# Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Darcy’s Flow in a Horizontal Channel

#### 2.1. Parallel Flow Regime

#### 2.2. Small Amplitude Perturbations

## 3. Stability Analysis

**Definition**

**1.**

**Definition**

**2.**

#### 3.1. Convective Instability

#### 3.2. Absolute Instability

**Theorem**

**1.**

- 1.
- There exists a unique saddle point of $\lambda \left(k\right)$, namely ${k}_{0}\in \mathbb{C}$;
- 2.
- There exists a path $\mathcal{C}$ in the complex k plane that crosses ${k}_{0}$ along a line of steepest descent of $\mathfrak{Re}\left(\lambda \right(k\left)\right)$;
- 3.
- The region of the complex plane between path $\mathcal{C}$ and the real axis, namely the line $\mathfrak{Im}\left(k\right)=0$, does not contain any singularity or branch cut of $\lambda \left(k\right)$ (see Figure 2).

- We find the saddle points ${k}_{0}\in \mathbb{C}$ by employing Equation (23) and by solving the algebraic equation ${\lambda}^{\prime}\left({k}_{0}\right)=0$.
- We check that the requirements stated in the thesis of Theorem 1 are satisfied by ${k}_{0}$.
- We determine the threshold condition between an unbounded time growth of $|{\mathcal{I}}_{n}(x,t)|$ and $|{\mathcal{J}}_{n}(x,t)|$ and an exponential decay to zero, expressed by the algebraic equation $\mathfrak{Re}\left(\lambda \left({k}_{0}\right)\right)=0$.

**Example**

**1.**

**Example**

**2.**

## 4. Results

## 5. A Matter of Scaling

## 6. Going Three-Dimensional

**Definition**

**3.**

**Definition**

**4.**

**Example**

**3.**

**Example**

**4.**

## 7. Conclusions

- The concepts of convective and absolute instability can be rigorously defined by applying the Fourier transform method to solve the perturbation equations. The Fourier transformed variable is the coordinate along the streamwise direction.
- The assessment of convective instability just deals with the time-growth, or time-decay, of the Fourier transformed perturbations. The study of the absolute instability, on the other hand, is focussed on the large-time behaviour of the perturbations themselves.
- The need to track the large-time behaviour of the perturbations, mathematically of the inverse Fourier transforms, leads to the central role played by the steepest-descent approximation.
- The two-dimensional analysis of convective instability does not yield a condition of convective instability different from that obtained with a three-dimensional analysis. The only difference is in the modes leading to the critical value of the Rayleigh number for the onset of convective instability, $R{a}_{c}$. The number of unstable Fourier modes at $Ra=R{a}_{c}$ gradually increases as the spanwise-to-vertical aspect ratio of the channel cross-section, $\tau $, increases.
- The onset of absolute instability occurs at a Rayleigh number $R{a}_{a}\ge R{a}_{c}$, where the equality $R{a}_{a}=R{a}_{c}$ holds if and only if the Péclet number, $Pe$, is zero. The threshold of absolute instability, $R{a}_{a}$, is a monotonic increasing function of $Pe$. When $Pe\gg 1$, $R{a}_{a}$ becomes approximately a linear function of $Pe$.
- For a given $Pe$, the threshold value of absolute instability, $R{a}_{a}$, obtained by a two-dimensional analysis coincides with that obtained by a three-dimensional analysis.
- A suitable scaling of the parameters turned out to map the study carried out in this paper to the analogous study of the Horton-Rogers-Lapwood problem, not only for the results of the convective instability study, but also for the analysis of the transition to absolute instability.

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**A sketch of the saddle point ${k}_{0}$ of $\mathfrak{Re}\left(\lambda \right(k\left)\right)$ in the complex k plane and of the line of steepest descent.

**Figure 3.**Threshold of absolute instability, $Ra=R{a}_{a}=22.9882$, with $Pe=10$, $n=1$: (

**a**) contours of $\mathfrak{Re}\left(\lambda \right(k\left)\right)$ in the complex k plane; (

**b**) contours of $\mathfrak{Im}\left(\lambda \right(k\left)\right)$ in the complex k plane. The red dots denote the saddle points ${k}_{0}=\pm 1.62238-i1.34154$; the yellow dots denote the singularities $k=\pm i\pi /2$; the black dashed contours are relative to $\lambda \left(k\right)=\lambda \left({k}_{0}\right)=\pm i20.1460$; the blue solid line is a possible path $\mathcal{C}$ that crosses the saddle points along a direction of steepest descent; the dotted blue line is the real axis, $\mathfrak{Im}\left(k\right)=0$.

**Figure 4.**Threshold of absolute instability, $Ra=R{a}_{a}=101.785$, with $Pe=50$, $n=1$: (

**a**) contours of $\mathfrak{Re}\left(\lambda \right(k\left)\right)$ in the complex k plane; (

**b**) contours of $\mathfrak{Im}\left(\lambda \right(k\left)\right)$ in the complex k plane. The red dots denote the saddle points ${k}_{0}=\pm 1.37471-i1.60524$; the yellow dots denote the singularities $k=\pm i\pi /2$; the black dashed contours are relative to $\lambda \left(k\right)=\lambda \left({k}_{0}\right)=\pm i113.262$; the blue solid line is a possible path $\mathcal{C}$ that crosses the saddle points along a direction of steepest descent; the dotted blue line is the real axis, $\mathfrak{Im}\left(k\right)=0$.

**Figure 5.**(

**a**) Curve in the complex $k/{\gamma}_{n}$ plane representing all possible saddle points of $\lambda \left(k\right)$, with $\mathfrak{Re}\left(\lambda \right(k\left)\right)=0$, for all real values of $(Ra,Pe)$, and for all natural numbers n. Yellow dots denote exclusions from the set of saddle points, while blue dots denote the saddle points with $Pe=0$; (

**b**) Curve in the complex k plane including all saddle points employed to evaluate $R{a}_{a}$ with either $Pe>0$ (lower plane) or $Pe<0$ (upper plane). The dashed blue line denotes the real axis, with blue dots denoting the case $Pe=0$, and yellow dots denoting the limiting cases $Pe\to \pm \infty $.

**Figure 6.**(

**a**) Transition to absolute instability: values of $Ra$ versus $Pe$ obtained for the saddle points ${k}_{0}$ with increasing values of $\mathfrak{Re}\left(\lambda \left({k}_{0}\right)\right)$, from negative to positive. The black line yields the threshold of absolute instability, namely $\mathfrak{Re}\left(\lambda \left({k}_{0}\right)\right)=0$; (

**b**) Map of the regions in the parametric plane $(Pe,Ra)$ relative to stability, convective instability, and absolute instability; the dashed line shows the asymptotic behaviour described by Equation (33).

**Figure 7.**Scaled convection streamlines of the perturbations for: (

**a**) the open upper boundary problem; (

**b**) the HRL problem.

**Table 1.**Threshold values of $Ra$ for the onset of absolute instability with increasing values of $Pe$. The relevant saddle points ${k}_{0}$ for the evaluation of $R{a}_{a}$ are reported.

$\mathbf{Pe}$ | ${\mathit{k}}_{0}$ | ${\mathbf{Ra}}_{\mathit{a}}$ |
---|---|---|

0 | $\pm \pi /2$ | ${\pi}^{2}$ |

5 | $\pm 1.69649-i0.94650$ | 14.4509 |

10 | $\pm 1.62238-i1.34154$ | 22.9882 |

15 | $\pm 1.53303-i1.47783$ | 32.4868 |

20 | $\pm 1.47785-i1.53301$ | 42.2506 |

25 | $\pm 1.44337-i1.56088$ | 52.1103 |

30 | $\pm 1.42029-i1.57729$ | 62.0136 |

35 | $\pm 1.40388-i1.58799$ | 71.9402 |

40 | $\pm 1.39166-i1.59548$ | 81.8806 |

45 | $\pm 1.38222-i1.60101$ | 91.8298 |

50 | $\pm 1.37471-i1.60524$ | 101.7851 |

55 | $\pm 1.36861-i1.60858$ | 111.7446 |

60 | $\pm 1.36355-i1.61129$ | 121.7072 |

65 | $\pm 1.35928-i1.61352$ | 131.6722 |

70 | $\pm 1.35565-i1.61539$ | 141.6391 |

75 | $\pm 1.35250-i1.61699$ | 151.6074 |

80 | $\pm 1.34976-i1.61836$ | 161.5768 |

85 | $\pm 1.34735-i1.61955$ | 171.5472 |

90 | $\pm 1.34521-i1.62060$ | 181.5183 |

95 | $\pm 1.34330-i1.62153$ | 191.4901 |

100 | $\pm 1.34159-i1.62236$ | 201.4625 |

$+\infty $ | $\pm 1.30970-i1.63663$ | $+\infty $ |

$\mathbf{Pe}$ | ${\mathbf{Ra}}_{\mathit{a},\mathbf{HRL}}{\phantom{\rule{3.33333pt}{0ex}}}^{\star}$ | ${\mathbf{Ra}}_{\mathit{a},\mathbf{HRL}}{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |
---|---|---|

10 | 57.8036 | 57.8036 |

20 | 91.9528 | 91.9528 |

50 | 208.441 | 208.441 |

100 | 407.140 | 407.140 |

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**MDPI and ACS Style**

Barletta, A.; Celli, M.
Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary. *Fluids* **2017**, *2*, 33.
https://doi.org/10.3390/fluids2020033

**AMA Style**

Barletta A, Celli M.
Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary. *Fluids*. 2017; 2(2):33.
https://doi.org/10.3390/fluids2020033

**Chicago/Turabian Style**

Barletta, Antonio, and Michele Celli.
2017. "Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary" *Fluids* 2, no. 2: 33.
https://doi.org/10.3390/fluids2020033