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Direct numerical simulations of Taylor--Couette turbulence: the effect of sand grain roughness

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 $Re_τ= 82$) to $Ta = 1.0\times 10^9$($Re_τ= 635$). 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.