Abstract
The Stokes–Brinkman coupling has been employed to investigate the flow through porous media composed of packed spheres. By matching the slip velocity using the Navier-slip condition, optimal values of the effective viscosity in the continuous stress condition and of the stress jump coefficient in the stress jump condition could be accurately determined. The correlations between the slip length (which has been accurately determined in Part 1) and the effective viscosity as well as the stress jump coefficient have been specified. The accuracy of these two optimal parameters (i.e., the effective viscosity and stress jump coefficient) has been assessed by comparing the velocity profiles in both the fluid and porous regions obtained from the Stokes–Brinkman coupling, with those obtained from the corresponding direct simulations. It is observed that the stress jump condition with the optimal stress jump coefficient yields a superior prediction in the velocity field. The optimal effective viscosity decreases as the solid volume fraction increases, whereas the stress jump coefficient increases with the solid volume fraction. By selecting the optimal parameters, the Stokes–Brinkman coupling with both the continuous stress condition and stress jump condition is applied to solve two example flow problems: a stick–slip–stick flow and a pressure-driven flow in a rectangular channel. Both the stress conditions in the Stokes–Brinkman coupling exhibit good performances in reproducing the velocity fields within the entire domain.
Similar content being viewed by others
References
Angot, P.: On the well-posed coupling between free fluid and porous viscous flows. Appl. Math. Lett. 24(6), 803–810 (2011)
Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid. Mech. 30(1), 197–207 (1967)
Breugem, W.: The effective viscosity of a channel-type porous medium. Phys. Fluids 19(10), 103104 (2007)
Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Flow Turbul. Combust. 1(1), 27 (1949)
Chandesris, M., Jamet, D.: Boundary conditions at a fluid–porous interface: an a priori estimation of the stress jump coefficients. Int. J. Heat Mass Transf. 50(17–18), 3422–3436 (2007)
Chellam, S., Wiesner, M.R.: Slip flow through porous media with permeable boundaries: implications for the dimensional scaling of packed beds. Water Environ. Res. 65(6), 744–749 (1993)
Esence, T., Bruch, A., Fourmigué, J., Stutz, B.: A versatile one-dimensional numerical model for packed-bed heat storage systems. Renew. Energy 133, 190–204 (2019)
Gasser, S., Paun, F., Bréchet, Y.: Absorptive properties of rigid porous media: application to face centered cubic sphere packing. J. Acoust. Soc. Am. 117(4), 2090–2099 (2005)
Goyeau, B., Lhuillier, D., Gobin, D., Velarde, M.G.: Momentum transport at a fluid–porous interface. Int. J. Heat Mass Trans. 46(21), 4071–4081 (2003)
Hwang, W.R., Advani, S.G.: Numerical simulations of Stokes–Brinkmanequations for permeability prediction of dual scale fibrous porous media. Phys. Fluids 22(11), 113101 (2010)
Hwang, W.R., Advani, S.G., Walsh, S.: Direct simulations of particle deposition and filtration in dual-scale porous media. Compos. Part A Appl. Sci. Manuf. 42(10), 1344–1352 (2011)
Jang, H.K., Kim, Y.J., Woo, N.S., Hwang, W.R.: Tensorial navier-slip boundary conditions for patterned surfaces for fluid mixing: numerical simulations and experiments. AIChE J. 62(12), 4574–4585 (2016)
Llorca, J., Martinez, J.L., Elices, M.: Reinforcement fracture and tensile ductility in sphere-reinforced metal-matrix composites. Fatigue Fract. Eng. Mater. Struct. 20(5), 689–702 (1997)
Lu, J., Jang, H.K., Lee, S.B., Hwang, W.R.: Characterization on the anisotropic slip for flows over unidirectional fibrous porous media for advanced composites manufacturing. Compos. Part A Appl. Sci. Manuf. 100, 9–19 (2017)
Lu, X., Xie, P., Ingham, D.B., Ma, L., Pourkashanian, M.: A porous media model for CFD simulations of gas–liquid two-phase flow in rotating packed beds. Chem. Eng. Sci. 189, 123–134 (2018)
Martys, N., Bentz, D.P., Garboczi, E.J.: Computer simulation study of the effective viscosity in Brinkman’s equation. Phys. Fluids 6(4), 1434–1439 (1994)
Min, J.Y., Kim, S.J.: A novel methodology for thermal analysis of a composite system consisting of a porous medium and an adjacent fluid layer. J. Heat Transf. 127(6), 648–656 (2005)
Neale, G., Nader, W.: Practical significance of Brinkman's extension of Darcy's law: coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Eng. 52(4), 475–478 (1974)
Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid—I. Theoretical development. Int. J. Heat Mass Transf. 38(14), 2635–2646 (1995a)
Ochoa-Tapia, J.A., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid—II. Comparison with experiment. Int. J. Heat Mass Transf. 38(14), 2647–2655 (1995b)
Saleh, S., Thovert, J.F., Adler, P.M.: Flow along porous media by partical image velocimetry. AIChE J. 39(11), 1765–1776 (1993)
Segurado, J., Llorca, J.: A numerical approximation to the elastic properties of sphere-reinforced composites. J. Mech. Phys. Solids 50(10), 2107–2121 (2002)
Starov, V.M., Zhdanov, V.G.: Effective viscosity and permeability of porous media. Colloids Surf. Physicochem. Eng. Asp. 192(1–3), 363–375 (2001)
Takhirov, A.: Stokes–Brinkmanlagrange multiplier/fictitious domain method for flows in pebble bed geometries. SIAM J. Numer. Anal. 51(5), 2874–2886 (2013)
Tamayol, A., Khosla, A., Gray, B.L., Bahrami, M.: Creeping flow through ordered arrays of micro-cylinders embedded in a rectangular minichannel. Int. J. Heat Mass Transf. 55(15–16), 3900–3908 (2012)
Tamayol, A., Yeom, J., Akbari, M., Bahrami, M.: Low Reynolds number flows across ordered arrays of micro-cylinders embedded in a rectangular micro/minichannel. Int. J. Heat Mass Transf. 58(1–2), 420–426 (2013)
Valdes-Parada, F.J., Ochoa-Tapia, J.A., Alvarez-Ramirez, J.: On the effective viscosity for the Darcy–Brinkman equation. Phys. A Stat. Mech. Appl. 385(1), 69–79 (2007)
Valdés-Parada, F.J., Alvarez-Ramírez, J., Goyeau, B., Ochoa-Tapia, J.A.: Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation. Transp. Porous Media 78(3), 439–457 (2009)
Varahasamy, M., Fand, R.M.: Heat transfer by forced convection in pipes packed with porous media whose matrices are composed of spheres. Int. J. Heat Mass Transf. 39(18), 3931–3947 (1996)
Vignesadler, M., Adler, P.M., Gougat, P.: Transport processes along fractals-The Cantor-Taylor brush. Physicochem. Hydrodyn. 8(4), 401–422 (1987)
Acknowledgements
The authors acknowledge financial supports from the National Research Foundation of Korea (NRF-2019R1A2C1003974) and from the Korea Agency for Infrastructure Technology Advancement grant funded by Ministry of Land, Infrastructure and Transport (17IFIP-B133614-01, The Industrial Strategic Technology Development Program).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lu, J.G., Hwang, W.R. On the Interfacial Flow Over Porous Media Composed of Packed Spheres: Part 2-Optimal Stokes–Brinkman Coupling with Effective Navier-Slip Approach. Transp Porous Med 132, 405–421 (2020). https://doi.org/10.1007/s11242-020-01398-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-020-01398-w