1,194 research outputs found
The dynamics of transition to turbulence in plane Couette flow
In plane Couette flow, the incompressible fluid between two plane parallel
walls is driven by the motion of those walls. The laminar solution, in which
the streamwise velocity varies linearly in the wall-normal direction, is known
to be linearly stable at all Reynolds numbers (). Yet, in both experiments
and computations, turbulence is observed for .
In this article, we show that for certain {\it threshold} perturbations of
the laminar flow, the flow approaches either steady or traveling wave
solutions. These solutions exhibit some aspects of turbulence but are not fully
turbulent even at . However, these solutions are linearly unstable and
flows that evolve along their unstable directions become fully turbulent. The
solution approached by a threshold perturbation could depend upon the nature of
the perturbation. Surprisingly, the positive eigenvalue that corresponds to one
family of solutions decreases in magnitude with increasing , with the rate
of decrease given by with
The critical layer in pipe flow at high Reynolds number
We report the computation of a family of traveling wave solutions of pipe
flow up to . As in all lower-branch solutions, streaks and rolls
feature prominently in these solutions. For large , these solutions develop
a critical layer away from the wall. Although the solutions are linearly
unstable, the two unstable eigenvalues approach 0 as at rates
given by and -- surprisingly, the solutions become
more stable as the flow becomes less viscous. The formation of the critical
layer and other aspects of the limit could be universal to
lower-branch solutions of shear flows. We give implementation details of the
GMRES-hookstep and Arnoldi iterations used for computing these solutions and
their spectra, while pointing out the new aspects of our method
Stable manifolds and homoclinic points near resonances in the restricted three-body problem
The restricted three-body problem describes the motion of a massless particle
under the influence of two primaries of masses and that circle
each other with period equal to . For small , a resonant periodic
motion of the massless particle in the rotating frame can be described by
relatively prime integers and , if its period around the heavier primary
is approximately , and by its approximate eccentricity . We give a
method for the formal development of the stable and unstable manifolds
associated with these resonant motions. We prove the validity of this formal
development and the existence of homoclinic points in the resonant region.
In the study of the Kirkwood gaps in the asteroid belt, the separatrices of
the averaged equations of the restricted three-body problem are commonly used
to derive analytical approximations to the boundaries of the resonances. We use
the unaveraged equations to find values of asteroid eccentricity below which
these approximations will not hold for the Kirkwood gaps with equal to
2/1, 7/3, 5/2, 3/1, and 4/1.
Another application is to the existence of asymmetric librations in the
exterior resonances. We give values of asteroid eccentricity below which
asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2
resonances for any however small. But if the eccentricity exceeds these
thresholds, asymmetric librations will exist for small enough in the
unaveraged restricted three-body problem
Travelling-waves consistent with turbulence-driven secondary flow in a square duct
We present numerically determined travelling-wave solutions for
pressure-driven flow through a straight duct with a square cross-section. This
family of solutions represents typical coherent structures (a staggered array
of counter-rotating streamwise vortices and an associated low-speed streak) on
each wall. Their streamwise average flow in the cross-sectional plane
corresponds to an eight vortex pattern much alike the secondary flow found in
the turbulent regime
- …