I've recently been learning about how to use Reynolds number to estimate if a fluid flow is laminar or turbulent while flowing through a pipe, and it has got me wondering if there is a similar way to tell if a flow is laminar or turbulent in a partially filled pipe.

I've tried googling a way to calculate this, but have not found anything helpful.

I'm not sure if this makes a difference, but I'm assuming the pipe is half filled and cylindrical.


Reynolds number for a full pipe is $\frac{\rho{V}{d}}{\mu}$, where $\rho$ is density, $V$ is velocity of the fluid, $d$ is diameter of the pipe, and $\mu$ is viscosity. How would this formula change for a half pipe?

  • $\begingroup$ Sure, if there is geometric similarity (pipes of the same cross-section shape, filled to the same fraction of the diameter) then transition to turbulence will occur at the same Reynolds number. $\endgroup$ Nov 24, 2019 at 21:41
  • $\begingroup$ @MaximUmansky, how would the Reynolds number be calculated for a partially filled pipe? Would the diameter be the diameter of the full pipe, the average diameter of the half pipe, or something else? $\endgroup$
    – 63677
    Nov 24, 2019 at 23:13
  • $\begingroup$ It does not really matter, as long as a consistent definition is used. You can use the full pipe diameter, that's convenient. $\endgroup$ Nov 25, 2019 at 19:53

2 Answers 2


Interesting question. I briefly skimmed through this paper which described this phenomenon.

They were able to model the flow by introducing a quantity known as the equivalent diameter, which was the diameter of a tube that would have the same cross sectional area as the "partially filled tube", given by:

$$D_\text{eq} = 2\sqrt{\frac{A}{\pi}}$$

where $A$ was the cross sectional area of the flow. Substituting in $A=\frac{1}{2}\pi R^2$ gives:

$$D_\text{eq} = \frac{d}{\sqrt{2}}$$

so we can say the Reynolds number is:

$$\text{Re} = \frac{\rho V d}{\sqrt{2}\mu}$$

for a half filled pipe.


Useful discussion and derivation. Just a note that you can extend this out to any partially full pipe scenario by inserting hydraulic radius (Area/Wetted Perimeter ) for the 'D' component in the Reynolds number expression. This is the normal approach that is used for open channel flow. The trick is that you need to multiply the hydraulic radius by 4 to get the same turbulence transition points as for pipe flow (traditionally, Reynolds number for open channel flow has different transition points to pipe flow due to the use of hydraulic radius).


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