Roll-cell instabilities in rotating laminar and trubulent channel flows

The stability of laminar and turbulent channel flow is examined for cases where Coriolis forces are introduced by steady rotation about an axis perpendicular to the plane of mean flow. Linearized equations of motion are derived for small disturbances of the Taylor type. Conditions for marginal stability in laminar Couette and Poiseuille flow correspond, in part, to the analogous solutions of buoyancy-driven convection instabilities in heated fluid layers, and to those of Taylor instabilities in the flow between rotating cylinders. In plane Poiseuille flow with rotation, the critical disturbance mode occurs at a Reynolds number of R ec = 88.53 and rotation number R o = 0.5. At higher Reynolds numbers, unstable conditions canexist over the range of rotation numbers given by 0 < R o < 3, provided the undisturbed flow remains laminar. A two-layer model is devised to investigate the onset of longitudinal instabilities in turbulent flow. The linear disturbance equations are solved essentially in their laminar form, whereby the velocity gradient of laminar flow is replaced by a numerically computed profile for the gradient of the turbulent mean velocity. The turbulent stress levels in the stable and unstable flow regions are represented by integrated averages of the eddy viscosity. Onset of instability for Reynolds numbers between 6000 and 35 000 is predicted to occur at R o = 0.022, a value in remarkable agreement with the experimentally observed appearance of roll instabilities in rotating turbulent channel flow.