Strong Spherical Shock Research Articles

Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents

Abstract In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust particle to the density of the mixture and van der Waals excluded volume.

An Investigation of Effective Parameters in Deuterium-Tritium Fuel Gains in Shock Ignition Method Driven by Laser Beams

Shock ignition (SI) is one of the methods considered in the concept of inertial confinement fusion (ICF). This two-step ICF process separates fuel assembly and ignition, relaxing the driver requirements and promising high gains. In SI scheme, a strong spherical shock wave converging at the end of the initial laser pulse ignites the pre-compressed fuel. In shock ignition, when the hot spot pressure is much higher than the surrounding cold fuel pressure, the fuel structure is considered non-isobaric. In this research, ignition conditions and fuel efficiency in shock ignition method were investigated. Then, the fuel efficiency correlations of total fuel energy, fuel gain and hot-spot radius in a non-isobaric model of fuel assembly were improved and compared with the numerical results of deuterium-tritium (DT) homogeneous fuel in SI scenario. Calculations showed that the hot spot formation conditions depend on the hot spot density and the surrounding cold fuel. Furthermore, using the improved energy efficiency equation, the role of physical parameters such as fuel mass and different ratios of hot spot pressure to the surrounding cold fuel pressure in energy efficiency were investigated.

Isothermal Strong Spherical Shock Waves in Uniform Medium

Isothermal Strong Spherical Shock Waves in Uniform Medium

Influence of Heat Transfer on Decreasing Intensity of a Spherical Explosion in Aqueous Foam

The two-phase model of aqueous foam behavior under strong spherical shock wave (SW) impact, is developed using equations of mixture conservation of momentum, mass and internal energy for each phase in Lagrange variables, taking into account bulk viscosity and interphase heat transfer. The numerical implementation of the model was carried out by the counting method, using the Neu-mann-Richtmyer viscosity and the Courant stability condition. The spherical explosion was modeled in the form of a SW, which has the same energy of charge of explosives, as used in experiments. A satisfactory agreement was obtained between the numerical solution, received by the proposed model, the analytical self-similar L.I. Sedov’s solution on a point spherical explosion in a gas and a new experimental data on the spherical explosion in aqueous foam. The causes, which lead to a significant decrease in amplitude and SW velocity in the studied media, are investigated in detail.

Self-Similar Motion of Strong Converging Cylindrical and Spherical Shock Waves in Non-Ideal Stellar Medium

A theoretical model for strong converging cylindrical and spherical shock waves in non-ideal gas characterized by the equation of state (EOS) of the Mie-Gruneisen type is investigated. The governing equations of unsteady one dimensional compressible flow including monochromatic radiation in Eulerian hydrodynamics are considered. These equations are reduced to a system of ordinary differential equations (ODEs) using similarity transformations. Shock is assumed to be strong and propagating into a medium according to a power law. In the present work, two different equations of state (EOS) of Mie-Gruneisen type have been considered and the cylindrical and spherical cases are worked out in detail. The complete set of governing equations is formulated as finite difference problem and solved numerically using MATLAB. The numerical technique applied in this paper provides a global solution to the problem for the flow variables, the similarity exponent α for different Gruneisen parameters. It is observed that increase in measure of shock strength β(ρ/ρ_0 ) has effect on the shock front. The velocity and pressure behind the shock front increases quickly in the presence of the monochromatic radiation and decreases gradually. A comparison between the results obtained for non-ideal and perfect gas in the presence of monochromatic radiation has been illustrated graphically.

On the Strong Spherical Shock Waves in a Two-Phase Gas–Particle Medium

In this paper, power series solutions for strong spherical shocks of time dependent variable energy propagating in a two-phase gas-particle medium are presented taking into consideration the power series solution technique (Sakurai in J Phys Soc Jpn 8:662-669,1953; Freeman in J Phys D Appl Phys 2(1):1697-1710,1968). Assuming the medium to be a mixture of a perfect gas and small solid particles, the power series solutions are obtained in terms of M^-2, where M is the upstream Mach number of shock. This investigation presents an overview of the effects due to an increase in (i) the propagation distance from the inner expanding surface and, (ii) the dust loading parameters on flow-field variables such as the velocity of fluid, the pressure, the density, and also on the speed of sound, the adiabatic compressibility of mixture and the change-in-entropy behind the strong spherical shock front.

Propagation of a strong spherical shock wave in a gravitating or non-gravitating dusty gas with exponentially varying density

The propagation of a strong spherical shock wave in a dusty gas with or without self-gravitational effects is investigated in the case of isothermal and adiabatic flows. The dusty gas is assumed to be a mixture of small solid particles and perfect gas. The equilibrium flow conditions are assumed to be maintained, and the density of the mixture is assumed to be varying and obeying an exponential law. Non-similarity solutions are obtained and the effects of variations of the mass concentration of solid particles in the mixture and the ratio of the density of solid particles to the initial density of the gas, and the presence of self-gravitational field on the flow variables are investigated at given times. Our analysis reveals that after inclusion of gravitational field effects surprisingly the shock strength increases and remarkable differences are found in the distribution of flow variables. An increase in time also, increases the shock strength. Further, it is investigated that the consideration of isothermal flow increases the shock strength, and removes the singularity in the density distribution. Also, the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and therefore the distance between the inner contact surface and the shock surface is reduced. The shock waves in self-gravitating dusty gas can be important for description of shocks in supernova explosions, in the study of central part of star burst galaxies, star formation and shocks in stellar explosion, nuclear explosion, in industry, rupture of a pressurized vessel and explosion in the ionosphere. Other potential applications of this study include analysis of data from exploding wire experiments and cylindrically symmetric hypersonic flow problems associated with meteors or re-entry of vehicles etc. A comparison is made between the solutions in the cases of the gravitating and the non-gravitating media. The obtained solutions are applicable for arbitrary values of time.

Law of convergence of strong cylindrical and spherical shock waves in a gas with a uniform density

A model of convergence of cylindrical and spherical shock waves in a gas with a uniform density has been considered. The partial differential equations of this model have been reduced to ordinary differential equations, from which the laws of convergence of such shock waves and the dependence α = f(γ, M) of their self-similarity index α on the heat capacity ratio γ of the gas and their Mach number M have been found. This dependence for cylindrical shock waves in monatomic (γ = 1.67) and diatomic (γ = 1.4) gases has been shown to agree with the experimental data within the measurement error.

Shock wave similarity invariants. I. Convergence laws of spherical and cylindrical shock waves in a constant-density gas

Using the similarity and dimensional methods, dimensionless similarity invariants of shock wave similarity are determined, using which convergence laws of strong spherical and cylindrical shock waves in a constant-density gas are determined.

Shock dynamics of strong imploding cylindrical and spherical shock waves with non-ideal gas effects

In this paper, the generalized analytical solution for one dimensional adiabatic flow behind the strong imploding shock waves propagating in a non-ideal gas is obtained by using Whitham’s geometrical shock dynamics theory. Landau and Lifshitz’s equation of state for non-ideal gas and Anand’s generalized shock jump relations are taken into consideration to explore the effects due to an increase in (i) the propagation distance from the centre of convergence, (ii) the non-idealness parameter and, (iii) the adiabatic index, on the shock velocity, pressure, density, particle velocity, sound speed, adiabatic compressibility and the change in entropy across the shock front. The findings provided a clear picture of whether and how the non-idealness parameter and the adiabatic index affect the flow field behind the strong imploding shock front.

Spherical shock wave generated by a moving piston in mixture of a non-ideal gas and small solid particles under a gravitational field

Self-similar solutions are obtained for one-dimensional isothermal and adiabatic unsteady flows behind a strong spherical shock wave propagating in a dusty gas. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of a non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The medium is under the influence of the gravitational field due to a heavy nucleus at the origin (Roche model). The effects of an increase in the mass concentration of solid particles, the ratio of the density of the solid particles to the initial density of the gas, the gravitational parameter and the parameter of non-idealness of the gas in the mixture, are investigated. It is shown that due to presence of gravitational field the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow-variables and the shock strength increase. A comparison has also been made between the isothermal and adiabatic flows. It is investigated that the singularity in the density and compressibility distributions near the piston in the case of adiabatic flow are removed when the flow is isothermal.

Shock dynamics of strong imploding cylindrical and spherical shock waves with real gas effects

Strong cylindrical and spherical shock implosion in a monatomic gas is considered. A simple solution is obtained by Whitham’s geometrical shock dynamics approach modified to account for the real gas effects. The real gas effects are introduced by jump relations over the shock and include several levels of ionization, Coulomb interaction as well as internal energy of the excited electrons. It is shown that ionization has a major effect on temperature and density behind the converging shock as well as on the shock acceleration. The temperature and acceleration being substantially reduced and density substantially increased as compared to the ideal nonionizing case. The ionization effect on the pressure behind the converging shock is less pronounced. It is also shown that for the considered test case of initial Mach number M0=8 the gas becomes completely ionized behind the spherical shock at approximately 1% of the initial radius from the focal point and its speed being decreased by a factor of 1.8 as compared to the ideal case.

Convergence of Strong Shock in a Van der Waals Gas

Strong cylindrical and spherical shock waves, collapsing at the center (or axis) of symmetry, are studied for a Van der Waals gas. The perturbation technique applied in this paper provides a global solution to the implosion problem, yielding the results for Guderley’s local self‐similar solution, which is valid only in the vicinity of the center/axis of implosion. The similarity exponents are found along with the corresponding amplitudes in the vicinity of the shock‐collapse. The flow parameters and the shock trajectory have been computed for different values of the adiabatic coefficient and the Van der Waals excluded volume.

Self-similar imploding relativistic shock waves

Self-similar solutions to the problem of a strong imploding relativistic shock wave are calculated. These solutions represent the relativistic generalization of the Newtonian Gouderley-Landau-Stanyukovich problem of a strong imploding spherical shock wave converging to a center. The solutions are found assuming that the preshocked flow has a uniform density and are accurate for sufficiently large times after the formation of the shock wave.

IMPLODING CYLINDRICAL AND SPHERICAL SHOCK WAVES IN A NON-IDEAL MEDIUM

Converging shock waves in an almost ideal medium are considered. The kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest. The approximate value of the similarity parameter obtained from there is compared with those derived from Whitham's Rule and the exact similarity solution at the instant of collapse of the shock wave. The above computation is carried out for different values of the parameter α, which depends on the internal volume of the gas molecules.

Propagation of Strong Spherical Shock Waves in a Dusty Gas

In this paper, we have examined one-dimensional unsteady self-similar adiabatic flow of a dusty gas behind a spherical shock wave with time dependent energy input. The dusty gas is assumed to be a mixture of small solid particles and perfect gas. The solid particles are continuously distributed in the perfect gas. In order to get some essential features of shock propagation, the solid particles are considered as a pseudo-fluid and it is assumed that the equilibrium flow condition is maintained in the flow-field, and viscous stress and heat conduction of the mixture are negligible. The presence of small solid particles affects the medium in two ways. On the one hand, the volume fraction of solid particles lowers the compressibility of the mixture; and on the other hand, the particle load increases the inertia of the mixture. It is found that if the density of the solid particles and that of the perfect gas are equal, by an increase in the mass fraction of solid particles in the mixture, the strength of the shock is decreases and its distance from the inner expanding surface is increased. Also, by an increase in the ratio of density of solid particles and that of the perfect gas in the mixture, the strength of the shock is increased and its distance from the inner surface is decreased. Further, an increase in the rate of energy input increases the shock velocity and decreases the distance between the shock front and the inner surface.

Self-similar spherical shock solution with sustained energy injection

We present the generalization of the Sedov-Taylor self-similar strong spherical shock solution for the case of a central energy source varying in time, $E=A t^k$, where $A$ and $k$ are constants. The known Sedov-Taylor solution corresponds to a particular adiabatic case of $k=0$ or \emph{instant shock} with an instant energy source of the shock, $E=A$. The self-similar hydrodynamic flow in the nonadiabatic $k\neq0$ case exists only under the appropriate local entropy (energy) input which must be supported by some radiative mechanism from the central engine. The specific case of $k=1$ corresponds to a permanent energy injection into the shock, or injection shock with a central source of constant luminosity, $L=A$, $E=A t$. The generalized self-similar shock solution may be applied to astrophysical objects in which the duration of central source activity is longer than the shock expansion time, e.g. the early phase of SN explosions, strong wind from stars and young pulsars, non-steady spherical outflow from black holes and collapsing dense stellar clusters with numerous neutron star collisions.

Structure and stability of a spherical shock wave in a Van der Waals gas

Strong spherical shock waves are studied for an imperfect gas, here modelled by a van der Waals equation of state. Similarity solutions of Guderley type are shown to exist, in which the radius of the shock is proportional to (-t) α , where t is time measured from the moment at which the shock focuses. The exponent a depends on both the ratio of specific heats, γ, and on the van der Waals excluded volume, b. For small b, the solution resembles the Guderley solution and is well described by the Chester-Chisnell-Whitham (CCW) approximation. A new branch of solutions, which the CCW approximation fails to locate, is shown to exist for larger b. The linear stability of the similarity solutions is examined directly, without the use of the CCW approximation. Normal modes grow (Re (β) 0) as (-t) αβ , where the 'growth rate', β, is a function of γ, b and n, the spherical harmonic wavenumber. No physically meaningful, discrete, spherically symmetric (n = 0) modes were found. This case was examined numerically and nonlinearly for shocks launched by a spherical 'piston'; no evidence of instability was discovered.

Strong spherical shock wave propagation in a self-gravitating RMHD rotating interplanetary medium

In the present paper, we have discussed the propagation of spherical shock waves in a radiating magnetohydrodynamic rotating interplanetary medium. The effects of force of self-gravitation have been taken into account with the assumption that at the equilibrium position the effects of rotation on the force of gravitation are negligible. The combined effects of rotation and gravitation on the variation of flow variables have been shown in tables. The particular cases of the problem have been discussed and compared by considering effects of rotation and gravitation separately.

Numerical investigation of three-dimensional shock focusing effects by an invariant difference scheme

Symmetry preserving difference schemes which are found on the basis of the Lie-group theory for partial differential equations applied to their first differential approximations are used to simulate the evolution and reflection of shock waves in a cubical cavity with rigid walls and walls with outflow boundary conditions. The explicit difference scheme was used for a shock capturing technique to find and trace the discontinuities in the flow field. The numerical model is based on the strong conservative form of the three-dimensional time-dependent Euler equations with an equation of state for an ideal gas. The three-dimensional time-dependent shock waves which are created by a centered gas volume under high pressure interact with the plane walls of the cavity and lead to focusing effects after the explosion. To follow the propagation of strong plane or spherical shock waves the shock capturing method was proved to be sufficient concerning the spatial resolution of the discontinuities. When interacting with the rigid walls of the cavity the shock waves are assumed to be reflected specularly and to interact with shock waves and contact discontinuities moving from the center to the edges and corners of the cavity. The symmetry preserving character of the invariant difference scheme under use was proved numerically by calculations over long time. No significant deviation of the symmetry character of the imposed initial distribution was found. Additionally, a numerical analysis of the conservation of energy in the cube was performed to find the behavior of the energy in time.