Hybrid integral transforms for flow development in ducts partially filled with porous media. (English) Zbl 1402.76127
Summary: A hybrid numerical-analytical solution is developed for laminar flow development in a parallel plate duct partially filled with porous media. The integral transform method is employed in combination with a single domain reformulation strategy for representing the heterogeneous media within the channel. A novel eigenfunction expansion basis is proposed, including abrupt spatial variations of physical properties due to the domain transitions. The introduction of the new basis allows for a solution with similar convergence rates as in previous applications with simpler formulations, as demonstrated through a careful convergence analysis of the expansions. The inherent automatic error control characteristic of the integral transforms approach then provides benchmark results for the developing velocity profile. Moreover, a physical analysis further verifies the consistency of both the proposed expansion and the mixed symbolic-numerical code developed. A detailed verification with a finite-element commercial code is also performed.
MSC:
| 76S05 | Flows in porous media; filtration; seepage |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76D08 | Lubrication theory |
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
Keywords:
integral transforms; single domain formulation; porous media; Navier-Stokes equations; channel flowReferences:
| [1] | Blessent, D.; Jørgensen, PR; Therrien, R., Comparing discrete fracture and continuum models to predict contaminant transport in fractured porous media, Groundwater, 52, 84-95, (2014) · doi:10.1111/gwat.12032 |
| [2] | Cotta, RM; Ungs, MJ; Mikhailov, MD, Contaminant transport in finite fractured porous medium: integral transforms and lumped-differential formulations, Ann. Nucl. Eng., 30, 261-285, (2003) · doi:10.1016/S0306-4549(02)00060-9 |
| [3] | Sandler, J.; Fowler, G.; Cheng, K.; Kovscek, AR, Solar-generated steam for oil recovery: reservoir simulation, economic analysis, and life cycle assessment, Energy Convers. Manag., 77, 721-732, (2014) · doi:10.1016/j.enconman.2013.10.026 |
| [4] | Marschewski, J.; Brenner, L.; Ebejer, N.; Ruch, P.; Michel, B.; Poulikakos, D., 3D-printed fluidic networks for high-power-density heat-managing miniaturized redox flow batteries, Energy Environ. Sci., 10, 780-787, (2017) · doi:10.1039/C6EE03192G |
| [5] | Lisboa, KM; Marschewski, J.; Ebejer, N.; Ruch, P.; Cotta, RM; Michel, B.; Poulikakos, D., Mass transport enhancement in redox flow batteries with corrugated fluidic networks, J. Power Sources., 359, 322-331, (2017) · doi:10.1016/j.jpowsour.2017.05.038 |
| [6] | Beavers, GS; Joseph, DD, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, 197-207, (1967) · doi:10.1017/S0022112067001375 |
| [7] | Nield, DA, Onset of convection in a fluid layer overlying a layer of a porous medium, J. Fluid Mech., 81, 513-522, (1977) · Zbl 0364.76032 · doi:10.1017/S0022112077002195 |
| [8] | Poulikakos, D.; Bejan, A.; Selimos, B.; Blake, KR, High Rayleigh number in a fluid overlying a porous bed, Int. J. Heat Fluid Flow., 7, 109-116, (1986) · doi:10.1016/0142-727X(86)90056-1 |
| [9] | Brinkman, HC, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, App. Sci. Res., A1, 27-34, (1949) · Zbl 0041.54204 · doi:10.1007/BF02120313 |
| [10] | Neale, G.; Nader, W., Practical significance of Brinkman extension of Darcy’s law: coupled parallel flows within a channel and a boundary porous medium, Can. J. Chem. Eng., 52, 472-478, (1974) · doi:10.1002/cjce.5450520407 |
| [11] | Hirata, SC; Goyeau, B.; Gobin, D.; Carr, M.; Cotta, RM, Linear stability of natural convection in superposed fluid and porous layers: influence of the interfacial modelling, Int. J. Heat Mass Transf., 50, 1356-1367, (2007) · Zbl 1118.80003 · doi:10.1016/j.ijheatmasstransfer.2006.09.038 |
| [12] | Hirata, SC; Goyeau, B.; Gobin, D.; Chandesris, M.; Jamet, D., Stability of natural convection in superposed fluid and porous layers: Equivalence of the one- and two-domain approaches, Int. J. Heat Mass Transf., 52, 533-536, (2009) · Zbl 1156.80330 · doi:10.1016/j.ijheatmasstransfer.2008.07.045 |
| [13] | Cotta, RM, Integral transforms in computational heat and fluid flow, (1993), CRC Press · Zbl 0974.35004 |
| [14] | Cotta, RM; Mikhailov, MD, Heat conduction: lumped analysis, integral transforms, symbolic computation, (1997), Wiley Interscience |
| [15] | Cotta, RM, The integral transform method in thermal, fluid sciences and engineering, (1998), Begell House · Zbl 0906.00025 |
| [16] | Cotta, RM; Knupp, DC; Naveira-Cotta, CP, Analytical heat and fluid flow in microchannels and microsystems, (2016), Springer |
| [17] | Celli, M.; Alves, LSDB; Barletta, A., Nonlinear stability analysis of Darcy’s flow with viscous heating, Proc. R. Soc. A, 472, 20160036, (2016) · Zbl 1371.76132 · doi:10.1098/rspa.2016.0036 |
| [18] | Carvalho, TMB; Cotta, RM; Mikhailov, MD, Flow development in entrance region of ducts, Commun. Numer. Methods Eng., 9, 503-509, (1993) · Zbl 0777.76067 · doi:10.1002/cnm.1640090608 |
| [19] | Machado, HA; Cotta, RM, Integral transform method for boundary layer equations in simultaneous heat and fluid flow problems, Int. J. Num. Meth. Heat Fluid Flow., 5, 225-237, (1995) · Zbl 0826.76067 · doi:10.1108/EUM0000000004118 |
| [20] | Bolivar, MAH; Cotta, RM; Lage, PLC, Integral transform solution of the laminar thermal boundary layer problem for flow past two-dimensional and axisymmetric bodies, Num. Heat Transf. Part A. Appl., 33, 779-797, (1998) · doi:10.1080/10407789808913966 |
| [21] | Lima, GGC; Santos, CAC; Haag, A.; Cotta, RM, Integral transform solution of internal flow problems based on Navier-Stokes equations and primitive variables formulation, Int. J. Num. Meth. Eng., 69, 544-561, (2007) · Zbl 1194.76224 · doi:10.1002/nme.1780 |
| [22] | Perez-Guerrero, JS; Cotta, RM, Integral transform solution for the lid-driven cavity flow problem in streamfunction-only formulation, Int. J. Num. Meth. Fluids., 15, 399-409, (1992) · Zbl 0753.76139 · doi:10.1002/fld.1650150403 |
| [23] | Perez-Guerrero, JS; Cotta, RM, Integral transform solution of developing laminar duct flow in Navier-Stokes formulation, Int. J. Numer. Meth. Fluids., 20, 1203-1213, (1995) · Zbl 0840.76075 · doi:10.1002/fld.1650201102 |
| [24] | Perez-Guerrero, JS; Cotta, RM, Benchmark integral transform results for flow over a backward-facing step, Comput. Fluids., 25, 527-540, (1996) · Zbl 0900.76427 · doi:10.1016/0045-7930(96)00005-9 |
| [25] | Lima, JA; Perez-Guerrero, JS; Cotta, RM, Hybrid solution of the averaged Navier-Stokes equations for turbulent flow, Comput. Mech., 19, 297-307, (1997) · Zbl 0892.76067 · doi:10.1007/s004660050178 |
| [26] | Pereira, LM; Perez-Guerrero, JS; Cotta, RM, Integral transformation of the Navier-Stokes equations in cylindrical geometry, Comput. Mech., 21, 60-70, (1998) · Zbl 0909.76018 · doi:10.1007/s004660050 |
| [27] | Leal, MA; Machado, HA; Cotta, RM, Integral transform solutions of transient natural convection in enclosures with variable fluid properties, Int. J. Heat Mass Transf., 43, 3977-3990, (2000) · Zbl 0982.76083 · doi:10.1016/S0017-9310(00)00023-5 |
| [28] | Perez-Guerrero, JS; Quaresma, JNN; Cotta, RM, Simulation of laminar flow inside ducts of irregular geometry using integral transforms, Comput. Mech., 25, 413-420, (2000) · Zbl 0966.76069 · doi:10.1007/s004660050488 |
| [29] | Ramos, R.; Perez-Guerrero, JS; Cotta, RM, Stratified flow over a backward facing step:- Hybrid solution by integral transforms, Int. J. Num. Meth. in Fluids., 35, 173-197, (2001) · Zbl 0990.76069 · doi:10.1002/1097-0363(20010130)35:2<173::AID-FLD88>3.0.CO;2-U |
| [30] | Silva, RL; Quaresma, JNN; Santos, CAC; Cotta, RM, Integral transforms solution for flow development in wavy wall ducts, Int. J. Numer. Meth. Heat Fluid Flow., 21, 219-243, (2011) · doi:10.1108/09615531111105416 |
| [31] | Matt, CFT; Quaresma, JNN; Cotta, RM, Analysis of magnetohydrodynamic natural convection in closed cavities through integral transforms, Int. J. Heat Mass Transf., 113, 502-513, (2017) · doi:10.1016/j.ijheatmasstransfer.2017.05.043 |
| [32] | Quaresma, JNN; Cotta, RM, Integral transform method for the Navier-Stokes equations in steady three-dimensional flow, 1, 281-287, (1997) |
| [33] | Knupp, DC; Naveira-Cotta, CP; Cotta, RM, Theoretical analysis of conjugated heat transfer with a single domain formulation and integral transforms, Int. Commun. Heat Mass. Transf., 39, 355-362, (2012) · doi:10.1016/j.icheatmasstransfer.2011.12.012 |
| [34] | Knupp, DC; Cotta, RM; Naveira-Cotta, CP, Heat transfer in microchannels with upstream-downstream regions coupling and wall conjugation effects, Num. Heat Transf. Part B. Fundam., 64, 365-387, (2013) · doi:10.1080/10407790.2013.810535 |
| [35] | Knupp, DC; Naveira-Cotta, CP; Cotta, RM, Conjugated convection-conduction analysis in microchannels with axial diffusion effects and a single domain dormulation, J. Heat Transfer., 135, 91401, (2013) · doi:10.1115/1.4024425 |
| [36] | Knupp, DC; Naveira-Cotta, CP; Cotta, RM, Theoretical-experimental analysis of conjugated heat transfer in nanocomposite heat spreaders with multiple microchannels, Int. J. Heat Mass Transf., 74, 306-318, (2014) · doi:10.1016/j.ijheatmasstransfer.2014.03.005 |
| [37] | Knupp, DC; Cotta, RM; Naveira-Cotta, CP; Kakaç, S., Transient conjugated heat transfer in microchannels: integral transforms with single domain formulation, Int. J. Therm. Sci., 88, 248-257, (2015) · doi:10.1016/j.ijthermalsci.2014.04.017 |
| [38] | Knupp, DC; Cotta, RM; Naveira-Cotta, CP, Fluid flow and conjugated heat transfer in arbitrarily shaped channels via single domain formulation and integral transforms, Int. J. Heat Mass Transf., 82, 479-489, (2015) · doi:10.1016/j.ijheatmasstransfer.2014.11.007 |
| [39] | Ochoa-Tapia, JA; Whitaker, S., Momentum transfer at the boundary between a porous medium and a homogeneous fluid - I. Theoretical development, Int. J. Heat Mass Transf., 38, 2635-2646, (1995) · Zbl 0923.76320 · doi:10.1016/0017-9310(94)00346-W |
| [40] | Pimentel, LCG; Pérez Guerrero, JS; Ulke, AG; Duda, FP; Heilbron Filho, PFL, Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions, Proc. R. Soc. A, 470, 20140021, (2014) · doi:10.1098/rspa.2014.0021 |
| [41] | Pereyra, V., PASVA3: An adaptive finite-difference FORTRAN program for first order nonlinear, ordinary boundary value problems, Lec Notes Comp. Sci., 76, 67-88, (1979) · Zbl 0434.65067 · doi:10.1007/3-540-09554-3_4 |
| [42] | COMSOL Multiphysics |
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