# Inclined porous medium convection at large Rayleigh number

## Abstract

High-Rayleigh-number ($Ra$$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle$$\unicode[STIX]{x1D719}$$of the layer satisfies$$0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($$\unicode[STIX]{x1D719}=0^{\circ }$$) case, except that as$$\unicode[STIX]{x1D719}$$is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number)$$Nu\sim CRa$$with a$$\unicode[STIX]{x1D719}$$-dependent prefactor$$C$$. When$$\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$$, however, where$$30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$$independently of$$Ra$$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large$$Ra$$and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when$$\unicode[STIX]{x1D719}\neq 0^{\circ }$$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for$$\unicode[STIX]{x1D719}=0^{\circ }$(Wenet al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.

- Authors:

- Publication Date:

- Research Org.:
- Energy Frontier Research Centers (EFRC) (United States). Center for Frontiers of Subsurface Energy Security (CFSES); Univ. of Texas, Austin, TX (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1538909

- DOE Contract Number:
- SC0001114

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Fluid Mechanics

- Additional Journal Information:
- Journal Volume: 837; Journal ID: ISSN 0022-1120

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Mechanics; Physics

### Citation Formats

```
Wen, Baole, and Chini, Gregory P.
```*Inclined porous medium convection at large Rayleigh number*. United States: N. p., 2018.
Web. doi:10.1017/jfm.2017.863.

```
Wen, Baole, & Chini, Gregory P.
```*Inclined porous medium convection at large Rayleigh number*. United States. doi:10.1017/jfm.2017.863.

```
Wen, Baole, and Chini, Gregory P. Fri .
"Inclined porous medium convection at large Rayleigh number". United States. doi:10.1017/jfm.2017.863.
```

```
@article{osti_1538909,
```

title = {Inclined porous medium convection at large Rayleigh number},

author = {Wen, Baole and Chini, Gregory P.},

abstractNote = {High-Rayleigh-number ($Ra$) convection in an inclined two-dimensional porous layer is investigated using direct numerical simulations (DNS) and stability and variational upper-bound analyses. When the inclination angle$\unicode[STIX]{x1D719}$of the layer satisfies$0^{\circ }<\unicode[STIX]{x1D719}\lesssim 25^{\circ }$, DNS confirm that the flow exhibits a three-region wall-normal asymptotic structure in accord with the strictly horizontal ($\unicode[STIX]{x1D719}=0^{\circ }$) case, except that as$\unicode[STIX]{x1D719}$is increased the time-mean spacing between neighbouring interior plumes also increases substantially. Both DNS and upper-bound analysis indicate that the heat transport enhancement factor (i.e. the Nusselt number)$Nu\sim CRa$with a$\unicode[STIX]{x1D719}$-dependent prefactor$C$. When$\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{t}$, however, where$30^{\circ }<\unicode[STIX]{x1D719}_{t}<32^{\circ }$independently of$Ra$, the columnar flow structure is completely broken down: the flow transitions to a large-scale travelling-wave convective roll state, and the heat transport is significantly reduced. To better understand the physics of inclined porous medium convection at large$Ra$and modest inclination angles, a spatial Floquet analysis is performed, yielding predictions of the linear stability of numerically computed, fully nonlinear steady convective states. The results show that there exist two types of instability when$\unicode[STIX]{x1D719}\neq 0^{\circ }$: a bulk-mode instability and a wall-mode instability, consistent with previous findings for$\unicode[STIX]{x1D719}=0^{\circ }$(Wenet al., J. Fluid Mech., vol. 772, 2015, pp. 197–224). The background flow induced by the inclination of the layer intensifies the bulk-mode instability during its subsequent nonlinear evolution, thereby favouring increased spacing between the interior plumes relative to that observed in convection in a horizontal porous layer.},

doi = {10.1017/jfm.2017.863},

journal = {Journal of Fluid Mechanics},

issn = {0022-1120},

number = ,

volume = 837,

place = {United States},

year = {2018},

month = {1}

}

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