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Chaotic and Periodic Natural Convection for Moderate and High Prandtl Numbers in a Porous Layer subject to Vibrations

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Abstract

The analysis of natural convection for moderate and high Prandtl numbers in a fluid-saturated porous layer heated from below and subject to vibrations is presented with a twofold objective. First, it aims at investigating the significance of including a time derivative term in Darcy’s equation when wave phenomena are being considered. Second, it is dedicated to reporting results related to the route to chaos for moderate and high Prandtl number convection. The results present conclusive evidence indicating that the time derivative term in Darcy’s equation cannot be neglected when wave phenomena are being considered even when the coefficient to this term is extremely small. The results also show occasional chaotic “bursts” at specific values (or small range of values) of the scaled Rayleigh number, \(R\), exceeding some threshold. This behavior is quite distinct from the case without forced vibrations, when the chaotic solution occupies a wide range of \(R\) values, interrupted only by periodic “bursts.” Periodic and chaotic solution alternate as the value of the scaled Rayleigh number varies.

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Abbreviations

\(Da\) :

Darcy number, defined by \({k_*}/H_*^2 \)

\({\hat{\mathbf{e}}}_x\) :

Unit vector in the \(x\) direction

\({\hat{\mathbf{e}}}_z \) :

Unit vector in the z direction

\({\hat{\mathbf{e}}}_n \) :

Unit vector normal to the boundary, positive outwards

\(g_{*}\) :

Acceleration due to gravity

\(H_{*}\) :

The height of the layer

\(k_*\) :

Permeability of the porous domain

\(K_f\) :

Thermal conductivity of the fluid phase

\(K_s\) :

Thermal conductivity of the solid phase

\(K_{e^*}\) :

Effective thermal conductivity, defined by \(\varphi K_f +\left( {1-\varphi } \right) K_s \)

\(L_{*}\) :

The length of the porous domain

\(L\) :

Reciprocal of aspect ratio, equals \({L_*}/{H_*}\)

\(p\) :

Reduced pressure (dimensionless)

\(Pr\) :

Porous media effective Prandtl number, equals \({\nu _*}/{\alpha _{e^*} }\)

Ra :

Porous media Rayleigh number, equals \({\beta _*\Delta T_c g_*k_*H_*}/{\alpha _{e^*} \nu _*}\)

\(R\) :

Scaled Rayleigh number, equals \({Ra}/{4 \pi ^{2}}\)

\(t\) :

Time (dimensionless)

\(T\) :

Dimensionless temperature, \({\left( {T-T_C } \right) }/{\left( {T_H -T_C } \right) }\)

\(T_C\) :

Cold wall temperature

\(T_H\) :

Hot wall temperature

\(u \) :

Horizontal \(x\) component of the filtration velocity

\(w \) :

Vertical \(z\) component of the filtration velocity

\(x \) :

Horizontal length co-ordinate

\(z\) :

Vertical co-ordinate

\(X\) :

Rescaled amplitude

\(Y\) :

Rescaled amplitude

\(Z\) :

Rescaled amplitude

\(\alpha \) :

A parameter related to the time derivative term in Darcy’s equation

\(\alpha _{e^{*}}\) :

Effective thermal diffusivity, defined by \({\left[ {\varphi K_f +\left( {1-\varphi } \right) K_s } \right] }/{\rho _f c_{p,f} }\)

\(\beta _{*}\) :

Thermal expansion coefficient

\(\gamma \) :

A parameter defined by \({L^{2}}/{\left( {L^{2}+1} \right) }\)

\(\phi \) :

Porosity

\(\nu _{*}\) :

Fluid’s kinematic viscosity

\(\psi \) :

Stream function

\(\Delta T_c\) :

Characteristic temperature difference, equals \(\left( {T_H -T_C } \right) \)

\(\omega \) :

Forced vibration frequency

\(\delta \) :

Forced vibration amplitude

\(*\) :

Dimensional values

cr :

Critical values

\(C\) :

Related to the cold wall

\(H\) :

Related to the hot wall

\(f\) :

Related to the fluid phase

\(s\) :

Related to the solid phase

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Acknowledgments

The University of Pretoria and Prof. JP. Meyer gave much support for this Project, both financially and intellectually. One of the authors (JJV) wishes to thank them for all the support

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Authors and Affiliations

  1. Department of Mechanical and Aeronautical Engineering, University of Pretoria, Lynnwood Rd, Hatfield, Pretoria, 0002, South Africa

    Johnathan J. Vadasz & Josua P. Meyer

  2. Gas Division EED, ESKOM, Arivia Business Park, Building 1, Simba Road, Sunninghill, Johannesburg, 2000, South Africa

    Saneshan Govender

Authors
  1. Johnathan J. Vadasz
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  2. Josua P. Meyer
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  3. Saneshan Govender
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Correspondence to Johnathan J. Vadasz.

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Vadasz, J.J., Meyer, J.P. & Govender, S. Chaotic and Periodic Natural Convection for Moderate and High Prandtl Numbers in a Porous Layer subject to Vibrations. Transp Porous Med 103, 279–294 (2014). https://doi.org/10.1007/s11242-014-0301-z

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