Primary and secondary motions in an annular plane Couette flow

Axisymmetric direct numerical simulations are carried out to study the hydrodynamics of a laminar, stationary, incompressible, Newtonian, single-phase flow in an Annular Plane Couette (APC) channel. This configuration is that of a Straight Plane Couette (SPC) flow but curved around itself to form an annulus. These simulations are validated by Particle Image Velocimetry measurements at different Reynolds numbers. The flow is analyzed using three dimensionless parameters: the channel aspect ratio $A_c$, which controls the effects of sidewall confinement, the channel curvature ratio $C_r$, which affects the centrifugal forces due to curvature, and the Reynolds number ${\rm Re}$. The rotation of the top annular plate generates a main flow in the azimuthal direction, while generating a secondary recirculation flow in the plane of the channel cross-section, due to the presence of a centrifugal force difference. As a result, the vertical profile of the azimuthal velocity deviates from the classical linear profile of the SPC flow, adopting an unexpected S-shape, similar to that observed in a turbulent plane Couette flow. Depending on the values of $A_c$ and $C_r$, the flow exhibits a wide range of behaviors, from a quasi-2D flow with a homogeneous shear rate at moderate ${\rm Re}$ with appropriate geometrical parameters ($A_c \gtrsim 5$, $C_r \lesssim 0.1$), to a complex 3D flow otherwise. Whereas it is laminar, the APC flow shares a strong analogy with a turbulent Taylor Couette (TC) flow. At large Reynolds numbers, the velocity gradients concentrate near the wall and the flow reaches an asymptotic regime where the torque scales as ${\rm Re}^\alpha$ and the flow structure becomes independent of ${\rm Re}$. Such properties make the APC flow an interesting configuration for fundamental investigations as a complement of the TC flow when shear and gravity are required to be in the same plane. In particular, it can be used to investigate the rheology of a dispersed two-phase mixture similarly to the work done by Yi et al. \cite{YI2021,Yi2022} in the TC device.