Dynamics in spectral solutions of Burgers equation

For low values of the viscosity coefficient, Burgers equation can develop sharp discontinuities, which are difficult to simulate in a computer. Oscillations can occur by discretization through spectral collocation methods, due to Gibbs phenomena. Under a dynamic point of view, these instabilities are related to bifurcations arising to the discretized equation. For different values of discretized points, herein a study is performed of the dynamics and bifurcations occurring in the spectral solutions of Burgers equation with symmetry. Several phenomena are observed, from limit cycles, strange attractors to the presence of bistability with two periodic attractors, with a periodic attractor and a strange attractor and with two strange attractors. Also, other stable equilibrium points can occur, diverse from the ones corresponding to the solution of Burgers equation.     

Facile and selective aliphatic C-H bond activation at ambient temperatures initiated by Cp*W(NO)(CH2CMe3)(eta(3)-CH2CHCHMe)

Thermolysis of Cp*W(NO)(CH2CMe3)(eta(3)-CH2CHCHMe) (1) at ambient temperatures leads to the loss of neopentane and the formation of the eta(2)-diene intermediate, Cp*W(NO)(eta(2)-CH2=CHCH=CH2) (A), which has been isolated as its 18e PMe3 adduct. In the presence of linear alkanes, A effects C-H activations of the hydrocarbons exclusively at their terminal carbons and forms 18e Cp*W(NO)(n-alkyl)(eta(3)-CH2CHCHMe) complexes. Similarly, treatments of 1 with methylcyclohexane, chloropentane, diethyl ether, and triethylamine all lead to the corresponding terminal C-H activation products. Furthermore, a judicious choice of solvents permits the C-H activation of gaseous hydrocarbons (i.e., propane, ethane, and methane) at ambient temperatures under moderately elevated pressures. However, reactions between intermediate A and cyclohexene, acetone, 3-pentanone, and 2-butyne lead to coupling between the eta(2)-diene ligand and the site of unsaturation on the organic molecule. For example, Cp*W(NO)(eta(3),eta(1)-CH2CHCHCH2C(CH2CH3)2O) is formed exclusively in 3-pentanone. When the site of unsaturation is sufficiently sterically hindered, as in the case of 2,3-dimethyl-2-butene, C-H activation again becomes dominant, and so the C-H activation product, Cp*W(NO)(eta(1)-CH2CMe=CMe2)(eta(3)-CH2CHCHMe), is formed exclusively from the alkene and 1. All new complexes have been characterized by conventional spectroscopic and analytical methods, and the solid-state molecular structures of most of them have been established by X-ray crystallographic analyses. Finally, the newly formed alkyl ligands may be liberated from the tungsten centers in the product complexes by treatment with iodine. Thus, exposure of a CDCl3 solution of the n-pentyl allyl complex, Cp*W(NO)(n-C5H11)(eta(3)-CH2CHCHMe), to I2 at -60 degrees C produces n-C5H11I in moderate yields.     

A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics

When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.     

Numerical study of modified Adomian's method applied to Burgers equation

The application of Adomian's decomposition method to partial differential equations, when the exact solution is not reached, demands the use of truncated series. But the solution's series may have small convergence radius and the truncated series may be inaccurate in many regions. In order to enlarge the convergence domain of the truncated series, Padé approximants (PAs) to the Adomian's series solution have been tested and applied to partial and ordinary differential equations, with good results. In this paper, PAs, both in x$x$ and t$t$ directions, applied to the truncated series solution given by Adomian's decomposition technique for Burgers equation, are tested. Numerical and graphical illustrations show that this technique can improve the accuracy and enlarge the domain of convergence of the solution. It is also shown in this paper, that the application of Adomian's method to the ordinary differential equations set arising from the discretization of the spatial derivatives by finite differences, the so-called method of lines, may reduce the convergence domain of the solution's series.     

Dynamics and synchronization of numerical solutions of the Burgers equation

For Chebyshev spectral solutions of the forced Burgers equation with low values of the viscosity coefficient, several bifurcations and stable attractors can be observed. Periodic orbits, quasiperiodic and strange ones may arise. Bistability can also be observed. Necessary conditions for these attractors to appear are discussed and justification for the non emerging of bistability for an example of a system symmetry break is presented. As an application for the dynamical behavior of spectral solutions of Burgers equation, the dynamics and synchronization of unidirectionally coupling of Chebyshev spectral solutions of Burgers equations by means of a linear coupling are described and discussed. Also, a nonlinear coupling is proposed and discussed.     

A new second‐moment closure approach for turbulent swirling confined flows

An improved anisotropic model for the dissipation rate—ε—of the turbulent kinetic energy (k), to be used together with a non‐linear pressure‐strain correlations model, is proposed. Experimental data from the open literature for two confined turbulent swirling flows are used to assess the performance of the proposed model in comparison to the standard ε transport equation and to a linear approach to model the pressure‐strain term that appears in the exact equations for the Reynolds‐stress tensor. For the less strongly swirling flow the predictions show much more sensitivity to the εtransport equation than to the pressure‐strain model. In opposition, for the more strongly swirling flow, the results show that the predictions are much sensitive to the pressure‐strain model. Nevertheless, the improved εtransport equation together with the non‐linear pressure strain model yield predictions in good agreement with experiments in both studied cases. Copyright © 2003 John Wiley & Sons, Ltd.