Mean free path effects in the shock-implosion problem

Goldsworthy, Mark; Pullin, D. I.

doi:10.1063/1.3075952

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…This also applies to the hydrodynamic case. 27 Second, for the hydrodynamic and magnetic Reynolds numbers we use density, velocity, and length scales as those immediately downstream of the shock, ρ and u, and the shock radius R s , respectively. Both Reynolds numbers are proportional to the product of these three scales.…”

Section: B Comparison Of Numerical Solution With Shock Dynamicsmentioning

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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Section: B Comparison Of Numerical Solution With Shock Dynamicsmentioning

confidence: 99%

Converging cylindrical shocks in ideal magnetohydrodynamics

et al. 2014

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“…The equation of state for non-ideal gas was considered as given by Landau and Lifshitz [35]. The general characteristic rule for the propagation velocity (23) - (27), respectively. The change in entropy  s across the imploding shock front in the non-ideal gas is given by the Eq.…”

Section: Resultsmentioning

confidence: 99%

“…The flow field behind the converging shock waves was investigated in van der Waals gas by Wu and Roberts [24,25] and Evans [26]. Using the direct simulation Monte Carlo (DSMC) method, Goldsworthy and Pullin [27] investigated the effects of finite Knudsen number in the problem of a cylindrically imploding shock wave and found that the solution of the dissipative flow field displays a scaling consistent with the Guderlay similarity exponent. Kjellander, Tillmark and Apazidis [28] studied strong cylindrical and spherical shock implosion in a monatomic real gas and shown that ionization has a major effect on temperature and density behind the converging shock as well as on the shock acceleration.…”

Section: Introductionmentioning

confidence: 99%

Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects

Anand

2013

Wave Motion

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Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects R K Anand Wave Motion (2013), http://dx.Abstract In this paper, the generalized analytical solution for one dimensional adiabatic flow behind the strong imploding shock waves propagating in a non-ideal gas is obtained by using Whitham's geometrical shock dynamics theory. Landau and Lifshitz's equation of state for non-ideal gas and Anand's generalized shock jump relations are taken into consideration to explore the effects due to an increase in (i) the propagation distance from the centre of convergence, (ii) the non-idealness parameter and, (iii) the adiabatic index, on the shock velocity, pressure, density, particle velocity, sound speed, adiabatic compressibility and the change in entropy across the shock front. The findings provided a clear picture of whether and how the non-idealness parameter and the adiabatic index affect the flow field behind the strong imploding shock front.Keywords Imploding shock waves, Non-ideal gas, Shock jump relations, Non-idealness parameter, Adiabatic index PACS numbers: 47.40-X, 45.55.kf Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects R K Anand Wave Motion (2013), http://dx.Shock wave phenomena also arise in astrophysics, hypersonic aerodynamics and hypervelocity impact. An understanding of the properties of the shock waves both in the near-field and the far-field is useful with regard to the characteristics such as shock strength, shock overpressure, shock speed, and impulse.The first study on converging shock waves was performed by Guderley [2], who presented his well-known self-similarity solution of strong converging cylindrical and spherical shock waves close to focus. The problem was also studied independently by Chester [3], Chisnell [4], and Whitham [5] with approximate methods, specifically geometrical shock dynamics. In their solutions, the exponent in the expression for Mach number as a function of shock radius for the spherical case is exactly twice that for the cylindrical case. This approximate result differs from the exact solution by less than one percent. The geometrical shock dynamics approach is both simple and intuitive while providing fairly accurate results. Sakurai [6,7], Sedov [8], Zel'dovich and Raizer [9], Lazarus and Richtmeyer [10], Van Dyke and Guttmann [11] and Hafner [12]presented high-accuracy results adopting alternative approaches for the investigation of the implosion problem under consideration. A numerical solution for a converging cylindrical shock wave was presented by Payne [13]. The problem of contracting spherical or cylindrical shock front propagation into a uniform gas at rest was studied by Stanyukovich [14]. The effects of overtaking disturbances on the motion of converging shock waves were studied by Yousaf [15]. The problems of implosion of a spherical shock wave in a gas and collapse of a spherical bubble in a liquid are discussed by Zel'dovich and Raizer [9] by using a self simila...

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Mean free path effects in the shock-implosion problem

Cited by 2 publications

References 9 publications

Converging cylindrical shocks in ideal magnetohydrodynamics

Converging cylindrical shocks in ideal magnetohydrodynamics

Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects

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