MULTIPLE STATES IN FLUID DYNAMICS:  STEADY OR TURBULENT FLOW IN A ROTATING CYLINDER

A long cylinder, rotating about its axis, provides a simple and dramatic visualization of multiple states in a fluid dynamical system.  The idea of multiple states plays a role in many problems:  thermal convection patterns, blocking in the Earth's atmosphere, regimes of climate, regimes of ocean circulation, and many others.

There are two long-time states of flow in a horizontal cylinder, half-filled with water, that rotates rapidly about its axis.  Which state is achieved depends on how the experiment is performed, i.e. on the initial conditions.  Basically:

a)  If the fluid starts out centrifuged to the wall, a stable regime of solid rotation with all the fluid out at the wall is attained.  In the rotating frame, this state is steady and axisymmetric and lasts forever (i.e. it is stable to small and even moderately large perturbations).

b)  If the fluid starts out at rest,  just lying at the bottom of the cylinder so its free surface is parallel to the horizontal axis, then the end state is unsteady, turbulent, three dimensional. 

The trick is to get to state a).  Getting to b) is easy. 

The apparatus consists of a long closed cylinder attached to a portable electric drill.

Fig. 1.  The long cylinder, constructed of 1/4 inch thick plexiglass, is half filled with water.  Food coloring or Kalliroscope may be added for additional visualization.  The optional aluminum base and bearings provide a more rugged demonstration.  The short vertically oriented cylinder shown in this photograph is used to illustrate the simple static equilibrium that accrues when gravity and centrifugal accelerations are perpendicular.

Fig. 2.  Sketch of the apparatus and the path to get to state a).  First hold the cylinder vertically.  Run the drill at its highest speed until the fluid is centrifuged out to the wall (see movies of static equilibria).  Once this happens slowly tip the cylinder to the horizontal position while keeping the rotation rate constant.  The end state will be steady solid rotation with the fluid sitting as illustrated.

To get to state b), simply hold the cylinder horizontally and initiate rotation (i.e. turn on the drill).  The fluid starting from this state is unstable, a turbulent rolls ("large eddies"), with a separation down the cylinder at distances about equal to the diameter may be observed.

Quick Time Movie of Both End States (small screen,  4.7MB)

Quick Time Movie of Both End States (large screen,  12MB)