Stochastic Drag
Eulerian-Lagrangian (EL) methods have gained substantial traction for modeling such strongly-coupled particle-laden flows due to a balance between speed and resolution. Existing drag force closures developed for EL methods typically capture the mean fluid-particle force experienced by an assembly of particles. Therefore, the variance in drag force, arising from neighbor-induced, sub-grid fluid velocity fluctuations (referred to as pseudo-turbulent kinetic energy; PTKE), is generally ignored. Neglecting higher-order drag statistics, resulting from neighbor effects, can detrimentally impact EL predictions for higher-order particle statistics (velocity variance and dispersion).
To arrive at efficient and accurate models for PTKE-induced drag force disturbances, I take a statistical approach. More specifically, the fluctuating drag statistics are modeled with an Ornstein-Uhlenbeck (OU) process:
Reconstructing the total drag force distribution, which is lost during coarse-graining in EL, dramatically improves the evolution of particle velocity variance when compared to fully-resolved simulations of homogeneous fluidization:
Due to the statistical nature of stochastic methods, sources and sinks to granular temperature, arising from neighbor-induced drag disturbances, may be derived for Euler-Euler (EE) frameworks. Since EE frameworks do not resolve discrete particles, the distribution of drag forces in acceleration-velocity phase space must be derived; with sources and sinks being quadrant-conditioned covariance integrals of said distribution:
However, heterogeneous particle distributions introduce large gradients in the quasi-steady drag force that may be significantly larger than the neighbor effect:
- Lattanzi, A., Tavanashad, V., Subramaniam, S., Capecelatro, J., 2020. Stochastic models for capturing dispersion in particle-laden flows. Journal of Fluid Mechanics 903.
- Lattanzi, A., Tavanashad, V., Subramaniam, S., Capecelatro, J., 2021a. A stochastic model for the hydrodynamic force in Euler-Lagrange simulations of particle-laden flows. arXiv preprint arXiv:2103.10581.
- Lattanzi, A., Tavanashad, V., Subramaniam, S., Capecelatro, J., 2021b. Fluid-mediated sources of granular temperature at finite Reynolds numbers. arXiv preprint arXiv:2108.01777.