On the inadmissibility of non-evolutionary shocks

In recent years, numerical solutions of the equations of compressible magnetohydrodynamic (MHD) flows have been found to contain intermediate shocks for certain kinds of problems. Since these results would seem to be in conflict with the classical theory of MHD shocks, they have stimulated attempts to reexamine various aspects of this theory, in particular the role of dissipation. In this paper, we study the general relationship between the evolutionary conditions for discontinuous solutions of the dissipation-free system and the existence and uniqueness of steady dissipative shock structures for systems of quasilinear conservation laws with a concave entropy function. Our results confirm the classical theory. We also show that the appearance of intermediate shocks in numerical simulations can be understood in terms of the properties of the equations of planar MHD, for which some of these shocks turn out to be evolutionary. Finally, we discuss ways in which numerical schemes can be modified in order to avoid the appearance of intermediate shocks in simulations with such symmetry

One-dimensional nonlinear stability of pathological detonations

In this paper we perform high-resolution one-dimensional time-dependent numerical simulations of detonations for which the underlying steady planar waves are of the pathological type. Pathological detonations are possible when there are endothermic or dissipative effects in the system. We consider a system with two consecutive irreversible reactions A[rightward arrow]B[rightward arrow]C, with an Arrhenius form of the reaction rates and the second reaction endothermic. The self-sustaining steady planar detonation then travels at the minimum speed, which is faster than the Chapman–Jouguet speed, and has an internal frozen sonic point at which the thermicity vanishes. The flow downstream of this sonic point is supersonic if the detonation is unsupported or subsonic if the detonation is supported, the two cases having very different detonation wave structures. We compare and contrast the long-time nonlinear behaviour of the unsupported and supported pathological detonations. We show that the stability of the supported and unsupported steady waves can be quite different, even near the stability boundary. Indeed, the unsupported detonation can easily fail while the supported wave propagates as a pulsating detonation. We also consider overdriven detonations for the system. We show that, in agreement with a linear stability analysis, the stability of the steady wave is very sensitive to the degree of overdrive near the pathological detonation speed, and that increasing the overdrive can destabilize the wave, in contrast to systems where the self-sustaining wave is the Chapman–Jouguet detonation

Rarefaction Shocks, Shock Errors and Low Order of Accuracy in ZEUS

We show that there are simple one dimensional problems for which the MHD code, ZEUS, generates significant errors, whereas upwind conservative schemes perform very well on these problems.Comment: 11 pages, accepted in ApJ Letter

The Fanno model for turbulent compressible flow

The paper considers the derivation and properties of the Fanno model for nearly unidirectional turbulent flow of gas in a tube. The model is relevant to many industrial processes. Approximate solutions are derived and numerically validated for evolving flows of initially small amplitude, and these solutions reveal the prevalence of localized large-time behaviour, which is in contrast to inviscid acoustic theory. The properties of large-amplitude travelling waves are summarized, which are also surprising when compared to those of inviscid theory

Erratum: The chemistry of transient molecular cloud cores

We assume that some, but not all, of the structure observed in molecular clouds is associated with transient features which are not bound by self-gravity. We investigate the chemistry of a transient density fluctuation, with properties similar to those of a core within a molecular cloud. We run a multipoint chemical code through a core's condensation from a diffuse medium to its eventual dispersion, over a period of ∼1 Myr. The dynamical description adopted for our study is based on an understanding of a particular mechanism, involving slow-mode wave excitation, for transient structure formation which so far has been studied in detail only with plane-parallel models in which self-gravity has not been included. We find a significant enhancement of the chemical composition of the core material on its return to diffuse conditions, whilst the expansion of the core as it disperses moves this material out to large distances from the core centre. This process transports molecular species formed in the high-density regions out into the diffuse medium. Chemical enrichment of the cloud as a whole also occurs, as other cores of various sizes, life-spans and separations evolve throughout. Enrichment is strongly affected by freeze-out on to dust grains, which takes place in high-density, high visual extinction regions. As the core disperses after reaching its peak density and the visual extinction drops below a critical value, grain mantles are evaporated back into the gas phase, initiating more chemistry. The influence of the sizes, masses and cycle periods of cores will be large both for the level of chemical enrichment of a dark cloud and ultimately for the low-mass star formation rate. The cores in which stars form are almost certainly bound by their self-gravity and are not transient in the sense that the cores on which most of our study is focused are transient. Obviously, enrichment of the chemistry of low-density material will not take place if self-gravity prevents the re-expansion of a core. We also consider the case of a self-gravitating core, by holding its peak density conditions for a further 0.4 Myr. We find that the differences near the peak densities between transient and gravitationally bound cores are generally small, and the resultant column densities for objects near the peak densities do not provide definitive criteria for discriminating between transient and bound cores. However, increases in fractional abundances due to reinjection of mantle-borne species may provide a criterion for detection of a non-bound core

Instability in Shocked Granular Gases

Shocks in granular media, such as vertically oscillated beds, have been shown to develop instabilities. Similar jet formation has been observed in explosively dispersed granular media. Our previous work addressed this instability by performing discrete-particle simulations of inelastic media undergoing shock compression. By allowing finite dissipation within the shock wave, instability manifests itself as distinctive high density non-uniformities and convective rolls within the shock structure. In the present study we have extended this work to investigate this instability at the continuum level. We modeled the Euler equations for granular gases with a modified cooling rate to include an impact velocity threshold necessary for inelastic collisions. Our results showed a fair agreement between the continuum and discrete-particle models. Discrepancies, such as higher frequency instabilities in our continuum results may be attributed to the absence of higher order effects.Comment: 6 pages, prepared for the proceedings of the APS Topical Group on Shock Compression of Condensed Matter in Seattle, Washington, from July 7th through July 12th, 201

Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes. The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method. Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow

Dark cloud chemistry in initially H-rich regions

The chemistry in dark regions of dense cores is explored as a function of the initial abundance ratio of H to H 2, on the assumption that some cores form on a timescale and are younger than the time required for the H :H 2 ratio to attain its equilibrium value. Observational diagnostics of non-equilibrium values of the initial H :H 2 ratio are identified. In initially H-rich material, the abundances of OH, NH 3, CN, and HNC are for some time higher than they are in initially H-poor material. In initially H-poor regions, the abundances of CO, species containing multiple carbon atoms in each molecule, and CS are larger for an (observationally significant) period than in initially H-rich material