  Direct numerical simulations (DNSs) are performed to analyze the global
transport properties of turbulent Taylor-Couette flow with inner rough wall up
to Taylor number $Ta=10^{10}$. The dimensionless torque $Nu_\omega$ shows an
effective scaling of $Nu_\omega \propto Ta^{0.42\pm0.01}$, which is steeper
than the ultimate regime effective scaling $Nu_\omega \propto Ta^{0.38}$ seen
for smooth inner and outer walls. It is found that at the inner rough wall, the
dominant contribution to the torque comes from the pressure forces on the
radial faces of the rough elements; while viscous shear stresses on the rough
surfaces contribute little to $Nu_\omega$. Thus, the log layer close to the
rough wall depends on the roughness length scale, rather than on the viscous
length scale. We then separate the torque contributed from the smooth inner
wall and the rough outer wall. It is found that the smooth wall torque scaling
follows $Nu_s \propto Ta_s^{0.38\pm0.01}$, in excellent agreement with the case
where both walls are smooth. In contrast, the rough wall torque scaling follows
$Nu_r \propto Ta_r^{0.47\pm0.03}$, very close to the pure ultimate regime
scaling $Nu_\omega \propto Ta^{1/2}$. The energy dissipation rate at the wall
of inner rough cylinder decreases significantly as a consequence of the wall
shear stress reduction caused by the flow separation at the rough elements. On
the other hand, the latter shed vortices in the bulk that are transported
towards the outer cylinder and dissipated. Compared to the purely smooth case,
the inner wall roughness renders the system more bulk dominated and thus
increases the effective scaling exponent.
