17,186 research outputs found
On the non-local geometry of turbulence
A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component three-dimensional fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. The characterization stage is based on the joint probability density function (p.d.f.), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the joint p.d.f. of each structure, that specify its location as a point in a multi-dimensional ‘feature space’. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This then allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry. Results are presented of a first application of this technique to a passive scalar field obtained from 5123 direct numerical simulation of scalar mixing by forced, isotropic turbulence (Reλ = 265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behaviour of passive scalar stirring and mixing by turbulence are discussed
Multi-scale geometric analysis of Lagrangian structures in isotropic turbulence
We report the multi-scale geometric analysis of Lagrangian structures in forced isotropic turbulence and also with a frozen turbulent field. A particle backward-tracking method, which is stable and topology preserving, was applied to obtain the Lagrangian scalar field φ governed by the pure advection equation in the Eulerian form ∂_tφ + u · ∇φ = 0. The temporal evolution of Lagrangian structures was first obtained by extracting iso-surfaces of φ with resolution 1024^3 at different times, from t = 0 to t = T_e, where T_e is the eddy turnover time. The surface area growth rate of the Lagrangian structure was quantified and the formation of stretched and rolled-up structures was observed in straining regions and stretched vortex tubes, respectively. The multi-scale geometric analysis of Bermejo-Moreno & Pullin (J. Fluid Mech., vol. 603, 2008, p. 101) has been applied to the evolution of φ to extract structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. In this multi-scale sense, we observe, for the evolving turbulent velocity field, an evolutionary breakdown of initially large-scale Lagrangian structures that first distort and then either themselves are broken down or stretched laterally into sheets. Moreover, after a finite time, this progression appears to be insensible to the form of the initially smooth Lagrangian field. In comparison with the statistical geometry of instantaneous passive scalar and enstrophy fields in turbulence obtained by Bermejo-Moreno & Pullin (2008) and Bermejo-Moreno et al. (J. Fluid Mech., vol. 620, 2009, p. 121), Lagrangian structures tend to exhibit more prevalent sheet-like shapes at intermediate and small scales. For the frozen flow, the Lagrangian field appears to be attracted onto a stream-surface field and it develops less complex multi-scale geometry than found for the turbulent velocity field. In the latter case, there appears to be a tendency for the Lagrangian field to move towards a vortex-surface field of the evolving turbulent flow but this is mitigated by cumulative viscous effects
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces
In this work, we are interested in the differential geometry of curves in the
simply isotropic and pseudo-isotropic 3-spaces, which are examples of
Cayley-Klein geometries whose absolute figure is given by a plane at infinity
and a degenerate quadric. Motivated by the success of rotation minimizing (RM)
frames in Euclidean and Lorentzian geometries, here we show how to build RM
frames in isotropic geometries and apply them in the study of isotropic
spherical curves. Indeed, through a convenient manipulation of osculating
spheres described in terms of RM frames, we show that it is possible to
characterize spherical curves via a linear equation involving the curvatures
that dictate the RM frame motion. For the case of pseudo-isotropic space, we
also discuss on the distinct choices for the absolute figure in the framework
of a Cayley-Klein geometry and prove that they are all equivalent approaches
through the use of Lorentz numbers (a complex-like system where the square of
the imaginary unit is ). Finally, we also show the possibility of obtaining
an isotropic RM frame by rotation of the Frenet frame through the use of
Galilean trigonometric functions and dual numbers (a complex-like system where
the square of the imaginary unit vanishes).Comment: 2 figures. To appear in "Tamkang Journal of Mathematics
Fast Fiber Orientation Estimation in Diffusion MRI from kq-Space Sampling and Anatomical Priors
High spatio-angular resolution diffusion MRI (dMRI) has been shown to provide
accurate identification of complex fiber configurations, albeit at the cost of
long acquisition times. We propose a method to recover intra-voxel fiber
configurations at high spatio-angular resolution relying on a kq-space
under-sampling scheme to enable accelerated acquisitions. The inverse problem
for reconstruction of the fiber orientation distribution (FOD) is regularized
by a structured sparsity prior promoting simultaneously voxelwise sparsity and
spatial smoothness of fiber orientation. Prior knowledge of the spatial
distribution of white matter, gray matter and cerebrospinal fluid is also
assumed. A minimization problem is formulated and solved via a forward-backward
convex optimization algorithmic structure. Simulations and real data analysis
suggest that accurate FOD mapping can be achieved from severe kq-space
under-sampling regimes, potentially enabling high spatio-angular dMRI in the
clinical setting.Comment: 10 pages, 5 figures, Supplementary Material
Fast, Scalable, and Interactive Software for Landau-de Gennes Numerical Modeling of Nematic Topological Defects
Numerical modeling of nematic liquid crystals using the tensorial Landau-de
Gennes (LdG) theory provides detailed insights into the structure and
energetics of the enormous variety of possible topological defect
configurations that may arise when the liquid crystal is in contact with
colloidal inclusions or structured boundaries. However, these methods can be
computationally expensive, making it challenging to predict (meta)stable
configurations involving several colloidal particles, and they are often
restricted to system sizes well below the experimental scale. Here we present
an open-source software package that exploits the embarrassingly parallel
structure of the lattice discretization of the LdG approach. Our
implementation, combining CUDA/C++ and OpenMPI, allows users to accelerate
simulations using both CPU and GPU resources in either single- or multiple-core
configurations. We make use of an efficient minimization algorithm, the Fast
Inertial Relaxation Engine (FIRE) method, that is well-suited to large-scale
parallelization, requiring little additional memory or computational cost while
offering performance competitive with other commonly used methods. In
multi-core operation we are able to scale simulations up to supra-micron length
scales of experimental relevance, and in single-core operation the simulation
package includes a user-friendly GUI environment for rapid prototyping of
interfacial features and the multifarious defect states they can promote. To
demonstrate this software package, we examine in detail the competition between
curvilinear disclinations and point-like hedgehog defects as size scale,
material properties, and geometric features are varied. We also study the
effects of an interface patterned with an array of topological point-defects.Comment: 16 pages, 6 figures, 1 youtube link. The full catastroph
Naked shell singularities on the brane
By utilizing non-standard slicings of 5-dimensional Schwarzschild and
Schwarzschild-AdS manifolds based on isotropic coordinates, we generate static
and spherically symmetric braneworld spacetimes containing shell-like naked
null singularities. For planar slicings, we find that the brane-matter sourcing
the solution is a perfect fluid with an exotic equation of state and a pressure
singularity where the brane crosses the bulk horizon. From a relativistic point
of view, such a singularity is required to maintain matter infinitesimally
above the surface of a black hole. From the point of view of the AdS/CFT
conjecture, the singular horizon can be seen as one possible quantum correction
to a classical black hole geometry. Various generalizations of planar slicings
are also considered for a Ricci-flat bulk, and we find that singular horizons
and exotic matter distributions are common features.Comment: REVTeX4, 13 pages, 6 figures, references and comments adde
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