Examining the Generality of ${\tau }_0/\mathrm{TKE}$ for Gravel and Cobble Beds with Sand Fill

Turbulence measurements over rough beds are used for a variety of purposes, including studies of habitat diversity for aquatic organisms, stream restoration efforts, and assessment of drag induced by vegetation. Turbulent kinetic energy (TKE) has been used to estimate bed shear stress by use of a proportionality constant of 0.19–0.21. The TKE method of bed shear stress estimation is less sensitive to flow misalignment relative to Reynolds stress or mean velocity measurements, but it has not been tested across a broad range of hydraulic conditions. In the work reported here, TKE was compared to bed shear stresses for gravel ($D_{50}=35\mathrm{mm}$) and cobble ($D_{50}=156\mathrm{mm}$) beds with varying discharges and levels of sand fill ($D_{50}=0.3\mathrm{mm}$). The ratio of bed shear stress (${\tau }_0$) with TKE was not strongly affected by mean velocity for either the gravel or cobble beds, but it was affected by bed roughness. The mean of the median values of ${\tau }_0/\mathrm{TKE}$ across three mean velocities was 0.24 for gravel with no sand in the bed and was reduced to 0.21 for sand fills of 62–216%. Sand fills of 0–39% for the cobble bed resulted in ${\tau }_0/\mathrm{TKE}=0.48$; increasing the sand fill to 46–65% produced ${\tau }_0/\mathrm{TKE}=0.40$. These results show that the roughness of the bed can affect the ratio of bed shear stress to TKE, which means that caution is required if TKE is to be used as a surrogate measurement for bed shear stress.