  Standing oscillations with multiple periods were found in a number of
atmospheric structures on the Sun. The ratio of the period of the fundamental
to twice the one of its first overtone, $P_1/2P_2$, is important in
applications of solar magneto-seismology. We examine how field-aligned flows
impact $P_1/2P_2$ of standing modes in solar magnetic cylinders. For coronal
loops, the flow effects are significant for both fast kink and sausage modes.
For kink ones, they reduce $P_1/2P_2$ by up to 17\% relative to the static case
even when the density contrast between the loop and its surroundings approaches
infinity. For sausage modes, the reduction in $P_1/2P_2$ due to flow is
typically $\lesssim 5.5\%$ compared with the static case. However, the
threshold aspect ratio, only above which can trapped sausage modes be
supported, may increase dramatically with the flow magnitude. For photospheric
tubes, the flow effect on $P_1/2P_2$ is not as strong. However, when applied to
sausage modes, introducing field-aligned flows offers more possibilities in
interpreting the multiple periods recently measured. We conclude that
field-aligned flows should be taken into account to help better understand what
causes the departure of $P_1/2P_2$ from unity.
