Third-order statistics and the dynamics of strongly anisotropic turbulent flows

Anisotropy is induced by body forces and/or mean large-scale gradients in turbulent flows. For flows without energy production, the dynamics of second-order velocity or second-order vorticity statistics are essentially governed by triple correlations, which are at the origin of the anisotropy that penetrates towards the inertial range, deeply altering the cascade and the eventual dissipation process, with a series of consequences on the evolution of homogeneous turbulence statistics: in the case of rotating turbulence, the anisotropic spectral transfer slaves the multiscale anisotropic energy distribution; nonlinear dynamics are responsible for the linear growth in terms of Ωt of axial integral length-scales; third-order structure functions, derived from velocity triple correlations, exhibit a significant departure from the 4/5 Kolmogorov law. We describe all these implications in detail, starting from the dynamical equations of velocity statistics in Fourier space, which yield third-order correlations at three points (triads) and allow the explicit removal of pressure fluctuations. We first extend the formalism to anisotropic rotating turbulence with ‘production’, in the presence of mean velocity gradients in the rotating frame. Second, we compare the spectral approach at three points to the two-point approach directly performed in physical space, in which we consider the transport of the scalar second-order structure function ?(δq)²?. This calls into play componental third-order correlations ?(δq)²δu?(r) in axisymmetric turbulence. This permits to discuss inhomogeneous anisotropic effects from spatial decay, shear, or production, as in the central region of a rotating round jet. We show that the above-mentioned important statistical quantities can be estimated from experimental planar particle image velocimetry, and that explicit passage relations systematically exist between one- and two-point statistics in physical and spectral space for second-order tensors, but also sometimes for third-order tensors that are involved in the dynamics.     

2-(芘-1-基)-1,8-萘啶的合成、理论计算及超快三阶非线性光学响应

合成了一个供体-受体(D-A)型含芘萘啶衍生物2-(芘-1-基)-1,8-萘啶(PN)。 通过核磁共振波谱(NMR)、傅里叶变换红外光谱(FTIR)、液-质联用仪(LC-MS)表征了其结构。 通过电子光谱和Z-扫描技术方法分别研究了化合物PN的线性光学性质和三阶非线性光学吸收,采用综合热分析方法测定了它的热稳定性。 结果表明,在532 nm、180 fs条件下,PN的三阶非线性吸收行为为反饱和吸收,其吸收系数为β=9.0×10^-14 m/W,显示出超快三阶非线性光学响应。 运用密度泛函理论方法计算了分子轨道能量、极化率和超极化率,结果表明电子转移能够在分子内部进行。 2-(芘-1-基)-1,8-萘啶的紫外光谱在450 nm以上无吸收,在非线性光学吸收、激光防护、吸收型光开关或双稳器件等方面可作备选材料。     

基于二茂铁的两个查尔酮衍生物的合成及超快三阶非线性光学响应

二茂铁是合成新颖有机功能材料的基本单元之一。 本文设计并合成了两个基于二茂铁的同分异构查尔酮衍生物:1-二茂铁基-3-(噻吩-2-基)丙烯酮(a)和1-二茂铁基-3-(噻吩-3-基)丙烯酮(b)。 采用超快激光Z-扫描技术(脉宽180 fs,波长532 nm)测定了化合物a和b的三阶非线性光学性质。 结果表明,化合物a吸收系数β=-2.1×10^-12 m/W,折射率n₂=1.9×10^-19 m²/W,分子超极化率γ=5.37×10^-32 esu;化合物b:β=-1.2×10^-13 m/W,n₂=2.0×10^-19 m²/W,γ=4.48×10^-32 esu。 说明在飞秒激光激发下,电荷转移能够在化合物a和b分子内部快速进行,二者均具有优异的超快三阶非线性光学响应。 在B3LYP/6-311+G(d,p)理论水平下,计算了化合物a和b分子轨道能量、极化率和各基团在前线分子轨道中的占有率。 理论计算结果显示,二茂铁基团在化合物a和b前线分子轨道中占有率分别为97%和98%,对两化合物的非线性光学性能起主导作用。     

高阶色散效应常系数Ginzburg-Landau方程自相似脉冲演化的解析分析

采用自相似分析方法,基于常系数高阶色散的Ginzburg-Landau方程,通过分离变量法得出了高阶色散效应自相似脉冲演化的解析解,给出了自相似脉冲的振幅、相位、啁啾以及脉冲宽度的一般表达式.研究表明,在增益光纤的二阶正常色散区域,同时考虑高阶色散和增益色散双重效应影响下演化的自相似孤子脉冲仍然保持线性啁啾;振幅解析解的三阶色散效应显著.这与数值计算的结果非常一致. 关键词： 三阶色散 Ginzburg-Landau方程 自相似脉冲 二阶正常色散     

一类三阶非线性非自治系统解的有界性

本文利用一为三阶非线性自治系统的李雅夫函数,给出了一类三阶非线性非自治系统解有界的充分性准则,推广了文「１－２」的结果。     

A first-second order splitting method for a third-order partial differential equation

A splitting of a third order partial differential equation into a first-order and a second-order one is proposed as the basis for a mixed finite element method to approximate its solution. A time-continuous numerical method is described and error estimates for its solution are demonstrated. Finally, a full discretization is described based on backward Euler finite differences in time, and error estimates for the resulting approximation are established. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 89–96, 1998     

Synthesis of ladder polymers containing polydiacetylene backbones connected with methylene chains and their optical properties

The polymers consisting of polydiacetylene (PDA) backbones were obtained from the novel monomer derivatives, R CC CC R′ CC CC R [where R =  (CH₂)₄OCONHCH₂COOC₄H₉, R′ =  (CH₂)_n ; n = 2, 4, 8] [4BCMU4A(n)], in which linear methylene chain is sandwiched between two diacetylene moieties by solid-state 1,4-addition reaction. The polymerization process was investigated in detail by using spectroscopic techniques such as solid-state ¹³C-NMR, visible absorption, and IR absorption spectra. It was estimated that the polymerization of 4BCMU4A(8) and 4BCMU4A(4) takes place by two consecutive 1,4-addition reactions to form two PDA backbones, which constitute the two poles of the respective ladders. The bridging methylene chain length in the monomer was found to play a vital role as far as the polymerization process is concerned. Thus, the monomers with eight or four methylene units could form the ladder–PDAs by a two-step process, whereas the monomer containing two methylene units could only undergo one-step of 1,4-addition reaction. Further, it was found that the crystallinity of the polymers depends on the methylene chain length in the monomers, 4BCMU4A(8) being the most crystalline of all. These structural features strongly affect their absorption spectra. The third-order nonlinear optical susceptibilities (χ⁽³⁾) for these polymers were measured using third-harmonic generation method. The largest χ⁽³⁾ value obtained was 3.4 × 10⁻¹¹ esu for the poly[4BCMU4A(8)] thin film in resonant region. © 1999 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 37: 3537–3548, 1999     

Computation of the oscillator strength and absorption coefficients for the intersubband transitions of the spherical quantum dot

The electronic structure and optical properties of one-electron Quantum Dot (QD) with and without an on-center impurity were investigated by assuming a spherically symmetric confining potential of finite depth. The energy eigenvalues and the state functions of QD were calculated by using a combination of Quantum Genetic Algorithm (QGA) and Hartree–Fock Roothan (HFR) method. We have calculated the binding energy for the states 1s,1p,1d,1f, oscillator strengths, the linear and third-order nonlinear optical absorption coefficients as a function of the incident photon energy and incident optical intensity for the 1s–1p, 1p–1d and 1d–1f transitions. The existence of the impurity has great influence on the optical absorption spectra and the oscillator strengths. Also we found that the magnitudes of the total absorption coefficients of the spherical QD increase for transitions between higher states.     

线性增长限制下一类三阶边值问题的可解性

考察了一类含一阶和二阶导数的三阶边值问题的可解性.非线性项f满足线性增长的限制.通过构造适当的Banach空间并利用Leray-Schauder非线性抉择证明了一个存在定理.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.