Error analysis of asymptotic solution of a heavy particle motion equation in fluid flows

Abstract

For weakly inertial particles subjected to volumetric forces and Stokes drag force in fluid flows, we can solve the simplified particle motion equation using the perturbation method. This method allows us to obtain a recursive formula for the nth-order correction of the asymptotic solution of particle velocity. We verified the error of the asymptotic solution under two typical flow fields: a time-varying uniform flow field with a volumetric force field and a two-dimensional non-uniform cellular flow field. In the former, the relative error of the asymptotic solution of particle velocity and position increases with the Stokes number, and we provided a quantitative analysis of the results. In the latter, we verify and analyze the asymptotic solution from two perspectives: the behavior of a single particle and the collective behaviors of many particles. For asymptotic solutions with maximum velocity and position errors of less than 5%, we select the solution with the lowest order correction and designate it as the optimal asymptotic solution. The order of the optimal asymptotic solution increases with increasing Stokes numbers and motion durations. However, in most cases, for weakly inertial particles [St ∼ O(10−3)], and the time t* ∼ O(10), the first-order asymptotic solution can achieve accuracy, where both St and t* are defined using the flow field characteristic time, Tf = 4π s. The results validate the rationale behind utilizing first-order asymptotic solutions in the fast Eulerian method for turbulent dispersion of weakly inertial particles.