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.
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.