Self-preservation in a zero pressure gradient rough-wall turbulent boundary layer

Abstract

A self-preservation (SP) analysis is carried out for a zero pressure gradient (ZPG) rough-wall turbulent boundary layer with a view to establishing the requirements of complete SP (i.e. SP across the entire layer) and determining if these are achievable. The analysis shows that SP is achievable in certain rough-wall boundary layers (irrespectively of the Reynolds number$Re$), when the mean viscous stress is zero or negligible compared to the form drag across the entire boundary layer. In this case, the velocity scale$u^{\ast }$must be constant, the length scale$l$should vary linearly with the streamwise distance$x$and the roughness height$k$must be proportional to$l$. Although this result is consistent with that of Rotta (Prog. Aeronaut. Sci., vol. 2 (1), 1962, pp. 1–95), it is derived in a more rigorous manner than the method employed by Rotta. Further, it is noted that complete SP is not possible in a smooth-wall ZPG turbulent boundary layer. The SP conditions are tested against published experimental data on both a smooth wall (Kulandaivelu, 2012, PhD thesis, The University of Melbourne) and a rough wall, where the roughness height increases linearly with$x$(Kamedaet al.,J. Fluid Sci. Technol., vol. 3 (1), 2008, pp. 31–42). Complete SP in a ZPG rough-wall turbulent boundary layer seems indeed possible when$k\propto x$.