The study examines the flow and heat transfer of a Newtonian fluid in a porous medium inside an enclosure with two straight boundaries and one curved boundary. This setup is important for heat storage and energy systems. The aim of this study is to solve the Brinkman-Forchheimer (BF) equation in an enclosure with two straight and one curved boundary. The research also looks to perform a thorough heat transfer analysis to improve the understanding of thermal behaviour in porous medium BF flow. Additionally, the study calculates the Nusselt number using a compatibility condition to ensure the results are physically consistent. Finally, it fits the Nusselt number as a function of the shape factor (s) and the Forchheimer number (F). This helps in capturing the trends in convective heat transfer behaviour within the medium.The main assumptions include a steady, fully developed flow in the z-direction with a constant axial pressure gradient -<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{dp}{dz}\)</EquationSource> </InlineEquation>, and zero axial velocity (w = 0) on all boundaries. The domain in three-dimensions is defined in cartesian coordinates (x, y, z), with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x^2+y^2 \le 1\)</EquationSource> </InlineEquation> on the curved boundary,ensuring the spatial constraint of the geometry. The quasi-linearisation method is used to linearise the governing equations, resulting in a system of linear algebraic equations that is subsequently solved using the alternate direction implicit (ADI) method with an accuracy of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({10}^{-4}\)</EquationSource> </InlineEquation>. The findings show that an increase in the shape factor (s) results in a plug flow behaviour and better heat retention, as in higher temperature profiles and centreline velocities. In contrast, higher Forchheimer numbers causes a drop in both velocity and temperature due to increased flow resistance. But as F goes up, the Nusselt number always increases, meaning heat is better transferred through convection. The study also shows that hot spots and heat islands form inside the enclosure, especially when the shape factor is higher, because the heat builds up more quickly when there is less resistance, which is an essential thing to think about for things like heat storage systems, where it is crucial to have better thermal efficiency.

Darcy–Forchheimer–Brinkman flow of a Newtonian fluid through an enclosure with two straight boundaries and one curved boundary

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发表日期：

2025年9月1日

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Neha Elizabeth Sam,
Smita S. Nagouda,
P. G. Siddheshwar

摘要

The study examines the flow and heat transfer of a Newtonian fluid in a porous medium inside an enclosure with two straight boundaries and one curved boundary. This setup is important for heat storage and energy systems. The aim of this study is to solve the Brinkman-Forchheimer (BF) equation in an enclosure with two straight and one curved boundary. The research also looks to perform a thorough heat transfer analysis to improve the understanding of thermal behaviour in porous medium BF flow. Additionally, the study calculates the Nusselt number using a compatibility condition to ensure the results are physically consistent. Finally, it fits the Nusselt number as a function of the shape factor (s) and the Forchheimer number (F). This helps in capturing the trends in convective heat transfer behaviour within the medium.The main assumptions include a steady, fully developed flow in the z-direction with a constant axial pressure gradient - \(\frac{dp}{dz}\) , and zero axial velocity (w = 0) on all boundaries. The domain in three-dimensions is defined in cartesian coordinates (x, y, z), with \(x^2+y^2 \le 1\) on the curved boundary,ensuring the spatial constraint of the geometry. The quasi-linearisation method is used to linearise the governing equations, resulting in a system of linear algebraic equations that is subsequently solved using the alternate direction implicit (ADI) method with an accuracy of \({10}^{-4}\) . The findings show that an increase in the shape factor (s) results in a plug flow behaviour and better heat retention, as in higher temperature profiles and centreline velocities. In contrast, higher Forchheimer numbers causes a drop in both velocity and temperature due to increased flow resistance. But as F goes up, the Nusselt number always increases, meaning heat is better transferred through convection. The study also shows that hot spots and heat islands form inside the enclosure, especially when the shape factor is higher, because the heat builds up more quickly when there is less resistance, which is an essential thing to think about for things like heat storage systems, where it is crucial to have better thermal efficiency.