47 research outputs found
Maximum Palinstrophy Growth in 2D Incompressible Flows
In this study we investigate vortex structures which lead to the maximum
possible growth of palinstrophy in two-dimensional incompressible flows on a
periodic domain. The issue of palinstrophy growth is related to a broader
research program focusing on extreme amplification of vorticity-related
quantities which may signal singularity formation in different flow models.
Such extreme vortex flows are found systematically via numerical solution of
suitable variational optimization problems. We identify several families of
maximizing solutions parameterized by their palinstrophy, palinstrophy and
energy and palinstrophy and enstrophy. Evidence is shown that some of these
families saturate estimates for the instantaneous rate of growth of
palinstrophy obtained using rigorous methods of mathematical analysis, thereby
demonstrating that this analysis is in fact sharp. In the limit of small
palinstrophies the optimal vortex structures are found analytically, whereas
for large palinstrophies they exhibit a self-similar multipolar structure. It
is also shown that the time evolution obtained using the instantaneously
optimal states with fixed energy and palinstrophy as the initial data saturates
the upper bound for the maximum growth of palinstrophy in finite time. Possible
implications of this finding for the questions concerning extreme behavior of
flows are discussed.Comment: 33 pages, 8 figures; to appear in "Journal of Fluid Mechanics
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
Computation of Steady Incompressible Flows in Unbounded Domains
In this study we revisit the problem of computing steady Navier-Stokes flows
in two-dimensional unbounded domains. Precise quantitative characterization of
such flows in the high-Reynolds number limit remains an open problem of
theoretical fluid dynamics. Following a review of key mathematical properties
of such solutions related to the slow decay of the velocity field at large
distances from the obstacle, we develop and carefully validate a
spectrally-accurate computational approach which ensures the correct behavior
of the solution at infinity. In the proposed method the numerical solution is
defined on the entire unbounded domain without the need to truncate this domain
to a finite box with some artificial boundary conditions prescribed at its
boundaries. Since our approach relies on the streamfunction-vorticity
formulation, the main complication is the presence of a discontinuity in the
streamfunction field at infinity which is related to the slow decay of this
field. We demonstrate how this difficulty can be overcome by reformulating the
problem using a suitable background "skeleton" field expressed in terms of the
corresponding Oseen flow combined with spectral filtering. The method is
thoroughly validated for Reynolds numbers spanning two orders of magnitude with
the results comparing favourably against known theoretical predictions and the
data available in the literature.Comment: 39 pages, 12 figures, accepted for publication in "Computers and
Fluids