Verifiable point-particle methods for two-way coupled particle-laden flows
Keywords: Mechanical Engineering.

This work is concerned with the computation of particle-laden flows, primarily in the dilute regime, using the point-particle approach. The point-particle method solves continuum transport equations for the fluid phase coupled to Lagrangian equations for each particle. The coupling is accomplished by modelled source terms (momentum, energy, mass) which function as surrogates for boundary conditions on particle surfaces. This dissertation begins by surveying some of the widely used source terms, especially drag models, and the conditions under which these drag models are applicable. The incorporation of a drag model in a numerical simulation in general requires knowledge of the undisturbed fluid velocity at the location of the particle, which is the fluid velocity the particle experiences at every point along its trajectory, in absence of the particle. Because this quantity is not directly available in coupled simulations of particle-fluid interaction, and because the undisturbed fluid velocity is difficult to interpret physically, most modellers neglect to model it and instead incorporate a disturbed fluid velocity at the particle location, which is found by interpolation of the fluid velocity to the particle location using standard interpolation schemes. In this dissertation, we will show that the difference between the disturbed and undisturbed fluid velocity can be large, that the difference scales with the ratio of the particle size to the grid spacing, and that estimating the undisturbed fluid velocity is necessary for successful verification of coupled point-particle methods. We develop a scheme motivated by Stokesian symmetries which estimates the undisturbed fluid velocity by correlating this quantity to an enhancement in fluid curvature created by point-particles. The scheme is found to well predict particle settling velocity at low and finite Reynolds numbers, while standard schemes used in literature greatly over predict particle settling velocity. By examining the total particle plus fluid energy equation, we find that accurate estimation of the undisturbed fluid velocity implies a correspondence principal--namely the correct prediction of dissipation rate consistent with the drag model chosen. We then explore the consequences of a verifiable point-particle method in forced and decaying homogeneous isotropic turbulence. In the former, the incorporation of the undisturbed fluid velocity prediction results in enhanced clustering of particles, especially at smaller separations compared with standard schemes. This is related to a broadening in acceleration probability density function when the undisturbed fluid velocity is used to calculate the drag force. In decaying turbulence, for several fluid and particle statistics, it is found that standard point-particle approaches do not converge with grid refinement, while incorporation of our proposed correction for the undisturbed fluid velocity can result in grid insensitive results for lower-order moments. Examination of higher order moments reveals grid dependence for all point-particle implementations which suggests that not all practical questions surrounding particle-laden flows are answerable with the point-particle method. In the next section, the point-particle method is directly compared against non-dimensionally identical simulations of resolved particles in decaying turbulence. We find that under certain conditions, specification of an appropriate drag model which accounts for finite particle Reynolds number and accurate computation of the undisturbed fluid velocity are necessary for successful validation of the point-particle method. Under these conditions, good agreement is found for both integral quantities such as fluid dissipation rate, but also in comparison of particle acceleration probability density functions. Interestingly, under the same conditions, it is found that using a less suitable drag model but where the undisturbed fluid velocity is accounted can yield better predictions of the point-particle method compared with when a more appropriate drag model is used without accounting for the undisturbed fluid velocity. Having spent a large portion of this work on examining momentum/energy coupling in the absence of heat transfer, we move toward the examination of problems where particles can move and exchange internal energy with the fluid. The heat transfer sources, analogous to the drag models discussed previously, depend on the undisturbed fluid temperature at the particle location. We develop a scheme to estimate the undisturbed temperature by correlating it to the measured disturbed curvature in the temperature field created by a point-particle. We then perform verification of the proposed procedure for a settling particle subject to radiation under low and finite heating conditions. The proposed correction for the undisturbed temperature, combined with the previous method for estimating the undisturbed fluid velocity, together significantly reduce the error in settling velocity and terminal temperature compared with standard point-particle schemes. Finally, we discuss some outstanding questions in particle-laden flows, and how the current methodologies presented can be extended. One such extension concerns calculation of undisturbed quantities on anisotropic grids which are often used in the neighborhood of walls. We outline a general approach to this problem using the method of discrete Green's functions.