  Progress in roughness research, mapping any given roughness geometry to its
fluid dynamic behaviour, has been hampered by the lack of accurate and direct
measurements of skin-friction drag, especially in open systems. The
Taylor--Couette (TC) system has the benefit of being a closed system, but its
potential for characterizing irregular, realistic, 3-D roughness has not been
previously considered in depth. Here, we present direct numerical simulations
(DNSs) of TC turbulence with sand grain roughness mounted on the inner
cylinder. The model proposed by Scotti (\textit{Phys. Fluids}, vol. 18, 031701,
2006) has been improved to simulate a random rough surface of monodisperse sand
grains, which is characterized by the equivalent sand grain height $k_s$.
Taylor numbers range from $Ta = 1.0\times 10^7$(corresponding to $Re_\tau =
82$) to $Ta = 1.0\times 10^9$($Re_\tau = 635$). We focus on the influence of
the roughness height $k_s^+$ in the transitionally rough regime, through
simulations of TC with rough surfaces, ranging from $k_s^+=5$ up to $k_s^+ =
92$, where the superscript `$+$' indicates non-dimensionalization in viscous
units. We find that the downwards shift of the logarithmic layer, due to
transitionally rough sand grains exhibits remarkably similar behavior to that
of the Nikuradse (\textit{VDI-Forschungsheft} 361, 1933) data of sand grain
roughness in pipe flow, regardless of the Taylor number dependent constants of
the logarithmic layer.
