Fully Correlated Stochastic Inter-Particle Collision Model for Euler–Lagrange Gas–Solid Flows

Abstract

In Lagrangian stochastic collision models, a fictitious particle is generated to act as a collision partner, with a velocity correlated to the velocity of the real colliding particle. However, most often, the fluid velocity seen by this fictitious particles is not accounted for in the generation of the fictitious particle velocity, leading to a de-correlation between the fictitious particle velocity and the local fluid velocity, which, after collision, leads to an unrealistic de-correlation of the real particle velocity and the fluid velocity as seen by the particle. This de-correlation, in turn, causes a spurious decrease of the particle kinetic energy, even though the collisions are assumed perfectly elastic. In this paper, we propose a new model in which the generated fictitious particle velocity is correctly correlated to both the real particle velocity and the local fluid velocity at the particle, hence preventing the spurious loss of the total particle kinetic energy. The model is suitable for small inertial particles. Two algorithms for integrating the collision frequency are also compared to each other. The models are validated using large eddy simulation (LES) of mono-dispersed particle-laden stationary homogeneous isotropic turbulence. Simulations are conducted with spherical particles with different turbulent Stokes number, $$St_t = [0.75 - 5.8]$$ S t t = [ 0.75 - 5.8 ] , and volume fractions, $$\alpha _p = [0.014 - 0.044]$$ α p = [ 0.014 - 0.044 ] , and are compared to the results of the LES using a deterministic discrete particle simulation model.

Study of Colliding Particle-Pair Velocity Correlation in Homogeneous Isotropic Turbulence

“…As the modulus of the RRV increases with Stokes number (Figure 13), the distance between both curves augments with particle inertia. Moreover, the results of the recent study of van Wachem et al [29] for R c are also depicted in Figure 12 as crosses, showing a good agreement with those obtained in the present simulations. However, in Reference [29], the volume fraction values were different for the four particle types analysed and ranged from 0.3% to 2%.…”

“…Moreover, the results of the recent study of van Wachem et al [29] for are also depicted in Figure 12 as crosses, showing a good agreement with those obtained in the present simulations. However, in Reference [29], the volume fraction values were different for the four particle types analysed and ranged from 0.3% to 2%. Another interesting fact that can be observed in the correlation functions in Figure 12 is that, for a fixed particle Stokes number, their values increase with increasing solid volume fraction.…”

“…The properties of the fictitious particle such as diameter or velocities are sampled randomly from previously constructed local distributions, and the probability of collision is computed according to kinetic theory. However, in this process, the fluctuating velocity correlation of both particles has to be taken into account [29,32] and such correlation depends on the particle response to the turbulent environment. In the original paper of Sommerfeld [31], it was modelled by the following decaying exponential:…”

“…This process is however very time consuming and its computational cost is, at least, proportional to N p (total number of tracked particles) at each Lagrangian time step. For instance, a recent study by van Wachem et al [29] mentions that the computational time needed for the direct detection approach is two orders of magnitude higher than that for a stochastic approach. Once a collision occurs, the change in translational and angular particle velocities is determined by solving the equations for the conservation of linear and angular momentum in connection with Coulomb's law of friction.…”

“…Stochastic inter-particle collision models [29,31,32], on the other hand, rely on the generation of fictitious collision partners and the calculation of the collision probability according to kinetic theory. The advantage of such models is that they do not require information on the location of the surrounding particles and hence they are also applicable if a sequential tracking of the particles is adopted, as it is usually done when applying the Euler-Lagrange approach to stationary flows.…”