Mach Stems

I am currently running a set of double Mach reflection runs at varying angles. No cooling.

Initial Thoughts (09/2010)

If the problem is symmetric—both bows facing upstream, with sizes and leading edges the same—then it seems to reduce to the so-called ​2D double Mach reflection problem.

This problem has a initially planar shock at a 60 degree angle incident to a reflecting boundary on the bottom. The point at which it initially intersects this bottom boundary is kept fixed (for some reason), and the top boundary is updated throughout the simulation with the analytic shock speed (cos(60) v_s).

As the simulation progresses, the structure develops multiple shocks, looking like this:

If in fact we can reduce the problem to the above, then there doubtless is literature on it.

Key Questions

1. What parameters determine the formation of the stems?
   • Presumably a function of impact angle. Mach number? Not much else involved.
2. In particular, how does a head-on collision compare to a side-by-side case?
   • Note that the velocity vectors "upstream" are different in these two cases!

Theory

Critical Angle

The theory of Mach stems shows up in Landau&Liftschitz (1959), p. 412. Below I base discussion on ​De Rosa et al., 1992, Phys.Rev.A, 45, 6130 (paper also attached to this page). Essentially, there exists a critical angle \phi\ge\phi_{crit}, depending on Mach number and \gamma, above which simple reflection does not happen. Instead "Mach reflection" occurs, i.e. the formation of the Mach stem.

From what I'm reading, they discuss a way to express the characteristics of a flow through an oblique shock, given an incoming angle \phi and an outgoing angle \theta, in a velocity coordinate system. The resulting geometric interpretation returns a solution which travels along a ​half-lemniscate (?).

A shock incident on a reflecting boundary may be considered a combination of incident shock I and reflecting shock R, with an intermediate region between. One may then use the above method to get the solution in the intermediate region, and from this the solution after the R shock. The above reference defines the critical angle \phi_{crit} in the strong-shock limit (M\rightarrow\infty) for the I shock (so-called "exact solution"). This limit reduces the lemniscates to circles, with the I circle straddling the vx-axis and the R circle intersecting at none, one, or two points,

The authors state that the transition to Mach reflection occurs when the R polar is tangential to the vx-axis. In the high-Mach-number limit for both R and I shocks (so-called "approximate solution"), they derive an expression which is a function of \gamma only:

\phi_{crit}=\arctan\left(\frac1{\gamma-1} \sqrt{\frac{\sqrt{\gamma^2-1}}{\gamma-\sqrt{\gamma^2-1}}} \times\left[1-\sqrt{\gamma^2-\gamma\sqrt{\gamma^2-1}}\right]\right)$

Tabulated values of critical angle

Here are some tabulated values of the critical angle (Mach reflection occurs for greater angles):

\gamma\quad	\phi_{crit}\quad
5/3	37.4o
1.6	38.2o
1.5	39.8o
1.4	41.7o
1.3	44.1o
1.2	47.7o
1.1	53.5o
1.0001	83.2o

Note that using the 'exact' expression leads to a value roughly 1-2^o less.

Bow shock shape

​Ostriker Lee 2001 give a "ballistic bow shock shape",

z(r) = -\left[ \frac13\left(\frac{r}{r_j}\right)^3-\frac{r}{r_j}+\frac23\right]\frac{v_s}{\beta c_s} r_j$

where z(r) is the axial distance upstream from the terminal bow shock as a function of radial distance r, r_j is the jet radius, v_s the bow shock velocity ($v_s=\frac{v_j}{1+\chi^{-1/2}})$, beta a factor of roughly 4 based on simulations, and c_s the preshock sound speed.

For typical parameters (todo:list), the incident angle is 90^o at r=1 and decreases rapidly, achieving 45^o at r~1.107, 30^o at r~1.18, 15^o at r~1.36, and 5^o at r~1.89.

(todo:include angle figure)

Thus, if for \gamma=5/3,\quad \phi_{crit}=37.4^o, a critical symmetric separation of bow shocks occurs at r\le1.138r_j.

I will run a couple of quick simulations to confirm.