Composite expansions for active and inactive motions in the streamwise Reynolds stress of turbulent boundary layers

The proper scaling and prediction of the streamwise Reynolds stress in turbulent boundary layers has been a controversial issue for more than a decade as its Reynolds Number dependence can not be removed by normal scaling. One issue that may explain the unusual behavior of the streamwise Reynolds stress is that it is affected by both active and inactive motions per the Townsend hypothesis. The goal of this research is to develop a composite expansion for the streamwise Reynolds stress in turbulent boundary layers that considers active and inactive motions, explains various Reynolds Number dependencies, and agrees with available data. Data for the Reynolds shear stress and the streamwise Reynolds stress from six sources are evaluated and as appropriate plotted on inner and outer scales. A new asymptotic representation for the Reynolds shear stress, +, that meets the requirements for a proper composite expansion is developed and applied. This new Reynolds shear stress composite expansion agrees with data and allows predictions of + for any Reynolds Number. The streamwise Reynolds stress, +, can be separated into active and inactive parts and the Reynolds shear stress can be used to represent the active part. The inactive streamwise Reynolds stress, #, is separated from the complete + in part of this work. An outer correlation equation with the correct asymptotic limits for the inactive streamwise Reynolds stress is developed and shown to fit the outer part of the # data. A separate inner correlation equation for inner inactive streamwise Reynolds stress is developed and fit to data. Together these two equations form a composite expansion for the inactive streamwise Reynolds stress for flat plate boundary layers. This composite expansion for the inactive streamwise Reynolds stress can be combined with the Reynolds shear stress expansion to produce predictions for + that agree with data. Thus a composite expansion for predicting the streamwise Reynolds stress in turbulent boundary layers is developed and shown to reproduce the correct trends, to agree with the available data, and to explain the Reynolds Number dependence of the streamwise Reynolds stress.