Rotating free-shear flows. I. Linear stability analysis

Abstract

Using linear stability analysis, the instability characteristics are examined of both planar wakes and mixing layers subjected to rigid-body rotation with axis of rotation perpendicular to the plane of the ambient flow. In particular, the tendency of rotation to stabilize or destabilize three-dimensional motions is addressed. In the inviscid limit the results are consistent with the criterion established by Pedley [J. Fluid Mech. 35, 97 (1969)] and Bradshaw [J. Fluid Mech. 36, 177 (1969)]. Cyclonic rotation and strong anticyclonic rotation tend to stabilize three-dimensional motions, whereas weaker anticyclonic rotation (Ro≳1) acts to destabilize these motions. This latter instability is in the form of streamwise rolls, similar to previous results obtained for boundary layer and channel flows. It is found that this instability is stronger than the coexisting Kelvin–Helmholtz instability for roughly the range 1.5<Ro<8, and its effect is maximum for Ro≂2. For the case of constant ambient shear, exact solutions are obtained which give further insight into the nature of the instability.

Instability patterns between counter-rotating disks

“…In addition to the Kelvin-Helmholtz instability, also present for linear non-rotating shear layers, centrifugal effect may occur, stabilizing or destabilizing the perturbation (Yanase et al, 1993;Liou, 1994). The extreme case where rotation dominates the dynamics (small Rossby number) is of great importance in geophysical flows (the so-called barotropic instability).…”

Transient growth and instability in rotating boundary layers

“…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.…”

“…Because the development is weak and because a great deal of closely related previous work 7,8,10,11,38 is based on temporal theory, this study adopted a temporal framework. However recent spatial analyses of ordinary boundary layer transient growth 39,40,41 are shown to be more realistic with respect to observations.…”

“…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. When the latter quantities are parallel (anti-parallel), the rotation is called cyclonic (anti-cyclonic).…”

“…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)). Such an instability mechanism is fundamentally inviscid and is reduced by the presence of viscosity.…”