# Nonlinear Convection in a Partitioned Porous Layer

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Method

## 4. Results and Discussion

#### 4.1. Linearised Theory

#### 4.2. Nonlinear Flows

## 5. Conclusions

## Conflicts of Interest

## Abbreviations

C | heat capacity |

g | gravity |

k | wavenumber ($\pi /{x}_{\mathrm{max}}$) |

${k}_{\mathrm{pm}}$ | thermal conductivity of the porous medium |

K | permeability |

L | height of one sublayer |

$\mathrm{Nu}$ | Nusselt number |

NX | number of intervals in the x-direction |

NZ | number of intervals in the z-direction |

p | pressure |

Ra | Darcy–Rayleigh number |

t | time |

T | dimensional temperature |

${T}_{c}$ | upper (cold) boundary temperature |

${T}_{h}$ | lower (hot) boundary temperature |

u | horizontal velocity |

w | vertical velocity |

x | horizontal coordinate |

${x}_{\mathrm{max}}$ | width of the computational domain |

z | vertical coordinate |

## $Greek\phantom{\rule{0.166667em}{0ex}}symbols$

CGIs | CpG islands |

β | expansion coefficient |

$\Delta T$ | temperature drop across a sublayer |

θ | temperature |

λ | value defined in Equation (21) |

μ | dynamic viscosity |

ρ | density |

σ | value defined in Equation (21) |

ψ | streamfunction |

## $Subscripts,\phantom{\rule{0.166667em}{0ex}}superscripts,\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}other\phantom{\rule{0.166667em}{0ex}}symbols$

c | critical value |

f | fluid |

pm | porous medium |

^ | dimensional quantity |

+ | immediately above the interface |

− | immediately below the interface |

## Appendix A. Accuracy Considerations

**Figure A1.**The value of $\mathrm{Nu}$ as a function of $\mathrm{Ra}$ for $k={k}_{c}$ using the three grids: $12\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}16$ (dotted), $24\times 32$ (dashed) and $48\times 64$ (continuous).

**Table A1.**Computed values of the Nusselt number at $\mathrm{Ra}=200$ and $k={k}_{c}$ as a function of the numerical grid.

NX × NZ | Nu |
---|---|

12 × 16 | 4.4238 |

24 × 32 | 4.5451 |

48 × 64 | 4.5720 |

**Table A2.**Extrapolated values of the critical Darcy–Rayleigh number and their errors as a function of the numerical grid.

NX × NZ | ${\mathbf{Ra}}_{\mathit{c}}$ | Error |
---|---|---|

12 × 16 | 27.762 | 2.45% |

24 × 32 | 27.265 | 0.62% |

48 × 64 | 27.129 | 0.15% |

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**Figure 1.**Depicting a porous layer split into two identical sublayers by an infinitesimally thin and impermeable interface.

**Figure 3.**

**Upper row**: streamlines (continuous) and isotherms (dashed) for the case ${x}_{\mathrm{max}}=\phantom{\rule{3.33333pt}{0ex}}1$ for the given values of $\mathrm{Ra}$.

**Lower row**: the corresponding isotherms for ${\theta}_{\mathrm{pert}}$ where continuous lines correspond to where ${\theta}_{\mathrm{pert}}>0$, dashed lines to where it is negative, the thick line to where it is zero, and the dotted line depicts the interface. In all cases, 20 equal intervals were taken for the isolines.

**Figure 4.**Streamlines, isotherms and perturbation isotherms for $\mathrm{Ra}=200$ and for the given values of ${x}_{\mathrm{max}}$. Isolines follow the same conventions as in Figure 3.

**Figure 5.**The variation of the Nusselt number with the Darcy–Rayleigh number for selected wavenumbers. The continuous lines correspond to $k=2.5$ (extreme left at $\mathrm{Nu}=1$), 3, $3.5$, 4, 5, 6, 7 and 8 (extreme right). The dotted line corresponds to $k=2$ and the dotted-dashed line to $k=1.5$.

**Figure 6.**The variation of the Nusselt number with the wavenumber for selected values of $\mathrm{Ra}$. The dashed line is the locus of points which correspond to $\mathrm{Nu}$ taking its maximum value. The dotted line illustrates how $\mathrm{Nu}$ changes when a disturbance with wavenumber k yields a flow with wavenumber $3k$ when $\mathrm{Ra}=150$; these curves have been suppressed for other values of $\mathrm{Ra}$ for clarity of presentation.

**Figure 7.**The value of k at which $\mathrm{Nu}$ is maximised as a function of $\mathrm{Ra}$. The continuous line corresponds to the standard $48\times 64$ grid, while the dashed line corresponds to a $24\times 32$ grid. More accurate computations using a $64\times 96$ grid are shown using the dotted line.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Rees, D.A.S.
Nonlinear Convection in a Partitioned Porous Layer. *Fluids* **2016**, *1*, 24.
https://doi.org/10.3390/fluids1030024

**AMA Style**

Rees DAS.
Nonlinear Convection in a Partitioned Porous Layer. *Fluids*. 2016; 1(3):24.
https://doi.org/10.3390/fluids1030024

**Chicago/Turabian Style**

Rees, D. Andrew S.
2016. "Nonlinear Convection in a Partitioned Porous Layer" *Fluids* 1, no. 3: 24.
https://doi.org/10.3390/fluids1030024