# External v.s. internal Reynolds number for cylindrical pipe

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?

• I don't understand your new situation after reorientation; is the fluid flowing over the cylinder? maybe you can give a quick sketch. In either way, the length scale used in the Reynolds equations needs to be the defining length scale of the situation. If there is no other length scale than the diameter than that is usually chosen as the defining length scale Feb 26, 2016 at 10:47
• You can only orient cylinder two ways that cover perpendicular and parralel situations in uniform flow, just visualize. Feb 26, 2016 at 21:01
• Note that $\nu$ is kinematic viscosity, not velocity. Aug 21, 2019 at 1:24
• You should also clarify if the flow is internal or external for the parallel case. Aug 21, 2019 at 1:29

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 $D$.