4.1. The main features of shock-wave interaction with local constriction

This study compares various local constriction geometries and their effects on flow and shock-wave evolution. The following description outlines the main features observed in our analysis of shock propagation, which are discussed in further detail in the next sections. A schematic description of the transient evolution of the interaction between a shock wave and a local constriction in a conduit is presented in figure 3 for both a smooth constriction (a–d) and for a sharp rectangular constriction (e–h). First, as the incident shock moves into the quiescent air and reaches the constriction at a uniform velocity $M_I$ , it impinges on the constriction. Immediately, the shock forms reflections at the walls, depending on the geometry. In the smooth case (figure 3 a), the reflection always begins as a weak Mach reflection (MR), and as the surface elevation increases, the reflection becomes stronger until reaching the maximum angle at $L/4$ . The maximum slope angle depends on the specific $L$ and $h$ of the model, as shown in table 1. For geometries in which the angle becomes sufficiently large, the reflection will transition to a regular reflection (RR). The transition occurs at an angle that depends on the Mach numbers, which can be calculated for the quasi-steady case (Ben-Dor Reference Ben-Dor2007), but, for the transient case, they differ significantly, with reports placing the transition angle for moderate Mach numbers of 1.2–1.8 at 52 $^\circ$ to 60 $^\circ$ (Ben-Dor Reference Ben-Dor2007; Geva et al. Reference Geva, Ram and Sadot2013). In this study we performed analysis for $M_I =$ 1.4 and 1.8. To allow direct comparison between cases, only $M_I=1.4$ is presented in the figures. The results for $M_I=1.8$ exhibit similar trends to those of $M_I=1.4$ . Reflected and transmitted shock-wave speeds for $M_I=1.8$ are given in figures 13 and 14. In the rectangular-geometry case, figure 3(e), when the shock impinges head-on on the flat 90 $^\circ$ face of the model, the reflected shock forms immediately, and a MR forms between the incident shock and the shock that propagates in the narrow region. In this case, the induced flow behind the shock wave interacts with the sharp upstream-facing corner, leading to immediate flow separation into a growing vortex. We have found that, in the sinusoidal, smooth geometry, the separation of flow begins in later stages, as further discussed in § 4.3. The shock waves that are reflected upstream expand, forming a reflected shock propagating upstream against the flow induced by the incident shock wave.

As the shock passes through the throat of the constriction, it begins to expand, forming a curved expanding transmitted shock. In the case of the sinusoidal geometry (figure 3 b), as long as the shock wave is moving along the surface of the expanding region, it does not form a reflection. However, when the surface angle begins to decrease (i.e. downstream of $3L/4$ ), a MR forms, which persists as the transmitted shock propagates downstream. In the rectangular constriction case, the transmitted shock wave expands along the vertical walls of the downstream expansion and impinges head-on onto the walls of the conduit, as depicted in figure 3( f). Since the angle between the transmitted expanding shock wave and the wall is small, a RR will form initially. Further downstream, the transmitted shock curvature will decrease, increasing the angle between the shock wave and the wall until the shock also transitions to a MR. A full description of this process is presented in Gichon et al. (Reference Gichon, Jose, Chandravamsi, Evron and Ram2024).

As the two primary shock waves, i.e. the transmitted and reflected shocks, continue to propagate away from the constrictions, the MRs will evolve so that the triple point will reverberate between the two walls of the conduit, forming multiple reflections behind them. In essence, the reflection pattern of the transmitted and reflected shock will become quite similar from this point on, as depicted in figures 3(c) and 3(g). Naturally, as the shock waves propagate away from the constriction, the repeated reflections behind them will subside, and the shock fronts will resume their normal, uniform state. The transmitted and reflected shocks will depend on the geometry and will be discussed in detail throughout the following sections. The geometry of the constriction will strongly affect the transient start-up processes within it as the primary transmitted and reflected shocks move away. At first, the reflected shocks from the transmitted and reflected shocks that form close to the constriction propagate back into it and interact with the flow, but these weaker shocks (not shown at all in the schematics of figure 3) subside rapidly. The flow in the constriction continues to evolve as the impulsively driven vortices expand, break up, and regions of flow separation form. During this time, the flow passing through the constriction accelerates, and oblique standing shock waves begin to form. In the sinusoidal geometry, flow separation occurs near the model’s peak height, whereas in the rectangular geometry, it occurs shortly after the separation point at the upstream-facing corner. Later, as the flow in the constriction reaches a more developed state, additional shock waves form behind them, forming a normal shock train. The exact nature of this standing shock wave depends on the geometry, and they typically form downstream of the constriction when the starting vortices break up and shear layers form that bound the flow behind the constriction, as shown in figure 3(d). In rectangular cases, they might first be preceded by a series of reflections from interactions between the first oblique shocks and the wall. In figures 3(d) and 3(h), the standing normal shocks are marked by SS $_2$ .

4.2. Validation of numerical results

The numerical simulations were validated against time-resolved schlieren imaging and wall-pressure measurements obtained using the shock-tube facility. The validation focuses on two representative geometries, both with BR $=0.5$ ( $h=10$ mm) and $\tilde {L}=\,1$ ( $L=40$ mm), representing two distinctly different cases: (i) an abrupt rectangular constriction with sharp corners and immediate separation at the upstream-facing edge and an abrupt expansion at the downstream side of the constriction, and (ii) a smooth sinusoidal constriction for which the post-shock flow remains attached initially and separation develops gradually as the impulsively driven flow accelerates through the contraction. These cases, therefore, test the ability of the solver to capture both the shock morphology and the start-up flow evolution that govern the reflected and transmitted shock waves and the resulting flow within the narrow region.

Figure 4.

Comparison of the LES results against experimental schlieren imaging and wall-pressure measurements for two constriction geometries with $h = 10$ mm (BR = 0.5) and $L = 40$ mm ( $\tilde {L}=1$ ). Panels (a,b) and ( f,g) compare experimental schlieren images with numerical schlieren computed from density gradients at different times. Panels (c, d, e) and (h, i, j) compare LES and experimental wall-pressure signals at three transducer locations: $x = -127.5\,\mathrm{mm}$ (upstream of the constriction), and $x = 42.5\,\mathrm{mm}$ and $x = 357.5\,\mathrm{mm}$ (downstream of the constriction). All pressure traces are shifted to $t = 0$ at the arrival of the incident shock at the most upstream transducer.

Figure 4. Long description

Panel A: Three schlieren images at different times (0.04 ms, 0.1 ms, 0.18 ms) showing experimental results with a constriction geometry. Panel B: Three numerical schlieren images at the same times as Panel A, computed from density gradients. Panel C: Line graph comparing LES and experimental wall-pressure signals at x = -127.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars. Panel D: Line graph comparing LES and experimental wall-pressure signals at x = 42.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars. Panel E: Line graph comparing LES and experimental wall-pressure signals at x = 357.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars. Panel F: Three schlieren images at different times (0.04 ms, 0.1 ms, 0.18 ms) showing experimental results with a different constriction geometry. Panel G: Three numerical schlieren images at the same times as Panel F, computed from density gradients. Panel H: Line graph comparing LES and experimental wall-pressure signals at x = -127.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars. Panel I: Line graph comparing LES and experimental wall-pressure signals at x = 42.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars. Panel J: Line graph comparing LES and experimental wall-pressure signals at x = 357.5 mm. The x-axis represents time in milliseconds, and the y-axis represents pressure in bars.

Figure 4 presents side-by-side comparisons between experimental schlieren images and numerical schlieren fields constructed from density gradients, together with the comparisons of local wall-pressure histories. For the rectangular constriction, schlieren comparisons are shown in figures 4(a) and 4(b), while the pressure histories are shown in figure 4(c–e). For the sinusoidal constriction, the same comparisons are shown in figures 4( f) and 4(g) and in figure 4(h–j), respectively.

The schlieren comparisons in figure 4 show that the computed shock locations and reflection patterns closely match the experimental results at each recorded time. The positions of the reflected and transmitted shocks are captured accurately, and the simulations reproduce the curvature and progressive straightening of the fronts as they propagate away from the constriction. In addition, the simulations capture the structure and evolution of reflections between the transmitted shock waves and the wall, the transition from RR to MR configurations, the trajectories of the triple points and the slip lines. The subsequent evolution of the reflected system, including the downstream motion of the triple points and the development of multiple reflected branches, is reproduced with high fidelity. The computed schlieren fields also capture the formation of vortical structures generated as the shock passes through the constriction, along with the ensuing shock–vortex interactions and the resulting shock pattern. These agreements are observed for both geometries, despite the clear difference between the sharp-cornered rectangular constriction and the smoothly contoured sinusoidal constriction.

The pressure comparisons in figure 4 provide a quantitative validation of both shock strength and timing. The upstream transducer (figures 4 c and 4 h, located at $x$ = −127.5 mm, where $x$ = 0 is set to be the entrance to the constriction) upstream of the constriction captures the incident and reflected shocks. In both geometries, the simulation reproduces the arrival time of the reflected shock and its pressure jump with very good agreement. Moreover, the simulation captures the subsequent pressure dynamics behind the reflected front, including the series of secondary compressions associated with reflections that reverberate between the conduit walls. Downstream of the constriction, the transducer closest to the model (figures 4 d and 4 i, at $x$ = 42.5 mm) captures the transmitted shock and the near-field unsteady processes that occur immediately after the expansion. Farther downstream (figures 4 e and 4 j, at $x$ = 357.5 mm), the pressure signals capture the transmitted shock after it has propagated into the uniform section and after the strongest near-field interactions have subsided. At both downstream locations, the simulation also faithfully predicts the arrival time of the transmitted shock, the magnitude of the primary pressure rise, and reproduces the pressure fluctuations induced by the reflection system that propagates behind it. The agreement is particularly important for the far-downstream and upstream transducers, indicating that the simulation correctly captures both the attenuation of the reflection patterns and the evolution of the transient flow field toward a quasi-steady mean pressure at late times.

Minor discrepancies between simulation and experiment occur at the near-downstream location $x$ = 42.5 mm shown in figures 4(d) and 4(i). This region is very close to the constriction and hence most sensitive to the details of the developing separated shear layers and to three-dimensional turbulence generated by shock–vortex and shock–flow interactions, vortex breakdown and shear-layer formation. The present computations are performed as an implicit LES and therefore do not fully resolve the smallest turbulent scales; accordingly, some differences in the local pressure trace are expected. Importantly, these differences are confined to the immediate vicinity of the constriction and have a small effect on the conclusions drawn from the simulations. The predicted reflected and transmitted shock strengths and their propagation speeds in the uniform upstream and downstream sections, which underpin the quantitative trends and regime distinctions reported below, are accurately captured, as demonstrated by the very good agreement at the upstream and downstream transducers.

Beyond validating shock strength and timing, figure 4 also shows that the simulations reproduce the formation and evolution of the local reflections and shock structures forming within the throat region. In addition, the simulations capture the formation of standing shocks downstream of the constrictions as the jet emerges from them, and the gradual rearrangement of the flow as it reaches steady conditions. Additional validation cases were examined across both shorter and longer models and multiple blockage ratios, and they all exhibit similar agreement: close correspondence in shock locations and reflection topology in schlieren, and accurate prediction of upstream and downstream pressure histories. This establishes the reliability of the simulations for the parametric analysis presented in the subsequent sections. Further validation of the numerical framework, including its ability to capture shock propagation, reflection patterns and pressure evolution in conduits with abrupt area changes, has also been presented in the recent work of Gichon et al. (Reference Gichon, Jose, Chandravamsi, Evron and Ram2024), where the same solver and modelling approach were shown to reproduce experimental measurements with comparable accuracy.

4.3. The transient start-up process at the constriction

Figure 5 presents line integral convolution visualisations and pressure fields obtained from the numerical simulations for a representative set of cases. The figure compares configurations with BR = 0.5 for (a) sinusoidal and (b) rectangular narrow-section geometries of length $\tilde {L}=0.25$ and $\tilde {L}=1$ , highlighting the transient evolution of the flow field within the constriction during the short time interval following the passage of the incident shock.

As described in figure 3, when the shock impinges on the upstream-facing wall of the rectangular constriction, it immediately forms a reflected shock wave that expands upstream while a portion of the shock propagates into the constriction. In the shorter case ( $\tilde {L}=0.25$ ), the shock quickly exits the constriction, expands into the downstream conduit, impinges on the walls and forms an RR. Shortly thereafter, as the shock expands into the downstream region, the reflection patterns transition to an MR. A similar course of events occurs in the longer rectangular case, except that the transmitted shock propagates within the narrow straight region for a longer duration. Moreover, examining the reflected shock pattern shows that its evolution during the first ${\sim}$ 0.1 ms after the reflected shocks exit the field of view is identical in both cases. Figure 5(b) also shows that the flow field that forms within the constriction immediately separates at the sharp corner of the leading edge, forms a small separated region inside the constriction, and a second separation at the trailing edge of the constriction. The large pressure gradient (discussed in detail in the next sections) that develops across the constriction induces the formation of a standing shock wave. Here, we begin to see significant differences between the cases; in the shorter case, the standing shock wave forms downstream of the constriction and is oblique, whereas in the longer case, it forms within the constriction and is a normal shock.

Figure 5.

The transient evolution of the shock–flow interaction within localised constrictions for cases with BR = 0.5, and different normalised lengths, $\tilde {L}=0.25,1$ . For each case, line integral convolution visualisations (a) illustrate the instantaneous flow structure, while the corresponding pressure fields (b) show the spatial pressure distribution.

Figure 5(a) shows that in the case of a sinusoidal constriction, the evolution of the flow field heavily depends on the constriction length, from the very first moments of the interaction. The length of the constriction determines the rate at which the slope of the wall changes and the maximum angle of the wall, which is reached at $L/4$ (see table 1). In the shorter sinusoidal case presented in figure 5(a) (BR = 0.5, $\tilde {L}=0.25$ ), the slope angle increases rapidly and reaches a maximum slope of $\theta _{max}=72^\circ$ . As expected, the reflection pattern that forms is quite similar to the short rectangular case shown in figure 5(b), exhibiting an immediate formation of a strong reflection expanding upstream. However, the gradual slope angle increase and the shallower maximum angle of the second case presented in figure 5(a) ( $\textrm{BR} = 0.5, \tilde {L} = 1$ , $\theta _{max} = 38^\circ$ ) leads to the formation of a weaker MR that is quickly dispersed into a series of pressure waves.

Figure 6(a) presents the shock reflection configurations that develop as the incident shock wave propagates into sinusoidal constrictions of various lengths and heights. In all cases, as the shock traverses the leading edge of the sinusoidal geometry, the smooth transition leads to the initial formation of a MR. In some of the geometries, the maximum slope angle $\theta _{max}$ is high enough to cause transition to a RR configuration. Those cases exhibiting a transition are enclosed by a red outline, whereas those remaining in the MR regime throughout are enclosed by a blue outline. The images in figure 6(a) show the reflection pattern when the shock propagates across the $\theta _{max}$ location. As one would expect, the figure shows that the transient build-up of the reflected shock at the constriction entrance depends strongly on the maximum slope. In the short cases where the slope rises rapidly and reaches high angles, a strong shock forms from the reflection point to the conduit wall. As $\theta _{max}$ decreases, the reflected shock no longer spans all the way to the wall and breaks up into a compression fan. With lower $\theta _{max}$ the reflection is further away from transitioning into a RR, and the reflected shock is weaker; in the lowest slopes presented in figure 6(a) there is virtually no reflected shock forming, and the interaction with the slope reduces to a kink in the shock wave.

Figure 6(b) places these observations against a classical detachment criterion (Ben-Dor Reference Ben-Dor2007). Treating the maximum wall inclination $\theta _{max}=\tan ^{-1}(\pi \tilde {h}/\tilde {L})$ as the reflecting-wedge angle of the incident shock, the pseudo-steady detachment criterion of two- and three-shock theory predicts the transition from MR to RR at $\theta _{max}\approx 48^\circ$ for $M_I=1.4$ , which maps to the straight boundary $\tilde {h}=(\tilde {L}/\pi )\tan \theta _{tr}$ separating the shaded regions. The criterion reproduces the observed ordering, with the short, strongly blocked geometries in the RR region and the long, gently sloped geometries in the MR region. The observed transition lies at a slightly smaller angle than the inviscid pseudo-steady value, consistent with a smooth, continuously turning wall (Ben-Dor et al. Reference Ben-Dor, Takayama and Kawauchi1980; Geva et al. Reference Geva, Ram and Sadot2013), for which the transition is non-stationary and lags the self-similar wedge result by an amount set by the local rate of turning. The single case near the boundary ( $\tilde {L}=1$ , $\tilde {h}=0.3$ , $\theta _{max}=43^\circ$ ) reflects this offset.

Figure 6.

(a) Snapshots of the incident shock wave as it propagates across the upstream side of the sinusoidal constrictions. The images show the shock as it passes $\theta _{max}$ . Cases that were identified to produce a RR are enclosed by a red outline, while those in which a MR are marked by a blue outline; (b) RR/MR phase diagram in the $(\tilde {L},\tilde {h})$ plane. The solid line is the pseudo-steady detachment-criterion transition at $M_I=1.4$ , $\tilde {h}=(\tilde {L}/\pi )\tan \theta _{tr}$ with $\theta _{tr}\approx 48^\circ$ , separating the RR region (red) from the MR region (blue).

The effects of the shock interaction with the upstream side of the constriction can be clearly seen in the time evolution of the pressure field presented in figure 7, which shows the instantaneous normalised pressure distribution along the conduit centreline in the vicinity of the constriction. The figure presents LES results for both rectangular and sinusoidal cases with a blockage ratio of BR = 0.5 and various lengths following the impingement of an incident shock wave with $M_I=1.4$ . The profiles capture the developing reflected and transmitted shock systems, as well as the pressure evolution within the constriction shortly after being impulsively driven by the incident shock wave.

Figure 7.

Centreline pressure distributions in the vicinity of the constriction during the early time following the impingement of the incident shock wave with $M_I = 1.4$ . Profiles are shown over the domain $\tilde {x}=-2.5$ to 4 at three successive times: (a) $t = 0.1\,\mathrm{ms}$ , (b) $t=0.2\,\text{ms}$ and (c) $t = 0.3\,\text{ms}$ . The results compare sinusoidal and rectangular constrictions with identical blockage ratio and three lengths ( $\tilde {L}=0.25,\ 1,\ 2$ ).

Figure 7. Long description

The image contains three line graphs labeled (a), (b), and (c), each showing pressure distributions over a domain from -2.5 to 3.75. The y-axis represents normalized pressure (p) ranging from 1 to 4. Panel A shows the pressure distribution at an early time after the shock wave impinges on the constriction. Panel B shows the distribution at a later time, and Panel C shows it at an even later time. Each graph compares sinusoidal and rectangular constrictions with identical blockage ratios and three different lengths (L = 0.25, 1, and 2). The lines are color-coded: yellow for L = 0.25 sin, red for L = 1 sin, green for L = 2 sin, blue for L = 0.25 rec, purple for L = 1 rec, and black for L = 2 rec. The graphs illustrate how the pressure distributions evolve over time and how different constriction geometries affect the pressure profiles.

At the earliest time shown ( $t=0.1\,\mathrm{ms}$ ), the pressure distribution generated by all rectangular cases is virtually identical, indicating minimal effects of the constriction length in early times. Conversely, in the case of a sinusoidal geometry, differences in pressure distribution occur immediately, in accordance with the different reflection configuration as described above. Figure 7 shows that for the shortest sinusoidal model ( $\tilde {L}=0.25$ ), a strong reflection occurs that is identical to the rectangular cases, but the transmitted shock wave is significantly reduced in strength. The longer case of $\tilde {L}=1$ creates a weaker reflected shock wave and a stronger transmitted shock wave, while the longest case with $\tilde {L}=2$ does not form a reflected shock wave at this time, but the transmitted shock wave is strong and still propagating within the constriction.

As time progresses, the reflected shock waves from the rectangular geometries remain approximately constant in strength, indicating that reflected shock strength remains independent of the constriction length. However, the transmitted shock waves begin to exhibit geometric effects arising from differences in reflection patterns and flow fields within the constrictions. For the sinusoidal cases, figures 7(b) and 7(c) show that, as the reflected shock waves build up to a normal shock, their strength increases, where the lower slopes are lagging behind. For the case of $\tilde {L}=2$ , we see that at time $t=0.3\,\rm ms$ , although the pressure waves propagating upstream steepen, they have not yet formed a shock wave. The multiple pressure jumps observed within and downstream of the constriction arise from repeated reverberations between the evolving reflection structures and the impulsively accelerated, separating flow within the constriction, as also evidenced by the separation and shear-layer development discussed in figure 5.

The reflection topology described above, however, explains only the upstream side of the interaction. The pathway by which the flow within the constriction itself develops toward its eventual configuration exhibits an equally strong, and less intuitive, dependence on the constriction length. Figure 8 examines the transient evolution of the flow within the throat during and immediately after the passage of the shock, overlaying numerical schlieren images with maps of the locally supersonic regions for four sinusoidal constrictions with BR = 0.5 and $\tilde {L}=0.25,\ 0.5,\ 0.75$ and $1$ .

In the longest case shown, $\tilde {L}=1$ , the impulsively accelerated flow initially remains attached to the diverging wall, as evidenced by the absence of separated structures in the schlieren images at early times. As the flow accelerates through the throat, a supersonic region first emerges from the wall itself at approximately $0.13\,\mathrm{ms}$ , visible in figure 8 as a supersonic pocket anchored to the diverging contour. This region grows and is terminated by a standing shock that assumes a lambda configuration at its foot. It is only through the adverse pressure gradient imposed by this shock that the boundary layer separates, and a detached vortex forms downstream of the lambda foot. In this pathway, therefore, the standing shock is the cause of separation: the transonic flow becomes supersonic while still attached, and separation follows as a consequence of the shock formation and growth.

Figure 8.

The transient evolution of the shock–flow interaction within sinusoidal constrictions with BR = 0.5 and normalised lengths $\tilde {L}=0.25,\ 0.5,\ 0.75,\ 1$ . The images present numerical schlieren from numerical simulations overlaid by corresponding maps of the local Mach number thresholded at $M=1$ , with supersonic regions shown in red, highlighting the geometry-dependent onset and location of the standing shock system.

In the shortest case, $\tilde {L}=0.25$ , the order of events is reversed. The schlieren images show that the flow detaches from the crest of the constriction almost immediately after the shock passes, well before any significant supersonic region appears, shedding vortices that bound the emerging jet. The subsequent acceleration to sonic conditions occurs not along the wall but within the core of this separated jet, as shown in figure 8, where the supersonic pocket is confined between the shear layers and detached from the geometry. The standing shock then forms at a later stage as a normal shock spanning the jet core between the two shear layers. Here, separation precedes and shapes the supersonic region: the effective aerodynamic contour is set by the separated shear layers rather than by the solid wall, and the standing shock forms within the flow structure they enclose.

The two intermediate lengths, $\tilde {L}=0.5$ and $0.75$ , show that these mechanisms are not mutually exclusive. In these cases, partial separation from the contour and wall-attached supersonic acceleration develop concurrently, and the resulting shock system carries signatures of both pathways, with supersonic pockets appearing both near the wall and within the developing jet. The distinction between the two limiting behaviours is therefore blurred at intermediate lengths, and the flow evolves through a mixed process in which neither mechanism clearly dominates.

The significance of this observation extends beyond the phenomenology of the start-up process. Figure 8 demonstrates that there are two fundamentally distinct paths by which the flow in the constriction approaches its steady configuration, one governed by shock-induced separation of an attached transonic flow and the other by the transonic acceleration of an already separated jet. Although both paths ultimately terminate in a quasi-steady choked state, the intermediate dynamics does not follow the same physical process and cannot be described by a common reduced-order framework. Modelling the transient start-up and predicting the final standing shock configuration therefore require different strategies depending on which pathway the flow follows, and at present there is no a priori criterion for determining which mechanism will govern a given combination of blockage ratio and constriction length.

Despite these divergent pathways, all cases share a common endpoint. The early-time reflection dynamics and the separation pathway jointly govern the flow initiation time in the constriction. Eventually, as the reflected and transmitted shock waves propagate away from the constriction, the strong reflections within the constriction will subside, and the flow will reach steady conditions on the upstream and downstream sides, marked by states 2 and 3 in figure 1, respectively. As this process proceeds, the reflected and transmitted shocks evolve toward well-defined steady propagation, providing the basis for examining their late-time strengths.

To highlight the temporal effects of geometry on the flow starting time, figure 9 shows the temporal evolution of the centreline pressure at the entrance to the constriction ( $x = 0$ ) for the sinusoidal geometries. The pressure is normalised by $P_2$ . Note that since $P_2$ depends on the blockage ratio, the normalised starting pressure at $t = 0$ differs across cases, whereas in reality it is the same for all ( $P_1$ ). This figure shows the time required for the pressure at the constriction entrance to reach its late-time steady level.

Figure 9.

Time histories of the normalised centreline pressure at the entrance to the constriction, $x = 0$ , for sinusoidal constrictions. The pressure is normalised by $P_2$ , the late-time pressure immediately upstream of the constriction (state 2). Each panel corresponds to a fixed blockage ratio BR, and compares three lengths $\tilde {L} = 0.25,\ 1,\ 2$ . The traces are shown after the passage of the incident shock (state 1), and illustrate the duration of the start-up process required for the flow to approach its late-time pressure behind the reflected shock wave $P_2$ . The vertical line in each panel marks an estimated end of the transient, determined by visual inspection.

Figure 9. Long description

The image contains four line graphs labeled (a), (b), (c), and (d). Each graph shows the time histories of the normalized centerline pressure at the entrance to a constriction for sinusoidal constrictions. The pressure is normalized by the late-time pressure immediately upstream of the constriction. Each panel corresponds to a fixed blockage ratio (BR) and compares three different lengths (L). The x-axis represents time (t) in milliseconds (ms), and the y-axis represents the normalized pressure (p). Panel (a) shows data for BR = 0.35 with lengths L = 0.25, 1, and 2. Panel (b) shows data for BR = 0.5 with lengths L = 0.25, 1, and 2. Panel (c) shows data for BR = 0.6 with lengths L = 0.25, 1, and 2. Panel (d) shows data for BR = 0.7 with lengths L = 0.25, 1, and 2. Each graph illustrates the duration of the start-up process required for the flow to approach its late-time pressure behind the reflected shock wave. The vertical line in each panel marks an estimated end of the transient, determined by visual inspection.

Figure 9 shows that more constricted geometries lead to shorter starting times for all lengths, and that for a given blockage ratio, sharper geometries produce stronger reflections that further reduce build-up times but increase pressure fluctuations. While this might seem surprising at first, it can be intuitively explained by the fact that shock waves moving through the constriction induce rapid changes in the flow configuration, thereby promoting faster adjustments of the flow to the geometry. Vertical black lines are plotted on figure 9 to show the time at which the pressure within the longest constrictions, i.e. $\tilde {L} = 2$ , roughly reaches steady-state pressure. Because the signals exhibit weak residual oscillations, an exact settling time cannot be accurately defined. We therefore estimate the end of the transient by visual inspection. The start-up process within the narrowing occurs on time scales that are approximately one to two orders of magnitude longer than the shock passage time. For $\tilde {L}=2$ , the $M_I$ = 1.4 shock wave passes through the constriction in ${\sim} 165\,\rm \mu \mathrm{s}$ and for $\tilde {L}=0.25$ the shock passes through the constriction in just ${\sim} 20\,\mu \mathrm{s}$ . The largest discrepancies occur for the least constricted cases (BR = 0.35), where the flow field remains transient with strong fluctuations for as much as ${\sim} 650\, \mu \mathrm{s}$ . As upstream processes subside, the flow field in the constriction reaches a relatively stable state, in which the pressure gradient across the constriction corresponds directly to the strengths of the reflected and transmitted shock waves, as further examined in the following section.

4.4. The late-time evolution of the reflected and transmitted shock waves

4.4.1. Rectangular constrictions

Figure 10.

(a) Numerical schlieren images at $t = 1\,\mathrm{ms}$ after the initial interaction of the incident shock with rectangular constrictions. Results are shown for four blockage ratios, BR = 0.35, 0.5, 0.6, 0.75, and three constriction lengths, $\tilde {L}=$ 0.25, 1, 2. (b) Pressure distribution along the centreline of the computational domain. The pressure profiles summarise the combined effects of blockage ratio and constriction length on the evolution of the transmitted and reflected shock waves for the rectangular geometries following the impingement of a $M_I$ = 1.4 shock wave. A magnified view of the transmitted shock front is shown in the inset.

Figure 10 summarises the late-time flow structures and centreline pressure distributions, 1 ms after the initial shock interaction with the upstream face of the rectangular constrictions. The figures show the steady-state flow conditions inside the constriction after both the transmitted and reflected shock waves have propagated away from it. Twelve representative cases are shown, corresponding to four blockage ratios (BR = 0.35, 0.5, 0.6, 0.75) and three constriction lengths ( $\tilde {L}$ = 0.25, 1, 2). Figure 10(b) shows the corresponding pressure distribution along the centreline of the computational domain, spanning a $20H$ section of the tunnel from $\tilde {x}=-7.5$ to $\tilde {x}=12.5$ .

As described above, in all rectangular-geometry cases, the initial shock–constriction interaction is qualitatively identical. When the incident shock impinges head-on on the upstream-facing vertical wall of the constriction, it immediately forms normal reflected shock sections that propagate back upstream. Simultaneously, the portion of the incident shock that passes through the opening continues into the constriction and propagates downstream, where it expands through a sharp $90^\circ$ corner. Since the inlet and outlet geometries of all the rectangular constrictions are the same, we see minor effects of the geometry length in late times, and the differences observed in figure 10 are driven primarily by the blockage ratio, which determines the late-time pressure gradient on the constriction. The schlieren images in figure 10(a) show that the shock configuration that develops inside the constriction after the initial start-up process has subsided is highly affected by the geometry. In all cases, the flow separates immediately at the sharp upstream-facing corners of the constriction, creating an effective contraction region. In all cases presented in figure 10 the flow through the constriction becomes choked, and as a consequence, shock waves form within the constriction. In some cases, we observe an oblique shock wave forming very close to the leading edge where the separated flow reattaches. This will occur if the constriction is significant, leading to a build-up of sufficient gradients across the constriction. However, if the constriction becomes longer, the resulting significant resistance to flow will slow it down at the entrance and delay the formation of a shock wave. In this case, the shock forms downstream, after the flow has adjusted to the geometry of the narrow region, resulting in a standing normal shock wave. Figure 10(a) shows that when BR = 0.35, normal shocks form in all cases, but as the blockage increases to cases where BR = 0.5, in the shorter cases $\tilde {L}$ = 0.25 and $\tilde {L}$ = 1 oblique shocks are formed, but not for $\tilde {L}$ = 2. For cases with $\tilde {L}$ = 2, we only see clear oblique shocks form in the most constricted case where BR = 0.75.

Examining the pressure distributions presented in figure 10(b) shows that, for a given blockage ratio, the reflected shock structure is largely insensitive to the length of the constriction. Again, highlighting that the initial shock–constriction interaction is fundamentally identical across all rectangular geometries with the same blockage ratio, the reflected shock strength depends primarily on the blockage ratio rather than on $\tilde {L}$ . This behaviour is expected and consistent with physical intuition, as the reflected shock is generated immediately by the head-on interaction of the incident shock with the upstream-facing vertical wall of the constriction. The agreement across different lengths further reinforces that the numerical simulations capture the underlying physics consistently and in a physically meaningful manner.

As expected, the transmitted shock shows a stronger dependence on both the blockage ratio and the constriction length, with the most open configurations generating the strongest transmitted shocks and increasing blockage leading to weaker downstream propagation. Superimposed on this trend, however, is a clear and systematic influence of constriction length. For a given blockage ratio, shorter constrictions consistently produce weaker transmitted shocks than longer ones. The magnified inlay plot in figure 10(b) shows the difference in the transmitted shock fronts more clearly. This effect is most evident in the more constricted configurations and can be understood by examining flow evolution within the constriction, as revealed by the schlieren images. Since longer constrictions allow better pressure matching between upstream and downstream, with fewer shocks, the flow exits the constriction as a stronger, more coherent jet, thereby supporting a stronger transmitted shock downstream.

The reflected shock-wave strength does not depend on the length of the narrow rectangular section, yet small differences are observed in the pressure field trailing the reflected shock as it propagates upstream. These differences become more pronounced as the blockage ratio increases. These differences can be associated with the interaction between the shock structures forming within the constriction and the reflected shock as it builds up and propagates upstream. In the most open cases, weaker reflections lead to a slower build-up of a fully developed normal reflected shock, allowing additional interaction between the internal shock system and the upstream-propagating reflected wave. This interaction influences the subsequent shock train that develops behind the reflected front. Aside from these minor variations, the reflected shock speed and the reverberations trailing it are governed almost exclusively by the blockage ratio.

4.4.2. Sinusoidal constrictions

Figure 11.

(a) Numerical schlieren images at $t = 2\,\mathrm{ms}$ after the initial interaction of the incident shock with sinusoidal constrictions. Results are shown for four blockage ratios, BR = 0.35, 0.5, 0.6, 0.75, and three constriction lengths, $\tilde {L}=$ 0.25, 1, 2. (b) Pressure distribution along the centreline of the computational domain. The pressure profiles summarise the combined effects of blockage ratio and constriction length on the evolution of the transmitted and reflected shock waves for the sinusoidal geometries following the impingement of a $M_I$ = 1.4 shock wave. A magnified view of the transmitted shock front is shown in the inset.

Figure 11. Long description

Panel A: A series of numerical schlieren images show the interaction of an incident shock with sinusoidal constrictions. The images are organized in a grid with varying blockage ratios (BR = 0.35, 0.5, 0.6, 0.75) and constriction lengths (0.25, 1, 2). Each row represents a different blockage ratio, and each column represents a different constriction length. Panel B: A line graph displays the pressure distribution along the centerline of the computational domain. The x-axis represents the position along the centerline, and the y-axis represents the pressure. Multiple lines, each in a different color, represent different combinations of blockage ratios and constriction lengths. The legend indicates the specific combinations of blockage ratio and constriction length for each line. An inset provides a magnified view of the transmitted shock front.

Figure 12.

A comparison of the centreline pressure distribution comparing the late-time shock wave between sinusoidal and rectangular constrictions after the impingement of an $M_I=1.4$ shock wave, $t=1$ ms after the shock enters the constriction.

Figure 11 compares numerical schlieren images and centreline pressure profiles for shock interaction with sinusoidal constrictions, using the same format as figure 10 for the rectangular geometries. Results are shown at $t = 2\,\text{ms}$ , after the flow within the constriction has reached a quasi-steady configuration and the dominant transients associated with the initial shock–geometry interaction have largely subsided. At these late times, the overall flow topology resembles that of the rectangular cases, but the smoother contraction and expansion in the sinusoidal geometries modify the internal shock structure. For the short constrictions ( $\tilde {L}=0.25$ ), no standing shocks form within the constriction. Instead, the shock pattern appears only downstream, within the core of the emerging jet. For longer constrictions ( $\tilde {L}=1$ ), standing shocks develop within the constriction, and the configuration depends on the blockage: a nearly normal shock is observed for the most open case, whereas more constricted cases exhibit an oblique shock system. Similar trends persist for $\tilde {L}=2$ , in contrast to the rectangular geometries, where the internal shock configuration shows a stronger tendency toward a nearly normal shock within the constriction.

Examining the centreline pressure traces in figure 11(b) shows that the transmitted shock strength is primarily governed by the blockage ratio, consistent with the rectangular cases. For the sinusoidal constrictions, however, the dependency on geometric details, both in the shock arrival time and in the pressure level behind the transmitted front, is noticeably weaker. This is consistent with the schlieren fields at late times, which indicate that the internal flow configuration is broadly similar across cases at a given blockage ratio. In addition, the smoother contraction and expansion allow the transmitted wave to exit the constriction more gradually, resulting in only minor differences in the downstream evolution. However, comparing the pressure plots in figure 11(b) with the rectangular cases reveals that, in the sinusoidal cases, the reflected shock wave is highly affected by the geometry. This effect is significant in all of the cases we have examined. In essence, these differences stem from the fact that the reflected shock wave does not form immediately upon its impingement on the constriction, as would have been the case in the rectangular geometry. This trend is further illustrated in figure 12, which directly compares the centreline pressure profiles for all six rectangular and sinusoidal cases with BR = 0.5 and $\tilde {L}$ = 0.25, 1 and 2. The comparison shows that rectangular geometries produce the strongest reflected shocks, thereby providing an upper bound on their strength for a given blockage ratio. In all sinusoidal cases, the reflected shock is weaker, consistent with the smoother contraction that reduces the impulsive compression at the entrance. Within the sinusoidal cases, the reflected shock strength decreases as $\tilde {L}$ increases, indicating that a longer, more gradual constriction further weakens the upstream compressive response. Hence, it is clear that there is a strong coupling between the short-time evolution of the shock propagation through the constriction and the build-up of transmitted and reflected waves.

Taken together, the results presented in this section reveal that the interaction between a shock wave and a localised constriction is governed by a highly complex and geometry-dependent transient start-up process, involving separation, vortex formation and multiple shock–structure interactions that evolve over time scales significantly longer than the shock passage itself. This early-time dynamics differs markedly between abrupt and smoothly contoured geometries and strongly influences the pathway by which the flow adjusts to the constriction. However, despite these pronounced differences in transient behaviour, the results consistently show that the final, late-time reflected and transmitted shock strengths collapse onto well-defined trends that depend primarily on the blockage ratio, with secondary sensitivity to geometric details.

This apparent contradiction highlights an important point: although the short-time evolution depends on the detailed physics of shock–geometry interactions, it does not provide a universal description of the system. A complete, predictive understanding of the transient dynamics would require resolving the full flow field using high-fidelity numerical simulations as done here. Nevertheless, the systematic behaviour observed at late times suggests that the essential outcome of this complex transient process can be represented through reduced-order descriptions.