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Coupled Prediction of Flows in Domains Containing a Porous Medium and a Free Stream

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Mathematical Models and Computer Simulations Aims and scope

Abstract

This study considers a model for coupled predictions of flows in domains containing a fluid-saturated porous medium and a free stream. The model is based on the generalized Navier–Stokes equations for an incompressible fluid, obtained by averaging over a representative elementary volume of the porous medium and written for the entire computational domain consisting of two subdomains differing in composition. To implement this model numerically, we develop a computational algorithm based on the finite element method and Newton’s method for solving nonlinear equations. It is implemented using the open computational platform FEniCS. The developed numerical technique is verified on the known numerical results of other authors. In addition, the model predictions of flows in an automobile catalytic converter are performed and discussed.

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RЕFERENCES

  1. A. Bejan, I. Dincer, S. Lorente, A.F. Miguel, and A.H. Reis, Porous and Complex Flow Structures in Modern Technologies (Springer, New York, 2004). https://doi.org/10.1007/978-1-4757-4221-3

  2. Handbook of Porous Media, Ed. by K. Vafai, 3rd ed. (CRC Press, Boca Raton, FL, 2015).

    MATH  Google Scholar 

  3. M. K. Das, P. P. Mukherjee, and K. Muralidhar, Modeling Transport Phenomena in Porous Media with Applications (Springer, Cham, 2018). https://doi.org/10.1007/978-3-319-69866-3

  4. R. Helmig, B. Flemisch, M. Wolff, and and B. Faigle, “Efficient modeling of flow and transport in porous media using multi-physics and multi-scale approaches,” in Handbook of Geomathematics, 2nd ed., Ed. by W. Freeden, M. Z. Nashed, and T. Sonar (Springer, Berlin, 2015), pp.703−749. https://doi.org/10.1007/978-3-642-54551-1_15

  5. H. P. G. Darcy, Les Fontaines Publiques de la Ville de Dijon (Victor Dalmont, Paris, 1856).

    Google Scholar 

  6. H. C. Brinkman, “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles,” Appl. Sci. Res., Ser. A 1 (1), 27−34 (1947). https://doi.org/10.1007/BF02120313

    Article  Google Scholar 

  7. P. Forchheimer, “Wasserbewegung durch Boden,” Z. Ver. Deutsch. Ing. 45 (50), 1782−1788 (1901).

    Google Scholar 

  8. S. Whitaker, The Method of Volume Averaging (Springer, Dordrecht, 1999), https://doi.org/10.1007/978-94-017-3389-2

  9. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed. (Springer, New York, 2012). https://doi.org/10.1007/978-1-4612-4254-3

  10. D. A. Nield and A. Bejan, Convection in Porous Media, 5th ed. (Springer, Cham, 2017). https://doi.org/10.1007/978-3-319-49562-0

  11. M. Nazari, Y. Mahmoudi, and K. Hooman, “Introduction to fluid flow and heat transfer in porous media,” in Convective Heat Transfer in Porous Media, Ed. by Y. Mahmoudi, K. Hooman, and K. Vafai (CRC Press, Boca Raton, FL, 2020), pp. 3−18. https://doi.org/10.1201/9780429020261-1

  12. B. Goyeau, D. Lhuillier, D. Gobin, and M. G. Velarde, “Momentum transport at a fluid−porous interface,” Int. J. Heat Mass Transfer 46 (21), 4071−4081 (2003). https://doi.org/10.1016/S0017-9310(03)00241-2

    Article  MATH  Google Scholar 

  13. D. Gobin and B. Goyeau, “Natural convection in partially porous media: a brief overview,” Int. J. Numer. Methods Heat Fluid Flow 18 (3/4), 465−490 (2008). https://doi.org/10.1108/09615530810853592

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Ehrhardt, J. Fuhrmann, E. Holzbecher, and A. Linke, “Mathematical modeling of channel-porous layer interfaces in PEM fuel cells,” WIAS Preprint No. 1375 (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2008).

    Google Scholar 

  15. M. Discacciati, “Coupling free and porous-media flows: Models and numerical approximation,” in Simulation of Flow in Porous Media: Applications in Energy and Environment, Ed. by P. Bastian, J. Kraus, R. Scheichl, and M. Wheeler (De Gruyter, Berlin, 2013), pp. 107−138. https://doi.org/10.1515/9783110282245.107

  16. P. Angot, B. Goyeau, and J. A. Ochoa-Tapia, “Asymptotic modeling of transport phenomena at the interface between a fluid and a porous layer: Jump conditions,” Phys. Rev. E 95 (6), 063302 (2017). https://doi.org/10.1103/PhysRevE.95.063302

    Article  MathSciNet  Google Scholar 

  17. E. Eggenweiler and I. Rybak, “Unsuitability of the Beavers−Joseph interface condition for filtration problems,” J. Fluid Mech. 892, A10 (2020). https://doi.org/10.1017/jfm.2020.194

    Article  MathSciNet  MATH  Google Scholar 

  18. I. Rybak, C. Schwarzmeier, E. Eggenweiler, and U. Rüde, “Validation and calibration of coupled porous-medium and free-flow problems using pore-scale resolved models,” Comput. Geosci. 25 (2), 621−635 (2021). https://doi.org/10.1007/s10596-020-09994-x

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Alazmi and K. Vafai, “Analysis of variants within the porous media transport models,” ASME J. Heat Transfer 122 (2), 303−326 (2000). https://doi.org/10.1115/1.521468

    Article  Google Scholar 

  20. C. T. Hsu and P. Cheng, “Thermal dispersion in a porous medium,” Int. J. Heat Mass Transfer, 33 (8), 1587−1597 (1990). https://doi.org/10.1016/0017-9310(90)90015-M

    Article  MATH  Google Scholar 

  21. M. J. S. de Lemos, Turbulence in Porous Media: Modeling and Applications, 2nd ed. (Elsevier, London, 2012).

    Google Scholar 

  22. FEniCS Project. https://fenicsproject.org/. Cited September 22, 2022.

  23. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Ed. by A. Logg, K.‑A. Mardal, and G. N. Wells (Springer, Berlin, 2012). https://doi.org/10.1007/978-3-642-23099-8

  24. R. Glowinski, “Finite element methods for incompressible viscous flow,” in Numerical Methods for Fluids (Part 3), Ed. by P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis, Vol. 9 (Elsevier, Amsterdam, 2003), pp. 3−1176. https://doi.org/10.1016/S1570-8659(03)09003-3

  25. V. John, Finite Element Methods for Incompressible Flow Problems (Springer, Cham, 2016). https://doi.org/10.1007/978-3-319-45750-5

  26. MUMPS: MUltifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr/. Cited September 22, 2022.

  27. Gmsh. https://gmsh.info/. Cited September 22, 2022.

  28. ParaView. https://www.paraview.org/. Cited September 22, 2022.

  29. D. K. Garthling, C. E. Hickox, and R. C. Givler, “Simulation of coupled viscous and porous flow problems,” Int. J. Comput. Fluid Dyn. 7 (1-2), 23−48 (1996). https://doi.org/10.1080/10618569608940751

    Article  MATH  Google Scholar 

  30. V. A. F. Costa, L. A. Oliveira, B. R. Baliga, and A. C. M. Sousa, “Simulation of coupled flows in adjacent porous and open domains using a control-volume finite-element method,” Numer. Heat Transfer, Part A: Appl. 45 (7), 675−697 (2004). https://doi.org/10.1080/10407780490424839

    Article  Google Scholar 

  31. L. Betchen, A. G. Straatman, and B. E. Thompson, “A nonequilibrium finite-volume model for conjugate fluid/porous/solid domains,” Numer. Heat Transfer, Part A: Appl. 49 (6), 543−565 (2006). https://doi.org/10.1080/10407780500430967

    Article  Google Scholar 

  32. M. Nordlund, M. Stanic, A. K. Kuczaj, E. M. A. Frederix, and B. J. Geurts, “Improved PISO algorithms for modeling density varying flow in conjugate fluid-porous domains,” J. Comput. Phys. 306, 199−215 (2016). https://doi.org/10.1016/j.jcp.2015.11.035

    Article  MathSciNet  MATH  Google Scholar 

  33. A. S. Kozelkov, S. V. Lashkin, V. R. Efremov, K. N. Volkov, and Yu. A. Tsibereva, “An implicit algorithm for solving Navier–Stokes equations to simulate flows in anisotropic porous media,” Comput. Fluids 160, 164−174 (2018). https://doi.org/10.1016/j.compfluid.2017.10.029

    Article  MathSciNet  MATH  Google Scholar 

  34. Z. Li, H. Zhang, Y. Liu, and J. M. McDonough, “Implementation of compressible porous–fluid coupling method in an aerodynamics and aeroacoustics code part I: Laminar flow,” Appl. Math. Comput. 364, 124682 (2020). https://doi.org/10.1016/j.amc.2019.124682

    Article  MathSciNet  MATH  Google Scholar 

  35. R. E. Hayes, A. Fadic, J. Mmbaga, and A. Najafi, “CFD modelling of the automotive catalytic converter,” Catal. Today 188 (1), 94−105 (2012). https://doi.org/10.1016//j.cattod.2012.03.015

    Article  Google Scholar 

  36. S. Porter, J. Saul, S. Aleksandrova, H. Medina, and S. Benjamin, “Hybrid flow modelling approach applied to automotive catalysts,” Appl. Math. Modell. 40 (19-20), 8435−8445 (2016). https://doi.org/10.1016/j.apm.2016.04.024

    Article  Google Scholar 

  37. H. A. Ibrahim, W. H. Ahmed, S. Abdou, and V. Blagojevic, “Experimental and numerical investigations of flow through catalytic converters,” Int. J. Heat Mass Transfer 127 (Part B), 546−560 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.052

  38. M. Hettel, E. Daymo, T. Schmidt, and O. Deutschmann, “CFD-Modeling of fluid domains with embedded monoliths with emphasis on automotive converters,” Chem. Eng. Process.: Process Intensif. 147, 107728 (2020). https://doi.org/10.1016/j.cep.2019.107728

    Article  Google Scholar 

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Funding

This study was funded by the Russian Foundation for Basic Research and the National Science Foundation of Bulgaria, project no. 20-51-18004.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    A. G. Churbanov, N. G. Churbanova & M. A. Trapeznikova

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  1. A. G. Churbanov
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  2. N. G. Churbanova
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  3. M. A. Trapeznikova
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Correspondence to A. G. Churbanov, N. G. Churbanova or M. A. Trapeznikova.

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Churbanov, A.G., Churbanova, N.G. & Trapeznikova, M.A. Coupled Prediction of Flows in Domains Containing a Porous Medium and a Free Stream. Math Models Comput Simul 15, 643–653 (2023). https://doi.org/10.1134/S207004822304004X

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