Active and Inactive Motions in the Streamwise Reynolds Stress of Boundary Layers.

The streamwise normal Reynolds stress \textit{$<$uu$>$(y, Re) } in wall layers does not scale simply with the friction velocity $u_{\ast}$ as many researchers have shown recently. For example, Degraaff and Eaton (JFM, \textbf{422, }p 319) and Metzger and Klewicki (P of F, \textbf{13}, p 692) propose the scaling $/(U_{0}/u_{\ast})$. In this talk we adopt Townsend's idea that the streamwise fluctuations consist of active motions, which produce the Reynolds shear stress \textit{$<$uv$>$} and inactive motions, which do not contribute to the Reynolds shear stress \textit{$<$uv$>$}. It has been proposed, Panton ( Phil. Trans. R. Soc. A (2007) \textbf{365}, p733) that these motions scale differently. It is assumed that the inactive correlation scales as $$/ ( $U_{0}/u_{\ast.})$ while all other parts of \textit{$<$uu$>$} , which in concept are related to the Reynolds shear stress \textit{$<$uv$>$}, scale with $u_{\ast }^{2}. $ Data for zero-pressure-gradient boundary layer flows is analyzed from this viewpoint. In order to extract the inactive correlation $$ it is assumed that the other motions are some constant fraction of the Reynolds shear stress $<$uv$>$. Fitted equations to the inner and outer regions allow a composite expansion to predict the streamwise Reynolds stress as a function of $y$ and \textit{Re}$_{\ast }$.