Abstract
In this paper we present numerical solutions for thermal boundary layers that are developed during forced convection in a porous medium located above a flat plate. The basic feature of such layers is that they are nonsimilar. In our study we consider thermal nonequilibrium between the two phases. Accordingly, each phase is endowed with its own energy equation. The boundary-layer equations are solved with the local nonsimilarity method. We examine convection of air and liquid water, while the solid matrix is supposed to be made of cast iron. According to our computations, there are significant differences between the temperature distributions of the two phases, especially at short and moderate distances from the edge of the flat plate. Also, due to the high conductivity of the solid matrix, the thermal boundary layers are much thicker than the hydrodynamic one. The profile of the local Nusselt number is quite sensitive on the Prandtl number and only far downstream it scales with the square root of the distance. Finally, the validity of the local thermal equilibrium assumption is assessed via a comparative study. According to it, this assumption leads to significant inaccuracies in the temperature profiles but yields reasonable estimates for the thickness of the thermal boundary layer of the fluid.
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References
Nield DA, Bejan A (2017) Convection in porous media, 5th edn. Springer, New York
Straughan B (2008) Stability and wave motion in porous media. Springer, New York
Vafai C (ed) (2015) Handbook of porous media. CRC Press, Boca Raton, FL
Das MK, Mukherjee PP, Muralidhar K (2018) Modeling transport phenomena in porous media with applications. Springer, Berlin
Vafai K, Tien CL (1981) Boundary and inertial effects on flow and heat transfer in porous media. Int J Heat Mass Transf 24:195–203
Vafai K, Thiyagaraja R (1987) Analysis of flow and heat transfer at the interface region of a porous medium. Int J Heat Mass Transf 2304:1391–1405
Beckermann C, Viskanta R (1987) Forced convection boundary layer flow and heat transfer along a flat plate embedded in a porous medium. Int J Heat Mass Transf 30:1547–1551
Nakayama A, Kokudai T, Koyama H (1990) Non-Darcian boundary layer flow and forced convective heat transfer over a flat plate in a fluid-saturated porous medium. J Heat Transf 112:157–162
Sparrow EM, Quack H, Boerner CJ (1970) Local nonsimilarity boundary-layer solutions. AIAA J 8:1936–1942
Hossain MA, Banu N, Nakayama A (1994) Non-Darcy forced convection boundary layer flow over a wedge embedded in a saturated porous medium. Numer Heat Transf A Appl 26:399–414
Celli M, Rees DAS, Barletta A (2010) The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media. Int J Heat Mass Transf 53:3533–3539
Gogate SP, Bharati MC, Misbah NE, Kudenatti RB (2023) Forced convection boundary layer flow and heat transfer along a flat plate embedded in a porous medium. Arch Appl Mech 93:551–569
Qawasmeh BR, Alrbrai M, Al-Dahidi S (2019) Forced convection heat transfer of Casson fluid in non-Darcy porous media. Int Commun Heat Mass 11:1–10
Farooq U, Ijaz MA, Khan MI, Isa SSPM, Lu DC (2020) Modeling and non-similar analysis for Darcy-Forchheimer-Brinkman model of Casson fluid in a porous media. Int Commun Heat Mass 119:104955
Farooq U, Hussain M, Ijaz MA, Khan WA, Farooq FB (2021) Impact of non-similar modeling on Darcy-Forchheimer-Brinkman model for forced convection of Casson nano-fluid in non-Darcy porous media. Int Commun Heat Mass 125:105312
Papalexandris MV (2023) Boundary-layer flow in a porous domain above a flat plate. J Eng Math 140:4
Sparrow EM, Yu HS (1971) Local nonsimilarity thermal boundary-layer solutions. ASME J Heat Transf 93:328–334
Papalexandris MV, Antoniadis PD (2015) A thermo-mechanical model for flows in superposed porous and fluid layers with interphasial heat and mass exchange. Int J Heat Mass Transf 88:42–54
Antoniadis PD, Papalexandris MV (2015) Dynamics of shear layers at the interface of a highly porous medium and a pure fluid. Phys Fluids 27:014104
Varsakelis C, Papalexandris MV (2017) On the well-posedness of the Darcy-Brinkman-Forchheimer equations for coupled porous media-clear fluid flow. Nonlinearity 30:1449–1464
Papalexandris MV (2021) Attenuation of gaseous detonations by porous media of fine microstructure. Combust Flame 232:111518
Hamtiaux V, Papalexandris MV (2023) Turbulent thermal convection in mixed porous-pure fluid domains. J Fluid Mech 961:27
Cowin SC (2013) Continuum mechanics of anisotropic materials. Springer, New York
Tanino Y, Nepf HM (2008) Laboratory investigation of mean drag in a random array of rigid, emergent cylinders. J Hydraul Eng 134:534–41
Sonnenwald F, Stovin VR, Guymer I (2019) Estimating drag coefficient for arrays of rigid cylinders representing emergent vegetation. J Hydraul Res 57:591–597
Tinoco RO, Cowen EA (2013) The direct and indirect measurement of boundary stress and drag on individual and complex arrays of elements. Exp Fluids 54:1–16
Incropera FP, DeWitt DP, Bergman TL, Lavine AS (2007) Fundamentals of heat and mass transfer, 6th edn. John Wiley, Hoboken
Görtler H (1957) A new series for the calculation of steady laminar boundary layers. J Math Mech 6:1–66
Schlichting H, Gersten K (2017) Boundary-layer theory, 9th edn. Springer, Berlin
Firnett PJ, Troesch BA (1974) Shooting-splitting method for sensitive two-point boundary value problems. In: Bettis DG (ed) Proceedings of the conference on the numerical solution of ordinary differential equations, pp 408–433. Springer, Berlin
Lebon DG, Jou D, Casas-Vázquez J (2008) Understanding non-equilibrium thermodynamics. Springer, Berlin
De Groot SR, Mazur P (2011) Non-equilibrium thermodynamics. Dover Publications Inc., New York
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Papalexandris, M.V. Thermal boundary-layer solutions for forced convection in a porous domain above a flat plate. J Eng Math 144, 3 (2024). https://doi.org/10.1007/s10665-023-10311-5
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DOI: https://doi.org/10.1007/s10665-023-10311-5
Keywords
- Darcy-Forchheimer law
- Local nonsimilarity method
- Nonsimilar boundary layers
- Porous media
- Thermal nonequilibrium