5.1. Effects of viscosity

The pressure field obtained in Block 0 at $\textit {Re}_w=10^4$ is shown in figure 4(a). The Navier–Stokes numerical solution is compared with the shock polars in figure 4(b). The notations ‘Block 0, MS’ and ‘Block 0, RS’ are used to label the distributions of the corresponding quantities behind the Mach stem and the reflected shock wave along the lines shown in figure 4(a). The notation ‘Block 3’ refers to the distributions of parameters behind the Mach stem and the reflected shock wave near the triple point on the third – maximum for this Reynolds number – level of nesting.

Figure 4. (a) Pressure contours at $\textit {Re}_w=10^4$. (b) Comparison of the Navier–Stokes numerical solution with the shock polars.

In accordance with the inviscid theory, the MS slope is expected to change from $90^\circ$ at the symmetry line to the angle predicted by the three-shock solution near the triple point, when moving upwards along the MS. Correspondingly, the flow deflection angle should change from zero to the value equal to the slope of the slip line in the three-shock solution. The numerical data in figure 4 show that when moving up the Mach stem, the angle increases at first, and in the $(\theta, p/p_\infty )$ plane, the data point moves exactly along the I-polar. Further up, closer to the three-shock intersection, the data point starts to move in the opposite direction along the I-polar, so that the flow deflection angle decreases and reaches a local minimum at some point B. When it starts to increase again, the point leaves the I-polar. Now both the pressure and the flow deflection angle increase, pass through local maxima, then start to decrease, and finally reach the R-polar at point D. However, point D does not coincide with the shock polar intersection point: it corresponds to a slightly higher value of the RS slope than that predicted by the three-shock solution. During further motion along the reflected shock wave, the data point moves along the R-polar, reaches the shock polar intersection point, and remains at this point until the reflected shock wave meets the first characteristic of the expansion fan EF. Further, the pressure and the flow deflection angle decrease rapidly.

The above-described behaviour of the numerical data in the $(\theta, p/p_\infty )$ plane is similar to what was observed previously by Khotyanovsky et al. (Reference Khotyanovsky, Bondar, Kudryavtsev, Shoev and Ivanov2009), Chen et al. (Reference Chen, Zhang and Liu2016) and Shoev & Ivanov (Reference Shoev and Ivanov2016) at low Reynolds numbers. As was noted above, similar deviations from the three-shock solution were also obtained in inviscid computations by Ben-Dor et al. (Reference Ben-Dor, Takayama and Needham1987), where they were caused by numerical viscosity inevitably present in shock-capturing schemes. In the present study, the deviations from the inviscid solution are caused by the sole influence of physical viscosity because the effects of numerical viscosity on the solution are made negligible by using sufficiently fine meshes resolving the internal structure of the shock waves. The deviation of the numerical data in Block 3 illustrates the influence of physical viscosity on the flow structure near the triple point, and confirms the existence of the non-Rankine–Hugoniot zone in the strong Mach reflection.

Let us consider the structure of this region in more detail. The numerical solution at $\textit {Re}_w=10^4$ obtained in Block 3 with complete resolution of the internal structure of shock waves is shown in figure 5. Figure 5(a) shows the Mach number field, where the black solid lines indicate the positions of the shock waves and slip line predicted by the three-shock solution. According to the three-shock solution, the predicted values of the Mach number behind the reflected shock wave and the Mach stem are 1.15 and 0.51, respectively. The viscosity leads to a small (within 3 %) decrease in both values.

Figure 5. Comparison of the viscous numerical solution with the inviscid solution near the triple point: (a) Mach number; (b) pressure contours; (c) numerical data and shock polars in the $(\theta, p/p_\infty )$ plane; (d) distributions of the flow deflection angle.

The pressure field is shown in figure 5(b). The inviscid three-shock solution predicts a constant-pressure region behind the triple point where $p/p_\infty =19.56$. In the viscous case, however, the constant-pressure region behind the triple point is not observed. There is a small region near point C where the pressure is approximately 10 % higher than that predicted by the three-shock theory. Further downstream from the shock waves, the pressure decreases, but there are some differences from the three-shock predictions even at a distance of $40\unicode{x2013} 60\lambda _\infty$. The points along the dashed line in figure 5(b) correspond to the points in the $(\theta, p/p_\infty )$ plane in figure 5(c). On the segment AB, the numerical data almost coincide with the I-polar, though the polar is passed in the opposite direction in this case. Starting from point B, the numerical data deviate significantly from the shock polars, and the flow deflection angle and the pressure increase. The pressure is maximum at point C, located immediately behind the shock wave intersection.

The numerical data return to the R-polar only at point D. The segment DE coincides with the R-polar; however, the data point does not reach the shock polar intersection because of the limited size of Block 3. Thus the numerical data lie on the segments AB and DE of the I- and R-polars, where one can speak about the existence of separate shock waves. The flow parameters behind these waves can be calculated from the Rankine–Hugoniot relations. At the same time, the slopes of these waves differ from those predicted by the three-shock solution. The segment BCD does not lie on any shock polar. Here, it is impossible to distinguish among the incident, reflected and Mach shock waves, so in the viscous case, a finite-size zone with an essentially two-dimensional flow replaces the inviscid triple point.

In the physical space, the segment BCD is situated behind the non-Rankine–Hugoniot zone introduced by Sternberg (Reference Sternberg1959) in order to resolve the von Neumann paradox. The size of BCD is approximately $10\unicode{x2013} 20 \lambda _\infty$, which is comparable with the shock wave thickness. The segments AB and DE lie inside the ‘equalisation zone’ in accordance with the terminology of Sternberg (Reference Sternberg1959). It is not the non-Rankine–Hugoniot zone, but it is still affected by the latter. The shock waves in the equalisation zone are clearly discernible. The parameters behind these waves in the equalisation zone can be calculated from the Rankine–Hugoniot (R–H) relations, but they differ from those predicted by the three-shock theory. The reason for this difference is fairly simple: the existence of the region of the essentially two-dimensional viscous flow alters the shapes of the adjacent shock waves. In particular, an inflection point appears on the Mach stem, and the wave above this point becomes convex with respect to the free stream. This phenomenon will be investigated in § 5.4.

Figure 5(d) shows the distributions of the flow deflection angle at three values, ${y/\lambda _\infty = 861}$, $867$ and $871$. In figure 5(b), they correspond to the horizontal lines passing through points B, C and D, respectively. The dashed line shows the flow deflection angle predicted by the three-shock theory. The distributions at $y/\lambda _\infty =861$ and $867$ reveal that the flow passing through the shock waves deflects up to values higher than that predicted by the three-shock theory. At $y/\lambda _\infty =871$, the flow first passes through the IS and deflects by $25^\circ$, which exactly corresponds to the wedge angle $\theta _w$. After that, the flow passes through the reflected shock RS and deflects back to a value lower than that predicted by the three-shock theory. Further downstream, the flow deflection gradually approaches the predicted value in all three cases.

Figure 6 shows the temperature distributions near the triple point. The maximum temperature is observed between the Mach stem and the shear layer (figure 6a). The temperature peak is clearly visible in figures 6(b,c) where the distributions along the lines $y/\lambda _\infty =850$, $860.4$ and $890$ are presented. They pass through the Mach stem below the temperature maximum, through the maximum itself, and through the incident and reflected shock waves above the maximum, respectively. The values predicted by the three-shock theory behind the shock waves are also shown and labelled as ‘IS, 3ST’, ‘MS, 3ST’ and ‘RS, 3ST’.

Figure 6. (a) Temperature contours near the triple point. (b) Distributions along the horizontal lines $y/\lambda _\infty =890$, $860.4$ and $850$ as compared to the three-shock solution. (c) Distributions along the horizontal lines $y/\lambda _\infty =860.4$ and $850$ as compared to the three-shock and normal shock solutions close to the temperature peak.

The distributions at $y/\lambda _\infty =850$ and $860.4$ demonstrate that the temperatures behind the Mach stem are higher than the value MS, 3ST. The distribution at $y/\lambda _\infty =890$ shows that the temperature behind the incident shock wave agrees well with the theoretical value IS, 3ST, while the temperature behind the reflected shock wave is slightly higher than RS, 3ST.

Figure 6(c) illustrates these differences in more detail. It is clearly seen that both distributions at $y/\lambda _\infty =850$ and $860.4$ display peaks exceeding the three-shock solution by 0.04 and 0.08, respectively. It should be noted that the temperature peak at ${y/\lambda _\infty =860.4}$ exceeds even the temperature behind the normal shock wave, ‘Normal shock, R–H’. A similar behaviour of temperature was also observed in the inviscid computation with a shock-capturing scheme (Ben-Dor et al. Reference Ben-Dor, Takayama and Needham1987) where numerical viscosity apparently acts similarly to physical viscosity.

A well-known specific feature of the viscous shock wave structure is the entropy maximum inside the wave. At the same time, the entropy behaviour inside the region where several shock waves merge has not been studied, though it is of considerable interest. Figure 7 shows the entropy contours and distributions near the triple point. All shock waves are clearly visible in figure 7(a), and the maximum value of entropy is observed inside the Mach stem. The entropy distributions along the horizontal lines $y/\lambda _\infty =850$, $870$ and $890$ reveal the entropy peaks inside each shock wave (figure 7c). At $y/\lambda _\infty =870$, two entropy peaks belonging to the IS and the RS are so close to each other that there is no entropy plateau between them. The entropy field inside the region where three shock waves meet is shown enlarged in figure 7(b). It is seen here that the contours inside the IS and the RS merge and transform into a single maximum inside the Mach stem. The entropy distributions along the horizontal lines through the region are plotted in figure 7(d). At $y/\lambda _\infty = 869$, it is still possible to distinguish separate peaks inside the incident (IS) and reflected (RS) shock waves. At lower values of $y/\lambda _\infty$, two entropy maxima merge into one corresponding to the Mach stem.

Figure 7. (a) Entropy contours near the triple point; (b) entropy contours in the enlarged region near the triple point; (c) distributions at various $y/\lambda _\infty$; (d) distributions at various $y/\lambda _\infty$ in the enlarged region near the triple point.

In order to quantify the size of the observed small viscous flow region in comparison to the macroscopic length scales of the flow, the density isolines in the whole computational domain (Block 0) are presented in figure 8(a). First, note that the three-shock solution at the macroscopic level clearly predict the angles of all shocks well. The angle of the slip line of the three-shock solution agrees well with that of the shear layer obtained in the computation, at least up to distances of approximately 10 % of the wedge length from the triple point. The area where the pressure and flow deflection angle are within a 1 % margin from the three-shock solution is shown in green. This is a significant region behind the reflected shock and above the shear layer. Its size, which is approximately 15 % of the wedge length, is nearly independent of the Reynolds number. There is another narrow portion of the flow field on the lower boundary of the shear layer where the parameters are close to the three-shock solution, yet it is much smaller than the upper one. The other part of the flow behind the Mach stem is not predicted by the three-shock solution. The fact that the intrinsically local solution is not valid here can be expected even without taking viscosity into account, because the Mach stem is curved and the flow behind it is subsonic. The high-pressure region behind the shock wave intersection discussed above is shown in red (again a 1 % margin has been chosen). It can be considered a region where the three-shock solution is clearly invalid due to viscosity effects. The region size is at least one order of magnitude smaller than that of the green area, and as will be shown below, is defined mainly by the mean free path length scale and therefore decreases with the Reynolds number. Note that we cannot speak strictly about the size of the regions, because the choice of 1 % margin is rather arbitrary.

Figure 8. (a) Density contours: green area indicates pressure and deflection angle are within 1 % of the three-shock solution; red area indicates pressure is higher than 1 % from the three-shock solution; white area indicates discontinuities in the three-shock solution. (b) Pressure and (c) flow deflection angle, in Block 3, with selected streamlines: green area indicates parameters are within 1 % of the three-shock solution; blue and red areas indicate parameters are lower or higher than 1 % from the three-shock solution.

One can look at this high-pressure region more closely in figure 8(b). The size of this region is approximately 100 mean free paths, and it is located mainly behind the reflected shock, although some part of it is located behind the shear layer. Farther from the shock intersection, the pressure is close to the three-shock solution behind both the Mach stem and the reflected shock, up to distances of approximately 150 mean free paths for the streamline passing through the triple point. A similar field for the deflection angle in figure 8(c) demonstrates that in the vicinity of the shock intersection, the flow deflection angle is lower than that predicted by the three-shock theory except for the streamline passing through the triple point, where it is greater than the three-shock theory value. It can also be demonstrated by comparing the positions of points A, B, C, D and E in the flow fields with their positions on the polar in figure 5(c). Note, on the one hand, that the direction of streamline passing through point C agrees well with the direction of the slip line in the three-shock solution; on the other hand, there is more than 1 % difference in the deflection angle: the flow in the shear layer is deflected more than is predicted by the shock wave solution. It corresponds to the position of point C on the polar. There is a very narrow region on the lower boundary of the shear layer where the flow direction is predicted by the inviscid theory, while this region is much bigger above the shear layer, which is in agreement with figures 8(a,b). Again comparing the positions of the points in the flow fields and on the polar, one can conclude that the non-Rankine–Hugoniot zone is significantly (almost by one order of magnitude) smaller than the equalisation zone where non-Rankine–Hugoniot relations are satisfied but the flow parameter values clearly contradict the three-shock theory. In particular, this is true for the segment DE and even for some part of the reflected shock above.

5.2. Passage to the inviscid limit

The free-stream mean free path $\lambda _\infty$ is the only independent length scale in the problem of the internal structure of the shock wave. For this reason, naturally, the solution obtained in coordinates normalised to $\lambda _\infty$ does not depend on the Reynolds number. In the irregular reflection of shock waves, there are additional length scales: the wedge length $w$, and the distance from the wedge trailing edge to the symmetry line $g$. In fact, the independent dimensionless parameters are the Reynolds number based on one of the length scales (e.g. $\textit {Re}_w$) and the ratio $g/w$. The following question is natural: does the flow field inside the region of shock wave interaction depend on these parameters? Of primary interest is the dependence on the Reynolds number $\textit {Re}_w$, which can vary within a wide range.

The flow fields near the triple point at $\textit {Re}_w=10^4$ and $10^{8}$ are compared in figure 9. Here, the coordinates are normalised to the mean free path of molecules in the free stream for the corresponding Reynolds number. The flow fields are shifted so that the shock wave intersection positions match each other. This shift is necessary because the Mach stem heights, and hence the triple point positions, are different at different Reynolds numbers.

Figure 9. Comparison of flow fields near the triple point at two Reynolds numbers: (a) pressure; (b) pressure in the enlarged region; (c) Mach number in the enlarged region; (d) flow deflection angle in the enlarged region.

Figure 9(a) shows the flow field near the triple point with size approximately ${100\lambda _\infty \times 100\lambda _\infty }$. It is seen clearly that the pressure contours inside the region with size approximately ${30\lambda _\infty \times 30\lambda _\infty }$ coincide completely both inside the shock waves and behind them. Noticeable differences in the pressure contours obtained at different $\textit {Re}_w$ are observed only at a larger distance downstream from the interaction region. The region with size approximately ${30\lambda _\infty \times 30\lambda _\infty }$ is zoomed out in figures 9(b,d). The zone of increasing pressure (figure 9b), which is not described by the inviscid three-shock theory, is observed even at $\textit {Re}_w=10^{8}$. In fact, figure 9(b) shows the vicinity of point C presented in figure 5(b), where a significant difference between the viscous and inviscid solutions is observed. Figures 9(c,d) show the Mach number and deflection angle flow fields, where the shear layer emanating from the shock wave interaction region is visible, in contrast to the pressure field. As is seen, the isolines coincide completely in a small vicinity of the triple point inside the shear layer.

For a more detailed quantitative comparison of flow fields at different Reynolds numbers, figure 10 shows the numerical data in the $(\theta, p/p_\infty )$ plane along with the shock polars (figure 10a) and the pressure distributions at various $y/\lambda _\infty$ along with the three-shock theory prediction (figure 10b). The pressure and deflection angle distributions along the black dashed curve in figure 9(a) are plotted in figure 10(a). It is seen clearly that the numerical data at all $\textit {Re}_w$ values agree well with each other. As concerns the pressure distributions along the horizontal lines, they also coincide at different $\textit {Re}_w$. In all cases, the pressure behind the shock waves is higher than that in the three-shock solution. Along the line $y/\lambda _\infty =870$ passing through the non-Rankine–Hugoniot zone, the pressure in the region close to point C in figure 10(a) exceeds the inviscid solution by almost 10 %. The results computed at the intermediate Reynolds number $\textit {Re}_w=10^6$ also confirm that the flow pattern around the triple point in coordinates normalised to $\lambda _\infty$ remains unchanged despite the change in the Reynolds number. The contours and the pressure distributions at $\textit {Re}_w=10^6$ are not shown because in the coordinates $(x/\lambda _\infty, y/\lambda _\infty )$, they are not distinguishable from the results presented for other Reynolds numbers.

Figure 10. Comparison of the numerical solutions at different Reynolds numbers: (a) the $(\theta, p/p_\infty )$ plane; (b) pressure distributions at various $y/\lambda _\infty$.

It should be emphasised that the mean free paths at $\textit {Re}_w=10^4$ and $10^{8}$ in non-normalised coordinates differ by $10^4$ times. The entire range of the $x$ coordinate over which the pressure distribution at $\textit {Re}_w=10^{8}$ is presented (figure 10b) is smaller than the mean free path at $\textit {Re}_w=10^4$. Nevertheless, being normalised to $\lambda _\infty$, they coincide completely.

Note that the maximum Reynolds number $10^8$ considered in the present study is somewhat close to the upper limit attainable in ground-based aerodynamic experiments.

5.3. Total enthalpy behaviour in the non-Rankine–Hugoniot zone

For a more detailed study of the non-Rankine–Hugoniot zone, we consider the total enthalpy distribution. According to the Rankine–Hugoniot relations, the total enthalpy is constant across the shock wave in a steady flow of an inviscid and non-heat-conducting fluid. For a viscous heat-conducting fluid, the total enthalpy changes inside the shock wave, but its values ahead of and behind the shock wave are equal (for more details, see e.g. Shoev, Timokhin & Bondar Reference Shoev, Timokhin and Bondar2020).

Figure 11(a) shows the total enthalpy field near the triple point. As shown in the previous subsection, the flow fields near the three-shock intersection in coordinates normalised to the mean free path $\lambda _{\infty }$ are identical at different Reynolds numbers, so the results reported in this subsection are relevant for a wide range of $\textit {Re}_w$. The total enthalpy increases inside the shock wave, reaches the maximum value, and then decreases; behind the shock wave, the total enthalpy is again equal to its free-stream value. A noticeable exception is the shear layer, where the viscosity effects lead to the non-uniform distribution of the total enthalpy.

Figure 11. Numerical solution at the Prandtl number $\textit {Pr} = 2/3$: (a) total enthalpy field near the triple point; (b) total enthalpy distributions at various $y/\lambda _\infty$.

The total enthalpy distributions along the lines $y/\lambda _\infty =850$, $870$ and $890$ are shown in figure 11(b). At $y/\lambda _\infty =890$, one can see perfect recovery of the total enthalpy behind the IS and RS because the line is far from the shock wave intersection. The characteristic peaks inside the IS and RS are separated by a distance much greater than the shock wave thickness. At $y/\lambda _\infty =870$, the total enthalpy behind the incident shock wave does not have enough space to return to its free-stream value; instead, it passes smoothly to the distribution inside the reflected shock wave. In the region of shock wave intersection, the distance along the $x$ axis between the total enthalpy peaks is comparable with the shock wave thickness. The total enthalpy behind the RS is not recovered because of the non-uniform shear layer flow. Further downstream, the line $y/\lambda _\infty =870$ leaves the shear layer because the shear layer is not aligned with the horizontal axis. As a result, the total enthalpy approaches its free-stream value at a distance of approximately $50\lambda _\infty$. At $y/\lambda _\infty =850$, the peak of the total enthalpy inside the MS is significantly higher than peaks inside other shock waves. It is explained by a greater shock strength of the MS. The total enthalpy is recovered behind the MS, then it changes in the shear layer.

According to the Navier–Stokes solution for the viscous shock transition, the total enthalpy is strictly constant everywhere, including the shock transition interior if ${\textit {Pr} = 3/4}$ (see also Appendix A). This property is used below to better visualise the location of the non-Rankine–Hugoniot zone.

The total enthalpy field computed with $\textit {Pr} = 3/4$ (all other parameters are the same) is shown in figure 12. The black solid contours (figure 12a) are the pressure isolines $p/p_\infty =1.01$, 9.1, 9.15 and 19.6, which approximately show the boundaries of the shock waves. The total enthalpy inside the shock waves is indeed constant everywhere except for the region of shock wave intersection. It is seen in figure 12(a) that the shear layer is formed inside the region where all shock waves merge together in the vicinity of the point $(x/\lambda _\infty, y/\lambda _\infty ) = (673, 867)$. The flow parameters behind this region cannot be predicted by the Rankine–Hugoniot relations because it is impossible to identify a separate shock wave. The one-dimensional viscous shock transition solution is not applicable in this region either because the flow here is essentially two-dimensional. Therefore, a deviation of the total enthalpy from a constant value could be expected. A noticeable change in the total enthalpy is observed inside the region of shock wave confluence between the pressure contours $p/p_\infty =1.01$ and $19.6$. In fact, it visualises the non-Rankine–Hugoniot zone. It is important to note that in contrast to the inviscid case, the shear layer in the Navier–Stokes computations does not originate from a point; instead, it emerges as a finite-size spot (approximately $10\lambda _\infty$; see figure 12a) immediately behind or even inside the region of shock wave interaction. The distributions at ${y/\lambda _\infty =850}$, $870$ and $890$ are presented in figure 12(b), where the total enthalpy is plotted on the same scale as for $\textit {Pr} = 2/3$. At $y/\lambda _\infty =890$, the total enthalpy is constant with high precision because this line is taken far from the region where the shock waves and the shear layer interact. At $y/\lambda _\infty =870$, the total enthalpy increases inside the reflected shock wave and the shear layer, then it decreases further downstream. Finally, at $y/\lambda _\infty =850$, the total enthalpy increases only slightly inside the Mach stem, while further downstream it first decreases and then increases again in the shear layer. It should be noted that the increase in the Prandtl number from $\textit {Pr} = 2/3$ to $3/4$ reduces the total enthalpy deviation inside the shear layer from the free-stream value. For example, at $y/\lambda _\infty = 850$, the minimum value is $-0.013$ at $\textit {Pr} = 2/3$ (figure 11b) and $-0.009$ at $\textit {Pr} = 3/4$ (figure 12b).

Figure 12. Numerical solution at the Prandtl number $\textit {Pr} = 3/4$. (a) Total enthalpy field near the triple point. The black solid contours indicate the shock wave boundaries. (b) Total enthalpy distributions at various $y/\lambda _\infty$. (c) Total enthalpy distributions along lines $L_1$, $L_2$ and $L_3$ crossing the shear layer.

The total enthalpy is not equal to its free-stream value inside the shear layer downstream from the non-Rankine–Hugoniot zone. The shear layer thickness grows downstream. The total enthalpy distribution across the shear layer is non-monotonic (figure 12c). It is seen that the shear layer consists of two parts, where the total enthalpy values are lower and higher than its free-stream value, respectively.

Let us consider in more detail the region where the shock waves meet (figure 13). Figures 13(a,c) additionally show the vertical blue lines corresponding to several $x/\lambda _\infty$ values, and the inclined pink lines $L_{b}$, $L_{c}$, $L_{t}$ along which the total enthalpy distributions are plotted in figures 13(b,d). The total enthalpy is constant along the line ${x/\lambda _\infty =669}$ in figure 13(b) because the shock waves do not yet interact there and are clearly separated from each other. However, slightly (only by $\lambda _\infty$) downstream, the total enthalpy noticeably deviates from the constant value: a small peak appears inside the incident shock wave in the distribution at $x/\lambda _\infty =670$. The minimum and maximum are visible in the next distribution at $x/\lambda _\infty =671$, inside the shock wave confluence. Further downstream ($x/\lambda _\infty =673$ and $676$), the total enthalpy deviates from its free-stream value more and more noticeably. As was demonstrated in figure 12, these peaks of the total enthalpy persist inside the shear layer far beyond the region of shock wave interaction. It should be noted that there is a local maximum in the total enthalpy field inside the region of shock wave interaction, which is evidenced by the presence of a closed isoline (through which the line $L_{c}$ passes; figure 13c).

Figure 13. Numerical solution at $\textit {Pr} = 3/4$. (a,c) Total enthalpy field in a small vicinity of the triple point. The black solid contours approximately indicate the shock wave boundaries. (b,d) Total enthalpy distributions along several vertical and inclined lines.

5.4. Curvature of the sonic line inside the Mach stem

Passing through the Mach stem, the supersonic flow is decelerated to a subsonic velocity so that the Mach stem in the inviscid case is also a sonic line. The slope of the Mach stem changes from $90^\circ$ near the symmetry line to the angle predicted by the three-shock theory at the triple point. In a viscous flow, the Mach stem is a finite-thickness shock transition, whereas the sonic line remains infinitely thin. The sonic line inclination can serve as a convenient quantity to characterise the change in the Mach stem shape from the symmetry line to the triple point.

It was shown above that viscous effects lead to noticeable deviations of the flow parameters behind the Mach stem from the I-polar (point B in figure 10a). Visualisation of the sonic line reveals an important feature of the viscous flow: the sonic line has an inflection close to the triple point (figure 14). Figure 14(a) shows the shape of the sonic line $x_{son}(y)$; the inflection point is the point where the second derivative vanishes, ${\partial ^2 x_{son}}/{\partial y^2} = 0$. The existence of the inflection point is hardly visible in figure 14(a); however, it is evident from figure 14(b), where ${\partial ^2 x_{son}}/{\partial y^2}$ is plotted. Here, point B is the same as in previous figures. As was shown by Tan, Ren & Wu (Reference Tan, Ren and Wu2006), the existence of the MS inflection near the triple point is more pronounced at other free-stream parameters.

Figure 14. (a) Deflection angle flow field near the triple point. The black solid curve is the sonic line. The streamlines are indicated by arrows. (b) Second derivative ${\partial ^2 x_{son}}/{\partial y^2}$ versus $y/\lambda _\infty$. (c,d) Fields of the mass flux components $\rho u$ (c) and $\rho v$ (d).

The fields of the mass flux components $\rho u$ and $\rho v$ (figures 14c,d) reveal two regions with closed contours of $\rho u$ separated by a line passing through the inflection point. At the same time, a closed isoline of $\rho v$ in figure 14(d) is located well above the inflection point and corresponds to the closed contour in the field of the flow deflection angle seen in figure 9(d). The inflection point on the MS observed in the Euler computations of Tan et al. (Reference Tan, Ren and Wu2006) was connected, most probably, with not physical but numerical viscosity, which is an inherent feature of shock-capturing schemes. Generally speaking, the form of the numerical viscosity differs from the viscous terms of the Navier–Stokes equations; nevertheless, the numerical viscosity still works in such a way that shock waves acquire an internal structure. Therefore, it can be assumed that the mechanisms of the emergence of the inflection point in these two cases are similar, and its existence is a specific feature of the irregular interaction of finite-thickness shock waves. This assumption can be verified by solving the Euler equations with a shock-fitting scheme, as was done e.g. by Ivanov et al. (Reference Ivanov, Bonfiglioli, Paciorri and Sabetta2010b) and Ivanov, Paciorri & Bonfiglioli (Reference Ivanov, Paciorri and Bonfiglioli2010c) for other flow conditions.

6.1. Effects of viscosity and passage to the inviscid limit

As was noted above, a numerical study of the weak reflection in a steady viscous flow was undertaken earlier by Ivanov et al. (Reference Ivanov, Shoev, Khotyanovsky, Bondar and Kudryavtsev2012), and the passage to the inviscid limit was investigated. In that paper, however, only the case where the three-shock solution does not exist (the von Neumann paradox conditions) was considered. In the present paper, we have performed similar computations for a more common situation when the I- and R-polars intersect, and the three-shock solution exists. The essential difference between the considered strong and weak reflections is that the flow behind the RS is subsonic in the latter case.

The computations are performed at the following values of the flow parameters: $M_\infty = 1.7$, $\gamma = 5 / 3$, $\theta _w = 8.5^\circ$, and various $\textit {Re}_w$. The shock polars for $\theta _w = 8.5^\circ$ are shown in figure 2(b); the three-shock solution corresponds to point B(C).

The pressure fields near the triple point at various Reynolds numbers (from ${\textit {Re}_w=4\times 10^3}$ to $\textit {Re}_w=2\times 10^8$) are shown in figure 15. The black solid lines show the locations of the shock waves and the slip line predicted by the three-shock theory. It is seen that the slopes of RS and MS at $\textit {Re}_w=4\times 10^3$ do not agree well with the inviscid solution. As the Reynolds number is increased, the agreement between the numerical solution and the three-shock theory becomes better and better. Despite a significant change in the shock wave slopes with an increase in the Reynolds number, there are only minor changes (${\sim }0.5\,\%$) in pressure behind these shock waves.

Figure 15. Pressure fields near the triple point at $M_\infty =1.7$, $\gamma =5/3$, $\theta _w=8.5^\circ$: (a) $\textit {Re}_w=4 \times 10^3$; (b) $\textit {Re}_w=4 \times 10^4$; (c) $\textit {Re}_w=8 \times 10^6$; (d) $\textit {Re}_w=2 \times 10^8$. The black solid lines show the shock wave locations predicted by the three-shock solution.

The fields of the flow deflection angle computed at $\textit {Re}_w=4 \times 10^3$ and $2 \times 10^8$ are shown in figures 16(a,b). As is seen clearly, in contrast to pressure, the flow deflection angle changes significantly: the maximum value increases from approximately $4.2^\circ$ to approximately $6.484^\circ$. At higher Reynolds numbers, the shear layer is thinner, and its slope tends to the slip line angle predicted by the three-shock theory. For a more detailed analysis of the flow near the triple point, we plot in figure 16(c) the flow deflection angle distributions along lines $L1$, $L2$, $L3$ (figure 16(a), $\textit {Re}_w=4\times 10^3$) and $L1$*, $L2$*, $L3$* (figure 16(b), $\textit {Re}_w=2\times 10^8$). The distributions at low and high Reynolds numbers are shown by solid curves and symbols, respectively.

Figure 16. Numerical solutions of the Navier–Stokes equations at different Reynolds numbers. Flow deflection angle field at (a) $\textit {Re}_w=4 \times 10^3$, and (b) $2 \times 10^8$. The black solid curves show the shock wave locations predicted by the three-shock theory. (c) Distributions of the flow deflection angle along several horizontal lines. The slope of the slip line predicted by the three-shock solution is shown as SL. (d) Numerical data plotted in the $(\theta, p/p_\infty )$ plane along with the shock polars.

Lines $L1$ and $L1$* pass through the IS and RS. It is seen clearly that inside the IS (this region is indicated as the ‘IS profile’) the distributions along $L1$ and $L1$* completely coincide when plotted in the $x/\lambda _\infty$ coordinates. As for the RS, the distributions inside it at different $\textit {Re}_w$ coincide up to a certain point. After that, the distributions diverge so that the flow deflection angle behind the RS is noticeably smaller at the lower Reynolds number. The curvature of the RS in the normalised coordinates is larger at low $\textit {Re}_w$. As a result, the slope of the RS at the intersection with $L1$ is noticeably different from that predicted by the three-shock solution. This difference is much smaller at higher $\textit {Re}_w$ since the RS shape is closer to the straight line.

Lines $L2$ and $L2$* pass through the region of shock wave confluence. The distributions along $L2$ and $L2$* already differ inside the IS. More exactly, the profiles also coincide up to a certain point and diverge rapidly behind it. At high Reynolds numbers, the distribution along $L2$* has a small peak, which almost reaches the theoretical value at the triple point labelled as SL in figure 16(c).

A similar behaviour is also observed for the distributions along lines $L3$ and $L3$* passing through the MS. At first, they almost coincide inside the MS; however, farther downstream they diverge significantly. As earlier, this is explained by a larger curvature of the MS (in the normalised coordinates) at low $\textit {Re}_w$.

It should be noted that the flow deflection angle at high Reynolds number is nearly the same in the whole region behind the shock waves. It is somewhat lower than SL value, the theoretical value at the triple point. A different behaviour is observed at low Reynolds number: behind the shock waves, the distributions along the lines $L1$ and $L2$ are fairly close to each other, whereas the distribution along $L3$ behind the MS is noticeably different from them.

The results computed at five different Reynolds numbers are compared in figure 16(d) in the $(\theta, p/p_\infty )$ plane. It is seen clearly that an increase in the Reynolds number leads to an increase in the maximum flow deflection angle, whereas the pressure remains nearly the same, which was actually demonstrated above in considering the isobars near the triple point. At $\textit {Re}_w = 2 \times 10^8$, the numerical data approach closely the shock polar intersection, which represents the three-shock solution. It can be expected that the inviscid solution will be reached in the limit $\textit {Re}_w \to \infty$. This behaviour is qualitatively consistent with that obtained previously in studying the weak reflection shock waves under the von Neumann paradox conditions. In both cases, the viscous solution tends to the inviscid configuration as the Reynolds number increases, with only one difference: this is a configuration corresponding to the four-wave solution in the case of the von Neumann paradox and a simpler, three-shock, configuration in the present case. On the other hand, the numerical simulations also show that the viscosity plays an important role in the formation of the flow structure near the triple point in a wide range of Reynolds numbers, and significantly changes the mechanism of interaction of three intersecting shock waves.

Figure 17 shows the density near the triple point at $Re_w = 2\times 10^8$ and five density distributions at various Reynolds numbers along lines $L_{t}$, $L_{c}$ and $L_{b}$. The lines are parallel to the slip line of the three-shock solution and chosen so that $L_{t}$ passes across the IS and RS, $L_{c}$ passes through the shock interaction region entering the shear layer, and $L_{b}$ passes across the MS. The $L_{c}$ line passes through the origin (the triple point of the three-shock solution), while $L_{t}$ and $L_{b}$ intercept the $y$ axis 40 mean free paths above and below the origin, respectively. The distributions along $L_{t}$ show that the density behind the RS tends to the three-shock solution as the Reynolds number increases. The deviation from the three-shock solution decreases with $Re_w$; in particular, it is less than 0.2 % at $Re_w > 8\times 10^5$. The density distributions along $L_{c}$ show that in the plotted region inside the shear layer, the numerical viscous solution at $Re_w > 8\times 10^5$ predicts almost a constant density value between the values predicted by the three-shock solution behind the RS and MS. At $Re_w < 8\times 10^5$, the density does not reach a plateau and decreases downstream from the triple point. In an inviscid flow, the density has a jump across the slip line likewise in a shock wave, therefore in a viscous flow, the density inside the shear layer has a value somewhere in between the values across the slip line. It is necessary to note that the pressure and deflection angle have no jumps across the slip line in an inviscid flow. That is why the numerical data in the $(\theta, p/p_\infty )$ plane tend to the shock polar intersection point (inviscid three-shock solution) as the Reynolds number grows. The density distribution along $L_{b}$ also shows convergence of the numerical viscous solution to the inviscid three-shock solution with an increase in the Reynolds number. More exactly, the deviation of density behind the MS is smaller than 0.4 % at $Re_w > 8\times 10^5$.

Figure 17. Density contours and distributions at various $Re_w$. (a) Density contours at $Re_w = 2\times 10^8$. The black solid lines show the three-shock solution. Lines $L_{t}$, $L_{c}$ and $L_{b}$ are parallel to the slip line of the three-shock solution. (b–d) Density distributions at various $Re_w$ along lines (b) $L_{t}$, (c) $L_{c}$, and (d) $L_{b}$. Lines labelled ‘Three-shock solution RS’, ‘RS’ and ‘Three-shock solution MS’, ‘MS’ show the density according to the three-shock solution behind the RS and MS, respectively. The dashed curves, showing the deviation from the three-shock solution, are labelled ‘Deviation’.