Our work on absolute and convective instabilities in confined shear flows is quite fundamental. This page contains a brief introduction to these instabilities and a brief description of our current work.
The stability of a flow can be determined by calculating the response to an impulse at, say x=0 and t=0 . If the amplitude of the response dies down in all space, the flow is stable. If it amplifies, a further distinction is necessary.
If the response is convected away from x=0, the flow is locally convectively unstable.
If the response expands around x=0, the flow is locally absolutely unstable.
Therefore, to determine whether a flow is convectively unstable or absolutely unstable, the part of the response that stays at x=0 is examined. This is the part with zero group velocity. If this dies away, the flow is convectively unstable. If this amplifies, it is absolutely unstable.
Local instability and global modes
The global behaviour of the flow depends on the competition between local instability and basic advection.
A single unconfined shear layer, for example, is locally convectively unstable. Perturbations at the start of the shear layer are selectively amplified. However, they are constantly transported away from the unstable region and this flow is globally linearly stable. Therefore, this flow behaves as an amplifier of extrinsic perturbations.
On the other hand, a double shear layer can be locally absolutely unstable. Resonances may occur in the absolutely unstable region and they are not transported away by the flow. This flow may be globally linearly unstable. It behaves as an oscillator rather than an amplifier. A common example of this is vortex shedding behind a bluff body, which is the non-linear development of such an instability.
We have discovered that confinement has a strong effect on transition between absolute and convective instability. This is particularly interesting when combined with other factors, such as swirl.
By confining a portion of a shear flow, one can create a region which is unambiguously absolutely unstable, surrounded by regions which are unambiguously convectively unstable, with abrupt transitions between the two. This is a useful case, both experimentally and numerically, for testing the relationship between local and global instability.
We are currently exploring these ideas further, both theoretically and experimentally.
Patrick Huerre and Peter Monkewitz (1990) Local and global instabilities in spatially developing flows
Annual Review of Fluid Mechanics22, 473 - 537
Jean-Marc Chomaz (2005) Global instabilities in spatially-developing flows: non-normality and nonlinearity
Annual Review of Fluid Mechanics37, 357 - 392