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Converging cylindrical shocks in ideal magnetohydrodynamics

Pullin, D. I. and Mostert, W. and Wheatley, V. and Samtaney, R. (2014) Converging cylindrical shocks in ideal magnetohydrodynamics. Physics of Fluids, 26 (9). Art. No. 097103. ISSN 1070-6631.

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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 R = √μ_0/p_0I/(2π) where I is the current, μ_0 is the permeability, and p_0 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Mostert, W.0000-0001-6251-4136
Samtaney, R.0000-0002-4702-6473
Additional Information:© 2014 AIP Publishing LLC. Received 9 May 2014; accepted 22 August 2014; published online 16 September 2014. This research was supported under Australian Research Council’s Discovery Projects funding scheme (Project No. DP120102378). Additionally, V. Wheatley is the recipient of an Australian Research Council Discovery Early Career Researcher Award (Project No. DE120102942). R. Samtaney was supported by baseline research funds at KAUST.
Funding AgencyGrant Number
Australian Research Council Discovery Projects funding schemeDP120102378
Australian Research Council Discovery Early Career Researcher AwardDE120102942
Issue or Number:9
Record Number:CaltechAUTHORS:20140929-093640759
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Official Citation: Converging cylindrical shocks in ideal magnetohydrodynamics D. I. Pullin, W. Mostert, V. Wheatley and R. Samtaney Phys. Fluids 26, 097103 (2014);
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:50089
Deposited By: Ruth Sustaita
Deposited On:29 Sep 2014 17:45
Last Modified:09 Mar 2020 13:19

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