Transition of oscillatory flow in tubes: an empirical model for application to Stirling engines

An empirical model for transition to turbulence in oscillatory flows in straight tubes is proposed. The model, fashioned after a correlation for transition of a boundary layer on a flat plate, yields the laminar flow momentum thickness Reynolds number that must be met before transition to turbulence will occur. The transition point is located by comparing this to the actual momentum thickness Reynolds number. Since in one-dimensional computation, as is typically employed in engine simulation codes, the momentum thickness Reynolds number cannot be computed, a scheme is proposed for estimating it in terms of the position within the cycle, the maximum value of the diameter Reynolds number within the cycle, Re_max, and the dimensionless frequency, Valensi number, Va. Another parameter required in the calculation of the point of transition is the turbulence intensity value within the core flow and external to the boundary layer. A means of evaluating this quantity, given the distance from the entrance end of the tube, the angular position within the cycle, and the two cycle parameters, Re_max and Va, is proposed. Results from an experimental study of oscillatory flow in a tube are employed to develop the model. When the flow is determined to be turbulent, it is proposed that a fully-developed, steady flow friction coefficient be applied. When the flow is laminar, the assumption of fully-developed flow cannot be made; thus, a method is suggested for estimating the friction factor. This method is parameterized on Re_max and Va.