Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling

Abstract

In Part I, simulations of strongly coupled fluid-particle flow in a vertical channel were performed with the purpose of understanding, in general, the fundamental physics of wall-bounded multiphase turbulence and, in particular, the roles of the spatially correlated and uncorrelated components of the particle velocity.The exact Reynolds-averaged (RA) equations for high-mass-loading suspensions were presented, and the unclosed terms that are retained in the context of fully developed channel flow were evaluated in an Eulerian-Lagrangian (EL) framework. Here, data from the EL simulations are used to validate a multiphase Reynolds-stress model (RSM) that predicts the wall-normal distribution of the two-phase, one-point turbulence statistics up to second order. It is shown that the anisotropy of the Reynolds stresses both near the wall and far away is a crucial component for predicting the distribution of the RA particle-phase volume fraction. Moreover, the decomposition of the phase-average (PA) particle-phase fluctuating energy into the spatially correlated and uncorrelated components is necessary to account for the boundary conditions at the wall. When these factors are properly accounted for in the RSM, the agreement with the EL turbulence statistics is satisfactory at first order (e.g., PA velocities) but less so at second order (e.g., PA turbulent kinetic energy). Finally, an algebraic stress model for the PA particle-phase pressure tensor and the Reynolds stresses is derived from the RSM using the weak-equilibrium assumption.

A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows

“…However, for the cases with two-way coupling, and especially with nonzero mean-slip velocity as in CIT, the correct prediction of U s is crucial for successful overall predictions. Interestingly, the steady-state model for U s given in (3.16) is relatively simple (compare, for example, the correlation used in (Capecelatro et al 2016b)), with the Lagrangian time scale T * L,1 playing a prominent role. In future work, it would be interesting to test (3.16) for CIT over a wide range of α p and ϕ values to determine whether the parameters in the model for T * L,1 should depend on these quantities.…”

“…In particular, for channel flows the correlated and uncorrelated particle velocity components generate separate spatial fluxes for all statistics. From the model developed in Capecelatro et al (2016b), it is known that, depending on the Stokes number, one or the other of these fluxes may be dominant. As a result, the wall-normal distribution of α p , as well as other statistics, is very sensitive to how the spatial fluxes are modelled.…”

“…Some remarks are in order concerning the fluid-phase Reynolds-stress model. Following Capecelatro et al (2016b), we have chosen the LRR-IP model, which is widely used and give reasonably good results, but any other realisable Reynolds-stress model could be chosen, if needed. The important point is that it has been demonstrated that a realisable Reynolds-stress model corresponds to each Lagrangian stochastic model for the fluid (Pope 1994).…”

“…( 1.15) This term, and several others, like triple correlations and fluxes, are evidently unclosed, which prevents us from solving the above set of RA equations. Recently, a model for the closure of these equations has been proposed (Capecelatro et al 2016b). The resulting picture obtained through the coarse graining is sketched in figure 1.…”

Direct comparison of Eulerian–Eulerian and Eulerian–Lagrangian simulations for particle‐laden vertical channel flow

“…In those studies, the fluid‐phase turbulence is governed primarily though mean‐shear production as what occurs in pure‐fluid flow in a channel at high Reynolds number 22 . In our prior work on particle‐laden channel flow at significant mass loading, 23,24 the channel Reynolds number was too low to observe transitions between CIT and fully developed turbulence. Here, our primary interest is whether the EE‐AG model can reproduce the turbulence statistics found in the EL simulations for the same high‐Reynolds‐number channel flow investigated in Capecelatro et al 15 For such cases, the computational cost is significant due to the grid requirements needed to resolve the turbulent boundary layers at the walls 22 …”