29 research outputs found
Nonautonomous control of stable and unstable manifolds in two-dimensional flows
We outline a method for controlling the location of stable and unstable
manifolds in the following sense. From a known location of the stable and
unstable manifolds in a steady two-dimensional flow, the primary segments of
the manifolds are to be moved to a user-specified time-varying location which
is near the steady location. We determine the nonautonomous perturbation to the
vector field required to achieve this control, and give a theoretical bound for
the error in the manifolds resulting from applying this control. The efficacy
of the control strategy is illustrated via a numerical example
Probing turbulent superstructures in Rayleigh-B\'{e}nard convection by Lagrangian trajectory clusters
We analyze large-scale patterns in three-dimensional turbulent convection in
a horizontally extended square convection cell by Lagrangian particle
trajectories calculated in direct numerical simulations. A simulation run at a
Prandtl number Pr , a Rayleigh number Ra , and an aspect ratio
is therefore considered. These large-scale structures, which are
denoted as turbulent superstructures of convection, are detected by the
spectrum of the graph Laplacian matrix. Our investigation, which follows
Hadjighasem {\it et al.}, Phys. Rev. E {\bf 93}, 063107 (2016), builds a
weighted and undirected graph from the trajectory points of Lagrangian
particles. Weights at the edges of the graph are determined by a mean dynamical
distance between different particle trajectories. It is demonstrated that the
resulting trajectory clusters, which are obtained by a subsequent -means
clustering, coincide with the superstructures in the Eulerian frame of
reference. Furthermore, the characteristic times and lengths
of the superstructures in the Lagrangian frame of reference agree
very well with their Eulerian counterparts, and ,
respectively. This trajectory-based clustering is found to work for times
. Longer time periods require a
change of the analysis method to a density-based trajectory clustering by means
of time-averaged Lagrangian pseudo-trajectories, which is applied in this
context for the first time. A small coherent subset of the pseudo-trajectories
is obtained in this way consisting of those Lagrangian particles that are
trapped for long times in the core of the superstructure circulation rolls and
are thus not subject to ongoing turbulent dispersion.Comment: 12 pages, 7 downsized figures, to appear in Phys. Rev. Fluid
Role of critical points of the skin friction field in formation of plumes in thermal convection
The dynamics in the thin boundary layers of temperature and velocity is the
key to a deeper understanding of turbulent transport of heat and momentum in
thermal convection. The velocity gradient at the hot and cold plates of a
Rayleigh-B\'{e}nard convection cell forms the two-dimensional skin friction
field and is related to the formation of thermal plumes in the respective
boundary layers. Our analysis is based on a direct numerical simulation of
Rayleigh-B\'{e}nard convection in a closed cylindrical cell of aspect ratio
and focused on the critical points of the skin friction field. We
identify triplets of critical points, which are composed of two unstable nodes
and a saddle between them, as the characteristic building block of the skin
friction field. Isolated triplets as well as networks of triplets are detected.
The majority of the ridges of line-like thermal plumes coincide with the
unstable manifolds of the saddles. From a dynamical Lagrangian perspective,
thermal plumes are formed together with an attractive hyperbolic Lagrangian
Coherent Structure of the skin friction field. We also discuss the differences
from the skin friction field in turbulent channel flows from the perspective of
the Poincar\'{e}-Hopf index theorem for two-dimensional vector fields
Lagrangian analysis of long-term dynamics of turbulent superstructures
In Rayleigh-Bénard convection, turbulent superstructures are large-scale patterns of circulation rolls created by hot ascending and cold descending thermal plumes. The evolution of these large-scale patterns happens on very large time scales τ [1]. Spectral clustering applied to Lagrangian particle trajectories on time intervals smaller than τ can be used to create clusters displaying a structure similar to the patterns detected in the Eulerian frame of reference [2]. However, this technique is unfeasible for the analysis of the evolution of turbulent superstructures due to turbulent dispersion. Therefore, we test the application of concepts of evolutionary spectral clustering [3] on Lagrangian particle trajectories to analyze the long-term dynamics of turbulent superstructures in the Lagrangian frame of reference
Network Measures of Mixing
Transport and mixing processes in fluid flows can be studied directly from
Lagrangian trajectory data, such as obtained from particle tracking
experiments. Recent work in this context highlights the application of
graph-based approaches, where trajectories serve as nodes and some similarity
or distance measure between them is employed to build a (possibly weighted)
network, which is then analyzed using spectral methods. Here, we consider the
simplest case of an unweighted, undirected network and analytically relate
local network measures such as node degree or clustering coefficient to flow
structures. In particular, we use these local measures to divide the family of
trajectories into groups of similar dynamical behavior via manifold learning
methods
Lagrangian heat transport in turbulent three-dimensional convection
Spatial regions that do not mix effectively with their surroundings and thus
contribute less to the heat transport in fully turbulent three-dimensional
Rayleigh-B\'{e}nard flows are identified by Lagrangian trajectories that stay
together for a longer time. These trajectories probe Lagrangian coherent sets
(CS) which we investigate here in direct numerical simulations in convection
cells with square cross section of aspect ratio , Rayleigh number
, and Prandtl numbers and . The analysis is
based on Lagrangian tracer particles which are advected in the
time-dependent flow. Clusters of trajectories are identified by a graph
Laplacian with a diffusion kernel, which quantifies the connectivity of
trajectory segments, and a subsequent sparse eigenbasis approximation (SEBA)
for cluster detection. The combination of graph Laplacian and SEBA leads to a
significantly improved cluster identification that is compared with the
large-scale patterns in the Eulerian frame of reference. We show that the
detected CS contribute by a third less to the global turbulent heat transport
for all investigated compared to the trajectories in the spatial
complement. This is realized by monitoring Nusselt numbers along the tracer
trajectory ensembles, a dimensionless local measure of heat transfer.Comment: 8 pages, 5 figure
Lagrangian studies of coherent sets and heat transport in constant heat flux-driven turbulent Rayleigh-B\'enard convection
We explore the mechanisms of heat transfer in a turbulent constant heat
flux-driven Rayleigh-B\'enard convection flow, which exhibits a hierarchy of
flow structures from granules to supergranules. Our computational framework
makes use of time-dependent flow networks. These are based on trajectories of
Lagrangian tracer particles that are advected in the flow. We identify coherent
sets in the Lagrangian frame of reference as those sets of trajectories that
stay closely together for an extended time span under the action of the
turbulent flow. Depending on the choice of the measure of coherence, sets with
different characteristics are detected. First, the application of a recently
proposed evolutionary spectral clustering scheme allows us to extract granular
coherent features that are shown to contribute significantly less to the global
heat transfer than their spatial complements. Moreover, splits and mergers of
these (leaking) coherent sets leave spectral footprints. Secondly, trajectories
which exhibit a small node degree in the corresponding network represent
objectively highly coherent flow structures and can be related to supergranules
as the other stage of the present flow hierarchy. We demonstrate that the
supergranular flow structures play a key role in the vertical heat transport
and that they exhibit a greater spatial extension than the granular structures
obtained from spectral clustering.Comment: 21 pages, 15 figure