3.1. Square interface

Figure 3 presents the experimental results of the square block impacted by a planar shock wave ( $M=1.17$ ). When the planar incident shock (IS) makes a head-on collision with the left surface of the volume, a regular refraction occurs because the incident angle ${\it\theta}$ is zero, forming a planar transmitted shock (TS) inside. As the incident shock propagates along the upper (or lower) interface of the polygon, an irregular refraction occurs where a Mach stem (MS) nearly normal to the interface is generated outside ( $t=91~{\rm\mu}\text{s}$ ). Meanwhile, the refracted shock (RS) is formed inside. The shock–shock interaction between the transmitted shock and the refracted shock induces the formation of new shocks (S1 and S2) and triple points (TP) inside the volume. Shortly after the incident shock impact, a small upstream vortex pair (UVR) is formed due to the baroclinic vorticity generation and deposition on the upper and lower interfaces. When the Mach stem propagates along the rightmost surface, it is then called the diffracted shock (DS). The diffracted shock refracts at the interface, making the wave patterns inside complicated ( $t=186~{\rm\mu}\text{s}$ ). As time proceeds, the upstream vortex pair grows constantly and a small downstream vortex pair emerges. Note that in our previous work of $\text{SF}_{6}/\text{N}_{2}/\text{SF}_{6}$ configuration (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), only one vortex pair is formed from the upstream corners and no obvious vortex pair is generated from the downstream corners. Subsequently, the interaction between two diffracted shocks happens at the outside of the downstream interface and results in a zone with higher pressures which force the downstream boundary moving inwards ( $t=282~{\rm\mu}\text{s}$ ). This phenomenon was also found in the interaction of a shock and a heavy bubble (Zhai et al. Reference Zhai, Si, Luo and Yang2011). Shortly afterwards, an outward jet, driven by the peak pressure caused by the complex interaction among shocks from the upper and lower half-planes inside the volume in the vicinity of the downstream boundary, is observed ( $t=401~{\rm\mu}\text{s}$ ) and grows with time. It is found that the pressure perturbation caused by the shock–shock interaction in heavy inhomogeneity plays an important role in the interface morphology. However, in the light polygons (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), no significant effect of pressure perturbation on interface morphology is observed because the transmitted shock moving faster than the other shocks is nearly unperturbed. As time elapses, the vortex pairs develop to large scales ( $t=877{-}1306~{\rm\mu}\text{s}$ ), intensifying the mixing between the two gases. Note that after the upstream interface passes through the thin pins located at the downstream corners, two bulges are found due to the effect of the thin pins. For light inhomogeneities (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), however, the thin pins seem to have limited influence on interface morphology, in which the bulge may be compensated by vortex rotation.

3.2. Rectangular interfaces

Two kinds of rectangle, named streamwise-rectangle and transverse-rectangle, with the same dimension but different orientations, as indicated in figure 1, are considered in the experiments. The corresponding interface developments are presented in figures 4 and 5, respectively.

For these two rectangles, the wave patterns at the early stage are similar to the square case and are therefore omitted here. As stated in the square case, the shock–shock interaction happens almost at the downstream interface, resulting in the formation of an outward jet. For the streamwise-rectangle, due to the large width–height ratio ( $L_{0}:h_{0}=2:1$ with $L_{0}$ and $h_{0}$ being the initial width and height, respectively), the interaction between the shocks from the upper and lower half-planes occurs almost at the volume centre ( $t=219~{\rm\mu}\text{s}$ in figure 4) and then these shocks separate from each other. Consequently, high pressures are produced almost at the volume centre and then diffuse with time. When the shocks approach the downstream interface, the pressures caused are not high enough and no obvious outward jet is observed ( $t=314{-}552~{\rm\mu}\text{s}$ in figure 4). For the transverse-rectangle, however, two small outward jets are observed at the downstream interface ( $t=361~{\rm\mu}\text{s}$ in figure 5). These two jets are driven by the pressures resulting from the interaction between the refracted shock, the transmitted shock, and the S1 and S2 at the upper or lower half-plane in the vicinity of the downstream boundary ( $t=218~{\rm\mu}\text{s}$ in figure 5). Due to the small width–height ratio ( $L_{0}:h_{0}=1:2$ ), the shocks from the upper half-plane cannot intersect with those from the lower half-plane when they meet the downstream boundary ( $t=147~{\rm\mu}\text{s}$ and $t=218~{\rm\mu}\text{s}$ in figure 5); the pressures generated in this case are lower than that produced in the square case and therefore the jets are smaller ( $t=504~{\rm\mu}\text{s}$ in figure 5). For these two rectangles, similar to the square, the upstream vortex pair dominates the flow at late stages ( $t=791{-}1505~{\rm\mu}\text{s}$ in figure 4 and $t=670{-}1051~{\rm\mu}\text{s}$ in figure 5). Unlike the square and the transverse-rectangle, where only one visible downstream vortex pair is produced, a second downstream vortex pair (SDVR) is generated in the vicinity of the downstream corners and gradually becomes prominent ( $t=1148~{\rm\mu}\text{s}$ ) for the streamwise-rectangle. The formation of the second downstream vortex pair may be attributed to the interaction of the shocks inside the volume with the interface. Moreover, because of the larger width, the upstream vortex pair has little influence on the second downstream vortex pair, allowing the latter to grow with time.

The evolution of a rectangular $\text{SF}_{6}$ block after shock acceleration was previously investigated experimentally and numerically by Bates et al. (Reference Bates, Nikiforakis and Holder2007). In their experiment, the rectangular block was formed by the microfilm membrane with the aid of wire meshes and profiled windows, and the flow was visualized by means of laser sheet illumination of the seeded $\text{SF}_{6}$ gas using a copper vapour laser. Comparing the present work with their results, qualitative agreement is achieved for both the wave pattern (the formation of a Mach stem and triple point inside the volume) and the interface morphology (the formation of the vortexes from the left-top and right-top corners). However, in their interface configuration, only the top interface is allowed to evolve after the shock impact and the interface is actually a continuous one since it is generated by virtue of the differing densities of the two gases. The characteristic of the continuous interface and the presence of the wire meshes may inhibit the interface motion and subsequently reduce the growth rate of the interface. Moreover, the shock refraction only happens on one side, decreasing the complexity of the shock–shock interaction inside the volume. From the present results, one can show that changes in width–height ratio will result in changes in shock behaviour and subsequently for interface morphologies.

3.3. Triangular interfaces

Figure 6 shows typical images of a planar shock wave ( $M=1.19$ ) interacting with the forward-triangle. Due to the direct transmission via the upstream cusp, the front of the transmitted shock inside the volume is so short that the refracted shocks from the two upstream oblique interfaces almost connect directly with each other ( $t=90~{\rm\mu}\text{s}$ ). After the incident shock passes through the downstream corners, shock diffraction occurs ( $t=185{-}233~{\rm\mu}\text{s}$ ) and a small downstream vortex pair can be found. Afterwards, a small jet is produced at the downstream interface centre, followed by shock transmission at the downstream interface ( $t=281{-}376~{\rm\mu}\text{s}$ ). The jet, similar to the square or the transverse-rectangle, may be induced by the high pressures resulting from the shock–shock interaction inside the volume. As time elapses, the downstream vortex pair and the upstream vortex pair grow gradually ( $t=685~{\rm\mu}\text{s}$ ). Moreover, the two initially oblique interfaces evolve, accompanied by the generation of many small vortical structures. At late stages, these small vortical structures become apparent and the vortex pair increases to a large scale ( $t=971{-}1257~{\rm\mu}\text{s}$ ). It is found that the interface morphology is completely different from what we have observed in the light forward-triangle (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), in which the remarkable feature is the formation of an inward jet from the upstream cusp and the eventual penetration of the jet through the downstream interface.

The process of a shock ( $M=1.19$ ) interacting with the backward-triangle is illustrated in figure 7. Analogous to the rectangular cases, the planar transmitted shock is first formed when the incident shock collides with the leftmost interface. The diffracted shock is subsequently generated when the incident shock propagates along the oblique interface and refraction of the diffracted shock at the oblique interface also occurs ( $t=82~{\rm\mu}\text{s}$ ). Under the action of the shock waves, the oblique interfaces deform slightly and a small upstream vortex pair is developed ( $t=82~{\rm\mu}\text{s}$ ). As the diffracted shock travels forwards, the directly transmitted shock becomes shorter and two S2 meet each other ( $t=177~{\rm\mu}\text{s}$ ). After the primary shocks leave the evolving interface ( $t=320~{\rm\mu}\text{s}$ ), the interaction among the shocks inside the volume drives the formation of an outward jet ( $t=439~{\rm\mu}\text{s}$ ). Note that the interaction of the diffracted shocks outside the volume near the apex also results in a high pressure zone, which restrains the rightward movement of the interface. Subsequently, the upstream vortex pair quickly exceeds the downstream interface, resulting in a hollow zone at the downstream cusp ( $t=630{-}797~{\rm\mu}\text{s}$ ). As time goes on, the upstream vortex pair increases to a large scale ( $t=1011{-}1320~{\rm\mu}\text{s}$ ) and ultimately dominates the flow field.

3.4. Diamond interface

A set of experimental schlieren photographs of a planar shock wave ( $M=1.19$ ) interacting with the diamond is presented in figure 8. The diamond can be considered as a combination of the forward-triangle and the backward-triangle volumes. For the upstream interfaces of the diamond, the wave patterns and interface morphology are similar to that of the forward-triangle ( $t=67{-}306~{\rm\mu}\text{s}$ ). For the downstream interfaces, the refracted shocks generated, respectively, from the incident shock and the diffracted shock interact with each other ( $t=163~{\rm\mu}\text{s}$ ), which is different from that in the backward-triangle where the refracted shock directly disturbs the transmitted shock. In addition, the interface morphology also behaves differently ( $t=567~{\rm\mu}\text{s}$ ), which may be attributed to the influences of the shock waves and the interface scales. As in other cases, small vortex pairs are generated at the cusp and the two apexes, and develop with time ( $t=234{-}1496~{\rm\mu}\text{s}$ ) owing to the vorticity production and deposition.

4.1. Wave patterns

Shock refraction at a fast/slow interface was systematically investigated in Abd-el Fattah & Henderson (Reference Abd-el Fattah and Henderson1978a ), where the wave patterns of regular refraction with a reflected shock (RRR) and irregular refraction of the Mach reflection type (MRR) were generally observed. Compared with the slow/fast case (Abd-el Fattah & Henderson Reference Abd-el Fattah and Henderson1978b ), the wave patterns of shock refraction at a fast/slow interface are less complicated. However, when a shock collides with a closed volume, the situation may be different. In our previous work of the light inhomogeneity (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), the complicated wave patterns occur outside the closed volume while the wave patterns inside the volume are quite simple. In the present study, in contrast, the wave patterns outside the volume are simple while the interaction of shocks inside the volume increases the complexity of the wave patterns.

For the forward-triangle, when the incident shock propagates along the inclined interface ( ${\it\theta}=60^{\circ }$ ), there is the regular group (RRR) which is characterized by a well-defined refraction point from which all the waves radiate along the straight rays, as presented in figure 9(a). The interaction of the refracted shock and the transmitted shock results in the formation of the S1 and the triple point inside the volume.

Figure 9. Wave patterns of a shock interacting with the forward-triangle at $t=90~{\rm\mu}\text{s}$ (a) and with the square at $t=91~{\rm\mu}\text{s}$ (b): ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ , the angles of the trajectories of the two triple points with the initial horizontal interface; ${\it\delta}$ , the angle of the refracted shock front with the initial horizontal interface. ISB and ISC are the incident shock along and away from the interface, respectively.

Figure 10. Wave patterns of a shock interacting with the backward-triangle at $t=82~{\rm\mu}\text{s}$ (a) and $t=177~{\rm\mu}\text{s}$ (b). The symbols are the same as those in figure 9.

Figure 11. Displacement histories of the incident shock along and away from the interface (ISB and ISC), the transmitted shock, and the first and second triple points (TP1 and TP2), for the square (a), streamwise-rectangle (b), transverse-rectangle (c), forward-triangle (d), backward-triangle (e) and diamond (f). Here ‘ISB-up’ and ‘ISB-down’ denote the incident shock along the upstream and downstream interfaces respectively for the diamond. The straight lines are from the linear fitting and all the quantities are measured based on their own reference points.

For the square, as the incident shock moves along the upper surface ( ${\it\theta}=90^{\circ }$ ), a Mach stem, nearly normal to the interface, intersects the incident shock and the reflected shock at a triple point above the interface, as shown in figure 9(b). Such refraction is referred to as MRR (Abd-el Fattah & Henderson Reference Abd-el Fattah and Henderson1978a ). The shock–shock interaction between the transmitted shock and the refracted shock inside the volume induces the formation of the S1, S2, two triple points and shock $n$ . Note that if we consider the transmitted shock as an incident shock, the wave pattern inside the volume is quite similar to the one occurring at a slow/fast interface which is called twin von Neumann refraction (TNR) (Abd-el Fattah & Henderson Reference Abd-el Fattah and Henderson1978b ). Under this circumstance, the refracted shock is regarded as the free precursor shock and the other shocks may have characteristics similar to corresponding ones at a slow/fast interface.

When the incident shock moves along the hypotenuse of the backward-triangle, the wave pattern of MRR occurs as presented in figure 10(a). The interaction of the transmitted shock and the refracted shock causes the emergence of the S1 and S2 inside the volume, as well as the appearance of the expansion waves at the interface. This type of wave system inside the volume has also been found at a slow/fast interface and is called free precursor refraction (FPR) (Jahn Reference Jahn1956; Abd-el Fattah et al. Reference Abd-el Fattah, Henderson and Lozzi1976; Abd-el Fattah & Henderson Reference Abd-el Fattah and Henderson1978b ). As the diffracted shock travels forwards, the wave pattern inside the volume changes, as shown in figure 10(b), which can be referred to as free precursor von Neumann refraction (FNR) at a slow/fast interface (Abd-el Fattah & Henderson Reference Abd-el Fattah and Henderson1978b ). Note that in the experimental work of Abd-el Fattah & Henderson (Reference Abd-el Fattah and Henderson1978b ), the wave pattern of bound precursor refraction (BPR) was found to be converted into FNR for a very weak group and into FPR for a weak group by changing the incident angle in each group. However, FNR and FPR were never observed within the same group even though they believed that FNR can convert into FPR. In our previous work (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), the wave patterns of FNR and FPR are experimentally observed in the same group and the transition from FNR to FPR was also found, which confirms the viewpoint proposed by Abd-el Fattah & Henderson (Reference Abd-el Fattah and Henderson1978b ). Moreover, in the numerical work of Henderson et al. (Reference Henderson, Colella and Puckett1991), the transitions of the wave patterns from BPR to FPR and then to FNR were observed within the same group. Unfortunately, there is no experimental support for this. In the present work, both FPR and FNR are also observed in the same case and FNR is found to be transformed from FPR, which provides the experimental evidence for the numerical results of Henderson et al. (Reference Henderson, Colella and Puckett1991). In other words, the reciprocal transformations between the wave patterns of FNR and FPR are experimentally found in our previous and present works. Note that in previous work, the different wave patterns were obtained by changing the incident angle for a fixed incident shock strength. In our work, however, the incident shock strength and the incident angle are both fixed, and the transition of the wave pattern happens during its evolution.

4.2. Velocity of characteristic waves

The positions of shocks and triple points are manually measured from the photographic results, and the corresponding $x{-}t$ diagrams are plotted in figure 11. The error bars are not included in the $x{-}t$ diagrams, because the errors of displacement are estimated to be less than 3 % for shocks and 8 % for triple points. For the square, the velocities of the incident shock along and away from the interface (ISB and ISC), the transmitted shock, as well as the two triple points (originating from the vertex of the interface) are indicated in figure 11(a). It is found that the ISC travels more quickly than the ISB which is disturbed by the expansion waves from the interface. The same treatments are carried out for two rectangular inhomogeneities and the corresponding results are presented in figures 11(b,c), respectively, and a similar conclusion to the square can be drawn. Figure 11(d) presents the velocity variations with time for the forward-triangle where there is only one triple point. The velocity of the ISB calculated from the linear fitting is $477.8~\text{m}~\text{s}^{-1}$ , which is exactly equal to the value computed from the expression of $V_{IS}/\!\cos (30^{\circ })$ , where $V_{IS}$ is the velocity of the incident shock and 30° is half of the forward-triangle interface vertex angle. This result confirms that the RRR does happen in this case. For the backward-triangle, due to the attenuation by the expansion waves, the velocity of the ISB is significantly decreased, as shown in figure 11(e). A similar feature to the backward-triangle is observed for the downstream interface of the diamond, as shown in figure 11( f).

For all cases, it is found that the initial shock speeds measured from the schlieren images coincide quite well with the values obtained by the piezoelectric pressure transducers, which indicates that the ambient air is almost not polluted by $\text{SF}_{6}$ . The physical properties of the ambient air ahead of and behind the shock are listed in table 1. The contamination of $\text{SF}_{6}$ by air inside the volume, however, must be considered in the experiments. For the square, rectangles and backward-triangle, the interaction of the planar shock with the left surface can be regarded as a one-dimensional (1D) problem, and therefore the transmitted shock speed ( $V_{TS}$ -theo) and Mach number ( $M_{TS}$ -theo) can be acquired by simply solving the 1D problem. The velocities of the transmitted shock for these four cases are first calculated assuming that both gases inside and outside the volume are pure. As listed in table 2, the theoretical values are all smaller than the measured counterparts, suggesting the impurity of the $\text{SF}_{6}$ inside the volume. Based on the experimental values of $V_{TS}$ -exp, 1D gas dynamics theory is adopted to calculate the transmitted shock strength $M_{TS}$ -exp, and subsequently the physical properties of the mixture, such as molecular weight, density and sound speed, neglecting the faint variation of the specific heat ratio of the mixture. These quantities are listed in table 3, from which one can find that the $\text{SF}_{6}$ inside the volume is severely polluted by air, especially for the backward-triangle.

Table 1. Physical properties of ambient air at $T_{0}=298~\text{K}$ and $p_{0}=101\,325~\text{Pa}$ . Here ${\it\rho}_{e}$ and ${\it\rho}_{2}$ are the densities of air ahead of and behind the incident shock, respectively, and $U_{f}$ is the post-shock flow velocity.

Table 2. Comparison of the velocities of the incident shock ( $V_{IS}$ ) and the transmitted shock ( $V_{TS}$ ) between experiment and theoretical analysis from the 1D problem. $M_{TS}$ is the Mach number of the transmitted shock. Note that the theoretical values are obtained based on the pure gases. The velocity unit is $\text{m}~\text{s}^{-1}$ .

Table 3. Physical properties of the test gas inside the volume at $T_{0}=298~\text{K}$ and $p_{0}=101\,325~\text{Pa}$ . Here $m\,\%$  A $+$ $n\,\%$  B denotes that the gas is the mixture of $m\,\%$  A and $n\,\%$  B (the mass fraction); $A^{+}$ is the post-shock Atwood number.

Time variations of the angles between the initial horizontal interface and the trajectories of two triple points ( ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ ), as well as the refracted shock front ( ${\it\delta}$ ) are given in figure 12 for the square, the rectangles and the forward-triangle, respectively. The schematic diagram of the refracted shock front and the trajectories of the triple points for the rectangular case are shown in figure 9(b). The error bars represent the uncertainty in manual measurements of the experimental results. It is found that for each case, the velocity and the moving direction of each triple point are nearly invariable and therefore the motion of the triple point can be regarded as self-similar or pseudo-stationary. In addition, the values of ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ in the present work approach the counterparts of the triple points that occur outside the $\text{N}_{2}$ volume in our previous work (Zhai et al. Reference Zhai, Wang, Si and Luo2014a ), which once again illustrates the similarity between them even though there are some disparities between the velocities of two triple points. Moreover, for square and rectangular geometries, the value of ${\it\delta}$ and the average values of ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ are quite similar. The discrepancies of the velocity and the angle among these three cases may be attributed to the different concentrations of the gas components inside the volume.

Figure 12. Variation of angles ${\it\alpha}_{1}$ , ${\it\alpha}_{2}$ and ${\it\delta}$ for the square (a), streamwise-rectangle (b), transverse-rectangle (c) and forward-triangle (d). The angles are defined in figure 9.

The angle ${\it\delta}$ can be deduced using the refractive index of the materials by $V_{IS}/\!\sin {\it\theta}=V_{RS}/\!\sin {\it\delta}$ , where $V_{RS}$ is the refracted shock speed (Henderson Reference Henderson1989). Approximately, the refracted shock has a speed equal to the vertical transmitted shock (Bates et al. Reference Bates, Nikiforakis and Holder2007). Based on the values measured in the experiments, one can estimate the angle ${\it\delta}$ . For the square and rectangular cases, the incident angle ${\it\theta}$ is considered to be 90°, and therefore ${\it\delta}=\arcsin (V_{RS}/V_{IS})$ . The values of angle ${\it\delta}$ are calculated to be 28°, 28.1° and 28.5° for the square, streamwise-rectangle and transverse-rectangle, respectively, in reasonable agreement with the experimental measurements. Based on the geometrical relationship, as indicated in figure 13(a), one can deduce that $V_{TS}=V_{TP2}\cos ({\it\alpha}_{2})$ , and $V_{RS}=V_{TP1}\sin ({\it\alpha}_{1}+{\it\delta})$ . From figures 11 and 12(a–c), it is found that the second triple point moves faster than the first triple point, and ${\it\alpha}_{1}+{\it\delta}<90^{\circ }-{\it\alpha}_{2}$ , which means that $V_{RS}<V_{TS}$ . In other words, the values of ${\it\delta}$ computed are slightly overestimated if the $V_{RS}$ is substituted by the $V_{TS}$ . From the former expression, the angle ${\it\alpha}_{2}$ is calculated to be 30.2°, 29.2° and 32.9° for the square, streamwise-rectangle and transverse-rectangle cases, respectively. There is an almost 2° difference compared with the manual measurements, which is within the range of experimental error. From the latter expression and the hypothesis that the refracted shock speed is equal to the transmitted shock speed, the values of ${\it\alpha}_{1}$ are calculated to be 33.1°, 33.9° and 30.5° for the square, streamwise-rectangle and transverse-rectangle, respectively. A larger discrepancy exists since each quantity is an approximation and the first two values are unsatisfactory because ${\it\alpha}_{1}$ must be smaller than ${\it\alpha}_{2}$ .

Figure 13. Sketch of angles of the top horizontal boundary with the refracted shock front ( ${\it\delta}$ ) and with the trajectories of the two triple points ( ${\it\alpha}_{1}$ and ${\it\alpha}_{2}$ ).  $V_{T}$ , the transmitted shock speed; $V_{R}$ , the refracted shock speed; $V_{TP}$ , the triple point speed.

Figure 14. Comparison of normalized movements of the distorted interface for (a) the square and the rectangles and (b) the triangles and the diamond. Normalized height of the interface structures for (c) the square and the rectangles and (d) the triangles and the diamond.

For the forward-triangle volume, regular refraction happens at the hypotenuse, and ${\it\alpha}$ is defined as the angle between the trajectory of the triple point and the hypotenuse, as shown in figure 13(b). According to the refraction law and geometrical relation, one can obtain that $V_{IS}/\!\sin {\it\theta}=V_{RS}/\!\sin {\it\delta}=V_{TP}\sin ({\it\alpha}+{\it\delta})/\!\sin {\it\delta}$ . It is found from the schlieren images that the triple point nearly propagates along the angle bisector, and therefore the angle ${\it\alpha}$ can be approximately considered to be 30°. Thus, using the experimental values of velocities, the refraction angle ${\it\delta}$ is computed from the above expression to be 26.6°, which is acceptable considering the experimental error. In fact, the assumption of ${\it\alpha}$ equal to 30° is practicable since the experimental value of ${\it\alpha}$ is measured to be about 28°.

5.1. Interface features

Figure 14 gives the normalized movements of the distorted interface and the normalized interface height obtained from the experiments; $x_{l}$ and $x_{r}$ denote the leftmost and rightmost boundaries of the interface, as indicated in figure 15, and are normalized as $x_{l}^{\ast }=(x_{l}-x_{0})/L_{0}$ and $x_{r}^{\ast }=(x_{r}-x_{0})/L_{0}$ , where $x_{0}$ is the initial leftmost position of the interface. As a result, $x_{l}^{\ast }=0$ and $x_{r}^{\ast }=1$ indicate the quiescence of the leftmost and rightmost boundaries, respectively. The interface height ( $h$ ) is normalized by its initial height ( $h^{\ast }=h/h_{0}$ ). The characteristic time $t_{0}$ is calculated using the post-shock flow velocity $U_{f}$ , that is, $t_{0}=L_{0}/U_{f}$ . For all cases, the errors of $x_{l}^{\ast },x_{r}^{\ast }$ and $h^{\ast }$ are estimated to be less than 3 % at early stages and 8 % at late stages, and the error bars are not included in the figures because of the data concentration.

Figure 15. Schematic sketch of the distances measured for $\text{SF}_{6}$ square inhomogeneity.

Table 4. Comparison of the normalized velocity of the leftmost boundary estimated from the linear fitting for six cases.

For all cases, the movements of the leftmost boundary ( $x_{l}^{\ast }$ ) are nearly steady for the range of time studied, as indicated in figures 14(a,b), and the corresponding normalized velocities are listed in table 4. It can be observed that the $x_{l}^{\ast }$ moves with nearly the same velocity for the square, the rectangles and the backward-triangle due to similar initial shapes. A similar velocity of $x_{l}^{\ast }$ is also found for the forward-triangle and the diamond. For the rightmost boundary ( $x_{r}^{\ast }$ ), they are stationary before the shock passes through it, and afterwards different behaviours are found for different interfaces. For the square and the rectangles, the rightmost boundaries also move with nearly the same velocity at early stages. However, for the streamwise-rectangle in the following development, the backward movement of the rightmost boundary is observed. Note that the $x_{r}^{\ast }$ always indicates the farthest position of the interface. Taking the square, for example, as illustrated in figure 15, before the upstream vortex pair catches up with the downstream one, the $x_{r}^{\ast }$ denotes the boundary of the downstream vortex pair. Otherwise, the $x_{r}^{\ast }$ represents the boundary of the upstream vortex pair. For the streamwise-rectangle, however, after the downstream vortex pair is generated and before the upstream vortex pair catches up with it, the second downstream vortex pair is produced and becomes prominent while the downstream vortex pair gradually diffuses and becomes obscure. The boundary of the second downstream vortex pair is subsequently considered to be the measuring position, leading to the reversed motion of the $x_{r}^{\ast }$ . The normalized movements of the leftmost and rightmost boundaries from the numerical results of Bates et al. (Reference Bates, Nikiforakis and Holder2007) are also included in the figures, represented by solid and dashed lines. Compared with the present work, it is found that either the leftmost or rightmost interface moves relatively slowly, which may be attributed to the presence of the wire meshes and the characteristics of the continuous interface described above. It can be observed from figure 14(b) that, after the compression stage, the $x_{r}^{\ast }$ of the forward-triangle moves with a nearly constant velocity due to the generation of the downstream vortex pair. For the backward-triangle and the diamond, however, the $x_{r}^{\ast }$ moves quite slowly after the compression stage compared with the forward-triangle, because the rightmost boundary does not change much at early times. When the upstream vortex pair surpasses the downstream structure and becomes the measuring position, the $x_{r}^{\ast }$ travels with a larger velocity.

The variations of the normalized height $h^{\ast }$ of the interface for all cases are presented in figures 14(c,d). It is found that $h^{\ast }$ increases steadily from the beginning for the square and rectangular geometries due to the continuous rotation of the upstream vortex pair, and the larger the width–height ratio, the faster the growth of $h^{\ast }$ . The variation of the normalized height from the computation of Bates et al. (Reference Bates, Nikiforakis and Holder2007), denoted by a solid line, is also shown here. Because the width–height ratio in the work of Bates et al. (Reference Bates, Nikiforakis and Holder2007) is 0.75, it is reasonable to find that its growth rate lies between the square and the transverse-rectangle (the width–height ratios for the square and the transverse-rectangle are 1 and 0.5, respectively). The rotation of the upstream vortex pair also leads to continuous increase of the $h^{\ast }$ for the backward-triangle. However, for the diamond and forward-triangle, due to the effect of the initial interface shape, the invariability of the $h^{\ast }$ is observed at the very early stage and then the $h^{\ast }$ drops, especially for the forward-triangle in which the $h^{\ast }$ always decreases within the time studied.

From the comparison of the movements of the distorted interface and the variations of the interface features, it can be concluded that the initial interface shape has a large influence on interface morphology and characteristics after the shock impact. For the square and rectangular geometries, although the interface shapes are similar, the differences in interface scale result in diversity of interface morphology, such as the numbers of outward jets. In addition, the variation of the normalized height indicates that the larger the initial interface height, the smaller the relative change in height. For the forward-triangle and the backward-triangle with the same scale but different orientations, greater differences are found for the variations of the interface displacement and height, where the compressive effect is quite different.

5.2. Vorticity

During the shock–interface interaction, the rotational flow evolves on much longer timescales than that of the shock passage through the local non-uniformity and can therefore have important long-term effects on the structure and property of the flow field. The equation that governs the evolution of vorticity ${\it\omega}$ in an inviscid fluid can be expressed by

(5.1) $$\begin{eqnarray}\frac{\text{D}{\it\omega}}{\text{D}t}=\frac{1}{{\it\rho}^{2}}(\boldsymbol{{\rm\nabla}}{\it\rho}\times \boldsymbol{{\rm\nabla}}p)+({\it\omega}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})v-{\it\omega}(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }v),\end{eqnarray}$$

where $\boldsymbol{{\rm\nabla}}{\it\rho}$ and $\boldsymbol{{\rm\nabla}}p$ stand for the density and pressure gradients, respectively, and $v$ is the fluid velocity. It has been proved that the misalignment of the local pressure and density gradients is the primary source of vorticity generation during the shock–interface interaction (Ranjan, Oakley & Bonazza Reference Ranjan, Oakley and Bonazza2011).

Figure 16 shows the diagrams of vorticity generation for the square, the forward-triangle and the diamond, where only the effects of the incident shock and the diffracted shock are emphasized for simplification. For the square, little vorticity is generated at the leftmost vertical interface where the pressure and density gradients are aligned. When the incident shock propagates along the horizontal upper interface, Mach reflection occurs, in which the Mach stem connects the incident shock with the interface. It is therefore the Mach stem that contributes the pressure gradient for the vorticity generation on the interface. However, the strength of the Mach stem is actually decaying as it moves forwards. If the variation of the Mach stem strength is neglected, a negative vorticity peak will be produced due to the near-orthogonality of the pressure and density gradients, and can be regarded as a constant value. As the diffracted shock moves along the rightmost interface, the vorticity amplitude is decreasing due to the gradual attenuation of the diffracted shock strength although the near-orthogonality of the pressure and density gradients is still maintained (Skews Reference Skews1967; Edwards Reference Edwards1983). For the upstream interfaces of the forward-triangle and the diamond, the pressure and density gradients are not orthogonal. At the downstream interfaces, although the diffracted shock strength decays, the orthogonality relationship is almost preserved. Therefore, it is difficult to estimate which vorticity amplitude is larger qualitatively. From the movements of the vortex pairs, it may be concluded that the vorticity deposited at the upstream and downstream interfaces is comparable.

Figure 16. Sketch maps of the vorticity generation on the boundaries for the square (a), forward-triangle (b) and diamond (c). $\boldsymbol{{\rm\nabla}}p$ , pressure gradient caused by the incident shock; $\boldsymbol{{\rm\nabla}}p^{\prime }$ , pressure gradient caused by the diffracted shock.

The circulation ${\it\Gamma}$ is an important quantity in the description of vortical flows. It is defined as a line integral of the velocity around a closed curve $L$ or is equal to the flux of vorticity through an open surface $A$ bounded by the curve, that is, ${\it\Gamma}=\oint _{L}\boldsymbol{U}\boldsymbol{\cdot }\text{d}s=\int _{A}{\it\omega}\boldsymbol{\cdot }\text{d}\boldsymbol{S}$ . In previous studies several theoretical models have been proposed to predict the circulation (Ranjan et al. Reference Ranjan, Oakley and Bonazza2011). In the following discussion, some models are adopted to estimate the circulation deposition on the interface and then the self-induced vortex speed. To calculate the circulation, the densities of the gases are needed in advance and therefore the forward-triangle and the diamond are ignored. Furthermore, for the backward-triangle, because the shock strength is greatly attenuated when the shock propagates along the hypotenuse and it is difficult to estimate the shock strength, the circulation deposition is also not considered here. For the square and the rectangles, as stated earlier, when the shock moves along the horizontal interface, the attenuation of the shock strength can be neglected. Consequently, only the circulation depositions on the horizontal interface in the square and rectangular cases are evaluated.

The circulation model provided by Picone & Boris (Reference Picone and Boris1988) (the PB model) considers the baroclinic term to be the only source term for the vorticity on the half-plane of the bubble, and is based on the initial properties of the shocked and unshocked ambient gases, and the geometry. For the current rectangular geometry, the PB model may be written as

(5.2) $$\begin{eqnarray}{\it\Gamma}_{PB}\approx \frac{L_{0}}{V_{IS}}\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}_{e}}\left(\frac{\boldsymbol{{\rm\nabla}}{\it\rho}}{{\it\rho}_{i}}\right),\end{eqnarray}$$

where ${\it\rho}_{i}$ and ${\it\rho}_{e}$ are the initial densities of the interior and exterior gases, respectively.

Yang, Kubota & Zukoski (Reference Yang, Kubota and Zukoski1994) numerically investigated the shock–cylinder interaction and proposed a circulation model (the YKZ model) assuming that the shock and the interface are non-deformable during shock–interface interaction. For the rectangular geometries in this study, the model can approximately be expressed by

(5.3) $$\begin{eqnarray}{\it\Gamma}_{YKZ}\approx \frac{2L_{0}}{V_{IS}}\frac{\boldsymbol{{\rm\nabla}}p}{{\it\rho}_{2}}\left(\frac{\boldsymbol{{\rm\nabla}}{\it\rho}}{{\it\rho}_{i}+{\it\rho}_{e}}\right),\end{eqnarray}$$

where ${\it\rho}_{2}$ is the post-shock flow density.

Based on the shock polar analysis, Samtaney & Zabusky (Reference Samtaney and Zabusky1994) proposed a model (the SZ model) to predict the deposited circulation on the assumption that the discontinuous interface is composed of many straight segments. The circulation ${\it\Gamma}$ can be written as

(5.4) $$\begin{eqnarray}{\it\Gamma}_{SZ}\approx \frac{2{\it\gamma}^{1/2}}{{\it\gamma}+1}(1-{\it\eta}^{-1/2})(1+M^{-1}+2M^{-2})(M-1)\sin {\it\theta},\end{eqnarray}$$

where ${\it\gamma}$ can be thought of as the average of the specific heat ratios of two gases, and ${\it\eta}$ is the density ratio of two gases.

From (5.2)–(5.4), the deposited circulation ${\it\Gamma}$ on the half-plane is estimated as listed in table 5. It is found that the values predicted by the PB model are the smallest ones among all the models, while the predictions by the YKZ and SZ models coincide well with each other. After the circulation ${\it\Gamma}$ is known, it is possible for us to deduce the self-induced velocity ( $V_{v}^{\prime }$ ) of the vortex pair through the simplified relation between $V_{v}^{\prime }$ and ${\it\Gamma}$ : $V_{v}^{\prime }={\it\Gamma}/(2{\rm\pi}L_{v})$ , where $L_{v}$ is the vortex spacing of the upstream vortex pair, as defined in figures 3–5. The vortex spacing $L_{v}$ and the horizontal position of the vortex core $X_{v}$ , both normalized by the initial width $L_{0}$ , are measured from the schlieren images and shown in figure 17, in which the error bars represent the uncertainty. The self-induced speeds of the vortex core, $V_{v}^{\prime }$ , obtained by these models are listed in table 5. It is found that both the maximum circulation and subsequently the maximum $V_{v}^{\prime }$ are achieved for the streamwise-rectangle while the minimum ones are achieved for the transverse-rectangle, which is closely associated with the length of the integral path.

Table 5. Comparison of the circulation ${\it\Gamma}$ and the self-induced velocity $V_{v}^{\prime }$ of the vortex pair for different models. The units of $L_{v}$ , ${\it\Gamma}$ and $V_{v}^{\prime }$ are mm, $\text{m}~\text{s}^{-2}$ and $\text{m}~\text{s}^{-1}$ , respectively.

Figure 17. Time variation of the vortex core $X_{v}$ in the streamwise direction, and the vortex spacing $L_{v}$ in the spanwise direction for the square (a), streamwise-rectangle (b), transverse-rectangle (c) and backward-triangle (d).

The vortex velocity $V_{v}$ , considering the post-shock flow velocity ( $U_{f}$ ) for heavy inhomogeneity, is defined as $V_{v}=U_{f}-V_{v}^{\prime }$ . Table 6 shows the vortex velocity predicted by these models, together with the experimental measured ones. It is found that these three models can reflect the circulation deposition to some extent although the models underestimate the values. Note that in experiments, the vortex velocity is measured from the upstream vortex pair. However, the self-induced velocity $V_{v}^{\prime }$ is obtained from the models based on the assumption that the vorticity generated at the surface is deposited in one vortex pair, which will definitely overestimate $V_{v}^{\prime }$ and consequently underestimate the vortex velocity (for heavy inhomogeneity, the velocity induced by the vorticity is in the opposite direction to the post-shock flow velocity). Therefore it is reasonable to find the smallest vortex velocity for the streamwise-rectangle and the largest value for the transverse-rectangle.

Table 6. Quantitative values of the vortex core speed $V_{v}$ measured from the experiments and circulation models for the square, streamwise-rectangle, transverse-rectangle and backward-triangle.

In addition, Rudinger & Somers (Reference Rudinger and Somers1960) proposed a model (the R–S model) to predict the vortex velocity by assuming that the gaseous cylinder is fully developed into a vortex pair after the shock acceleration. In the R–S model, the vortex velocity $V_{v}$ is defined as $V_{v}=U_{f}(1-0.203A^{+})$ , where $A^{+}$ is the post-shock Atwood number. The velocities of the vortex pair predicted by the R–S model for the square, two rectangles and the backward-triangle are listed in table 6 (gas contamination is considered). It is found that the R–S model also underestimates the vortex velocity for the same reason stated above. For the backward-triangle, the R–S model provides a better prediction for vortex speed because only one developed vortex pair is generated in this case. From the comparison, one can conclude that the circulation models mentioned above and the R–S model can, at least for current study, provide a rough prediction of the vortex velocity.