Generalized logarithmic law and its consequences

There has been considerable controversy during the past few years concerning the validity of the universal logarithmic law that describes the mean velocity profile in the overlap region of a turbulent wall-bounded flow. Alternative Reynolds-number-dependent power laws have been advanced. We propose herein an extension of the classical two-layer approach to higher-order terms involving the Karman number and the dimensionless wall-normal coordinate. The inner and outer regions of the boundary layer are described using Poincare expansions, and asymptotic matching is applied in the overlap zone. Because of the specific sequence of gauge functions chosen, the resulting profile depends explicitly on powers of the reciprocal of the Karman number. The generalized law does not exhibit a pure logarithmic region for large but finite Reynolds numbers. On the other hand, the limiting function of all individual Reynolds-number-dependent profiles described by the generalized law shows a logarithmic behavior. As compared to either the simple log or power law, the proposed generalized law provides a superior fit to existing high-fidelity data.