Gaussian Weighted Trajectory Method Ⅳ： No Rainbow Effect in Practice

The Gaussian weighted trajectory method （GWTM） is a practical implementation of classical S matrix theory （CSMT） in the random phase approximation, CSMT being the first and simplest semi-classical approach of molecular collisions, developped in the early seventies. Though very close in spirit to the purely classical description, GWTM accounts to some extent for the quantization of the different degrees-of-freedom involved in the processes. While CSMT may give diverging final state distributions, in relation to the rainbow effect of elastic scattering theory, GWTM has never led to such a mathematical catastrophe. The goal of the present note is to explain this finding.

Six-DOF trajectory optimization for reusable launch vehicles via Gauss pseudospectral method

To be close to the practical flight process and increase the precision of optimal trajectory, a six-degree-offreedom（6-DOF） trajectory is optimized for the reusable launch vehicle（RLV） using the Gauss pseudospectral method（GPM）. Different from the traditional trajectory optimization problem which generally considers the RLV as a point mass, the coupling between translational dynamics and rotational dynamics is taken into account. An optimization problem is formulated to minimize a performance index subject to 6-DOF equations of motion, including translational and rotational dynamics. A two-step optimal strategy is then introduced to reduce the large calculations caused by multiple variables and convergence confinement in 6-DOF trajectory optimization. The simulation results demonstrate that the 6-DOF trajectory optimal strategy for RLV is feasible.

Quasi-Classical Trajectory Study on O＋＋DH（v=0,j=0）→OD＋＋H Reaction at Different Collision Energy

The quasi-classical trajectory calculations O＋＋DH（v=0,j=0）→OD＋＋H reactions on the RODRIGO potential energy surface have been carried out to study the isotope effect on stereo-dynamics at the collision energies of 1.0, 1.5, 2.0, and 2.5 eV. The distributions of dihedral angle P（~r） and the distributions of P（Or） are discussed. Furthermore, the angular distributions of the product rotational vectors in the form of polar plot in θr and φr are calculated. The differential cross section shows interesting phenomenon that the reaction is dominated by the direct reaction mechanism. Reaction probability and reaction cross section are also calculated. The calculations indicate that the stereo-dynamics properties of the title reactions are sensitive to the collision energy.

Quasi-Classical Trajectory Study of the Chemical Reaction Ca+CH_3I

The Ca＋CH3I→CaI＋CH3 reaction system has been studied with the quasi-classical trajectory method on the extended Lond-Eyring-Polanyi-Sato（LEPS） potential energy surface. At collision energy Ecol=10.78 kJ/mol, the calculated results show that the CaI vibrational population peaks are located at v=2. The calculated cross section decreases slowly with the collision energy increasing. The angle product distributions tend toward backward scattering. The calculated （P2（J^1·K）） values deviate slightly from-0.5 and decrease with increasing collision energy. The Quasiclassical trajectory calculation（QCT） results are in reasonable agreement with experimental data. Moreover, the dynamics of the reaction has been discussed.

Use of the Method of Guidance by a Required Velocity in Control of Spacecraft Attitude

We apply the method of guidance by a required velocity for solving the optimal control problem over spacecraft’s reorientation from known initial attitude into a required final attitude.We suppose that attitude control is carried out by impulse jet engines.For optimization of fuel consumption,the controlling moments are calculated and formed according to the method of free trajectories together with principle of iterative control using the quaternions for generating commands to actuators.Optimal solution corresponds to the principle“acceleration-free rotation-separate corrections-free rotation-braking”.Rotation along a hitting trajectory is supported by insignificant correction of the uncontrolled motion at discrete instants between segments of acceleration and braking.Various strategies of forming the correction impulses during stage of free motion are suggested.Improving accuracy of achievement of spacecraft's final position is reached by terminal control using information about current attitude and angular velocity measurements for determining an instant of beginning of braking(condition for start of braking based on actual motion parameters is formulated in analytical form).The described method is universal and invariant relative to moments of inertia.Developed laws of attitude control concern the algorithms with prognostic model,the synthesized control modes are invariant with respect to both external perturbations and parametric errors.Results of mathematical modeling are presented that demonstrate practical feasibility and high efficiency of designed algorithms.

Distribution Characteristics of Ozone and the Relationship with Typhoon Processes in the Northwest Pacific Ocean

Based on the vertical ozone reanalysis data and total ozone column data derived from the European Centre for Medium-Range Weather Forecasts,the spatial and temporal distribution characteristics of ozone on each isobaric surface in the troposphere over the Northwest Pacific Ocean were analyzed,and the backward trajectory method was used to track the influence of typhoon on the distribution of ozone.The results show that the updraft near the typhoon center transported the air with low O_(3)content in the lower layer to the upper layer,which reduced O_(3)content in the upper layer and formed a low-value area of O_(3).The variation trend of total ozone column in the regions where typhoons"Megi"and"Fengshen"occurred was analyzed by the case analysis method.It is found that there was a low-value area of total ozone column anomaly near the typhoon center,and there was a certain correlation between typhoon intensity PDI and total ozone column anomaly at the development and maturity stages of typhoons.

A semi-analytical multi-harmonic balance method on full-3D contact model for dynamic analysis of dry friction systems

Dry friction damping structures are widely-used in aero-engines to mitigate vibration.The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately and efficiently predict the forced response of frictionally damped structures.The state-of-the-art Multi-Harmonic Balance Method(MHBM)on quasi-3D contact model in engineering cannot precisely capture the kinematics on the friction interface although the efficiency is high.The full-3D contact model can describe the constitutive relationship of the interface in a more accurate manner;however,the efficiency and convergence are not guaranteed for large-scale models.In this paper,a semi-analytical MHBM on full-3D contact model is proposed.The original Trajectory Tracking Method(TTM)for evaluating the contact force is reformulated to make the calculation more concise and the derivation of the Analytical Jacobian Matrix(AJM)feasible.Based on the chain rule of derivation,the AJM which is the core to upgrade the performance is deduced.Through a shrouded blade finite element model,the accuracy and efficiency of the proposed method are compared with both the MHBM on full-3D contact model with numerical Jacobian matrix and the MHBM on quasi-3D contact model with AJM.The results show that the AJM improves significantly the efficiency of the MHBM on full-3D contact model.The time cost of the proposed method is in the same order of magnitude as that of the MHBM on quasi-3D contact model.We also confirm that the full-3D contact model is necessary for the dynamic analyses of shrouded blades.If one uses the quasi-3D model,the estimation relative error of damping can even reach 31.8%in some cases.In addition,the AJM also brings benefits for stability analysis.It is highly recommended that engineers use the MHBM on full-3D contact model for the dynamic analysis and design of shrouded blades.

Theoretical Study on Stereodynamics of Reactions of N（^2D）＋H2→NH＋H and N（^2D）＋D2→ND＋D

The vector correlations between products and reagents for the title reactions have been calculated by the quasi-classical trajectory method at a collision energy of 21.32 kJ/mol on an accurate potential energy surface of Ho et al. （J. Chem. Phys. 119, 3063 （2003））. The peaks of the product angular distribution are found to be in both backward and forward directions for the two title reactions. The product rotational angular momentum is not only aligned, but also oriented along the negative direction of y-axis. These theoretical results are in good agreement with recent experimental findings for the two title reactions. The isotopic effect is also revealed and primarily attributed to the difference of the mass factor in the two title reactions.

Influence of rotational excitation and collision energy on the stereo dynamics of the reaction:N(~4S)+H_2 (v=0,j=0,2,5,10)→NH(X^3Σ^-)+H

The N＋H2 reaction has attracted a great deal of attention from both the experimental and the theoretical community, and most of the attention has been paid to the first excited state N （2 D） atoms in collisions with hydrogen molecules and the scalar properties of the reaction. In this paper, we study the stereo dynamical properties and calculate the reaction cross sections of the N（4S） ＋ H2 （v=0, j=0, 2, 5, 10） → NH（X3∑-） ＋ H using the quasi-classical trajectory （QCT） method on an accurate NH2 potential energy surface （PES） reported by Poveda and Varandas [Poveda L A and Varandas A J C 2005 Phys. Chem. Chem. Phys. 7 28671, in a collision energy range of 25 kcal.mol-1 -140 kcal.mol-1. Results indicate that the reactant rotational excitation and initial collision energy both have a considerable influence on the distributions of the k-j＇ correlation, the k-k＇-j＇ correlation and k-k＇ correlation. The differential cross section is found to be sensitive to collision energy.

Stereodynamics study of reactions N（2D）＋HD→NH＋D and ND＋H

The product polarizations of the title reactions are investigated by employing the quasi-classical trajectory （QCT） method. The four generalized polarization-dependent differential cross-sections （PDDCSs） （2π/σ）（dσ00/dωt）, （2π/σ）（dσ20/dωt）, （2π/σ）（dσ22＋/dωt）, and （2π/σ）（dσ21-/dωt） are calculated in the centre-of-mass frame. The distribution of the angle between κ and j＇, P（θr）, the distribution of the dihedral angle denoting κ-κ＇-j＇ correlation, P（Фr）, as well as the angular distribution of product rotational vectors in the form of polar plots P（θr, Фr） are calculated. The isotope effect is also revealed and primarily attributed to the difference in mass factor between the two title reactions.

Theoretical study of the stereodynamics of the He+HD^+ reaction

This paper investigates the stereodynamics of the reaction He＋HD^＋ by the quasi-classical trajectory （QCT） method using the most accurate AQUILANTI surface [Aquilanti et al 2000 Mol. Phys. 98 1835]. The distribution P（Фτ） of dihedral angle and the distribution P（θτ） of angle between k and j＇ have been presented at three different collision energies. Four generalized polarization-dependent differential cross-sections （2π/σ）（dσ00/dωt）, （2π/σ）（dσ20/dωt）, （2π/σ）（dσ22/dωt）, （（2π/σ）（dσ21-/dωt） are also calculated. Some interesting results are obtained from the comparison of the stereodynamics of the title reaction at different collision energies.

The effect of collision energy on the stereodynamics of the reaction H(~2S)+NH(X^3∑^-,v = 0,j = 0)→ N(~4S)+H_2

The quasi-classical trajectory （QCT） is calculated to study the stereodynamics properties of the title reaction H（^2S） ＋ NH （X^3 ∑^-, v = 0, j = 0）→ N（^4S） ＋ H2 on the ground state ^4A″ potential energy surface （PES） constructed by Zhai and Han [2011 Jr. Chem. Phys. 135 104314]. The calculated QCT reaction probabilities and cross sections are in good agreement with the previous theoretical results. The effects of the collision energy on the k-kt distribution and the product polarization of H2 are studied in detail. It is found that the scattering direction of the product is strongly dependent on the collision energy. With the increase in the collision energy, the scattering directions of the products change from backward scattering to forward scattering. The distribution of P（Or） is strongly dependent on the collision energy below the lower collision energy （about 11.53 kcal/mol）. In addition, the P（（Pr） distribution dramatically changes as the collision energy increases. The calculated QCT results indicate that the collision energy plays an important role in determining the stereodynamics of the title reaction.

Effects of collision energy and rotational quantum number on stereodynamics of the reactions:H(~2S)+NH(v=0,j=0,2,5,10)→N(~4S)+H_2

The stereodynamical properties of H（^2S） ＋ NH（v = 0,j = 0,2,5,10）→N（^4S） ＋ H2 reactions are studied in this paper by using the quasi-classical trajectory（QCT） method with different collision energies on the double many-body expansion（DMBE） potential energy surface（PES）（Poveda L A and Varandas A J C 2005 Phys.Chem.Chem.Phys.7 2867）.In a range of collision energy from 2 to 20 kcal/mol,the vibrational rotational quantum numbers of the NH molecules are specifically investigated on v = 0 and j = 0,2,5,10 respectively.The distributions of P（θr）,P（φr）,P（θr,φr）,（2π/σ）（dσ（00）/dωt）differential cross-section（DCSs） and integral cross-sections（ICSs） are calculated.The ICSs,computed for collision energies from 2 kcal/mol to 20 kcal/mol,for the ground state are in good agreement with the cited data.The results show that the reagent rotational quantum number and initial collision energy both have a significant effect on the distributions of the k-j＇,the k-k＇-j＇,and the k-k＇ correlations.In addition,the DCS is found to be susceptible to collision energy,but it is not significantly affected by the rotational excitation of reagent.

Theoretical study of stereodynamics for the N+H_2/D_2/T_2 reactions

The effects of isotopic variants on stereodynamic properties for the title reactions have been investigated using a quasi-classical trajectory method based on the first excited state NH2（I^2A＇） potential energy surface [Li Y Q and Varandas A J C 2010 J. Phys. Chem. A 114 9644]. The forward–backward symmetry scattering of the differential cross section can be observed, which demonstrates that all these reactions follow the insertion mechanism. Three angle distribution functions P（θr）, P（φr）, and P（θr, φr） with different collision energies and target molecules H2/D2/T2 are calculated. It is shown that the product rotational angular momentum is not only aligned, but also oriented along the direction perpendicular to the scattering plane. The title reaction is mainly governed by the ＂in-plane＂ mechanism through the calculated distribution function P（θr, φr）. The observable influences on the rotational polarization of the product by the isotopic substitution of H/D/T can be demonstrated.

Stereodynamics study of the exchange reaction O(~3P)+CH_4 →H+OCH_3

A new London-Eyring-Polanyi-Sato potential energy surface is employed in this work to study the stereo properties of the O （^3p） ＋ CH4 → H ＋ OCH3 reaction in its rovibrationally ground state using the quasiclassical trajectory method （QCT）. Our calculations are performed at a range of collision energies, Ec = 1.5 eV^-3.5 eV, and the excitation function obtained by the QCT method accords well with the experimental data. The product rotational polarization is calculated, and the product shows a strong rotational polarization in the centre-of-mass coordinate system. The orientation of the product rotational angular momenta is sensitive to the increase in collision energy, and the alignment of the product rotational angular momenta shows some of the properties of the heavy heavy-light mass combination reactions. In the isotopic substituted reaction study, when the H atoms in methane are replaced by D atoms, the rotational polarization is obviously reduced. The polarization-dependent differential cross section is also studied by this QCT calculation to provide detailed information about the rotational alignment and orientation of the product.

Stereodynamics of the O(~3P) with H_2(D_2) (ν=0,j=0) reaction

Quasi-classical trajectory theory is used to study the reaction of O（3p） with H2 （D2） based on the ground 3A″ potential energy surface （PES）. The reaction cross section of the reaction O＋H2→＋OH＋H is in excellent agreement with the previous result. Vector correlations, product rotational alignment parameters （P2（j′. k）） and several polarizeddependent differential cross sections are further calculated for the reaction. The product polarization distribution exhibits different characteristics that can be ascribed to different motion paths on the PES, arising from various collision energies or mass factors.

Effect of reagent vibrational excitation and isotope substitution on the stereo-dynamics of the Ba+HF→BaF+H reaction

Based on an extended London-Eyring-Polanyi-Sato （LEPS） potential energy surface （PES）, the Ba q-HF reaction has been studied by the quasi-classical trajectory （QCT） method. The reaction integral cross section as a function of collision energy for the Ba q- HF --＊ BaF q- H reaction is presented and the influence of isotope substitution on the differential cross sections （DCSs） and alignments of the product＇s rotational angular momentum have also been studied. The results suggest that the integral cross sections increase with increasing collision energy, and the vibrational excitation of the reagent has great influence on the DCS. In addition, the product＇s rotational polarization is very strong as a result of heavy-heavy-light （HHL） mass combination, and the distinct effect of isotope substitution on the stereodynamics is also revealed.

Two-phase micro-and macro-time scales in particle-laden turbulent channel flows

The micro- and macro-time scales in two-phase turbulent channel flows are investigated using the direct nu- merical simulation and the Lagrangian particle trajectory methods for the fluid- and the particle-phases, respectively. Lagrangian and Eulerian time scales of both phases are cal- culated using velocity correlation functions. Due to flow anisotropy, micro-time scales are not the same with the theo- retical estimations in large Reynolds number （isotropic） tur- bulence. Lagrangian macro-time scales of particle-phase and of fluid-phase seen by particles are both dependent on particle Stokes number. The fluid-phase Lagrangian inte- gral time scales increase with distance from the wall, longer than those time scales seen by particles. The Eulerian inte- gral macro-time scales increase in near-wall regions but de- crease in out-layer regions. The moving Eulerian time scales are also investigated and compared with Lagrangian integral time scales, and in good agreement with previous measure- ments and numerical predictions. For the fluid particles the micro Eulerian time scales are longer than the Lagrangian ones in the near wall regions, while away from the walls the micro Lagrangian time scales are longer. The Lagrangian integral time scales are longer than the Eulerian ones. The results are useful for further understanding two-phase flow physics and especially for constructing accurate prediction models of inertial particle dispersion.

Stereodynamics in reaction O(~1D) + CH_4 →OH + CH_3

A theoretical study of the stereodynamics for reaction O(1D) + CH4→OH + CH3 has been carried out using the quasiclassical trajectory method(QCT) on a potential energy surface structured by Gonzalez et al. The integral cross sections(ICSs), differential cross sections(DCSs) and product rotational angular momentum polarization have been calculated. With the collision energy increasing, the ICS decreases. There is no threshold energy, because no barrier is found on the minimum energy path. The DCS results show that the backward and forward scatterings exist at the same time. With the collision energy increasing, the dominant rotation of the product changes from the right-handed direction to the left-handed direction in planes parallel to the scattering plane. In the isotopic effect study, the decrease of the mass factor weakens the polarization degree of the rotational angular momentum vectors of the products.

Quantum and quasiclassical dynamics of C(^(3)P)+H_(2)(^(1Σ_(g)^(+)))→H(^(2)S)+CH(^(2)Ⅱ)reaction:Coriolis coupling effects and stereodynamics

The dynamics of C+H_(2)→H+CH reaction is theoretically studied using the quasiclassical trajectory and quantum mechanical wave packet methods.The analysis of reaction probabilities,integral cross sections,and rate coefficients reveal the essential Coriolis coupling effects in the quantum mechanical wave packet calculations.The calculated polarizationdependent differential cross section,P(θ_(r))and P(Φ_(r))show that the j'of product rotational angular momentum is not only aligned along the y axis and the direction of the vector x+z,but also strongly oriented along the positive y axis.