Shock Theory Research Articles

Waves and nonclassical shocks in a scalar conservation law with nonconvex flux

The propagation of waves in the nonlinear equation¶¶\(\frac{\partial V}{\partial t} + \left(\Gamma V + \frac{\Lambda V^{2}}{2}\right) \frac{\partial V}{\partial\hat{x}} = \hat{\nu}\frac{\partial^{2}V}{\partial\hat{x}^{2}} + \hat{\beta}\frac{\partial^{3}V}{\partial\hat{x}^{3}}, \quad \hat{\nu}>0, \quad\hat{\beta}\Lambda>0,\)¶¶generates undercompressive shocks in the hyperbolic limit with dispersion and dissipation balanced. These shocks are undercompressive in type and the diversity of phenomena possible is illustrated for three different initial conditions: a propagating shock through a wave fan, a square pulse and a periodic pulse constructed from constant states. The rich variety of wave phenomena exhibited:- shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations produce waves that have no counterpart in classical shock theories. A mechanism for the formation of a nonclassical shock from a classical shock by wavefan interaction is presented.

Modern theory of Fermi acceleration: A new challenge to plasma physics

One of the main features of astrophysical shocks is their ability to accelerate particles to extremely high energies. The leading acceleration mechanism, the diffusive shock acceleration, is reviewed. It is demonstrated that its efficiency critically depends on the injection of thermal plasma into acceleration which takes place at the subshock of the collisionless shock structure that, in turn, can be significantly smoothed by energetic particles. Furthermore, their inhomogeneous distribution provides free energy for magnetohydrodynamic (MHD) turbulence regulating the subshock strength and injection rate. Moreover, the MHD turbulence confines particles to the shock front controlling their maximum energy and bootstrapping acceleration. Therefore, the study of the MHD turbulence in a compressive plasma flow near a shock is a key to the understanding of the entire process. The calculation of the injection rate became part of the collisionless shock theory. It is argued that the further progress in diffusive shock acceleration theory is impossible without a significant advance in these two areas of plasma physics.

Interface imprinting by a rippled shock using an intense laser

Perturbation imprinting at a flat interface by a rippled shock has been observed in a laser hydrodynamics experiment. A strong shock was driven through a three-layer target, with the first interface rippled, and the second flat. The chosen thickness of the second layer gave instability growth with opposite phases at the two interfaces, consistent with two-dimensional simulations and rippled shock theory.

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.

Ideal shocks in 2-layer flow Part II: Under a passive layer

In this second part of the study, ideal shock theory in two-layer stratified flow is extended toinclude a third passive layer (i.e., a two and a half layer system). With the presence of a passivelayer, two linear wave modes and “viscous tail modes” exist, complicating the solubility conditionsand uniqueness proofs for two layer shocks. It is found however, that shocks can beunambiguously classified as external or internal based on the states of criticality that theyconnect. The steepening condition, while still necessary, provides a less restrictive constraintthan it did with a rigid lid. Thus, we have to rely more on solutions to the full viscous shockequations to establish shock existence. The detailed structure, momentum exchange, andBernoulli loss in a viscous shock are examined using an analytical weak shock solution and aset of numerical solutions for shocks with finite amplitudes. A shock regime diagram (F1 by F2)is constructed based on the numerical integration of the full viscous shock equations. For strongexternal jumps, a cusp region (i.e., in the sense of catastrophe theory) is identified on the regimediagram. For pre-shock states within the cusp, three end states are possible and two of theseare realizable. The cusp has several physical implications. It indicates that an equal distributionof dissipation between the two layers in shocks is mathematically possible but physically inaccessible.It also allows hysteresis in time varying flows, and promotes the occurrence of doubleshocks (i.e., closely spaced shocks of different character). The results are compared with classicalshock solutions and a set of time dependent numerical experiments.

Shock at the millennium. I. Walter B. Cannon and Alfred Blalock.

Present management of shock derives, in part, from the classic investigations of Walter B. Cannon and Alfred Blalock. The intersections of their professional lives as recorded in the professional literature and in personal correspondence suggest that Blalock's pivotal studies of experimental shock were fueled, at least in part, by Cannon's inability to resolve the inconsistencies of the then-popular toxic theory of shock. Cannon appears to have substantially shaped Blalock's thought and work, initially as authority and competitor and later as colleague and friend. Blalock's experimental proof that injury precipitated obligatory locoregional fluid losses, the effects of which could be ameliorated by vigorous restoration of plasma volume, became a cornerstone of shock theory and therapy.

Lattice Boltzmann BGK simulation of nonlinear sound waves: the development of a shock front

The application of the lattice Boltzmann model to simulating nonlinear propagative acoustic waves is considered. The lattice Boltzmann model, and its application to the study of nonlinear sound propagation, are discussed. Lattice Boltzmann simulations of the development of a shock front are performed when a sound wave is emitted from a high-amplitude sinusoidal source. For a number of parameters, representing different physical situations, the wave development is compared with inviscid shock theory and with the solution of Burgers' equation for a fully viscous fluid. The simulations show good agreement with Burgers' equation and with the inviscid theory when propagation at high Reynolds number is considered. These results suggest that the lattice Boltzmann model is a useful technique for studying a range of problems in nonlinear acoustics.

An approximate solution for spherical and cylindrical piston problem

A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks.

An analytical study of Mach reflection in nonequilibrium steady flows

In the present work, we study the influence of both thermal and chemical relaxation phenomena on steady Mach reflections. To characterize the dependency of the scales of the phenomenon upon the relaxation we develop a quasi-one-dimensional model for a nonequilibrium diatomic gas coupled with the three shock theory. The results show that Mach reflections are strongly affected by vibrational relaxation and to a lesser extent by chemical excitation. The Mach stem height and its location, and the extent of the subsonic pocket are found to depend both on the geometrical setup and on the relaxation length, and do not scale linearly with the geometrical size as found in the ideal case. In addition, the relaxation is found to cause a forward displacement of the Mach stem thus counteracting Mach reflection instabilities. The nonlinearity has also been confirmed by a perturbation analysis of the equations for small Damköhler numbers.

Thermal shock properties of chemically toughened borosilicate glass

This paper describes an experimental test of the unified theory of thermal shock. A borosilicate glass (78.1%SiO 2–13.3%B 2O 3–4.7%Na 2O–3.4%Al 2O 3) subjected to ion-exchange in molten KNO 3, was used. The theory predicts the critical temperature difference for fracture initiation by thermal shock; however, it overestimates the number of cracks which propagate simultaneously. Despite that fact, taken into account the numerous assumptions of the theoretical model and the possible experimental errors, our results indicate a fair agreement between theory and experiment.

Solar Wind at 6.8 Solar Radii from UVCS Observation of Comet C/1996Y1

The comet C/1996Y1, a member of the Kreutz family of Sun-grazing comets, was observed with the Ultraviolet Coronagraph Spectrometer (UVCS) aboard the Solar and Heliospheric Observatory (SOHO) satellite. The Lyα line profile and spatial distribution are interpreted in terms of the theory of bow shocks driven by mass-loading. At the time of the observation, the comet was 6.8 R☉ from the Sun in a region of high-speed wind, a region difficult to observe directly with the SOHO instruments but an important region for testing models of solar wind acceleration and heating. We find a solar wind speed below 640 km s-1 and a constraint on the combination of solar wind speed and proton temperature. The total energy per proton at 6.8 R☉ is 50%-75% of the energy at 1 AU, indicating that significant heating occurs at larger radii. The centroid and width of the Lyα line generally confirm the predictions of models of the cometary bow shock driven by mass-loading as cometary molecules are ionized and swept up in the solar wind. We estimate an outgassing rate of 20 kg s-1, which implies an active area of the nucleus only about 6.7 m in diameter at 6.8 R☉. This is likely to be the size of the nucleus, because any inert mantle would have probably been blown off during the approach to the Sun.

The collective culture shock in transition countries ‐ theoretical and empirical implications

Individual culture shock is a well known and evidenced phenomenon. It describes the psychological and also physical reactions of a person staying abroad. These reactions are the result of confrontation with a foreign culture. According to the authors such reactions also exist on the level of society as a whole. This “collective culture shock” influences management and business relations and causes problems. A theory of the collective culture shock is developed that analyses these problems and provides a framework for solutions. This theory suggests that cultural processes and features in transition countries that are usually attributed to the communist heritage are the result of collective culture shock.

High-order azimuthal instabilities on a cylindrical liquid jet driven by temporal and spatial perturbations

A method has been developed to drive a cylindrical liquid jet unstable for deformations with axial wavelengths shorter than the circumference of the jet and azimuthal mode numbers greater than 0. The benefit of this method is that a cylindrical liquid jet can be broken into a spray with an average diameter smaller than the diameter of the initial jet. The higher-order instabilities were created by establishing initial conditions for the jet in space and time at the nozzle. An electromechanical transducer creates the applied temporal initial condition which is a sinusoidally varying velocity perturbation added to the steady velocity of the jet. The amplitude of the velocity perturbation can be as large as the jet’s steady velocity and the energy in the applied velocity perturbation drives the instability. The spatial perturbation is created by placing perturbations in the circumference of the nozzle. As the velocity perturbation travels on the jet, its leading edge steepens and the trailing edge broadens in a manner analogous to the steepening of a pressure pulse in a compressible gas. If the driven velocity perturbation is sufficiently large, a shock or jump forms on the leading edge of the velocity pulse and the jet may break up into higher-order modes. A theoretical analysis of the breakup process, based on an adaptation of compressible fluid shock theory, is used to derive a fundamental lower bound on the spray’s Sauter mean diameter as a function of the velocity perturbation amplitude. Techniques for approaching the theoretical minimum spray diameter by using the higher-order modes to atomize liquid jets are discussed.

A syndrome produced by diverse nocuous agents. 1936.

A syndrome produced by diverse nocuous agents. 1936.

The atmospheric cosmic- and solar energetic particle radiation environment at aircraft altitudes

Galactic cosmic rays interact with the solar wind, the earth's magnetic field and its atmosphere to produce hadron, lepton and photon fields at aircraft altitudes. In addition to cosmic rays, energetic particles generated by solar activity bombard the earth from time to time. These particles, while less energetic than cosmic rays, also produce radiation fields at aircraft altitudes which have qualitatively the same properties as atmospheric cosmic rays. We have used a code based on transport theory to calculate atmospheric cosmic-ray quantities and compared them with experimental data. Agreement with these data is seen to be good. We have then used this code to calculate equivalent doses to aircraft crews. We have also used the code to calculate radiation doses from several large solar energetic particle events which took place in 1989, including the very large event that occurred on September 29 th and 30 th of that year. The spectra incident on the atmosphere were determined assuming diffusive shock theory.

Analytical results on MHD intermediate shocks

The admissibility of MHD intermediate shocks has been a matter of much debate over the years. Though the legitimacy of such shocks was shown in a recent series of investigations by Wu, rigorous analytical proof of the results was considered being unsettled. In this paper, as a step towards developing such proof, we present a theory of MHD shocks. On the basis of global analysis, we develop a shock admissibility condition. Using this theory, we explain how intermediate shocks and compound waves are generated.

What do we really know about collisionless shocks?

The things that we really know about collisionless shocks are summarized; examples are given of measurements which confirm the theoretical foundations. Typical expected and observed phenomena is described. Some shock theory which is apparently not incorporated into current analysis is also presented.

Analytical Prediction of Hole Diameter in Thin Plates Due to Hypervelocity Impact of Spherical Projectiles

A first-principles-based model is presented for calculating the hole diameter resulting from the normal hypervelocity impact of a spherical aluminum projectile on a thin aluminum plate. One-dimensional shock theory is used to predict the creation and attenuation of Hugoniot pressures along the plate surface. Pressures are translated into the plate thickness by calculating intersecting positions of advancing shock fronts and centered-fan rarefaction waves. The radial position at which the shock pressure equals a predetermined value is defined to be the hole diameter. The model was calibrated by determining this critical value for aluminum-an-aluminum impacts using several hundred data points. A residuals analysis indicated some inherent problems with the model. Two empirical factors were added to account for thin plate and two-dimensional shock dissipation effects. The predictions of the adjusted model are shown to compare well with predictions of several empirical hole diameter models.

Analytical Prediction of Hole Diameter in Thin Plates Due to Hypervelocity Impact of Spherical Projectiles

A first-principles-based model is presented for calculating the hole diameter resulting from the normal hypervelocity impact of a spherical aluminum projectile on a thin aluminum plate. One-dimensional shock theory is used to predict the creation and attenuation of Hugoniot pressures along the plate surface. Pressures are translated into the plate thickness by calculating intersecting positions of advancing shock fronts and centered-fan rarefaction waves. The radial position at which the shock pressure equals a predetermined value is defined to be the hole diameter. The model was calibrated by determining this critical value for aluminum-an-aluminum impacts using several hundred data points. A residuals analysis indicated some inherent problems with the model. Two empirical factors were added to account for thin plate and two-dimensional shock dissipation effects. The predictions of the adjusted model are shown to compare well with predictions of several empirical hole diameter models.

On simple waves and weak shock theory for the equations of alluvials river hydraulics

This paper considers the hydraulic equations governing alluvial river dynamics. The shock-wave classes for this system were described and fully analysed by Needham & Hey (1991). The complementary expansion wave classes are determined in this paper. In addition, more general simple waves are considered and a weak-shock theory is established. This is used to consider several practical examples, including the dynamics of dredged river sections, the propagation of sediment slugs and the effects of sediment blocking and extraction.