Abstract
Macroscopic balance equations of mass, momentum and energy for compressible Newtonian fluids within a thermoelastic solid matrix are developed as the theoretical basis for wave motion in multiphase deformable porous media. This leads to the rigorous development of the extended Forchheimer terms accounting for the momentum exchange between the phases through the solid-fluid interfaces. An additional relation presenting the deviation (assumed of a lower order of magnitude) from the macroscopic momentum balance equation, is also presented. Nondimensional investigation of the phases' macroscopic balance equations, yield four evolution periods associated with different dominant balance equations which are obtained following an abrupt change in fluid's pressure and temperature. During the second evolution period, the inertial terms are dominant. As a result the momentum balance equations reduce to nonlinear wave equations. Various analytical solutions of these equations are described for the 1-D case. Comparison with literature and verification with shock tube experiments, serve as validation of the developed theory and the computer code.
A 1-D TVD-based numerical study of shock wave propagation in saturated porous media, is presented. A parametric investigation using the developed computer code is also given.
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Attenborough, K.: Acoustical characteristics of porous materials, Phys. Reports 82(3) (1982), 179-227.
Bachmat, Y. and Bear, J.: Macroscopic modeling of transport phenomena in porous media: 1. The continuum approach, Transport in Porous Media 1 (1986), 213-240.
Baer, M. R.: Numerical studies of dynamic compaction of inert and energetic granular materials, ASME J. Appl. Mech. 55 (1988), 36-43.
Baer, M. R.: A numerical study of shock wave reflections on low density foam, Shock Waves 2 (1992), 121-124.
Baer, M. R. and Nunziato, J. W.: Atwo-phase mixture theory for the deflagration to detonation transition (DDT) in reactive granular materials, Int. J. Multiphase Flow 12 (1986), 861-889.
Bear, J. and Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.
Bear, J. and Sorek, S.: Evolution of governing mass and momentum balance equations following an abrupt pressure impact in a porous medium, Transport in Porous Media 5 (1990), 169-185.
Bear, J., Sorek, S., Ben-Dor,G. and Mazor, G.: Displacementwaves in saturated thermoelastic porous media: I. Basic equations, Fluid Dynam. Res. 9 (1992), 155-164.
Ben-Dor, G.: Shock Wave Reflection Phenomena, Springer-Verlag, New York, 1991.
Ben-Dor, G., Mazor, G., Igra, O., Sorek, S. and Onodera, H.: Shock wave interaction with cellular materials: II. Open cell foams; Experimental and numerical results, ShockWaves 3(3) (1994), 167-179.
Ben-Dor, G. and Zaretsky, E. B.: Head-on interaction of planar shockwaves with polyurethane foams-A semi-empirical model, Archivem Appl. Mech. 64 (1994), 1-8.
Biot, M. A.: Theory of propagation of elastic waves in fluid-saturated porous solid, J. Acoust. Soc. Amer. 28 (1956), 168-191.
Corapcioglu, M. Y.: Wave propagation in porous media-a review, In: J. Bear and M. Y. Corapcioglu (eds), Transport Processes in Porous Media, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991, pp. 373-469.
Degrande, G. and de Roeck, G.: FFT-based spectral analysis methodology for one-dimensional wave propagation in poroelastic media, Transport in Porous Media 9 (1992), 85-97.
Du Plessis, J. P. and Masliyan, J. H.: Mathematical modeling of flow through consolidated isotropic porous media, Transport in Porous Media 3 (1988), 145-164.
Du Plessis, J. P., Montillet, A., Comiti, J. and Legrand, J.: Pressure drop prediction for flow through high porosity metallic foams, Chem. Engg. Sci. 49(21) (1994), 3545-3553.
Gelfand, B. E., Gubanov, A. V. and Timofeev, E. I.: Interaction between shock waves and porous screen, Sov. Phys.-Fluid Mech. 4 (1983), 79-84.
Gibson, L. J. and Ashby M. F.: Cellular Solids-Structure & Properties, Program Press, Headington Hill Hall, Oxford, England, 1988.
Glass, I. I. and Hall, G. J.: Shock Tubes, Handbook of Supersonic Aerodynamics, Sec. 18, Navond Report 1488, Vol. 6, 1959, p. 66.
van der Grinten, J. G. M., Smits, M. A., van der Kogel, H. and van Dongen, M. E. H.: Shock-induced wave propagation in and reflection from porous column partially saturated with water, In: H. Gronig (ed.), Shock Tubes and Waves, VCH Verlagsgesellschaft mbH, Weinheim, Germany, 1988, pp. 357-362.
Gvozdeva, L. G., Faresov, Yu. M. and Fokeev, V. P.: Interaction between air shock wave and porous compressible material, Sov. Phys. Appl. Math. Tech. Phys. 3 (1985), 111-115.
Gvozdeva, L. G. and Faresov, Yu. M.: Approximate calculation of steady state shock wave parameters in porous compressible materials, J. Appl. Mech. Tech. Phys. (English translation of PMTF, Zh. Prik. Mekh. Tekh. Fiz.), 27 (1986), 107-111.
Harten, A.: High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), 357-393.
Henderson, L. F., Virgona, R. J., Di, J. and Gvozdeva, L. G.: Refraction of a normal shock wave from nitrogen into polyurethane foam, In: Y. W. Kim (ed.), Current Topics in Shock Waves, American Institute of Physics, New York, U.S.A., 1990, pp. 814-818.
Hsu, C. T. and Cheng, P.: Thermal dispersion in a porous medium, Int. J. Heat Mass Transfer 33(8) (1990), 1587-1597.
Igra, O. and Ben-Dor, G.: Dusty shock waves, ASME Appl. Mech. Rev. 41 (1988), 379-437.
Krylov, A., Sorek, S., Levy, A. and Ben-Dor, G.: Simple waves in saturated porous media: I. The isothermal case, JSME Int. J. Series B 39(2) (1996), 294-298.
Landau, L. D. and Lifshitz, E. M.: Fluid Mechanics, 2nd edn, Pergamon Press, 1987, pp. 378-385.
Levy, A.: Wave Propagation in a Saturated Porous Media, PhD Thesis, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel, 1996, (in Hebrew).
Levy, A., Ben-Dor, G., Skews, B. W. and Sorek, S.: Head-on collision of normal shock waves with rigid porous materials, Experiments in Fluids 15 (1993), 183-190.
Levy, A., Ben-Dor, G., Sorek, S. and Bear, J.: Jump conditions across strong compaction waves in gas saturated rigid porous media, Shock Waves 3(2) (1993), 105-111.
Levy, A., Sorek, S., Ben-Dor, G. and Bear, J.: Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature, Transport in Porous Media 21 (1995), 241-268.
Levy, A., Sorek, S., Ben-Dor, G. and Skews B. W.: Waves propagation in saturated rigid porous media: analytical model and comparison with experimental results, Fluid Dynam. Res. 17 (1996), 49-65.
Levy A., Ben-Dor G., and Sorek, S.: Numerical investigation of the propagation of shock waves in rigid materials: development of the computer code and comparison with experimental results, J. Fluid Mech. 324 (1996), 163-179.
Li, H. and Ben-Dor, G.: Head-on interaction of weak planar shock waves with flexible porous materials-analytical model, Int. J. Multiphase Flow 21(5) (1995), 941-947.
Li, H., Levy, A. and Ben-Dor, G.: Analytical prediction of regular reflection over porous surfaces and comparison with experiments, J. Fluid Mech. 282 (1995), 219-232.
Liepmann, H. W. and Roshko, A.: Elements of Gas Dynamics, Wiley, New York, U.S.A., 1957.
Mazor, G., Ben-Dor, G., Igra, O. and Sorek, S.: Shockwave interaction with cellular materials: Part I. Analytical investigation and governing equations, Shock Waves 3 (1994), 159-165.
Nield, D. A.: The limitations of the Brinkman-Forchheimer equation in modelling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow 12 (1991), 269-272.
Nield, D. A.: Modelling high speed flow of compressible fluid in a saturated porous medium, Transport in Porous Media 14 (1994), 85-88.
Nigmatulin, R. I. and Gubaidullin, A. A.: Linear waves in saturated porous media, Transport in Porous Media 9 (1992), 135-142.
Nikolaevskij, V. M.: Mechanics of Porous and Fractured Media, World Scientific, Singapore, 1990.
Olim, M., van Dongen, M. E. W., Kitamura, T. and Takayama, K.: Numerical simulation of the propagation of shock waves in compressible open-cell porous foams, Int. J. Multiphase Flow 20(3) (1994), 557-568.
Powers, J. M., Stewart, D. S. and Krier, H.: Analysis of steady compaction waves in porous materials, ASME J. Appl. Mech. 56 (1989), 15-24.
Sandusky, H. W. and Liddiard, T. P.: Dynamic Compaction of Porous Beds, NSWCTR 83-256, NAVAL Surface Weapons Center, White Oak, Md., U.S.A., 1985.
Skews, B. W.: The reflected pressure field in the interaction of weak shock waves with a compressible porous foams, Shock Waves 1 (1991), 205-211.
Skews, B. W., Atkins, M. D. and Seitz, M. W.: The impact of shock wave on porous compressible foams, J. Fluid Mech. 253 (1993), 245-265.
Smeulders, D. M. J., de La Rosette, S. P. M. and Van Dongen, M. E. H.: Waves in partially saturated porous media, Transport in Porous Media 9 (1992), 25-37.
Sorek, S., Bear, J., Ben-Dor, G. and Mazor, G.: Shock waves in saturated thermoelastic porous media, Transport in Porous Media 9 (1992), 3-13.
Sorek, S., and Levi-Hevroni, D.: A model for solute transport following an abrupt pressure impact in nondeformable porous media, 2nd Inter. Conf. on Water Pollution Milan 1993, pp. 77-84.
Sorek, S., Krylov, A., Levy, A. and Ben-Dor, G.: Simple waves in saturated porous media: II. The nonisothermal case, JSME Int. J. Series B 39(2) (1996), 299-304.
Sorek, S.: A model for solute transport following an abrupt pressure impact in saturated porous media, Transport in Porous Media 22 (1996), 271-285.
Zaretsky, E. B. and Igra, O.: Head-on collision of a normal shock wave with a polyester foam, The 19th Int. Symp. Shock Waves, Marseille, France, 1993.
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Sorek, S., Levy, A., Ben-dor, G. et al. Contributions to Theoretical/Experimental Developments in Shock Waves Propagation in Porous Media. Transport in Porous Media 34, 63–100 (1999). https://doi.org/10.1023/A:1006553206369
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DOI: https://doi.org/10.1023/A:1006553206369