Laminar-to-Turbulence Transition Revealed Through a Reynolds Number Equivalence

Abstract

Flow transition from laminar to turbulent mode (and vice versa)—that is, the initiation of turbulence—is one of the most important research subjects in the history of engineering. Even for pipe flow, predicting the onset of turbulence requires sophisticated instrumentation and/or direct numerical simulation, based on observing the instantaneous flow structure formation and evolution. In this work, a local Reynolds number equivalence γ (ratio of local inertia effect to viscous effect) is seen to conform to the Universal Law of the Wall, where γ = 1 represents a quantitative balance between the abovementioned two effects. This coincides with the wall layer thickness (y+ = 1, where y+ is the dimensionless distance from the wall surface defined in the Universal Law of the Wall). It is found that the characteristic of how the local derivative of γ against the local velocity changes with increasing velocity determines the onset of turbulence. For pipe flow, γ ≈ 25, and for plate flow, γ ≈ 151.5. These findings suggest that a certain combination of γ and velocity (nonlinearity) can qualify the source of turbulence (i.e., generate turbulent energy). Similarly, a re-evaluation of the previous findings reveals that only the geometrically narrow domain can act locally as the source of turbulence, with the rest of the flow field largely being left for transporting and dissipating. This understanding will have an impact on the future large-scale modeling of turbulence. Keywords: Local Reynolds number equivalence, Flow transition from laminar to turbulent mode, Universal Law of the Wall, Pipe flow, Plate flow, Modeling