1. Introduction
A shock wave is a thin, compressive and nonlinear disturbance that propagates through a medium, producing abrupt and finite increases in pressure, temperature and density. It is characterised by a discontinuous drop in flow velocity, a loss in total pressure and an irreversible increase in entropy, all occurring over a very small spatial region governed by dissipative molecular effects. The interaction of shock waves with multiphase interfaces, such as gas–liquid boundaries, occurs in several important phenomena, including cavitation bubble collapse, droplet breakup in supersonic flows and underwater or oversea explosions. Understanding these shock interactions over multiphase interfaces plays a crucial role in the development of blast wave mitigation strategies. A key aspect of such interactions is the response of the shock wave to its collision at the interface of the two media with different acoustic impedances,
$Z$
, defined as the product of the density and speed of sound of the medium. During this interaction, the shock wave undergoes reflection and refraction, resulting in changes to its propagation angle and strength.
Existing research has primarily focused on the interaction of blast waves over a water interface (Borisov, Kogarko & Lyubimov Reference Borisov, Kogarko and Lyubimov1965; Flores & Holt Reference Flores and Holt1982; Arun Kumar, Rajesh & Jagadeesh Reference Arun Kumar, Rajesh and Jagadeesh2022) and shock wave interactions over a water column (Igra & Takayama Reference Igra and Takayama2001; Chen & Liang Reference Chen and Liang2008; Sembian et al. Reference Sembian, Liverts, Tillmark and Apazidis2016), which can be three-dimensional and unsteady in nature. These scenarios involve continuous variations in shock wave strength and/or interface inclination angles, complicating the analysis of the shock refraction/reflection patterns. To address such complexities, it is crucial to first understand the fundamental physics of two-dimensional pseudosteady shock refraction/reflection phenomena at fixed interface inclination angles. For air–water interfaces, this phenomenon is typically referred to as the shock refraction problem over a water wedge.
Extensive research on pseudosteady shock reflections on solid wedges (see figure 1
a) has yielded substantial insights into different reflection configurations and the transition criteria for various types of reflections (Ben-Dor Reference Ben-Dor2007). Furthermore, thorough investigations have been conducted on the phenomenon of shock refraction at different gas–gas interfaces under pseudosteady conditions (Abd-El-Fattah & Henderson Reference Abd-El-Fattah and Henderson1976, Reference Abd-El-Fattah and Henderson1978a
,
Reference Abd-El-Fattah and Hendersonb
; Henderson Reference Henderson1970, Reference Henderson1989; Henderson et al. Reference Henderson, Colella and Puckett1991, Reference Henderson, Puckett and Colella2004; Lieberman & Shepherd Reference Lieberman and Shepherd2007). However, in contrast to these gas–gas studies, the phenomenon of pseudosteady shock refraction at multiphase interfaces, such as air–water interfaces, remains comparatively under-explored. Although shock refraction classifications for gas–gas and gas–liquid interfaces appear similar, gas–liquid systems exhibit markedly different physical and analytical behaviours. The substantial acoustic impedance mismatch in gas–liquid interfaces, such as air–water interfaces (
$Z_{w\textit{ater}} / Z_{\textit{air}} \approx 3670$
), results in dominant shock reflection, restricted transmitted shock formation and distinct patterns such as free precursor refractions accompanied by transmitted acoustic pressure waves (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). The disparity in the acoustic impedance between air and water, which exceeds three orders of magnitude, plays a central role in governing the reflection-dominant behaviour observed in gas–liquid refraction. Table 1 provides representative values of density, speed of sound and resulting impedance for air and water at 300 K. These values quantitatively illustrate the large disparity in acoustic properties, which underlies the distinct refraction patterns observed at gas–liquid interfaces and supports the characteristic differences in shock behaviour in air–water interfaces. Such behaviours are rarely observed in gas–gas configurations and arise only under limited conditions. Furthermore, due to the near incompressibility of liquids and the altered wave interactions, standard analytical frameworks for gas–gas refraction are inadequate and must be reformulated. These considerations motivate a dedicated investigation into the distinct refraction dynamics at gas–liquid interfaces, which the present study undertakes as a natural extension of our earlier work (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024).
Figure 1. Schematic diagrams of a shock wave moving at a constant shock Mach number interacting with (a) a solid wedge and (b) a material interface: i – incident shock, r – reflected shock, t – transmitted shock, mm – interface, mm’ – deflected interface.
Table 1. Thermophysical properties of air and water at 300 K relevant to shock refraction studies.
Before delving into the existing literature on pseudosteady shock refraction at multiphase interfaces, it is essential to first establish a fundamental understanding of how shock waves interact with an interface. The characteristics of a pseudosteady shock refraction are determined by the strength of the incident shock wave,
$\zeta$
, and the inclination angle,
$\beta$
, which defines the orientation of the shock wave relative to the interface separating the two media, as shown in figure 1(b). The strength of the shock wave, defined as the pre-shock to post-shock pressure ratio, can be expressed indirectly using the shock Mach number
$M_S$
. Over the years, investigations into shock refraction phenomena have focused on two types of interfaces: slow–fast (s / f) and fast–slow ( f / s), which are categorised based on the relative acoustic speeds
$(a)$
of the media involved. Specifically, s / f interfaces correspond to
$a_1 \lt a_2$
, while f / s interfaces correspond to
$a_1 \gt a_2$
. In the above scenarios, 1 and 2 represent the first and second media, and the moving shock wave originates within the first medium and propagates toward the second medium. Consequently, the present study, which focuses on the shock refraction phenomenon over a water wedge, is classified under the category of an s / f interface.
Takayama & Ben-Dor (Reference Takayama and Ben-Dor1989) investigated the phenomenon of shock reflection over a water wedge, focusing on the detachment condition (Ben-Dor Reference Ben-Dor2007),
$\theta _D$
, that defines the transition between regular reflection (RR) to Mach reflection (MR). The study explored the transition angle variations observed at different shock Mach numbers between the water wedge and the solid wedge. The experiments of Takayama & Ben-Dor also provided interferometric evidence of distinct shock refraction patterns in the water wedge, which were later classified as free precursor refraction with regular reflection (FPR) (Nourgaliev et al. Reference Nourgaliev, Sushchikh, Dinh and Theofanous2005) and free precursor refraction with Mach reflection (FMR) (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). Subsequently, Nourgaliev et al. (Reference Nourgaliev, Sushchikh, Dinh and Theofanous2005) performed high-resolution numerical simulations to study shock refraction patterns and sequences at both s / f (air–water) and f / s (water–air) interfaces. The water was modelled as a stiffened gas, and the simulations were conducted using the novel adaptive characteristics-based matching method in conjunction with adaptive mesh refinement to accurately resolve the shock refraction patterns. For an s / f air–water interface, they identified the sequence of refraction patterns, with increasing inclination angle, for very weak incident shocks to be: regular refraction with a reflected shock wave (RRR)
$\rightarrow$
FPR
$\rightarrow$
free precursor von-Neumann refraction (FNR). Conversely, for the f / s water–air interface, where only strong shocks are possible, the numerical results confirmed a sequence comprising regular refraction with reflected expansion
$\rightarrow$
anomalous refraction with reflected expansion.
Further experimental and numerical studies by Wan et al. (Reference Wan, Jeon, Deiterding and Eliasson2017) investigated shock reflections on water wedges, with a focus on the detachment condition and its deviations compared with solid wedges over a broader range of shock Mach numbers. Their findings highlighted significant mismatches in detachment conditions between water and solid wedges for almost all shock Mach numbers, with the detachment condition in water wedges occurring at lower wedge angles than in solid wedges, particularly at higher shock Mach numbers. The numerical simulations of Wan et al. (Reference Wan, Jeon, Deiterding and Eliasson2017) indicated the occurrence of irregular refraction with Mach reflection (IRMR). An IRMR configuration consists of an MR in air, accompanied by an oblique transmitted shock wave in water, which is anchored to the foot of the Mach stem at the refraction point on the interface. For
$ M_S = 3$
and
$ M_S = 4$
, at wedge angles (
$\theta _w$
, refer to figure 1 for definition) of
$ 40^\circ$
and
$ 45^\circ$
, IRMR configurations were observed featuring a double Mach reflection (DMR) structure in air. This reflection pattern involves two triple points: the first triple point (
$ T$
) marks the intersection of the incident shock (
$ i$
), reflected shock (
$ r$
) and Mach stem (
$ m$
), generating the primary slip line. The second triple point (
$ T'$
) forms farther downstream at the junction of the reflected shock (
$ r$
), a secondary Mach stem (
$ m'$
) and a secondary reflected wave (
$ r'$
), producing a secondary slip line. The trajectories
$ \chi$
and
$ \chi '$
represent the paths traced by these triple points over time and are typically expressed as angles relative to the wedge surface. For water wedges, these trajectories were shown to be smaller than those observed for solid wedges at corresponding shock Mach numbers and wedge angles.
In a recent study by Anbu Serene Raj et al., a comprehensive investigation was conducted into the pseudosteady shock refraction phenomenon across water wedges, focusing specifically on the very weak incident shock strength group. Experiments performed using a tiltable vertical shock tube confirmed the refraction sequence for this group as RRR
$\rightarrow$
(bound precursor refraction, BPR)
$\rightarrow$
FPR
$\rightarrow$
FMR. Notably, FMR, free precursor refraction with MR, previously misidentified as an FNR (Nourgaliev et al. Reference Nourgaliev, Sushchikh, Dinh and Theofanous2005), was validated through a time-resolved schlieren imaging technique and corroborated using numerical simulations. In addition, the study employed shock polar analysis and analytical shock relations for water, derived from the stiffened gas equation of state, to establish analytical transition conditions for the air–water interfaces across various shock Mach numbers and inclination angles. The analytical study facilitated the proper classification of shock strength groups into very weak, weak and strong categories for the s / f air–water interfaces, along with the associated shock refraction sequences. These sequences are summarised in table 2, along with the respective shock strength group ranges. It should be noted that the refraction patterns mentioned within brackets in table 2 are transition patterns.
Table 2. Summary of previous studies on pseudosteady shock refraction over water wedges.
Apart from the numerical simulations by Wan et al., there is a significant lack of data for shock refraction over water wedges at higher
$M_S$
and lower
$\theta _w$
. Furthermore, the anticipated refraction sequence, as documented by Anbu Serene Raj et al., for the weak and strong incident shock strength categories necessitates both experimental and numerical validations. Besides, the transition dynamics of shock reflection and refraction patterns over water wedges, especially beyond the regular-to-Mach reflection (RR–MR) transition, are significantly under-explored in comparison with shock reflection patterns on solid wedges, highlighting a crucial gap in the comprehension of these intricate phenomena. In an effort to address these knowledge gaps, this work builds upon our previous study (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024) and aims to provide a comprehensive understanding of the pseudosteady shock refraction phenomenon over water wedges at higher shock Mach numbers using numerical simulations.
The objectives of the present work involve:
-
(i) numerical simulations at higher shock Mach numbers, primarily focusing on refraction patterns and sequences within the previously unexplored weak and strong incident shock strength groups;
-
(ii) proposing a modified transition criterion for the bound precursor refraction pattern in the weak incident shock strength group to account for the observed deviations; and
-
(iii) comparing the shock structures and the transition of single Mach reflection to transitional Mach reflection (SMR
$\rightarrow$
TMR) between solid and water wedges at higher
$M_S$
and lower
$\theta _w$
.
The current paper is organised as follows. The numerical methods used to simulate the water-wedge and solid-wedge scenarios are outlined in § 2. Section 3 presents the findings for the weak incident shock strength group, including the revised transition criteria for IRMR
$\rightarrow$
FMR. The refraction sequence associated with the strong incident shock strength group is discussed in § 4. Finally, § 5 provides qualitative and quantitative comparisons between the water-wedge simulations and the solid-wedge simulations.
2. Numerical methodology
2.1. Water-wedge simulations
The compressible multiphase simulations are conducted using the BlastFoam solver (BlastFoam 2020), which is built on the OpenFOAM framework. The governing equations of the compressible multifluid BlastFoam solver are based on the five-equation model (Shyue Reference Shyue1999; Zheng et al. Reference Zheng, Shu, Chew and Qin2011), which describes the conservation of mass, momentum, energy and a transport equation for
$\alpha$
, the volume fraction of the fluids in a two-phase flow. The system of equations is closed using appropriate equations of state for the respective fluids, i.e. the ideal gas equation of state for air and the stiffened gas equation of state for water. The stiffened gas equation of state is represented in its temperature form as
\begin{align} p(\rho ,T) &= \rho (\gamma -1)c_{\mathrm{v}}T - p_{\infty }, \end{align}
\begin{align} a &= \sqrt {\frac {\gamma (p + p_{\infty })}{\rho }}, \end{align}
where
$p$
,
$\rho$
,
$T$
,
$c_v$
,
$\gamma$
and
$p_\infty$
are the pressure, density, temperature, specific heat capacity at constant volume, specific heat capacity ratio and stiffening pressure, respectively. The relation for the speed of sound of a stiffened gas is given by (2.2). In the current study, the stiffened gas parameters for water are defined as
$\gamma = 2.8$
and
$p_\infty = 850\,\mathrm{MPa}$
. For a detailed description of the numerical methodology and the use of stiffened gas parameters, refer to the work of Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). The current study employs the finite volume method with Harten-Lax-van Leer-Contact (HLLC) flux evaluation (Toro, Spruce & Speares Reference Toro, Spruce and Speares1994) and a fourth-order strong stability-preserving Runge–Kutta (SSPRK) method (Spiteri & Ruuth Reference Spiteri and Ruuth2002) for temporal discretisation. The validation of the solver is detailed in Appendix A.
Figure 2. Numerical domain for water-wedge simulations: green line – inlet, red line – moving incident shock wave, pink line – outlet, brown line – air/water interface, black line – walls.
The computational domain is a rectangular region with dimensions
${L} \times {H} = 60\,\textrm {mm}\times 38\,\textrm {mm}$
, as illustrated in figure 2. The domain is discretised using a uniform Cartesian grid with a resolution of
$\delta x = \delta y = \,{L/2400}$
, corresponding to the finest grid resolution established in our previous study (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). For these simulations, water is initialised on the right-hand side of the domain at a specified inclination angle,
$\beta = 90^\circ -\theta _w$
. At
$t = 0\,\textrm {s}$
, a moving shock wave is initialised
$5\,\textrm {mm}$
ahead of the leading edge of the air–water interface. The pre-shock and post-shock conditions are set to match the desired shock Mach number.
2.2. Solid-wedge simulations
Numerical simulations of solid-wedge pseudosteady shock interactions are carried out using an in-house two-dimensional inviscid shock-capturing finite volume code (Reshma et al. Reference Reshma, Vinoth, Rajesh and Ben-Dor2021; Paramanantham, Janakiram & Gopalapillai Reference Paramanantham, Janakiram and Gopalapillai2022; Baby, Paramanantham & Rajesh Reference Baby, Paramanantham and Rajesh2024; Vishnu Prasad et al. Reference Vishnu Prasad, Anbu Serene Raj, Paramanantham, Athira and Rajesh2025) that employs fifth-order targeted essentially non-oscillatory (TENO) reconstruction (Fu Reference Fu2019) for improved accuracy. The TENO reconstruction enhances the performance of traditional weighted essentially non-oscillatory (WENO) schemes by adaptively selecting stencils that offer the optimal balance between accuracy in smooth regions and robustness near discontinuities. During the reconstruction process, a cutoff filter is employed to detect shocks within the stencil, determining whether a central reconstruction or a WENO-based approach should be applied. The reconstruction is performed in characteristic space, which ensures stability and robustness when dealing with strong shocks or discontinuities. The reconstructed values at the cell interfaces are then used to compute the fluxes using the HLLC approximate Riemann solver, which is well suited for capturing shocks and contact discontinuities. Finally, time integration is carried out using the explicit Third-order Strong Stability Preserving Runge-Kutta (SSPRK3) scheme (Gottlieb et al. Reference Gottlieb, Shu and Tadmor2001), ensuring stability while maintaining high temporal accuracy. The modified numerical domain for the solid-wedge case is illustrated in figure 3. In the domain, part of the bottom wall is elevated and fixed at the required wedge angle (
$\theta _w$
). The shock wave is initialised at the same location as that of the water-wedge case. Appendix B provides a detailed account of the grid independence study conducted for the in-house TENO code.
Figure 3. Numerical domain for solid-wedge cases: green line – inlet, red line – moving incident shock wave, pink line – outlet, black line – walls.
3. Weak incident shock strength group
From table 2, the range of shock Mach numbers for the weak incident shock strength group is identified to be
$2.725 \lt M_S \lt 4.325$
. As generating shock Mach numbers within this range was not feasible in the tiltable vertical shock tube discussed earlier, numerical simulations were conducted for selected values of
$M_S = [3,\,3.4,\,3.7,\,4,\,4.1,\,4.2]$
to verify the refraction sequence predicted by Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). Within this range, the refraction pattern transitions from RRR to IRMR, eventually evolving into an FMR through the bound precursor refraction with an MR (BPMR) transition pattern. The detailed refraction pattern for a representative case is illustrated and analysed in this section.
3.1. Refractions with transmitted shock
3.1.1. Regular refraction with a reflected shock wave: RRR
Figure 4 shows the refraction pattern obtained at an inclination angle of
$35^\circ$
for the shock Mach number
$M_S = 3.4$
, corresponding to a
$\zeta$
of 0.075. The contour shows the gradient of
$\log \,p\times \log \,\rho$
in the streamwise (
$x$
) direction. As predicted by Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024), the RRR configuration is evident at lower inclination angles. The RRR pattern, by definition, represents a three-shock system comprising the incident shock (
$i$
), the reflected shock (
$r$
) and the transmitted shock (
$t$
). In this configuration, the transmitted shock is an oblique shock wave that remains attached to the refraction point (
$R$
) on the interface, where all three shock waves meet. In the weak incident shock strength group, the RRR pattern persists until the detachment condition is reached (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). In contrast, within the very weak incident shock strength group, the refraction pattern transitions from an RRR to a free precursor refraction (FPR) through a BPR before the RR in air can evolve into an MR (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024). Therefore, in the weak incident shock strength group, as the inclination angle increases, the refraction pattern shifts into the IRMR regime, illustrated by the schematics drawn in the frame of reference attached to the triple point in figure 5.
Figure 4. Numerical contour of x-gradient of
$\log \,p\times \log \,\rho$
showing an RRR configuration occurring at
$M_S = 3.4$
,
$\beta = 35^\circ , \theta _w = 55^\circ$
: i – incident shock, r – reflected shock, t – transmitted shock, R – refraction point.
3.1.2. Irregular refraction with a Mach reflection: IRMR
The IRMR pattern encompasses all configurations with an irregular or an MR in air (or the slower medium, in general), namely, a DMR, TMR or an SMR and an attached oblique transmitted shock, as illustrated in the schematics in figure 5. From figure 6(a), it is observed that, at an inclination angle of
$\beta = 45^\circ$
, the IRMR refraction pattern exhibits a MR in air alongside a transmitted shock wave in water (refer to figure 5
a for the schematic diagram). It is observed that the secondary reflected shock wave (
$r$
’) and the slip line interact with the interface. The resultant disturbance is transmitted into the liquid at an angle different from that of the primary transmitted shock from the foot of the Mach stem,
$ t$
, leading to the generation of a secondary transmitted shock wave (
$t'$
). Figure 7 shows the extracted pressure profile along a horizontal line through the transmitted waves, clearly demonstrating that both
$ t$
and
$ t'$
are shock waves, as indicated by sharp discontinuous pressure jumps. A dark blue region is observed just ahead of
$ t'$
in both
$x$
-momentum flux density (inset of figure 6
a) and pressure contour plot (figure 7). The extracted pressure profile reveals that this feature corresponds to a localised pressure drop, characteristic of an expansion region immediately following the transmitted shock. Similar features have also been noted in Wan et al., where a local temperature dip was observed just upstream of
$ t'$
. These shock waves subsequently coalesce within the water, away from the interface. Upon reaching the bottom wall of the computational domain in the water, the coalesced shock wave interacts with the leading-edge corner-generated wave, resulting in a significant curvature. This interaction ensures the wall-normal condition of the transmitted shock wave inside the water. Furthermore, as the inclination angle increases, the reflection in the air within the IRMR pattern starts transitioning from a DMR to a TMR (refer to figure 5
b for the schematic diagram). This is evident from the observed weakening of
$r$
’ and the initiation of a smooth kink formation – the slope discontinuity at the sharp second triple point
$T$
’ is gradually smoothing out into a continuous curvature, referred to as the kink
$k$
. Nonetheless, at the inclination angle of
$55^\circ$
, the reflection is still a DMR, as a secondary reflected shock
$r$
’ is still present, as shown in the inset of figure 6(b) displaying the x-momentum flux density contour,
$\rho u$
, where
$u$
is the x-component of velocity. For even higher shock Mach numbers, an IRMR can occur with an SMR as shown in figure 5(c).
Figure 5. Wave schematics of the possible IRMR configurations over water wedges: i – incident shock, r – reflected shock, r’ – secondary reflected shock, m – Mach stem, m’ – secondary Mach stem, t – primary transmitted shock, t’ – secondary transmitted shock, R – refraction point, T – first triple-point, T’ – second triple-point, k – kink. (a) An IRMR with a double Mach reflection (DMR) in air, (b) An IRMR with a TMR in air, (c) An IRMR with a SMR in air.
Figure 6. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
display the IRMR configurations occurring at shock Mach number
$M_S = 3.4$
over the water interface, with insets showing contours of x-momentum flux density,
$\rho u$
(min: 0, max: 15 000): i – incident shock, r – reflected shock, r’ – secondary reflected shock, m – Mach stem, s – slip line, m’ – secondary Mach stem, t – primary transmitted shock, t’ – secondary transmitted shock, R – refraction point, T – first triple point, T’ – second triple point. (a)
$\beta=45^{\circ}, \theta_{w} = 45^{\circ} $
, (b)
$\beta=55^{\circ}, \theta_{w} = 35^{\circ} $
.
Figure 7. Numerical pressure contour of an IRMR at
$M_S = 3.4, \beta =45^\circ$
along with the extracted pressure data plot along a horizontal line: dashed white line – pressure data extraction line.
3.1.3. Bound precursor refraction with a Mach reflection: BPMR
In air–water systems, a bound precursor refraction is a transition pattern between refraction patterns with a transmitted oblique shock wave and a transmitted free precursor acoustic pressure wave. It consists of a transmitted shock wave attached to the refraction point on the interface, making a shock angle
$\phi _b=90^\circ$
. Appendix C details the steps involved in measuring the transmitted shock angle and determining the errors associated with the calculation of angles from numerical contours.
Figure 8. Numerical contour of x-gradient of
$\log \,p\times \log \,\rho$
showing the BPMR configuration occurring at
$\beta = 60^\circ , \theta _w = 30^\circ$
for
$M_S = 3.4$
over the water interface, with the inset showing contours of x-momentum flux density,
$\rho u$
(min: 0, max: 15 000): i – incident shock, r – reflected shock, m – Mach stem, s – slip line, t – transmitted shock, R – refraction point, T – first triple point, k – kink.
Figure 9. Wave schematics of BPMR: m–m – undeflected interface, m–m’ – deflected interface, i – incident shock, r – reflected shock, t – transmitted shock, m – Mach stem, R – refraction point.
As the inclination angle increases, a transition from IRMR to FMR is anticipated. It is important to note that, prior to the current study, the occurrence of an FMR at the air–water interface for higher shock Mach numbers in the weak incident shock strength group was neither recognised nor was the transition between these patterns identified. Given that the reflection in air has already transitioned to an MR, it is expected, based on the work of Anbu Serene Raj et al., that the bound precursor transition should occur with an MR in air. Figure 8 reveals this transition at an inclination angle of
$60^\circ$
, marking the shift towards the FMR pattern.
Figure 10. Bound Precursor refraction with a Mach Reflection occurring at different inclination angles for shock Mach numbers 3.7 and 4 in the weak incident shock strength group, displayed in numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
: i – incident shock, r – reflected shock, m – Mach stem, s – slip line, t – transmitted shock, R – refraction point, T – first triple point, k – kink. (a)
$M_S = 3.7, \beta = 70^{\circ} (\theta_{w}= 20^{\circ}) $
and (b)
$M_S = 4, \beta = 80^{\circ} (\theta_{w}= 10^{\circ}) $
.
To distinguish this refraction pattern from the bound precursor refraction observed in the very weak incident shock strength group (BPR), it will be designated as BPMR, a pattern unique to the weak incident shock strength group in air/water interfaces. The schematic of a BPMR refraction pattern is depicted in figure 9. In the transformed frame of reference, the incident flow Mach number in water ahead of the transmitted shock is marginally supersonic, i.e.
$M_b = 1 + \epsilon$
, where
$\epsilon$
is a very small value. Based on current numerical simulations, the complementary wedge angle (
$\theta _w^c = \theta _w + \chi$
) for the BPMR transition pattern at
$M_S = 3.4$
is measured to be
$38.49^\circ$
. It is worth highlighting that, although the free precursor refractions occur when
$ M_b \lt 1$
, the conditions
$ \phi _b = 90^\circ$
and
$ M_b = 1$
lie in close proximity. Nevertheless, the sonic criterion based on
$ M_b$
is conventionally used to define the bound precursor transition (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024).
Figure 11. Shock strength groups and transition lines for various shock refraction patterns in (
$M_S,\theta _w^c$
) plane for air–water (s /f) interface: RRR – regular refraction with a reflected shock wave, FPR – free precursor refraction with an RR, FMR – free precursor Mach refraction, BPMR – bound precursor refraction with a Mach reflection, IRMR – irregular refraction with a Mach reflection,
$NRD$
– no reflection domain; I – very weak shock group, II – weak shock group, III – strong shock group,
$M_b^T=1$
– sonic line of the incident flow Mach number in the triple point’s frame of reference (dashed red line) theorised in Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024),
$M_b^R=1$
– sonic line of the incident flow Mach number in the refraction point’s frame of reference (solid red line) proposed in the present study.
The BPMR transition pattern is a critical feature of the weak incident shock strength group. To investigate further, numerical simulations are performed for other shock Mach numbers within this group to determine their respective BPMR transition angles (see figure 10). It is observed that the measured BPMR transition angles differ from the previously predicted analytical values of Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024), as illustrated by the dashed red line in figure 11. This discrepancy in the transition angle highlights the need for further refinement in the analytical modelling developed by Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024).
The following analysis builds on the observed discrepancy in the transition criteria in the weak shock strength group, providing a more refined representation of the transition criteria for bound precursor refraction patterns. In the very weak shock strength group, the bound precursor refraction occurs between an RRR and an FPR, where the reflection in the air remains as an RR. In this case, the incident flow Mach numbers in both air and water are calculated from the frame of reference attached to the refraction point,
$R$
, on the interface. However, in our previous study (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024), the transition condition for refraction patterns exhibiting irregular reflections (
$\chi \neq 0$
) in air, such as FMR, BPMR and IRMR, was calculated using the incident flow Mach number in the air relative to the triple-point frame of reference (
$M_0^T$
). Subsequently, the incident flow Mach number in water (
$M_b$
) calculated based on
$M_0^T$
was used to determine the bound precursor refraction transition angles over the air–water interface.
In irregular refraction patterns, it is the Mach stem, rather than the incident shock, that interacts with the interface and transmits into water. Consequently, the incident flow Mach number in water,
$M_b$
, should be determined by the speed of the refraction point
$R$
, i.e. the foot of the Mach stem. Currently, we have two Mach numbers:
$M_0^T$
, which corresponds to the flow Mach number in the air ahead of the incident shock wave in the triple-point frame of reference, and
$M_0^R$
, which is the flow Mach number in the air ahead of the incident shock wave in the refraction point frame of reference (refer to figure 9). Assuming that the Mach stem remains straight for the entire duration of the shock travel over the wedge, these two flow Mach numbers are related by the triple-point trajectory angle
$\chi$
as follows:
\begin{equation} M_0^R = M_0^T \cos \chi = \left (\frac {M_S}{\cos \theta _w^c}\right ) \cos \chi . \end{equation}
From (3.1),
$M_b$
is calculated as
\begin{equation} M_b^R = M_0^R \left (\frac {a_0}{a_b}\right )\!, \end{equation}
where
$a_0$
and
$a_b$
are the speeds of sound of undisturbed air and water, respectively. With the above equation for
$M_b$
, the sonic line obtained analytically (
$M_b^R=1$
: solid red line in figure 11) matches well with the numerically obtained
$\theta _w^c$
values for the BPMR cases (refer to the markers in figure 11). As anticipated, the sonic line of
$M_b^R$
begins to deviate from that of
$M_b^T$
following the detachment condition, wherein the influence of non-zero
$\chi$
from MRs becomes significant. This modified approach provides a more accurate characterisation of the transition behaviour, particularly for the irregular refraction patterns in the weak shock strength group. It is essential to recognise that the modified transition line does not cross the
$M_1=1$
line (green line in figure 11), where
$M_1$
is the Mach number of the flow behind the incident shock wave. This line marks the boundary of the no-reflection domain in the (
$M_S,\theta _w^c$
)-plane, which separates regions where reflection patterns are not possible (Vasilev et al. Reference Vasilev, Elperin and Ben-Dor2008).
The analytically obtained BPMR transition angles of
$M_S = (3,\,3.4,\,3.7,\,4,\,4.1,\,4.2)$
and the corresponding refraction patterns observed numerically are listed in table 3. It should be noted that
$\beta$
is the inclination angle defined as
$90^\circ - \theta _w$
. From the data presented in table 3, it is evident that the observed BPMR inclination angles show slight deviations from the analytically predicted
$\beta _{M_b^R=1}$
inclination angles. This discrepancy can be attributed to two factors: the variations in the first triple-point trajectory angle observed in the simulations and the assumption of a straight Mach stem. These arise due to the need for a known value of
$ \chi$
to construct the analytical
$ M_b^R=1$
transition line (refer to (3.1)). Unlike solid wedges, there is currently no analytical model available that can predict the shock reflection/refraction process in water wedges and determine the value of
$\chi$
. Consequently, in this study,
$\chi$
is computed using the analytical formulation developed for shock reflection over solid wedges (Ben-Dor Reference Ben-Dor2007), effectively assuming no shock transmission through the water wedge. A detailed quantification of this deviation from the analytical three-shock theory (3ST) predictions is provided in § 5.
Table 3. Numerically observed BPMR inclination angles compared with the analytical BPMR transition angles and their corresponding refraction patterns in the weak incident shock strength group: t-
$w$
ave – transmitted free precursor pressure wave,
$t$
– transmitted shock wave.
3.2. Refraction with transmitted precursor
$w$
ave
For inclination angles exceeding the BPMR transition angle, a free precursor refraction pattern, specifically the FMR configuration, is observed, as depicted in figure 12. At these inclination angles, the transmitted wave is seen to have detached from the refraction point R (the foot of the Mach stem) on the interface. For
$M_S=3.4$
, all
$\beta$
values above
$60^\circ$
consistently exhibit the FMR configuration as the refraction pattern, with the MR transitioning from TMR
$\rightarrow$
SMR, as seen in figure 12(b).
Figure 12. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
showing the FMR configuration occurring at higher inclination angles (a)
$\beta = 65^{\circ}$
,
$\theta_{w} = 25^{\circ}$
and (b)
$\beta = 80^\circ$
,
$\theta_{w} = 10^{\circ}$
for
$M_S = 3.4$
over the water interface, with the inset showing contour of x-momentum flux density,
$\rho u$
(min: 0, max: 15 000) : i – incident shock, r – reflected shock, m – Mach stem, s – slip line, t-
$w$
ave – transmitted pressure wave, R – refraction point, T – first triple point, k – kink.
Compared with the shock Mach numbers in the very weak shock strength group considered in the prior study by Anbu Serene Raj et al. (Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024), such as
$M_S=1.46,\,1.7,\,2.2$
, the development of the Richtmyer–Meshkov instability (RMI) at the interface is significantly more pronounced in the weak shock strength group. Furthermore, based on the framework for inclined interface RMI (McFarland et al. Reference McFarland, Reilly, Creel, McDonald, Finn and Ranjan2014), the perturbations considered in this study are nonlinear, as all the inclination angles exceed
$22^\circ$
. As a result, the leading edge of the interface rapidly curls upon shock wave traversal, resulting in the characteristic spike formation associated with RMI, where the heavier fluid penetrates into the lighter fluid. For inclination angles beyond
$\beta =55^\circ$
, the spike and mushroom-like structures form so rapidly that a localised shock wave is generated behind the roll-up, as highlighted in figure 12(b). The localised shock formation is caused by the interaction between the accelerated post-shock flow that reaches supersonic velocities and the vortical structures at the interface formed from RMI development. The Mach number contour of air shown in figure 13 reveals that a local supersonic patch bounded by the sonic line (black line) is located adjacent to the vortical region, and it is clear that this interaction demands the generation of a shock wave to appropriately turn the flow. The localised shock is more pronounced at lower wedge angles, such as
$\beta =80^\circ$
. The interface dynamics is distinctly captured in the numerical simulations, highlighting the impact of higher shock strengths on pseudosteady multiphase shock refraction phenomena.
Figure 13. Numerical Mach number contour for the gas phase (air) of an FMR at
$M_S = 3.4, \beta =70^\circ$
.
The analysis of the results corresponding to
$M_S=3.4$
(as shown in figures 4, 6, 8, 12) confirms numerically that the refraction sequence within the weak incident shock strength group is consistent with the analytical prediction reported in our earlier study (Anbu Serene Raj et al. Reference Anbu Serene Raj, Vishnu Prasad, Rajesh and Sameen2024) as RRR
$\rightarrow$
IRMR
$\rightarrow$
(BPMR)
$\rightarrow$
FMR. However, the shock Mach number
$(M_S)$
range for the weak incident shock strength group presented in the current work has been determined from the intersection of the sonic line corresponding to
$M_b^R$
with the detachment condition of the solid wedges,
$\theta _D$
(refer to figure 11). In the absence of an analytical model for the detachment condition in water wedges, the solid-wedge detachment criterion has been used as an approximation. Previous experimental investigation by Takayama & Ben-Dor (Reference Takayama and Ben-Dor1989) and numerical studies by Wan et al. (Reference Wan, Jeon, Deiterding and Eliasson2017) have demonstrated that the detachment condition for water wedges is lower than that for solid wedges. This discrepancy implies that the true lower bound of the weak shock strength group is likely to be higher than the value estimated using the solid-wedge model. To investigate this further, additional numerical simulations were performed over the range
$M_S = 2.7 {-} 3$
in increments of 0.05, covering all wedge inclination angles. These simulations revealed that, up to
$M_S = 2.9$
, the refraction pattern followed the sequence associated with very weak incident shocks (RRR
$\rightarrow$
BPR
$\rightarrow$
FPR
$\rightarrow$
FMR). At
$M_S = 2.9$
the BPR, FPR and FMR transitions occurred at
$ \beta = 42^\circ$
,
$ 42.5^\circ$
and
$ 43^\circ$
, respectively. For
$M_S = 3.0$
, however, BPMR was clearly observed at
$ \beta = 46.25^\circ$
, confirming the onset of the weak incident shock strength group (see figure 14). This discrepancy exposes a critical gap: the lack of an analytical model for the detachment condition in water wedges.
These observations substantiate two key insights:
-
(i) the actual boundary between the very weak and weak incident shock strength groups lies somewhere between
$ M_S = 2.9$
and
$ M_S = 3$
, higher than the analytically predicted threshold of
$ M_S = 2.725$
, calculated from the solid-wedge detachment criteria; and -
(ii) there exists a downward shift in the detachment angle
$ \theta _D$
for water wedges at higher shock Mach numbers, a trend originally hypothesised by Wan et al. (Reference Wan, Jeon, Deiterding and Eliasson2017) through their numerical studies (refer to figure 11).
Resolving this discrepancy and accurately defining the group boundary requires an analytical model for air–water refraction systems that incorporates energy loss mechanisms during shock transmission into water.
Figure 14. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
showing the bound precursor configurations occurring near the actual very weak –weak incident shock strength group boundary: i – incident shock, r – reflected shock, m – Mach stem, m’ – secondary Mach stem, r’ – secondary reflected shock, T – first triple point, T’ – second triple point. (a)
$\textit{BPR}: M_S = 2.9, \beta = 42^{\circ}, \theta = 48^{\circ}) $
and (b)
$\textit{BPMR}: M_S = 3, \beta = 46.25^{\circ}, \theta_{w} = 43.75^{\circ}) $
.
It is also important to emphasise that the instability development in these simulations reflects the inviscid RMI, neglecting the effects of viscosity, surface tension and water phase change. In experimental scenarios, these factors may significantly influence interface growth and evolution and their interaction with the shock refraction pattern. However, the time scales associated with the refraction pattern are small; the inviscid RMI development will be sufficient to accurately characterise the refraction pattern. Future research may include additional diffusional parameters to accurately simulate the RMI, thereby enhancing current findings and providing a more thorough understanding of instability dynamics in the weak shock strength group.
4. Strong incident shock strength group
Table 2 reveals that, for any shock Mach number higher than 4.325, the shock refraction sequence over an air–water interface will be RRR
$\rightarrow$
IRMR. To verify this prediction, numerical simulations are performed for a chosen shock Mach number of 4.4 across a range of inclination angles.
Figure 15. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
and inset of x-momentum flux density,
$\rho u$
(min: 0, max: 15 000), for an incident shock with shock Mach number
$M_S = 4.4$
refracting over water interface at different inclination angles: i – incident shock, r – reflected shock, r’ – secondary reflected shock, m – Mach stem, m’ – secondary Mach stem, s – slip line, t – primary transmitted shock, t’ – secondary transmitted shock,
$t_r$
– reflected transmitted-shock, R – refraction point, T – first triple point, T’ – second triple point, k – kink. (a)
$\beta=35^{\circ}, \theta_{w} = 55^{\circ} $
, (b)
$\beta=45^{\circ}, \theta_{w} = 45^{\circ} $
, (c)
$\beta=55^{\circ}, \theta_{w} = 35^{\circ} $
, (d)
$\beta=60^{\circ}, \theta_{w} = 30^{\circ} $
, (e)
$\beta=70^{\circ}, \theta_{w} = 20^{\circ} $
, (f)
$\beta=85^{\circ}, \theta_{w} = 5^{\circ} $
.
The resulting shock refraction patterns are presented in figure 15. As observed in figure 15(a), at an inclination angle of
$\beta =35^\circ$
, the refraction pattern is an RRR configuration. Notably, the transmitted shock exhibits an almost normal orientation to the bottom wall, devoid of any curvature, unlike the pattern observed in figure 4. The transmitted shock angle is calculated to be approximately
$ 34.65^\circ \pm 0.5^\circ$
, making its inclination with respect to the bottom wall
$ 89.65^\circ \pm 0.5^\circ$
, confirming the near-normal orientation. Additionally, the speed of the reflection point of the transmitted shock on the bottom wall (along the
$x$
-direction) is approximately 1516 m/s, which is marginally higher than the speed of sound in water. This causes the transmitted shock to reach the wall before the corner-generated wave. For a given
$ M_S$
, there exists a threshold inclination angle at which the transmitted shock angle
$ \phi _b$
becomes equal to the wedge angle
$ \theta _w$
. When the inclination angle is below this threshold, the corner-generated wave always overtakes the transmitted shock, leading to shock curvature near the wall. In contrast, when the inclination angle exceeds this threshold, the corner-generated wave never catches up to the transmitted shock, and the transmitted shock angle increases in such a way that it reflects obliquely from the bottom wall.
At
$\beta =45^\circ ,55^\circ ,60^\circ$
, the refraction pattern has transitioned into an IRMR with a DMR in air, as shown in figures 15(b), 15(c), 15(d), respectively. For inclination angles exceeding
$55^\circ$
, the transmitted shock angle increases to the extent that it reflects off the bottom wall, forming a discernible reflected–transmitted shock denoted as
$t_r$
. This reflection constitutes an RR of an oblique shock within the water. Figure 16 shows the extracted pressure plot at two different
$y$
locations of an IRMR occurring at
$\beta = 70^\circ$
. It is observed that, as the reflected–transmitted shock
$t_r$
reaches the water–air interface, it is significantly attenuated and exhibits a smooth profile, thereby degenerating into a compression wave. With further increases in
$\beta$
, the refraction pattern remains an IRMR. However, the reflection pattern evolves in air, transitioning into a TMR at
$70^\circ$
and subsequently into an SMR at
$85^\circ$
.
These findings confirm that the shock strength in the strong incident shock group is sufficiently high to prevent the occurrence of free precursor refractions. Regardless of the inclination angle, a shock wave is transmitted into the water. Therefore, the refraction sequence for the strong incident shock strength group is verified to be RRR
$\rightarrow$
IRMR. The increased shock strength in this group results in significantly higher baroclinic vorticity deposition along the interface. Consequently, the development of the RMI is accelerated and more pronounced compared with the very weak and weak incident shock strength groups. Interface roll-up begins even at small inclination angles, exhibiting the localised shock wave generations even from
$60^\circ$
and becomes apparent at higher inclination angles, such as that seen in figure 15( f).
Figure 16. Numerical contour of an IRMR at
$M_S = 4.4, \beta =70^\circ$
along with the extracted pressure data plot along a horizontal line: dashed lines – pressure data extraction lines at two different
$y$
locations.
5. Water wedge vs solid wedge
In § 3.1.3, deviations of triple-point trajectories in water-wedge cases from the analytical predictions based on the 3ST (of the solid wedge) were observed. To quantify these deviations, this section presents a comparative study of the interactions of a pseudosteady shock with both solid and water wedges. It should be noted that the trajectory angles of the triple point (
$\chi$
) and also the kink (
$\chi '$
) are measured from the line joining the leading edge of the water wedge (of the undisturbed interface) and the respective points (triple point
$T$
or kink
$k$
). The first point at which the reflected shock
$r$
slope changes is taken to be the kink, as illustrated in figures 5 and 17.
Figure 17. The numerical x-gradient contours of
$\log \,p\times \log \,\rho$
of
$M_S = 4$
,
$\theta _w = 20^\circ$
for (a) a solid-wedge case showcasing TMR and (b) a water-wedge case (brown) showcasing an IRMR and their corresponding magnified views (c) and (d) showing the kink and triple-point locations in detail: i – incident shock, r – reflected shock, m – Mach stem, m’ – secondary Mach stem, t – transmitted shock, T – first triple point, k – kink, s – slip line.
Initially, to compare the shock structures between a solid wedge and a water wedge, a shock Mach number of 4 and a wedge angle of
$\theta _w = 20^\circ$
are chosen. The results extracted at
$t = 26\,{\unicode{x03BC}} \textrm {s}$
for both solid and water wedges are shown in figure 17 to highlight the differences in the shock structures. In the solid-wedge case, a TMR is observed, with a kink (
$k$
) on the reflected shock wave. The trajectory of the triple point
$\chi$
is measured to be
$13.2^\circ$
, while the kink’s trajectory
$\chi '$
is measured at
$16^\circ$
. For the water-wedge case, an IRMR is observed, as shown in figure 17(b). The IRMR comprises a transmitted shock wave (
$t$
) within water attached at the foot of the Mach stem on the interface, alongside an MR in air (in this case, a TMR configuration), featuring a triple-point trajectory and kink angles of
$\chi = 12.7^\circ$
and
$\chi ' = 15.3^\circ$
, respectively. It is noteworthy that the Mach stem appears more curved at its foot near the interface compared with that observed for the solid wedge. The slight deviation in triple point and kink trajectories reflects the influence of the interface dynamics. Since the shock Mach number for the present case belongs to the strong shock strength group, the shock impingement will deposit a large amount of baroclinic vorticity to the air–water interface, initiating the RMI. The growth of RMI causes the leading edge of the water wedge to curl up into distinct spikes, as explained in the previous sections. This deformation coincides with the upstream displacement of the secondary Mach stem,
$m'$
, compared with the solid-wedge case, where it remains closely anchored to the leading edge, as evident from figure 17. While such features are indicative of RMI growth, other instability mechanisms may also contribute in the early stages. A detailed instability analysis, which remains underexplored in the literature, is needed to provide further clarity. This effect, coupled with energy loss due to transmission (as discussed by Wan et al. (Reference Wan, Jeon, Deiterding and Eliasson2017)), may be attributed to the reason why the triple-point trajectories of the water-wedge cases differ significantly from those observed in the solid-wedge cases.
To obtain a clearer understanding of the differences between the shock structures from a solid wedge and a water wedge, other shock Mach numbers ranging from 2.4 to 4 are also considered. Figures 18 and 19 compare the shock patterns obtained for shock Mach numbers 2.4, 2.7, 3 and 3.4, 3.7, respectively, over a solid and water wedge. The comparisons for solid and water wedges are carried out at the same time instance for a particular shock Mach number. It should also be noted that the time instance of
$t = 26\,{\unicode{x03BC}} \textrm {s}$
is specific to the
$M_S = 4$
case and does not apply uniformly to other shock Mach numbers. Given that the incident shock speeds vary significantly across the range of
$M_S = 2.4$
to
$M_S = 4$
, a fixed time instance would not provide a meaningful comparison. To ensure clarity in visualising the shock reflection/refraction process, the time instance for each case has been carefully chosen such that the shock wave is approximately at the midpoint of the solid or water wedge. For a wedge angle of
$20^\circ$
, the solid wedge produces an SMR at
$M_S$
= 2.4, 2.7 (figure 18
a, c), whereas it produces a TMR, identified by the presence of a kink that marks the inflexion point on the reflected shock wave, at
$M_S$
= 3, 3.4 and 3.7 (figures 18
e, 19
a, 19
c). The water-wedge cases, namely, figures 18(b), 18(d), 18( f), 19(b), all exhibit the same refraction pattern: a free precursor refraction with a Mach reflection (FMR). It should be noted that the FMR cases with a lower shock Mach number produce a refraction pattern where the irregular reflection in the air is farther away from the free precursor wave inside water along the interface. As the shock Mach number increases, the distance between the refraction point
$R$
and the transmitted free precursor wave reduces. For
$M_S$
= 3.7, a BPMR is observed as shown in figure 19(d). As the incident shock gets stronger, the shock refraction pattern transitions from a free precursor refraction eventually into an IRMR with a transmitted shock wave inside water (figure 17
b).
Figure 18. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
and inset of x-momentum flux density,
$\rho u$
(min:0, max:7000), showing incident shocks with various shock Mach numbers refracting over solid wedges of
$\theta _w = 20^\circ$
and water wedges of
$\theta _w = 20^\circ$
,
$\beta = 70^\circ$
: i – incident shock, r – reflected shock, m – Mach stem, m’ – secondary Mach stem, T – first triple point, k – kink, s – slip line. (a) Solid wedge:
$M_{S}=2.4$
, (b) Water wedge:
$M_{S}=2.4$
, (c) Solid wedge:
$M_{S}=2.7$
, (d) Water wedge:
$M_{S}=2.7$
, (e) Solid wedge:
$M_{S}=3$
, (f) Water wedge:
$M_{S}=3$
.
Figure 19. Numerical contours of x-gradient of
$\log \,p\times \log \,\rho$
showing incident shocks with various shock Mach numbers refracting over solid wedges of
$\theta _w = 20^\circ$
and water wedges of
$\theta _w = 20^\circ$
,
$\beta = 70^\circ$
: i – incident shock, r – reflected shock, m – Mach stem, m’ – secondary Mach stem, t – transmitted shock, T – first triple point, k – kink, s – slip line. (a) Solid wedge:
$M_{S}=3.4$
, (b) Water wedge:
$M_{S}=3.4$
, (c) Solid wedge:
$M_{S}=3.7$
, (d) Water wedge:
$M_{S}=3.7$
.
A comparison of the first triple-point trajectory between the solid and water wedges was conducted for various shock Mach numbers to examine the nature of the deviation at different shock strengths. For the same wedge angle, the variation of the first triple-point trajectory
$\chi$
with respect to shock Mach number
$M_S$
is plotted in figure 20, along with associated errors (refer to Appendix D for error propagation calculations). The results for the solid wedge are compared with analytical 3ST predictions (Ben-Dor Reference Ben-Dor2007). The present simulations reveal a deviation in triple-point trajectories for solid wedges that increases with shock strength, consistent with the findings of Wan et al. for higher wedge angles. This deviation has been attributed to the inherent assumptions used in closing the 3ST analytical models (Law & Glass Reference Law1971). In contrast, the
$\chi$
values for the water wedge exhibit better agreement with analytical 3ST predictions. As expected, for both solid and water wedges, the discrepancy from analytical
$\chi$
increases with shock Mach number.
Figure 20. Comparison of the variation of the first triple-point trajectory (
$\chi$
) with respect to shock Mach number (
$M_S$
) for solid wedge (
$\theta _w = 20^\circ$
), water wedge (
$\beta = 70^\circ$
) and analytical 3ST (Ben-Dor Reference Ben-Dor2007).
Figure 21. Comparison of
$\omega _{ir}$
for solid wedge (
$\theta _w = 20^\circ$
) and water wedge (
$\beta = 70^\circ$
) with respect to various shock Mach numbers (
$M_S$
).
In addition to the change in the triple-point trajectory angles, the solid and water wedges exhibit minor differences in the transition of the reflection pattern observed in the air. In shock reflection studies,
$\omega _{ir}$
is defined as the angle between the incident shock
$i$
and the reflected shock
$r$
. A TMR is said to have transitioned into an SMR when
$\omega _{ir}$
shifts from an obtuse angle to an acute angle (Ben-Dor Reference Ben-Dor2007). The solid-wedge TMR
$\rightarrow$
SMR transition condition is specified as
$\omega _{ir} = 90^\circ$
(refer to the solid orange line in figure 11). In water wedges, the interplay between the diminished
$\chi$
and the RMI evolution influences the shock structure and alters
$\omega _{ir}$
. Consequently, the transition from TMR to SMR is affected by the water wedge. It should be noted that, in solid-wedge reflections, these quantities are known to be interdependent through analytical models. While no such model yet exists for air–water shock refractions, the observed trends suggest a similar coupling. Establishing a quantitative analytical framework to describe this coupling in multiphase systems remains an important direction for future research.
From the simulated results,
$\omega _{ir}$
is extracted for both solid and water wedges and is plotted against the shock Mach number in figure 21. It is observed that, when compared with the solid wedges, water wedges result in a lower
$\omega _{ir}$
across all shock Mach numbers, with the difference increasing with increasing shock strength. Specifically, for
$\theta _w=20^\circ$
, the TMR to SMR transition occurred at a shock Mach number of 2.7 for the solid wedge, while it occurred at a higher shock Mach number of 2.8 for the water wedge.
These results emphasise the significant distinctions between the shock reflection and refraction patterns observed for a given shock Mach number and wedge angle in solid and water wedges. The differences in triple-point trajectories, shock reflection transitions and the impact of interface instability in water wedges highlight the unique dynamics at play in these two scenarios.
6. Conclusion
The pseudosteady shock refraction sequence over a water wedge predicted in our previous study for the weak shock strength group is numerically verified to be RRR
$\rightarrow$
IRMR
$\rightarrow$
(BPMR)
$\rightarrow$
FMR at different shock Mach numbers ranging from 3 to 4.2. The characteristic transition pattern of the weak shock strength group is identified to be the BPMR. The results indicate that the incident flow Mach number in air with reference to the refraction point (
$M_0^R$
), rather than the triple point (
$M_0^T$
), should be used to compute the incident flow Mach number inside water (
$M_b$
) to obtain proper transition angles. This approach incorporates the impact of the triple-point trajectory, resulting in a modified sonic line based on
$M_b^R$
for the bound precursor to free precursor refraction transition. In the strong shock strength group, numerical simulations confirm the refraction sequence to be RRR
$\rightarrow$
IRMR, where the strength of the incident shock is sufficient to eliminate free precursor refractions. Furthermore, a detailed parametric sweep with narrow spacing in
$ M_S$
and
$ \beta$
was conducted near the analytically predicted lower bound of the weak incident shock strength group to validate the transition threshold. The results revealed that the actual boundary deviates from the analytical estimate of
$ M_S = 2.725$
, instead lying between
$ M_S = 2.9$
and
$ M_S = 3$
. This deviation arises primarily from the use of solid-wedge detachment criteria in the absence of an analytical model for water wedges.
Computations show that the first triple-point trajectory of the shock reflection patterns from the solid wedge is higher than that of the refraction patterns over the water wedge for all shock Mach numbers due to the loss of energy into transmission. Interestingly, the water-wedge
$ \chi$
values follow the analytical 3ST predictions more closely than those of the solid wedge, corroborating trends observed in previous numerical studies in the literature. The finding highlights the need for developing dedicated analytical frameworks tailored to air–water shock refraction to more accurately define regime boundaries and transition lines. Due to the stronger nature of the shock waves, the interface is seen to quickly develop into a mushroom-like spike structure, pushing the reflected wave farther upstream. This phenomenon significantly affects the angle between the incident and the reflected shock wave, leading to a delayed SMR
$\rightarrow$
TMR transition over water wedges.
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Appendix A. BlastFoam: solver validation
The BlastFoam Euler solver is validated with a shock wave–water column interaction problem studied experimentally by Igra & Takayama (Reference Igra and Takayama2001). Chen & Liang (Reference Chen and Liang2008) performed numerical simulations and validated the results against the interferogram images of Igra & Takayama. In their numerical study, water is modelled as a stiffened gas with
$\gamma = 1.932$
and
$p_\infty = 1.1645\times 10^9\,\textrm {Pa}$
. The current simulation adopts a computational grid resolution of
$\delta x = \delta y = 0.025\,\mathrm{mm}$
, consistent with the methodology of Chen & Liang. The problem set-up is illustrated in figure 22.
Figure 22. Two-dimensional shock–single water column interaction test case pressure contour at
$t=0\,\textrm {s}$
: black solid line represents the air–water interface (
$\alpha =0.5$
).
This two-dimensional test case consists of a moving shock wave with a shock Mach number of 1.47, which is initialised
$4\,\mathrm{mm}$
from the centre of a water column with a diameter of
$4.8\,\mathrm{mm}$
. The numerical domain is a square region of side length
$40\,\mathrm{mm}$
, extending in the range
$[-20,20]$
, with the water column positioned at the origin
$(0,0)$
. The pre- and post-shock conditions, based on the location of the shock wave, are initialised within the domain as follows:
\begin{align} \left (\begin{array}{c} \alpha \\ p \\ \rho \\ \gamma \\ p_\infty \end{array}\right )_{0,\textit{air}}&=\left (\begin{array}{c} 0 \\ 10^5 \mathrm{\,Pa} \\ 1.2 \mathrm{\,kg}\, \mathrm{m^{-3}} \\ 1.4 \\ 0 \mathrm{\,Pa} \end{array}\right )\!, \end{align}
\begin{align} \left (\begin{array}{c} \alpha \\ p \\ \rho \\ \gamma \\ p_\infty \end{array}\right )_{1,\textit{air}}&=\left (\begin{array}{c} 0 \\ 2.354\times 10^5 \mathrm{\,Pa} \\ 2.212 \mathrm{\,kg}\, \mathrm{m^{-3}} \\ 1.4 \\ 0 \mathrm{\,Pa} \end{array}\right )\!, \end{align}
\begin{align} \left (\begin{array}{c} \alpha \\ p \\ \rho \\ \gamma \\ p_\infty \end{array}\right )_{0,w\textit{ater}}&=\left (\begin{array}{c} 1 \\ 10^5 \mathrm{\,Pa} \\ 1053 \mathrm{\,kg} \, \mathrm{m^{-3}} \\ 2.8 \\ 8.5\times 10^8 \mathrm{\,Pa} \end{array}\right )\!. \end{align}
The instantaneous plots at times
$t=0,5,15,23,43\,{\unicode{x03BC}} \textrm {s}$
are shown in figure 23. The left panel displays pressure contours, while the right panel shows contours of the gradient of
$\log \,p\times \log \,\rho$
. The pressure contour plots are compared with those presented by Chen & Liang at each time step. In these plots, the interface between air and water is marked as a solid black line, corresponding to a volume fraction value of 0.5.
Figure 23. Comparison of a moving shock wave of
$M_S=1.47$
interacting with a two-dimensional water column at various instances: RS – reflected shock wave, TW – transmitted Wave, DS – diffracted shock, SL – slip Line, VP – vortex Pair, M1, M2, TP1, TP2 are the first and second Mach stems and triple Point, respectively. (a)
$t = 5\,\mu\textrm{s} $
, (b)
$t = 15\,\mu\textrm{s} $
, (c)
$t = 23\,\mu\textrm{s} $
, (d)
$t = 43\,\mu\textrm{s} $
.
At the initial interaction of the incident shock wave with the water column, part of the shock wave is reflected as a reflected shock wave (RS), while the remainder is transmitted into the water column, propagating as a transmitted wave (TW). The transmitted precursor wave (TW) travels faster than the incident shock wave, as shown in figure 23(a). As the incident shock wave progresses beyond the centre of the water column and reaches the convex surface, it diffracts into a curved shock wave, referred to as the diffracted shock wave (DS), as illustrated in figure 23(b). By
$t=15\,{\unicode{x03BC}} \textrm {s}$
, a vortex pair (VP) begins to form on the rear side of the water column. The diffraction of the incident shock wave produces an MR configuration with a triple point (TP1).
At
$t=23\,{\mu}\textrm{s}$
, the diffracted shocks from the top and bottom surfaces of the water column interact, leading to the formation of another MR configuration, as shown in figure 23(c). By
$t=43\,{\mu}\textrm{s}$
, as the incident shock continues to propagate to the right, the second MR configuration, including TP2 and M2, grows substantially along the centreline. Simultaneously, the transmitted wave reflects back and forth within the water column as compression and expansion waves, creating alternating regions of high and low pressure as time progresses.
Although the current study employs a different combination of (
$\gamma$
,
$p_\infty$
), the results show excellent agreement with those reported by Chen & Liang (Reference Chen and Liang2008).
Appendix B. In-house TENO code: Grid independence study
Grid independence study was conducted for the solid-wedge case with shock Mach number
$M_S=3$
and wedge angle
$\theta _w=20^\circ$
, representative of the configurations used for comparison throughout the study. Four mesh resolutions were considered: 0.1, 0.05, 0.025 and 0.010 mm. The pressure contour for the 0.025 mm case is shown in figure 24(a). To quantitatively assess grid convergence, pressure profiles were extracted along a horizontal probe line at y = 0.015 mm. The resulting plots of pressure versus horizontal distance are shown in figure 24(b). The zoomed-in pressure plots near the reflected shock in figure 24(c) and Mach stem in figure 24(d) clearly demonstrate that both the 0.025 and 0.010 mm meshes resolve the shock structures with nearly identical accuracy, while the coarser meshes show higher dissipation. These results confirm that the 0.025 mm resolution used throughout the study is sufficient to capture the essential shock features. (a) Numerical pressure contour of the solid wedge:
$M_{S} = 3 ,\theta_{w}=20^{\circ}$
, (b) Extracted pressure plot at
$y=0.015\,\text{mm} $
of four different grids. (c) The magnified view of the pressure profiles near the reflected shock, (d) The magnified view of the pressure profiles near the Mach stem.
Figure 24. Grid independence study for the in-house TENO code used for solid-wedge simulations. (a) Numerical pressure contour of the solid wedge:
$M_{S} = 3 ,\theta_{w}=20^{\circ}$
, (b) Extracted pressure plot at
$y=0.015\,\text{mm} $
of four different grids. (c) The magnified view of the pressure profiles near the reflected shock, (d) The magnified view of the pressure profiles near the Mach stem.
Appendix C. Error propagation:
$\boldsymbol{\phi}_{\boldsymbol{b}}, \boldsymbol{\omega}_{\boldsymbol{ir}}$
calculation
The following methodology describes the steps involved in determining the errors associated with the calculation of angles from numerical contours that involve three points or pixel locations, such as the transmitted shock angle (
$\phi _b$
) and the angle between the incident and the reflected shocks (
$\omega _{ir}$
).
In a BPMR, the perpendicularity of the transmitted shock at the point of refraction
$R$
is quantitatively evaluated using the geometric relationship between the interface and the tangent of the transmitted shock front at
$R$
, as illustrated in figure 25. In this figure, the red dashed line represents the undisturbed interface, and the blue dashed line denotes the tangent to the transmitted shock front, both intersecting at the refraction point
$R$
. The angle
$ \phi _b$
is defined as the angle between these two lines and is calculated using three points:
-
(i)
$ A$
on the interface; -
(ii)
$ B$
on the tangent to the transmitted shock; and -
(iii)
$ R$
, the point of refraction on the interface.
Figure 25. Numerical contour of a BPMR at
$M_S = 3.7, \beta =70^\circ$
: dashed red line - undisturbed interface, dashed blue line - tangent to the transmitted shock wave
$t$
at the point of refraction
$R$
.
If these points correspond to pixel coordinates
$(x_2, y_2)$
,
$(x_3, y_3)$
and
$(x_1, y_1)$
, respectively, the transmitted shock angle
$ \phi _b$
is then computed using the dot product formula
\begin{equation} \phi _b = \arccos \left ( \frac {(x_2 - x_1)(x_3 - x_1) + (y_2 - y_1)(y_3 - y_1)}{\sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2} \boldsymbol{\cdot }\sqrt {(x_3 - x_1)^2 + (y_3 - y_1)^2}} \right ). \end{equation}
To assess the accuracy of this method, we have derived a complete error propagation expression (refer to (C2)) for
$ \phi _b$
, which accounts for pixel-level uncertainties in the extracted point coordinates
\begin{equation} \boldsymbol{\nabla }\phi _b = \sqrt { \left (\! \frac {\partial \phi _b}{\partial x_1} \boldsymbol{\nabla }x \!\right )^2 \!\!+\! \left ( \!\frac {\partial \phi _b}{\partial x_2} \boldsymbol{\nabla }x \!\right )^2\!\! + \!\left ( \!\frac {\partial \phi _b}{\partial x_3} \boldsymbol{\nabla }x \right )^2 \!\!+ \!\left ( \!\frac {\partial \phi _b}{\partial y_1} \boldsymbol{\nabla }y \!\right )^2 \!\!+ \!\left ( \!\frac {\partial \phi _b}{\partial y_2} \boldsymbol{\nabla }y \right )^2 \!\!+ \!\left ( \!\frac {\partial \phi _b}{\partial y_3} \boldsymbol{\nabla }y \!\right )^2 }. \end{equation}
In (C2),
$\boldsymbol{\nabla }\phi _b$
,
$\boldsymbol{\nabla }x$
and
$\boldsymbol{\nabla }y$
denote the error in transmitted shock angle and pixel uncertainties in the
$x,y$
coordinates, respectively. Based on this formulation, it is found that a conservative estimate of pixel error of
$ \boldsymbol{\nabla }x = \boldsymbol{\nabla }y = \pm 2$
pixels (as the transmitted shock inside water is slightly thicker owing to its weak nature) results in an angular uncertainty of approximately
$ \boldsymbol{\nabla }\phi _b = \pm 0.6^\circ$
, and always within
$ \pm 1^\circ$
for all the cases examined. This methodology ensures a systematic and reproducible way of verifying the condition
$ \phi _b = 90^\circ$
, which marks the transition between IRMR and FMR in the weak incident shock strength group.
In a similar way, error propagation in
$\omega _{ir}$
can also been found using three points:
-
(i)
$ A$
on the incident shock, -
(ii)
$ B$
on the reflected shock (before the kink/second triple point, in case of a TMR/DMR), and -
(iii)
$ T$
at the triple point.
Appendix D. Error propagation:
$\boldsymbol{\chi}, \boldsymbol{\chi'}$
calculation
The following methodology describes the steps involved in determining the errors associated with the calculation of angles from numerical contours that involve two points or pixel locations. Let
$\theta$
be the angle of a line joining two points
$P_1$
and
$P_2$
with respect to the horizontal. These points are defined by the pixel locations (
$x_1, y_1$
) and (
$x_2, y_2$
), respectively. Therefore, the angle the line makes with the horizontal is defined by
\begin{equation} \theta = \tan ^{-1} (z) = \tan ^{-1} \left (\frac {{\rm d}y}{{\rm d}x} \right ) = \tan ^{-1} \left (\frac {y_2 - y_1}{x_2 - x_1} \right ). \end{equation}
If the error in pixel location is given as
$\Delta x_1$
,
$\Delta y_1$
,
$\Delta x_2$
,
$\Delta y_2$
for
$x_1$
,
$y_1$
,
$x_2$
and
$y_2$
respectively, then errors of
${\rm d}x$
and
${\rm d}y$
are given by
\begin{equation} {\Delta {\rm d}x}^2 = {\Delta x_1}^2 + {\Delta x_2}^2, {\Delta {\rm d}y}^2 = {\Delta y_1}^2 + {\Delta y_2}^2, \end{equation}
respectively. Therefore, the error of
$z={\rm d}y/{\rm d}x$
is calculated by
\begin{equation} {\Delta z}^2 = z^2 \left ( \left (\frac {\Delta {\rm d}y}{y} \right )^2 + \left (\frac {\Delta {\rm d}x}{x}\right )^2 \right ). \end{equation}
With the error of
$z$
now known, the error associated with
$\theta$
can be determined as
\begin{equation} \Delta \theta = \frac {\Delta z}{1 + z^{2}}. \end{equation}
The above methodology outlines the steps for calculating errors in angle measurements from numerical contours, taking into account pixel location uncertainties to produce error bars.
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