Three-dimensional numerical simulations of free convection in a layered porous enclosure

Guerrero-Martínez, Fernando J.; Younger, Paul L.; Karimi, Nader; Kyriakis, Sotirios A.

doi:10.1016/j.ijheatmasstransfer.2016.10.072

Cited by 31 publications

(20 citation statements)

References 21 publications

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“…Using these variables, the dimensionless steady state flow and transport equations become

\nabla^{2} boldψ - italicRa ((), \frac{\partial c}{\partial Y} e_{x} bold-italic- \frac{\partial c}{\partial X} e_{y}) bold-italic= 0,

((), \nabla \times boldψ) . \nabla c - \nabla^{2} c = 0,

where

italicRa = \frac{italicgK ((), ρ_{1} - ρ_{0}) H}{μ \overset{D}{true} ε}

is the Rayleigh number, which expresses the ratio of buoyancy driven to diffusion‐driven salt fluxes. This system of equations is similar to the one encountered in the problem of natural convection in cubic box (Guerrero‐Martínez et al, ; Luz Neto et al, ).…”

Section: The Fourier Series Solutionmentioning

confidence: 80%

“…With ∇ ⋅ ψ = ∂ψ x / ∂X + ∂ψ y / ∂Y + ∂ψ z / ∂Z = 0, one gets ∂ψ z / ∂Z = 0. As shown in Luz Neto et al () and Guerrero‐Martínez et al (), the vector potential impervious BCs become

\begin{matrix} lefttrue \\ \frac{\partial ψ_{x}}{\partial X} = ψ_{y} = ψ_{z} = 0, at X = 0, 1, \\ \frac{\partial ψ_{y}}{\partial Y} = ψ_{x} = ψ_{z} = 0, at Y = 0, 1, \\ \frac{\partial ψ_{z}}{\partial Z} = ψ_{x} = ψ_{y} = 0, at Z = 0, 1 . \end{matrix}

…”

Section: The Fourier Series Solutionmentioning

confidence: 86%

“…A solenoidal vector potential can be used as shown in Guerrero‐Martínez et al () and Luz Neto et al (), leading to ∇ ⋅ φ = 0. Therefore, in this case, equation reduces to

\nabla \times \nabla \times boldφ bold-italic= - \nabla^{2} boldφ .

…”

Section: The Fourier Series Solutionmentioning

confidence: 99%

“…At steady state, the velocity field ( q ) admits a vector potential ( φ ), such that (Guerrero‐Martínez et al, ; Hirasaki & Hellums, ; Luz Neto et al, )

boldq = \nabla \times boldφ .

…”

Section: The Fourier Series Solutionmentioning

confidence: 99%

“…In fact, contrary to 2‐D DDF processes in porous enclosures, which are well understood, relatively less investigation is available for 3‐D processes. The most extensively studied 3‐D configuration is the vertical density gradient where two different concentrations (or temperatures) are imposed at the domain top and bottom surfaces (e.g., Guerrero‐Martínez et al, ; Johannsen et al, ; Oswald & Kinzelbach, ; Pau et al, ; Voss et al, ). However, Fajraoui et al () and Nield and Bejan () reported some important situations (geological storage of carbon dioxide and geothermal reservoirs) in which the density gradient can be horizontal.…”

Section: Introductionmentioning

confidence: 99%

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A 3‐D Semianalytical Solution for Density‐Driven Flow in Porous Media

Shao

Fahs

Hoteit

et al. 2018

Water Resources Research

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Existing analytical and semianalytical solutions for density-driven flow (DDF) in porous media are limited to 2-D domains. In this work, we develop a semianalytical solution using the Fourier Galerkin method to describe DDF induced by salinity gradients in a 3-D porous enclosure. The solution is constructed by deriving the vector potential form of the governing equations and changing variables to obtain periodic boundary conditions. Solving the 3-D spectral system of equations can be computationally challenging. To alleviate computations, we develop an efficient approach, based on reducing the number of primary unknowns and simplifying the nonlinear terms, which allows us to simplify and solve the problem using only salt concentration as primary unknown. Test cases dealing with different Rayleigh numbers are solved to analyze the solution and gain physical insight into 3-D DDF processes. In fact, the solution displays a 3-D convective cell (actually a vortex) that resembles the quarter of a torus, which would not be possible in 2-D. Results also show that 3-D effects become more important at high Rayleigh number. We compare the semianalytical solution to research (Transport of RadioACtive Elements in Subsurface) and industrial (COMSOL Multiphysics®) codes. We show cases (high Raleigh number) where the numerical solution suffers from numerical artifacts, which highlight the worthiness of our semianalytical solution for code verification and benchmarking. In this context, we propose quantitative indicators based on several metrics characterizing the fluid flow and mass transfer processes and we provide open access to the source code of the semianalytical solution and to the corresponding numerical models. SHAO ET AL.10,094

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“…Using these variables, the dimensionless steady state flow and transport equations become

\nabla^{2} boldψ - italicRa ((), \frac{\partial c}{\partial Y} e_{x} bold-italic- \frac{\partial c}{\partial X} e_{y}) bold-italic= 0,

((), \nabla \times boldψ) . \nabla c - \nabla^{2} c = 0,

where

italicRa = \frac{italicgK ((), ρ_{1} - ρ_{0}) H}{μ \overset{D}{true} ε}

Section: The Fourier Series Solutionmentioning

confidence: 80%

\begin{matrix} lefttrue \\ \frac{\partial ψ_{x}}{\partial X} = ψ_{y} = ψ_{z} = 0, at X = 0, 1, \\ \frac{\partial ψ_{y}}{\partial Y} = ψ_{x} = ψ_{z} = 0, at Y = 0, 1, \\ \frac{\partial ψ_{z}}{\partial Z} = ψ_{x} = ψ_{y} = 0, at Z = 0, 1 . \end{matrix}

…”

Section: The Fourier Series Solutionmentioning

confidence: 86%

“…A solenoidal vector potential can be used as shown in Guerrero‐Martínez et al () and Luz Neto et al (), leading to ∇ ⋅ φ = 0. Therefore, in this case, equation reduces to

\nabla \times \nabla \times boldφ bold-italic= - \nabla^{2} boldφ .

…”

Section: The Fourier Series Solutionmentioning

confidence: 99%

“…At steady state, the velocity field ( q ) admits a vector potential ( φ ), such that (Guerrero‐Martínez et al, ; Hirasaki & Hellums, ; Luz Neto et al, )

boldq = \nabla \times boldφ .

…”

Section: The Fourier Series Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A 3‐D Semianalytical Solution for Density‐Driven Flow in Porous Media

Shao

Fahs

Hoteit

et al. 2018

Water Resources Research

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Numerical investigation of natural convection in a square enclosure partially filled with horizontal layers of a porous medium

Abdulwahed

Ali

2022

Heat Trans

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Two‐dimensional buoyancy‐induced flow and heat transfer inside a square enclosure partially occupied by copper metallic foam subjected to a symmetric side cooling and constant heat flux bottom heating was tested numerically. Finite Element Method was employed to solve the governing partial differential equations of the flow field and the Local Thermal Equilibrium model was used for the energy equation. The system boundaries were defined as lower heated wall by constant heat flux, cooled lateral walls, and insulated top wall. The three parameters elected to conduct the study are heater length (7 ≤ ζ ≤ 20 cm), constant heat flux (150 ≤ q″ ≤ 600 W m−2), and porous material thickness (5 ≤ H ≤ 20 cm). The porous material used was the copper metal foam of porosity = 0.9 and pore density PPI = 10, and saturated with a fluid of Prandtl number = 0.7. On the basis of the results obtained, it was concluded that at the porous layer thickness = 5 cm, the rate of heat transferred was (74.6%) higher than when the layer height was 20 cm (the cavity is fully filled) and at the same thickness it was found that the heat rate is (51.4%) higher than when using the half filling (H = 10 cm). Further, the local and mean Nusselt number is maximum when using the largest heater size and smallest porous layer thickness. Finally, better circulation and convective modes were observed at high values of heat flux.

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Internal Natural Convection: Heating from Below

Nield

Bejan

2017

Convection in Porous Media

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Three-dimensional numerical simulations of free convection in a layered porous enclosure

Cited by 31 publications

References 21 publications

A 3‐D Semianalytical Solution for Density‐Driven Flow in Porous Media

A 3‐D Semianalytical Solution for Density‐Driven Flow in Porous Media

Numerical investigation of natural convection in a square enclosure partially filled with horizontal layers of a porous medium

Internal Natural Convection: Heating from Below

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