Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions

Abstract

The acceleration of an inertial particle in a gas-solid flow arises from the particle's interaction with the gas and from interparticle interactions such as collisions. Analytical treatments to derive a particle acceleration model are difficult outside the Stokes flow regime, but for moderate Reynolds numbers (based on the mean slip velocity between gas and particles) particle-resolved direct numerical simulation (PR-DNS) is a viable tool for model development. In this study, PR-DNS of freely-evolving gas-solid suspensions are performed using the particle-resolved uncontaminated-fluid reconcilable immersed-boundary method (PUReIBM) that has been extensively validated in previous studies. Analysis of the particle velocity variance (granular temperature) equation in statistically homogeneous gas-solid flow shows that a straightforward extension of a class of mean particle acceleration models (drag laws) to their corresponding instantaneous versions, by replacing the mean particle velocity with the instantaneous particle velocity, predicts a granular temperature that decays to zero, which is at variance with the steady particle granular temperature that is obtained from PR-DNS. Fluctuations in particle velocity and particle acceleration (and their correlation) are important because the particle acceleration-velocity covariance governs the evolution of the particle velocity variance (characterized by the particle granular temperature), which plays an important role in the prediction of the core annular structure in riser flows. The acceleration-velocity covariance arising from hydrodynamic forces can be decomposed into source and dissipation terms that appear in the granular temperature evolution equation, and these have already been quantified in the Stokes flow regime using a combination of kinetic theory closure and multipole expansion simulations. From PR-DNS data we show that the fluctuations in the particle acceleration that are aligned with fluctuations in the particle velocity give rise to a source term in the granular temperature evolution equation. This approach is used to quantify the hydrodynamic source and dissipation terms of granular temperature from PR-DNS results for freely-evolving gas-solid suspensions that are performed over a wide range of solid volume fraction (0.1 φ 0.4), Reynolds number based on the slip velocity between the solid and the fluid phase (10 Re m 100) and solid-to-fluid density ratio (100 ρ p /ρ f 2000). The straightforward extension of drag law models does not give rise to any source in the granular temperature due to hydrodynamic effects. This motivates the development of † Email address for correspondence: shankar@iastate.edu ‡ Present address: CD-adapco, Lebanon, NH 03766, USA. 696S. Tenneti, M. Mehrabadi and S. Subramaniam better Lagrangian particle acceleration models that can be used in Lagrangian-Eulerian formulations of gas-solid flow. It is found that a Langevin equation for the increment in the particle velocity reproduces PR-DNS results for the stationary particle velo...

Development of a gas–solid drag law for clustered particles using particle-resolved direct numerical simulation

“…This family of curves asymptotes to a Heaviside function in the Reynolds number up to 100. For this case, the particle granular temperature is approximately in the range of 0.064m 2 /s 2 to 2.728m 2 /s 2 (Tenneti et al, 2016). If we assume that the Hamaker constant is A = 10 −19 J, the solidphase density ratio is 400, and the minimum particle separation is d 0 = 2 Angstrom, then the minimum and maximum values of Ha for the particles suspended in an air flow would be Ha min = 3×10 −5 and Ha max = 1.2×10 −3 , respectively.…”

“…Now if we assume that in Eq. 4 the maximum overlap δ is 1% of the particle diameter d p and also assume that the granular temperature in gas-solid flow scales as T / | W | 2 ∼ 10 −2 (Tenneti et al, 2016), it can be concluded from Eq. 4 that the contact to fluid timescales ratio is…”

Importance of the fluid-particle drag model in predicting segregation in bidisperse gas-solid flow

“…This method is shown to be accurate and numerically convergent (Garg et al, 2010;Tenneti et al, 2011). In addition, PUReIBM has been successfully used to simulate fixed particle assemblies (Tenneti et al, 2010(Tenneti et al, , 2011Sun et al, 2015) and freely evolving suspensions of monodisperse gas-solid flows (Subramaniam et al, 2014;Mehrabadi et al, 2015;Tenneti et al, 2016).…”

Analysis of gas-solid flow using particle-resolved direct numerical simulation: flow physics and modeling

“…This method is shown to be accurate and numerically convergent (Garg et al, 2010c;Tenneti et al, 2011). In addition, PUReIBM has been successfully used to simulate fixed particle assemblies (Tenneti et al, 2010(Tenneti et al, , 2011Sun et al, 2015) and freely evolving suspensions of monodisperse gas-solid flows (Subramaniam et al, 2014;Mehrabadi et al, 2015;Tenneti et al, 2016). The extension of the PUReIBM formulation to account for polydisperse gas-solid suspensions is straightforward (see C).…”

“…The difference between the net mean velocity of the two particle classes contributes to the relative particle mass flux that is the signature of particle segregation. On the other hand, the interaction of particles with flow structures gives rise to generation of particle velocity fluctuations (Tenneti et al, 2016) that are characterized by a non-zero level of kinetic energy. If the particle suspension is dense, then the probability of particle collisions increases with increase of particle velocity fluctuations.…”