Variational approach to the closure problem of turbulence theory

A new method is proposed to solve the closure problem of turbulence theory and to drive the Kolmogorov law in an Eulerian framework. Instead of using complex Fourier components of velocity field as modal parameters, a complete set of independent real parameters and dynamic equations are worked out to describe the dynamic states of a turbulence. Classical statistical mechanics is used to study the statistical behavior of the turbulence. An approximate stationary solution of the Liouville equation is obtained by a perturbation method based on a Langevin-Fokker-Planck (LFP) model. The dynamic damping coefficient eta of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution; this leads to a convergent integral equation for eta to replace the divergent response equation of Kraichnan's direct-interaction (DI) approximation, thereby solving the closure problem without appealing to a Lagrangian formulation. The Kolmogorov constant Ko is evaluated numerically, obtaining Ko = 1.2, which is compatible with the experimental data given by Gibson and Schwartz, (1963).

Thickness-dependent molecular chain and lamellar crystal orientation in ultrathin poly(di-n-hexylsilane) films

The molecular chain and lamellar crystal orientation in ultrathin films (thickness < 100 nm) of poly(di-n-hexylsilane) (PDHS) on silicon wafer substrates have been investigated by using transmission electronic microscopy, wide-angle X-ray diffraction, atomic force microscopy, and UV absorption spectroscopy. PDHS showed a film thickness-dependent molecular chain and lamellar crystal orientation. Lamellar crystals grew preferentially in flat-on orientation in the monolayer ultrathin films of PDHS, i.e., the silicon backbones were oriented along the surface-normal direction. By contrast, the orientation of lamellar crystals was preferentially edge-on in ultrathin films thicker than ca. 13 nm, i.e., the silicon backbones were oriented parallel to the substrate surface. We interpret the different orientations of molecular chain and lamellar crystal as due to the reduction of the entropy of the polymer chain near the substrate surface and the particularity of the crystallographic (001) plane of flat-on lamellae, respectively. A remarkable influence of the orientations of the silicon backbone on the UV absorption of these PDHS ultrathin films was observed due to the one-dimensional nature of sigma-electrons delocalized along the silicon backbone.