Fully developed turbulent flow in a pipe - An intermediate layer

The crucial limiting processes for a turbulent flow are found from the analysis of incomplete equations in the ordered space at large Reynolds numbers, and the matching procedure is clarified by showing that for incomplete equations, the classical matching principle is not sufficient and that the situation is rescued by Millikan's (1938) argument. Various limiting processes are investigated to obtain a crucial intermediate limit whose transverse dimension is of the order of the geometric mean of the transverse dimensions of the classical inner and outer layers. The momentum equations and the boundary conditions in the outer, intermediate, and inner layers are analyzed and matched in their overlap domains where velocity distribution is logarithmic but slopes can be different. The measurements show that substantial log regions do exist in the two overlap domains and that the ratio of their slopes is 2.03. The predictions for Reynolds stress and turbulent energy production are in excellent agreement with the data for almost the entire turbulent flow region.