# Reynolds number and characteristic length for a rectangular object in a pipe

I'm working on the problem of a steady 2-dimensional incompressible viscous flow through a straight delimited fluid pipe, in which a rectangular object is found,

This problem is aimed to be solved numerically by the stream-vorticity method. However I have a problem to understand how to choose the characteristic length $L$ for the system. In a document its proposed that the characteristic length of the system is $2W$(the length of the barrier perpendicular to the flow), however they don't explain why did they choose it. I haven't found any reference for this problem, thus some references are appreciated. In this document the Reynolds number for this problem is :

$$Re = \frac{2Wv_o}{\nu}$$

where $v_0$ is the velocity of the flow very far from the rectangular object and $\nu$ is the dynamic viscosity. Why they choose this characteristic length for this system ?, Why not $T$ ?

• I don't know of many cases of interest where W would be zero. But you can certainly look at a case where T is vanishing thin, and obtain a relevant answer. That seems to be the motivation for choosing 2W as the characteristic dimension for the Re. – Chet Miller Nov 10 '16 at 3:00
• Very good observation, for the case of T approaching zero, would be the case of a plate immerse in a fluid. Whereas the case of W approaching zero is not of any additional interest to the flow with no barrier. – Keith Nov 10 '16 at 3:05
• Actually you must take into account all relevant parameters and form all possible dimensionless groups (say using Pi theorem). That is to say $T/W,H/W$ must also enter the problem. My answer physics.stackexchange.com/questions/286285/… might help. – Deep Nov 10 '16 at 5:36