# On Moderate-Rayleigh-Number Convection in an Inclined Porous Layer

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Dynamics at Moderate Ra

#### 3.1. Numerical Simulation Results

#### 3.2. Steady Convective States

#### 3.3. Secondary Stability Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dimensionless geometry and background/basic state for 2D convection in inclined Rayleigh–Darcy domains. (

**a**) Closed domain; (

**b**) L-periodic domain. x and z are the wall-parallel and wall-normal coordinates, respectively, and g is the (dimensional) acceleration of gravity. For $\varphi >{0}^{\xb0}$, the basic-state temperature field (realized in the absence of convection) varies only in z, as in the horizontal case ($\varphi ={0}^{\xb0}$). The basic-state velocity field is nonzero, however, with the background shear flow strengthening as the inclination angle $\varphi $ is increased. The flow in (

**a**) represents the basic unicellular shear flow observed in experiments in a closed domain. In an infinitely extended layer (i.e., $L\to \infty $), the unicellular base state becomes x-independent and reduces to a laminar unidirectional shear flow, as shown in (

**b**).

**Figure 2.**Snapshots of isotherms (left) and corresponding streamlines (right) from numerical simulations at $Ra=100$ and $L=2$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={10}^{\xb0}$; (

**c**) $\varphi ={25}^{\xb0}$. The streamlines of the natural (positive $\psi $) and antinatural (negative $\psi $) rolls are shown in red and blue, respectively. For a range of $\varphi $ values, the flow takes the form of stable steady convective rolls. As $\varphi $ is increased, the natural roll becomes more vigorous (see the colorbar limits) and more tightly attached to the walls, while the antinatural roll is suppressed and becomes detached from the walls.

**Figure 3.**Snapshots of isotherms (left) and corresponding streamlines (right) from numerical simulations at $Ra=300$ and $L=2$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={10}^{\xb0}$; (

**c**) $\varphi =17.5\xb0$; (

**d**) $\varphi ={25}^{\xb0}$. In this case, the flows in (

**a**,

**d**) are steady; in (

**b**,

**c**), the upper and lower boundary layers of the antinatural rolls (negative $\psi $) become unstable. At $\varphi \approx {25}^{\xb0}$, the small proto-plumes generated from the boundary-layer instabilities of the antinatural rolls split the unsteady two-cell convection (with one natural and one antinatural roll) into a steady four-cell convective state, thereby reducing the aspect ratio of each roll.

**Figure 4.**Snapshots of isotherms (left) and corresponding streamlines (right) from numerical simulations at $Ra=500$ and $L=2$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={10}^{\xb0}$; (

**c**) $\varphi ={15}^{\xb0}$. For $\varphi <{15}^{\xb0}$, the convection appears in the form of unsteady rolls (

**a**,

**b**). However, as $\varphi $ is increased, the boundary-layer instability of the antinatural roll (negative $\psi $) becomes stronger and splits the unsteady two-cell convective state into the steady four-cell convection pattern shown in (

**c**).

**Figure 5.**Snapshots of isotherms from numerical simulations at $\varphi ={35}^{\xb0}$ and $L=10$. (

**a**–

**d**) $Ra=100$; (

**e**) $Ra=300$; (

**f**) $Ra=500$. Sub-plots (

**a**–

**d**) show steady convective states obtained using different initial conditions, while (

**e**) and (

**f**) show snapshots of time-dependent states. Although the basic-state shear flow is linearly stable for $\varphi >{\varphi}_{t}\approx {31.3}^{\xb0}$ in 2D, convection nevertheless may be realized by initializing with sufficiently large-amplitude disturbances; i.e., sub-critical instabilities are possible in this parameter regime. The spatially-localized convective states evident in (

**a**,

**b**), observed here for the first time in single-species porous medium convection, are one manifestation of this sub-critical instability.

**Figure 6.**Isotherms (left) and streamlines (right) of steady convective states at $Ra=500$ and $L=2$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={10}^{\xb0}$; (

**c**) $\varphi ={20}^{\xb0}$; (

**d**) $\varphi ={30}^{\xb0}$. As the inclination angle is increased, the natural roll (positive $\psi $) becomes more vigorous (see the colorbar limits) and more tightly attached to the walls, while the antinatural roll (negative $\psi $) is suppressed and becomes detached from the walls. At $\varphi ={30}^{\xb0}$, the antinatural roll makes contact with the upper and lower walls only at certain localized intervals in x.

**Figure 7.**Magnitude of ${\psi}_{m}$ for steady convective states as a function of $\varphi $ at $Ra=500$ and $L=2$. ${\psi}_{m}$ denotes the extremum $\psi $ value corresponding to the natural roll with max($\psi $) (positive) and antinatural roll with min($\psi $) (negative). As $\varphi $ is increased, the natural-roll motion is intensified, while the antinatural-roll motion is suppressed.

**Figure 8.**Variation of the maximum growth rate, ${\sigma}_{m}$, with $\varphi $ at moderate $Ra$, ${L}_{s}=2$ and $\beta =0$. At $Ra=300$, the steady state is marginally stable for $\varphi <{10}^{\xb0}$ and becomes weakly unstable at $\varphi ={10}^{\xb0}$. The same branch of steady states is not obtained at large $\varphi $ for $Ra=500$ and 792 using the present numerical scheme.

**Figure 9.**The fastest-growing 2D temperature eigenfunctions at $Ra=500$, ${L}_{s}=2$ and $\beta =0$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={20}^{\xb0}$. For the horizontal case, reflection symmetry is satisfied and both the natural and antinatural rolls are equally unstable. However, as $\varphi $ is increased, the natural roll is stabilized and the instability of the antinatural roll is intensified.

**Figure 10.**The leading eigenvalues, $\sigma =\lambda /Ra$, at $Ra=500$, ${L}_{s}=2$ and $\beta =0$. (

**a**) $\varphi ={0}^{\xb0}$; (

**b**) $\varphi ={20}^{\xb0}$. All of the unstable modes for both the horizontal and inclined cases exhibit a similar structure as that of the corresponding fastest-growing mode in Figure 9.

**Table 1.**Approximate angle $\varphi $ at which the flow transitions from two-cell convection to four-cell convection in numerical simulations at moderate $Ra$ and $L=2$.

$\mathit{Ra}$ | 300 | 500 | 792 | 998 |
---|---|---|---|---|

$\varphi $ | ${25}^{\xb0}$ | ${15}^{\xb0}$ | ${10}^{\xb0}$ | ${5}^{\xb0}$ |

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**MDPI and ACS Style**

Wen, B.; Chini, G.P.
On Moderate-Rayleigh-Number Convection in an Inclined Porous Layer. *Fluids* **2019**, *4*, 101.
https://doi.org/10.3390/fluids4020101

**AMA Style**

Wen B, Chini GP.
On Moderate-Rayleigh-Number Convection in an Inclined Porous Layer. *Fluids*. 2019; 4(2):101.
https://doi.org/10.3390/fluids4020101

**Chicago/Turabian Style**

Wen, Baole, and Gregory P. Chini.
2019. "On Moderate-Rayleigh-Number Convection in an Inclined Porous Layer" *Fluids* 4, no. 2: 101.
https://doi.org/10.3390/fluids4020101