A Functional Relationship for Modeling Laminar to Turbulent Flow Transitions

Abstract

A dimensional analysis which 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, laminar and turbulent. 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 capture the effects of initial conditions (mean flow and turbulence) on the variation of the characteristic variable. The union of these two independent relationships is formed leveraging the concept of flow intermittency to yield a generic functional relationship that incorporates transitional flow effects and fully encompasses solutions spanning the laminar to turbulent flow regimes. Empirical models to several common flows are formed to demonstrate the engineering potential of the proposed functional relationship.