3.1. Stabilized reflection of ODWs with ζ = 0

The reflection patterns of ODWs in the combustor at different flight Mach numbers with the reflection point located precisely at the expansion corner (i.e. ζ = 0) are analysed first. The time-dependent simulation results suggest that the flow structures with ζ = 0 all remain stabilized in the combustor at the investigated Mach numbers (9, 9.5, 10 and 10.5). Taking Ma = 9 and 10 as examples, the stabilized flow structures are depicted in figure 3 by shadowgraphs, temperature contours, H₂ mass fraction contours and pressure isolines. When the high-speed premixed H₂–air mixture flows into the combustor, an OSW first forms over the leading tip of the cowl, inducing combustion of the mixture downstream. Away from the cowl wall, the combustion zone approaches the OSW front and eventually interacts with the shock-wave front over a short distance. Consequently, the flame front is coupled tightly with the OSW front, and the shock-wave angle increases, implying the formation of an ODW. Across this ODW, the flow temperature abruptly increases to approximately 3000 K, and the H₂ mass fraction drops to a rather low level; these phenomena are the other two typical features of a detonation wave. Then, the ODW reflects at the expansion corner within the combustor, forming a reflected shock wave (RfSW), which is weak due to the parallel geometry of the flow channel behind the expansion corner. Downstream in the combustor, the RfSW reflects repeatedly between the internal walls and further weakens gradually before it meets another expansion fan at the outlet of the combustor. Finally, the high-temperature combustion products exit the combustor and expand and accelerate in the nozzle.

Figure 3. Flow fields of the stabilized reflection of ODWs with ζ = 0 at Ma = 9 (left) and Ma = 10 (right): (a) shadowgraphs, (b) temperature contours and (c) H₂ mass fraction contours.

A comparison of the flow structures between these two Mach numbers in figure 3 reveals that the initiation length under Ma = 10 is shorter than that under Ma = 9. A transverse wave (TW) connecting to the OSW–ODW transition point exists in the case of Ma = 9, implying an abrupt OSW–ODW transition pattern at this relatively low Mach number, while no obvious TW can be observed within the initiation zone in the case of Ma = 10, meaning the OSW–ODW transition pattern at this relatively high Mach number is smooth. These findings regarding the ODW initiation structure coincide with those found in previous numerical studies (Teng & Jiang Reference Teng and Jiang2012; Teng et al. Reference Teng, Ng and Jiang2017, Reference Teng, Tian, Zhang, Zhou and Ng2021). Notably, as compared to the geometric scales of the combustor, the OSW–ODW transition zones are rather small, implying that the overall flow fields in the combustor are in near-equilibrium states, although some non-equilibrium effects still occur locally. Further, a higher Mach number leads to a smaller oblique detonation angle, and thus the leading tip of the cowl occurs farther upstream from the expansion corner for ζ = 0 (with the same height of the combustor). Additionally, the RfSW angles and its further reflected shocks are smaller for higher Mach numbers. Moreover, the combustion temperature in the combustor at Ma = 10 is slightly higher than that at Ma = 9, which is attributed to the higher total temperature but the same deflection angle of the inflow entering the combustor.

3.2. Stabilized Mach reflection of ODWs with small ζ > 0

When the ideal reflection point of the ODW is located on the wall before the expansion corner (i.e. for ζ > 0), the Mach reflection pattern of the ODW coincides with the formation of an MS. For small ζ > 0, the flow field under Ma = 10 and ζ = 10/60 is taken as an example. Figure 4 depicts the evolution of the flow field in this case by shadowgraphs overlaid with sonic lines. To quantitatively exhibit the evolution of the formed MS and the stabilization characteristics of the Mach reflection of the ODW, the temporal evolutions of the location and corresponding motion speed of the pressure-jump point along the wall before the expansion corner are plotted in figure 5, where the location of the pressure-jump point is evaluated according to its distance from the expansion corner. Obviously, this pressure-jump point represents the location of the ODW reflection point for a regular reflection pattern or the location of the MS for a Mach reflection pattern. Therefore, figure 5 also presents the temporal evolutions of the formed MS when a Mach reflection pattern appears in the late period of the evolution of the flow field. In addition, points at different time instants corresponding to the shadowgraphs shown in figure 4 are also marked on the motion speed curve in figure 5 for convenience.

Figure 4. Evolution of the flow field (shadowgraphs) of the stabilized Mach reflection of an ODW at Ma = 10 and ζ = 10/60: (a) t = 0.068 ms, (b) t = 0.101 ms, (c) t = 0.116 ms, (d) t = 0.131 ms, (e) t = 0.208 ms, ( f) t = 0.760 ms and (g) t = 1.581 ms.

Figure 5. Evolutions of the location and corresponding motion speed of the pressure-jump point along the wall before the expansion corner of the stabilized Mach reflection of ODWs at Ma = 10 and ζ = 10/60.

As indicated in figure 4(a), after the ODW has formed over the cowl's internal wall but before it becomes fully established, its reflection point is located on the wall behind the expansion corner, and no pressure jump occurs along the wall before the expansion corner. At this moment, the location of the pressure-jump point and its corresponding motion speed are simply set to zero (point ‘a’ in figure 5). Hence, when the ODW reflects precisely at the expansion corner later in its evolution (figure 4b), the motion speed of the pressure-jump point abruptly rises (point ‘b’ in figure 5). Next, a regular reflection pattern occurs over the wall before the expansion corner (figure 4c) because of the relatively small oblique detonation angle of the tail of the ODW during its evolution, and the reflection point moves upstream along the wall continuously with a gradually decreasing motion speed (point ‘c’ in figure 5). Then, the regular reflection pattern of the ODW transits to a Mach reflection pattern (figure 4d), and the motion speed of the pressure-jump point reaches another obvious peak (point ‘d’ in figure 5) during the transition process. As depicted in figures 4(e) and 4( f), an MS forms and grows thereafter while continuously moving upstream along the wall, which is again accompanied by a decreasing motion speed. In addition to the formation of the MS and RfSW connected to the triple point, an SL is also emitted from the triple point in the Mach reflection pattern of the ODW, thereby separating the flows across the ODW and MS. When the RfSW of the ODW reflects off the cowl wall, a secondary reflected shock wave (RfSW2) forms and reflects later over the SL, forming another reflected shock wave and, notably, a transmitted MS (tMS) between the SL and the lower wall of the combustor. As indicated in figure 5, after the formation of the Mach reflection pattern, the motion speed of the formed MS rapidly drops and then gradually decreases towards zero. After a sufficiently long period, the motion speed of the MS finally drops to zero, with the MS finally reaching its most upstream location. As a result, all the flow structures remain stabilized in the combustor, and the flow field reaches its steady state, as shown in figure 4(g).

The temperature and H₂ mass fraction contours along with the pressure isolines of the final steady flow field in this case are exhibited in figure 6. As indicated in figures 4(g) and 6, in addition to the ODW, a high temperature reaching approximately 3500 K and a rather low H₂ mass fraction are also featured just behind the MS, implying the tight coupling between the flame front and the MS. In other words, the MS is an overdriven NDW. According to Zhang et al. (Reference Zhang, Wen, Zhang, Liu and Jiang2021b), two stabilized detonation combustion modes, namely the oblique detonation combustion mode and the normal detonation combustion mode, simultaneously exist in the ODE combustor in this case. Moreover, a subsonic zone capable of propagating pressure waves upstream exists behind the MS and extends to the expansion fan emitting from the expansion corner (figure 4g). However, no obvious downstream pressure wave acting on or propagating in the subsonic zone is observed, and hence the MS is not disturbed and remains stabilized at a certain position before the expansion corner.

Figure 6. Contours of the temperature and H₂ mass fraction of the stabilized Mach reflection of an ODW (Ma = 10, ζ = 10/60) at t = 1.581 ms with different mesh sizes: (a) temperature contour (Δx = 0.1 mm), (b) H₂ mass fraction contour (Δx = 0.1 mm), (c) temperature contour (Δx = 0.05 mm) and (d) H₂ mass fraction contour (Δx = 0.05 mm).

To exclude the dependence of the simulation results on the grid size, grid resolution tests are carefully carried out by halving the sizes of the quadrilateral grid cells in both directions, leading to characteristic grid sizes of 0.05 mm in the main combustion zone for the fine grids and 0.1 mm for the coarse grids. As a result, the total number of fine grid cells is almost four times that of coarse grid cells. Taking the case of Ma = 10 and ζ = 10/60 as an example again, comparisons of the flow fields obtained with different grid sizes are shown in figure 6, while comparisons of the flow parameter distributions are depicted in figure 7. Differences in the flow structures are difficult to distinguish by using different grid sizes. Furthermore, the distribution curves of the various flow parameters obtained with different grid sizes almost overlap with each other, especially the locations of the jumps or drops of the flow parameters, implying that the simulations accurately capture the locations of key flow structures such as detonation fronts and shock waves. Therefore, a coarse grid with a characteristic grid size of 0.1 mm is sufficient to resolve the key flow features and is adopted in this study.

Figure 7. Flow parameter distributions for the stabilized Mach reflection of an ODW in the combustor (Ma = 10, ζ = 10/60) at t = 1.581 ms with different mesh sizes: (a) along the cowl wall (y = 0.06 m), (b) along the central line (y = 0.03 m) and (c) along the lower walls (i.e. $\overline {\textrm{AO}} $, $\overline {\textrm{OE}} $ and $\overline {\textrm{EF}} $ in figure 2).

It should be noted that, although the overall flow structures concerned in this paper are shown to be independent of grid size approximately when a characteristic grid size of 0.1 mm is used, this grid size is still not fine enough to precisely predict the induction length of an ODW or capture the potential cellular structures on the detonation surface. The grid-size requirement for predicting the aforementioned detonation features generally needs to ensure that there are at least 20 grid points within the half reaction zone of the corresponding Chapman–Jouguet detonation wave (Teng, Jiang & Ng Reference Teng, Jiang and Ng2014; Shen & Parsani Reference Shen and Parsani2017; Shi et al. Reference Shi, Uy and Wen2020). For example, for the inflow conditions of Ma = 10 listed in table 1, the length of the half reaction zone is approximately 0.4 mm, implying a minimum grid-size requirement of 0.02 mm, which is significantly smaller than that used in the present study (0.1 mm). Fortunately, the precise prediction of the induction length of an ODW does not affect the overall flow structures in the combustor, and the cellular characteristics of the detonation surfaces are not the main focus of this study; hence, a much finer grid is not employed to save computational resources.

For the other cases with small positive ζ values (i.e. ζ = 2/60 and 5/60) at Ma = 10, the flow fields of the Mach reflection of the ODW are also stabilized, and similar flow structures can be observed, as shown in figure 8. In other words, an MS, an SL, an RfSW and a subsonic zone form near the reflection point of the ODW. Downstream of the ODW, the RfSW reflects off the cowl wall to form an RfSW2, and the RfSW2 further reflects over the SL, forming another reflected shock wave and a tMS between the SL and the lower wall of the combustor. As ζ increases, i.e. as the designed reflection point of the ODW moves upstream, the final stabilized location of the MS also moves upstream; additionally, the length of the MS increases, as does the size of the subsonic zone behind the MS. Further, the locations of the RfSW2 and tMS downstream of the subsonic zone move farther upstream and approach the subsonic zone with increasing ζ. For Ma = 9.5 or 10.5, the Mach reflection patterns of the ODW similarly become stabilized at small positive ζ values, as summarized in figure 9, and stabilized flow structures similar to those discussed in this section can be obtained.

Figure 8. Shadowgraphs of the stabilized Mach reflection patterns of ODWs at Ma = 10: (a) ζ = 2/60 and (b) ζ = 5/60.

Figure 9. Summary of the stabilization characteristics of the Mach reflection of ODWs at different Ma and with different ζ values. The dashed line corresponds to the critical Mach number (i.e. Ma_cri = 9.3) predicted by the theory constructed in § 3.5.

3.3. Destabilized Mach reflection of ODWs with large ζ > 0

Again, Ma = 10 is taken as an example for discussion in this section. Different from the stable cases with small ζ, when ζ further increases to a larger value, for example ζ = 20/60 or 30/60, the flow field of the Mach reflection of the ODW becomes destabilized in the ODE combustor, as depicted in figure 9. Similar to figure 5, the temporal evolutions of the location of the pressure-jump point along the wall before the expansion corner and its corresponding motion speed in the cases of ζ = 20/60 and 30/60 at Ma = 10 are shown in figure 10. According to the above discussion in § 3.2, these curves represent the evolutions of the formed MS, where the MS forms at the time instant of the second peak in the motion speed curve. In the early period, the motion speed of the MS, which continuously moves upstream, is greater than zero but decreases, rapidly at first and then gradually. Accordingly, the upstream motion of the MS approaches zero and seemingly stops at a certain position, and the MS appears to be stabilized. However, as the flow field continues to evolve, at a certain time instant (approximately t = 0.9 ms for ζ = 20/60 or t = 0.7 ms for ζ = 30/60), the motion speed of the MS suddenly begins to increase, after which the MS accelerates while moving upstream. Finally, the MS moves upstream far from the expansion corner, indicating that the MS has moved out of the combustor, and the MS continues moving upstream with a very large speed. In other words, the Mach reflection of the ODW becomes destabilized in the combustor at a large value of ζ, which may lead to the engine becoming unstart. Therefore, to help design the geometry of the combustor, it is imperative to determine the mechanisms responsible for the destabilized Mach reflection of ODWs as well as the relevant influencing factors.

Figure 10. Evolutions of the location and corresponding motion speed of the pressure-jump point along the wall before the expansion corner of the destabilized Mach reflection of ODWs at Ma = 10: (a) ζ = 30/60 and (b) ζ = 20/60.

Taking ζ = 30/60 as an example, the evolution of the flow field of the destabilized Mach reflection of an ODW at Ma = 10 is illustrated in figure 11 by shadowgraphs along with sonic lines. The time instants corresponding to the exhibited flow fields are also marked along the motion speed curve of the pressure-jump point in figure 10 for convenience. After the Mach reflection pattern of the ODW is established (figure 11a), flow structures similar to the stabilized structures in § 3.2, including the MS, SL, RfSW, RfSW2, tMS and subsonic zone, also appear in the flow field of this destabilized case. In the early period of evolution of this destabilized Mach reflection of the ODW, the tMS is located relatively far downstream from the subsonic zone behind the MS (figure 11a). As time proceeds, the upstream motion of the MS drives the following flow structures (e.g. the RfSW, RfSW2 and tMS) upstream; consequently, the location of the tMS approaches the subsonic zone behind the MS (figure 11b). As indicated in figure 10(a), the flow structures tend to stabilize since the motion speed decreases to zero during this period. However, at approximately t = 0.585 ms (figure 11c), the tMS acts on the subsonic zone before the flow structures are fully stabilized. Thereafter, the pressure rise caused by the tMS rapidly propagates upstream through this subsonic zone, as shown in figures 11(d) and 11(e). Note that the symbol ‘tMS’ in these two figure panels is enclosed in parentheses, implying that the pressure rise caused by the tMS is no longer as sharp as that of a shock wave due to the diffusion effects of propagating through a subsonic zone.

Figure 11. Shadowgraphs of the flow field of the destabilized Mach reflection of an ODW (Ma = 10, ζ = 30/60) at different time instants: (a) t = 0.369 ms, (b) t = 0.534 ms, (c) t = 0.585 ms, (d) t = 0.623 ms, (e) t = 0.665 ms, ( f) t = 0.713 ms, (g) t = 1.080 ms, (h) t = 1.345 ms and (i) t = 1.582 ms.

The upstream propagation of the pressure rise caused by the tMS is also clearly visualized in figure 12, showing the evolution of the pressure distributions along the wall before the expansion corner (after the tMS acts on the subsonic zone). As shown in figure 12, the pressure rise caused by the tMS propagates upstream through the subsonic zone and finally arrives at the detonation front of the MS at approximately t = 0.713 ms (see also figure 11f), resulting in an increase in the pressure behind the MS and consequently the pressure ratio across the MS. Notably, the pressure ratio across an overdriven detonation wave must match its propagation speed relative to the incoming flow, and vice versa. Therefore, this increase in the pressure ratio across the MS results in an increase in its relative propagation speed; that is, the MS accelerates while moving upstream, as indicated by the sudden increase in the motion speed of the MS (point ‘f’) in figure 10(a). Then, the upstream motion of the MS enters a so-called ‘positive feedback’ period: the more upstream the MS travels, the closer the distance between the MS and RfSW2 (figure 11g), the higher the pressure ratio across the MS induced by the RfSW2 pressure rise and, consequently, the larger the upstream motion speed of the MS. As depicted in figure 10(a), the motion speed of the MS increases continuously and reaches a large value after the pressure rise produced by the tMS arrives at the detonation front. At approximately t = 1.345 ms (figure 11h), the MS reaches the leading tip of the cowl, and the ODW vanishes, resulting in only an NDW existing at the entrance of the combustor. Subsequently, the detonation front moves out of the combustor; instead, a bow detonation wave (BDW) forms and continues moving upstream, as shown in figure 11(i). In this destabilized case of the Mach reflection of an ODW (Ma = 10, ζ = 30/60), because the detonation front that serves as the main combustion organization structure in the ODE moves out of the combustor and thereafter continues to move upstream with a large motion speed, the combustor (and thus the engine) cannot work normally and may even fail.

Figure 12. Pressure distributions along the wall before the expansion corner of the destabilized Mach reflection of an ODW (Ma = 10, ζ = 30/60) at different time instants.

For ζ = 20/60 at the same Ma, analyses of the instantaneous flow fields in combination with the evolutions of the MS location and its motion speed suggest that the sudden upstream acceleration of the MS during its apparent stabilization (figure 10b) can also be attributed to the action of the tMS on the subsonic zone behind the MS (as shown in figure 13), which causes a pressure rise that propagates upstream to the MS and subsequently results in the incompatibility of the pressure ratio across the MS and its relative propagation speed. Then, the upstream motion of the MS enters a positive feedback period, and the detonation front moves out of the combustor. At other flight Mach numbers (i.e. Ma = 9.5 and 10.5), destabilized Mach reflection patterns of ODWs also occur at large ζ values, as shown in figure 9, and similar analyses suggest that the relevant destabilization mechanisms are also similar. Since this kind of destabilization phenomenon is induced by the action of a shock wave (the tMS in this section), this process is referred to as wave-induced destabilization in this paper.

Figure 13. Shadowgraphs of the flow field of the destabilized Mach reflection of an ODW (Ma = 10, ζ = 20/60) at different time instants: (a) t = 0.826 ms and (b) t = 0.982 ms.

3.4. Inherently destabilized Mach reflection of ODWs at low Ma

As depicted in figure 9, for a relatively large Mach number (Ma = 9.5, 10 or 10.5), when the ζ (>0) value is small (i.e. the designed ODW reflection point is located close to the expansion corner), the flow structures of the Mach reflection of an ODW remain stabilized in the combustor, whereas the detonation front moves out of the combustor upstream for large ζ values (i.e. the designed ODW reflection point is located far upstream from the expansion corner), implying the destabilization of the Mach reflection pattern of the ODW. However, a special phenomenon of stabilization/destabilization occurs for a relatively low Mach number, for example Ma = 9, as shown in figure 14. For Ma = 9 and ζ = 12/60, the Mach reflection of an ODW is destabilized in the combustor, and the evolution of the MS is similar to the evolutionary characteristics in the destabilized cases of Ma = 10. After the Mach reflection pattern of an ODW forms (corresponding to the first peak on the speed curve in figure 14a for this case), the upstream motion speed of the MS initially drops rapidly and then decreases gradually. During the stabilization of the MS, the motion speed of the MS suddenly rises, and the MS accelerates upstream. As a result, the MS moves out of the combustor to a location far upstream from the expansion corner. According to the previous analyses, one may expect to stabilize the Mach reflection of the ODW in the combustor by decreasing the ζ value, i.e. by setting the designed ODW reflection point closer to the expansion corner. However, while this guideline prevails for Ma = 9.5, 10 and 10.5, this solution fails for Ma = 9. By continuously decreasing the ζ value from 12/60 to 6/60, 3/60 or even 1/60, the motion speed of the MS still exhibits a sudden increase, eventually leading to the destabilization of the flow field. One of the key differences among the different ζ values is the time instant at which the motion speed of the MS rises; in other words, the time instant of the turning point occurs later for smaller ζ values. Notably, the designed reflection point is only 1 mm upstream of the expansion corner for a 6 cm high combustor with ζ = 1/60, but the destabilization of the Mach reflection of an ODW still occurs for this extremely small value of ζ. Remarkably, the flow structures at this low Ma are stable with ζ = 0, where the ODW reflects precisely at the expansion corner and the Mach reflection pattern of an ODW does not form. The above analyses suggest that destabilization is inherent in the Mach reflection of an ODW at this low Ma, when the ODW reflects off the wall before the expansion corner.

Figure 14. Evolutions of the location and corresponding motion speed of the pressure-jump point along the inlet wall of the destabilized Mach reflection of ODWs at Ma = 9: (a) ζ = 12/60, (b) ζ = 6/60, (c) ζ = 3/60 and (d) ζ = 1/60.

To determine the inherent mechanism causing the sudden increase in the motion speed of the MS in these low-Ma cases that directly destabilizes the Mach reflection of ODWs, the instantaneous flow fields and pressure distributions along the wall before the expansion corner are analysed near the turning time instant, as shown in figures 15 and 16, respectively, taking Ma = 9 and ζ = 12/60 as an example. Figure 15(a) demonstrates that the RfSW2 crosses the SL directly and then reflects off the lower wall at this low Ma rather than reflecting over the SL and forming a tMS at a relatively high Ma (e.g. figure 11a). Although slightly different flow structures are formed at this low Ma, the RfSW2 acts on the subsonic zone behind the MS at approximately t = 0.446 ms (figure 15b) as the MS moves upstream continuously. Thereafter, the pressure rise caused by the RfSW2 propagates upstream through this subsonic zone towards the detonation front of the MS, as shown in figures 15(c–e) and 16. After this pressure wave arrives at the detonation front of the MS at approximately t = 0.560 ms, the pressure behind the MS and the pressure ratio across the MS both increase; the resulting incompatibility between the pressure ratio across the MS and its relative propagation speed leads to a sudden increase in the motion speed of the MS (point ‘e’ in figure 14a), ultimately causing the destabilization of the detonation front in the combustor (figures 15f and 15g).

Figure 15. Shadowgraphs of the flow field of the destabilized Mach reflection of an ODW (Ma = 9, ζ = 12/60) at different time instants: (a) t = 0.396 ms, (b) t = 0.446 ms, (c) t = 0.479 ms, (d) t = 0.499 ms, (e) t = 0.560 ms, ( f) t = 0.618 ms and (g) t = 1.110 ms.

Figure 16. Pressure distributions along the wall before the expansion corner of the destabilized Mach reflection of an ODW (Ma = 9, ζ = 12/60) at different time instants.

It is suggested that the Mach reflection of an ODW before an expansion corner (ζ > 0) can be directly destabilized by the action of the RfSW2 or its tMS on the subsonic zone behind the MS regardless of the flight Mach number. Indeed, this is easy to understand with the above analyses. As the MS moves upstream, the RfSW2 and/or its tMS moves upstream and approaches the MS, and when the MS moves sufficiently far upstream from the expansion corner, the RfSW2 and/or its tMS acts on the subsonic zone and eventually causes destabilization. For a relatively high Ma (e.g. Ma = 10), the stabilized distance of the MS from the expansion corner increases as ζ increases. Consequently, a small ζ value results in a short distance of the MS from the expansion corner, which prevents the tMS of the RfSW2 from acting on the subsonic zone. On the other hand, a potentially long distance of the MS from the expansion corner is implied with a large ζ value, leading to destabilization. However, destabilization always occurs at Ma = 9, even with an extremely small ζ value of 1/60, implying that a stabilized MS location may not exist for this low Mach number and consequently that the MS may continuously move upstream once the Mach reflection pattern of the ODW forms. In other words, destabilization may be inherent in the Mach reflection of an ODW at a low Ma, and the action of the RfSW2 or its tMS on the subsonic zone behind the MS may only accelerate the upstream motion of the MS. Hence, this kind of destabilization phenomenon at a low Ma is referred to as inherent destabilization in this paper, and an in-depth theoretical analysis is further conducted to understand the relevant mechanisms.

3.5. Formulating the stabilized location of the MS

In the following formulation, the effects of the RfSW2 and its tMS are neglected since they are found to only accelerate the inherent upstream motion of the MS at a low Ma, as discussed in § 3.4. According to the above analyses of the instantaneous flow fields, the SL is inclined downward relative to the incoming flow, as is schematically shown in figure 17. Hence, the subsonic flow crossing the MS accelerates towards becoming sonic in the contracted flow path bounded by the SL and the lower wall, and the geometric throat is positioned at the expansion corner (point O). Assume that the MS is a straight line perpendicular to the lower wall and that the SL is also a straight line with an inclination angle of α relative to the incoming flow. According to the geometric relationship exhibited in figure 17, the length of the MS can be calculated by

(3.1)\begin{equation}\overline {\textrm{MS}} = (\overline {\textrm{MO}} - \overline {\textrm{RO}} ) \tan \beta ,\end{equation}

where β is the oblique detonation angle of the ODW relative to the incoming flow. The distance from the expansion corner to the SL, namely $\overline {\textrm{OL}} $, which is considered as the width of the geometric throat, can be expressed as

(3.2)\begin{equation}\overline {\textrm{OL}} = \overline {\textrm{ON}} \cos \alpha = [(\overline {\textrm{MO}} - \overline {\textrm{RO}} ) \tan \beta - \overline {\textrm{MO}} \tan \alpha] \cos \alpha ,\end{equation}

where $\overline {\textrm{ON}} $ is a segment parallel to MS. Therefore, the contraction ratio of the flow path from the MS to the geometric throat $\overline {\textrm{OL}} $ appears as

(3.3)\begin{equation}\begin{array}{ccccc} \dfrac{{\overline {\textrm{MS}} }}{{\overline {\textrm{OL}} }} & = \dfrac{{(\overline {\textrm{MO}} - \overline {\textrm{RO}} ) \tan \beta }}{{[(\overline {\textrm{MO}} - \overline {\textrm{RO}} ) \tan \beta - \overline {\textrm{MO}} \tan \alpha ] \cos \alpha }}\\ & = {\left\{ {\left[ {1 - \dfrac{{\tan \alpha }}{{(\textrm{1} - {{\overline {\textrm{RO}} } / {\overline {\textrm{MO}} }}) \tan \beta }}} \right] \cos \alpha } \right\}^{ - 1}}. \end{array}\end{equation}

Equation (3.3) reveals that as the S point (the triple point) of the MS moves upstream along the ODW (i.e. the triple point is always located on the ODW), the ratio of $\overline {\textrm{RO}} \textrm{/}\overline {\textrm{MO}} $ (<1) decreases, and the contraction ratio of the flow path, $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} $, decreases accordingly. If the MS is located close to the expansion corner, the contraction ratio of the flow path behind the MS is large enough that a sonic state may be achieved before the geometric throat $\overline {\textrm{OL}} $, as schematically shown in figure 17(a) (the location of the carmine dashed line corresponds to the contraction ratio of the flow path of the sonic state). Notably, the streamlines behind the MS are generally curved to adjust the flow direction to conform to the flow path between the SL and the lower wall. Further contraction of the flow path behind the sonic line would directly choke the flow. As a result, the MS would move upstream to adjust the flow behind the MS to ensure that the sonic state occurs at the geometric throat $\overline {\textrm{OL}} $. Obviously, the geometric converging effects of the flow path before the expansion corner contribute to sustaining the MS. In contrast, if the MS is located far upstream from the expansion corner (figure 17b), the contraction ratio becomes so small that the local Mach number at the geometric throat $\overline {\textrm{OL}} $ is still smaller than one. In other words, under this circumstance, the flow behind the MS may be subsonic throughout the flow path before and across the expansion corner. Then, the geometric diverging effects of the flow path behind the expansion corner could propagate upstream through left-running characteristic lines across the expansion corner and arrive at the MS, which weakens the geometric converging effects of the flow path before the expansion corner. As a result, the MS would be blown away by the upstream supersonic flow. It appears that the stabilized location of the MS (i.e. $\overline {\textrm{MO}} $) corresponds to a critical condition in which the local Mach number at the geometric throat $\overline {\textrm{OL}} $ is equal to one, which is similar to the Kantrowitz limit in the unstart problems of scramjet inlets (Curran & Murthy Reference Curran and Murthy2001). In such a problem, the Kantrowitz-limit contraction ratio is determined by the one-dimensional isentropic area ratio that accelerates a subsonic flow (behind a normal shock wave) to a sonic flow at the throat. In the present problem regarding the stabilization of the Mach reflection of an ODW, the similar Kantrowitz-limit contraction ratio, CR_Kantrowitz, can be evaluated for a specific NDW, and it can be employed to determine the theoretical stabilization location of the MS.

Figure 17. Geometric relationship of the Mach reflection of an ODW before an expansion corner: (a) $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \mathrm{\ > }C{R_{Kantrowitz}}$ and (b) $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \mathrm{\ < }C{R_{Kantrowitz}}$.

From the above analysis, the critical condition for the stabilization of the MS appears as

(3.4)\begin{equation}Ma{|_{\overline {\textrm{OL}} }} = 1,\end{equation}

or

(3.5)\begin{equation}\frac{{\overline {\textrm{MS}} }}{{\overline {\textrm{OL}} }} = C{R_{Kantrowitz}}.\end{equation}

Substituting (3.3) into (3.5) yields the formula for calculating the stabilized location of the MS, expressed as

(3.6)\begin{equation}\frac{{\overline {\textrm{MO}} }}{{\overline {\textrm{RO}} }} = \frac{{(\cos \alpha - CR_{Kantrowitz}^{ - 1}) \tan \beta }}{{(\cos \alpha - CR_{Kantrowitz}^{ - 1}) \tan \beta - \sin \alpha }}.\end{equation}

Obviously, with the parameters α, β and CR_Kantrowitz being known, the stabilized location of MS (i.e. $\overline {\textrm{MO}} $) corresponding to the designed reflection location of ODW (i.e. $\overline {\textrm{RO}} $) can be evaluated directly from (3.6).

To evaluate the oblique detonation angle of the ODW, β, the inclination angle of the SL, α, and the Kantrowitz-limit contraction ratio behind the MS, CR_Kantrowitz, for different cases, the chemical equilibrium hypothesis is employed in the present study to account for changes in the chemical compositions and gas thermodynamic properties and to more accurately predict the heats of chemical reactions; that is, the chemical reaction rates are assumed to be infinitely fast, and chemical equilibrium states are assumed to be achieved everywhere in the flow field behind the leading shock/detonation fronts (i.e. behind the ODW and MS). Further, a two-step cyclic iterative solution method proposed by the authors (Zhang Reference Zhang2020a; Zhang et al. Reference Zhang, Wen, Zhang, Liu and Jiang2021a) is adopted to solve the chemical equilibrium solutions of different processes with high accuracy, including the chemical equilibrium normal shock relationship, the chemical equilibrium strong and weak oblique shock relationships (where ‘strong’ and ‘weak’ refer to the strong and weak branches of the oblique shock solutions, respectively) and the chemical equilibrium isentropic relationship. For clarity, the solution processes of the above chemical equilibrium relationships are briefly presented in Appendices A–E; for the detailed derivations and evaluations of the solution accuracy, the reader may refer to Zhang (Reference Zhang2020a) and Zhang et al. (Reference Zhang, Wen, Zhang, Liu and Jiang2021a). Specifically, for different cases, the oblique detonation angle of the ODW, β, can be directly evaluated by the chemical equilibrium weak oblique shock relationship; the predicted values of β in different cases agree well with those observed in the numerical simulations in the present study. For the evaluation of the Kantrowitz-limit contraction ratio behind the MS, CR_Kantrowitz, the flow state behind the MS is first obtained through the chemical equilibrium normal shock relationship; then, to evaluate the MS–throat area ratio through the chemical equilibrium isentropic relationship, the subsonic flow state needs to accelerate to the sonic state in the contracted flow path between the SL and the lower wall, during which the chemical compositions change due to shifts in chemical equilibrium. Moreover, to determine the inclination angle of the SL, α, the three-shock theory (Ben-Dor Reference Ben-Dor2007) is applied to the neighbourhood of the triple point, as schematically shown in figure 18. The MS is slightly inclined with respect to the perpendicular direction of the incoming flow in the neighbourhood of the triple point S, and hence the flow state behind this small segment of the MS needs to be evaluated through the chemical equilibrium strong oblique shock relationship, while the change in the flow state across the RfSW (from region 2 to region 3, also based on the assumption of chemical equilibrium) is calculated by the chemical equilibrium weak oblique shock relationship. Thereafter, to determine the inclination angle of the SL, α, the pressure balance between both sides of the SL, namely between region 3 behind the RfSW that deflects the flow by an angle of $\theta - \alpha$ (θ = 25° in this paper) and region 4 behind the inclined MS that deflects the flow by an angle of α, i.e. p ₃ = p ₄ (figure 18), needs to be satisfied. With the chemical equilibrium strong and weak shock relationships, this pressure-balanced state can be solved by utilizing a simple dichotomizing search algorithm.

Figure 18. Illustration of the flows in the neighbourhood of the triple point of a Mach reflection pattern.

From the above analyses, the parameters α, β and CR_Kantrowitz can be uniquely determined (based on the assumption of chemical equilibrium) according to the inflow conditions before the combustor, which depend uniquely on Ma, as shown in figure 19. The specific values of these parameters for the stabilized flight Mach numbers (i.e. Ma = 9.5, 10 and 10.5) are summarized in table 2. As Ma decreases, both the chemical equilibrium inclination angle of the SL, α, and the chemical equilibrium oblique detonation angle of the ODW, β, increase, whereas the chemical equilibrium Kantrowitz-limit contraction ratio behind the MS, CR_Kantrowitz, decreases. Because the flow Mach number behind the chemical equilibrium MS is closer to one at a lower Ma, it is easier for this subsonic flow to accelerate to the sonic state, leading to a smaller limiting contraction ratio. Substituting these parameters into (3.6), the predicted stabilized locations of the MS (i.e. $\overline {\textrm{MO}} $) at Ma = 9.5, 10 and 10.5 as a function of the designed ODW reflection location (i.e. $\overline {\textrm{RO}} $) are plotted as the dashed lines in figure 20. For comparison, the real values of $\overline {\textrm{MO}} $ observed in the numerical simulations are also presented in figure 20, revealing that (3.6) underestimates the stabilized locations of the MS to a large extent compared with the numerical simulations.

Figure 19. Changes in (a) the inclination angle of the SL, α, (b) the oblique detonation angle of the ODW, β, and (c) the Kantrowitz-limit contraction ratio behind the MS, CR_Kantrowitz, as functions of Ma based on the chemical equilibrium hypothesis.

Figure 20. Changes in the stabilized location of the MS (i.e. $\overline {\textrm{MO}} $) as a function of the designed ODW reflection location (i.e. $\overline {\textrm{RO}} $).

Table 2. Summary of the geometric parameters in the stabilized Mach reflection of ODWs.

These discrepancies in the stabilized locations of the MS between the theoretical predictions by (3.6) and the real observations via numerical simulations result from the following factors. First, the real SL between the triple point and the geometric throat is not strictly straight, as shown in figure 21(a) (taking the case of Ma = 10 and ζ = 5/60 as an example). Due to the limited space in the neighbourhood of the triple point compared with the chemical reaction characteristic length, the chemical equilibrium state is not fully achieved near the triple point, resulting in the local inclination angle of the SL being larger than the corresponding chemical equilibrium value, as depicted in figure 21(b). As the fluid flows downstream, the flow state varies to the chemical equilibrium state, and hence the local inclination angle of the SL decreases gradually to the corresponding chemical equilibrium value. However, the segment of the SL close to the geometric throat $\overline {\textrm{OL}} $ becomes increasingly inclined downward under the effects of the expansion fan emitted from the expansion corner (figure 21a), leading to an increase in the local inclination angle of the SL and another deviation from its chemical equilibrium value (figure 21b). Influenced by these two effects, the average inclination angle of the SL from the triple point to the geometric throat is undoubtedly larger than that predicted by the chemical equilibrium theory. Second, due to the local chemical non-equilibrium effects of the flow behind the MS, the change in the flow from the MS to the throat does not exactly follow the chemical equilibrium isentropic process. Furthermore, due to the special one-sided expansion geometry of the expansion corner and the finite-length contracted flow path in the present problem, the one-dimensional assumption in the traditional Kantrowitz limit is no longer satisfied (Curran & Murthy Reference Curran and Murthy2001), and hence the flow at the geometric throat $\overline {\textrm{OL}} $ is not exactly at the sonic state; in fact, the Mach number of this flow is smaller than one and varies along $\overline {\textrm{OL}} $, as shown, for example, in figures 4(g) and 8. As a result, the real contraction ratio from the MS to the geometric throat $\overline {\textrm{OL}} $, i.e. $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} $, deviates slightly from the chemical equilibrium Kantrowitz-limit contraction ratio, CR_Kantrowitz.

Figure 21. Numerical shadowgraph of the flow field near the reflection point of the ODW (a) and the spatial distribution of the local inclination angle of the SL (b) in the case of Ma = 10 and ζ = 5/60.

To consider the above factors causing the discrepancies in the theoretical predictions of the stabilized locations of the MS compared to the numerical observations, a corrected parameter, κ, is introduced to the present theoretical formulation, and the critical condition for the stabilization of the MS, i.e. (3.5), is rewritten as

(3.7)\begin{equation}\kappa \frac{{\overline {\textrm{MS}} }}{{\overline {\textrm{OL}} }} = C{R_{Kantrowitz}}.\end{equation}

Then, a corrected formula for calculating the stabilized location of the MS is obtained by substituting (3.3) into (3.7), expressed as

(3.8)\begin{equation}\frac{{\overline {\textrm{MO}} }}{{\overline {\textrm{RO}} }} = \frac{{(\cos \alpha - \kappa CR_{Kantrowitz}^{ - 1}) \tan \beta }}{{(\cos \alpha - \kappa CR_{Kantrowitz}^{ - 1}) \tan \beta - \sin \alpha }}.\end{equation}

To evaluate the corrected parameter, κ, (3.8) is rearranged into

(3.9)\begin{equation}\kappa = \left[ {\cos \alpha - \frac{{\sin \alpha }}{{(1 - \overline {\textrm{RO}} /\overline {\textrm{MO}} ) \tan \beta }}} \right] C{R_{Kantrowitz}},\end{equation}

which is substituted with the numerical values of $\overline {\textrm{MO}} $ in the stabilized cases at Ma = 9.5, 10 and 10.5 summarized in table 2. The value of κ ranges from approximately 1.19 to 1.33 for these stabilized Mach reflection cases. Although the corrected parameter, κ, does not change much between different cases, it appears to be a function of the flight Mach number, Ma, and the dimensionless reflection location, ζ, i.e. κ = κ(Ma, ζ), as depicted in figure 22.

Figure 22. Changes in the corrected parameter, κ, as a function of the dimensionless reflection location, ζ.

One purpose of this section is to provide theoretical predictions of the stabilized locations of the MS at Ma = 9 and small ζ values, especially its asymptotic behaviour as ζ goes to zero at this relatively low Ma. Since the values of Ma and ζ do not change much compared to those in the stabilized cases summarized in table 2, a second-order Taylor expansion approximation of the function of $f(Ma,\zeta ) \equiv {(\kappa \text{--}1)^{ - 1}}$ can be applied to fit the corrected parameter, κ, as follows:

(3.10)\begin{align}\begin{array}{ccccc} {(\kappa - 1)^{ - 1}} & = f(Ma,\zeta ) \approx {f_0} + \dfrac{{\partial f}}{{\partial Ma}}(Ma - M{a_0}) + \dfrac{{\partial f}}{{\partial \zeta }}(\zeta - {\zeta _0}) + \dfrac{{{\partial ^2}f}}{{\partial M{a^2}}}{(Ma - M{a_0})^2}\\ & \quad + \dfrac{{{\partial ^2}f}}{{\partial {\zeta ^2}}}{(\zeta - {\zeta _0})^2} + \dfrac{{{\partial ^2}f}}{{\partial Ma\partial \zeta }}(Ma - M{a_0})(\zeta - {\zeta _0}). \end{array}\end{align}

After rearranging the above equation, it can be rewritten as

(3.11)\begin{equation}\kappa (Ma,\zeta ) = 1 + {({a_1} + {a_2} Ma + {a_3} \zeta + {a_4} M{a^2} + {a_5} {\zeta ^2} + {a_6} Ma \zeta )^{ - 1}},\end{equation}

where a ₁ = 187.3, a ₂ = −35.23, a ₃ = 67.63, a ₄ = 1.680, a ₅ = −5.364 and a ₆ = −5.937, as determined by the least squares method using the data of κ given in table 2. The fitting curves of κ = κ(Ma, ζ) by (3.11) are presented in figure 22. The fitting curves and the numerical data of the stabilized cases exhibit good agreement. Further, the corrected theoretical predictions of the stabilized locations of the MS (i.e. $\overline {\textrm{MO}} $) using (3.8) and (3.11) are also presented in figure 20 as the solid lines, demonstrating good consistency between the corrected theoretical predictions and the numerical observations of the stabilized cases.

For Ma = 9, the chemical equilibrium oblique detonation angle of the ODW and the inclination angle of the SL are β = 55.34° and α = 13.16°, respectively, while the corresponding chemical equilibrium Kantrowitz-limit contraction ratio of the subsonic flow behind the MS is CR_Kantrowitz = 1.3790. For ζ = 1/60 (i.e. $\overline {\textrm{RO}} = 1\;\textrm{mm}$), the predicted value of the corrected parameter using (3.11) is κ = 1.151. Substituting all the above parameters into (3.8), the theoretical prediction of the ratio of $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} $ is equal to −7.454, which is smaller than zero. This result obviously contradicts the geometric relationship depicted in figure 17, i.e. $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} > 1 > 0$, implying that a stabilized location of the MS (i.e. $\overline {\textrm{MO}} $) cannot be achieved in this low-Ma case. Figure 23 shows the changes in the value of κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ as the MS moves upstream at ζ = 1/60 (i.e. $\overline {\textrm{RO}} = 1\;\textrm{mm}$) for different Ma, revealing that the value of κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ is greater than zero when $\overline {\textrm{MO}} $ is small for all Ma. According to the previous analysis of figure 17 but with the corrected critical condition of (3.7), when κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}} > \textrm{ }0$, the flow becomes choked in the contracted flow path bounded by the SL and the lower wall, resulting in the MS moving upstream. In the cases of Ma = 9.5, 10 and 10.5, κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ approaches its zero point as $\overline {\textrm{MO}} $ increases, and, finally, the MS remains stabilized at the corresponding location. In contrast, if the change in κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ passes through its zero point to a negative value in the cases of low Ma (Ma = 9.5, 10 and 10.5), the MS becomes attenuated downstream under the geometric diverging effects of the expansion corner; as a result, κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ returns to its zero point, and the MS remains stabilized. Therefore, the existence of zero points of κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ for these Ma values implies the potential stabilization of the MS at a specific location. However, the value of κ $\overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ is always greater than zero, and there is no zero point in the evolution of $\overline {\textrm{MO}} $ in the case of Ma = 9. Therefore, the MS grows and moves upstream continuously after it forms, leading to the destabilization of the Mach reflection of the ODW in this low-Ma case. Further, as depicted in figure 20, the predicted stabilized location of the MS at Ma = 9 is always negative for ζ > 0, implying that the Mach reflection of ODWs is inherently destabilized at this low Ma due to flow choking, when the ODW reflects before an expansion corner, i.e. ζ > 0, as discussed in § 3.4.

Figure 23. Changes in the value of $\kappa \overline {\textrm{MS}} \textrm{/}\overline {\textrm{OL}} \text{--}C{R_{Kantrowitz}}$ as a function of $\overline {\textrm{MO}} $ for $\overline {\textrm{RO}} = 1\;\textrm{mm}$ (i.e. ζ = 1/60) at different Ma.

According to the previous analyses, the stabilization of the Mach reflection of an ODW before an expansion corner is feasible at small ζ because destabilization would be directly triggered by the action of the RfSW2 or its tMS on the subsonic zone behind the MS at large ζ. Due to the limited spatial resolution of numerical simulations, it is impossible to infinitely decrease ζ to check whether the Mach reflection of ODWs before an expansion corner at a specific Ma is inherently destabilized. However, the inherent stabilization or destabilization characteristics at a specific Ma can be confirmed by the asymptotic behaviour of $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} $ when $\zeta \to 0 +$, as shown in figure 24. The limit of $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} $ is greater than zero when Ma is greater than approximately 9.3, implying that when ζ is small enough that the RfSW2 or its tMS does not act on the subsonic zone behind the MS, the Mach reflection of an ODW before the expansion corner can be stabilized. For Ma smaller than approximately 9.3, the limit of $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} $ is smaller than zero, implying that the Mach reflection of an ODW before an expansion corner is inherently destabilized regardless of how small ζ (>0) is. In other words, the critical Mach number of inherently stabilized/destabilized Mach reflection of ODWs before an expansion corner, Ma_cri, predicted by the present theory, is approximately 9.3. The present theoretical prediction is consistent with the numerical observations summarized in figure 9, in which the Mach reflections of ODWs can be stabilized with a small ζ at Ma ≥ 9.5 but exhibit destabilization with all ζ > 0 at Ma = 9. The numerical prediction of Ma_cri appears between 9 and 9.5.

Figure 24. Asymptotic change in the value of $\overline {\textrm{MO}} \textrm{/}\overline {\textrm{RO}} $ as a function of Ma when $\overline {\textrm{RO}} \to 0 + $ $(\textrm{or}\;\zeta \to 0 + )$.

Further, figure 19 shows that as Ma decreases, the inclination angle of the SL, α, increases, resulting in an increase in the real contraction extent of the flow path bounded by the SL and the lower wall, whereas the corresponding Kantrowitz-limit contraction ratio of the subsonic flow behind the MS decreases. Hence, the subsonic flow behind the MS is easily choked before the expansion corner at a low Ma. In other words, this subsonic flow may accelerate to the sonic state before the expansion corner and be choked by further area contraction, resulting in the MS continuously moving upstream and inherent destabilization, as shown in figure 17(a). In summary, the theoretical results presented in this section demonstrate that the inherent destabilization of the Mach reflection of an ODW before an expansion corner at a low Ma is attributed to the geometric flow choking phenomenon caused by the large flow path contraction behind the MS.