Correlation of Fluctuating Vorticity in Turbulent Wall Layers

It is commonly known that Reynolds shear stress $$ scales with the friction velocity $u_{\ast }^{2}$. On the other hand, Degraaff and Eaton ( JFM, \textbf{422, }p 319 ) and Metzger and Klewicki ( P of F, \textbf{13}, p 6 92 ) have shown that the streamwise Reynolds stress \textit{$<$uu$>$} scales more nearly as $U u_{\ast }$. Townsend proposed that motions were ``active'' if they contributed to the Reynolds shear stress and ``inactive'' otherwise. Here, Townsend's definition is modified to say that motions are ``active'' if they scale with $u_{\ast }$; the same scaling as the Reynolds stress. A fluctuation that does not scale with $u_{\ast }$ is ``inactive.'' Vorticity profiles from the DNS (described in the various papers of Del Alamo, Jimenez, Zandonade, Moser, and Hoyas (P of F \textbf{15}, L-41; JFM, \textbf{500},p135, P of F, \textbf{18}, 011702) ) are reviewed. It is found that, in the limit of high Reynolds number, the outer region is free of vorticity. In the inner region the vortcity $<${\_}$_{y}${\_}$_{y}>$ is active with no inactive component. The other components, $<${\_}$_{x}${\_}$_{x}>$ and$<${\_}$_{z}${\_}$_{z}>$, have active components that scale as $<${\_}{\_}{\_}${\rm g}{\rm v}{\rm g}{\rm o} u_{\ast }^{4}$\textit{/{\_}}$^{{\rm y}}) $and inactive components that scale as $<${\_}{\_}{\_}${\rm g}{\rm v}{\rm g}{\rm o}{\rm o} u_{\ast }$\textit{/{\_}${\rm p}$}$^{{\rm y}} u_{\ast }U. $Since at the wall the vorticity and shear stress are proportional, the wall stress fluctuations are found to be: $<${\_}$_{x{\rm w}}{\rm g}${\_}$_{x{\rm w}}>$ / ${\rm o}$\textit{{\_}}$^{{\rm y}} u_{\ast }^{3}U)=0.007$and $<${\_}$_{z{\rm w}}{\rm g}${\_}$_{z{\rm w}}>$ / ${\rm o}$\textit{{\_}}$^{{\rm y}} u_{\ast }^{3}U)=$ 0.0038.