Analytical Criterion for the Turbulent-to-Laminar Transition in Channel Flow

Abstract

An analytical criterion for the disappearance of turbulent flow in a channel is developed within the Alexeev hydrodynamic equations (AHE). The analysis builds on an analytical representation of the streamwise velocity as a superposition of a parabolic component and a nonlinear turbulent component coupled to the transverse velocity field. The critical Reynolds number is obtained from the balance of viscous dissipation of kinetic energy associated with the turbulent component. The resulting critical Reynolds number is expressed analytically as \(Re_c=f(δ,γ,n)\), where \(δ\) is the Reynolds-independent AHE similarity parameter, \(γ\) is the weighting coefficient of the turbulent component, and \(n\) is the transverse-velocity mode number. The criterion describes the disappearance of the turbulent solution as the Reynolds number is decreased and therefore addresses the turbulent-to-laminar transition rather than the linear instability of the laminar solution. Unlike a criterion based on a single universal critical Reynolds number, the AHE criterion predicts a transition threshold that depends on the parameters characterizing the flow. For representative channel-flow parameters, the predicted critical Reynolds number is approximately \(Re_c=1000\text{--}1200\). The predicted threshold is compared with experimental observations of transition and of the persistence of laminar flow in channel flows. The analysis suggests that viscous dissipation provides a mechanism limiting the persistence of the turbulent solution below a critical Reynolds number.%