The objective of this study is to improve Eulerian-Eulerian models of particle-laden turbulent flow. We begin by understanding the behavior of two existing models-one proposed by Simonin [von Kármán Institute of Fluid Dynamics Lecture Series, 1996], and the other by Ahmadi [Int. J. Multiphase Flow16, 323 (1990)]-in the limiting case of statistically homogeneous particle-laden turbulent flow. The decay of particle-phase and fluid-phase turbulent kinetic energy (TKE) is compared with direct numerical simulation results. Even this simple flow poses a significant challenge to current models, which have difficulty reproducing important physical phenomena such as the variation of turbulent kinetic energy decay with increasing particle Stokes number. The model for the interphase TKE transfer time scale is identified as one source of this difficulty. A new model for the interphase transfer time scale is proposed that accounts for the interaction of particles with a range of fluid turbulence scales. A new multiphase turbulence model-the equilibration of energy model (EEM)-is proposed, which incorporates this multiscale interphase transfer time scale. The model for Reynolds stress in both fluid and particle phases is derived in this work. The new EEM model is validated in decaying homogeneous particle-laden turbulence, and in particle-laden homogeneous shear flow. The particle and fluid TKE evolution predicted by the EEM model correctly reproduce the trends with important nondimensional parameters, such as particle Stokes number. The objective of this study is to improve Eulerian-Eulerian models of particle-laden turbulent flow. We begin by understanding the behavior of two existing models-one proposed by Simonin ͓von Kármán Institute of Fluid Dynamics Lecture Series, 1996͔, and the other by Ahmadi ͓Int. J. Multiphase Flow 16, 323 ͑1990͔͒-in the limiting case of statistically homogeneous particle-laden turbulent flow. The decay of particle-phase and fluid-phase turbulent kinetic energy ͑TKE͒ is compared with direct numerical simulation results. Even this simple flow poses a significant challenge to current models, which have difficulty reproducing important physical phenomena such as the variation of turbulent kinetic energy decay with increasing particle Stokes number. The model for the interphase TKE transfer time scale is identified as one source of this difficulty. A new model for the interphase transfer time scale is proposed that accounts for the interaction of particles with a range of fluid turbulence scales. A new multiphase turbulence model-the equilibration of energy model ͑EEM͒-is proposed, which incorporates this multiscale interphase transfer time scale. The model for Reynolds stress in both fluid and particle phases is derived in this work. The new EEM model is validated in decaying homogeneous particle-laden turbulence, and in particle-laden homogeneous shear flow. The particle and fluid TKE evolution predicted by the EEM model correctly reproduce the trends with important nondimensional parameter...
Experiments indicate that particle clusters that form in fluidized-bed risers can enhance gas-phase velocity fluctuations. Direct numerical simulations (DNS) of turbulent flow past uniform and clustered configurations of fixed particle assemblies at the same solid volume fraction are performed to gain insight into particle clustering effects on gas-phase turbulence, and to guide model development.The DNS approach is based on a discrete-time, direct-forcing immersed boundary method (IBM) that imposes no-slip and no-penetration boundary conditions on each particle's surface. Results are reported for mean flow Reynolds number Re p = 50 and the ratio of the particle diameter d p to Kolmogorov scale is 5.5. The DNS confirm experimental observations that the clustered configurations enhance the level of fluid-phase turbulent kinetic energy (TKE) more than the uniform configurations, and this increase is found to arise from a lower dissipation rate in the clustered particle configuration. The simulations also reveal that the particle-fluid interaction results in significantly anisotropic fluid-phase turbulence, the source of which is traced to the anisotropic nature of the interphase TKE transfer and dissipation tensors. This study indicates that when particles are larger than the Kolmogorov scale (d p > η), modeling the fluid-phase TKE alone may not be adequate to capture the underlying physics in multiphase turbulence because the Reynolds stress is anisotropic. It also shows that multiphase turbulence models should consider the effect of particle clustering in the dissipation model.
The interphase transfer of turbulent kinetic energy (TKE) is an important term that affects the evolution of TKE in fluid and particle phases in particle-laden turbulent flow. This work shows that the interphase TKE transfer terms must obey a mathematical constraint, which in the limiting case of statistically homogeneous flow with zero mean velocity in both phases, requires these terms be equal and opposite. In the single-point statistical approach called the two-fluid theory, the interphase TKE transfer terms are unclosed and need to be modeled. Multiphase turbulencemodels that satisfy this constraint of conservative interphase TKE transfer admit a term-by-term comparison with true direct numerical simulations (DNS) that enforce the exact velocityboundary condition on each particle's surface. Analysis of three models reveals that not all models satisfy the requirement of conservative interphase TKE transfer. DNS that invoke the point-particle assumption also do not obey this principle of conservative interphase TKE transfer, and this precludes the comparison of model predictions of TKE budgets in each phase with point-particle DNS. This study motivates the development of multiphase turbulencemodels based on the insights revealed by this analysis, leading to a meaningful comparison of TKE budgets with true DNS. The interphase transfer of turbulent kinetic energy ͑TKE͒ is an important term that affects the evolution of TKE in fluid and particle phases in particle-laden turbulent flow. This work shows that the interphase TKE transfer terms must obey a mathematical constraint, which in the limiting case of statistically homogeneous flow with zero mean velocity in both phases, requires these terms be equal and opposite. In the single-point statistical approach called the two-fluid theory, the interphase TKE transfer terms are unclosed and need to be modeled. Multiphase turbulence models that satisfy this constraint of conservative interphase TKE transfer admit a term-by-term comparison with true direct numerical simulations ͑DNS͒ that enforce the exact velocity boundary condition on each particle's surface. Analysis of three models reveals that not all models satisfy the requirement of conservative interphase TKE transfer. DNS that invoke the point-particle assumption also do not obey this principle of conservative interphase TKE transfer, and this precludes the comparison of model predictions of TKE budgets in each phase with point-particle DNS. This study motivates the development of multiphase turbulence models based on the insights revealed by this analysis, leading to a meaningful comparison of TKE budgets with true DNS.
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