Lagrangian motion

In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model for Rayleigh-Benard convection [31], which is given by the stream function ... 

Matsuno, T., Lagrangian motion of air parcels in the stratosphere in the presence of planetary waves. Pure Appl Geophys 118, 189, 1980. 

Notice that if the ratio p/a is equal to some (p/a)Lj with j = 1, 2, 3, 4 or 5, the point Lj only corresponds to the associated circular Eulerian or Lagrangian motion. Hence, with (21), the division of phase space is the following ... 

The optimal pulse separation between two images is influenced by a number of parameters. Two main types of error affect the choice of pulse separation (At), that is, random error and acceleration error. Random error arises from noise during recording of images and subsequent interrogation of the particle images. Acceleration error arises from approximation of Lagrangian motion of tracer particle to local Eulerian velocity based on small particle displacement ... 

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... 

The relationship between H and vibrational frequencies can be made clear by recalling the classical equations of motion in the Lagrangian formulation ... 

Applying the Lagrangian equations to this form for L gives the equations of motion of the qj coordinates ... 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

Figure 2.10. (a) An Eulerian x-t diagram of a shock wave propagating into a material in motion. The fluid particle travels a distance ut, and the shock travels a distance Uti in time ti. (b) A Lagrangian h-t diagram of the same sequence. The shock travels a distance Cti in this system. 

The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

In the foregoing treatments of pressure feedback, the simulation volume retains its cubic form, so changes consist of uniform contractions and expansions. The method is readily extended to the case of a simulation region in which the lengths and directions of the edges are allowed to vary independently. Parrinello and Rahman [31] and Nose and Klein [32] extended the Andersen method to the case of noncubic simulation cells and derived a new Lagrangian for the extended system. Though their equations of motion are... 

The equations of motion follow directly from the Lagrangian formulation containing all constraints. The result is... 

In practice, particle tracking is usually performed in a Lagrangian frame of reference, and the motion of a particle is governed by... 

Note that for general parameterizations this metric matrix is neither skew diagonal nor constant-, see below. The equations of motion expressed in Eq. (2.6) are obtained by using the Principle of Stationary Action, 5A = 0, with Lagrangian... 

The equation for S is recognized as a Hamilton-Jacobi equation for a mechanical action. The solution can be written in terms of a Lagrangian L, for the nuclear motions, introducing the path Q t, Qin,Q), starting initially at positions Qin and ending at Q at time t, and the corresponding generalized velocities Q. The result is... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

These equations of motion can be integrated by many standard ensembles constant energy, constant volume, constant temperature and constant pressure. More complex forms of the extended Lagrangian are possible and readers are referred to Ref. [17] for a Lagrangian that allows intermolecular charge transfer. 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

The motion of a particle in the flow field can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the fluid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton s second law as... 

The computational code used in solving the hydrodynamic equation is developed based on the CFDLIB, a finite-volume hydro-code using a common data structure and a common numerical method (Kashiwa et al., 1994). An explicit time-marching, cell-centered Implicit Continuous-fluid Eulerian (ICE) numerical technique is employed to solve the governing equations (Amsden and Harlow, 1968). The computation cycle is split to two distinct phases a Lagrangian phase and a remapping phase, in which the Arbitrary Lagrangian Eulerian (ALE) technique is applied to support the arbitrary mesh motion with fluid flow. 

The Euler Lagrangian approach is very common in the field of dilute dispersed two-phase flow. Already in the mid 1980s, a particle tracking routine was available in the commercial CFD-code FLUENT. In the Euler-Lagrangian approach, the dispersed phase is conceived as a collection of individual particles (solid particles, droplets, bubbles) for which the equations of motion can be solved individually. The particles are conceived as point particles which move... 

In the two-fluid formulation, the motion or velocity field of each of the two continuous phases is described by its own momentum balances or NS equations (see, e.g., Rietema and Van den Akker, 1983 or Van den Akker, 1986). In both momentum balances, a phase interaction force between the two continuous phases occurs predominantly, of course with opposite sign. Two-fluid models therefore belong to the class of two-way coupling approaches. The continuum formulation of the phase interaction force should reflect the same effects as experienced by the individual particles and discussed above in the context of the Lagrangian description of dispersed two-phase flow. 

As a matter of fact, in comparison with the Euler-Lagrangian approach, the complete Eulerian (or Euler-Euler) approach may better comply with denser two-phase flows, i.e., with higher volume fractions of the dispersed phase, when tracking individual particles is no longer doable in view of the computational times involved and the computer memory required, and when the physical interactions become too dominating to be ignored. Under these circumstances, the motion of individual particles may be overlooked and it is wiser to opt for a more superficial strategy that, however, still has to take the proper physics into account. 

In addition, it is dubious whether this new correlation due to Brucato et al. (1998) should be used in any Euler-Lagrangian approach and in LES which take at least part of the effect of the turbulence on the particle motion into account in a different way. So far, the LES due to Derksen (2003, 2006a) did not need a modified particle drag coefficient to attain agreement with experimental data. Anyhow, the need of modifying particle drag coefficient in some way illustrates the shortcomings of the current RANS-based two-fluid approach of two-phase flow in stirred vessels. 

---