Reflection Transition of Converging CylindricalShock Wave Segments

B.J. Gray and B.W. Skews

1 Introduction

An investigation was carried out into the transition between various types of reflec-tion of converging cylindrical shock wave segments over various wedges, and theeffect of the wave incidence angle and the Mach number of the incident wave on thereflection type. Such a reflection differs significantly from those of planar waves,as both the Mach number and the incident angle of the shock wave are time de-pendant. The transition conditions were examined in terms of transition criteria thathave been suggested for planar waves in the literature.

In general, current research on the behaviour of curved waves is limited to cylin-drical and spherical waves. In particular, the only previous work that related to re-flection of curved waves that could be found were those relating to the reflectionof blast waves generated by explosions at a certain height above the ground. Thisincluded the work of Takayama and Sekiguchi [1], Hu and Glass [2], and Liang etal.[3, 4]. However, there is no research to be found in the literature on the reflec-tion of a free curved shock segment that encounters an inclined surface, despite thepossible significance such research may have in a number of fields.

2 Review of Transition Criteria for Planar Waves

When an oblique shock wave encounters a solid boundary, reflection needs to occurin order for the flow to meet boundary conditions. For a plane wave reflecting offan inclined surface, reflection in steady conditions will be either a Mach reflection(MR) or regular reflection (RR), and the reflection pattern that occurs is a functionof the wave incident angle and Mach number [5]. For a given Mach number, thereis a critical incident angle (known as the maximum-deflection angle) above which

B.J. Gray · B.W. SkewsFlow Research Unit, School of Mechanical, Industrial and Aeronautical Engineering,University of the Witwatersrand, Johannesburg, 2050, South Africa

996 B.J. Gray and B.W. Skews

no reflected wave exists that is capable of deflecting the flow so that it runs parallelto the boundary. If the incident angle of the shock wave exceeds the maximumdeflection angle, Mach reflection must occur according to the maximum-deflectioncondition.

The sonic condition infers that in order for a Mach reflection to occur, some formof length scale needs to be transmitted to the reflection point in order for the Machstem to form [6]. This is not possible if the flow behind the reflected wave is super-sonic relative to the reflection point. Thus the sonic condition states that transitionfrom regular reflection to Mach reflection will occur when the flow behind the re-flected wave becomes sonic relative to the reflection point. The transition anglespredicted by the maximum-deflection and sonic criteria typically vary by fractionsof a degree, and it is difficult to experimentally distinguish the difference betweenthe two criteria.

In unsteady flows, the above mentioned criteria require a discontinuous pressurechange in the region behind the reflected shock wave. Such a discontinuous changewould cause unsteady pressure waves to form in the system in order to maintainmechanical equilibrium. Consequently, a new transition criterion was proposed byHenderson and Lozzi [7], which stated that the transition would occur at a criticalangle at which the pressure behind the reflected shock can vary continuously as thereflection transition takes place, which would occur when the flows predicted bytwo- and three-shock theories coincide. For a Mach reflection at this particular in-cident angle, the flow behind the reflected shock is parallel to the reflection surface.This represents the limit at which direct Mach reflection (DiMR) can occur. Be-low this angle, any Mach reflection that occurs must be an inverse Mach reflection(IMR), which is an unstable reflection in which the triple point moves towards thereflection surface, eventually terminating in transitioned regular reflection (TRR).Below a Mach number of 2.23, the mechanical equilibrium criterion coincides withthe maximum deflection criterion.

Ben-Dor [5] describes some analytical models for predicting the transition fromMR to TRR for a planar wave impinging on a concave cylindrical surface. The basisof these models is that the Mach stem in a TRR represents the limit which cornersignals carrying information about the length scale of the system can reach. How-ever, they are independent of the radius of curvature of the wall, yet experimentalobservations have shown that the radius does play a role in determining the tran-sition angle. Ben-Dor states that a model taking the radius into account [8] is stillincomplete.

3 Numerical Computation of Cylindrical Shock Waves overInclined Surfaces

Numerical simulations were carried out in FLUENT R© using an inviscid, secondorder implicit density based solver. The flow was modelled using a third orderMUSCL scheme with Roe’s Flux-Difference Splitting scheme, with least squares,cell-based spatial discretisation. The domain consisted of a curved pressure inlet,

Reflection Transition of Converging Cylindrical Shock Wave Segments 997

two converging walls, and an inclined wedge, which was meshed using a mappedquadrilateral meshing scheme. The pressure at the inlet was set such that a cylin-drical shock wave would be generated so as to have a specific Mach number atthe wedge apex. Initially, simulations were run using three different mesh densities,these being 100, 150, and 200 cells across the inlet. In addition, the mesh was adap-tively refined up to 6 times in regions where the density gradient exceeded 2% ofthe maximum density gradient in the domain. The general behaviour of the shockwave for all three meshes was found to be similar, but for further investigation, themesh density of 200 cells across the inlet was used as this offered better resolutionof flow features which formed during the reflection transition.

Computations were carried out for shock waves with a radius of 100 mm, andMach numbers at the wedge apex of 1.29, 1.43, 1.57 and 1.70 impinging on wedgesof angles of 20◦, 25◦, 30◦, 35◦, 40◦, 45◦ and 50◦to the original flow direction. Ini-tially, the reflection pattern that resulted was a direct Mach reflection (DiMR), withthe shear layer starting at the the triple point and sloping towards the surface, andwith the triple point trajectory directed away from the surface. This transitionedinto an inverse Mach reflection (IMR) in which the triple point trajectory turns backtoward the reflection surface, and the shear layer slopes from the triple point awayfrom the surface. The Mach reflection eventually terminates in a transitioned regularreflection (TRR), which is characterised by a Mach stem which follows the reflec-tion point up the wedge, but at a slower velocity. An example of the evolution of thereflection structure is shown in Figure 1.

4 Discussion

The sequence of reflection patterns of DiMR→StMR→IMR→TRR is similar to thesequence that takes place in the analogous case of a planar shock impinging on acurved wall, and that of the reflection of an oblique shock in a wind tunnel with con-tinuously varying upstream geometry. However, the cases investigated differ fromthose situations in that the Mach number of the shock wave and the conditions be-hind the shock vary continuously.

In Figures 1(d) to (f), a Kelvin-Helmholtz instability is seen to form in the shearlayer. Such instabilities are typically greatly exaggerated in high resolution solu-tions to the Euler equations (such as the one carried out here), and are usually muchsmaller or not present at all in experimental results or solutions to the full Navier-Stokes equations, as discussed by Sun and Takayama [9]. However, the instabilitiesonly form after the transition to TRR, and thus do not affect the validity of theresults. Furthermore, the Kelvin-Helmholtz instability is not expected to have a sig-nificant effect on the shape of the shock structure.

For planar shock waves in pseudosteady flows (such as those which occur inshock tubes), the criterion which is considered to come closest to predicting transi-tion is the sonic condition. Figure 2 shows a plot of the transition angle for a planarwave predicted by the sonic condition against Mach number. Cases that lie above thetransition curve would be expected to show a Mach reflection, while cases below the

998 B.J. Gray and B.W. Skews

(a) Shock immediately beforeencountering the wedgeat t=0μs

(b) Direct Mach reflection(DiMR) at t=34μs

(c) Stationary Machreflection (StMR) att=68μs

(d) Inverse Mach reflection(IMR) at t=77μs

(e) Triple point collidingwith the reflectionsurface at t=82μs

(f) Transitioned regularreflection (TRR) att=89μs

Fig. 1 Evolution of the shock reflection structure for a shock reflecting off a 30◦ wedge witha Mach number of 1.43 and shock radius of 100 mm at the wedge apex, showing contours ofconstant density. A 6x zoomed image of the reflection is shown below each image

curve would be expected to show regular reflection. On the same axes, the reflectiontype obtained in this study immediately after the shock encounters the wedge apexare plotted.

A Mach stem and shear layer was evident in 22 of the 28 cases tested, allowingthose reflection patterns to be identified as Mach reflections. The remaining 6 casesshowed no visible Mach stem or shear layer, and were classified as regular reflec-tions. Comparing the results to the transition line predicted by the sonic criterion, itis apparent that 5 cases which demonstrated a Mach reflection after the wedge apexshould have shown a regular reflection. The fact that Mach reflection could occurbelow the theoretical transition line implies that the transition criteria are different

Reflection Transition of Converging Cylindrical Shock Wave Segments 999

1.2 1.3 1.4 1.5 1.6 1.7 1.835

40

45

50

55

60

65

70

75

Shock Mach number at wedge apex

Inci

dent

ang

le (

°)

regular reflectionsMach reflectionSonic criterion

Fig. 2 Initial reflection patterns obtained immediately behind the wedge apex, including thetransition line predicted by the sonic transition criterion

1.3 1.4 1.5 1.6 1.7 1.8 1.9 215

20

25

30

35

40

45

Shock Mach number at transition to TRR

Inci

dent

ang

le (

°)

20° wedge25° wedge30° wedge35° wedge40° wedge45° wedge

Fig. 3 Variation of shock incident angle at the point of transition to TRR

for curved waves, although whether this is due to the radius of curvature of the shockwave or the rate of amplification of the Mach number is unclear.

Figure 3 shows the points at which transition to TRR occurred. These pointscannot be directly compared to the first three of the transition criteria described insection 2, as these criteria were developed for application to the RR→MR transitionof a planar shock with a constant Mach number. Ben-Dor’s models for predictingTRR transition make a number of assumptions which make a direct comparisoninvalid in this case — he assumes that the incident shock is planar and propagatesat a constant velocity; the reflected shock has negligible strength compared to theincident shock; the conditions behind the shock are uniform; and that information

1000 B.J. Gray and B.W. Skews

about the length scale needs to be transmitted to the reflection point along the wedge.The results show that the first three of these assumptions do not hold. In addition, thelength scale need not be carried to the reflection point by corner generated signals,as information about a length scale is inherently carried to the transition point in theMach stem of the IMR (although this information is lost at the transition point).

5 Conclusion

MR was found to occur when the shock encountered the wedge in most cases, evenin some cases where the sonic condition predicts RR for planar waves at the sameincident angle and Mach number. Thus, shock wave curvature has some impact onwhich reflection pattern occurs. When the transition from MR to TRR is consid-ered, there is significant disagreement between the transition points predicted bythe results of the numerical simulations of converging cylindrical waves, and thosepredicted by the transition criteria for planar waves. A more accurate transition cri-terion would have to take into account the radius of the shock wave and the timetaken for the triple point to return to the surface once IMR occurs. A unique facilityis currently being constructed to validate the above findings experimentally.

References

1. Takayama, K., Sekiguchi, H.: Triple-point trajectory of a strong spherical shock wave.AIAA Journal 19, 815–817 (1981)

2. Hu, T., Glass, I.: Blast wave reflection trajectories from a height of burst. AIAA Journal 24,607–610 (1986)

3. Liang, S.M., Hsu, J.L., Wang, J.S.: Numerical study of cylindrical blast-wave propagationand reflection. AIAA Journal 39(6), 1152–1158 (2001)

4. Liang, S.M., Wang, J.S., Chen, H.: Numerical study of spherical blast-wave propagationand reflection. Shock Waves 12(1), 59–68 (2002)

5. Ben-Dor, G.: Shock wave reflection phenomena, 2nd edn. Springer, Heidelberg (2007)6. Hornung, H.: Regular and Mach Reflection of Shock Waves. Annual Review of Fluid

Mechanics 18(1), 33–58 (1986)7. Henderson, L.F., Lozzi, A.: Experiments on transition of Mach reflexion. Journal of Fluid

Mechanics 68, part 1, 139–155 (1975)8. Ben-Dor, G., Takayama, K.: The dynamics of the transition from Mach to regular reflec-

tion over concave cylinders. Israel Journal of Technology 23, 71–74 (1986/1987)9. Sun, M., Takayama, K.: A note on numerical simulation of vortical structures in shock

diffraction. Shock Waves 13, 25–32 (2003)