Hydrodynamic instabilities

Taylor-Couette instability

Introduction
The geometry that consists in 2 concentric cylinders which border a fluid, is well known in fluid mechanics. Indeed it is a famous exemple of exact solution of the Navier-Stockes equations.These results are reminded here. The shape of the velocity field is orthoradial. But for certain cases some instabilities appear with a shape of toroidal rolls.
• If the interior cylinder is maintained fix and the exterior one moves, the flow is stable until the velocity becomes high and the flow clearly turbulent.
• If the exterior cylinder is fixed and then the interior starts to turn, some values of rotation speed give another kind of instabilities. Some tore rolls appear around the interior cylinder.

The first flow was studied by M. Couette in1901 while G.I. Taylor discovered the instabilities for the first time in 1923. This problem looks like the Rayleigh-Benard problem. In this problem rolls appear because of the buoyancy forces due to the instable stratification of density.
Here the velocity gradients stand for the temperature gradients. Then the variation of kinetic momentum create a radial gradient of centrifuge force.

An adimensionnal number can be defined such as the Rayleigh number. Here it is the Taylor number defined as:

where Omega is the rotation speed, R is the radius of the interior cylinder, a the distance between both cylinders and Nu the kinematic viscosity of the fluid.

The aim of this study is to show Taylor Couette instabilities with a CFD code like Fluent.
Exact Navier-Stockes solutions
Couette flow is a basic example of exact Navier-Stockes solution in a laminar case. Some results can be reminded.
With the boundary conditions v(r=R1)=Omega1*R1  and  v(r=R2)=Omega2*R2, the velocity field is:
V(r)=0
V(x)=0
V(theta)= a*r+b/r with the constants a=(Omega2*R22 - Omega1*R12 )/(R22 -R12) and b=R12 R22(Omega1-Omega2)/(R22 -R12)
The velocity field in a laminar case is purely orthoradial.