2007-2008
Compact Courses
Dynamics of transition and turbulence in shear flows
by
Prof. Divakar Viswanath
University of Michigan, USA
March 2008
at
L H - 1, Department of Mathematics
Indian Institute of Science, Bangalore
Abstract
The statistical study of turbulence is primarily interested in space and time averages formed in many different ways. A dynamical study would be interested not just in averages but also in the way in which the whole velocity field, thought of as a point in phase space, evolves over time. The dynamical point of view is natural in some instances as evidenced by certain striking coherent structures in the near-wall region of turbulent shear flows that were first detected in the 1960s. Although understanding the dynamics of transition and turbulence appear to be very difficult problems, it appears as if a little light on that problem may have been shed by some computations and experiments carried out in the last ten years or so.
These lectures will provide an introduction to the dynamical study of transition and turbulence. We will show how to reformulate the Navier-Stokes equation for channels and pipes to get rid of pressure so that the equation becomes a dynamical system. From that reformulation, we will derive numerical methods for computing periodic solutions, relative periodic solutions, steady solutions, and traveling waves. Some of the periodic solutions reproduce the breakup and re-formation of streaks observed in the near-wall region. Both recent experiments and computations suggest that lower-branch traveling waves could be important to transition to turbulence. That connection, along with other related developments, will be clarified and explained.
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