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In fluid dynamics, when I have to compute Reynolds number, I was taught to differentiate between two cases: $$ 1)\space \frac{\rho\cdot v\cdot d}{\mu}$$ or $$2)\space \frac{v\cdot d}{\mu}$$

but, why do I have to do this? Why shouldn't I simply use the first formula? what is the criteria to use in order to include or omit the density (i.e $\rho$)?

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  • $\begingroup$ Is diameter always the characteristic length? what happens if the duct is rectangular? $\endgroup$
    – Solar Mike
    Commented Jun 25, 2023 at 13:41
  • $\begingroup$ @SolarMike It doesn't matter, diameter can be circular or rectangular, I guess $\endgroup$
    – Gabriel Burzacchini
    Commented Jun 25, 2023 at 13:42
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    $\begingroup$ Have you checked out any books like Mechanics of Fluids by Massey? $\endgroup$
    – Solar Mike
    Commented Jun 25, 2023 at 14:01
  • $\begingroup$ yes, but it uses only the first formula, it always includes density $\endgroup$
    – Gabriel Burzacchini
    Commented Jun 25, 2023 at 14:14
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    $\begingroup$ Are you sure the symbol in the denominator in the second case is $\mu$ and not $\nu$? $\endgroup$
    – Daniel Hatton
    Commented Jun 25, 2023 at 16:36

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I've never seen the second formula. It's incorrect, unless it's a specific application like water where density is numerically 1 g/cc and they just don't write it.

The first formula

$$\mathrm{Re} =\frac{\rho vd}{\mu}$$

is the definition of Reynolds number and is always true. But you may sometimes see it written in terms of kinematic viscosity $\nu$ which is equal to $\mu/\rho$, and so you will see:

$$\mathrm{Re} =\frac{vd}{\nu}$$

In other more specialized geometries like a porous bed or perforated plate, you may see the Reynolds number defined in terms of other parameters like the mass flow rate and some dimension. But for flow in a pipe, the first formula is always true.

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