Streamwise Reynolds Stress Partitioned into Active and Inactive Correlations.

The streamwise Reynolds stress \textit{$<$uu$>$ } in wall layers has a peak near the wall that is an order of magnitude higher that the shear stress <$u$v>. Townsend attributed this to an ``inactive'' swirling motion $u-w$ that does not contribute to the shear stress. The peak continues to increase with Reynolds number. Degraaff and Eaton ( JFM, \textbf{422, }p 319 ) and Metzger and Klewicki ( P of F, \textbf{13}, p 692 ) have essentially shown that the peak increases as \textit{$<$uu$>$}$_{MAX}/ u_{\ast }^{2}$\textit{ $\sim $ U}$_{0}/u_{\ast . }$ With these facts in mind, it is proposed that the inactive motion $u_{I}$ scales with ($U_{0}u_{\ast })^{1/2}$ , and the active motion $u_{A}$ scales with $u_{\ast }$. With these ideas, an asymptotic expansion for the streamwise stress consists of three terms with gauge functions 1, $(u_{\ast }/U_{0}$\textit{(Re*))}$^{1/2},$ and$ u_{\ast }/U_{0}$\textit{(Re*)}: \textit{$<$uu$>$/(u}$_{\ast }U_{0}$\textit{) $\sim $ f}$_{0}(y)+ f_{1}(y)(u_{\ast }/U_{0})^{1/2}+ f_{2}(y)(u_{\ast }/U_{0}). ^{ }$ The terms,$ f_{0}$\textit{(y) {\&} f}$_{2}(y),$represent the autocorrelations of the inactive and active motions respectively, while the$ f_{2}(y)$ term represents the cross-correlation of those motions. This form appears to be valid for both inner and outer regions. Data for channel flow was analyzed by making crude approximations for $ f_{1}$\textit{(y) {\&} f}$_{2}(y) $and solving for $f_{0}(y). $Equations fitted to $f_{0}(y)$ allow a composite expansion to predict the streamwise Reynolds stress as a function of $y$ and \textit{Re}$_{\ast }$.