The statistics of velocity and temperature fluctuations in axisymmetric laminar-to-turbulent transitions

The procedures for studying continuous phase transitions of thermodynamic equilibrium systems are extended to discuss the laminar-to-turbulent transitional flows in circular tubes. The flow in the transition range is treated as a composition of the laminar and turbulent flows assuming that the radial fluctuating velocity has the same value as that of the fully turbulent flow. The composite ratio of the turbulent flow is used as an order parameter to define the composite flow. The fluctuations of the composite ratios are introduced, and the criterion of minimum entropy production is used to derive an equation which can describe the transition behaviors. The convective heat transfer characteristics in the transition range in a heated circular tube are also discussed adopting the same procedures. Similar and separate processes for the transitions of the flow and convective heat transfer types are allowed in the heated circular tube. The macroscopic fluctuations in the transition range show both probabilistic and deterministic characteristics simultaneously. The agreements with measurements are given including those obtained in flow and heat transfer experiments.