Convergence of strong shock waves in an ideal gas with dust particles (original) (raw)
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Propagation of strong shock waves in a non-ideal gas
Acta Astronautica, 2019
We studied the problem of converging cylindrical and spherical strong shock waves collapsing at the axis/center of symmetry for a non-ideal gas with constant density. We have applied the perturbation series technique which provides us a global solution to the implosion shock wave problem yielding the results of Guderley's local self-similar solution, which is valid only in the vicinity of the axis/center of implosion. We analyzed the flow parameters by expanding the solution in powers of time and found the similarity exponents as well as the corresponding amplitudes in the vicinity of the shock-collapse. The flow parameters and the shock trajectory have been drawn in the region extending from the piston to the center of collapse for different values of adiabatic coefficient and the non-ideal parameter.
Strong shock wave propagation in a mixture of a gas and dusty particles with gravitational force
Zamm Zeitschrift Fur Angewandte Mathematik Und Mechanik, 1981
The presence of gravitational force in the considered problem makes it generally impossible to obtain similarity solutions. However, when the shock wave is strong and the shock speed has a large value, the gravitational force effect will be small because the Froude number will be very large. In this case, it is possible to use perturbation theory based on the similarity solution reported by Pai et al. (1980) to find the first-order effect of gravitational force on the strong shock propagation in a mixture of a gas and dusty particles. Except for the inclusion of the gravitational force terms, the present investigation is similar to the study conducted by Pai et al. (1980). It is found that the consideration of the gravitational force leads to some fundamental changes in the pressure, density, and velocity histories behind the propagating strong shock.
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.
Converging shock wave in a dusty gas through nonstandard analysis
Ain Shams Engineering Journal, 2012
A problem of propagation of strong plane and converging shock wave is studied in a mixture of a gas and small solid particles. It is assumed that the solid particles are continuously distributed in the gas. Jump conditions for plane and converging shock waves are derived using the nonstandard analysis. It is also assumed that the shock thickness occurs at infinitesimal interval and jump functions are smooth across this interval. The distribution of flow parameters across the shock wave are presented in terms of Heaviside functions and Dirac Delta measures.
Meccanica, 2014
A self-similar solution for the flow behind a strong shock wave propagating in a mixture of a nonideal gas and small solid particles in which the density remains constant and radiation flux is important, has been obtained. The solid particles are considered as a pseudo-fluid and it is assumed that the equilibrium flow condition is maintained. The radiative flux is calculated from the conservation equations without applying any restriction on optical properties of the medium. The effects of the non-idealness of gas b, the mass concentration of solid particles k p and the ratio of density of solid particles to the initial density of gas G 1 on the shock and on the flow-field behind it are investigated. It is shown that the effects of the nonidealness of the gas on the shock strength and on the flow-profiles in the flow-field behind the shock are reduced by the presence of solid particles in the gas.
Converging shock waves in a Van der Waals gas of variable density
The Quarterly Journal of Mechanics and Applied Mathematics, 2020
Summary The converging problem of cylindrically or spherically symmetric strong shock wave collapsing at the axis/centre of symmetry, is studied in a non-ideal inhomogeneous gaseous medium. Here, we assume that the undisturbed medium is spatially variable and the density of a gas is decreasing towards the axis/centre according to a power law. In the present work, we have used the perturbation technique to the implosion problem and obtained a global solution that also admits Guderley’s asymptotic solution in a very good agreement which holds only in the vicinity of the axis/centre of implosion. The similarity exponents together with their corresponding amplitudes are determined by expanding the flow parameters in powers of time. We also refined the leading similarity exponents near the axis/centre of convergence. We compared our calculated results with the already existing results and found them in good agreements up to two decimal places. Shock position and flow parameters are analy...
Self-similar solutions and converging shocks in a non-ideal gas with dust particles
International Journal of Non-Linear Mechanics
In this paper, we have used the Lie group of transformations and obtained the whole range of self-similar solutions to the problem of propagation of shock waves through a non-ideal, dusty gas. The conditions essential for the existence of similarity solutions for a strong shock are discussed. The problem of imploding (converging) shock wave is also worked out and the effects of the mass concentration of the dust particles, ratio of the density of solid particles to that of initial density of the medium, the relative specific heat and the effect of the non-ideal parameter, on the shock formation has been studied in detail.
A group theoretic method is used to obtain an entire class of similarity solutions to the problem of shocks propagating through a non-ideal gas and to characterize analytically the state dependent form of the medium ahead for which the problem is invariant and admits similarity solutions. Different cases of possible solutions, known in the literature, with a power law, exponential or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular case of collapse of imploding cylindrically and spherically symmetric shock in a medium in which initial density obeys power law is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables behind the shock, and comparison is made with the known results.
Fluid Dynamics Research, 1990
The governing equations describing the propagation of a moderate planar normal shock wave into a homogeneous dust-gas suspension were formulated and solved numerically using the flux-corrected transport (FCT) technique. The numerically predicted attenuation of the shock wave was compared with the experimental results of Sommerfeld. Good agreement was obtained. It was found that the attenuation of moderate normal planar shock waves propagating into dusty gases with relatively high loading ratios of solid particles can be described by a general attenuation law.