Non-self-similar characteristics of weak Mach reflection: the von Neumann paradox

2004 ◽  
Vol 35 (4) ◽  
pp. 275-286 ◽  
Author(s):  
Susumu Kobayashi ◽  
Takashi Adachi ◽  
Tateyuki Suzuki
Keyword(s):  
Mach Reflection ◽  
Von Neumann ◽  
Von Neumann Paradox ◽  
Self Similar

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

The persistence of regular reflection during strong shock diffraction over rigid ramps

2001 ◽  
Vol 431 ◽  
pp. 273-296 ◽  
Author(s):  
L. F. HENDERSON ◽  
K. TAKAYAMA ◽  
W. Y. CRUTCHFIELD ◽  
S. ITABASHI
Keyword(s):  
Stokes Equations ◽  
Mach Reflection ◽  
Strong Shock ◽  
Navier Stokes ◽  
Regular Reflection ◽  
Navier Stokes Equations ◽  
Von Neumann ◽  
Initial Appearance ◽  
Self Similar ◽  
Initial Shock

We report on calculations and experiments with strong shocks diffracting over rigid ramps in argon. The numerical results were obtained by integrating the conservation equations that included the Navier–Stokes equations. The results predict that if the ramp angle θ is less than the angle θe that corresponds to the detachment of a shock, θ < θe, then the onset of Mach reflection (MR) will be delayed by the initial appearance of a precursor regular reflection (PRR). The PRR is subsequently swept away by an overtaking corner signal (cs) that forces the eruption of the MR which then rapidly evolves into a self-similar state. An objective was to make an experimental test of the predictions. These were confirmed by twice photographing the diffracting shock as it travelled along the ramp. We could get a PRR with the first exposure and an MR with the second. According to the von Neumann perfect gas theory, a PRR does not exist when θ < θe. A viscous length scale xint is a measure of the position on the ramp where the dynamic transition PRR → MR takes place. It is significantly larger in the experiments than in the calculations. This is attributed to the fact that fluctuations from turbulence and surface roughness were not modelled in the calculations. It was found that xint → ∞ as θ → θe. Experiments were done to find out how xint depended on the initial shock tube pressure p0. The dependence was strong but could be greatly reduced by forming a Reynolds number based on xint. Finally by definition, regular reflection (RR) never interacts with a boundary layer, while PRR always interacts; so they are different phenomena.

The effects of thermal conductivity and viscosity of argon on shock waves diffracting over rigid ramps

1997 ◽  
Vol 331 ◽  
pp. 1-36 ◽  
Author(s):  
L. F. HENDERSON ◽  
W. Y. CRUTCHFIELD ◽  
R. J. VIRGONA
Keyword(s):  
Boundary Conditions ◽  
Numerical Scheme ◽  
Stokes Equations ◽  
Mach Reflection ◽  
Ideal Gas ◽  
Slip Boundary Conditions ◽  
Viscous Boundary Layer ◽  
Von Neumann ◽  
Slip Boundary ◽  
Self Similar

Experiments were done with strong shocks diffracting over steel ramps immersed in argon. Numerical simulations of the experiments were done by integrating the Navier–Stokes equations with a higher-order Godunov finite difference numerical scheme using isothermal non-slip boundary conditions. Adiabatic, slip boundary conditions were also studied to simulate cavity-type diffractions. Some results from an Euler numerical scheme for an ideal gas are presented for comparison. When the ramp angle θ is small enough to cause Mach reflection MR, it is found that real gas effects delay its appearance and that the trajectory of its shock triple point is initially curved; it eventually becomes straight as the MR evolves into a self-similar system. The diffraction is a regular reflection RR in the delayed state, and this is subsequently swept away by a corner signal overtaking the RR and forcing the eruption of the Mach shock. The dynamic transition occurs at, or close to, the ideal gas detachment criterion θe. The passage of the corner signal is marked by large oscillations in the thickness of the viscous boundary layer. With increasing θ, the delay in the onset of MR is increased as the dynamic process slows. Once self-similarity is established the von Neumann criterion is supported. While the evidence for the von Neumann criterion is strong, it is not conclusive because of the numerical expense. The delayed transition causes some experimental data for the trajectory to be subject to a simple parallax error. The adiabatic, slip boundary condition for self-similar flow also supports the von Neumann criterion while θ < θe, but the trajectory angle discontinuously changes to zero at θe, so that θe is supported by the numerics, contrary to experiments.

On the von Neumann paradox of weak Mach reflection

1989 ◽  
Vol 4 (5-6) ◽  
pp. 333-345 ◽  
Author(s):  
Akira Sakurai ◽  
L F Henderson ◽  
Kazuyoshi Takayama ◽  
Zbigniew Walenta ◽  
Philip Colella
Keyword(s):  
Mach Reflection ◽  
Von Neumann ◽  
Von Neumann Paradox

Corrigendum to “On the von Neumann paradox of weak Mach reflection” [Fluid Dyn. Res. 4 (1989) 333]

2001 ◽  
Vol 29 (2) ◽  
pp. 135-135
Author(s):  
Akira Sakurai ◽  
L F Henderson ◽  
Kazuyoshi Takayama ◽  
Zbighiew Walenta ◽  
Philip Colella
Keyword(s):  
Mach Reflection ◽  
Von Neumann ◽  
Von Neumann Paradox

Reconsideration of the So-Called von Neumann Paradox in the Reflection of a Shock Wave over a Wedge

2007 ◽  
Vol 566 ◽  
pp. 1-8
Author(s):  
Eugene I. Vasilev ◽  
Tov Elperin ◽  
Gabi Ben-Dor
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Triple Point ◽  
Mach Reflection ◽  
Weak Shock ◽  
Weak Shock Wave ◽  
Plane Shock ◽  
Von Neumann ◽  
Von Neumann Paradox ◽  
Guderley Reflection

Numerous experimental investigations on the reflection of plane shock waves over straight wedges indicated that there is a domain, frequently referred to as the weak shock wave domain, inside which the resulted wave configurations resemble the wave configuration of a Mach reflection although the classical three-shock theory does not provide an analytical solution. This paradox is known in the literature as the von Neumann paradox. While numerically investigating this paradox Colella & Henderson [1] suggested that the observed reflections were not Mach reflections but another reflection, in which the reflected wave at the triple point was not a shock wave but a compression wave. They termed them it von Neumann reflection. Consequently, based on their study there was no paradox since the three-shock theory never aimed at predicting this wave configuration. Vasilev & Kraiko [2] who numerically investigated the same phenomenon a decade later concluded that the wave configuration, inside the questionable domain, includes in addition to the three shock waves a very tiny Prandtl-Meyer expansion fan centered at the triple point. This wave configuration, which was first predicted by Guderley [3], was recently observed experimentally by Skews & Ashworth [4] who named it Guderley reflection. The entire phenomenon was re-investigated by us analytically. It has been found that there are in fact three different reflection configurations inside the weak reflection domain: • A von Neumann reflection – vNR, • A yet not named reflection – ?R, • A Guderley reflection – GR. The transition boundaries between MR, vNR, ?R and GR and their domains have been determined analytically. The reported study presents for the first time a full solution of the weak shock wave domain, which has been puzzling the scientific community for a few decades. Although the present study has been conducted in a perfect gas, it is believed that the reported various wave configurations, namely, vNR, ?R and GR, exist also in the reflection of shock waves in condensed matter.

Transition from regular to Mach reflection of shock waves Part 2. The steady-flow criterion

1982 ◽  
Vol 123 ◽  
pp. 155-164 ◽  
Author(s):  
H. G. Hornung ◽  
M. L. Robinson
Keyword(s):  
Shock Waves ◽  
Steady Flow ◽  
Mach Reflection ◽  
Strong Shock ◽  
Hysteresis Effect ◽  
Neumann Condition ◽  
Flow Transition ◽  
Von Neumann ◽  
Mach Numbers ◽  
Flow Criterion

It is shown experimentally that, in steady flow, transition to Mach reflection occurs at the von Neumann condition in the strong shock range (Mach numbers from 2.8 to 5). This criterion applies with both increasing and decreasing shock angle, so that the hysteresis effect predicted by Hornung, Oertel & Sandeman (1979) could not be observed. However, evidence of the effect is shown to be displayed in an unsteady experiment of Henderson & Lozzi (1979).

Analytical reconsideration of the so-called von Neumann paradox in the reflection of a shock wave over a wedge

Shock Waves ◽  
2009 ◽  
pp. 1353-1358
Author(s):  
Eugene I. Vasilev ◽  
Tov Elperin ◽  
Gabi Ben-Dor
Keyword(s):  
Shock Wave ◽  
Von Neumann ◽  
Von Neumann Paradox
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