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Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

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Abstract

Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics®. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.

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Abbreviations

\( {A}_{\text{disp}}^{XX} \) :

Coefficient of the non-dimensional dispersion tensor

\( {A}_{\text{disp}}^{XZ} \) :

Coefficient of the non-dimensional dispersion tensor

\( {A}_{\text{disp}}^{ZZ} \) :

Coefficient of the non-dimensional dispersion tensor

\( {A}_{\text{L}} \) :

Non-dimensional longitudinal dispersion coefficient

\( B_{m,n} \) :

Fourier series coefficient—stream function

\( \tilde{B}_{g,h} \) :

Matrix coefficient defined in “Appendix A”

\( C_{r,s} \) :

Fourier series coefficient—temperature

\( {\text{Er}}_{{\overline{\text{Nu}} }} \) :

Relative error on Nusselt number

\( {\text{Er}}_{{\theta^{\text{top}} }} \) :

Relative error on temperature at the top wall

\( {\text{Er}}_{{U^{\text{top}} }} \) :

Relative error on horizontal velocity at the top wall

\( {\text{Er}}_{{V^{\text{top}} }} \) :

Relative error on vertical velocity at the top wall

\( F^{{\text{Disp}}} \) :

Function including all dispersion terms

\( g \) :

Gravitational acceleration

\( H \) :

Square size

\( h \) :

Local convection coefficient

\( {\mathbf{I}} \) :

Identity tensor

K :

Permeability

N m :

Number of Fourier modes in z—stream function

\( N_{n} \) :

Number of Fourier modes in x—stream function

\( N_{p} \) :

Number of integration points

\( N_{r} \) :

Number of Fourier modes in z—temperature

\( N_{s} \) :

Number of Fourier modes in x—temperature

\( {\text{Nu}} \) :

Local Nusselt number

\( \overline{\text{Nu}} \) :

Average Nusselt number

\( n_{A} \) :

Polynomial degree for \( {A}_{\text{L}} \)

\( n_{R} \) :

Polynomial degree for \( R_{{\alpha_{\text{disp}} }} \)

\( p \) :

Fluid pressure

\( P_{i,j} \) :

Polynomial coefficient of the scaling relation

\( {\text{Ra}} \) :

Rayleigh number

\( R^{\rm F} \) :

Residual of the flow equation

\( R^{\rm H} \) :

Residual of the heat equation

\( R_{g,h}^{\rm F} \) :

Residual of the spectral flow equation

\( R_{g,h}^{\rm H} \) :

Residual of the spectral heat equation

\( R_{{\alpha_{\text{disp}} }} \) :

Ratio of the transverse to longitudinal dispersion

T :

Temperature

\( T_{\text{h}} \) :

Hot temperature at the left wall

\( T_{\text{c}} \) :

Cold temperature at the right wall

\( u \) :

Horizontal velocity component

\( U \) :

Non-dimensional horizontal velocity

\( U^{\text{top}} \) :

Non-dimensional horizontal velocity—top wall

\( U^{\hbox{max} } \) :

Maximum horizontal velocity

\( v \) :

Vertical velocity

\( V \) :

Non-dimensional vertical velocity

\( V^{\text{hot}} \) :

Non-dimensional vertical velocity—hot wall

\( V^{\hbox{max} } \) :

Maximum non-dimensional vertical velocity

\( \overrightarrow {{\mathbf{V}}} \) :

Darcy’s velocity

\( \left| {\overrightarrow {{\mathbf{V}}} } \right| \) :

Magnitude of velocity vector

\( \text{Wp}_{i} \) :

Weight integration function

\( x \), \( X \) :

Abscissa, non-dimensional abscissa

\( z \), \( Z \) :

Elevation, non-dimensional elevation

\( \text{Xp}_{i} ,\text{Zp}_{i} \) :

Coordinates of integration points

\( \alpha_{\text{L}} ,\alpha_{\text{T}} \) :

Longitudinal and transverse dispersion

\( \alpha_{m} \) :

Effective thermal diffusivity

\( {\varvec{\upalpha}}_{{{\mathbf{disp}}}} \) :

Dispersion tensor

\( \beta \) :

Thermal expansion

\( \beta_{g,m,r}^{\text{I}} \) :

Matrix coefficient defined in “Appendix A”

\( \beta_{g,m,r}^{\text{II}} \) :

Matrix coefficient defined in “Appendix A”

\( \delta_{i,j} \) :

Kronecker delta function

\( \varepsilon^{g} \) :

Vector coefficient defined in “Appendix A”

\( \theta \) :

Non-dimensional temperature

\( \theta^{\text{top}} \) :

Non-dimensional temperature at the top wall

\( \varTheta \) :

Shifted non-dimensional temperature

\( \lambda_{\text{surf}} \) :

Thermal conductivity

\( \varLambda_{h,s} \) :

Matrix coefficient defined in “Appendix A”

\( \mu \) :

Fluid viscosity

\( \rho \) :

Fluid density

\( \rho_{\text{c}} \) :

Fluid density at the cold temperature

\( \sigma \) :

Ratio of heat capacity porous material to fluid

\( \varphi \) :

Stream function

\( \varPhi \) :

Non-dimensional stream function

\( \chi_{h,n,s}^{\text{I}} \) :

Matrix coefficient defined in “Appendix A”

\( \chi_{h,n,s}^{\text{II}} \) :

Matrix coefficient defined in “Appendix A”

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Authors and Affiliations

  1. Laboratoire d’Hydrologie et Geochemie de Strasbourg, University of Strasbourg/EOST/ENGEES, CNRS, 1 Rue Blessig, 67084, Strasbourg, France

    Marwan Fahs & Anis Younes

  2. Institute of Fluid Mechanics and Environmental Physics in Civil Engineering, Leibniz Universität Hannover, Appelstr. 9A, 30167, Hannover, Germany

    Thomas Graf & Tuong Vi Tran

  3. Department of Civil Engineering, Sharif University of Technology, PO Box 11155-9313, Tehran, Iran

    Behzad Ataie-Ashtiani

  4. National Centre for Groundwater Research and Training, College of Science and Engineering, Flinders University, GPO Box 2100, Adelaide, SA, 5001, Australia

    Behzad Ataie-Ashtiani & Craig. T. Simmons

  5. IRD UMR LISAH, 92761, Montpellier, France

    Anis Younes

  6. Laboratoire de Modélisation en Hydraulique et Environnement, Ecole Nationale d’Ingénieurs de Tunis, Tunis, Tunisia

    Anis Younes

Authors
  1. Marwan Fahs
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  2. Thomas Graf
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  3. Tuong Vi Tran
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  4. Behzad Ataie-Ashtiani
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  5. Craig. T. Simmons
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  6. Anis Younes
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Correspondence to Marwan Fahs.

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Appendices

Appendix A: Coefficients of the Spectral System

The coefficients of Eqs. (27) and (28) are defined as follows:

$$ \varLambda_{i,j} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if}}\,{\text{i}} = {\text{j}}} \hfill \\ {\frac{{1 - ( - 1)^{i + j} }}{i + j} + \frac{{1 - ( - 1)^{i - j} }}{i - j}} \hfill & {{\text{if}}\,{\text{i}} \ne {\text{j}}} \hfill \\ \end{array} } \right. $$
(A.1)
$$ \varepsilon^{g} = \left\{ {\begin{array}{*{20}l} 2 \hfill & {g = 0} \hfill \\ 1 \hfill & {g \ne 0} \hfill \\ \end{array} } \right. $$
(A.2)
$$ \tilde{B}_{g,h} = \left\{ \begin{aligned} B_{g,h} \quad {\text{if}}\,g \le {\text{Nm}}\,{\text{and}}\,h \le {\text{Nn}} \hfill \\ 0 \hfill \\ \end{aligned} \right. $$
(A.3)
$$ \begin{aligned} \beta_{g,m,r}^{\text{I}} & = \delta_{g,r - m} + \delta_{g,m - r} + \delta_{g,m + r} \\ \beta_{g,m,r}^{\text{II}} & = \delta_{g,r - m} - \delta_{g,r + m} + \delta_{g,m - r} \\ \chi_{h,n,s}^{\text{I}} & = \delta_{h,s + n} - \delta_{h,s - n} + \delta_{h,n - s} \\ \chi_{h,n,s}^{\text{II}} & = \delta_{h,n + s} - \delta_{h,n - s} + \delta_{h,s - n} \\ \end{aligned} $$
(A.4)

Appendix B: Physical Parameters Used in TRACES, COMSOL and OGS

See Table 2.

Table 2 Dimensional parameters for the six test cases
Full size table

Appendix C: Coefficients for the Polynomial Correlations of the Average Nusselt Number

See Table 3.

Table 3 Coefficients of the polynomials used for the correlation between the average Nusselt number and the dispersion coefficients (\( {A}_{\text{L}} \) and \( R_{{\alpha_{\text{disp}} }} \))
Full size table

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Fahs, M., Graf, T., Tran, T.V. et al. Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series. Transp Porous Med 131, 537–568 (2020). https://doi.org/10.1007/s11242-019-01356-1

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  • DOI: https://doi.org/10.1007/s11242-019-01356-1

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