Modeling Interphase Turbulent Kinetic Energy Transfer in Lagrangian-Eulerian Spray Computations

Abstract

Modeling turbulent multiphase flows, such as sprays, is a major challenge owing to droplet (or solid-particle) interactions with a wide range of turbulence length and time scales. In a broad class of Lagrangian-Eulerian models, the instantaneous Lagrangian dispersedphase velocity evolves on a time scale that is proportional to the particle response time 2 ()/(18) p d p f f d. Numerical simulations of a model from this class reveal a nonmonotonic and unphysical increase of the turbulent kinetic energy (TKE) in the dispersed phase k d that is not seen in direct numerical simulations (DNS) of decaying, homogeneous turbulence laden with solid particles. Analysis of this class of models shows that for a linear drag law corresponding to the Stokes regime, the entire class of models will predict an anomalous increase in k d for decaying turbulent flow laden with solid particles or droplets. Even though the particle response time is the appropriate time scale to characterize momentum transfer between sub-Kolmogorov-size dispersed-phase particles and the smallest turbulent eddies (for droplet/particle Reynolds number of < 1), it is incapable of capturing the range of time-and length-scale interactions that are reflected in the evolution of k d. A new model that employs a time scale based on a multiscale analysis is proposed. This model succeeds in capturing the dispersed-phase TKE and fluid-phase TKE evolution observed in DNS. The model also correctly predicts the trends of TKE evolution in both phases for different Stokes numbers.

Two-way coupled stochastic model for dispersion of inertial particles in turbulence

“…(4.13) and (4.14)). In accordance with the EoE concept, and to introduce the capability of capturing the multiscale nature of a turbulent two-phase mixture into CSM, the time scales τ 2 and τ 4 are chosen to be equal to τ π = τ int /C π , where τ int is a multiscale interaction time scale for interphase TKE transfer proposed by Pai & Subramaniam (2006). It was shown in that study that the new time scale accurately captures the dependence of the interphase TKE transfer on St η .…”

“…The constant C π is chosen to be 1.0 in this study. Details on the development of the time scale are given in Pai & Subramaniam (2006) and Xu & Subramaniam (2006). Only important results are reviewed here for the sake of completeness.…”

“…of u can be computed directly from the solution. However, if u is assumed to obey a joint normal distribution with zero mean and covariance σ 2 f δ ij , where σ 2 f = (2/3)k f and δ ij is the Kronecker delta, as is done in recent studies (Pai & Subramaniam 2006;Xu & Subramaniam 2006)…”

“…and whose positions evolve according to 2) are incapable of capturing the observed trends that were noted earlier in the decay of TKE with Stokes number St η = τ p /τ η , when tested in the canonical problem of particle-laden decaying homogeneous turbulence (Pai & Subramaniam 2006). In the above equation, V p is the instantaneous particle velocity, U f is the instantaneous gas-phase velocity (also sometimes referred to as the gas-phase velocity 'seen' by the particles), τ p = (ρ d d 2 p )/(ρ f 18ν f ) is the particle response time scale, X p is the particle position and F add represents additional terms that include lift and body forces.…”

“…It was shown in Pai & Subramaniam (2006) that when τ p is employed as the time scale for interphase TKE transfer, it is incapable of capturing the multiple time scale interactions of the dispersed phase with the carrier-phase turbulence. When τ p is replaced by a multiscale interaction time scale, predicted trends of TKE decay from the LE model match DNS results (Pai & Subramaniam 2006), thereby validating the choice of the multiscale interaction time scale as the appropriate time scale for interphase TKE transfer in such statistical models.…”