Asymptotic closure model for inertial particle transport in turbulent boundary layers

Transport equations for heavy inertial particles in turbulent boundary layers may be derived from an underlying phase-space probability density function (PDF) equation. These equations, however, are unclosed, and the standard closure approach is to use a quasi-Normal approximation (QNA) in which the fourth moments are approximated as behaving as if the velocities were Normally distributed. Except for particles with weak inertia, the QNA leads to large quantitative errors, and is not consistent with the known asymptotic predictions of Sikovsky (Flow Turbulence Combust, vol. 92, 2014, pp. 41-64) for the moments of the PDF in the viscous sublayer. We derive a new closure approximation based on an asymptotic solution to the transport equations in regions where the effect of particle inertia is significant. The new closure is consistent with the asymptotic predictions of Sikovsky, but applies even outside the viscous sublayer. Comparisons with direct numerical simulations (DNS) show that the new closure gives similar results to the QNA (with the QNA results in slightly better agreement with the DNS) when the viscous Stokes number is $St<10$, but for $St>10$ the new model is in far better agreement with the DNS than the QNA. While the predictions from the new model leave room for improvement, the results suggest that this new closure strategy is a very effective alternative to the traditional QNA approach, and the closure could be refined in future work.