# The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{c}= 4π

^{2}and k

_{c}= π. This porous medium analogue of the much older Rayleigh-Bénard problem shares many attributes of the latter. The neutral curve, which describes the onset of convection, is unimodal with one minimum in both cases, and weakly nonlinear analyses also show that two-dimensional rolls form the preferred pattern immediately post-onset (see Rees and Riley [3], Newell and Whitehead [4]). The porous medium configuration, as studied by Horton and Rogers [1] and Lapwood [2], also has the advantage that the analysis proceeds analytically even within the weakly nonlinear range, and therefore it forms a good pedagogical introduction to the study of weakly nonlinear theory.

_{c}= 12 and k

_{c}= 0 (Nield [5]). If the porous medium is layered, then it is possible to have bimodal convection, where the neutral curve has two minima, and also to have convection with a square planform immediately post onset (McKibbin and O’Sullivan [6], Rees and Riley [7]). If the layer has a constant vertical throughflow of fluid (Sutton [8]), then the bifurcation to convective flow is subcritical (Pieters and Schuttelaars [9], Rees [10]), meaning that strongly nonlinear flow exists at Rayleigh numbers below the linear threshold.

## 2. Governing Equations

_{∞}+ ΔTcos(ωt), where T

_{∞}is the ambient temperature far from the heated surface. The normal velocity at the surface is zero.

_{∞}), $\widehat{g}$ gravity, β the coefficient of cubical expansion, σ the heat capacity ratio of the saturated medium to that of the fluid, and α the thermal diffusivity of the saturated medium. The heated surface has a spatially uniform but time-varying surface temperature distribution, T = T

_{∞}+ ΔTcos(ωt), where ΔT is the maximum temperature difference between the wall and the ambient medium.

## 3. Basic State

## 4. Linear Stability Analysis

## 5. Numerical Solutions

#### 5.1. Numerical Method

#### 5.2. Numerical Accuracy

_{max}= 10. It was found that further increases in z

_{max}demonstrated that more than six significant figures of accuracy had been achieved, and therefore the disturbance was well-contained by the computational domain. In addition, at least four significant figures of accuracy were demonstrated by grid refinement (interval halving), and therefore there was no need to use either smaller spatial steps or a larger computational domain.

^{−4}. The equivalent error for the first order method is 1.0 × 10

^{−2}. We therefore conclude that the use of the second order method with dt = 0.005 will give solutions of more than acceptable accuracy. It will be noticed that the accuracy of the solutions reduces a little when k takes greater values. This happens because the disturbance tends to occupy a smaller region when k takes larger values (so that the disturbance retains an approximately unit aspect ratio), and therefore the spatial resolution decreases. It is will shown later that the smallest value of Ra is in the vicinity of k = 1 where the accuracy is very good indeed.

#### 5.3. Neutral Curves

_{c}= 1.1878, Ra

_{c}= 40.2889,

#### 5.4. Disturbance Profiles

_{max}and f

_{min}were to be plotted, then it will be clear that the disturbance has a period of 2, even though the basic state has a period of 1. Indeed it was found that f

_{max}(t) = f

_{min}(t + 1) for all t. Even more clearly, the gradient of the reduced temperature disturbance, gʹ(t), has a period of 2. It is therefore of great interest to determine physically why this should arise.

_{max}, in order to indicate clearly where the disturbances are concentrated at each instant of time; the values of ±|Θ|

_{max}may be found in Table 3. It is also worth noting that the associated streamfunction corresponds (for t = 0) to two whole counter-rotating cells centred in the vertical direction roughly where the temperature disturbances are.

## 6. Conclusions

_{c}= 40.2889 and k

_{c}= 1.1878. Alongside the growth and decay of the disturbance over one forcing period, it was found that disturbances, having been generated at the bounding surface when it is relatively warm, then rise away from the surface and follow the motion of the potentially unstable regions of the basic state. This latter provides the mechanism for cells to reverse direction after one period, and it yields a disturbance that has double the forcing period, a subharmonic instability.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

f | reduced streamfunction |

g | reduced temperature |

$\widehat{g}$ | gravity |

k | wavenumber |

k_{c} | critical wavenumber |

K | permeability |

L | length scale |

p | pressure |

$\overline{p}$ | pressure (dimensional) |

Ra | Darcy-Rayleigh number |

Ra_{c} | critical Darcy-Rayleigh number |

t | time |

T | temperature (dimensional) |

T_{∞} | temperature (ambient) |

T_{w} | temperature (wall) |

u | velocity in x-direction |

$\overline{u}$ | velocity in x-direction (dimensional) |

w | velocity in z-direction |

$\overline{w}$ | velocity in z-direction (dimensional) |

x | horizontal coordinate |

$\overline{x}$ | horizontal coordinate (dimensional) |

z | vertical coordinate |

$\overline{z}$ | vertical coordinate (dimensional) |

Greek letters | |

α | thermal diffusivity |

β | thermal expansion coefficient |

ΔT | temperature difference |

θ | temperature (nondimensional) |

Θ | temperature disturbance |

λ | exponential growth rate |

μ | dynamic viscosity |

ρ | reference density |

σ | heat capacity ratio |

ψ | streamfunction |

Ψ | streamfunction disturbance |

ω | thermal forcing frequency |

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**Figure 3.**Disturbance isotherms corresponding to critical conditions, namely k = k

_{c}and Ra = Ra

_{c}, and over one forcing period.

**Figure 4.**Variation with time of the surface rate of heat transfer of the disturbance over two forcing periods: k = 0.5 (blue); k = 1; k = k

_{c}(dashed); k = 1.5; k = 2.0; k = 2.5; k = 3.0; and k = 3.5 (red); where Ra takes the corresponding value from the neutral curve.

k | Order 1 | Order 2 | ||||
---|---|---|---|---|---|---|

dt = 0.02 | dt = 0.01 | dt = 0.005 | dt = 0.02 | dt = 0.01 | dt = 0.005 | |

1.0 | 39.5114 | 40.2806 | 40.6998 | 41.0355 | 41.1143 | 41.1341 |

2.0 | 42.0992 | 44.9746 | 46.5697 | 47.8046 | 48.1649 | 48.2544 |

3.0 | 52.6382 | 59.2835 | 63.3727 | 66.0481 | 67.7715 | 68.1918 |

**Table 2.**The dependence of the critical wavenumber and Rayleigh number on the numerical method and the timestep.

dt | 1st Order Centred | 2nd Order | ||
---|---|---|---|---|

k_{c} | Ra_{c} | k_{c} | Ra_{c} | |

0.02 | 1.2912 | 37.9437 | 1.1923 | 40.1568 |

0.01 | 1.2356 | 39.1009 | 1.1889 | 40.2560 |

0.005 | 1.2109 | 39.6914 | 1.1881 | 40.2807 |

**Table 3.**Values of ${\left|\mathit{\theta}\right|}_{\mathit{m}\mathit{a}\mathit{x}}$ for each thermal disturbance field shown in Figure 3.

time, t | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
---|---|---|---|---|---|---|---|---|---|---|---|

${\left|\mathit{\theta}\right|}_{\mathit{m}\mathit{a}\mathit{x}}$ | 0.930 | 4.133 | 14.350 | 22.633 | 19.083 | 11.489 | 5.912 | 2.849 | 1.348 | 0.643 | 0.930 |

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**MDPI and ACS Style**

Bidin, B.; Rees, D.A.S.
The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium. *Fluids* **2016**, *1*, 41.
https://doi.org/10.3390/fluids1040041

**AMA Style**

Bidin B, Rees DAS.
The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium. *Fluids*. 2016; 1(4):41.
https://doi.org/10.3390/fluids1040041

**Chicago/Turabian Style**

Bidin, Biliana, and D. Andrew S. Rees.
2016. "The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium" *Fluids* 1, no. 4: 41.
https://doi.org/10.3390/fluids1040041