stem growth becomes independent of the wedge motion as $M_{t}/M_{\infty }$ decreases, converging to a constant non-zero magnitude (see table 1). This highlights once more the inherent transient character of the transition process. The duration of such growth is also affected by the selected value of $2g/w$ as this ratio influences the target steady-state Mach stem height. This is illustrated in figure 6(a) where the evolution of the Mach stem height with respect to the measured lower flow deflection $\unicode[STIX]{x1D717}_{2}$ for the slowest $\text{RI}\rightarrow \text{MI}$ case, represented by a solid line and labelled as P04 in table 1, shows an abrupt change in trend in the vicinity of point $d$. This occurrence segregates the transient process into a segment mostly related to the $\text{RI}\rightarrow \text{MI}$ transition and a subsequent segment related to the quasi-steady evolution of the MI due to the progressive wedge motion. For larger rotational velocities, as the wedge-motion velocity scale (characterised by $M_{tip}/M_{\infty }$) becomes of the order of the characteristic Mach stem growth rate $(\text{d}h_{ms}/\text{d}t)/u_{\infty }$, this change in trend becomes less abrupt. Another characteristic feature associated with the $\text{RI}\rightarrow \text{MI}$ transition is the fact that the free stream Mach number felt by the Mach stem temporarily increases due to its relative motion towards the free stream flow. This causes an instantaneous overshoot in the time evolution of the entropy rise across the shock system, see figure 9(a), that accentuates for fast rotations.

Figure 6. Evolution of (a) the Mach stem height, and (b) the pressure jump across the wave system with respect to the measured lower flow deflection $\unicode[STIX]{x1D717}_{2}$ in the vicinity of the interaction location. Here, —— indicates $\text{RI}\rightarrow \text{MI}$ transition; - - - - $\text{MI}\rightarrow \text{RI}$ transition (see, respectively, cases P04 and P08 in table 1); and in (b) the colours blue and orange denote pressure measurements obtained downstream of the upper and lower triple point, respectively (the solid blue line is hardly visible as it falls below the solid orange line). Theoretical pressure jumps predicted by steady-state shock polar theory are additionally included in (b) as

for the $\text{RI}\rightarrow \text{MI}$ transition, with

 indicating the theoretical value at detachment, and

 and

 the theoretical $\text{MI}\rightarrow \text{RI}$ pressure evolution behind the upper and lower triple point, respectively. The red arrow in (b) points towards the pressure peak observed when the Mach stem collapses at the interaction location.

Figure 7. Sequence of instantaneous density gradient magnitude (non-dimensionalised with $w/\unicode[STIX]{x1D70C}_{\infty }$) corresponding to points a–d in figure 6(a). Solid yellow lines denote the sonic condition $M=1$; red arrows point at the kink in both reflected shocks as a consequence of the interaction with the pressure wave generated during $\text{RI}\rightarrow \text{MI}$ transition.

To illustrate the overall flow topology in the $\text{RI}\rightarrow \text{MI}$ transition process, figure 7(a–d) includes a sequence of flow visualizations corresponding to points a–d in figure 6(a). Upon first glance, some characteristic unsteady features, such as the increase in spatial extent of the subsonic pocket, embedded within the yellow line denoting sonic conditions, and the associated Mach stem growth, are clearly visible. Of particular interest is the flow field depicted in figure 7(a), which shows a pressure wave that emanates from the interaction location during the $\text{RI}\rightarrow \text{MI}$ transition, similar to that reported by Felthun & Skews (Reference Felthun and Skews2004) for a symmetric interaction. This emerging wave results in a kink in both reflected shocks (see the red arrows in figure 7a) that segregate, as indicated by the sonic contour, the RI strong-shock solution from the emerging MI weak-shock solution. Continuous pressure measurements behind the shock system, included as solid blue and orange lines in figure 6(b), confirm this occurrence and show approximately a 23 % pressure drop across the wave. This is in good agreement with the theoretical pressure drop between detachment and von Neumann conditions given by the steady-state shock polar analysis in figure 1(c). Note that the propagation velocity of the pressure wave differs above and below the emerging slipline pair due to the distinct flow properties in these regions. In the absence of both PME, the Mach stem would grow monotonically until unstarting the whole computational domain. However, the interaction with the expansion rays results in a converging–diverging slipline configuration that permits the acceleration beyond Mach unity of the enclosed flow (notice the clear sonic throat in figures 7c and 7d). A sixth-order polynomial fit to each slipline allows an estimation of the instantaneous stream-duct inlet-to-throat ratio $A/A^{\ast }$ (based on the Mach stem height and the minimum distance between sliplines) and reveals a shift from $A/A^{\ast }=1.65$ to $A/A^{\ast }=1.43$ between figures 7(c) and (d). The latter value agrees well with the theoretical estimate $A/A^{\ast }=1.39$ given by steady one-dimensional isentropic nozzle flow theory at an inlet Mach number corresponding to that after a normal shock at $M_{\infty }=3$. Such agreement suggests once more that the transient phase related to $\text{RI}\rightarrow \text{MI}$ transition is completed, in line with the change in trend around point $d$ of the Mach stem growth in figure 6(a). This is further supported by other unsteady flow features vanishing as the MI develops. For instance, a shift in the Mach stem curvature from a forward to a backward bend is observed as predicted by steady-state shock polar theory. A consequence of this process is the generation of weak acoustic waves that reach both sliplines and promote the formation of Kelvin–Helmholtz instabilities. However, once the Mach stem is fully established (figure 7d), these acoustic waves are no longer present and so slipline instabilities clearly develop further downstream.

Regarding the slowest $\text{MI}\rightarrow \text{RI}$ transition case at $2g/w=0.84$, case P08 in table 1, the corresponding Mach stem height and pressure jump evolution are also included as dashed lines in figures 6(a) and 6(b). As expected from numerical computations absent of free stream disturbances, the interaction hysteresis first hypothesised by Hornung et al. (Reference Hornung, Oertel and Sandeman1979) becomes apparent. We define the characteristic velocity scale of the $\text{MI}\rightarrow \text{RI}$ transition process as the largest Mach stem shrink rate in absolute magnitude $\Vert \text{d}h_{ms}/\text{d}t\Vert /u_{\infty }$. Values are included in table 1 for all rotational velocities considered, again showing a convergence towards a non-zero value as $M_{tip}/M_{\infty }$ decreases. This is a direct consequence of the Mach stem being finite at the precise instant the MI becomes unstable, which also explains the clear change in trend in the Mach stem height evolution (dashed line in figure 6a) around $h_{ms}/w=0.05$. Therefore, regardless of the wedge rotation rate, the Mach stem collapse at the interaction location is inherently unsteady and leads to an instantaneous over-pressure at transition (indicated with a red arrow in figure 6b). Since a mismatch with the pressure level associated with the equivalent RI solution appears, this is accommodated by a weak pressure wave that emanates from the interaction location in a similar fashion as in the $\text{RI}\rightarrow \text{MI}$ case. Instantaneous flow impressions included in figures 8(a) and 8(b) illustrate the aforementioned. Similar to what is reported by Felthun & Skews (Reference Felthun and Skews2004), we also observe a slight increase in the Mach stem height (prior to the monotonic shrinking) due to a weak expansion wave generated at the lower domain boundary when the shock foot motion is initiated.

Figure 8. Instantaneous density gradient magnitude illustrating the collapse of the Mach stem during $\text{MI}\rightarrow \text{RI}$ transition, see case P08 in table 1. Solid yellow lines denote the sonic condition $M=1$.

Figure 9. Time evolution of (a) the maximum entropy jump $\unicode[STIX]{x0394}s$ across the wave system, and (b) ——, the $x$-coordinate, and – ⋅ – ⋅ –, the $y$-coordinate of the interaction location for case P01 in table 1. The quasi-inertial region is shaded in grey and the time instant at which $\text{RI}\rightarrow \text{MI}$ transition occurs is indicated by - - - -.

Before concluding this section, we would like to highlight a particular feature revealed when considering the spatial evolution of the interaction location over time. As shown in figure 9(b) for the fastest rotational velocity $M_{tip}/M_{\infty }=0.1$, case P01 in table 1, the interaction location moves along $C_{1}$ with essentially constant velocity before transition occurs, see the shaded area in the figure. Since this was observed to be the case for all $\text{RI}\rightarrow \text{MI}$ transitions triggered by the lower wedge pitch, we could incorporate the shock motion in current shock polar theory by conducting a coordinate transformation to the (moving) frame of reference of the RI interaction location. Its velocity in both $x$ and $y$ directions was taken as the slope of the linear least squares regression to each curve in figure 9(b) within the shaded area, respectively. Corrected detachment conditions were recalculated and then transformed back into the original frame of reference of the computational domain in order to allow comparison with numerical data. Results for all rotational velocities are included as red circles in figure 5 and under $\unicode[STIX]{x1D717}_{2,c}^{d}$ in table 1. As observed, the agreement with numerical data improves significantly. This highlights the impact of rapid rotation on the shock system and the inability of steady shock polar theory to properly predict transition in the absence of unsteady considerations. We attribute the quantitative discrepancy for large rotational velocities to the complex unsteady motion of $C_{2}$, which also involves pitching due to the progressive change in shock strength. Thus, although the interaction location moves along $C_{1}$ with an apparent constant velocity, flow unsteadiness is still present leading to an overall under-predicted transitional $\unicode[STIX]{x1D717}_{2}$ as observed in figure 5. For the $\text{MI}\rightarrow \text{RI}$ transition cases, a similar analysis could not be conducted due to the substantial triple point acceleration.

3.2 Oscillation of lower wedge deflection

The second excitation mechanism investigated is the sinusoidal oscillation of the lower wedge deflection. We consider two values for the mean lower wedge deflection $\unicode[STIX]{x1D717}_{2,i}$; one within the theoretical DSD and one outside it, in the RI domain. For the former, $\unicode[STIX]{x1D717}_{2,i}=15.78^{\circ }$ is selected with an oscillation amplitude of $2^{\circ }$ so that the boundaries of the steady-state DSD in terms of flow deflection ($\unicode[STIX]{x1D717}_{2}^{n}=14.14^{\circ }$ and $\unicode[STIX]{x1D717}_{2}^{d}=17.43^{\circ }$; see figure 1c) are crossed during every period. The influence of initializing the solution with either a converged RI and MI is examined. For the case outside the steady-state DSD in the RI domain, $\unicode[STIX]{x1D717}_{2,i}=13.89^{\circ }$ is set with an oscillation amplitude of $4^{\circ }$. Regarding the excitation frequency, we use the available time scale of our set-up, $w/u_{\infty }$, scaled by a factor based on previous work on SWTBLI. In particular, we were inspired by the LES computations of Matheis & Hickel (Reference Matheis and Hickel2015) at a free stream Mach number $M_{\infty }=3$ for which the incoming turbulent boundary layer thickness to shock generator hypotenuse ratio, $\unicode[STIX]{x1D6FF}/w$, is 78.82. A base excitation frequency of $f_{1}=0.125u_{\infty }/w$ is therefore chosen to obtain good agreement with the low-frequency dynamics of the separation shock in their computations. In order to assess the effect of increasing excitation frequency in the response of the wave system, higher frequencies $f_{2}=2f_{1}=0.25u_{\infty }/w$ and $f_{3}=4f_{1}=0.5u_{\infty }/w$ are additionally investigated.

Table 2. Summary of relevant parameters for the oscillation of the lower wedge deflection: $\unicode[STIX]{x1D717}_{2,i}$ corresponds to the mean wedge deflection in degrees; $A$ denotes the amplitude of oscillation in degrees; $f$ is the oscillation frequency; $\unicode[STIX]{x1D711}$ is the phase of the sinusoid in degrees and; subscripts $+$ and $-$ denote, respectively, the maximum and minimum value recorded during the first period of oscillation. See table 1 for additional remarks.

Figure 10. Results for the lower wedge deflection oscillation around $\unicode[STIX]{x1D717}_{2,i}=15.78^{\circ }$ with an amplitude of $2^{\circ }$ and excitation frequencies of: (a–d) $f_{1}=0.125u_{\infty }/w$, and (e–h) $f_{2}=0.25u_{\infty }/w$. Solid and dash-dotted lines refer to initial RI and MI, respectively (the opacity of the latter is additionally set to 50 % for clarity). The start time of oscillation is denoted by $t_{0}$, and the time axis is non-dimensionalised with the excitation frequency $f$. Pressure measurements downstream of the upper and lower triple points are included in blue and orange, respectively. Dashed horizontal blue lines highlight the values at steady-state detachment (upper) and von Neumann (lower) conditions for $M_{\infty }=3$ and $\unicode[STIX]{x1D717}_{1}=25^{\circ }$.

For the oscillatory motion initiated within the steady-state DSD, the motion direction is set such as to bring the wave pattern towards its stability boundary; i.e., the wedge deflection initially increases for a starting RI and decreases for an MI. In a similar fashion as in § 3.1, we define the characteristic velocity scales associated with the $\text{RI}\rightarrow \text{MI}$ and $\text{MI}\rightarrow \text{RI}$ transition as the maximum Mach stem growth, $(\text{d}h_{ms}^{+}/\text{d}t)/u_{\infty }$, and maximum Mach stem shrink rate, $(\text{d}h_{ms}^{-}/\text{d}t)/u_{\infty }$ (i.e., maximum absolute value of negative Mach stem growth). In this case, they are both measured within the first period of oscillation of the lower flow deflection $\unicode[STIX]{x1D717}_{2}$ at the interaction point. A summary of relevant parameters is given in table 2 including the corresponding maximum wedge tip speed in terms of $M_{tip}/M_{\infty }$.

Figure 10(a–d) shows the time evolution of the Mach stem height, $\unicode[STIX]{x1D717}_{2}$, static pressure and entropy rise across the shock system for $\unicode[STIX]{x1D717}_{2,i}=15.78^{\circ }$ and $f_{1}=0.125u_{\infty }/w$, labelled as case D01 in table 2. All quantities are measured in the vicinity of the interaction location as discussed in § 2.3, and solid and dashed lines refer to an initially converged RI and MI wave pattern, respectively. In the case of a starting RI (solid line), the Mach stem appears during the first period of oscillation and never disappears thereafter. This observation can be explained as follows. First, the lower wedge oscillation initially brings the RI configuration outside its stability boundary. Close to the peak deviation from the mean deflection $\unicode[STIX]{x1D717}_{2,i}$, the RI is most unstable and $\text{RI}\rightarrow \text{MI}$ transition occurs. For the considered excitation frequency $f_{1}$, the resulting Mach stem growth is accentuated with $\text{d}h_{ms}^{+}/\text{d}t$ exceeding the corresponding maximum wedge tip speed $M_{tip}/M_{\infty }$; see table 2. The Mach stem keeps growing until the MI is no longer stable, which, due to the existence of the DSD, occurs exclusively when the lower flow deflection $\unicode[STIX]{x1D717}_{2}$ at the interaction location is close to its minimum. If such flow deflection prevails over a prolonged time, the Mach interaction monotonically shrinks as observed in § 3.1. However, because of the periodic excitation, stable conditions for the MI are recovered before the Mach stem collapses, which prevents $\text{MI}\rightarrow \text{RI}$ transition and allows the Mach stem to grow again. As a result, the mean Mach stem height increases progressively over several periods until a mean steady state identical to that obtained for an initial MI is reached (compare solid and dashed lines in figure 10a). Pressure and entropy measurements included in figures 10(c) and 10(d), respectively, confirm this occurrence and illustrate the characteristic dissipative nature of an MI.

A sequence of instantaneous impressions of the density gradient magnitude corresponding to the first period of oscillation of case D01 is included in figure 11(a–d)(see also Movie 1 available as supplementary material at https://doi.org/10.1017/jfm.2020.28). Time instances are marked as red squares in figure 10(a–d). As the oscillation progresses, an alternation between a convergent and divergent slipline orientation is observed in line with the characteristic streamwise (upstream and downstream) motion of the Mach stem and the associated Mach stem growth. In a similar fashion as discussed in § 3.1 for case P04, a kink in both reflected shocks appears after $\text{RI}\rightarrow \text{MI}$ transition occurs due to the presence of a pressure wave accommodating the pressure mismatch between the RI and the emerging MI; see figure 11(a). It is interesting to note, however, that the steady-state pressure level at detachment is not reached prior to the $\text{RI}\rightarrow \text{MI}$ transition (maximum pressure excursion does not lead to the uppermost dashed blue line in figure 10c). As emphasised in the previous section, this is a consequence of the inability of current steady-state shock polar theory to characterise time-dependent interactions.

Figure 11. Sequence of instantaneous density gradient magnitude for the lower wedge deflection oscillation of $2^{\circ }$ amplitude around $\unicode[STIX]{x1D717}_{2,i}=15.78^{\circ }$ for an initially steady RI. Excitation frequencies correspond to: (a–d) $f_{1}=0.125u_{\infty }/w$; and (e–h) $f_{2}=0.25u_{\infty }/w$. Time instances are marked sequentially as red squares in figures 10(a–c) and 10(d–f), respectively, for $f_{1}$ and $f_{2}$. The solid yellow line denotes the sonic condition.

Our results thus demonstrate that, for mean flow conditions within the steady-state DSD, periodic flow deflection perturbations across the $C_{2}$ incident shock are capable of (i) triggering $\text{RI}\rightarrow \text{MI}$ transition, and (ii) sustain the MI thereafter. This is a result of the Mach stem dynamics during a period of excitation, exhibiting a large initial growth rate that prevents it from collapsing. A sensitivity study on the excitation frequency reveals, however, a change in the shock interaction response as the excitation frequency increases. For an excitation frequency twice as large as the base frequency, $f_{2}=2f_{1}=0.25u_{\infty }/w$ (case D02 in table 2), $\text{RI}\rightarrow \text{MI}$ transition still occurs but the collapse of the Mach stem at the interaction location (and thus transition to RI) follows; see figure 10(e–h). In this case, a slower (less impulsive) Mach stem growth than that for case D01 is measured. As documented in table 2, the maximum growth $(\text{d}h_{ms}^{+}/\text{d}t)/u_{\infty }$ no longer exceeds the corresponding maximum wedge tip speed $M_{tip}/M_{\infty }$ nor the maximum shrink rate $(\text{d}h_{ms}^{-}/\text{d}t)/u_{\infty }$. Thus, the fraction of the excitation period for which the Mach stem shrinks is sufficient to permit its collapse. The resulting $\text{RI}\rightarrow \text{MI}\rightarrow \text{RI}$ alternation observed within the first period of oscillation becomes periodic with the MI manifesting approximately 46 % of the excitation period. A sequence of instantaneous impressions of the density gradient magnitude is included in figure 11(e–h) to illustrate the corresponding flow topology. Time instances of the flow impressions are, respectively, indicated by red squares in figure 10(e–h) during the first period of oscillation. In a similar fashion as for cases P05–P10 considered in § 3.1, the maximum Mach stem shrink rate $\text{d}h_{ms}^{-}/\text{d}t$ is recorded right before $\text{MI}\rightarrow \text{RI}$ transition occurs, which leads to the formation of a similar pressure wave that accommodates the pressure difference between both shock configurations (see figures 11g and 11h).

It is worth mentioning that in cases D04–D06 featuring an initial MI, transition to RI never occurred within the simulated time. In fact, the Mach stem remains in all cases far from collapsing at the interaction location, with the maximum deviation from the mean decreasing with increasing excitation frequency. This clearly shows that a much lower excitation frequency is required to enforce $\text{MI}\rightarrow \text{RI}$ transition than the opposite (Kudryavtsev et al. Reference Kudryavtsev, Khotyanovsky, Ivanov, Hadjadj and Vandromme2002; Khotyanovsky et al. Reference Khotyanovsky, Kudryavtsev and Ivanov2006), and thus suggests that any single-time event allowing the Mach stem to fully develop would lead to a sustained MI thereafter. Results for the excitation frequency $f_{3}$ are only included in table 2 for brevity, as they exhibit qualitatively the same behaviour as those for $f_{2}$.

Figure 12. Evolution of the Mach stem height $h_{ms}$ and lower flow deflection below the $C_{2}$–$C_{4}$ intersection for the lower wedge deflection oscillation of $4^{\circ }$ amplitude around the mean $\unicode[STIX]{x1D717}_{2,i}=13.89^{\circ }$ outside the DSD. Here, $\cdots \cdots$, denotes numerical results for the excitation frequency $f_{1}=0.125u_{\infty }/w$; ——, for $f_{2}=0.25u_{\infty }/w$; and – ⋅ – ⋅ –, for $f_{3}=0.5u_{\infty }/w$, respectively. The start time of oscillation is $t_{0}$, and the time axis is non-dimensionalised with the excitation frequency $f$. Dashed horizontal blue lines highlight the values at detachment (upper) and von Neumann (lower) conditions.

In view of the above, it is furthermore investigated whether a similar periodic disturbance can also sustain an MI for mean flow conditions that are below the steady-state von Neumann boundary, as found in the simulations by Matheis & Hickel (Reference Matheis and Hickel2015) for an SWTBLI at $M_{\infty }=2$. In order to investigate if a similar scenario can be supported by unsteady inviscid simulations, additional computations were conducted for the current baseline conditions together with a lower mean wedge deflection $\unicode[STIX]{x1D717}_{2,i}=13.89^{\circ }$ slightly below the steady-state von Neumann condition ($\unicode[STIX]{x1D717}_{2}^{n}=14.14^{\circ }$, see figure 1c). The excitation frequencies considered correspond to those defined earlier in this section but combined with an amplitude of oscillation of $4^{\circ }$ instead of $2^{\circ }$, such that the maximum instantaneous wedge deflection exceeds the steady-state detachment boundary $\unicode[STIX]{x1D717}_{2}^{d}=17.43^{\circ }$. Corresponding time scales are listed in table 2 for all the excitation frequencies, with the resulting Mach stem height and $\unicode[STIX]{x1D717}_{2}$ evolution over time presented in figures 12(a) and 12(b), respectively. As shown, none of the cases result in a sustained MI over the integrated time. Instead, a periodic $\text{RI}\rightarrow \text{MI}\rightarrow \text{RI}$ alternation develops similar to that obtained for cases D02 and D03.

3.3 Streamwise oscillation of lower wedge without pitch

Table 3. Summary of relevant parameters for the streamwise oscillation of the lower wedge without pitch: $\unicode[STIX]{x1D717}_{2,i}$ corresponds to the fixed wedge deflection in degrees; $A$ denotes the amplitude of oscillation in terms of the wedge hypotenuse $w$ and; subscripts $+$ and $-$ denote, respectively, the maximum and minimum value recorded during the first period of oscillation for which the entropy level characteristic of an MI is reached. See table 1 for additional remarks.

Both excitation mechanisms considered above are based on the variation of the lower wedge deflection around its trailing edge, point O in figure 3, which remained fixed throughout the computations. We now consider the periodic oscillation of the lower wedge location while its deflection remains unaltered. This excitation mechanism may also be seen as a surrogate of the low-frequency motion of the separation shock foot in SWTBLI (Touber & Sandham Reference Touber and Sandham2011). Baseline conditions together with $\unicode[STIX]{x1D717}_{2}=15.78^{\circ }$ are considered in a similar fashion as in § 3.2, with both the RI and MI as starting wave patterns. A starting RI located outside the steady-state DSD with $\unicode[STIX]{x1D717}_{1}=13.89^{\circ }$ is additionally investigated. The motion of the lower wedge is governed by a sinusoidal oscillation of its trailing edge along the lower boundary of the domain with the mean streamwise location equal to the upper wedge trailing edge $x_{ut}$. The excitation frequencies used to characterise the motion also correspond to those defined in § 3.2, and the impact of two different amplitudes of oscillation, $0.05w$ and $0.10w$, is assessed. The former amplitude leads to a maximum streamwise Mach number ratio $M_{tip}/M_{\infty }$ of 0.039, 0.079 and 0.157 for $f_{1}$, $f_{2}$ and $f_{3}$, respectively, whilst those corresponding to the largest oscillation amplitude $0.10w$ are twice as large; see table 3. Oscillation time scales for this case are also defined as in § 3.2 but measured within the first period of oscillation for which the entropy jump through the wave system reaches characteristic values of an MI at $M_{\infty }=3$.

Figure 13. Results for the streamwise oscillation of the lower wedge at an amplitude of $0.05w$ and: (a–d) an initial RI pattern, and (e–h) an initial MI pattern; both starting within the steady-state DSD ($\unicode[STIX]{x1D717}_{2}=15.78^{\circ }$). Here, $\cdots \cdots$, denotes numerical results for the excitation frequency $f_{1}=0.125u_{\infty }/w$; ——, for $f_{2}=0.25u_{\infty }/w$; and – ⋅ – ⋅ –, for $f_{3}=0.5u_{\infty }/w$, respectively. Panels (c) and (f) show the average pressure between measurements taken downstream of both triple points. See caption of figures 10 and 12 for additional remarks.

Results for the $0.05w$ oscillation amplitude are shown in figure 13(a–h) for all excitation frequencies. When the initial shock pattern corresponds to an RI, see figure 13(a–d), all except the lowest excitation frequency trigger transition to an MI. It is interesting to note the particular Mach stem dynamics exhibited in case S03 (dash-dotted lines in figure 13a–d). As shown, an MI configuration is eventually sustained but only after seven periods of oscillation in which the $\text{RI}\rightarrow \text{MI}\rightarrow \text{RI}$ alternation occurs. Such alternation, however, is not periodic (the time interval for which the MI is present increases monotonically over consecutive oscillations) which indicates the presence of hysteresis in the flow (an animation for this particular case, Movie 2, is also available in the supplementary material). The evident asymmetry in the evolution of $\unicode[STIX]{x1D717}_{2}$ at the interaction location (figure 13b) further suggests that the streamwise oscillation of the shock generator results in much more complex $C_{2}$ shock dynamics than the corresponding oscillation of its incidence. This is additionally supported by the larger deviations in flow deflection and static pressure from the theoretical estimates based on a steady-state shock polar analysis. Under these circumstances, transition appears to be governed by the instantaneous $M_{\infty }$–$\unicode[STIX]{x1D717}_{1}$–$\unicode[STIX]{x1D717}_{2}$ combination in the (moving) interaction frame of reference no longer able to accommodate an RI configuration. If such conditions are met, and the Mach stem growth rate is sufficiently large, the MI solution materialises and prevails thereafter. Our results thus demonstrate that low amplitude longitudinal oscillations are also capable of triggering $\text{RI}\rightarrow \text{MI}$ transition, and that they are more effective at retaining the MI pattern for large excitation frequencies than deflection oscillations.

Additional computations with a $0.10w$ oscillation amplitude were conducted for the same excitations frequencies, with the resulting time evolution of the Mach stem height shown in figure 14 for all cases. Numerical data in figure 14(a) involves an initial RI embedded within the steady-state DSD, for the same initial conditions as in the $0.05w$ case, whereas that in figure 14(b) corresponds to an initial MI. As it can be observed, the MI was unambiguously sustained for all perturbations except for the highest frequency of oscillation in figure 14(a). Such a rapid change of boundary conditions prevents the MI from establishing, which leads to a continuous $\text{RI}\rightarrow \text{MI}\rightarrow \text{RI}$ alternation during every period. This same alternation manifests for all amplitudes and excitation frequencies considered when the initial wave pattern corresponds to an RI outside the steady-state DSD ($\unicode[STIX]{x1D717}_{2,0}=13.89^{\circ }$).

Figure 14. Non-dimensional Mach stem height ($h_{ms}/w$) evolution for an amplitude of oscillation of $0.10w$ and: (a) an initial RI pattern; and (b) an initial MI pattern, both embedded within the theoretical DSD ($\unicode[STIX]{x1D717}_{2}=15.78^{\circ }$). Here, $\cdots \cdots$, denotes numerical results for the excitation frequency $f_{1}=0.125u_{\infty }/w$; ——, for $f_{2}=0.25u_{\infty }/w$; and – ⋅ – ⋅ –, for $f_{3}=0.5u_{\infty }/w$, respectively.