Dissipation in Lie–Poisson systems and the Lorenz-84 model

The relation between dissipation and the symplectic structure of the momentum-space is studied in so(3) Lie algebra and in 2D fluid dynamics. Three kinds of dissipative mechanisms are discussed and a general bracket formalism is introduced. A chaotic dynamical system due to Lorenz, and largely studied in low-dimensional models of geophysical fluid dynamics, is analysed in its geometric and dynamical features, by means of the formalism previously introduced. A mechanism of energy transfer for this low-order model is discussed.     

Directional integration on unstructured meshes via supermesh construction

Unstructured meshes are in widespread use throughout computational physics, but calculating diagnostics of simulations on such meshes can be challenging. For example, in geophysical fluid dynamics, it is frequently desirable to compute directional integrals such as vertical integrals and zonal averages; however, it is difficult to compute these on meshes with no inherent spatial structure. This is widely regarded as an obstacle to the adoption of unstructured mesh numerical modelling in this field. In this paper, we describe an algorithm by which one can exactly compute such directional integrals on arbitrarily unstructured meshes. This is achieved via the solution of a problem of computational geometry, constructing the supermesh of two meshes. We demonstrate the utility of this approach by applying it to a classical geophysical fluid dynamics system: the thermally driven rotating annulus. This addresses an important objection to the more widespread use of unstructured mesh modelling.     

Lattice mesoscopic model of dynamically heterogeneous fluids

We introduce a mesoscopic three-dimensional lattice Boltzmann model which attempts to mimic the physical features associated with cage effects in dynamically heterogeneous fluids. To this purpose, we extend the standard lattice Boltzmann dynamics with self-consistent constraints based on the nonlocal density of the surrounding fluid. The resulting dynamics exhibits typical features of dynamic heterogeneous fluids, such as non-Gaussian density distributions and long-time relaxation. Because of its intrinsically parallel dynamics, and absence of statistical noise, the method is expected to compute significantly faster than molecular dynamics, Monte Carlo, and lattice glass models.     

区域谱元正压大气模式台风移动数值模拟试验

针对地图投影坐标系下的正压原始方程组,将计算区域按三角形元进行分解,在三角形元内用三角形截断的勒让德多项式的积为插值函数对变量进行谱分解,发展出区域正压谱元大气模式.采用固定边界条件,以2006年5月15日08时500 hPa位势高度和风场为初值,在勒让德多项式最高阶数为3和7这两种情形下开展0601号台风"珍珠"移动的数值模拟试验.结果表明,数值模式模拟的风压场关系合理,数值模式的实现是成功的.     

Towards a Landau–Ginzburg-Type Theory for Granular Fluids

In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau–Ginzburg (LG) models for critical and unstable fluids. The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field. Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.     

Constrained molecular dynamics in the isothermal-isobaric ensemble and its adaptation for adiabatic free energy dynamics

The implementation of holonomic constraints within measure-preserving integrators for molecular dynamics simulations in the isothermal-isobaric ensemble is considered. We review the basic methodology of generating measure-preserving integrators for the microcanonical, canonical, and isothermal-isobaric ensembles and proceed to show how the standard SHAKE and RATTLE algorithms must be modified for the isothermal-isobaric ensemble. Comparison is made between constrained and unconstrained simulations employing multiple time scale integration techniques. Finally, we describe a temperature accelerated version of the isothermal-isobaric molecular dynamics approach, in which the cell matrix is adiabatically decoupled from the particles and maintained at a high temperature as a means of exploring polymorphism in molecular crystals. We demonstrate that constraints can be easily adapted for this new approach and, again, we compare the performace of this temperature-accelerated scheme with and without bond constraints.     

Rings: Semicoherent oceanic features

Solitary wave or isolated eddy models are often invoked to explain the longevity of strong nonlinear features in oceans and atmospheres. But when we look at the physics in detail, we find that models of isolated eddies often hinge either on an oversimplification of the dynamics or on constraints which are not appropriate for the observed eddies. In a more complete model, as this study demonstrates, rings (and probably other nonlinear geophysical eddies as well) will interact with their surroundings via Rossby wave radiation, primarily in the barotropic mode. Such wave generation leads to a slow decay of the eddy as energy leaks into the wave field.     

Corrected potential energy functions for constrained molecular dynamics

Atomic oscillations present in classical molecular dynamics restrict the step size that can be used. Multiple time stepping schemes offer only modest improvements, and implicit integrators are costly and inaccurate. The best approach may be to actually remove the highest frequency oscillations by constraining bond lengths and bond angles, thus permitting perhaps a 4-fold increase in the step size. However, omitting degrees of freedom produces errors in statistical averages, and rigid angles do not bend for strong excluded volume forces. These difficulties can be addressed by an enhanced treatment of holonomic constrained dynamics using ideas from papers of Fixman (1974) and Reich (1995, 1999). In particular, the 1995 paper proposes the use of “flexible” constraints, and the 1999 paper uses a modified potential energy function with rigid constraints to emulate flexible constraints. Presented here is a more direct and rigorous derivation of the latter approach, together with justification for the use of constraints in molecular modeling. With rigor comes limitations, so practical compromises are proposed: simplifications of the equations and their judicious application when assumptions are violated. Included are suggestions for new approaches.     

Surface-directed spinodal decomposition and enrichment in fluid mixtures: Molecular dynamics simulations

We present results from molecular dynamics simulations for the non-equilibrium evolution of a binary fluid in the presence of a wetting surface. We study the pattern dynamics which results when a homogeneous fluid mixture is quenched to temperatures both above and below the critical temperature. Our extensive computer simulation results are in agreement with arguments based on Ginzburg-Landau theory.     

Simultaneous effects of magnetic field and space porosity on compressible Maxwell fluid transport induced by a surface acoustic wave in a microchannel

Peristaltic motion induced by a surface acoustic wave of a viscous, compressible and electrically conducting Maxwell fluid in a confined parallel-plane microchannel through a porous medium is investigated in the presence of a constant magnetic field. The slip velocity is considered and the problem is discussed only for the free pumping case. A perturbation technique is employed to analyze the problem in terms of a small amplitude ratio. The phenomenon of a “backward flow” is found to exist in the center and at the boundaries of the channel. In the second order approximation, the net axial velocity is calculated for various values of the fluid parameters. Finally, the effects of the parameters of interest on the mean axial velocity, the reversal flow, and the perturbation function are discussed and shown graphically. We find that in the non-Newtonian regime, there is a possibility of a fluid flow in the direction opposite to the propagation of the traveling wave. This work is the most general model of peristalsis created to date with wide-ranging applications in biological, geophysical and industrial fluid dynamics.     

Geometric, variational discretization of continuum theories

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincaré systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes.     

Comparison of nonequilibrium molecular dynamics with experimental measurements in the nonlinear shear-thinning regime

Nonequilibrium molecular dynamics of a low-molecular-weight fluid (squalane) are compared with experimental measurements in both the linear (Newtonian) and nonlinear (non-Newtonian) regimes. The experimental and simulation data are shown to follow the same time-temperature superposition master curve. This represents the first comparison of the nonlinear rheology predicted by nonequilibrium molecular dynamics with experiment, and is thus the first experimental test of nonequilibrium molecular dynamics simulations in the nonlinear regime.     

微通道中流体流动的分子动力学模拟

本文建立了周期外力作用下流体在微通道间流动模型.以此模型为基础,模拟计算了氩在亲水性和憎水性平行平板壁面间的流动过程,并对速度分布、温度分布等进行了统计.模拟结果表明在憎水性壁面附近,易观察到速度滑移和温度阶跃现象,而亲水性壁面上则反之.与N-S方程和能量方程的解进行了比较,亲水性通道速度和温度的模拟结果与分析解吻合较好,憎水性通道的模拟结果与分析解相差较大.同时,对300℃下不同密度水的流动进行了模拟,随着密度的增大,在壁面附近流体流动特性发生了变化.     

Study of thermal properties of the metastable supersaturated vapor with the restricted ensemble

The pressure and internal energy data at different densities of the supersaturated Argon vapor at the reduced temperatures 0.7 and 0.8 are obtained by the restricted canonical ensemble Monte Carlo simulation method [D.S. Corti, P. Debenedetti, Chem. Eng. Sci. 49 (1994) 2717]. In order to maintain the system in its one-phase state, different constraints on the density fluctuations have been imposed, varying with densities approaching and beyond the spinodal. The results are comparable with a molecular dynamics simulation study [A. Linhart, C.C. Chen, J. Vrabec, H. Hasse, J. Chem. Phys. 122 (2005) 144506] on the same system. The current study verifies the conclusion drawn by the simulation work that clustering of Argon atoms exists even in the vicinity of the spinodal. Compared with molecular dynamics simulations, our method can give the equilibrium properties of a metastable fluid, for example internal energies.     

Stochastic-dynamical thermostats for constraints and stiff restraints

A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nosé-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermostats in the setting of simulating systems with holonomic constraints. The approaches described are easily implemented and facilitate the recovery of correct canonical averages with minimal disturbance of the underlying dynamics. For the purpose of illustrating our results, we examine the numerical application of these methods to a simple atomic chain, where a Fixman term is required to correct the thermodynamic ensemble.     

Detecting coherent structures using braids

The detection of coherent structures is an important problem in fluid dynamics, particularly in geophysical applications. For instance, knowledge of how regions of fluid are isolated from each other allows prediction of the ultimate fate of oil spills. Existing methods detect Lagrangian coherent structures, which are barriers to transport, by examining the stretching field as given by finite-time Lyapunov exponents. These methods are very effective when the velocity field is well-determined, but in many applications only a small number of flow trajectories are known, for example when dealing with oceanic float data. We introduce a topological method for detecting invariant regions based on a small set of trajectories. In this method, we regard the two-dimensional trajectory data as a braid in three dimensions, with time being the third coordinate. Invariant regions then correspond to trajectories that travel together and do not entangle other trajectories. We detect these regions by examining the growth of hypothetical loops surrounding sets of trajectories, and searching for loops that show negligible growth.