Can a turbulent boundary layer become independent of the Reynolds number?

Abstract

This paper examines the Reynolds number ($Re$) dependence of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) which develops over a two-dimensional rough wall with a view to ascertaining whether this type of boundary layer can become independent of $Re$. Measurements are made using hot-wire anemometry over a rough wall that consists of a periodic arrangement of cylindrical rods with a streamwise spacing of eight times the rod diameter. The present results, together with those obtained over a sand-grain roughness at high Reynolds number, indicate that a $Re$-independent state can be achieved at a moderate $Re$. However, it is also found that the mean velocity distributions over different roughness geometries do not collapse when normalised by appropriate velocity and length scales. This lack of collapse is attributed to the difference in the drag coefficient between these geometries. We also show that the collapse of the $U_{\unicode[STIX]{x1D70F}}$-normalised mean velocity defect profiles may not necessarily reflect $Re$-independence. A better indicator of the asymptotic state of $Re$ is the mean velocity defect profile normalised by the free-stream velocity and plotted as a function of $y/\unicode[STIX]{x1D6FF}$, where $y$ is the vertical distance from the wall and $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness. This is well supported by the measurements.