Converging cylindrical shocks in ideal magnetohydrodynamics

Handle URI:
http://hdl.handle.net/10754/346726
Title:
Converging cylindrical shocks in ideal magnetohydrodynamics
Authors:
Pullin, D. I.; Mostert, W.; Wheatley, V.; Samtaney, Ravi ( 0000-0002-4702-6473 )
Abstract:
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 = √μ0/p0 I/(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.
KAUST Department:
Mechanical Engineering Program; Physical Sciences and Engineering (PSE) Division
Citation:
Converging cylindrical shocks in ideal magnetohydrodynamics 2014, 26 (9):097103 Physics of Fluids
Publisher:
American Institute of Physics
Journal:
Physics of Fluids
Issue Date:
Sep-2014
DOI:
10.1063/1.4894743
Type:
Article
ISSN:
1070-6631; 1089-7666
Additional Links:
http://scitation.aip.org/content/aip/journal/pof2/26/9/10.1063/1.4894743
Appears in Collections:
Articles; Physical Sciences and Engineering (PSE) Division; Mechanical Engineering Program

Full metadata record

DC FieldValue Language
dc.contributor.authorPullin, D. I.en
dc.contributor.authorMostert, W.en
dc.contributor.authorWheatley, V.en
dc.contributor.authorSamtaney, Ravien
dc.date.accessioned2015-03-17T06:06:07Zen
dc.date.available2015-03-17T06:06:07Zen
dc.date.issued2014-09en
dc.identifier.citationConverging cylindrical shocks in ideal magnetohydrodynamics 2014, 26 (9):097103 Physics of Fluidsen
dc.identifier.issn1070-6631en
dc.identifier.issn1089-7666en
dc.identifier.doi10.1063/1.4894743en
dc.identifier.urihttp://hdl.handle.net/10754/346726en
dc.description.abstractWe 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 = √μ0/p0 I/(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.en
dc.publisherAmerican Institute of Physicsen
dc.relation.urlhttp://scitation.aip.org/content/aip/journal/pof2/26/9/10.1063/1.4894743en
dc.rightsArchived with thanks to Physics of Fluidsen
dc.titleConverging cylindrical shocks in ideal magnetohydrodynamicsen
dc.typeArticleen
dc.contributor.departmentMechanical Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.identifier.journalPhysics of Fluidsen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionGraduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USAen
dc.contributor.institutionSchool of Mechanical and Mining Engineering, University of Queensland, Queensland 4072, Australiaen
dc.contributor.institutionSchool of Mechanical and Mining Engineering, University of Queensland, Queensland 4072, Australiaen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorSamtaney, Ravien
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.