Lagrangian approaches

Although we have derived the desired equations, we have at the same time generated new dependent variables u iu kc )J,k= 1,2,3. If we generate additional equations for these variables, we find that still more dependent variables appear. The closure problem becomes even worse if a nonlinear chemical reaction is occurring. If the single species decays by a second-order reaction, then the term (R) in (18.4) becomes -k((c)2 + (c 2)), where (c 2) is a new dependent variable. If we were to derive an equation for (c 2), we would find the emergence of new dependent variables ( jc 2), (c 3), and (0c /0x -0c /0xy). It is because of the closure problem that an Eulerian description of turbulent diffusion will not permit exact solution even for the mean concentration (c). 

The Lagrangian approach to turbulent diffusion is concerned with the behavior of representative fluid particles.4 We therefore begin by considering a single particle that is at location x at time i in a turbulent fluid. The subsequent motion of the particle can be described by its trajectory, X[x, t /], that is, its position at any later time t. Let [/(xi, x2, x3, t)dx dx2 dx3 = v /(x, t)dx = probability that the particle at time t will be in volume element x to 1 + dx, x2 to x2+dx2, and x3 to x3 + dx3, that is, that x X[ x + dx, and so on. Thus i /(x, t) is the probability density function (pdf) for the particle s location at time t. By the definition of a probability density function 

The probability density of finding the particle at x at t can be expressed as the product of two other probability densities  

The probability density that if the particle is at x at t1, it will undergo a displacement to x at t. Denote this probability density Q(x. t x, t ) and call it the transition probability density for the particle. 

The probability density that the particle was at x at t, v(/(x, t ), integrated over all possible starting points x. Thus 

Although we have derived the desired equations, we have at the same time generated new dependent variables j,k = 1,2, 3. If we generate additional equations for these 

The Lagrangian approach to turbulent diffusion is concerned with the behavior of representative fluid particles. We therefore begin by considering a single particle that is at location x at time t in a turbulent fluid. The subsequent motion of the particle can be described by its trajectory, X[x, r], that is, its position at any later time t. Let 

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For simulating computationally the spatial and temporal evolution of both physical and chemical processes in mixing devices operated in a turbulent singlephase mode, two essentially different approaches are available the Lagrangian approach and the Eulerian technique. These will be explained briefly. 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase flow in stirred tanks in a RANS context may be treated in two ways Euler-Lagrange or Euler-Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 

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... 

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 simulating physical operations carried out in stirred vessels, generally one has the choice between a Lagrangian approach and a Eulerian description. While the former approach is based on tracking the paths of many individual fluid elements or dispersed-phase particles, the latter exploits the continuum concept. The two approaches offer different vistas on the operations and require different computational capabilities. Which of the two approaches is most... 

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. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

Aerosol production and transport over the oceans are of interest in studies concerning cloud physics, air pollution, atmospheric optics, and air-sea interactions. However, the contribution of sea spray droplets to the transfer of moisture and latent heat from the sea to the atmosphere is not well known. In an effort to investigate these phenomena, Edson et al.[12l used an interactive Eulerian-Lagrangian approach to simulate the generation, turbulent transport and evaporation of droplets. The k-e turbulence closure model was incorporated in the Eulerian-Lagrangian model to accurately simulate... 

Generally, 3-D models are essential for calculating the radial distributions of spray mass, spray enthalpy, and microstructural characteristics. In some applications, axisymmetry conditions may be assumed, so that 2-D models are adequate. Similarly to normal liquid sprays, the momentum, heat and mass transfer processes between atomization gas and metal droplets may be treated using either an Eulerian or a Lagrangian approach. 

Now we turn to the Eulerian approach to obtain the same results as we have just obtained by the Lagrangian approach. 

Finally, Lagrangian approaches to source apportionment have been used in some airsheds. For example,... 

Typical Lagrangian approaches include the deterministic trajectory method and the stochastic trajectory method. The deterministic trajectory method neglects all the turbulent transport processes of the particle phase, while the stochastic trajectory method takes into account the effect of gas turbulence on the particle motion by considering the instantaneous gas velocity in the formulation of the equation of motion of particles. To obtain the statistical... 

Lagrangian, respectively. At first we note that the total energy is conserved in both the dynamics, with oscillations orders of magnitude smaller than the oscillations of the potential energy. The latter presents on the other hand a behavior that is quite different in the two cases. For the case in which the charges are equilibrated at each step, the oscillations are quite large, of the order of 3.5 x 10-3 au, and they last for the whole trajectory. On the other hand, for the extended Lagrangian approach, after an initial period... 

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