2,167 research outputs found
Koopman analysis of the long-term evolution in a turbulent convection cell
We analyse the long-time evolution of the three-dimensional flow in a closed
cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction
analysis. A data-driven basis derived from diffusion kernels known in machine
learning is employed here to represent a regularized generator of the unitary
Koopman group in the sense of a Galerkin approximation. The resulting Koopman
eigenfunctions can be grouped into subsets in accordance with the discrete
symmetries in a cubic box. In particular, a projection of the velocity field
onto the first group of eigenfunctions reveals the four stable large-scale
circulation (LSC) states in the convection cell. We recapture the preferential
circulation rolls in diagonal corners and the short-term switching through roll
states parallel to the side faces which have also been seen in other
simulations and experiments. The diagonal macroscopic flow states can last as
long as a thousand convective free-fall time units. In addition, we find that
specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced
oscillatory fluctuations for particular stable diagonal states of the LSC. The
corresponding velocity field structures, such as corner vortices and swirls in
the midplane, are also discussed via spatiotemporal reconstructions.Comment: 32 pages, 9 figures, article in press at Journal of Fluid Mechanic
On some electroconvection models
We consider a model of electroconvection motivated by studies of the motion
of a two dimensional annular suspended smectic film under the influence of an
electric potential maintained at the boundary by two cylindrical electrodes. We
prove that this electroconvection model has global in time unique smooth
solutions
Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection
By utilizing diffusion maps embedding and transition matrix analysis we
investigate sparse temperature measurement time-series data from
Rayleigh--B\'enard convection experiments in a cylindrical container of aspect
ratio between its diameter () and height (). We consider
the two cases of a cylinder at rest and rotating around its cylinder axis. We
find that the relative amplitude of the large-scale circulation (LSC) and its
orientation inside the container at different points in time are associated to
prominent geometric features in the embedding space spanned by the two dominant
diffusion-maps eigenvectors. From this two-dimensional embedding we can measure
azimuthal drift and diffusion rates, as well as coherence times of the LSC. In
addition, we can distinguish from the data clearly the single roll state (SRS),
when a single roll extends through the whole cell, from the double roll state
(DRS), when two counter-rotating rolls are on top of each other. Based on this
embedding we also build a transition matrix (a discrete transfer operator),
whose eigenvectors and eigenvalues reveal typical time scales for the stability
of the SRS and DRS as well as for the azimuthal drift velocity of the flow
structures inside the cylinder. Thus, the combination of nonlinear dimension
reduction and dynamical systems tools enables to gain insight into turbulent
flows without relying on model assumptions
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
Subdivision Directional Fields
We present a novel linear subdivision scheme for face-based tangent
directional fields on triangle meshes. Our subdivision scheme is based on a
novel coordinate-free representation of directional fields as halfedge-based
scalar quantities, bridging the finite-element representation with discrete
exterior calculus. By commuting with differential operators, our subdivision is
structure-preserving: it reproduces curl-free fields precisely, and reproduces
divergence-free fields in the weak sense. Moreover, our subdivision scheme
directly extends to directional fields with several vectors per face by working
on the branched covering space. Finally, we demonstrate how our scheme can be
applied to directional-field design, advection, and robust earth mover's
distance computation, for efficient and robust computation
Laplacian Projection Based Global Physical Prior Smoke Reconstruction
We present a novel framework for reconstructing fluid dynamics in real-life scenarios. Our approach leverages sparse view images and incorporates physical priors across long series of frames, resulting in reconstructed fluids with enhanced physical consistency. Unlike previous methods, we utilize a differentiable fluid simulator (DFS) and a differentiable renderer (DR) to exploit global physical priors, reducing reconstruction errors without the need for manual regularization coefficients. We introduce divergence-free Laplacian eigenfunctions (div-free LE) as velocity bases, improving computational efficiency and memory usage. By employing gradient-related strategies, we achieve better convergence and superior results. Extensive experiments demonstrate the effectiveness of our method, showcasing improved reconstruction quality and computational efficiency compared to existing approaches. We validate our approach using both synthetic and real data, highlighting its practical potential
Model-reduced variational fluid simulation
We present a model-reduced variational Eulerian integrator for incompressible fluids, which combines the efficiency gains of dimension reduction, the qualitative robustness of coarse spatial and temporal resolutions of geometric integrators, and the simplicity of sub-grid accurate boundary conditions on regular grids to deal with arbitrarily-shaped domains. At the core of our contributions is a functional map approach to fluid simulation for which scalar- and vector-valued eigenfunctions of the Laplacian operator can be easily used as reduced bases. Using a variational integrator in time to preserve liveliness and a simple, yet accurate embedding of the fluid domain onto a Cartesian grid, our model-reduced fluid simulator can achieve realistic animations in significantly less computational time than full-scale non-dissipative methods but without the numerical viscosity from which current reduced methods suffer. We also demonstrate the versatility of our approach by showing how it easily extends to magnetohydrodynamics and turbulence modeling in 2D, 3D and curved domains
Kernel Analog Forecasting: Multiscale Test Problems
Data-driven prediction is becoming increasingly widespread as the volume of
data available grows and as algorithmic development matches this growth. The
nature of the predictions made, and the manner in which they should be
interpreted, depends crucially on the extent to which the variables chosen for
prediction are Markovian, or approximately Markovian. Multiscale systems
provide a framework in which this issue can be analyzed. In this work kernel
analog forecasting methods are studied from the perspective of data generated
by multiscale dynamical systems. The problems chosen exhibit a variety of
different Markovian closures, using both averaging and homogenization;
furthermore, settings where scale-separation is not present and the predicted
variables are non-Markovian, are also considered. The studies provide guidance
for the interpretation of data-driven prediction methods when used in practice.Comment: 30 pages, 14 figures; clarified several ambiguous parts, added
references, and a comparison with Lorenz' original method (Sec. 4.5
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