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2006 - 2007
announces
compact course on
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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