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For laminar flow in a pipe or annulus of diameter $D$, there is a region from the circle of entry where velocity profile slowly develops until it is fully developed and no longer changes with the axial direction of the pipe.

The length from the start of the pipe till the end of varying velocity distribution is the entrance length $L_e$. Several empirical correlations show that $L_e = f(Re,D)$, as given by most fluid mechanics textbooks. However, none cover why this is a function of Reynold’s number and Diameter from a dimensional analysis and mass balance perspective. How can this expression be proven mathematically?

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  • $\begingroup$ I suggest that most relationships like this are proven empirically, not mathematically $\endgroup$
    – Tiger Guy
    Apr 4, 2023 at 4:38

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We had an experiment with a long tube and an axial fan where the flow profile was measured with a pitot tube across the diameter at various points along the length.

As the developing profile can be plotted and the fully developed profile as well, then the results can be compared with other fluids with dimensional analysis - something you @TheTenthBox can check out (Rayleigh and Buckingham also).

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  • $\begingroup$ My main question revolves around why entrance length is a function of $Re,L$ and how it can be proven. $\endgroup$ Apr 4, 2023 at 10:50
  • $\begingroup$ @TheTenthBox because the Reynolds number defines the characteristic of the flow, as in Laminar, Critical or Turbulent. The velocity profile develops for each. The expressions come from work by Reynolds, Colebrooke White and many others. there is SO much to read from the names I have given you which will expand the knowledge. $\endgroup$
    – Solar Mike
    Apr 4, 2023 at 10:58
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It may have to do something with random nature of turbulence, which is connected to Reynolds number. You basically need to "destroy" the information about the entrance for the flow to be developed and turbulence may help you with that.

In case of laminar flow, the main mechanism may be different. You could take element of the pipe with the fluid far from the entrance and determine equilibrium velocity profile. Any different profile should gravitate to that one and rate of the change will be resisted by inertia and helped by viscous forces, the ratio of which is the Reynolds number.

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First, make a list (by inspection of the Navier-Stokes equations) of what variables the entrance length $L_{\textrm{e}}$ might depend on. One comes up with fluid density $\rho$, dynamic viscosity $\eta$, (mean) velocity $u$, and pipe diameter $D$. Hence there are five variables altogether. Those variables contain three base dimensions (mass, length, and time). Hence, whatever the relationship is between them, the Buckingham $\pi$ theorem tells us it can be expressed as a relationship between $5-3 = 2$ dimensionless variables; that is to say, if we can construct two distinct dimensionless groupings of the variables, then the value of one of those groupings depends only on the value of the other. We construct dimensionless groupings $L_{\textrm{e}}/D$ and $\rho u D/\eta$ (the second one being the Reynolds number). We know those groupings are distinct, because the first one has no involvement of $u$ and the second one has no involvement of $L_{\textrm{e}}$ (the "non-repeating variables"). Hence, we know that $L_{\textrm{e}}/D$ depends only on the Reynolds number. An algebraic way of writing that statement is $L_{\textrm{e}}/D = f\left(\mathit{Re}\right)$, or equivalently $L_{\textrm{e}} = Df\left(\mathit{Re}\right)$ (actually a stronger statement than the one you were looking to prove).

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  • $\begingroup$ Having had overnight to think about this argument, it involves a couple of assumptions about the surface finish of the inside of the pipe and about the spatial distribution of velocity over the entrance, which could broadly be put under the heading of "geometrical similarity". $\endgroup$ Apr 29 at 12:58

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