An empiric profile for a turbulent flow in a circular pipe

 
Abstract

Here I propose an empiric profile for the turbulent flow in a circular pipe. The profile is a simple continuous function, similar but different to the power law equation. It's only parameter is fixed based on the stress on the wall and a flat profile in the center.

Equation development

We all know that there is not an analytical solution for the velocity profile in a circular pipe in turbulent flow. There is, of course, for developed laminar flow: 

\[v(r) = v_\text{c} \left[ 1 - \left(r \over R\right)^2 \right] \tag{1} \]

where \(v_\text{c}\) is the velocity at the center, \(r\) is the radial coordinate, \(R\) the radius of the tube. On the other hand, there's a well-know empirical average profile for turbulent flow, which is know as the power-law profile:

\[\bar{v}(r) = \bar{v}_\text{c} \left( 1 - {r \over R} \right)^n \tag{2} \]

where \(n\) is an empirical parameter, usually around 1/7. This model represents acceptably well the average profile for the average velocity, managing to define a correct dependence of the flow with the pressure gradient and flow integrated in the flow area. But it has a drawback: it gives wrong values of stress at the wall (\(\tau_\text{w} \to \infty\)) and at the center, (\(\tau_\text{c} \neq 0\)).

In order to overcome these flaw, I propose the following empiric profile:

\[\bar{v}(r) = \bar{v}_\text{c} \left[ 1 - \left(r \over R\right)^m \right] \tag{3} \]

where \(m\) is another empirical constant. Eq. (3) is obviously based on the laminar profile, Eq. (1), but with an augmented power. Naturally, the profile exhibit by Eq. (3) is completely flat for \(r = 0\), which implies \(\tau_\text{c} = 0\). On the other hand, there are many ways to set parameter \(m\) like, for instance, fitting the average velocity profile. Here I will follow a different approach: match the stress at the wall prediceted by Eq. (3) with that given for a turbulent flow based on the friction factor.

We know that, at any point, the stress in the (assumed newtonian) fluid is:

\[ \tau (r,m) = \mu {{\operatorname{d}v}\over{\operatorname{d}r}} \tag{4} \]

where \(\mu\) is the dynamic viscocity. In case of Eq. (3),

\[ \tau (r,m) = -{{m \mu \bar{v}_\text{c}}\over{R}}\left(r \over R\right)^{m-1}  \tag{5} \]

Evaluating Eq. (5) at \(r = R\) we obtain the predicted stress at the wall,

\[ \tau_\text{w} = -{{m \mu \bar{v}_\text{c}}\over{R}}  \tag{6} \]

which is a linear function of \(m\). At the same time, the wall stress from momentum balance is

 \[ \tau_\text{w} = -{{\Delta P}\over L}{D \over 4}  \tag{7} \]

Matching Eqs. (6) and (7), from the pressure gradient and the average velocity in the center line, the value of \(m\) can be obtained for different Reynolds numbers, \(\text{Re}\).

Now, there is something that we should care for this to be useful: in addition to giving real values of \(\tau_\text{w}\), Eq. (3) it must provide a real value (or approximately) of the averaged speed defined as:

\[ \langle v \rangle = {\dot V \over  A} \tag{8} \]

where \(\dot V\) is the volume flow through the pipe, which can be evaluated from:

\[ \dot V = \int_A {(\bar{\mathbf{v}} \cdot \breve{\mathbf{n}}) {\operatorname{d}A}} = 2 \pi \int_0^R { \bar{v}_\text{c} \left[ 1 - \left(r \over R\right)^m \right] r {\operatorname{d}r}} = { \pi m R^2 \bar{v}_\text{c} \over 2+m}  \tag{9}\]

and \(A = \pi D^2/4 = \pi R^2\); therefore,

\[\langle v \rangle = {m \bar{v}_\text{c} \over{2+m}} \tag{10} \]

Now, we may compute the pressure gradient, \({\Delta P\over L}\) directly from the loss of mechanical energy:

\[h_\text{L} = f {L \over D}{\alpha \langle v \rangle^2 \over 2 g}{\Delta P\over \gamma} \tag{11} \]

where \(h_\text{L}\) is the height loss due to the fluid viscosity, \(\gamma\) is the specific weight, \(f\) is the friction factor, and \(\alpha\) is the kinetic energy correction coefficient, which is defined as:

\[\alpha = { \int_A {\bar{v}(r)^2 (\bar{\mathbf{v}} \cdot \breve{\mathbf{n}}) {\operatorname{d}A}} \over \langle v \rangle^3 A} \tag{12} \]

 and, for Eq. (3):

\[\alpha = {3 (2+m)^2 \over (2+3m)(m+1)} \tag{13} \]

Matching Eqs. (6) and (7) and replacing the pressure gradient from (11):

\[ {{f {L \over D}{\alpha \langle v \rangle^2 \gamma \over 2 g} }\over L} {D \over 4} = {f \delta \alpha \langle v \rangle^2 \over 8 } = {{m \mu \bar{v}_\text{c}}\over{R}} \tag{14}\]

where \(\delta = \gamma / g\) is the mass density. (Yes, I prefer \(\delta\) instead of \(\rho\)). Now, replacing \(\bar{v}_\text{c}\) from Eq. (10) we obtain:

\[ {f \delta \alpha \langle v \rangle \over 8 } = {{ (2 +m) \mu }\over{R}} \tag{15}\]

Substituting \(\alpha\) from Eq. (13):

 \[ {f \delta 3 (2+m) \langle v \rangle \over 8 (2 + 3m)(m+1) } = {{\mu }\over{R}} \tag{16}\]

Acknowledging that \(\text{Re} ={ \delta  \langle v \rangle D / \mu}\), we may write Eq. (16) as:

\[0 = 32 - 6 f \text{Re}+(80-3 f \text{Re}) m+48 m^2 \tag{17}\]

what allows us to obtaing \(m\) from the Baskara equation:

\[m = {3 f \text{Re}-80 \pm \sqrt{(3 f \text{Re}-80)^2-192(32 - 6 f \text{Re})} \over 96} \tag{18} \]

Now, we have a way to define \(m\), matching the stress in the wall as well the height loss without the need of correlating anything. Now, it's turn to evaluate this function.

Example

Let's give some number, say, for 0.1 m^3/s of water flowing in pipe of 20 cm of internal diameter. Assume \(\delta\) = 1000 kg/m3 and \(\mu\) = 10-3 Pa·s. Average speed is:

\[\langle v \rangle= {4 \times 0.1 [\text{m}^3/\text{s}] \over \pi 0.2^2 [\text{m}^2]} = 0.3183 \left[\text{m}\over \text{s}\right] \]

\[\text{Re} = {1000 [\text{kg}/\text{m}^3] \times 0.3183 [\text{m}/ \text{s}] \times 0.2 [\text{m}] \over 10^{-3} [\text{Pa·s}]} = 63660 \]

and we may calculate \(f\) from the following correlation for a flat pipe:

\[f = {1 \over \left[\ln \left(\text{Re} \sqrt{f} \over 2.51 \right) \right]^2} \]

from which we obtain \(f\) = 0.0154. With this, we now have \[m = 61.59\]

which is a rather large power, meaning a really flat, piston-like, profile; and

\[\bar{v}_\text{c} = 0.3286 \left[\text{m}\over \text{s}\right] \]

So, the velocity at the center of the pipe is just a 3% larger than the average speed. 

Figure 1 depicts the proposed profile. For the sake of comparing. Figure 1 also shows the power-law profile with \(n = \sqrt{f} = 0.124 \). As can be seen, Eq (3) provides a too flat profile, when compared with the power-low equation.

Figure 1. Comparison between velocity profiles from Eq. (2) (power-law) and (3) potential. For the power-lay profile, \(n = \sqrt{f}\) and \(\bar{v}_\text{c} = (n+1)(n+2) \langle v \rangle/2 \).

Partial conclusions

The profile is too flat, although it will be shown that it represents experimental velocity near the wall better than the linear profile; and that it starts to fail when the logarithmic regimen start