An Asymptotic Description of the Attached, Turbulent, Oscillatory Boundary Layer

Abstract

The attached, temporally-oscillating turbulent boundary layer is investigated by use of asymptotic matching techniques, valid for the limit of large Reynolds numbers. Much of the analysis is applicable to generally accepted turbulence models (which satisfy a few basic assumptions as detailed in the paper), and this is then applied in particular to two well established turbulence models, namely the k - ε transport model and the Baldwin-Lomax mixing-length model. As in the laminar case, the steady-streaming Reynolds number is found to be an important parameter, although in the turbulent case this is important at leading (rather than second) order. In particular, the time dependence of the wall shear (and the displacement thickness) is found to leading order to be independent of the specific closure model, but just differs by a multiplicative constant dependent on the particular model. Results are also compared with previous computational and experimental data; the agreement is encouraging. In addition to describing the oscillatory flow above a flat wall, these leading order results are applicable to flow past general bodies, provided the amplitude of oscillation is small compared to the surface radius of curvature. In the case of the Baldwin-Lomax model, the nature of the higher-order terms, including the steady streaming caused by the interaction of curvature and inertia effects is also investigated. This analysis suggests some limitations on the applicability of the model to the finer details of the flow, due to the occurrence of discontinuities (and singularities) in the higher-order asymptotic solution, particularly when inertia effects are taken into account.