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.

inclined convection
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 ~ Pr1/4 Ra1/4,   Re ~ Pr –1/2 Ra1/2  for Pr≪1 and Nu ~ Pr 0 Ra1/4,   Re ~ Pr –1 Ra1/2 for Pr≫1. DNS show that the transition between the regimes takes place for Pr around 10 –1.

vertical convection
(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).

Further reading

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.