Second Log-Wake Law from Pipe Symmetry and its Applications in Symmetric and Antisymmetric Channel Flows

The velocity distribution of turbulent pipe flow is often described by the Coles log-wake law that is a superposition of the law of the wall due to the wall shear stress and the law of the wake due to the free turbulence at the centerline. Yet, the log-wake law does not meet the pipe symmetry principle because it includes only the effects of the wall-induced turbulence at the invert (the log law) and the free turbulence at the pipe centerline (the wake law), it misses the effect of the wall-induced turbulence at the obvert. This research then adds an additional log term due to the pipe obvert to the Coles log-wake law and innovates a second log-wake law. The slight modification of the Coles log-wake law results in a significant change in the eddy viscosity law that leads to an accurate antisymmetric velocity distribution law (including a sine integral wake law) for turbulent Couette channel flow. These laws agree with data from pipes, symmetric channel flow, and antisymmetric channel flow. Furthermore, this research can lead to a new path to solving open channel flow, which is discussed in a separate paper.