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

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* 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 (Kameda et 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 alpha x.