Fully-developed internal turbulent flow

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<math>-\frac{d\bar{p}}{dx}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}</math>
<math>-\frac{d\bar{p}}{dx}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}</math>
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where
where
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<math>{{\tau }_{app}}=\rho (\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial y}</math>
<math>{{\tau }_{app}}=\rho (\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial y}</math>
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is the apparent or total shear stress and y is the distance measured from the wall (<math>y={{r}_{o}}-r</math>). The apparent shear stress is equal to <math>{{\tau }_{w}}</math> at the wall and zero at the centerline. Integrating eq. (5.272) from the centerline to the wall yields,
is the apparent or total shear stress and y is the distance measured from the wall (<math>y={{r}_{o}}-r</math>). The apparent shear stress is equal to <math>{{\tau }_{w}}</math> at the wall and zero at the centerline. Integrating eq. (5.272) from the centerline to the wall yields,
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<math>-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}</math>
<math>-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}</math>
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Integrating eq. (5.274) in the interval of (0, r), one obtains:
Integrating eq. (5.274) in the interval of (0, r), one obtains:
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<math>{{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)</math>
<math>{{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)</math>
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where y is measured from the tube wall (<math>y={{r}_{0}}-r</math>). Equation (5.275) shows that the shear stress is a linear function for internal turbulent flow. Close to the wall where r is near r0 (or y is near 0), the apparent shear stress is nearly a constant, i.e., <math>{{\tau }_{app}}\approx {{\tau }_{w}}</math>. The law of the wall resulting from the two-layer turbulent model (see Section 4.11.2) can be applied near the wall to yield:
where y is measured from the tube wall (<math>y={{r}_{0}}-r</math>). Equation (5.275) shows that the shear stress is a linear function for internal turbulent flow. Close to the wall where r is near r0 (or y is near 0), the apparent shear stress is nearly a constant, i.e., <math>{{\tau }_{app}}\approx {{\tau }_{w}}</math>. The law of the wall resulting from the two-layer turbulent model (see Section 4.11.2) can be applied near the wall to yield:
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<math>{{u}^{+}}=2.5\ln {{y}^{+}}+5.5</math>
<math>{{u}^{+}}=2.5\ln {{y}^{+}}+5.5</math>
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which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in Section 4.11.2 and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as
which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in Section 4.11.2 and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as
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<math>{{u}^{+}}=\frac{{\bar{u}}}{{{u}_{\tau }}},\text{ }{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }</math>
<math>{{u}^{+}}=\frac{{\bar{u}}}{{{u}_{\tau }}},\text{ }{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }</math>
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where
where
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<math>{{u}_{\tau }}=\sqrt{\frac{{{\tau }_{w}}}{\rho }}</math>
<math>{{u}_{\tau }}=\sqrt{\frac{{{\tau }_{w}}}{\rho }}</math>
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It should be pointed out that the Nikuradse equation (5.276) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (5.276) is a finite value, not zero as it should be.  In addition, eq. (5.276) also implies that <math>{{\varepsilon }_{M}}/\nu =0</math> at the centerline, which is also not true because the centerline is also in the fully turbulent region. Reichardt (1951) suggested the following empirical correlation for the eddy diffusivity:  
It should be pointed out that the Nikuradse equation (5.276) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (5.276) is a finite value, not zero as it should be.  In addition, eq. (5.276) also implies that <math>{{\varepsilon }_{M}}/\nu =0</math> at the centerline, which is also not true because the centerline is also in the fully turbulent region. Reichardt (1951) suggested the following empirical correlation for the eddy diffusivity:  
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<math>\frac{{{\varepsilon }_{M}}}{\nu }=\frac{\kappa {{y}^{+}}}{6}\left( 1+\frac{r}{{{r}_{o}}} \right)\left[ 1+2{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]</math>
<math>\frac{{{\varepsilon }_{M}}}{\nu }=\frac{\kappa {{y}^{+}}}{6}\left( 1+\frac{r}{{{r}_{o}}} \right)\left[ 1+2{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]</math>
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which becomes <math>{{\varepsilon }_{M}}/\nu =\kappa {{y}^{+}}</math> near the wall – a result that coincides with the mixing length theory. Equation (5.279) produces finite eddy diffusivity at the centerline. Assuming <math>{{\varepsilon }_{M}}\gg \nu </math>, eq. (5.273) becomes
which becomes <math>{{\varepsilon }_{M}}/\nu =\kappa {{y}^{+}}</math> near the wall – a result that coincides with the mixing length theory. Equation (5.279) produces finite eddy diffusivity at the centerline. Assuming <math>{{\varepsilon }_{M}}\gg \nu </math>, eq. (5.273) becomes
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<math>{{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}</math>
<math>{{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}</math>
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Substituting eqs. (5.275) and (5.279), and integrating the resultant equation, the following velocity profile is obtained
Substituting eqs. (5.275) and (5.279), and integrating the resultant equation, the following velocity profile is obtained
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<math>{{u}^{+}}=2.5\ln \left\{ \frac{3(1+r/{{r}_{o}})}{2[1+2{{(r/{{r}_{o}})}^{2}}]}{{y}^{+}} \right\}+5.5</math>
<math>{{u}^{+}}=2.5\ln \left\{ \frac{3(1+r/{{r}_{o}})}{2[1+2{{(r/{{r}_{o}})}^{2}}]}{{y}^{+}} \right\}+5.5</math>
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which becomes identical to eq. (5.276) near the wall and produces zero slope at the centerline.  
which becomes identical to eq. (5.276) near the wall and produces zero slope at the centerline.  
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<math>{{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}</math>
<math>{{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}</math>
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where <math>{{\bar{u}}_{m}}</math> is the mean velocity over the cross-section of the duct. For axisymmetric flow in a circular tube, it is obtained by
where <math>{{\bar{u}}_{m}}</math> is the mean velocity over the cross-section of the duct. For axisymmetric flow in a circular tube, it is obtained by
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<math>{{\bar{u}}_{m}}=\frac{2}{r_{o}^{2}}\int_{0}^{{{r}_{o}}}{\bar{u}rdr}</math>
<math>{{\bar{u}}_{m}}=\frac{2}{r_{o}^{2}}\int_{0}^{{{r}_{o}}}{\bar{u}rdr}</math>
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The definition of the friction factor, eq. (5.282), can be rewritten as
The definition of the friction factor, eq. (5.282), can be rewritten as
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<math>{{\left( \frac{{{\tau }_{w}}}{\rho } \right)}^{1/2}}={{\bar{u}}_{m}}{{\left( \frac{{{c}_{f}}}{2} \right)}^{1/2}}</math>
<math>{{\left( \frac{{{\tau }_{w}}}{\rho } \right)}^{1/2}}={{\bar{u}}_{m}}{{\left( \frac{{{c}_{f}}}{2} \right)}^{1/2}}</math>
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For moderate Reynolds number, the velocity profile in the entire tube can be approximated as (Kays et al., 2005)
For moderate Reynolds number, the velocity profile in the entire tube can be approximated as (Kays et al., 2005)
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<math>{{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}</math>
<math>{{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}</math>
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At the centerline where <math>\bar{u}={{\bar{u}}_{c}}\text{ and }y={{r}_{0}}</math>, the centerline velocity satisfies:  
At the centerline where <math>\bar{u}={{\bar{u}}_{c}}\text{ and }y={{r}_{0}}</math>, the centerline velocity satisfies:  
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<math>\frac{{{{\bar{u}}}_{c}}}{\sqrt{{{\tau }_{w}}/\rho }}=8.6{{\left( \frac{{{r}_{o}}\sqrt{{{\tau }_{w}}/\rho }}{\nu } \right)}^{1/7}}</math>
<math>\frac{{{{\bar{u}}}_{c}}}{\sqrt{{{\tau }_{w}}/\rho }}=8.6{{\left( \frac{{{r}_{o}}\sqrt{{{\tau }_{w}}/\rho }}{\nu } \right)}^{1/7}}</math>
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The velocity at any radius is related to the centerline velocity by
The velocity at any radius is related to the centerline velocity by
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<math>\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}</math>
<math>\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}</math>
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Substituting eq. (5.287) into eq. (5.283), a relationship between the mean velocity and centerline velocity is obtained:
Substituting eq. (5.287) into eq. (5.283), a relationship between the mean velocity and centerline velocity is obtained:
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<math>{{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}</math>
<math>{{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}</math>
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Substituting eq. (5.284) and (5.288) into eq. (5.286) and considering the definition of Reynolds number <math>{{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu </math>, the friction coefficient can be obtained as
Substituting eq. (5.284) and (5.288) into eq. (5.286) and considering the definition of Reynolds number <math>{{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu </math>, the friction coefficient can be obtained as
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<math>{{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}</math>
<math>{{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}</math>
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which agreed with the experimental data very well up to <math>{{\operatorname{Re}}_{D}}=5\times {{10}^{4}}</math>. For even higher Reynolds number, the following empirical correlation works better for smooth tubes (see Fig. 5.32):
which agreed with the experimental data very well up to <math>{{\operatorname{Re}}_{D}}=5\times {{10}^{4}}</math>. For even higher Reynolds number, the following empirical correlation works better for smooth tubes (see Fig. 5.32):
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<math>{{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}</math>
<math>{{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}</math>
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Instead of the one-seventh law, eq. (5.285), the law of the wall, eq. (5.276), can be used to obtain the following correlation:
Instead of the one-seventh law, eq. (5.285), the law of the wall, eq. (5.276), can be used to obtain the following correlation:
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<math>c_{f}^{-1/2}=1.737\ln (c_{f}^{1/2}{{\operatorname{Re}}_{D}})-0.396</math>
<math>c_{f}^{-1/2}=1.737\ln (c_{f}^{1/2}{{\operatorname{Re}}_{D}})-0.396</math>
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which is referred to in the literature as the Kármán-Nikuradse relation. Equation (5.291) is valid up to ReD = 106.
which is referred to in the literature as the Kármán-Nikuradse relation. Equation (5.291) is valid up to ReD = 106.
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<math>{{\operatorname{Re}}_{k}}=\frac{{{k}_{s}}{{u}_{\tau }}}{\nu }=\frac{{{k}_{s}}{{({{\tau }_{w}}/\rho )}^{1/2}}}{\nu }</math>
<math>{{\operatorname{Re}}_{k}}=\frac{{{k}_{s}}{{u}_{\tau }}}{\nu }=\frac{{{k}_{s}}{{({{\tau }_{w}}/\rho )}^{1/2}}}{\nu }</math>
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where ks is the roughness. When the roughness Reynolds number is greater than 70, the friction coefficient is no longer a strong function of the Reynolds number and becomes a constant, which is referred to as a fully rough surface. In the fully rough surface regime, the roughness size excees the order of the magnitude of what would have been the thickness of the viscous sublayer for a smooth surface. The friction coefficient for the fully rough surface regime can be obtained from the following empirical correlation:
where ks is the roughness. When the roughness Reynolds number is greater than 70, the friction coefficient is no longer a strong function of the Reynolds number and becomes a constant, which is referred to as a fully rough surface. In the fully rough surface regime, the roughness size excees the order of the magnitude of what would have been the thickness of the viscous sublayer for a smooth surface. The friction coefficient for the fully rough surface regime can be obtained from the following empirical correlation:
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<math>{{c}_{f}}\simeq {{\left( 1.74\ln \frac{D}{{{k}_{s}}}+2.28 \right)}^{-2}}</math>
<math>{{c}_{f}}\simeq {{\left( 1.74\ln \frac{D}{{{k}_{s}}}+2.28 \right)}^{-2}}</math>
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Revision as of 02:52, 8 July 2010

Since the turbulent boundary layer grows much faster than the laminar boundary layer, the lengths of hydrodynamic and thermal entrances for turbulent internal flow are also much shorter than those for laminar flow. When the Prandtl number of the fluid is on the order of 1 (e.g., air or water), the lengths of the hydrodynamic and thermal entrances are about 10 times of the diameter of the tube, i.e.,

\frac{{{L}_{H}}}{D}\sim \frac{{{L}_{T}}}{D}\sim 10

(1)

The internal turbulent flow becomes fully developed after x > LH or LT. Similar to the laminar internal flow, we have \bar{v}=0\text{, and }\partial \bar{u}/\partial x=0 for fully developed flow, and the momentum eq. (5.267) becomes

-\frac{d\bar{p}}{dx}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}

(2)

where

{{\tau }_{app}}=\rho (\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial y}

(3)

is the apparent or total shear stress and y is the distance measured from the wall (y = ror). The apparent shear stress is equal to τw at the wall and zero at the centerline. Integrating eq. (5.272) from the centerline to the wall yields,

-\frac{d\bar{p}}{dx}=2\frac{{{\tau }_{w}}}{{{r}_{o}}}

which can be substituted into eq. (5.272) to yield

-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}

(4)

Integrating eq. (5.274) in the interval of (0, r), one obtains:

{{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)

(5)

where y is measured from the tube wall (y = r0r). Equation (5.275) shows that the shear stress is a linear function for internal turbulent flow. Close to the wall where r is near r0 (or y is near 0), the apparent shear stress is nearly a constant, i.e., {{\tau }_{app}}\approx {{\tau }_{w}}. The law of the wall resulting from the two-layer turbulent model (see Section 4.11.2) can be applied near the wall to yield:

u + = 2.5lny + + 5.5

(6)

which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in Section 4.11.2 and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as

{{u}^{+}}=\frac{{\bar{u}}}{{{u}_{\tau }}},\text{ }{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }

(7)

where

{{u}_{\tau }}=\sqrt{\frac{{{\tau }_{w}}}{\rho }}

(8)

It should be pointed out that the Nikuradse equation (5.276) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (5.276) is a finite value, not zero as it should be. In addition, eq. (5.276) also implies that {{\varepsilon }_{M}}/\nu =0 at the centerline, which is also not true because the centerline is also in the fully turbulent region. Reichardt (1951) suggested the following empirical correlation for the eddy diffusivity:

\frac{{{\varepsilon }_{M}}}{\nu }=\frac{\kappa {{y}^{+}}}{6}\left( 1+\frac{r}{{{r}_{o}}} \right)\left[ 1+2{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]

(9)

which becomes {{\varepsilon }_{M}}/\nu =\kappa {{y}^{+}} near the wall – a result that coincides with the mixing length theory. Equation (5.279) produces finite eddy diffusivity at the centerline. Assuming {{\varepsilon }_{M}}\gg \nu , eq. (5.273) becomes

{{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}

(10)

Substituting eqs. (5.275) and (5.279), and integrating the resultant equation, the following velocity profile is obtained

{{u}^{+}}=2.5\ln \left\{ \frac{3(1+r/{{r}_{o}})}{2[1+2{{(r/{{r}_{o}})}^{2}}]}{{y}^{+}} \right\}+5.5

(11)

which becomes identical to eq. (5.276) near the wall and produces zero slope at the centerline. The friction factor for internal turbulent flow is defined as

{{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}

(12)

where {{\bar{u}}_{m}} is the mean velocity over the cross-section of the duct. For axisymmetric flow in a circular tube, it is obtained by

{{\bar{u}}_{m}}=\frac{2}{r_{o}^{2}}\int_{0}^{{{r}_{o}}}{\bar{u}rdr}

(13)

The definition of the friction factor, eq. (5.282), can be rewritten as

{{\left( \frac{{{\tau }_{w}}}{\rho } \right)}^{1/2}}={{\bar{u}}_{m}}{{\left( \frac{{{c}_{f}}}{2} \right)}^{1/2}}

(14)

For moderate Reynolds number, the velocity profile in the entire tube can be approximated as (Kays et al., 2005)

{{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}

(15)

At the centerline where \bar{u}={{\bar{u}}_{c}}\text{ and }y={{r}_{0}}, the centerline velocity satisfies:

\frac{{{{\bar{u}}}_{c}}}{\sqrt{{{\tau }_{w}}/\rho }}=8.6{{\left( \frac{{{r}_{o}}\sqrt{{{\tau }_{w}}/\rho }}{\nu } \right)}^{1/7}}

(16)

The velocity at any radius is related to the centerline velocity by

\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}

(17)

Substituting eq. (5.287) into eq. (5.283), a relationship between the mean velocity and centerline velocity is obtained:

{{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}

(18)

Substituting eq. (5.284) and (5.288) into eq. (5.286) and considering the definition of Reynolds number {{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu , the friction coefficient can be obtained as

Friction factor for duct flow
Friction factor for duct flow.

{{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}

(19)

which agreed with the experimental data very well up to {{\operatorname{Re}}_{D}}=5\times {{10}^{4}}. For even higher Reynolds number, the following empirical correlation works better for smooth tubes (see Fig. 5.32):

{{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}

(20)

Instead of the one-seventh law, eq. (5.285), the law of the wall, eq. (5.276), can be used to obtain the following correlation:

c_{f}^{-1/2}=1.737\ln (c_{f}^{1/2}{{\operatorname{Re}}_{D}})-0.396

(21)

which is referred to in the literature as the Kármán-Nikuradse relation. Equation (5.291) is valid up to ReD = 106. For fully-developed turbulent flow in a non-circular tube, eq. (5.291) is still applicable provided the hydraulic diameter is used in the definition of the Reynolds number. In this case, the friction coefficient, cf, is defined based on the perimeter-averaged wall shear stress because the shear stress is no longer uniform around the periphery of the cross-section.

When the inner surface of the tube is not smooth, the friction will significantly increase with roughness (see Fig. 5.32). The effect of the surface roughness can be measured by the roughness Reynolds number defined as:

{{\operatorname{Re}}_{k}}=\frac{{{k}_{s}}{{u}_{\tau }}}{\nu }=\frac{{{k}_{s}}{{({{\tau }_{w}}/\rho )}^{1/2}}}{\nu }

(22)

where ks is the roughness. When the roughness Reynolds number is greater than 70, the friction coefficient is no longer a strong function of the Reynolds number and becomes a constant, which is referred to as a fully rough surface. In the fully rough surface regime, the roughness size excees the order of the magnitude of what would have been the thickness of the viscous sublayer for a smooth surface. The friction coefficient for the fully rough surface regime can be obtained from the following empirical correlation:

{{c}_{f}}\simeq {{\left( 1.74\ln \frac{D}{{{k}_{s}}}+2.28 \right)}^{-2}}

(23)