3,353 research outputs found
Invariant Solution underlying Oblique Stripe Patterns in Plane Couette Flow
When subcritical shear flows transition to turbulence, laminar and turbulent
flow often coexists in space, giving rise to turbulent-laminar patterns. Most
prominent are regular stripe patterns with large-scale periodicity and oblique
orientation. Oblique stripes are a robust phenomenon, observed in experiments
and flow simulations, yet their origin remains unclear. We demonstrate the
existence of an invariant equilibrium solution of the fully nonlinear 3D
Navier-Stokes equations that resembles the oblique pattern of turbulent-laminar
stripes in plane Couette flow. We uncover the origin of the stripe equilibrium
and show how it emerges from the well-studied Nagata equilibrium via two
successive symmetry-breaking bifurcations
Studying edge geometry in transiently turbulent shear flows
In linearly stable shear flows at moderate Re, turbulence spontaneously
decays despite the existence of a codimension-one manifold, termed the edge of
chaos, which separates decaying perturbations from those triggering turbulence.
We statistically analyse the decay in plane Couette flow, quantify the breaking
of self-sustaining feedback loops and demonstrate the existence of a whole
continuum of possible decay paths. Drawing parallels with low-dimensional
models and monitoring the location of the edge relative to decaying
trajectories we provide evidence, that the edge of chaos separates state space
not globally. It is instead wrapped around the turbulence generating structures
and not an independent dynamical structure but part of the chaotic saddle.
Thereby, decaying trajectories need not cross the edge, but circumnavigate it
while unwrapping from the turbulent saddle.Comment: 11 pages, 6 figure
Snakes and ladders: localized solutions of plane Couette flow
We demonstrate the existence of a large number of exact solutions of plane
Couette flow, which share the topology of known periodic solutions but are
localized in space. Solutions of different size are organized in a
snakes-and-ladders structure strikingly similar to that observed for simpler
pattern-forming PDE systems. These new solutions are a step towards extending
the dynamical systems view of transitional turbulence to spatially extended
flows.Comment: submitted to Physics Review Letter
Edge states control droplet break-up in sub-critical extensional flows
A fluid droplet suspended in an extensional flow of moderate intensity may
break into pieces, depending on the amplitude of the initial droplet
deformation. In subcritical uniaxial extensional flow the non-breaking base
state is linearly stable, implying that only a finite amplitude perturbation
can trigger break-up. Consequently, the stable base solution is surrounded by
its finite basin of attraction. The basin boundary, which separates initial
droplet shapes returning to the non-breaking base state from those becoming
unstable and breaking up, is characterized using edge tracking techniques. We
numerically construct the edge state, a dynamically unstable equilibrium whose
stable manifold forms the basin boundary. The edge state equilibrium controls
if the droplet breaks and selects a unique path towards break-up. This path
physically corresponds to the well-known end-pinching mechanism. Our results
thereby rationalize the dynamics observed experimentally [Stone & Leal, J.
Fluid Mech. 206, 223 (1989)
Modulation equations near the Eckhaus boundary: the KdV equation
We are interested in the description of small modulations in time and space
of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*}
\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi
|\Psi|^2, \end{align*} near the Eckhaus boundary, that is, when the wave train
is near the threshold of its first instability. Depending on the parameters , a number of modulation equations can be derived, such as
the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau
based amplitude equations. Here we establish error estimates showing that the
KdV approximation makes correct predictions in a certain parameter regime. Our
proof is based on energy estimates and exploits the conservation law structure
of the critical mode. In order to improve linear damping we work in spaces of
analytic functions.Comment: 44 pages, 8 figure
Single ions trapped in a one-dimensional optical lattice
We report on three-dimensional optical trapping of single ions in an optical
lattice formed by two counter-propagating laser beams. We characterize the
trapping parameters of the standing wave using the ion as a sensor stored in a
hybrid trap consisting of a radio-frequency (rf), a dc, and the optical
potential. When loading ions directly from the rf into the standing-wave trap,
we observe a dominant heating rate. Monte Carlo simulations confirm rf-induced
parametric excitations within the deep optical lattice as the main source. We
demonstrate a way around this effect by an alternative transfer protocol which
involves an intermediate step of optical confinement in a single-beam trap
avoiding the temporal overlap of the standing wave and the rf field.
Implications arise for hybrid (rf/optical) and pure optical traps as platforms
for ultra-cold chemistry experiments exploring atom--ion collisions or quantum
simulation experiments with ions, or combinations of ions and atoms.Comment: 5 pages, 4 figure
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