13,519 research outputs found
The instanton method and its numerical implementation in fluid mechanics
A precise characterization of structures occurring in turbulent fluid flows
at high Reynolds numbers is one of the last open problems of classical physics.
In this review we discuss recent developments related to the application of
instanton methods to turbulence. Instantons are saddle point configurations of
the underlying path integrals. They are equivalent to minimizers of the related
Freidlin-Wentzell action and known to be able to characterize rare events in
such systems. While there is an impressive body of work concerning their
analytical description, this review focuses on the question on how to compute
these minimizers numerically. In a short introduction we present the relevant
mathematical and physical background before we discuss the stochastic Burgers
equation in detail. We present algorithms to compute instantons numerically by
an efficient solution of the corresponding Euler-Lagrange equations. A second
focus is the discussion of a recently developed numerical filtering technique
that allows to extract instantons from direct numerical simulations. In the
following we present modifications of the algorithms to make them efficient
when applied to two- or three-dimensional fluid dynamical problems. We
illustrate these ideas using the two-dimensional Burgers equation and the
three-dimensional Navier-Stokes equations
Structure-Preserving Discretization of Incompressible Fluids
The geometric nature of Euler fluids has been clearly identified and
extensively studied over the years, culminating with Lagrangian and Hamiltonian
descriptions of fluid dynamics where the configuration space is defined as the
volume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewed
as a consequence of Noether's theorem associated with the particle relabeling
symmetry of fluid mechanics. However computational approaches to fluid
mechanics have been largely derived from a numerical-analytic point of view,
and are rarely designed with structure preservation in mind, and often suffer
from spurious numerical artifacts such as energy and circulation drift. In
contrast, this paper geometrically derives discrete equations of motion for
fluid dynamics from first principles in a purely Eulerian form. Our approach
approximates the group of volume-preserving diffeomorphisms using a finite
dimensional Lie group, and associated discrete Euler equations are derived from
a variational principle with non-holonomic constraints. The resulting discrete
equations of motion yield a structure-preserving time integrator with good
long-term energy behavior and for which an exact discrete Kelvin's circulation
theorem holds
Flow networks: A characterization of geophysical fluid transport
We represent transport between different regions of a fluid domain by flow
networks, constructed from the discrete representation of the Perron-Frobenius
or transfer operator associated to the fluid advection dynamics. The procedure
is useful to analyze fluid dynamics in geophysical contexts, as illustrated by
the construction of a flow network associated to the surface circulation in the
Mediterranean sea. We use network-theory tools to analyze the flow network and
gain insights into transport processes. In particular we quantitatively relate
dispersion and mixing characteristics, classically quantified by Lyapunov
exponents, to the degree of the network nodes. A family of network entropies is
defined from the network adjacency matrix, and related to the statistics of
stretching in the fluid, in particular to the Lyapunov exponent field. Finally
we use a network community detection algorithm, Infomap, to partition the
Mediterranean network into coherent regions, i.e. areas internally well mixed,
but with little fluid interchange between them.Comment: 16 pages, 15 figures. v2: published versio
Machine Learning for Fluid Mechanics
The field of fluid mechanics is rapidly advancing, driven by unprecedented
volumes of data from field measurements, experiments and large-scale
simulations at multiple spatiotemporal scales. Machine learning offers a wealth
of techniques to extract information from data that could be translated into
knowledge about the underlying fluid mechanics. Moreover, machine learning
algorithms can augment domain knowledge and automate tasks related to flow
control and optimization. This article presents an overview of past history,
current developments, and emerging opportunities of machine learning for fluid
mechanics. It outlines fundamental machine learning methodologies and discusses
their uses for understanding, modeling, optimizing, and controlling fluid
flows. The strengths and limitations of these methods are addressed from the
perspective of scientific inquiry that considers data as an inherent part of
modeling, experimentation, and simulation. Machine learning provides a powerful
information processing framework that can enrich, and possibly even transform,
current lines of fluid mechanics research and industrial applications.Comment: To appear in the Annual Reviews of Fluid Mechanics, 202
Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents
There are two main strategies for improving the projection-based reduced
order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to
the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM
bases that yield more accurate ROMs. In this paper, we use the latter. We
propose new Lagrangian inner products that we use together with Eulerian and
Lagrangian data to construct new Lagrangian ROMs. We show that the new
Lagrangian ROMs are orders of magnitude more accurate than the standard
Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to
construct the ROM basis. Specifically, for the quasi-geostrophic equations, we
show that the new Lagrangian ROMs are more accurate than the standard Eulerian
ROMs in approximating not only Lagrangian fields (e.g., the finite time
Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction).
We emphasize that the new Lagrangian ROMs do not employ any closure modeling to
model the effect of discarded modes (which is standard procedure for
low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase
in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian
inner products used to build the Lagrangian ROM basis
Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media
Numerical micropermeametry is performed on three dimensional porous samples
having a linear size of approximately 3 mm and a resolution of 7.5 m. One
of the samples is a microtomographic image of Fontainebleau sandstone. Two of
the samples are stochastic reconstructions with the same porosity, specific
surface area, and two-point correlation function as the Fontainebleau sample.
The fourth sample is a physical model which mimics the processes of
sedimentation, compaction and diagenesis of Fontainebleau sandstone. The
permeabilities of these samples are determined by numerically solving at low
Reynolds numbers the appropriate Stokes equations in the pore spaces of the
samples. The physical diagenesis model appears to reproduce the permeability of
the real sandstone sample quite accurately, while the permeabilities of the
stochastic reconstructions deviate from the latter by at least an order of
magnitude. This finding confirms earlier qualitative predictions based on local
porosity theory. Two numerical algorithms were used in these simulations. One
is based on the lattice-Boltzmann method, and the other on conventional
finite-difference techniques. The accuracy of these two methods is discussed
and compared, also with experiment.Comment: to appear in: Phys.Rev.E (2002), 32 pages, Latex, 1 Figur
- …