Steady Shock Research Articles

Estimate of Temperature Distribution in Steady Shock Wave Front. 1st Report. Theoretical Approach.

Estimate of Temperature Distribution in Steady Shock Wave Front. 1st Report. Theoretical Approach.

Orientation Dependence of Shock Structure with Melting in L-J Crystal from Molecular Dynamics

We use molecular dynamics to study the structure of steady shock waves with melting transition traveling along the �100�, �110�, and �111� directions in the Lennard-Jones (L-J) perfect fcc crystal. Unlike shock waves in gases andfluid s, solidshocks exhibit oscillatory behaviour of profiles within the front that persists even in the strong shocks with melting. Surprisingly, shock wave along the �100� direction compresses the L-J crystal to the final overheatedsolidstate in contrast to �110� and �111� cases wherein the crystal meltedat the same shock velocity us =4 .39 km/s. Moreover, the �110� and �111� melting shock waves differ widely in the front structure: the �111� profile without oscillations is closely similar to that of the fluidshocks, whereas the �110� shock exhibits a steady precursor of solitary wave train. The evolution of velocity andpair d istribution functions across the shock layer are explored to study the shock-induced structure transformations and melting transition occuring within the region of length roughly (10 ÷ 30) σ, σ =3 .405 ˙ A.

Material properties experiments using the awe high power laser, Helen

Summary In this report we discuss two experimental programmes recently carried out using the AWE 1 TW Nd-glass laser, HELEN, to determine the properties of materials under shock loading conditions relevant to hypervelocity impacts. In the first experiment a laser heated hohlraum is employed to generate a uniform steady planar shock which is used to derive high accuracy equation of state data using the impedance match method at pressures in excess of those generally available by conventional means and hitherto only achieved in the vicinity of nuclear explosions. The results are shown to be self consistent and reproducible and thus represent a significant new tool in the material modeler's armoury. In the second experiment direct laser illumination at 10 13 W.cm −2 is used to generate an attenuating shock in an aluminium target. Direct evidence of a spall layer is obtained by means of x-ray backlighting for the first time in such laser driven experiments. A cumulative damage model is required to predict the spallation and a preliminary analysis shows that the Johnson model, with parameters derived from conventional experiments at much lower strain rates, can be used to predict the spall thickness at the higher rates associated with the laser experiment.

Shock Wave Structure in Lennard-Jones Crystal via Molecular Dynamics

The molecular dynamics (MD) method is applied to study the internal structure of strong steady shock waves in a Lennard-Jones crystal. We develop two different approaches to MD shock simulation which enable us to calculate fine-grained profiles of flow variables and their fluctuations in close detail. In contrast to shock waves in gases and fluids, solid shocks exhibit oscillatory profiles within the shock front. We study in detail these oscillations by exploring the evolution of velocity distribution, pair distribution, and potential energy distribution functions across the shock layer.

Application of Particle Image Velocimetry to a Small-Scale de Laval Nozzle

The application of particle image velocimetry (PIV) to a Mach 1.4 de Laval expansion nozzle is described. Velocity measurements are taken inside and outside the nozzle and are compared to schlieren data. The PIV technique is shown to resolve a normal shock inside the nozzle and underexpanded shock cell structures outside the nozzle in the open jet with a maximum spatial resolution of 0.54 mm, which is higher than has previously been achieved. A Reynolds-averaged computational fluid dynamics (CFD) simulation is also described and used to provide a priori data for a PIV optimization method previously published by the authors. The instantaneous PIV measurement of the steady normal shock, inside the nozzle, is shown to achieve reasonable agreement with the CFD prediction within experimental limits. Problems encountered at the recording stage with respect to seeding contamination of the viewing area are also highlighted, and a solution using an alternative seeding method is described

Optimum overtaking compression shocks with restrictions imposed on the total flow-deflection angle

The problem of optimization of gasdynamic variables behind a system of two steady oblique compression shocks with restrictions imposed on the flow-defection angle is considered. The intervals of input parameters, in which this system turns out to be more effective than one shock, are determined. On the basis of an analysis of the system optimal for the static pressure, the physical meaning of the transition from one type of the reflected discontinuity to another is explained for the problem of interaction of overtaking oblique compression shocks.

Characteristic analysis of a complex two-dimensional magnetohydrodynamic bow shock flow with steady compound shocks

A simple, compact, and systematic derivation is given of the characteristic properties of the magnetohydrodynamic (MHD) equations with two independent variables (time-dependent MHD in the xt plane and steady MHD in the xy plane), based on the symmetrizable Galilean invariant form of the equations and using a matrix approach. A numerically obtained stationary planar field-aligned MHD bow shock flow with interacting shocks is then analyzed in terms of hyperbolic and elliptic regions, steady xy characteristics, limiting lines, and allowed shock transitions. With the help of this analysis, a wave structure present in the bow shock flow can be interpreted as a double steady compound shock. This interpretation is based on the complete analogy demonstrated in our analysis, between the xy characteristic structure of this novel steady compound shock and the xt characteristic structure of the well-known time-dependent MHD compound shock.

Steady and Unsteady Shock Waves on Overlapping Grids

The interaction between steady and unsteady shock waves and interfaces of overlapping grids is studied. Conditions permitting shock penetration through an interface and ensuring solution uniqueness are derived. Easily usable corrections ensuring shock/interface penetration and convergence are devised.

Pseudo-Shock in Straight Duct with Porous Wall and Bypass System.

As the basic study of the passive control of the shock oscillation, steady pseudo shock in the rectangular ducts with the porous wall and the bypass system was investigated experimentally and numerically. The expriments were carried out in a blow-down supersonic wind tunnel at a free stream Mach number of 2.0. Numerical simulations were made with the 2-D compressible Navier Stokes code. In sloving the equations, 2nd-order accurate Harten-Yee's upwind TVD scheme and κ-ε turbulence model were used. It is shown that the boundary layer thickness under the first shock wave of the pseudo-shock is controlled by the passive control method without noticeable changes of the pseudo-shock strcuture. But the total pressure loss downstream of the pseudo-shock is reduced by this method. And the effects of some parameters influencing the passive control method are studied numerically.

Unstable Godunov Discrete Profiles for Steady Shock Waves

Given a system of n conservation laws ut+f(u)x=0, the steady shock waves, when processed by the Godunov scheme, admit rather simple discrete profiles. One shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an (n-1)-dimensional space. Applying our theory to the gas dynamics with the perfect gas law p=(\ga-1)\rho e$, we construct unstable profiles for values of $\ga$ between 1 and $\ga^*$ and rather strong shocks. Since $\ga^*\in]7/5,5/3[$, our result applies to the air but not to monoatomic gases. Finally, we illustrate this analysis by some numerical experiments, both with $\ga \ga^*$.

Prediction Method of Unsteady Combustion Around Hypersonic Projectile in Stoichiometric Hydrogen-Air

An effective nondimensional parameter, referred to as the first Damkohler parameter, is proposed to quantitatively classify the unsteady flow regime of shock-induced combustion around a hypervelocity spherical projectile in the stoichiometric hydrogen-air mixture. The parameter consists of the ratio of the fluid characteristic timescale to the chemical characteristic timescale. The fluid characteristic time is defined as a projectile diameter divided by a speed of sound behind the normal segment of the steady bow shock. The chemical characteristic time is defined as a temperature behind the normal segment of the steady bow shock over the maximum temperature increase per unit time for the exothermicity. The temperature increase for the exothermicity is estimated by the time integration of the species equations in the zero dimension in space. The proposed first Damkohler number can be analytically computed by the chemical and fluid characteristics only, without ballistic range experiments or expensive simulations on high-performance computers. The parameter is quantitatively characterized from the two distinct flow regimes observed in the experimental and numerical results. Also, features of the transition between two regimes are clarified by changing the projectile diameter, the projectile speed, and the test gas pressure.

Scattering as a mechanism for structured shock waves in metals

The scattering of wave energy during shock propagation through heterogenous media is examined as an alternative to visco-plasticity as the physics underlying the formation of structured steady shock waves in polycrystalline metals. A theory based on a quasi-harmonic representation of scattered acoustic energy in solids is pursued and used to develop continuum constitutive relations to describe nonlinear wave propagation in heterogeneous solids. Resulting constitutive models are compared with shock wave profile data for metals.

Fragmentation of a Shock‐compressed Sheet of Primordial Gas

We estimate the fragment mass scale of a primordial gas sheet compressed by steady state shock waves with a wide variety of initial parameters. We estimate the perturbation growth timescale τMGR(~τff) and consider that fragmentation occurs when τcool = τMGR, then calculate the Jeans mass at the fragmentation. It is shown that there are two fragment mass scales depending on the values of shock velocity and preshock density because of the trough of the cooling coefficient at T ~ 8000 K. As an application, we consider the disklike collapse of the protocloud just after the turnaround. We assume the standard cold dark matter model with h = 0.8 and Ωbh2 = 0.01 and estimate the typical shock velocity and preshock density for each mass scale at the collapse. Compared with our steady state shock results, there is found to be a critical mass ~108 M☉ which assorts the subsequent evolution of protoclouds. We suggest that this critical mass scale provides the observed minimum mass of the bright galaxy sequence and the mass scale of the first luminous objects.

Simulating Electron Transport and Synchrotron Emission in Radio Galaxies: Shock Acceleration and Synchrotron Aging in Axisymmetric Flows

We introduce a simple and economical but effective method for including relativistic electron transport in multidimensional simulations of radio galaxies. The method is designed to follow explicitly diffusive acceleration at shocks, and, in smooth flows, second-order Fermi acceleration, plus adiabatic and synchrotron losses for electrons in the energy range responsible for radio emission in these objects. We are able to follow both the spatial and the energy (momentum) distributions of the electrons, so that direct synchrotron emission properties can be modeled in time-dependent simulated flows of this type for the first time. That feature is essential if simulations are to bridge successfully the fundamental physical gap between flow dynamics and observed emissions. As an initial step toward that goal, we present results from some axisymmetric MHD simulations of Mach 20 light jet flows. These explicitly explore the effects of shock acceleration, as well as adiabatic expansion and synchrotron aging in smooth flows. The simulations demonstrate the importance of the fact that even for steady inflows jet terminal shocks are not simple, steady plane structures. Most importantly, this should play a very major role in determining the properties of synchrotron emission within the terminal hot spot and in the lobes generated by the jet back flow. In fact, the outflows are inherently complex, because of the basic "driven" character of a jet flow. Consequently, the nonthermal electron population emerging from the jet may encounter a wide range of shock types and strengths, as well as of magnetic field environments. We may expect to find a complex range in synchrotron spectral and brightness patterns associated with terminal hot spots and lobes. These include the possibility of steep spectral gradients (of either sign) within hot spots, the potential in lobes for islands of flat-spectrum electrons within steeper spectral regions (or the reverse), and spectral gradients coming from the dynamical history of a given flow element rather than from synchrotron aging of the embedded electrons. Finally, synchrotron "aging" in the lobes tends to proceed more slowly than one would estimate from regions of high emissivity. This is a consequence of the fact that those regions are ordinarily places where the magnetic fields are the strongest, so that the instantaneous rates of energy loss are atypical of the full history of the electron population. This feature supports earlier suggestions that nonuniform field structures may help to explain why dynamical ages of FR II sources often seem to be greater than the apparent age of the electrons radiating in the lobes, as measured in terms of spectral steepening, or the absence thereof.

Dust grains and the structure of steady C-type magnetohydrodynamic shock waves in molecular clouds

I examine the role of dust grains in determining the structure of steady, cold, oblique C-type shocks in dense molecular gas. Gas pressure, the inertia of the charged components and changes in ionization are neglected. The grain charge and rate coefficients for electron-neutral and grain-neutral elastic scattering are assumed constant at values appropriate to the shock interior. An MRN size distribution is accounted for by estimating an effective grain abundance and Hall parameter for single-size grains. A one-parameter family of intermediate shocks exists for each shock speed vs between the intermediate signal speed vA cos θ and v2vA cot θ, where vA is the pre-shock Alfvén speed and θ is the angle between the pre-shock magnetic field and the normal to the shock front. In addition, there is a unique fast shock for each vs > vA. If the pre-shock density nH ≳ 105 cm−3 and the pre-shock magnetic field satisfies B(mG)/nH(105 cm−3 ≲ 1, grains are partially decoupled from the magnetic field and the field and velocity components within fast shocks do not lie in the plane containing the pre-shock field and the shock normal. The resulting shock structure is significantly thinner than in models that do not take this into account. Existing models systematically underestimate the grain-neutral drift speed and the heating rate within the shock front. At densities in excess of 108 cm −3 these effects may be reduced by the nearly equal abundances of positive and negative grains.

Approximate analytical calculation of the mach configuration of steady shock waves in a plane constricting channel

An approximate analytical model for calculation of the parameters of a steady gas flow inside a plane constricting channel formed by two symmetrically positioned wedges is suggested. A Mach configuration of shock waves (triple point) is formed in the channel when the wedge angles are larger than some critical value. The flow calculation in a constricting channel reduces to the solution of the iterative problem for a system of nonlinear algebraic equations. The configurations of shock waves, the slipstream, and the sonic line are described by the proposed model of a gas flow. A comparison of the results obtained using this model allows a fairly accurate calculation of the Mach stem and the length of the subsonic-flow region.

Prediction method of unsteady combustion around hypersonic projectile in stoichiometric hydrogen-air

An effective nondimensional parameter, referred to as the first Damkohler parameter, is proposed to quantitatively classify the unsteady flow regime of shock-induced combustion around a hypervelocity spherical projectile in the stoichiometric hydrogen-air mixture. The parameter consists of the ratio of the fluid characteristic timescale to the chemical characteristic timescale. The fluid characteristic time is defined as a projectile diameter divided by a speed of sound behind the normal segment of the steady bow shock. The chemical characteristic time is defined as a temperature behind the normal segment of the steady bow shock over the maximum temperature increase per unit time for the exothermicity. The temperature increase for the exothermicity is estimated by the time integration of the species equations in the zero dimension in space. The proposed first Damkohler number can be analytically computed by the chemical and fluid characteristics only, without ballistic range experiments or expensive simulations on high-performance computers. The parameter is quantitatively characterized from the two distinct flow regimes observed in the experimental and numerical results. Also, features of the transition between two regimes are clarified by changing the projectile diameter, the projectile speed, and the test gas pressure.

Study of Accuracy and Stability of the Particle Method for Compressible Viscous Flows in Unsteady Motion and Its Application.

A grid-free particle method is developed for solving Navier-Stokes equations of one dimensional viscous compressible fluids in unsteady motion. The particles, representing fluid density, are moved at the acceleration derived from the motion equation, and the total energy of each moving particle is increased/decreased according to the energy equation. A shock wave and an expansion wave in a shock tube are simulated as numerical experiments to study the accuracy and the stability of the present method. The structures of a steady shock wave of Argon simulated by our method, the finite difference method and the direct-simulation Monte Carlo method were compared.

The Wardle Instability in Interstellar Shocks. II. Gas Temperature and Line Emission

We have modeled the gas temperature structure in unstable C-type shocks and obtained predictions for the resultant CO and H2 rotational line emissions, using numerical simulations of the Wardle instability presented in Paper I. Our model for the thermal balance of the gas includes ion-neutral frictional heating; compressional heating; radiative cooling due to rotational and ro-vibrational transitions of the molecules CO, H2O, and H2; and gas-grain collisional cooling. We obtained results for the gas temperature distribution in—and H2 and CO line emission from—shocks of neutral Alfvénic Mach number 10 and velocity 20 or 40 km s-1 in which the Wardle instability has saturated. Both two- and three-dimensional simulations were carried out for shocks in which the preshock magnetic field is perpendicular to the shock propagation direction, and a two-dimensional simulation was carried out for the case in which the magnetic field is obliquely oriented with respect to the shock propagation direction. Although the Wardle instability profoundly affects the density structure behind C-type shocks, most of the shock-excited molecular line emission is generated upstream of the region where the strongest effects of the instability are felt. Thus the Wardle instability has a relatively small effect on the overall gas temperature distribution in—and the emission-line spectrum from—C-type shocks, at least for the cases that we have considered. In none of the cases that we have considered thus far did any of the predicted emission-line luminosities change by more than a factor of 2.5, and in most cases the effects of instability were significantly smaller than that. Slightly larger changes in the line luminosities seem likely for three-dimensional simulations of oblique shocks, although such simulations have yet to be carried out and lie beyond the scope of this study. Given the typical uncertainties that are always present when model predictions are compared with real astronomical data, we conclude that Wardle instability does not imprint any clear observational signature on the shock-excited CO and H2 line strengths. This result justifies the use of one-dimensional steady shock models in the interpretation of observations of shock-excited line emission in regions of star formation. Our three-dimensional simulations of perpendicular shocks revealed the presence of warm filamentary structures that are aligned along the magnetic field, a result that is of possible relevance to models of water maser emission from C-type shocks.

Monte Carlo analysis of the hysteresis phenomenon in steady shock wave reflections

Monte Carlo analysis of the hysteresis phenomenon in steady shock wave reflections