  A numerical study of two-dimensional flow past a confined circular cylinder
with slip wall is performed. A dimensionless number, Knudsen number ($Kn$) is
used to describe the slip length of cylinder wall. The Reynolds number ($Re$)
and Knudsen number ($Kn$) ranges considered are $Re = [1, 180]$ and $Kn = [0,
\infty)$, respectively. Time-averaged flow separation angle ($\bar{\theta_s}$),
dimensionless recirculation length ($\bar{L_s}$) and the tangential velocity
($\bar{u_{\tau}}$) distributed on the cylinder's wall, drag coefficient
($\bar{C_d}$) and drag reduction ($DR$) are investigated. The time-averaged
tangential velocity distribution on the cylinder's wall fit well with the
formula $\bar{u_{\tau}} =
[\frac{\alpha}{1+{\beta}e^{-{\gamma}({\pi}-{\theta})}}+{\delta}]sin(\theta) $,
where the coefficients ($\alpha$, $\beta$, $\gamma$, $\delta$) are related with
$Re$ and $Kn$. Several scaling-laws are found,
$log(\bar{u_{{\tau}max}})\sim{log(Re)}$ and $\bar{u_{{\tau}max}}\sim{Kn}$ for
low $Kn$, ($\bar{u_{{\tau}max}}$ is the maximum tangential velocity on the
cylinder's wall), $log(DR)\sim{log(Re)}$ ($Re\leq45$ and $Kn\leq0.1$),
$log(DR)\sim{log(Kn)}$ ($Kn\leq0.05$). At low $Re$, $DR_v$ (the friction drag
reduction) is the main source of $DR$. However, $DR_p$ (the differential
pressure drag reduction) contributes the most to $DR$ at high $Re$
($Re>\sim60$) and $Kn$ over a critical number. $DR_v$ is found almost
independent to $Re$.

Fluid Dynamics