Velocity field Lagrangian

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. 

A third way of looking at the velocity field is from a Lagrangian perspective (i.e., following a fluid particle see Section 6.7 for more details). In this case, the position of the fluid particle x+(r) varies with time. The resulting Lagrangian velocity... 

A Lagrangian description of the velocity field can be used to find the location X(f) of the fluid element at time 0 < t that started at X(0). In the Lagrangian description, (3.3) implies that the scalar field associated with the fluid element will remain unchanged, i.e., (X(f), 0 = (X(0), 0). 

If Eq. (14) holds then anomalous diffusion may appear only for D = 0 and very strong Lagrangian velocity correlations. The latter condition can be realized— for example, in time periodic velocity fields where the Lagrangian phase space has a complicated self-similar structure of islands and cantori [30]. Here superdiffusion is due to the almost trapping, for arbitrarily long time, of the ballistic trajectories close to the cantori, which are organized in complicated selfsimilar structures. 

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 ... 

Each droplet draws its own trajectory in space that depends on its radius r. One could easily get a Lagrangian model for the droplet ensemble if the air velocity field would be exactly known. The problem consists however in the fact that the latter field V(x, z) itself is dependent on all the moving obstructions and their complex velocity field V(x,y z). Interpreting droplets as a continuous medium, as a gas of droplets , and substituting the individual derivatives by local ones, one gets the following Eulerian model for it ... 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

In stochastic Lagrangian particle models, the evolution of the concentration field is computed in a two-step process. First, the Eulerian velocity field in the region of interest must be calculated, either by solution of the Navier-Stokes equations or via an approximate method that satisfies mass consistency. The solution must also provide the local statistics of the velocity field. Individual particles are then released, and their position is updated over a time increment dt using an equation of the form (Wilson and Sawford 1996)... 

Later Taylor began to search for more suitable means for the the description of turbulence [165]. The statistical approach to the study of turbulence was initiated by a paper by Taylor [158]. In the work of Taylor [158] on turbulent transport the important role of the Lagrangian correlation function (i.e., the one point time correlation) of the velocity field was first demonstrated. Taylor showed that the turbulent diffusion of particles starting from a point depends on the correlation between the velocity of a fluid particle at any instant and that of the same particle after a certain correlation time interval. 

Before we finish this subsection, we would like to discuss the practical limitations of the Poincare sections, which require Lagrangian particle tracking for extended times. In reality. Fig. 2 presents stroboscopic images of the same four particles passing through thousands of mixing block boundaries. This has two basic implications. First, numerical calculation of the Poincare sections requires either analytical solutions or high-order accurate discretizations of the velocity field. Otherwise, the results may suffer from numerical diffusion and dispersion errors, and the KAM boundaries may not be identified accurately. Second, it is experimentally difficult, if not impossible, to track particles (in three-dimensions) beyond a certain distance allowed by the field of view of the microscopy technique. Despite these... 

From the Lagrangian map, the velocity field can be easily recovered by differentiation with respect to time. The inverse procedure, obtaining the Lagrangian map from the velocity field, requires integration of (2.4) and it is usually not possible doing it other than numerically. In any case, one can express the formal solution of equation (2.2) in terms of the solution of (2.3) and the inverse of the Lagrangian map 1 as... 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

Equation (4.13) is a particular case. Ci is a set of concentrations interacting through the reaction terms Ri and vx = (t)x is the, possibly time-dependent, transverse velocity field pointing towards the center of the filament located at x = 0. As in Sect. 2.7.1 this is to be understood as a local Lagrangian description. 

Lagrangian inverse methods also rely on a balance equation. In this case, the principle is to consider individual marked fluid particles which (with sufficient knowledge of the velocity field) can be followed as they move. The balance equation in a Lagrangian framework is simply... 

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