How is the relationship between Entrance Length and Reynold’s number proven?

Question

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?

Answer 1

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).

Answer 2

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.

Answer 3

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).

Answer 4

Let's start by figuring out what the "end of varying velocity distribution" means. It is fully developed flow. It is the length down the pipe where the boundary layers which form at the wall meet in the center of the pipe. When the boundary layer converges at the center, the mean velocity profile will not change.

There are several methods to see this relationship:

Buckingham Pi:

This method essentially argues that non-dimensional variables must be functions of non-dimensional variables

So, if $L_e/D$ is non-dimensional, it should be a function of other non dimensional variables. Looking at what may be important to the problem:

$ V $ - velocity

$ L_{char}$ - some characteristic length (geometry, like diameter D)

$ \nu $ - kinematic viscosity

$\epsilon$ - surface roughness

a fairly natural combination involved forms $ Re = \frac{V D}{\nu}$

From the Navier Stokes Equations:

The non-dimensional form of the boundary layer equations (simplified from the Navier Stokes equations) are

$ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{dP}{dx} + \frac{1}{Re} \frac{\partial^2u}{\partial y^2} $

Here Reynold's number shows the dominance of momentum diffusion within the flow. This diffusion is what leads to the development of the boundary layers which transition to fully developed flow. Almost all of the correlations you have for laminar flow, and even most turbulent flow correlations, are derived from integral solutions to the boundary layer equations. Naturally, these will have Reynold's number within them

From the engineering perspective:

Reynold's number is the balance between momentum diffusivity $\nu$ and inertia $V$. This balance will determine how quickly the no-slip condition at the wall will be felt at the center of the pipe. Low inertia + high diffusivity = almost instantly feel effects of the wall and vise versa. So, we should expect Re to appear as a variable in our correlations. (That being said, a turbulent boundary layer will develop much faster than a laminar boundary layer due to mixing and the resultant momentum transfer)