Smooth-pipe regime

When the pipe surface is smooth (the "smooth pipe" curve in Figure 2), the friction factor's variation with Re can be modeled by the Kármán–Prandtl resistance equation for turbulent flow in smooth pipes^[4] with the parameters suitably adjusted

${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=1.930\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.537.}$

The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data.^[8] The product Re√f_D (called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of Re√f_D, the friction factor is also fixed.

In the Kármán–Prandtl resistance equation, f_D can be expressed in closed form as an analytic function of Re through the use of the Lambert W function:

${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}={\frac {1.930}{\ln(10)}}W\left(10^{\frac {-0.537}{1.930}}{\frac {\ln(10)}{1.930}}\mathrm {Re} \right)=0.838\ W(0.629\ \mathrm {Re} )}$

In this flow regime, many small vortices are responsible for the transfer of momentum between the bulk of the fluid to the pipe wall. As the friction Reynolds number Re√f_D increases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in Blasius boundary layer theory.

Rough-pipe regime

When the pipe surface's roughness height ε is significant (typically at high Reynolds number), the friction factor departs from the smooth pipe curve, ultimately approaching an asymptotic value ("rough pipe" regime). In this regime, the resistance to flow varies according to the square of the mean flow velocity and is insensitive to Reynolds number. Here, it is useful to employ yet another dimensionless parameter of the flow, the roughness Reynolds number^[9]

${\displaystyle R_{*}={\frac {1}{\sqrt {8}}}\left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\,\right){\frac {\varepsilon }{D}}}$

where the roughness height ε is scaled to the pipe diameter D.

Figure 3. Roughness function B vs. friction Reynolds number R_∗. The data fall on a single trajectory when plotted in this way. The regime R_∗ < 1 is effectively that of smooth pipe flow. For large R_∗, the roughness function B approaches a constant value. Phenomenological functions attempting to fit these data, including the Afzal^[10] and Colebrook–White^[11] are shown.

It is illustrative to plot the roughness function B:^[12]

${\displaystyle B(R_{*})={\frac {1}{1.930{\sqrt {f_{\mathrm {D} }}}}}+\log \left({\frac {1.90}{\sqrt {8}}}\cdot {\frac {\varepsilon }{D}}\right)}$

Figure 3 shows B versus R_∗ for the rough pipe data of Nikuradse,^[9] Shockling,^[13] and Langelandsvik.^[14]

In this view, the data at different roughness ratio ε/D fall together when plotted against R_∗, demonstrating scaling in the variable R_∗. The following features are present:

• When ε = 0, then R_∗ is identically zero: flow is always in the smooth pipe regime. The data for these points lie to the left extreme of the abscissa and are not within the frame of the graph.
• When R_∗ < 5, the data lie on the line B(R_∗) = R_∗; flow is in the smooth pipe regime.
• When R_∗ > 100, the data asymptotically approach a horizontal line; they are independent of Re, f_D, and ε/D.
• The intermediate range of 5 < R_∗ < 100 constitutes a transition from one behavior to the other. The data depart from the line B(R_∗) = R_∗ very slowly, reach a maximum near R_∗ = 10, then fall to a constant value.

Afzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in R_∗ that ensures proper behavior for 1 < R_∗ < 50 (the transition from the smooth pipe regime to the rough pipe regime):^[10][15][16]

${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.305R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}$

and

${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.930\log \left({\frac {1.90}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.34R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}$

This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for R_∗ → ∞ along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.

The friction factor for another analogous roughness becomes

 : ${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.0\,\log _{10}\left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.305R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}$

and

: ${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.93\,\log _{10}\left({\frac {1.91}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.34R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}$

This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for R_∗ → ∞ along with one further parameter, 26, to govern the transition from smooth to rough flow.

The Colebrook–White relation^[11] fits the friction factor with a function of the form

${\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+{\frac {R_{*}}{3.3}}\right)\right).}$^{[lower-alpha 4]}

This relation has the correct behavior at extreme values of R_∗, as shown by the labeled curve in Figure 3: when R_∗ is small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin.^[13] Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed.

Direct calculation when friction loss S is known

In typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity g and the kinematic viscosity of the fluid ν are known, as are the diameter of the pipe D and its roughness height ε. If as well the head loss per unit length S is a known quantity, then the friction factor f_D can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation for √f_D,

${\displaystyle {\sqrt {f_{\mathrm {D} }}}={\frac {\sqrt {2gSD}}{\langle v\rangle }}}$

we can now express Re√f_D:

${\displaystyle \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}={\frac {1}{\nu }}{\sqrt {2g}}{\sqrt {S}}{\sqrt {D^{3}}}}$

Expressing the roughness Reynolds number R_∗,

${\displaystyle {\begin{aligned}R_{*}&={\frac {\varepsilon }{D}}\cdot \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\cdot {\frac {1}{\sqrt {8}}}\\&={\frac {1}{2}}{\frac {\sqrt {g}}{\nu }}\varepsilon {\sqrt {S}}{\sqrt {D}}\end{aligned}}}$

we have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor f_D, the flow velocity ⟨v⟩, and the volumetric flow rate Q.