The Transition to Turbulence via the Edge of Chaos in Shear Flows

Dr. Lina Kim
Postdoctoral Researcher and Lecturer
University of California Santa Barbara

The transition to turbulence in linearly stable shear flows is one of the most intriguing and  outstanding  problems  in  classical  physics. Turbulent  dynamics  are  readily observed  at  flow  speeds  where  the  laminar  state  remains  stable  under  infinitesimal perturbations. Moreover,  for  a  smaller  class  of  shear  flows,  such  as  plane  Couette and pipe flows, linear stability theory predicts that the laminar state remains stable for all  Reynolds  numbers. However,  numerical  simulations  and  experiments  provide evidence  that  these  flows  exhibit  turbulence,  which  can  decay  spontaneously,  for sufficiently high Reynolds numbers and perturbations.

These observations are in agreement with a chaotic saddle coexisting with the laminar state in the system's state space. Analyzing the 'edge of chaos' between laminar and turbulent  dynamics  for  these  systems  have  generated  additional  insight  into  the transition  mechanism. This  boundary  contains  'edge  states'  which  are  too  weak  to become  turbulent  and  too  strong  to  laminarize  can  be  either  dynamically  simple  or complex structures. To identify the edge states, we employ an iterated edge tracking algorithm which is implemented for several refinements until we find a trajectory which neither becomes turbulent or laminarizes. We show that for edge states with a stable manifold of codimension-1, the edge tracking algorithm  will converge to it. We study the  edge  states  for  various  channel  sizes  and  compare  their  dynamics  to  those  of edge states for other shear flows.

Lina  Kim  received  a  BS  in  Mechanical  Engineering  from  the  University  of California Riverside  in  1999,  an  MS  and  PhD  in  Mechanical  Engineering  from  the University  of  California  Santa  Barbara,  respectively  in  2003  and  2009. Her  current research  interests  focus  on  investigating  the  transition  to  turbulence  via  edge  state mechanisms  in  shear  flows  such  as  sinusoidal  shear  flow,  plane  Couette  flow,  pipe
flow,  and  plane  Poiseuille  flow. Current  topics  of  interest  include  the  role  of  linear  transient  energy  growth  in  the  transition  to  turbulence,  turbulence  control,  applying edge  of  chaos  ideas  to  large-scale  interconnected  and  biological  systems. She  is  currently  a  postdoctoral researcher  and  lecturer  at  the  University  of  California  Santa Barbara.