The stability of imploding detonations in the geometrical shock dynamics (CCW) model

1992 ◽  
Vol 4 (4) ◽  
pp. 835-844 ◽  
Author(s):  
C. Richard DeVore ◽  
Elaine S. Oran
Keyword(s):  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
The Stability

Geometrical shock dynamics for magnetohydrodynamic fast shocks

2016 ◽  
Vol 811 ◽  
Author(s):  
W. Mostert ◽  
D. I. Pullin ◽  
R. Samtaney ◽  
V. Wheatley
Keyword(s):  
Mach Number ◽  
Two Dimensions ◽  
Mhd Equations ◽  
Relative Pressure ◽  
Gas Dynamic ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
Ideal Magnetohydrodynamic ◽  
The Stability ◽  
Spatially Varying

We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$, where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.

A numerical scheme for shock propagation in three dimensions

1988 ◽  
Vol 416 (1850) ◽  
pp. 179-198 ◽  
Keyword(s):  
Numerical Scheme ◽  
Three Dimensional ◽  
Experimental Results ◽  
Three Dimensions ◽  
Shock Propagation ◽  
Two Dimensional ◽  
Shock Surface ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
Three Space

A numerical scheme for shock propagation in three space dimensions is presented. The motion of the leading shock surface is calculated by using Whitham’s theory of geometrical shock dynamics. The numerical scheme is used to examine the focusing of initially curved shock surfaces and the diffraction of shocks in a pipe with a 90° bend. Numerical and experimental results for the corresponding two-dimensional or axi-symmetrical cases are used to compare with the new and more complicated three-dimensional results.

scholarly journals Numerical shock propagation using geometrical shock dynamics

1986 ◽  
Vol 171 (-1) ◽  
pp. 519 ◽  
Author(s):  
W. D. Henshaw ◽  
N. F. Smyth ◽  
D. W. Schwendeman
Keyword(s):  
Shock Propagation ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics

scholarly journals Oblique shock reflection from an axis of symmetry: shock dynamics and relation to the Guderley singularity

2001 ◽  
Vol 438 ◽  
pp. 231-245 ◽  
Author(s):  
H. G. HORNUNG ◽  
D. W. SCHWENDEMAN
Keyword(s):  
Oblique Shock ◽  
Shock Reflection ◽  
Mathematical Form ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
Self Similar ◽  
Axis Of Symmetry ◽  
Weak Shocks ◽  
Oblique Shock Reflection ◽  
Good Agreement

Oblique shock reflection from an axis of symmetry is studied using Whitham's theory of geometrical shock dynamics, and the results are compared with previous numerical simulations of the phenomenon by Hornung (2000). The shock shapes (for strong and weak shocks), and the location of the shock-shock (for strong shocks), are in good agreement with the numerical results, though the detail of the shock reflection structure is, of course, not resolved by shock dynamics. A guess at a mathematical form of the shock shape based on an analogy with the Guderley singularity in cylindrical shock implosion, in the form of a generalized hyperbola, fits the shock shape very well. The smooth variation of the exponent in this equation with initial shock angle from the Guderley value at zero to 0.5 at 90° supports the analogy. Finally, steady-flow shock reflection from a symmetry axis is related to the self-similar flow.

On converging shock waves of spherical and polyhedral form

2002 ◽  
Vol 454 ◽  
pp. 365-386 ◽  
Author(s):  
DONALD W. SCHWENDEMAN
Keyword(s):  
Shock Waves ◽  
Numerical Study ◽  
Stability Result ◽  
Shock Propagation ◽  
Initial Shape ◽  
Regular Polyhedra ◽  
Converging Shock Waves ◽  
Geometrical Shock Dynamics ◽  
The Stability ◽  
Spherical Shock

The behaviour of converging spherical shock waves is considered using Whitham's theory of geometrical shock dynamics. An analysis of converging shocks whose initial shape takes the form of regular polyhedra is presented. The analysis of this problem is motivated by the earlier work on converging cylindrical shocks discussed in Schwendeman & Whitham (1987). In that paper, exact solutions were reported for converging polygonal shocks in which the initial shape re-forms repeatedly as the shock contracts. For the polyhedral case, the analysis is performed both analytically and numerically for an equivalent problem involving shock propagation in a converging channel with triangular cross-section. It is found that a repeating sequence of shock surfaces composed of nearly planar pieces develops, although the initial planar surface does not re-form, and that the increase in strength of the shock at each iterate in the sequence follows the same behaviour as for a converging spherical shock independent of the convergence angle of the channel. In this sense, the shocks are stable and the result is analogous to that found in the two-dimensional case. A numerical study of converging spherical shocks subject to smooth initial perturbations in strength shows a strong tendency to form surfaces composed of nearly planar pieces suggesting that the stability result is fairly general.

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