  We numerically simulate turbulent Taylor-Couette flow for independently
rotating inner and outer cylinders, focusing on the analogy with turbulent
Rayleigh-B\'enard flow. Reynolds numbers of $Re_i=8\cdot10^3$ and
$Re_o=\pm4\cdot10^3$ of the inner and outer cylinders, respectively, are
reached, corresponding to Taylor numbers Ta up to $10^8$. Effective scaling
laws for the torque and other system responses are found. Recent experiments
with the Twente turbulent Taylor-Couette ($T^3C$) setup and with a similar
facility in Maryland at very high Reynolds numbers have revealed an optimum
transport at a certain non-zero rotation rate ratio $a = -\omega_o / \omega_i$
of about $a_{opt}=0.33-0.35$. For large enough $Ta$ in the numerically
accessible range we also find such an optimum transport at non-zero
counter-rotation. The position of this maximum is found to shift with the
driving, reaching a maximum of $a_{opt}=0.15$ for $Ta=2.5\cdot10^7$. An
explanation for this shift is elucidated, consistent with the experimental
result that $a_{opt}$ becomes approximately independent of the driving strength
for large enough Reynolds numbers. We furthermore numerically calculate the
angular velocity profiles and visualize the different flow structures for the
various regimes. By writing the equations in a frame co-rotating with the outer
cylinder a link is found between the local angular velocity profiles and the
global transport quantities.
