An Extended Rouse Model of Inertial Particles Settling in Turbulent Boundary Layers

Abstract

The settling of inertial particles in turbulence boundary layers plays an essential role in many meteorological, industrial and environmental processes, and is governed by multifarious mechanisms. First, turbulence alters the settling velocity of inertial particles through different effects, like preferential sweeping mechanism, loitering effect and vortex trapping. Second, the existence of a wall introduces extra effects that can influence particle settling, such as turbophoresis. The Rouse model was the most famous model in predicting particle settling in vertical wall-bounded settling. Nevertheless, it is only valid for inertia-less particles in the logarithmic region. A theory by Bragg et al., based on phase-space probability density theory, incorporates particle inertia into the Rouse model, and quantifies the contributions from the aforementioned mechanisms to the particle vertical velocity. The theory is valid for all particle Stokes numbers, yet it still lacks a closed form.In this work, one way to close the equations presented by Bragg et al. (the extended Rouse model) was examined. Using a central differencing scheme combined with an iterative method, the nonlinear second-order differential equation of the variance of vertical particle velocity was solved. The predictions of the variance of vertical particle velocity S and the particle concentration PDF ρ by the model were studied and compared to DNS. The comparison indicates that the extended Rouse model is able to predict many features of S and ρ, like the accumulation of particles close to the wall and turbophoretic drift. However, the quantitative agreement between the predictions by the model and DNS is poor. There are two probable reasons for the discrepancies between the predictions and DNS. First, the closure of the term in the equation may be a source of errors. Second, the lower boundary condition, whose validity is suspicious for particles with weak inertia, can be a reason for the discrepancies. In order to investigate the cause for the disagreement, three different boundary conditions (zero-gradient condition, asymptotic matching, iterative condition) were examined. The results indicate that the boundary conditions have a very limited influence on the predictions. As a result, the closure of the terms is more likely to be responsible for the discrepancies.