Abstract
Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations in the porous medium, which is of high interest in the design of granular shock isolators.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Ahmadi, M.M., Mohammadi, S., Nemati Hayati, Ali: Analytical solution for shock wave propagation in granular materials. In: 13th Annual and 2nd International Fluid Dynamics Conference, FD-2010-26-28, Shiraz, Iran (2010)
Ahmadi M.M., Mohammadi S., Nemati Hayati Ali: Analytical derivation of tortuosity and permeability of mono-sized spheres: a volume averaging approach. Phys. Rev. E 83, 026312 (2011)
Bear J., Bachmat Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer, Dordrecht (1990)
Ben-Dor G., Levy A., Sorek S.: Numerical investigation of the propagation of shock waves in rigid porous materials: solution of the Riemann problem. Int. J. Numer. Methods Heat Fluid Flow 7(8), 801–813 (1997a)
Ben-Dor G., Britan A., Elperin T., Igra O., Jiang J.P.: Experimental investigation of the interaction between weak shock waves and granular layers. Exp. Fluids 22, 432–443 (1997b)
Guinot V.: Wave Propagation in Fluids: Models and Numerical Techniques. 1st edn. Wiley, New York (2008)
Hoffmann, K. A., Chiang, S.T.: Computational Fluid Dynamics, 4th edn. Engineering Education System (2000). ISBN–10:0962373117, ISBN–13:978–0962373114
Juanes R., Patzek T.W.: Analytical solution to the Riemann problem of threephase flow in porous media. Transp. Porous Media 55(1), 47–70 (2004)
Juanes R.: Determination of the wave structure of the three-phase flow Riemann problem. Transp. Porous Media 60, 135–139 (2005)
Krylov A., Sorek S., Levy A., Ben-Dor G.: Simple waves in saturated porous media. I. The isothermal case. Jpn. Soc. Mech. Eng. Int. J. B 39(2), 294–298 (1996)
Laney C.B.: Computational Gasdynamics. 1st edn. Cambridge University Press, Cambridge (1998)
Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, Philadelphia (1973)
Levy A., Ben-Dor G., Sorek S., Bear J.: Jump conditions across strong compaction waves in gas saturated rigid porous media. Shock Waves 3(2), 105–111 (1993a)
Levy A., Ben-Dor G., Skews B.W., Sorek S.: Head-on collision of normal shock waves with rigid porous materials. Exp. Fluids 15, 183–190 (1993b)
Levy A., Sorek S., Ben-Dor G., Bear J.: Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature. Transp. Porous Media 21, 241–268 (1995a)
Levy A., Sorek S., Ben-Dor G., Skews B.: Waves propagation in saturated rigid porous media: analytical model and comparison with experimental results. Fluid Dyn. Res. 17, 49–65 (1995b)
Li H., Levy A., Ben-Dor G.: Analytical prediction of regular reflection over rigid porous surfaces in pseudo-steady flows. J. Fluid Mech. 282, 219–232 (1995)
Löhner R.: Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods. 2nd edn. Wiley, New York (2008)
Rogg B., Hermann D., Adomeit G.: Shock-induced flow in regular arrays of cylinders and packed beds. Int J. Heat Mass Transf. 28, 2285–2297 (1985)
Sod G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)
Sonntag R.E., Borgnakke C., Van Wylen G.J.: Fundamentals of Thermodynamics. 6th edn. Wiley, New York (2003)
Sorek S., Krylov A., Levy A., Ben-Dor G.: Simple waves in saturated porous media. II. The nonisothermal case. Jpn. Soc. Mech. Eng. Int. J. B 39, 299–304 (1996)
Toro E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. 3rd edn. Springer, Berlin (2009)
van der Grinten J.G.M, van Dongen M.E.H., van der Kogel H.: A shock-tube technique for studying pore-pressure propagation in dry and water-saturated porous medium. J. Appl. Phys. 58, 2937–2942 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hayati, A.N., Ahmadi, M.M. & Mohammadi, S. Analytical Solution for Isothermal Flow in a Shock Tube Containing Rigid Granular Material. Transp Porous Med 93, 13–27 (2012). https://doi.org/10.1007/s11242-012-9940-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-012-9940-0