18 research outputs found
Multiresolution approximation of the vector fields on T^3
Multiresolution approximation (MRA) of the vector fields on T^3 is studied.
We introduced in the Fourier space a triad of vector fields called helical
vectors which derived from the spherical coordinate system basis. Utilizing the
helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a
synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3
and the Beltrami decomposition that decompose the space of solenoidal vector
fields into the eigenspaces of curl operator. In the course of proof, a general
construction procedure of the divergence-free orthonormal complete basis from
the basis of scalar function space is presented. Applying this procedure to MRA
of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity
and regularity of vector wavelets. It is conjectured that the solenoidal
wavelet basis must break r-regular condition, i.e. some wavelet functions
cannot be rapidly decreasing function because of the inevitable singularities
of helical vectors. The localization property and spatial structure of
solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's
wavelet) are also investigated numerically.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy
Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations
We address the problem in Navier-Stokes isotropic turbulence of why the
vorticity accumulates on thin sets such as quasi-one-dimensional tubes and
quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst,
Kerstein, Kerr and Gibson, who observed that the vorticity vector
{\boldmath\omega} aligns with the intermediate eigenvector of the strain
matrix , we study this problem in the context of both the three-dimensional
Euler and Navier-Stokes equations using the variables \alpha =
\hat{{\boldmath\xi}}\cdot S\hat{{\boldmath\xi}} and {\boldmath\chi} =
\hat{{\boldmath\xi}}\times S\hat{{\boldmath\xi}} where
\hat{{\boldmath\xi}} = {\boldmath\omega}/\omega. This introduces the
dynamic angle , which lies between
{\boldmath\omega} and S{\boldmath\omega}. For the Euler equations a
closed set of differential equations for and {\boldmath\chi} is
derived in terms of the Hessian matrix of the pressure . For
the Navier-Stokes equations, the Burgers vortex and shear layer solutions turn
out to be the Lagrangian fixed point solutions of the equivalent
(\alpha,{\boldmath\chi}) equations with a corresponding angle .
Under certain assumptions for more general flows it is shown that there is an
attracting fixed point of the (\alpha,\bchi) equations which corresponds to
positive vortex stretching and for which the cosine of the corresponding angle
is close to unity. This indicates that near alignment is an attracting state of
the system and is consistent with the formation of Burgers-like structures.Comment: To appear in Nonlinearity Nov. 199
Local or Nonlocal? Orthonormal Divergence-free Wavelet Analysis of Nonlinear Interactions in Turbulence
"Using orthonorma1 divergence-free vector wavelet, we analyzed nonlinear transfer in a three-dimensional homogeneous, isotropic turbulence. Analogous Fourier analysis is also carried out. It is verified that energy is transferred locally. We also find that, in terms of wavelet analysis and geometrically partitioned Fourier band average analysis, the contribution of local interactions to the transfer is not negligible compared to that of nonlocal ones. On the other hand, Iinearly partitioned Fourier band average analysis drastically changes the qualitative appearance of transfer process, i.e. the result shows the predominance of nonlocal interactions.