Converging shock wave in a dusty gas through nonstandard analysis (original) (raw)
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Shock jump relations for a dusty gas atmosphere
Astrophysics and Space Science, 2014
This paper presents generalized forms of jump relations for one dimensional shock waves propagating in a dusty gas. The dusty gas is assumed to be a mixture of a perfect gas and spherically small solid particles, in which solid particle are continuously distributed. The generalized jump relations reduce to the Rankine-Hugoniot conditions for shocks in an idea gas when the mass fraction (concentration) of solid particles in the mixture becomes zero. The jump relations for pressure, density, temperature, particle velocity, and change-in-entropy across the shock front are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the initial volume fraction of the solid particles. The computations have been performed for various values of mass concentration of the solid particles and for the ratio of density of solid particles to the constant initial density of gas. Tables and graphs of numerical results are presented and discussed.
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.
Convergence of strong shock waves in an ideal gas with dust particles
Physics of Fluids, 2022
In this paper, the authors study the problem of an imploding strong cylindrical/spherical shock wave collapsing at the axis/center of a cylindrical/spherical piston that is filled with a dusty gas of uniform density. The dusty gas is assumed to be a mixture of an ideal gas and a large number of dust particles. The dust particles are of a micrometric size and uniformly distributed in the mixture. A mathematical model using a system of hyperbolic partial differential equations is presented for the considered problem. The perturbation series method is used to solve the implosion problem, providing a global solution and yielding accurately the results of Guderley's local similarity solution, which holds only in the neighborhood of the axis/center of implosion. The values of all possible real similarity exponents and the corresponding amplitudes are determined in the vicinity of the shock collapse by extending the flow variables and shock location in the Taylor series in time t. Furthermore, the obtained values of similarity exponents have been compared with the existing results and numerical results obtained by the other methods. The effects of the adiabatic exponent c, the wavefront curvature a, and various dusty gas parameters such as r, K p , and G 0 on the shock trajectory and flow variables have been graphically analyzed.
Nonstandard analysis and shock wave jump conditions in a one-dimensional compressible gas
2007
Nonstandard analysis is a relatively new area of mathematics in which infinitesimal numbers can be defined and manipulated rigorously like real numbers. This report presents a fairly comprehensive tutorial on nonstandard analysis for physicists and engineers with many examples applicable to generalized functions. To demonstrate the power of the subject, the problem of shock wave jump conditions is studied for a one-dimensional compressible gas. It is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval. To use conservations laws, smooth pre-distributions of the Dirac delta measure are applied whose supports are contained within the shock thickness. Furthermore, smooth pre-distributions of the Heaviside function are applied which vary from zero to one across the shock wave. It is shown that if the equations of motion are expressed in nonconservative form then the relationships between the jump functions for the flow parameters may be found unambiguously. The analysis yields the classical Rankine-Hugoniot jump conditions for an inviscid shock wave. Moreover, non-monotonic entropy jump conditions are obtained for both inviscid and viscous flows. The report shows that products of generalized functions may be defined consistently using nonstandard analysis; however, physically meaningful products of generalized functions must be determined from the physics of the problem and not the mathematical form of the governing equations.
Interaction of a weak discontinuity wave with a characteristic shock in a dusty gas
Theoretical and Computational Fluid Dynamics, 2008
In this paper, the evolution of a characteristic shock in a dusty gas is investigated and its interaction with a weak discontinuity wave is studied. The transport equation for the amplitude of the weak discontinuity wave, which is of Bernoulli type, is obtained. The amplitudes of the reflected and transmitted waves after interaction of the weak discontinuity with the characteristic shock are evaluated by using the results of the general theory of wave interaction.
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.
The distribution of particles in a shock-induced boundary layer of a dusty gas over a solid surface
Acta Mechanica Sinica, 1991
The laminar boundary layer behind a constant-speed shock wave moving through a dusty gas along a solid surface is studied. The Saffman lift force acting on a spherical particle in a gas boundary layer is taken into account. A method for calculating the density profile of dispersed phase near the wall is proposed and some numerical results are given. It is shown that behind the shock wave, there exists a curved thin layer where the density of particles is many times higher than the original one. This dust collection effect may be of essential importance to the problem of dust explosion in industry.
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.
A parametric study of the head-on collision of normal shock waves in dusty gases
Fluid Dynamics Research, 1988
The effect of the various flow parameters, namely: the diameter of the solid particles, the material density of the solid particles, and the loading ratio of the solid particles on the flow field which is obtained when two normal shock waves collide head-on in a two phase dust-gas suspension has been investigated numerically, using the modified random choice method (ReM). The results were compared with those appropriate to the dust physical parameters used recently by Elperin, Ben-Dor and Igra in their study of the head-on collision of normal shock waves in dusty gases.
Head-on collision of normal shock waves in dusty gases
International Journal of Heat and Fluid Flow, 1987
The head-on collision of normal shock waves in dusty gases has been investigated numerically, using the modified random-choice method. The results concerning the various flow field properties as well as the waves configuration were compared with those of a pure gas case.