Numerical shock propagation using geometrical shock dynamics

Henshaw, William D.; Smyth, Noel F.; Schwendeman, Donald W.

doi:10.1017/s0022112086001568

Cited by 63 publications

(71 citation statements)

References 8 publications

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“…From our knowledge, this is the first time that a comprehensive solution of the GSD Riemann problem is given, together with a link to the p-system. Nevertheless, one can mention the works of Henshaw, Smyth and Schwendeman [15] and Schwendeman [20] where the Whitham GSD equations are rewritten in conservative form. Schwendeman [20] also mentions the fact that the Riemann problem admits simple solutions, but does not give full details.…”

Section: Solutions Of the Riemann Problem In Dimensionmentioning

confidence: 99%

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A fast-marching like algorithm for geometrical shock dynamics

Noumir

Guilcher

Lardjane

et al. 2015

Journal of Computational Physics

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We develop a new algorithm for the computation of the geometrical shock dynamics model (GSD). The method relies on the fast-marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid. The proposed algorithm is based on a second order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure. Reference solutions are built for a smooth radial configuration and for the 2D Riemann problem. The link between the GSD model and p-systems is given. Numerical experiments demonstrate the accuracy and the ability of the scheme to handle singularities.

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Section: Solutions Of the Riemann Problem In Dimensionmentioning

confidence: 99%

“…(i) Lagrangian, or front-tracking, methods have first been experimented [15,18]. In such an approach, the shock front is explicitly discretized by markers evolved in time and regularly resampled.…”

Section: Introductionmentioning

confidence: 99%

A fast-marching like algorithm for geometrical shock dynamics

Noumir

Guilcher

Lardjane

et al. 2015

Journal of Computational Physics

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“…If the principal interest lies in the shock physics and flow conditions immediately behind the shock, then this reduction of dimensionality is appealing. Geometrical shock dynamics has proved effective for a variety of shock-dynamics problems in multi-dimensional flows including shock focusing in gas dynamics, 9 propagation of shock waves along walls and in channels, 10 shock-wave stability, 7 and shock propagation in non-uniform media. 11 When applied to the symmetrically converging shock of the Guderley type, GSD provides an extremely accurate approximation to power-law exponents and in fact can describe an almost universal shock collapse process from an infinitesimal wave at infinity to a strong-shock state near the point of convergence.…”

Section: Introductionmentioning

confidence: 99%

Converging cylindrical shocks in ideal magnetohydrodynamics

et al. 2014

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We consider a cylindrically symmetrical shock converging onto an axis within the framework of ideal, compressible-gas non-dissipative magnetohydrodynamics (MHD). In cylindrical polar co-ordinates we restrict attention to either constant axial magnetic field or to the azimuthal but singular magnetic field produced by a line current on the axis. Under the constraint of zero normal magnetic field and zero tangential fluid speed at the shock, a set of restricted shock-jump conditions are obtained as functions of the shock Mach number, defined as the ratio of the local shock speed to the unique magnetohydrodynamic wave speed ahead of the shock, and also of a parameter measuring the local strength of the magnetic field. For the line current case, two approaches are explored and the results compared in detail. The first is geometrical shock-dynamics where the restricted shock-jump conditions are applied directly to the equation on the characteristic entering the shock from behind. This gives an ordinary-differential equation for the shock Mach number as a function of radius which is integrated numerically to provide profiles of the shock implosion. Also, analytic, asymptotic results are obtained for the shock trajectory at small radius. The second approach is direct numerical solution of the radially symmetric MHD equations using a shock-capturing method. For the axial magnetic field case the shock implosion is of the Guderley power-law type with exponent that is not affected by the presence of a finite magnetic field. For the axial current case, however, the presence of a tangential magnetic field ahead of the shock with strength inversely proportional to radius introduces a length scale \documentclass[12pt]{minimal}\begin{document}$R=\sqrt{\mu _0/p_0}\,I/(2\,\pi )$\end{document}R=μ0/p0I/(2π) where I is the current, μ0 is the permeability, and p0 is the pressure ahead of the shock. For shocks initiated at r ≫ R, shock convergence is first accompanied by shock strengthening as for the strictly gas-dynamic implosion. The diverging magnetic field then slows the shock Mach number growth producing a maximum followed by monotonic reduction towards magnetosonic conditions, even as the shock accelerates toward the axis. A parameter space of initial shock Mach number at a given radius is explored and the implications of the present results for inertial confinement fusion are discussed.

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“…Two numerical approaches have been used for a wide range of problems: the method of characteristics 12,13 and front-tracking methods. 8 Characteristics methods are typically more cumbersome than finite-difference methods and difficult to extend to three dimensional problems, and thus their numerical application has been fairly limited. In front-tracking methods points along the shock front are advanced along rays normal to the front according to the shock Mach number.…”

Section: Methodsmentioning

confidence: 99%

“…Henshaw et al, 8 using a front-tracking method to solve the equations of shock dynamics, analyzed shock focusing and compared calculated results with the experiments of Sturtevant and Kulkarny. 1 They showed that shock dynamics qualitatively reproduces the weak-, moderate-and strongshock behavior illustrated in Fig.…”

Section: Introductionmentioning

confidence: 99%

Shock wave focusing using geometrical shock dynamics

Cates

Sturtevant

1997

Physics of Fluids

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A finite-difference numerical method for geometrical shock dynamics has been developed based on the analogy between the nonlinear ray equations and the supersonic potential equation. The method has proven to be an efficient and inexpensive tool for approximately analyzing the focusing of weak shock waves, where complex nonlinear wave interactions occur over a large range of physical scales. The numerical results exhibit the qualitative behavior of strong, moderate, and weak shock focusing observed experimentally. The physical mechanisms that are influenced by aperture angle and shock strength are properly represented by geometrical shock dynamics. Comparison with experimental measurements of the location at which maximum shock pressure occurs shows good agreement, but the maximum pressure at focus is overestimated by about 60%. This error, though large, is acceptable when the speed and low cost of the method is taken into consideration. The error is primarily due to the under prediction of disturbance speed on weak shock fronts. Adequate resolution of the focal region proves to be particularly important to properly judge the validity of shock dynamics theory, under-resolution leading to overly optimistic conclusions. © 1997 American Institute of Physics.

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Numerical shock propagation using geometrical shock dynamics

Cited by 63 publications

References 8 publications

A fast-marching like algorithm for geometrical shock dynamics

A fast-marching like algorithm for geometrical shock dynamics

Converging cylindrical shocks in ideal magnetohydrodynamics

Shock wave focusing using geometrical shock dynamics

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