Euler-Lagrangian

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 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. 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

The task of modeling binary droplet collisions in Euler-Lagrangian simulations of spray flows was first taken up by O Rourke and coworkers. Their model in [83] first estimates the coalescence efficiency, which is the probability that coalescence occurs after the collision, once it has taken place ... 

Probability Density Function model (PDF) Discrete Ordinate model (DO) Euler-Lagrangian model... 

The aim of this numerical study is to analyze the effects of the grouping/clustering of droplets in sprays on the heat transfer. In this study an Euler-Lagrangian algorithm is used to simulate the droplet-gas interactions. The droplets are assumed spherical with a radius smaller than the smallest length scale of the turbulence and a density much larger than that of the ambient gas. The heat transfer of individual droplets is characterized by the Nusselt number, which is defined by the... 

The clustering of droplets within twin-fluid atomizers sprays has been investigated by means of LES. The point particle method has been used to track the dispersed phase within the Euler-Lagrangian framework. The simulated structures show agreement with Particle-lmage-Velocimetry measurements. The simulation could quantitatively be verified by evaluating the dispersed phase velocity profiles... 

By applying the first-order condition for optimization, that is, the first derivative with respect to the control variable, and Lagrange multipliers should disappear resulting in Euler-Lagrangian differential equations given below... 

Formulate the maximum distillate problem using the calculus of variations. Solution Since this problem contains equality constraints, we need to use the Euler-Lagrangian formulation. First, all three equality constraints (Equations 5.47 to 5.49) are augmented to the objective function to form a new objective function... 

In the Euler—Lagrangian approach of two-phase flow (see, e.g., Crowe et al, 1996), the particles are treated as point particles the finite volume of the particles is not considered and the flow around the particles is not resolved. The motion of the particle is simulated by means of Newton s second law and that is why the fluid—particle interaction force is needed and the empirical correlations enter. Although the flow around the particles is not resolved, any empirical correlation does reflect the hydrodynamics of the canonical case involved. The use of the Euler—Lagrange approach, or point-particle method, is usually restricted to the more dilute gas—solid and liquid—soHd systems. Ignoring the mutual interaction of particles is therefore not too serious a simplification. The fluid-particle interaction can be treated in the simpler one-way mode or according to the more complicated two-way coupling mode in which the particles also affect the carrier phase flow field (Decker and Sommerfeld, 2000 Derksen, 2003 Derksen et al, 2008). 

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