Mach stem papers

There are only 2 relevant papers on ADS with "Mach stem" in the title:

  1. My HEDLA conference paper: ​http://adsabs.harvard.edu/abs/2014arXiv1412.6495H
  1. Kris' hysteresis paper: ​http://adsabs.harvard.edu/abs/2013HEDP....9..251Y

More results are found by searching for "Mach reflection":

  1. Kemm 2014, interesting paper on double Mach stem problem: ​http://adsabs.harvard.edu/abs/2014arXiv1404.6510K
  1. Ivanov 2002, nice reference on regular vs. Mach reflection, experimental and numerical: ​http://adsabs.harvard.edu/abs/2002ESASP.487..341I

Other references refer to optics, water waves, detonation waves, etc. I don't think these are very applicable to our scenario. Papers 1-4 might have enough references within for me to use. Paper 4 has led me to some purely fluid mechanics papers:

  1. Chpoun 1995, wind tunnel experiments regular vs. Mach reflection, ​http://adsabs.harvard.edu/abs/1995JFM...301...19C
  1. Hornung 1982, more on transition from regular to Mach reflection: ​http://adsabs.harvard.edu/abs/1982JFM...123..155H
  1. Ivanov 1995, more on hysteresis effects: ​http://adsabs.harvard.edu/abs/1995PhFl....7..685I

There are a few books that give good descriptions of shock reflection phenomena which I have used as references in my previous paper as well:

  1. Ben-Dor: Shock Wave Reflection Phenomena
  1. Courant and Friedrichs: Supersonic Flows and Shock Waves in Applied Mathematical Sciences
  1. Landau and Lifshitz: Fluid Mechanics

There is also a derivation of the critical angle in this paper:

  1. de Rosa 1992: ​http://adsabs.harvard.edu/abs/1992PhRvA..45.6130D

Ben-Dor is definitely seems to be a key player in shock-wave reflection. He has a lengthy review paper just on shock wave reflection in different regimes:

  1. Ben-Dor 1988: ​http://adsabs.harvard.edu/abs/1988PrAeS..25..329B


I've skimmed through these papers, but I will be reading them more closely over the next few days as I start to write my paper. Some key things that I have learned:

Historically, there have been two minimum critical angles for Mach stem formation based on two different criterion. The one that I have been using is known as the "detachment" criterion (alpha_d), and it is commonly cited in text books. The other criterion is the "von Neumann" or "mechanical equilibrium" criterion (alpha_n), and it is a pressure argument which leads to an angle less than the detachment criterion. Many papers talk about a dual-solution region where the angle is alpha_n < alpha < alpha_d. In this region, both regular reflection and Mach reflection are theoretically possible. The figure below illustrates this point:

Ben-Dor's review paper suggests that for steady flows, the mechanical equilibrium criterion is the correct one to use, and that regular reflection only occurs above alpha_n in what he calls psuedo-steady flows. One complication that I see is that alpha_n is much more dependent on M than alpha_d. Thus, strong shock approximations might not be appropriate, and I would have to use a critical angle formula that is also dependent on M. I have yet to find a good formula for that like I did for alpha_d.

Ben-Dor also has an region showing no reflection in his figures, but there doesn't seem to be a good explanation of the transition criterion or a useful formula anywhere.

Experimental results vary. Some agree with alpha_n, and some agree with alpha_d, while others show hysteresis effects and support the dual-solution region argument.

In conclusion, I might be quoting a minimum critical angle of ~ 40 deg, when in actuality it could be more like 30 deg. The issue could be avoided for Mach numbers ~ 2.2, since alpha_n = alpha_d at this M. However, then I would have to use the exact formula for alpha_d that contains M since a strong shock approximation would no longer be valid for M=2.2.

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