Cylindrical Shock Waves Research Articles

Exact Solution for an Unsteady Isothermal Flow Behind a Cylindrical Shock Wave in a Rotating Perfect Gas with an Axial Magnetic Field and Variable Density

An exact solution of the problem on the propagation of a cylindrical shock wave in a rotating perfect gas with an axial magnetic field in the case of an isothermal flow is obtained. The initial density, magnetic field strength, and the initial angular velocity in the ambient medium are assumed to vary according to the power law. An exact similarity solution obtained by the McVittie method for an isothermal flow in a rotating gas is first reported. Similarity transformations are used to transform a system of partial differential equations into a system of ordinary differential equations, and then the product solution of McVittie is used to obtain the exact solution. The effects of the values of the gas specific heat ratio, rotational parameter, and of the strength of the initial magnetic field are discussed. It is shown that the shock velocity increases and the shock strength decreases with increase in the values of these parameters. The effect of variation in the value of the initial density index is also studied. The obtained solutions show that the radial fluid velocity, density, pressure, and the magnetic field strength tend to zero as the axis of symmetry is approached.

Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method

The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.

Approximate analytical solution for ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas: isothermal flow

The propagation of an ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas under isothermal flow condition with an azimuthal magnetic field is investigated. The electrical conductivity is assumed to be negligible in the medium ahead of the shock wave, which after the passage of the shock wave becomes infinitely large. The magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are assumed to be varying according to the power law with distance from the axis of symmetry in the undisturbed medium. The zeroth and first-order approximations are discussed by the aid of the power series method. Solutions for the zeroth-order approximation are constructed in analytical form. Distributions of hydrodynamical quantities are discussed. The effect of flow parameters, namely, shock wave Cowling number c∗, adiabatic exponent γ, rotational parameter L, and gas non-idealness parameter [Formula: see text] are studied on the flow variables. Due to the consideration of a rotating medium or due to the presence of magnetic field, the total energy of the disturbance increases, while with an increase in adiabatic exponent γ the total energy of the disturbance decreases. Density and pressure vanish near the axis of symmetry, thus forming a vacuum there.

Shock wave propagation in long laboratory sparks under negative switching impulses

A series of 1.27 m air gap discharge tests under negative switching impulses were carried out to characterize shock wave expansion behavior along leader channels. A high-speed Schlieren system was used to identify shock wave propagation characteristics near the high-voltage electrode. According to the Schlieren images, three distinct types of shock waves were recognized during the leader propagation and after the air gap breakdown: (1) spherical shock wave (SSW), (2) cylindrical shock wave (CSW), and (3) bow shock wave (BSW). To the best of our knowledge, it is the first time to recognize SSWs at the point where the space leader meets the main channel and BSWs along the leader branch after the main leader bridges the air gap. According to the proposed velocity calculation method, the propagation velocity of SSW and CSW was calculated as a function of the shock radius, as well as the shock wave head of BSW. Moreover, a transient magnetohydrodynamics model was developed to predict shock wave front propagation characteristics of both straight and bending channels after breakdown. The predicted shock wave propagation velocity and shock radius showed good agreement with Schlieren images.

Blast waves propagation in magnetogasdynamics: power series method

Abstract Blast waves are produced when there is a sudden deposition of a substantial amount of energy into a confined region. It is an area of pressure moving supersonically outward from the source of the explosion. Immediately after the blast, the fore-end of the blast wave is headed by the shock waves, propagating in the outward direction. As the considered problem is highly nonlinear, to find out its solution is a tough task. However, few techniques are available in literature that may give us an approximate analytic solution. Here, the blast wave problem in magnetogasdynamics involving cylindrical shock waves of moderate strength is considered, and approximate analytic solutions with the help of the power series method (or Sakurai’s approach [1]) are found. The magnetic field is supposed to be directed orthogonally to the motion of the gas particles in an ideal medium with infinite electrical conductivity. The density is assumed to be uniform in the undisturbed medium. Using power series method, we obtain approximate analytic solutions in the form of a power series in ( a 0 / U ) 2 ${\left({a}_{0}/U\right)}^{2}$ , where a 0 and U are the velocities of sound in an undisturbed medium and shock front, respectively. We construct solutions for the first-order approximation in closed form. Numerical computations have been performed to determine the flow-field in an ideal magnetogasdynamics. The numerical results obtained in the absence of magnetic field recover the existing results in the literature. Also, these results are found to be in good agreement with those obtained by the Runge–Kutta method of fourth-order. Further, the flow variables are illustrated through figures behind the shock front under the effect of the magnetic field. The interesting fact about the present work is that the solutions to the problem are obtained in the closed form.

Similarity solutions for the strong shock waves in magnetogasdynamics with the effect of monochromatic radiation

In this paper, the method of invariance of Lie group is used to find the similarity solutions of the cylindrical shock waves for one-dimensional unsteady flow in magnetogasdynamics in the presence of monochromatic radiation. The density is assumed to be non-uniform ahead of the shock. The radiation flux is assumed to move through the gas. It is also shown that how the radiation parameter, ambient density exponent, ratio of specific heats (adiabatic constant) and the initial magnetic field strength affect the profiles of the flow variables. It is found that the shock wave decays in the presence of strong magnetic field and shows reverse behavior in the case of monochromatic radiation, i.e., the shock wave grows faster in the presence of radiation. To obtain the similarity solutions and the profiles of the flow variables, numerical calculations are performed using the software package MATHEMATICA.

Cylindrical shock wave in a self-gravitating perfect gas with azimuthal magnetic field via Lie group invariance method

In this paper, we have studied the propagation of cylindrical shock waves in a self-gravitating perfect gas under the influence of azimuthal magnetic field. The method of Lie group invariance is used to construct some special class of self-similar solutions in the presence of the azimuthal magnetic field. The different cases of solutions with a power law and exponential law shock paths are obtained with the choice of arbitrary constants appearing in the expressions for the infinitesimal generators. The similarity solution for cylindrical shock wave with power law shock path is discussed in detail. The effects of variation of Alfven-Mach number, gravitation parameter, initial density variation index and adiabatic exponent on the flow variables are analyzed graphically. It is obtained that the increase in the values of Alfven-Mach number, gravitation parameter and adiabatic exponent have decaying effect on the shock strength. Also, the shock strength increases with an increase in the values of initial density variation index. A comparison is also made between the solutions in gravitating and non-gravitating cases in the presence of magnetic field.

Shock-wave propagation in supercritical CO2 induced by nanosecond-pulsed arc plasma

Shock waves generated by arc plasma in supercritical (SC) CO2 have the potential to create novel reaction fields. However, there have been few studies of shock-wave characteristics in SC-CO2. This study provides the results of visualization of shock-wave propagation in SC-CO2 generated by nanosecond-pulsed arc plasma. A propagating cylindrical shock wave originating from a discharge channel was observed using time-resolved shadowgraph imaging. The shock wave separated from the cylindrical dark-region induced by pulsed arc plasma within 64 nanoseconds. The Mach numbers of the shock waves were investigated against the medium density of the CO2 ranging from a high-pressure gas phase to the SC phase. The Mach number reached a local maximum at the critical CO2 density level. The anomaly in the Mach number can be explained by the local maximum of the specific heat capacity ratio of the CO2 at close to the critical condition of SC-CO2.

Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating ideal gas using Lie Group theoretic method: Isothermal flow

The propagation of cylindrical shock wave under the influence of axial magnetic field in rotating medium under isothermal flow condition is investigated. The density, magnetic field and azimuthal and axial components of fluid velocity are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for infinitesimals of the Local Lie group of transformations bring about three different cases of solutions, i.e. with power law shock path, exponential law shock path and a particular case of power law shock path. Numerical solutions are obtained in the cases of power law and exponential law shock paths. Distribution of gasdynamical quantities are discussed through figures. The effects of variation in values of Alfven-Mach number [Formula: see text], ambient azimuthal velocity index [Formula: see text] and ambient density index [Formula: see text] are studied on flow variables and on shock strength. The numerical integration is done using software Mathematica. It is obtained that magnetic field has a decaying effect on shock strength. Also, increase in value of ambient density or ambient azimuthal velocity variation index in the case of power law shock path and increase in value of ambient density variation index in case of exponential law shock path have the decaying effect on shock strength.

Shock Wave Driven Out by a Piston in a Mixture of a Non-ideal Gas and Small Solid Particles Under the Influence of Azimuthal or Axial Magnetic Field

One-dimensional self-similar unsteady flows behind a spherical or cylindrical shock wave driven out by a piston moving with time according to a power law in a dusty gas is investigated. The medium is assumed to be the mixture of small solid particles of micro-size and a non-ideal gas. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by the inner surface (piston). It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both cases, when the flow between the shock and the piston is isothermal or adiabatic. The shock waves in non-ideal dusty gas can be important for the description of shocks in supernova explosions, in the study of a flare produced shock in the solar wind, the central part of starburst galaxies, nuclear explosion, rupture of the pressurized vessel, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or re-entry vehicles, etc. The findings of the present work provided a clear picture of whether and how the non-idealness of the gas and the presence of the magnetic field affect the propagation of shock and the flow behind it. It is interesting to note that in the presence of azimuthal magnetic field, the pressure and density vanish at the piston and hence a vacuum is formed at the center of symmetry for both the isothermal and adiabatic flows, which is in excellent agreement with the laboratory condition to produce the shock wave.

Spherical and cylindrical shocks in a non-ideal dusty gas with magnetic field under the action of heat conduction and radiation heat flux

In this article, the propagation of spherical or cylindrical shock waves in a mixture of small solid particles of microsize and a non-ideal gas with conductive as well as radiative heat fluxes are studied under the influence of an azimuthal or axial magnetic field. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by a piston. It is assumed that the equilibrium flow conditions are maintained and the moving piston continuously supplies the variable energy input. The density of the undisturbed medium is assumed to be constant in order to obtain the self-similar solutions. Heat conduction is expressed in terms of Fourier’s law, and the radiation is considered to be of diffusion type for an optically thick gray gas model. The thermal conductivity and the absorption coefficient are assumed to vary with temperature and density. Numerical calculations have been performed to obtain the flow profiles of variables. The effects of different values of the non-idealness parameter, the strength of the magnetic field, the mass concentration, the ratio of the density of solid particles to the initial density of the gas, the piston velocity index, and the adiabatic index are shown in detail. It is interesting to note that in the presence of an azimuthal magnetic field, the pressure and density vanish at the piston, and hence, a vacuum is formed at the center of symmetry, which is in excellent agreement with the laboratory condition to produce the shock wave. The author of the article agrees to the retraction of the article effective AUGUST 20, 2021.

One-dimensional cylindrical shock waves in non-ideal gas under magnetic field

In the present paper, we analyze the evolutionary behavior of imploding strong shock waves propagating through a non-ideal gas in the presence of axial magnetic field. An evolution equation has been constructed by using the method based on the kinematics of one-dimensional motion of shock waves. The values of similarity exponents have been calculated by using the first order truncation approximation which describes the decay behavior of strong shocks. The approximate values of the similarity exponents are compared with the similarity exponents calculated by the CCW approximation, the exact similarity solution and perturbation technique.

Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter $$\alpha $$ related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when $$\alpha $$ is small. When $$\alpha $$ is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

Similarity solutions using Lie group theoretic method for cylindrical shock wave in self-gravitating perfect gas with axial magnetic field: isothermal flow

Propagation of cylindrical shock wave in a self-gravitating perfect gas under the influence of axial magnetic field using Lie group of transformation method is investigated. The flow is considered to be isothermal. Density and magnetic field are assumed to be varying in the undisturbed medium. Two different cases of solutions are brought out by the arbitrary constants appearing in the expressions of infinitesimals of local Lie group of transformations. One is with a power law shock path and the other one is with an exponential law shock path. Numerical solutions are obtained for both the cases of power law and exponential law shock paths. The effects of variation in Alfven-Mach number, gravitational parameter and ambient density variation index for power law shock path and effects of variation in Alfven-Mach number, gravitational parameter and ambient magnetic field variation index on the flow variables in the case of exponential law shock path are studied. Also the effects of increase in value of gravitational parameter and in the strength of ambient magnetic field on the shock strength are investigated. The increase in value of Alfven-Mach number leads to the increase in the density ratio which infers to the decrease in shock strength.

Propagation of shock wave in a rotational axisymmetric ideal gas with density varying exponentially and azimuthal magnetic field: isothermal flow

In the present paper, the non-similarity solution for unsteady isothermal flow behind the cylindrical shock wave in a rotational axisymmetric perfect gas in the presence of azimuthal magnetic field is investigated. The ambient medium is assumed to have axial, azimuthal and radial components of fluid velocity. Solutions are obtained for MHD shock in a rotating medium with the vorticity vector and its components in one-dimensional flow case. The numerical solutions are obtained using Mathematica software and Runge–Kutta method of the fourth order. The Alfven Mach number, time and adiabatic exponent effects are worked out in detail. It is obtained that in the presence of magnetic field at the piston (inner expanding surface), the pressure and density vanish and hence a vacuum is formed at the line of symmetry, which is an excellent conformity with conditions to produce the shock wave in laboratory. Also, without magnetic field, the shock strength increases with an increase in time, whereas time has reverse affects on the shock strength in the presence of magnetic field. Our solutions are valid for arbitrary values of time. A comparison is also made between the behavior of non-rotating and rotating medium solutions in the presence or absence of magnetic field.

Isothermal Motion of Hydromagnetic Cylindrical Shock Waves in Self Gravitating Rotating Gas

In this paper the Chisnell2 Chester1 Whitham3 (CCW1-3) method has been used to study the isothermal propagation of cylindrical hydromagnetic diverging shock waves in self gravitating rotating gas in presence of constant axial magnetic field. Assuming density decreasing atmosphere and exploring magnetic field as weak and strong, the analytical expressions have been derived for both weak and strong shocks and finally the effects of overtaking disturbances behind the flow on the motion of shock have also been included

Approximate analytical solution for shock wave in rotational axisymmetric perfect gas with azimuthal magnetic field: Isothermal flow

The propagation of cylindrical shock wave in rotational axisymmetric perfect gas under isothermal flow condition with azimuthal magnetic field is investigated. Distributions of gas dynamical quantities are discussed. The density, magnetic pressure, azimuthal fluid velocity and axial fluid velocity are assumed to be varying according to power law with distance from the axis of symmetry in the undisturbed medium. Approximate analytical solutions are obtained by expanding flow variables in power series. Zeroth-order and first-order approximations are discussed with the aid of power series method. Solutions for zeroth-order approximation are constructed in approximate analytical form. The effect of flow parameters namely: shock Cowling number $$C_{0}$$, ambient density variation index q and adiabatic exponent $$\gamma $$ are studied on the flow variables. Consideration of magnetic pressure increases the total energy of disturbance of zeroth order while with increase in ambient density variation index or adiabatic exponent, the total energy of disturbance decreases.

Similarity Solutions of Strong Shock Waves for Isothermal Flow in an Ideal Gas

Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

Dynamic enlargement of a hole in a sheet: Crater formation and propagation of cylindrical shock waves

Predicting the shape of a crater formed by high velocity impact is of interest in several fields. It can aid in design of more efficient protective structures, in forensic analysis of bullet holes, and in understanding the effects of meteorite impact in both space systems and in extreme geological events. In this paper we present, for the first time, a complete theoretical solution of the dynamic plane-stress problem. We consider the steady-state expansion of a cylindrical hole in a strain hardening elastoplastic sheet and find that a self-similar field emerges if the ‘specific cavitation energy’ is constant. It is shown that at the quasistatic limit this solution reduces to available classical solutions, while at high expansion velocities shock waves can appear. Investigation of the constitutive sensitivities of the expansion field is conducted and compared with available results for the spherical field which is commonly applied to predict resistance to high velocity penetration. It is shown that shock waves appear at significantly lower expansion velocities, in the plane-stress deformation pattern, for which material compressibility is found to have a negligible effect. This insensitivity can be taken advantage of in the future for design of light weight protective layers by incorporating porosity.

Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie’s symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie’s group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer’s rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley’s rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles.