  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.
