ORCID Profile
0000-0001-6251-4136
Current Organisation
University of Oxford
Does something not look right? The information on this page has been harvested from data sources that may not be up to date. We continue to work with information providers to improve coverage and quality. To report an issue, use the Feedback Form.
Publisher: Elsevier BV
Date: 09-2021
Publisher: AIP Publishing
Date: 09-2014
DOI: 10.1063/1.4894743
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 \\documentclass[12pt]{minimal}\\begin{document}$R=\\sqrt{\\mu _0 _0}\\,I/(2\\,\\pi )$\\end{document}R=μ0 0I/(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 erging 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.
Publisher: Cambridge University Press (CUP)
Date: 16-03-2016
DOI: 10.1017/JFM.2016.138
Abstract: We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius $r$ , as $r\\rightarrow 0$ , of the shock Mach number $M(r)$ and pressure behind the shock $p(r)$ as a function of the magnetic field power-law exponent ${\\it\\mu}\\geqslant 0$ , where ${\\it\\mu}=0$ gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both $M(r)$ and the time evolution on the shock, as a function of $r$ , constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for $M(r)$ and $p(r)$ are obtained over a range of ${\\it\\mu}$ for general ${\\it\\gamma}$ , and for both small and large $r$ . In addition, numerical solutions of the GSD equations are performed over a large range of $r$ , for selected parameters using ${\\it\\gamma}=5/3$ . The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of ${\\it\\mu}$ . For $0\\leqslant {\\it\\mu} /13$ , $M$ and $p$ both approach unity at shock impact $r=0$ owing to the dominance of the strong magnetic field over the lifying effects of geometrical convergence. When ${\\it\\mu}\\geqslant 0.816$ (for ${\\it\\gamma}=5/3$ ), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular $M(r)$ and $p(r)$ , $r\\rightarrow 0$ . For $4/13 {\\it\\mu}\\leqslant 0.816$ three distinct regions of $M(r)$ variation are identified. For each of these $p(r)$ is singular at the axis.
Publisher: American Physical Society (APS)
Date: 23-07-2018
Publisher: Cambridge University Press (CUP)
Date: 08-05-2018
DOI: 10.1017/JFM.2018.263
Abstract: We present an analysis that predicts the time to development of a singularity in the shape profile of a spatially periodic perturbed, planar shock wave for ideal gas dynamics. Beginning with a formulation in complex coordinates of Whitham’s approximate model geometrical shock dynamics (GSD), we apply a spectral treatment to derive the asymptotic form for the leading-order behaviour of the shock Fourier coefficients for large mode numbers and time. This is shown to determine a critical time at which the coefficients begin to decay, with respect to mode number, at an algebraic rate with an exponent of $-5/2$ , indicating loss of analyticity and the formation of a singularity in the shock geometry. The critical time is found to be inversely proportional to a representative measure of perturbation litude $\\unicode[STIX]{x1D716}$ with an explicit analytic form for the constant of proportionality in terms of gas and shock parameters. To leading order, the time to singularity formation is dependent only on the first Fourier mode. Comparison with results of numerical solutions to the full GSD equations shows that the predicted critical time somewhat underestimates the time for shock–shock (triple-point) formation, where the latter is obtained by post-processing the numerical GSD results using an edge-detection algorithm. Aspects of the analysis suggest that the appearance of loss of analyticity in the shock surface may be a precursor to the first appearance of shock–shocks, which may account for part of the discrepancy. The frequency of oscillation of the shock perturbation is accurately predicted. In addition, the analysis is extended to the evolution of a perturbed planar, fast magnetohydrodynamic shock for the case when the external magnetic field is aligned parallel to the unperturbed shock. It is found that, for a strong shock, the presence of the magnetic field produces only a higher-order correction to the GSD equations with the result that the time to loss of analyticity is the same as for the gas-dynamic flow. Limitations and improvements for the analysis are discussed, as are comparisons with the analogous appearance of singularity formation in vortex-sheet evolution in an incompressible inviscid fluid.
Publisher: American Physical Society (APS)
Date: 26-01-2017
Publisher: AIP Publishing
Date: 10-2015
DOI: 10.1063/1.4932110
Abstract: The effects of seed magnetic fields on the Richtmyer-Meshkov instability driven by converging cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Two different seed field configurations at various strengths are applied over a cylindrical or spherical density interface which has a single-dominant-mode perturbation. The shocks that excite the instability are generated with appropriate Riemann problems in a numerical formulation and the effect of the seed field on the growth rate and symmetry of the perturbations on the density interface is examined. We find reduced perturbation growth for both field configurations and all tested strengths. The extent of growth suppression increases with seed field strength but varies with the angle of the field to interface. The seed field configuration does not significantly affect extent of suppression of the instability, allowing it to be chosen to minimize its effect on implosion distortion. However, stronger seed fields are required in three dimensions to suppress the instability effectively.
Publisher: AIP Publishing
Date: 12-2014
DOI: 10.1063/1.4902432
Abstract: The effects of various seed magnetic fields on the dynamics of cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Here, we present a fundamental investigation of this problem utilizing cylindrical and spherical Riemann problems under three seed field configurations to initialize the implosions. The resulting flows are simulated numerically, revealing rich flow structures, including multiple families of magnetohydrodynamic shocks and rarefactions that interact non-linearly. We fully characterize these flow structures, examine their axi- and spherisymmetry-breaking behaviour, and provide data on asymmetry evolution for different field strengths and driving pressures for each seed field configuration. We find that out of the configurations investigated, a seed field for which the implosion centre is a saddle point in at least one plane exhibits the least degree of asymmetry during implosion.
Publisher: Cambridge University Press (CUP)
Date: 12-12-2016
DOI: 10.1017/JFM.2016.767
Abstract: 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.
Location: United Kingdom of Great Britain and Northern Ireland
Location: United States of America
Location: United States of America
No related grants have been discovered for Wouter Mostert.