Extending the logarithmic velocity profile in turbulent channel flow

The logarithmic velocity profile is a key feature of wall-bounded turbulence, typically observed in a narrow near-wall region. In this study, we revisit the mixing length (lm) foundation of the logarithmic law in turbulent channel flow and propose a method to extend the logarithmic velocity profile. The derivation follows a bottom-up approach without relying on the uniform shear stress assumption or Prandtl's linear mixing length model. We demonstrate that the logarithmic velocity profile emerges where Prandtl's parabolic model for lm aligns with its true value. Extending this overlap region for lm correspondingly broadens the logarithmic profile. Based on this principle, we propose an enhanced mixing length model with a wider overlap region compared to the existing models. When applied to conventional velocity transformations—such as the IC-type for incompressible flow, and the Van Driest (VD-type) and Trettel and Larsson (TL-type) transformations for compressible flow—enhanced versions of these transformations are obtained accordingly. The enhanced model and corresponding transformations are evaluated using a series of direct numerical simulation data, covering a wide range of Mach and Reynolds numbers. In both incompressible and compressible turbulent channel flows, the predicted mixing length closely follows the theoretical distribution throughout the outer layer. The extended logarithmic profile reaches the channel center, with the corresponding diagnostic function deviating by no more than ±7% from the reference value. Compared to conventional logarithmic and defect laws, the extended logarithmic profile provides a more consistent description of velocity distribution across the outer layer.