Fa WUHuihui DAIDexing KONG

1.Department of Mathematics，Zhejiang University，Hangzhou 310027，China;2.Department of Mathematics，City University of Hong Kong，83 Tat Chee Avenue，Kowloon Tong，Hong Kong;3.Department of Mathematics，Zhongyuan University of Technology，Zhengzhou 450007，ChinaE-mail:wufa85zju.edu.cn;mahhdai@cityu.edu.hk;dkong@zju.edu.cn

MECHANISM FOR THE TRANSITION FROM A REGULAR REFLECTION TO A MACH REFLECTION OR A VON NEUMANN REFLECTION?

1.Department of Mathematics，Zhejiang University，Hangzhou 310027，China;
2.Department of Mathematics，City University of Hong Kong，83 Tat Chee Avenue，Kowloon Tong，Hong Kong;
3.Department of Mathematics，Zhongyuan University of Technology，Zhengzhou 450007，China
E-mail:wufa85zju.edu.cn;mahhdai@cityu.edu.hk;dkong@zju.edu.cn

In this paper，by taking into account the thickness of the incident shock as well as the influence of the boundary layer，we point out that even in a regular reflection there should be present a contact discontinuity.By using the smallest energy criterion，the inclined angle of this contact discontinuity can be determined for differen incident angle.Then，with this inclined contact discontinuity，together with the law of conservation of mass，the mechanism for the transition from a regular reflection to a Mach reflection or a von Neumann reflection becomes clear.The important roles played by the leftest point in the reflected shock polar are identified.

steady flows，shock wave;shock polar;regular reflection;Mach reflection;von Neumann reflection

2010 MR Subject Classification35L65;76L05

1　Introduction

It is well-known that the reflection of a plane shock at a rigid wall will be a regular reflection（RR）（some time also called a simple reflection）if its angle of incidence is comparatively small，but that it will be a more complicated irregular（for example，Mach reflection（MR）or von Neumann reflection（NR））for a larger incident angle.There are a lot of works on the transition between regular reflection and irregular one，in particular，Mach reflection and von Neumann reflection（see e.g.，［1，2，5，7］）.The criterion commonly used to predict the transition RR←→MR or NR between the two wave systems makes use of the boundary condition that the flow downstream of the reflection must be parallel to the wall.Henderson and Lozzi［5］presented detailed experimental data on the transition between regular and Mach reflections.Datawere obtained for steady，pseudo-steady and unsteady flows.Experimental data also included a study of the continuous and discontinuous transitions predicted by other researchers.

In the literature，the discussions on the transition of a RR to a MR or NR are usually based on that a shock is thickless.Here，we shall consider the thickness of a shock in the reflection process.Due to such a thickness，naturally there is a“geometrical stem”in the wave configuration（cf.Fig.12）.Also，the interaction between the reflected shock and the boundary layer is taken into account to a certain extent.Then，we offer the explanation why there is a contact discontinuity even in a regular reflection.Evidences of experimental results done by others which support our conclusion are also provided.The orientation of this contact discontinuity is dependent on the angle of the incident shock.A theorem is established to determine the orientation of this contact discontinuity for the energy being smallest.It is found that in a regular reflection（the incident angle is smaller than αe，the extreme incident angle αe），this contact discontinuity is parallel to the wall.If the incident angle is larger than αe，then this contact discontinuity has to decline an angle β（＞0）towards the wall，where β corresponds the leftest point in the R-polar（cf.Figs.11 and 9）.Then，the geometrical stem，the contact discontinuity and the boundary layer form a triangle.As there are fluids entering this triangle through the geometrical stem，the triple point（or，the top point of this triangle）has to start moving upward，so the wave configuration of a Mach or von Neumann reflection is formed.The importance of the position of the leftest point is made clear by us.

The paper is organized as follows.In Section 2，we briefly recall classical results for thickless shocks.The arguments that a contact discontinuity is present in a regular reflection are provided in Section 3.Transition mechanism from a regular reflection to a Mach reflection is discussed in Section 4，where we also prove a theorem which can be used to determine the inclined angle of the contact discontinuity.In Section 5，we give the explanation for the formation of a von Neumann reflection.Finally，based on the position of the leftest point of the R-polar，two conjectures are made.

2　Some Main Results of Standard Shock Dynamics

In the theory of standard shock dynamics，a shock is treated as a thickless line across which physical quantities can experience jumps.Also，when there presents a solid boundary，usually the boundary layer effect is also neglected.Here，we review some classical results.A more comprehensive review can be found in Hornung［7］.

2.1The Jump Conditions on Shock and Shock Polars

Throughout this paper we study the stationary compressible flow in R2.The corresponding Euler system in two-dimensional space readswhere（u，v），p，ρ and S represent the velocity，pressure，density and entropy，respectively.In the case of polytropic gas considered in the present paper，the state equation is

in which A（S）is a positive function of S and γ＞1 is the adiabatic exponent.In a steady flow through an oblique shock wave（see Fig.1），the conservation equations may be used to relate the state downstream of the shock（subscript 2）to the state upstream（subscript 1）:

where M1is the flow Mach number in the state 1，γ is the adiabatic exponent of the gas.

Fig.1　An oblique shock wave

For fixed γ and M1＞1，by（2.1）-（2.4）we can draw the shock polars in the（θ，α）-plane and the（θ，p）-plane;see Figs.2-3.

Fig.2　Shock polar in the（θ，α）-plane

Fig.3　Shock polar in the（θ，p）-plane

Property 2.1Fig.2 shows that the deflection angle θ is a function of the shock angle α. As α increases from arcsindecreases from M1to the value 1（sonic point）just before θ reaches the maximum value（Maximum-deflection point）.Further increase of α causes M2and θ to decrease until（normal shock）.

Remark 2.1Fig.2 comes from（2.4）directly.Eliminating α between（2.2）and（2.4），we can draw Fig.3.Both curves in Figs.2-3 are symmetrical about θ=0.

Remark 2.2For the details on the jump conditions and shock polars，we refer the classical papers［7，8］and book［3］.

2.2Regular Reflection

Consider a plane shock being reflected off a wall or symmetry plane parallel to→q1;see Figs.4-5.

Fig.4　Regular reflection in a steady flow

Fig.5　shock polar in the（θ，p）-plane for a regular reflection

Obviously，the incident shock I deflects the flow to an angle θ2，and the reflected shock R is needed to deflect it back to θ3=0.It is better to represent this situation in the（θ，p）-plane: the state 2 lies on the incident shock polar.From this new state we may draw the second（reflected）shock polar with M=M2from the state 2:（θ2，p2）.Thus，the deflection is taken from 0 to θ2by the incident shock I，and from θ2to 0 by the reflected shock R.When θ2is not so large（i.e.，α is not so large），there are two intersection points，denoted by 3 and?3 respectively，of the reflected shock polar with the p-axis.The point 3 stands for the state given by the weak reflected shock，while?3 stands for the state given by the strong reflected shock. Physically，the weak reflected shock is stable，but the strong one is unstable.

When the incident angle α is increased（in this case，θ2is also increased），there exists a value α=αe（correspondingly，there is a value θe）such that the two intersection points 3 and ?3 coincide（i.e.，the leftest point in the R-polar is in contact with the vertical axis），and beyond which the reflected shock polar does not reach the p-axis any more;see Fig.6.

Fig.6　The extreme case of the regular reflection:θ2=θe

The configuration mentioned above is called a regular reflection;αeis called the extreme incident angle of the regular reflection，while θeis said the extreme deflection angle of the regular reflection.The physical picture of the regular reflection is given in Fig.4.

2.3Mach Reflection

As mentioned above，when the incident angle α is increased and becomes larger than αe（correspondingly，the deflection angle θ2is also increased and becomes larger than θe），in this case we observe that the reflected shock polar does not reach the p-axis any more;see Fig.7.

Fig.7　Mach reflection in the（θ，p）-plane

Fig.8　Mach reflection in the physical plane

In the present situation，the deflection in the region 3 is still toward the wall.In order to accommodate the transition from θ＞0 to θ=0，there is also a contact discontinuity CD and a near-normal shock，called the Mach stem and denoted by MS，which allows the triple shock point P to move away from the wall.The density，velocity and the entropy are discontinuous across CD，but the pressure and the streamline deflection are continuous.As a result，in the shock polar of the（θ，p）-plane，the points representing the states 3 and 4 coincide with each other.The upper intersection of I and R in Fig.8 gives the triple point condition，and the curved Mach stem MS is represented by the part of 4-5 of the curve I.

The reflection configuration mentioned above is called a Mach reflection.By the way，the wall-jetting effect in the Mach reflection is investigated by Henderson et al.［6］.

2.4von Neumann Reflection

Fig.9?。╬，θ）-solution of a von Neumann reflection

Fig.10　The wave configuration of a von Neumann reflection

Besides the Mach reflection，for some M1and a suitable incident angle α，another kind of irregular reflection，the so-called von Neumann reflection can also appear（see［2］）.For this reflection，the wave configuration consists of an incident shock wave，i，a band of selfsimilar reflected compression wave，c，a reflected shock wave，r，a Mach stem，m，and a contact discontinuity，s;see Fig.10.In this case，the shock polar corresponding to the reflected shock wave，the R-polar，does not intercept either the-axis or the upper branch of the incident shock polar，the I-polar;see Fig.9.However，since the boundary condition at the wall must be satisfied（which needs θ=0），the R-polar should be somehow connected to the upper branch of the I-polar（which intercepts the-axis，i.e.，θ=0）.This can be achieved through a C-polar of a compression wave which starts at a point in the R-polar，where this point is，in our view，still an open issue.Here，we take the point of view that it should start at the leftest point of the R-polar;see Fig.9.The justification will be given in Section 5.

3　The Presence of a Contact Discontinuity in a Regular Reflection

Currently，there are no satisfactory explanations why the wave configuration of a regular reflection should becomes that of a Mach reflection or a von Neumann reflection if the incident Mach number and the incident angle α vary.One important difference between a regular reflection（as we understand it as far）and a Mach reflection or a von Neumann reflection is that the latter two have a contact discontinuity in the wave configurations while the former does not have one.Here we argue that there should be a contact discontinuity even in a regular reflection.The explanation is given below.

As mentioned earlier，in the theory of standard shock dynamics，a shock is treated as a thickless line and the boundary layer effect near the wall is also neglected.Here，we shall consider the thickness of a shock purely from the geometrical point of view and further somehow take into account the boundary layer effect1The thickness assumption on the shock results in the viscosity in the ideal gas.In other words，the thickness assumption on the shock implies the assumption on the viscosity in the gas（at least near boundary layer），namely，the thickness of the shock is due to the viscosity in the gas..When an incident shock with a geometrical thickness is reflected from the wall，then automatically a stem（we call it a geometrical stem）is formed above the boundary layer;see Fig.12.Denote the contact point of the geometrical stem and the boundary layer by C.Then，at C，the normal velocity vCshould be zero.However，its tangential velocity uCis unknown.But，it is clear that uCwill depend on the viscosity of the gas;say，uC=uC（ν），where ν is the fluid viscosity.Incidently，it was observed in the experiment by Dewey and McMillin［4］that the foot of the Mach stem was thicker.This could imply that the influence of the viscosity on the point C is important.

We use A to denote the point near the triple point on the side of the reflected shock.If there is no any kind discontinuity between the point A and the point C，due to the closeness of A and C（the geometrical stem is very short），we can regard that

where vAand uAare the normal and tangential velocities at the point A，respectively.However，by using the jump conditions（or the shock polar）and（3.1），uAcan be determined completelyin terms of the incident Mach number M1and the incident angle α，so uAis independent of the fluid viscosity.This then shows the contradiction of（3.2）.As a result，there should be some kind of discontinuity between the point A and the point C.A natural choice is that there is a contact discontinuity between the point A and the point C such that（3.1）still holds but（3.2）does not need to be satisfied.

We also point out that in many experiments one can observe that the thickness of a contact discontinuity is much less than that of a shock.Thus，there should be the geometrical space for a contact discontinuity lying in between the point A and the point C.

It should be noted that the geometrical stem is very short due to the very smallness of the thickness of a shock.Because of this and also because the contact discontinuity could be short itself，this contact discontinuity is very difficult to be observed in experiments.Nevertheless，there are some experimental evidences to support our argument that there is a contact discontinuity in a regular reflection.The wave configuration of a regular reflection in our theory is equivalent to that of a Mach reflection with an almost zero-length Mach stem and a contact discontinuity parallel to the wall.Such a wave configuration has been observed in the experiments of the shock reflection over a concave wall，as reported in a presentation given by Sun et al.in the 23th International Symposium on Shock Waves（Fort Worth，Tenax，July 22-27，2001）.The full paper can be found in the website:http://ceres.ifs.tohoku.ac.jp/～swrc/papers/issw23/55889.pdf. Recently，Sudani et al.［9］studied the irregular effects on the transition from regular to Mach reflection.In the picture taken by them（see Figs.13（a）and 13（h）in［9］），a dark line in the middle，in the case of a regular reflection，was clearly visible，which appeared to be a contact discontinuity.Sudani and Hornug et al.gave a presentation in the 15th International Mach Reflection Symposium（Aachen，Germany，September 15-19，2002）.Their work was not formally published.Interested readers can find the slides of the presentation in the website: www.galcit.caltech.edu/～hans/slides.pdf.In one experiment，they inserted a needle near the reflection point.From the picture taken by them in the regular reflection，it appeared that there was a contact discontinuity.

With the presence of a contact discontinuity in a regular reflection，then it is easy to explain the transition to the wave configuration of a Mach reflection and that of a von Neumann reflection.

4　Mechanism of the Transition to a Mach Reflection

Note that the solution discussed in this paper is piecewise constant in the domains corresponding to the states 1-3.So，to understand the transition mechanism，we need to determine the inclined angle βcof the contact discontinuity with the horizontal direction in a regular reflection.The deflection angle across the reflected shock is θ2-βc;see Fig.13.We impose that θ2-βc≤θ2（i.e.，βc≥0）such that the transition to an inverted Mach reflection（one needs to impose some restrictions downstream in order for it to appear）will not happen.The angle βcwill depend on the incident angle α of the incident shock.

Theorem 4.1If α≤αe，the energy in the state 3 is the smallest for βc=0，where αeis the detachment angle;if α＞αe，the energy in the state 3 is the smallest for βc=β，where β is the smallest angle in the reflected R-polar;see Fig.11.

Fig.11　The definition of β

Fig.12　Physical picture at the moment before the Mach reflection appears

Fig.13　Physical picture at the moment when the Mach reflection appears

Proof of Theorem 4.1For simplicity，we consider the perfect gas，that is，the state equation of gas reads

where p is the pressure，ρ is the density，S is the entropy，A＞0 is a constant，γ＞1 is the adiabatic exponent，and cν＞0 is the specific heat capacity.

By（2.1）-（2.4），p2，ρ2，M2in the state 2 and the deflection angle θ2can be solved once p1，ρ1，M1in the state 1 and the incident angle α are given.Denote the pressure，the density，the entropy and Mach number in the state 3 by p，ρ，S，M.Similar to（2.1）-（2.4），on the reflected shock R we have

where φ is the incident angle in the state 2，i.e.，the deflection angle in the state 3;see Fig.13.

We now calculate the entropy S in the state 3.

By（4.2）-（4.3），it follows from（4.1）that

where φ2is the incident angle in the state 2 corresponding to the caseCombining（4.6）and（4.8），we observe that the entropy S is an increasing function of x，that is，S is an increasing function of sinφ and then it is also an increasing function ofOn one hand，it follows from（4.5）that φ is an increasing function of（also see Fig.2）.Summarizing the above arguments leads to the following fact:the entropy S is an increasing function ofOn the other hand，it is well-known that the physical entropy，denoted by Sphysics，satisfies

and then Sphysicsis a decreasing function ofWe note that if α≤αe，as βc≥0，the largest value ofTherefore the physical entropy takes its minimum at βc=0.This implies that the energy of gas in the state 3 is the smallest when βc=0.If α＞αe（in this case β＞0），as βc≥β，the largest value offor which βc=β.This implies that the energy of gas is the smallest.Thus，the proof is completed.

If we use the smallest energy as the selection criterion for the angle of the contact discontinuity，from the above theorem，we can conclude that:

1.If the incident angle α≤αe，the contact discontinuity is horizontal（parallel to the wall）;

2.If the incident angle α＞αe，the contact discontinuity will decline an angle β towards the wall.

Consider a process that the incident angle is gradually increased from a value less than αeto a value larger that αe.When α≤αe，the wave configuration is one with an incident shock，a reflected shock and a horizontal contact discontinuity.After α＞αe（we are considering the onset of the transition），the contact discontinuity starts declining an angle β towards the wall. Then，for the triangle bounded by the geometrical stem，the boundary layer and the contact discontinuity，as there are fluid particles entering this triangle through the geometrical stem，to conserve the mass，the triple point has to start moving.As a result，a Mach reflection will be formed.

It should be noted that the geometrical stem should not be treated as a shock stem as it is too close to the boundary layer.Thus，in the onset of the transition the state at the point E（see Fig.13）and the state 1 should not in the same shock polar.However，as the triple point is moving away from the wall and the stem becomes taller and taller，the influence of the boundary layer on the top portion of the stem should become smaller and smaller，and that part of the stem can then be treated as a shock stem（i.e.，the Mach stem）.

5　Mechanism of the Transition to a von Neumann Reflection

Consider the case that in a regular reflection the R-polar does not intersect the I-polar. Then，the incident angle is gradually increased to a value larger than the detachment angle.At this stage，the R-polar and I-polar is connected by a C-polar of a compression wave;see the discussion in the Subsection 2.4.In this section，we shall give the justification where the C-polar should intersect the R-polar and give an explanation why the original horizontal contact discontinuity in the regular reflection has to decline an angle towards the wall.

The physical picture of a von Neumann reflection is shown in Fig.14.As for the Mach reflection，we use the smallest energy as the selection criterion.In Fig.9，the state 3 and state 4 are connected by a compression wave，across which the entropy does not change.On the other hand，the state 3 is on the R-polar，so Theorem 4.1 still applies.Thus，for the smallest energy state，the deflection angle from the state 2 to the state 3 should be θ2-β，i.e.，the state 3 corresponds the leftest point of the R-polar.So，the velocity at the state 3 is not horizontal but instead has an angle β with the horizontal line.Across the compression wave，i.e.，from the state 3 to the state 4，for a simple gas，by（2.2）-（2.4）we obtain

As p4＞p3，we have θ4＞θ3.Thus，the velocity at the state 4 has an angle（i.e.，β?in Fig.9）with the horizontal line，which is larger than β.This angle is also the angle between the contact discontinuity and the horizontal line.

From the above discussion，we can see that once the incident angle is increased to a value larger than the detachment angle αe，a compression wave will appear and at the meantime the contact discontinuity（which is horizontal for θ≤αe）will decline an angle β?（＞β＞0）with the horizontal line.Then，as there are fluid particles crossing the geometrical stem to enter the triangle bounded by the contact discontinuity，the boundary layer and the geometrical stem，due to the conservation of mass，the triple point has to start moving，so the von Neumann reflection is formed.

Fig.14　The mechanism of formation of von Neumann reflection

6　Mach Reflection or von Neumann Reflection？

In experiments，for a given incident Mach number and incident angle，sometimes a Mach reflection is observed and sometimes a von Neumann reflection is observed;see［8］and［2］.Here，based on the results of Sections 4-5，where the importance of the leftest point of the R-polar has been made clear，we make some conjectures on when a Mach reflection or von Neumann reflection can appear.Clearly，if this leftest point is at the left side of the vertical axis，there is a regular reflection，and if this leftest point is at the right side of the vertical axis，there is an irregular reflection.Here，we consider the latter situation.

Fig.15　Mach reflection

Fig.16　Mach reflection-continuity

Conjecture 1After the incident angle is larger than αe，if the leftest point of the R-polar is above/on the upper branch of the I-polar，only a Mach reflection can appear;see Figs.15-16.

We take the view that the wave configuration should be as simple as possible.In this case，the R-polar must intersect the I-polar，and there is no need for a C-polar（representing a compression wave）to connect the R-polar and the I-polar，so a von Neumann reflection will not appear.

Conjecture 2After the incident angle is larger than αe，if the leftest point of the R-polar is below the upper branch of the I-polar and the R-polar also intersect the I-polar，then both a Mach reflection and a von Neumann reflection can appear;see Figs.17-18.

Fig.17　Mach reflection and von Neumann reflection

Fig.18　Mach reflection and von Neumann reflection-continuity

In this case，from the point of view of the smallest energy criterion，a von Neumann reflection is preferred.On the other hand，from the point of view of a simpler wave configuration，a Mach reflection is preferred.Here，we take the view that both are possible.

In the case that the leftest point of the R-polar is below the upper branch of the I-polar and the R-polar does not intersect the I-polar，as discussed in Subsection 2.4 and Section 5，a von Neumann reflection will appear.

References

［1］Ben-Dor Gabi.Shock Wave Reflection Phenonmena.New York:Springer-Verlag，1992

［2］Colella P，Henderson L F.The von Neumann paradox for the diffraction of weak shock waves.J Fluid Mech，1990，213:71-94

［3］Courant R，F(xiàn)riedrichs K O.Supersonic Flow and Shock Waves.New York:Interscience Publishers，1948

［4］Dewey J M，McMillin D J.Observation and analysis of the Mach reflection of weak uniform plane shock waves.Part1.Observations.J Fluid Mech，1985，152:49-66

［5］Henderson L F，Lozzi A.Experiments on transition of Mach reflection.J Fluid Mech，1975，68:139-155

［6］Henderson L F，Vasilev E I，Ben-Dor G，Elperin T.The wall-jetting effect in Mach reflection:theoretical consideration and numerical investigation.J Fluid Mech，2003，479:259-286

［7］Hornung H.Regular and Mach reflection of shock waves.Ann Rev Fluid Mech，1986，18:33-58

［8］von Neumann.Oblique Reflection of Shocks.Explos Res Rep 12，Navy Dept，Bureau of Ordnace，Washington，DC，USA，1943

［9］Sudani N，Sato M，Karasawa T，Noda J，Tate A，Watanabe M.Irregular effects on the trasition regular to Mach reflection of shock waves in wind tunnel flows.J Fluid Mech，2002，459:167-185

?January 18，2015;revised July 2，2015.Wu and Kong are supported by the NNSF of China（11271323，91330105）and the Zhejiang Provincial Natural Science Foundation of China（LZ13A010002）;Dai is supported by a GRF grant（CityU 11303015）from the Research Grants Council of Hong Kong SAR，China.

?Huihui DAI.