Roll-cell instabilities in rotating laminar and trubulent channel flows

Abstract

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 Rec= 88.53 and rotation number Ro= 0.5. At higher Reynolds numbers, unstable conditions canexist over the range of rotation numbers given by 0 < Ro< 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 Ro= 0.022, a value in remarkable agreement with the experimentally observed appearance of roll instabilities in rotating turbulent channel flow.

Secondary instability and tertiary states in rotating plane Couette flow

“…For the majority of this paper we describe rotation in terms of Ro; however, in §11 we use Ω , in accordance with experimental preference. Lezius & Johnston (1976) have shown that RPCF loses stability in the rotation parameter range 0 < Ro < 1, to spanwise periodic disturbances with critical wavenumber β ≈ 1.5585. The neutral curve takes the form…”

Transient growth and instability in rotating boundary layers

“…In RB flow, the condition for instability (24) found to re-stabilize channel flows, 8,24 at Ro values consistent with the vorticity criterion (24).…”

“…2b), that depends on the value of Ro, in agreement with previous results for other rotating shear flows. 7,8,9,10 The extent of the rotationally unstable region grows rapidly with rotation, and is quite large at Ro = 300 which is the strongest rotation depicted in figure Fig. 2a.…”

“…On the contrary, weak anti-cyclonic rotation results in unstable streamwise vortices. All these properties have been verified for rotating Poiseuille flow, 7,8,9 rotating planar wakes and rotating mixing layers. 10 In addition, instability thresholds have been determined in terms of Rossby and Reynolds numbers (the Reynolds number Re and Rossby number Ro are defined explicitly in §III(A)).…”

“…This instability arises due to the imbalance of pressure and Coriolis forces. 1 Rotating channel flows have been shown to exhibit this instability, in both theory and experiment, 7,8,9 while rotating free shear flows have been predicted to be likewise unstable. 10 However, this modal instability is present only for a certain range of rotations rates and depends on the direction of the frame rotation vector (here the z-axis) with respect to the vorticity vector of the base flow.…”