Wall friction measurement in the absence of mean shear

The dimensionless frictional force $f$ between a pipe wall and a flowing turbulent fluid is $f=\nu \overline {s} /U^2$, where $U$ is mean flow speed in the $x$-direction, $\nu$ is kinematic viscosity, and $\overline s=\frac{\partial u }{\partial y }$, where the $y$ axis is perpendicular to the flow direction. The derivative is evaluated at the wall, $y$ = 0. Described here a scheme for measuring $f$ in a turbulent fluid where $\overline{s}$ is close to zero. Hence the source of frictional dissipation is from $fluctuations$ in the shear about its mean, namely $\overline {s^2}$. This type of shear is encountered in turbulence in a closed container such as a food mixer. The scheme, which involves photon correlation spectroscopy, averages the shear rate over a laser spot size $w$ $\simeq$ 100 $\mu$m or smaller. The scheme yields the probability density function (PDF) of components of the shear rate tensor and the moments of of the PDF. The theory will be described briefly and measurements will be presented where $\overline{s} \simeq$ 0. In that limit $f$ is redefined to be $f =u' \overline{s_{ij}}/\nu$, where $\overline{s_{ij}}$ is the dominant component being measured, and $u'$ is the rms fluctuations of the velocity.