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

We investigate the flow structure and dynamics of moderate-Rayleigh-number ( R a ) thermal convection in a two-dimensional inclined porous layer. High-resolution numerical simulations confirm the emergence of O ( 1 ) aspect-ratio large-scale convective rolls, with one ‘natural’ roll rotating in the counterclockwise direction and one ‘antinatural’ roll rotating in the clockwise direction. As the inclination angle ϕ is increased, the background mean shear flow intensifies the natural-roll motion, while suppressing the antinatural-roll motion. Our numerical simulations also reveal—for the first time in single-species porous medium convection—the existence of spatially-localized convective states at large ϕ , which we suggest are enabled by subcritical instability of the base state at sufficiently large inclination angles. To better understand the physics of inclined porous medium convection at different ϕ , we numerically compute steady convective solutions using Newton iteration and then perform secondary stability analysis of these nonlinear states using Floquet theory. Our analysis indicates that the inclination of the porous layer stabilizes the boundary layers of the natural roll, but intensifies the boundary-layer instability of the antinatural roll. These results facilitate physical understanding of the large-scale cellular flows observed in the numerical simulations at different values of ϕ .

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

We investigate the flow structure and dynamics of moderate-Rayleigh-number ($Ra$) thermal convection in a two-dimensional inclined porous layer. Direct numerical simulations (DNS) confirm the emergence of $\mathit{O}(1)$ aspect-ratio large-scale convective rolls, with one `natural' roll rotating in the counterclockwise direction and one `antinatural' roll rotating in the clockwise direction. As the inclination angle $\phi$ is increased, the background mean shear flow intensifies the natural-roll motion, while suppressing the antinatural-roll motion. Moreover, our DNS reveal---for the first time in single-species porous medium convection---the existence of \emph{spatially-localized} convective states at large $\phi$, which we suggest are enabled by subcritical instability of the base state at sufficiently large inclination angles. To better understand the physics of inclined porous medium convection at different $\phi$, we numerically compute steady convective solutions using Newton iteration and then perform secondary stability analysis of these nonlinear states using Floquet theory. Our analysis indicates that the inclination of the porous layer stabilizes the boundary layers of the natural roll, but intensifies the boundary-layer instability of the antinatural roll. These results facilitate physical understanding of the large-scale cellular flows observed in the DNS at different values of $\phi$.

Inclined porous medium convection at large Rayleigh number

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 $\unicode[STIX]{x1D719}=0^{\circ }$ (Wen et 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.

Scaling bounds on dissipation in turbulent flows

We propose a new rigorous method for estimating statistical quantities in fluid dynamics such as the (average) energy dissipation rate directly from the equations of motion. The method is tested on shear flow, channel flow, Rayleigh–Bénard convection and porous medium convection.

Structure and stability of steady porous medium convection at large Rayleigh number

A systematic investigation of unstable steady-state solutions of the Darcy–Oberbeck–Boussinesq equations at large values of the Rayleigh number $\mathit{Ra}$ is performed to gain insight into two-dimensional porous medium convection in domains of varying aspect ratio $L$. The steady convective states are shown to transport less heat than the statistically steady ‘turbulent’ flow realised at the same parameter values: the Nusselt number $\mathit{Nu}\sim \mathit{Ra}$ for turbulent porous medium convection, while $\mathit{Nu}\sim \mathit{Ra}^{0.6}$ for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-flux-maximising solutions shrinks roughly as $L\sim \mathit{Ra}^{-0.5}$, reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech., vol. 737, 2013, pp. 205–231) by treating a base convective state, and secondary stability modes, that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small-aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical $\mathit{Ra}$-dependent aspect ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically steady porous medium convection results from a balance between the competing effects of these two types of instability.