Shock Wave Of Arbitrary Strength Research Articles

Shock waves in Euler equations for compressible medium

Shock waves of arbitrary strength in the Euler equations for compressible media are studied. The admissibility condition for a shock wave is shown to be equivalent to its formation according to the entropy production criterion. The Riemann problem with large data has a unique admissible solutions. These quantitative results are based on the exact global expressions for the basic physical variables as the states move along the Hugoniot and wave curves.

Propagation of one-dimensional planar and nonplanar shock waves in nonideal radiating gas

The present study seeks to investigate a quasilinear hyperbolic system of partial differential equations which describes the unsteady one-dimensional motion of a shock wave of arbitrary strength propagating through a nonideal radiating gas. We have derived an infinite hierarchy of the transport equation which is based on the kinematics of one-dimensional motion of shock front. By using the truncation approximation method, an infinite hierarchy of transport equations, which governs the shock strength and the induced discontinuities behind it, is derived to study the kinematics of the shock front. The first three transport equations (i.e., first, second, and third-orders) are used to study the growth and decay behavior of shocks in van der Waals radiating gas. The decay laws for weak shock waves in nonradiating gas are entirely recovered in the second-order truncation approximation. The results obtained by the first three approximations for shock waves of arbitrary strength are compared with the results predicted by the characteristic rule. Also, the effect of nonideal parameters and radiation on the evolutionary behavior of shock waves are discussed and depicted pictorially.

Kinematics of spherical shock waves in an interstellar ideal gas clouds with dust particles

In this article, a system of partial differential equations governing the one‐dimensional motion of inviscid, self‐gravitating, spherical dusty gas cloud is considered. The evolutionary behavior of spherical shock waves of arbitrary strength in an interstellar dusty gas cloud is examined. By utilizing the method based on the kinematics of the one‐dimensional motion of shock waves, we derived an infinite set of transport equations governing the strength of shock waves and induced discontinuity behind it. By applying the truncation procedure to the infinite set of transport equations, we get an efficient system of finite number ordinary differential equations describing shock propagation, which can be regarded as a good approximation of the infinite hierarchy of the system. The truncated equations describing the shock strength and the induced discontinuity are used to analyze the growth and decay behavior of shock waves of arbitrary strength in the dusty gas medium. We considered the first two truncation approximations and the obtained results for the exponent from the successive approximation and compared our results with Guderley's exact similarity solution and the characteristic rule.

One-dimensional spherical shock waves in an interstellar dusty gas clouds

Abstract A system of partial differential equations describing the one-dimensional motion of an inviscid self-gravitating and spherical symmetric dusty gas cloud, is considered. Using the method of the kinematics of one-dimensional motion of shock waves, the evolution equation for the spherical shock wave of arbitrary strength in interstellar dusty gas clouds is derived. By applying first order truncation approximation procedure, an efficient system of ordinary differential equations describing shock propagation, which can be regarded as a good approximation of infinite hierarchy of the system. The truncated equations, which describe the shock strength and the induced discontinuity, are used to analyze the behavior of the shock wave of arbitrary strength in a medium of dusty gas. The results are obtained for the exponents from the successive approximation and compared with the results obtained by Guderley’s exact similarity solution and characteristic rule (CCW approximation). The effects of the parameters of the dusty gas and cooling-heating function on the shock strength are depicted graphically.

Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation.

CURRENT THEORIES OF DISSIPATION IN THE RELATIVISTIC REGIME SUFFER FROM ONE OF TWO DEFICITS: either their dissipation is not causal or no profiles for strong shock waves exist. This paper proposes a relativistic Navier-Stokes-Fourier-type viscosity and heat conduction tensor such that the resulting second-order system of partial differential equations for the fluid dynamics of pure radiation is symmetric hyperbolic. This system has causal dissipation as well as the property that all shock waves of arbitrary strength have smooth profiles. Entropy production is positive both on gradients near those of solutions to the dissipation-free equations and on gradients of shock profiles. This shows that the new dissipation stress tensor complies to leading order with the principles of thermodynamics. Whether higher order modifications of the ansatz are required to obtain full compatibility with the second law far from the zero-dissipation equilibrium is left to further investigations. The system has exactly three a priori free parameters χ,η,ζ, corresponding physically to heat conductivity, shear viscosity and bulk viscosity. If the bulk viscosity is zero (as is stated in the literature) and the total stress-energy tensor is trace free, the entire viscosity and heat conduction tensor is determined to within a constant factor.

Kinematics of a shock wave of arbitrary strength in a non-ideal gas

Singular surface theory is used to study the evolutionary behaviour of an unsteady three-dimensional motion of a shock wave of arbitrary strength propagating through a non-ideal gas. The dynamical coupling between the shock front and the induced discontinuities behind it is investigated by considering an infinite system of transport equations governing the strength of a shock wave and the induced discontinuities behind it. This infinite system, when subjected to a truncation approximation, efficiently describes the shock motion. Disturbances propagating on the shock and the onset of shock-shocks are briefly discussed. For a two-dimensional shock motion, our transport equations bear a structural resemblance to those of geometrical shock dynamics. Attention is drawn to the connection between the transport equation obtained by using the truncation rule and the one obtained by using the characteristic rule. The effects of van der Waals’ excluded volume and wavefront geometry on the evolutionary behaviour of shocks are discussed.

Particle acceleration and coronal mass ejection driven shocks: Shocks of arbitrary strength

There is substantial evidence suggesting that energetic particles observed in “gradual” solar energetic particle events are accelerated at shock waves driven out of the corona by coronal mass ejections (CMEs). We present a model of particle acceleration at interplanetary shock waves, assumed to be driven by CMEs, in which the upstream wave intensity, driven by the accelerated particles, is calculated self‐consistently using the steady‐state solution to the wave growth equation. This then allows for the self‐consistent calculation of the momentum dependent spatial diffusion coefficient which ultimately governs both the acceleration and subsequent evolution of the energetic particles. The model is consequently applicable to shock waves of arbitrary strength, a significant improvement on previous models which were generally only valid for very strong shock waves. The model is able to calculate minimum and maximum particles energies as the shock propagates out into the solar wind and can determine time‐dependent downstream spectra. The spectra of particles escaping into the relatively undisturbed upstream medium is also calculated and in future will be used as input to a detailed transport model to determine upstream spectra and intensity profiles. Although we do not compare the results with any individual observations, the model is able to reproduce some of the observed features of “gradual” SEP events. The self‐consistent calculation of the upstream wave intensity will in future allow this model to be extended to consider the acceleration of particles of various charge states and masses.

Particle acceleration at CME driven shock waves

A dynamical time-dependent model of particle acceleration at coronal mass ejection (CME) driven shock waves is presented. We consider first a one-dimensional model applicable to shock waves of arbitrary strength. The model includes the determination of the particle's injection energy, the maximum energy of particles accelerated at the shock, energetic particle spectra at all spatial and temporal locations, and the dynamical distribution of particles that are trapped in the post shock flow and those that escape upstream and downstream of the evolving shock complex. We then consider the extension of this model to two dimensions. The 2-dimensional model computes the variation of the shock strength both temporally and spatially and illustrates how this effects the evolution of the energetic particle spectra and intensities.

Rupture-disk-less shock-tube with compression tube driven by free piston

A new technique is proposed for a shock tube driven by a freely moving piston. In a conventional free-piston-driven shock tube, a rupture disk is employed between the compression tube and the shock generation tube. In the present method, however, the conventional rupture disk is replaced by a newly developed fast action valve which is activated by the compressed gas generated in the compression tube. The present method enables us to generate high Mach number shock waves of arbitrary strength with good reproducibility. The performance of the new method is demonstrated experimentally. This also enables us to be carefree to scattering of fragments of the rupture disk.

Three dimensional shock wave propagation in an ideal gas

A systematic study is made of an unsteady three dimensional motion of a shock wave of arbitrary strength propagating through an ideal gas. The dynamical coupling between the shock front and the rearward flow is investigated by considering an infinite system of transport equations for the variation of jumps in pressure and its space derivatives across the shock. This infinite system is then truncated to get a closed system of coupled differential equations, which efficiently describes the shock motion. Disturbances propagating on the shock and the onset of shock-shocks are briefly discussed. In the limit of vanishing shock strength, the first order truncation approximation leads to an exact description of acceleration waves. Asymptotic decay laws for the weak shocks and rearward precursor disturbances are exactly recovered. In the strong shock limit, the first order approximation leads to a propagation law for imploding shocks, which is in agreement with the Guderley's exact similarity solution. Attention is drawn to the connection between the transport equations along shock rays obtained here and the corresponding results obtained from an alternative method, using the theory of generalized functions.

On one-dimensional planar and nonplanar shock waves in a relaxing gas

The paper examines the evolutionary behavior of shock waves of arbitrary strength propagating through a relaxing gas in a duct with spatially varying cross section. An infinite system of transport equations, governing the strength of a shock wave and the induced discontinuities behind it, are derived in order to study the kinematics of the shock front. The infinite system of transport equations, when subjected to a truncation approximation, provides an efficient system of only finite number of ordinary differential equations describing the shock propagation problem. The analysis, which accounts for the dynamical coupling between the shock fronts and the flow behind them, describes correctly the nonlinear steepening effects of the flow behind the shocks. Effects of relaxation on the evolutionary behavior of shocks are discussed. The first-order truncation approximation accurately describes the decay behavior of weak shocks; the usual decay laws for weak shocks in a nonrelaxing gas are exactly recovered. The results concerning shocks of arbitrary strength are compared with the characteristic rule. In the limit of vanishing shock strength, the transport equation for the first-order induced discontinuity leads to an exact description of an acceleration wave. In the strong shock limit, the second-order truncation criterion leads to a propagation law for imploding shocks which is in agreement (within 5% error) with the Guderley’s exact similarity solution.

Uniformly valid analytical solution to the problem of a decaying shock wave

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

Study of non-self similar flows behind shock waves in a dusty gas

Non-self similar solutions for plane, cylindrical and spherical unsteady flows of dusty gas (mixture of gas and small solid particles) behind shock waves of arbitrary strength initiated by the instantaneous release of finite energy and propagating into a uniform medium at rest are investigated. Two methods based upon the assumptions, namely, (i) power law density and (ii) linear velocity, are employed to study the effects of the parameters characterising the initial volume fraction Z o , mass concentration of the solid particles K p in the dusty gas and the ratio of the density of the solid particles to that of the initial density of the gas G on shock trajectory and the flow field behind the shock. Comparative study of the results obtained by the two assumptions is also made. A simple formula is derived to study the effect of the solid particles on the shock strength in the strong shock regime.

Stability of Shock Waves of Arbitrary Strength With Viscosity and Heat Conduction

The problem of the stability of shock waves with viscosity and heat conduction has been previously formulated as an eigenvalue problem involving a set of linear ordinary differential equations in a finite domain with what are shown to be regular singular points at the ends of the domain. By means of a computer-aided Frobenius type of analysis, it is shown that the (continuous) eigenvalue spectrum is such that the shock waves will be stable for all values of the shock-strength parameter. Some actual solutions of the disturbance equations are shown.

The diffraction of a shock wave by an inclined body of revolution

The interaction between a plane normal shock wave of arbitrary strength and a slender body of revolution, held with its axis at a small angle α to the direction of propagation of the shock, is considered. The pressure field is determined by using the method of matched asymptotic expansions and the general results are applied to the problem of diffraction by an inclined circular cone.

The diffraction of a shock wave by a slender body

The problem of the interaction of a plane normal shock wave of arbitrary strength with a general slender body is treated by the method of matched asymptotic expansions. An inner solution valid near the body, an outer solution valid away from the body together with an inner-shock solution valid in the region which is both near the body and near the shock are required. It is shown how the pressure near the body can be obtained if the solution for steady flow past the same body is known, and the lateral forces acting on the body can be determined by using the results of Ward [7]. The general results are applied to the problem of diffraction by a circular cone, of semi-vertex angle τ, and the presence of the inner-shock solution is shown to contribute a term of (τ3) to the expression for the drag coefficient.

The propagation of non-uniform slow shock waves

A linearized analysis, based on small entropy variations, is used to determine the perturbation generated when an initially uniform slow magneto-hydrodynamic shock wave of arbitrary strength, propagating along a tube of uniform cross-sectional area, encounters a section of non-uniform cross-sectional area. Although this problem was solved previously, an error in the numerical computations has recently been discovered. This note serves to correct the error.

Diffraction of shock waves by a moving thin wing

An analytical solution is obtained for the flow field due to the impinging of a plane shock wave of arbitrary strength by a thin wing moving in the opposite direction. The planform and the thickness distribution of the wing can be arbitrary and the speed of the wing can be either supersonic or subsonic relative to the undisturbed stream ahead of the shock or to that behind the shock. The solution is a generalization of the previous solution of Ting & Ludloff for the diffraction of shock wave by a two-dimensional stationary airfoil to a three-dimensional wing moving with supersonic or subsonic speed relative to the stream ahead of or behind the shock. The solution is employed for the analysis of the changes in aerodynamic forces when an airplane encounters a blast wave or a shock wave of another airplane. It is also used to study the diffraction of a shock wave or anN-wave advancing over flat terrains.

Propagation of a shock wave of arbitrary strength in two half planes containing a free surface

The problem of propagation of an initially plane shock wave along a fluid interface is solved numerically, using the method of characteristics in conjunction with Whitham's technique. The results are limited to differences in shock velocity in the two fluid media of about 30 %. Greater degrees of shock diffraction will be accompanied by wave reflexions which have not been considered in Whitham's technique. The diffracted shock is found to be remarkably uniform, except in the vicinity of the free surface and the undisturbed shock, where it undergoes large changes in curvature.

Waves of finite amplitude in a hot plasma

A study is made of a plane shock wave of arbitrary strength propagating in a hot rarefied plasma across the magnetic field. The question of the propagation of nonstationary waves of finite but small amplitude under these conditions is examined.