Correction scheme for point-particle models applied to a nonlinear drag law in simulations of particle-fluid interaction

Drag laws for particles in fluids are often expressed in terms of the undisturbed fluid velocity, defined as the fluid velocity a particle sees before its disturbance develops in the fluid. In two-way coupled point-particle simulations the information from the undisturbed state is not available and must be approximated using the disturbed velocity field. ([Horwitz, J. A. K. & Mani, A. 2016 Accurate calculation of stokes drag for point-particle tracking in two-way coupled flows. Journal of Computational Physics318, 85-109]) recently developed a procedure to estimate the undisturbed velocity for particles moving at low Reynolds number and obeying the linear Stokes drag law. Using this correction, convergence of numerical simulations was demonstrated to match the expected physical behavior for a range of canonical settings. In this paper we examine this correction scheme for particles moving at finite Reynolds number, by considering the nonlinear Schiller-Naumann drag law. Our results indicate that a linear correction can significantly improve prediction of drag force up to particle Reynolds number of about 10. Additionally, we present an investigation of the impact of this correction when applied to simulations of forced homogeneous turbulent particle-laden flows to further demonstrate the importance of modelling the undisturbed fluid velocity. While the shapes of particle acceleration and Reynolds number pdfs are not sensitive to correcting for the undisturbed fluid velocity, we show that an uncorrected scheme can result in significant under-prediction of the mean particle Reynolds number and standard deviation of particle acceleration. Furthermore, examination of particle radial distribution function reveals modest enhancements in preferential concentration predicted by the correction scheme (for St > 1). Our investigations of turbulent flows indicate the undisturbed fluid velocity correction to be more important for particles with larger size, while the Stokes number dependence is more complicated. The correction procedure shows a greater difference in particle slip velocity at higher Stokes numbers but more enhancement in preferential concentration at lower Stokes numbers. Finally, we propose a regime diagram to guide scheme selection for point-particle modelling.