Session 9

Thursday, 19 May 2016, 11:00 AM to 12:30 PM

20 minutes per talk, 10 minutes for questions and speaker change

From	To	Name	Affiliation	Title
10:30am	11:00am	Themis Sapsis	Massachusetts Institute of Technology	A minimization principle for the description of modes associated with finite-time instabilities
11:00am	11:30am	Sanjeeva Balasuriya	University of Adelaide	Transport between two fluids across their mutual flow interface: the streakline approach
11:30am	12:00pm	Nicholas Ouellette	Stanford University	Hyperbolic Neighborhoods in Unsteady Flow

 

Session 9 Abstracts

A minimization principle for the description of modes associated with finite-time instabilities

Themis Sapsis (Massachusetts Institute of Technology)
We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent,
orthonormal basis that captures directions of the phase space associated with transient instabilities. While
these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics
through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to
extreme responses. However, their essentially transient character makes their description a particularly
challenging task. We develop a minimization framework that focuses on the optimal approximation of the
system dynamics in the neighbourhood of the system state. This minimization formulation results in
differential equations that evolve a time-dependent basis so that it optimally approximates the most
unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear
systems, including the advection–diffusion operator in a strongly non-normal regime as well as the Orr–
Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low-dimensional system with
transient instabilities and the vertical jet in cross-flow. We demonstrate that the time-dependent subspace
captures the strongly transient non-normal energy growth (in the short-time regime), while for longer
times the modes capture the expected asymptotic behavior.
Joint work with Hessam Babaee.

Transport between two fluids across their mutual flow interface: the streakline approach

Sanjeeva Balasuriya (University of Adelaide)
Mixing between initially coherent blobs of two different fluids must be initiated by fluid transporting across
the mutual fluid interface. In general, there is no necessity for the physical flow barrier between the fluids
to be associated with extremal or exponential attraction as might be revealed by applying Lagrangian
coherent structures, finite-time Lyapunov exponents or other methods on the fluid velocity. Crossinterface
transport can be achieved by imposing unsteady velocity agitations in the interface region. It is
shown that streaklines are key to understanding the breaking of the interface, and a theory for locating
the relevant streaklines is presented. The streaklines are numerically verified in several examples. The
relationship to the unsteady advective transport between the two fluids is established.

Hyperbolic Neighborhoods in Unsteady Flow

Nicholas Ouellette (Stanford University)
Hyperbolic points and their unsteady generalization (hyperbolic trajectories) drive the exponential
stretching that is the hallmark of nonlinear and chaotic flow. In infinite-time flows, the stable and unstable
manifolds attached to each hyperbolic trajectory mark fluid elements that asymptote towards the
hyperbolic trajectory, and which will therefore eventually experience exponential stretching. In an
unsteady finite-time flow, however, hyperbolic trajectories (which move around in the flow) need not
remain hyperbolic for all time. We introduce a new way to determine their region of influence, which we
term a hyperbolic neighborhood, which marks fluid elements whose dynamics are instantaneously
dominated by the hyperbolic trajectory. Fluid elements traversing a flow experience exponential boosts
in stretching while within these time-varying regions. We demonstrate our method with several analytical
examples, as well as with experimental data from a quasi-two-dimensional laboratory flow.