A circular pipe has a diameter of $1$ m, bed slope of $1$ in $1000$, and Manning’s roughness coefficient equal to $0.01$. It may be treated as an open channel flow when it is flowing just full, i.e., the water level just touches the crest. The discharge in this condition is denoted by $Q_{\text{full}}$. Similarly, the discharge when the pipe is flowing half-full, i.e., with a flow depth of $0.5$ m, is denoted by $Q_{\text{half}}$. The ratio $Q_{\text{full}} / Q_{\text{half}}$ is
1. $1$
2. $\sqrt{2}$
3. $2$
4. $4$