Optimal regular reflection of shock and blast waves

The regular reflection of an oblique steady shock in supersonic gas flow is considered. The static pressure extremum conditions after the point of reflection of the shock with fixed strength depending on oncoming flow Mach number are determined analytically. The obtained results are applied to solution of the mechanically equivalent problem of the reflection of a propagating shock from an inclined surface. Non-monotonic variation of the mechanical loads on the obstacle with respect to its inclination angle is shown; the obstacle slope angles that correspond to pressure minima downwards of the unsteady shock reflection point are determined analytically.


Introduction
The regular reflection of steady and unsteady shocks from a rigid surface or a flow symmetry plane is well-studied phenomenon. If the ratio  of the gas specific heats in the steady supersonic flow is known and fixed, then the reflected shock parameters depend solely on the oncoming flow Mach number М (Fig. 1a) and one additional parameter of the incident shock (for example, its strength 1 problem corresponds to variation in the wedge angle  in the problem of unsteady (quasisteady) reflection of propagating shock of the same strength. Thus, the non-monotonic dependence of the mechanical loads behind reflected unsteady shock from an inclined surface appears from the non-monotonic pressure variation downstream the point of steady reflection.

Governing equations for the strength of the reflected shock
The oncoming flow parameters (Mach number М , pressure p and the gas specific heat ratio,  ) and the shock slope angle 1  determined its strength [6][7][8]

Limiting values for the strength of the incident shock
Satisfactory solution of (1) exists, if the incident shock strength obeys the condition, ( Here   M J d is the shock strength that corresponds to the so-called detachment criterion [1,9] Since solutions derived for steady flows will be applied later to unsteady shock reflections, we consider now the entire range (2) of incident shock parameters.

The equivalent unsteady flow with regular reflection of the propagating shock
After the inversion of motion, the reflection of the moving shock of the same strength, that propagates in a stagnant gas media and reflects from a wedge having deflection angle Fig. 1b) is mechanically equivalent to the regular steady reflection shown in Fig. 1a [13]. Reflection point R (Fig. 1b)  The amplitude of the incident shock is: the amplitude of the reflected wave is equal to , and the overpressure after the reflection point is It is therefore apparent that the reflected wave amplitude The pressure ratio across the reflected wave, 2 K characterizes its amplitude relative to the incident shock: Both coefficients are minimal ( in the case of a sliding shock and the reflected shock disappears (degenerates to a weak disturbance, can be evaluated using the Crussard-Izmaylov formula [14]: with corresponding values of the factors of pressure (5) and amplification (6).

Conclusion.
Flow Mach numbers and parameters of the incident shocks that achieve minima in the static pressure downstream of the reflected shocks obey the cubic algebraic equation in "shock strength -oncoming flow Mach number" variables. The analogous equation defines the obstacle slope angles that yield a minimum pressure after the reflection of the propagating shock having any amplitude. Those optimal angles exist at all theoretically possible parameters of the incident shock. The optimal shock reflection differs both from the normal one and from limiting case of shock reflection transition. The geometrical optimization of shock/blast wave interaction with obstacles and constructions allows to diminish the impulse mechanical loads significantly. The obtained theoretical results can be applied, at least, for design of blast-resistant constructions, blast protection equipment, in aviation and rocket engineering, and in supersonic aerodynamics.