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When calculating Reynolds number for cylindrical pipe, we were taught:
$Re=vd/\nu$, where $v$ is the fluid velocity, $d$ is diameter, and $\nu$ is kinematic velocity.
If you reorient the cylinder so now the flow hits the side of the cylinder, does the Reynolds number change? How do you justify this?
It depends on the diameter to length ratio $D/L$ of the cylinder.
If the flow is parallel to the length of the cylinder and $D/L\ll1$ then the length scale in the Reynolds number is the distance from the leading edge of the cylinder. In this case the diameter and length of the cylinder play no role. See e.g. boundary layer growth over a flat plate. If $D/L\sim1$ then besides the Reynolds number (which contains either $D$ or $L$, it doesn't matter), the value of the ratio also determines the flow.
If the flow is perpendicular and $D/L\ll1$ then the length scale is $D$. This is easiest visualized by reducing it to 2d which reduces the cylinder to a circle which has only one length scale $D$.
Assuming that the parallel flow case you are considering is internal flow through the pipe, then the length scale in both parallel and perpendicular scenarios will be the pipe diameter, D. However, even though the Reynolds Number may be the same in both cases, the nature of the flows is entirely different, so trying to somehow 'equate' or compare them through the Reynolds number is not meaningful.
