10,225 research outputs found
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
A numerical study of viscous vortex rings using a spectral method
Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass
The analysis and simulation of compressible turbulence
Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible, (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence
VFISV: Very Fast Inversion of the Stokes Vector for the Helioseismic and Magnetic Imager
In this paper we describe in detail the implementation and main properties of
a new inversion code for the polarized radiative transfer equation (VFISV: Very
Fast inversion of the Stokes vector). VFISV will routinely analyze pipeline
data from the Helioseismic and Magnetic Imager (HMI) on-board of the Solar
Dynamics Observatory (SDO). It will provide full-disk maps (40964096
pixels) of the magnetic field vector on the Solar Photosphere every 10 minutes.
For this reason VFISV is optimized to achieve an inversion speed that will
allow it to invert 16 million pixels every 10 minutes with a modest number
(approx. 50) of CPUs. Here we focus on describing a number of important
details, simplifications and tweaks that have allowed us to significantly speed
up the inversion process. We also give details on tests performed with data
from the spectropolarimeter on-board of the Hinode spacecraft.Comment: 23 pages, 9 figures (2 color). Submitted for publication to Solar
Physic
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