A Functional Scaling Relationship for Laminar to Turbulent Flow Regimes

A dimensional analysis that is based on the scaling of the two-dimensional Navier-Stokes equations is presented for correlating bulk flow characteristics arising from a variety of initial conditions. The analysis yields a functional relationship between the characteristic variable of the flow region and the Reynolds number for each of the two independent flow regimes. A linear relationship is realized for the laminar regime, while a nonlinear relationship is realized for the turbulent regime. Both relationships incorporate mass-flow profile characteristics to fully capture the effects of initial conditions on the variation of the characteristic variables. The union of these two independent relationships is formed utilizing the concept of flow intermittency to further expand into a generic scaling relationship that incorporates transitional flow effects to fully encompass solutions spanning the laminar to turbulent flow regimes. The results of the analysis are discussed within the context of several flow phenomena (e.g. pipe flow, jet flow & separated flow) resulting from various initial and boundary conditions.