Principle of TaylorCouette instability
Manifestations of TaylorCouette instability
Principle of TaylorCouette instability
The geometry consists in two concentric cylinders. A fluid is confined between the two cylinders.
Scheme of the domain for TaylorCouette instability
If the interior cylinder is mentained fix and the
exterior one moves with velocity ,
the flow is stable and becomes clearly turbulent for one certain value
of but there is no transition
and no instability.
On the other hand, if the exterior cylinder is maintained
fix and the interior one moves with velocity ,
for a critical value of rotation speed ,
instabilities can appear : there is a stacking of contrarotative cells
and we speak also about tore rolls around the interior cylinder inside
the fluid. The shape of the velocity field inside fluid is then orthoradial.



Tore rolls in TaylorCouette instability
If the rotative speed of the exterior cylinder increases
tore rolls begin to deform to give turbulence.


Tore rolls for a higher rotation speed of the exterior cylinder
This firt flow was studied by Couette in 1901 and Taylor discovered instabilities for the first time in 1923 with the second flow.
This problem looks like the RayleighBenard one ; it is also an instability with threshold.
This instability is due to the destabilizing effect of the centrifugal force, and there is a competition between this effect and the stabilizing effect of viscous drag force.
The gradient of centrifugal force due to variation of kinetic momentum gives velocity gradients so rolls appear inside the fluid if centrifugal force is superior to viscous drag force.
An adimensionnal number can be found such as the Rayleigh number. Here it is the Taylor number which represents the rate between the centrifugal force and the viscous drag force :
where is the rotation velocity of the interior cylinder, R the average radius of cylinders ( ), d the distance between both cylinders ( ), the kinematic viscosity of the fluid.
The critical value of Taylor number to develop TaylorCouette instability is :
Like for BenardMarangoni instability, manifestations of TaylorCouette instability look like manisfestations of RayleighBenard instability.(See Manifestations of RayleighBenard instability).
Moreover, it is not easy to see with our eyes such instability because it is an internal instability and our eyes don't detect movements of fluid inside a big mass of this fluid if there is no colouring or tracer.
Simulation of TaylorCouette instability