Shear and Buoyancy

In inclined convection a fluid layer is tilted with respect to the gravity direction, and thus, not only buoyancy, but also shear drives the flow in this case. An inclination angle of β=0 corresponds to Rayleigh–Bénard convection, i.e. a fluid heated from below and cooled from above, and β=π/2 corresponds to vertical convection, where a fluid layer is confined between vertically aligned heating and cooling plates. The heat flux dependence on the inclination angle is not universal and is a also a complicated non-monotonic function of the Rayleigh and Prandtl number. Moreover, a slight cell tilt may not only stabilise a large scale circulation but can also enforce one for cases where the preferred Rayleigh–Bénard state is a more complicated multiple roll state.

In liquid metals a strong heat flux increase is observed in the case of inclined convection (see figure below). For liquid sodium the increase is almost 20-30% at the optimal inclination angle compared to RBC case and even for vertical convection (β=π/2) the heat flux is incresed by about 10%. This strong increase is only observed in low-Prandtl number fluids, like liquid sodium. The increase for the case of Pr=1, Ra = 10⁹ is only approximately 6% at optimal inclination angle.

Normalised Nusselt number Nu versus the inclination angle β for DNS at Ra=1.67⋅107 (diamonds), LES at Ra=1.5⋅107 (triangles), experiments at Ra=1.42⋅107 (open circles) all for Pr≈0.009, DNS at Ra=1.67⋅106, Pr=0.094 (crosses) and DNS at Ra=109, Pr=1 (squares). — Normalised Nusselt number Nu versus the inclination angle β for DNS at Ra=1.67⋅10⁷ (diamonds), LES at Ra=1.5⋅10⁷ (triangles), experiments at Ra=1.42⋅10⁷ (open circles) all for Pr≈0.009, DNS at Ra=1.67⋅10⁶, Pr=0.094 (crosses) and DNS at Ra=10⁹, Pr=1 (squares).

Normalised Nusselt number Nu versus the inclination angle β for DNS at Ra=1.67⋅10⁷ (diamonds), LES at Ra=1.5⋅10⁷ (triangles), experiments at Ra=1.42⋅10⁷ (open circles) all for Pr≈0.009, DNS at Ra=1.67⋅10⁶, Pr=0.094 (crosses) and DNS at Ra=10⁹, Pr=1 (squares).

The reason for the strong increase in inclined liquid sodium convection is the suppression of the so-called twisting and sloshing modes (see the video below). In the case of RBC the phase of the large-scale circulation close to the heated plate shows a anti-correlation compared to the phase at the cooled plate. This comes along with a strongly varying instantaneous heat flux. However, inclination suppresses these modes, which leads to a less varying and more efficient heat flux.

These two videos at inclination angles of 36° (top) and 72° (bottom) show the temperature isosurfaces for the DNS at Ra=1.67⋅10⁷, Pr=0.0094 from two perspectives, the inclined plane view (left) and a view that is facing the hot ascending plumes (right). While the first case shows the sloshing and twisting, it is suppressed in the second case.

Isosurfaces of the instantaneous temperature fields in inclined convection in cylindrical containers filled with a fluid of Pr=1, for Ra = 106, 107, 108. The sizes of the convection cells reflect the proportions between the different Ra. Adopted from Shishkina & Horn, J. Fluid Mech. 790 (2016), R3. — Isosurfaces of the instantaneous temperature fields in inclined convection in cylindrical containers filled with a fluid of Pr=1, for Ra = 10⁶, 10⁷, 10⁸. The sizes of the convection cells reflect the proportions between the different Ra. Adopted from Shishkina & Horn, *J. Fluid Mech. **790** (2016), R3*.

Isosurfaces of the instantaneous temperature fields in inclined convection in cylindrical containers filled with a fluid of Pr=1, for Ra = 10⁶, 10⁷, 10⁸. The sizes of the convection cells reflect the proportions between the different Ra. Adopted from Shishkina & Horn, *J. Fluid Mech. **790** (2016), R3*.

Instantaneous temperature fields in inclined convection cell (β=0.4π) filled with a fluid of Pr=1, for Ra=10⁹.

In the case of laminar vertical convection (β=π/2) it is possible to derive the dependence of the Reynolds number Re and the Nusselt number Nu on the Rayleigh number Ra and the Prandtl number Pr by advancing the Prandtl boundary layer theory, see Shishkina, PRE 93 (2016). This yields two limiting scaling regimes: Nu ~ Pr^1/4 Ra^1/4, Re ~ Pr^–1/2 Ra^1/2 for Pr≪1 and Nu ~ Pr⁰ Ra^1/4, Re ~ Pr^–1 Ra^1/2 for Pr≫1. DNS show that the transition between the regimes takes place for Pr around 10^–1.

(a), (c) Ra-dependences and (b), (d) Pr-dependences of (a), (b) the Nusselt number and (c), (d) the Reynolds number, as obtained in the DNS of vertical convection (β=π/2) for (a), (c) Pr=0.1 (dotted circle), Pr=1 (upward triangle), Pr=10 (dotted square) and for (b), (d) Ra=10 6 (dotted diamond) and Ra=10 7 (downward triangle). Adopted from Shishkina, Phys. Rev. E 93 (2016). — (a), (c) Ra-dependences and (b), (d) Pr-dependences of (a), (b) the Nusselt number and (c), (d) the Reynolds number, as obtained in the DNS of vertical convection (β=π/2) for (a), (c) Pr=0.1 (dotted circle), Pr=1 (upward triangle), Pr=10 (dotted square) and for (b), (d) Ra=10⁶ (dotted diamond) and Ra=10⁷ (downward triangle). Adopted from Shishkina, *Phys. Rev. E 93 (2016)*.

(a), (c) Ra-dependences and (b), (d) Pr-dependences of (a), (b) the Nusselt number and (c), (d) the Reynolds number, as obtained in the DNS of vertical convection (β=π/2) for (a), (c) Pr=0.1 (dotted circle), Pr=1 (upward triangle), Pr=10 (dotted square) and for (b), (d) Ra=10⁶ (dotted diamond) and Ra=10⁷ (downward triangle). Adopted from Shishkina, *Phys. Rev. E 93 (2016)*.

Lukas Zwirner, Ruslan Khalilov, Ilya Kolesnichenko, Andrey Mamykin, Sergei Mandrykin, Alexander Pavlinov, Alexander Shestakov, Andrei Teimurazov, Peter Frick, Olga Shishkina. The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection. J. Fluid Mech. 884 (2020), A18.

Lukas Zwirner and Olga Shishkina. Confined inclined thermal convection in low-Prandtl-number fluids. J. Fluid Mech. 850 (2018), 984–1008.

Olga Shishkina. Momentum and heat transport scalings in laminar vertical convection. Phys. Rev. E 93 (2016), 051102 (R) .

Olga Shishkina, Susanne Horn. Thermal convection in inclined cylindrical containers. J. Fluid Mech. 790 (2016), R3.

Shear and Buoyancy

Further reading