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CATEGORIES:Fluid Mechanics (DAMTP)
SUMMARY:Verifying global stability of fluid flows despite
transient growth of energy - David Goluskin\, Univ
ersity of Victoria
DTSTART;TZID=Europe/London:20220218T160000
DTEND;TZID=Europe/London:20220218T170000
UID:TALK168443AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/168443
DESCRIPTION:Verifying nonlinear stability of a laminar fluid f
low against all perturbations is a classic challen
ge in fluid dynamics. All past results rely on mon
otonic decrease of a perturbation energy or a simi
lar quadratic generalized energy. This "energy met
hod" cannot show global stability of any flow in w
hich perturbation energy may grow transiently. For
the many flows that allow transient energy growth
but seem to be globally stable (e.g. pipe flow an
d other parallel shear flows at certain Reynolds n
umbers) there has been no way to mathematically ve
rify global stability. After explaining why the en
ergy method was the only way to verify global stab
ility of fluid flows for over 100 years\, I will d
escribe a different approach that is broadly appli
cable but more technical. This approach\, proposed
in 2012 by Goulart and Chernyshenko\, uses sum-of
-squares polynomials to computationally construct
non-quadratic Lyapunov functions that decrease mon
otonically for all flow perturbations. I will pres
ent a computational implementation of this approac
h for the example of 2D plane Couette flow\, where
we have verified global stability at Reynolds num
bers above the energy stability threshold. This en
ergy stability result for 2D Couette flow had not
been improved upon since being found by Orr in 190
7. The results I will present are the first verifi
cation of global stability – for any fluid flow –
that surpasses the energy method. This is joint wo
rk with Federico Fuentes (Universidad Católica de
Chile) and Sergei Chernyshenko (Imperial College L
ondon).
LOCATION:MR2\, Centre for Mathematical Sciences\, Wilberfor
ce Road\, Cambridge
CONTACT:Prof. Jerome Neufeld
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