# Fully-developed internal turbulent flow

(Difference between revisions)
 Revision as of 02:51, 8 July 2010 (view source) (Created page with '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 …')← Older edit Current revision as of 09:15, 27 July 2010 (view source) (4 intermediate revisions not shown) Line 8: Line 8: |{{EquationRef|(1)}} |{{EquationRef|(1)}} |} |} - The internal turbulent flow becomes fully developed after $x>{{L}_{H}}\text{ or }{{L}_{T}}$. 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 + The internal turbulent flow becomes fully developed after $x>{{L}_{H}}\text{ or }{{L}_{T}}$. 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 equation becomes {| class="wikitable" border="0" {| class="wikitable" border="0" Line 15: Line 15: $-\frac{d\bar{p}}{dx}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}$ $-\frac{d\bar{p}}{dx}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}$ - |{{EquationRef|(1)}} + |{{EquationRef|(2)}} |} |} where where Line 24: Line 24: ${{\tau }_{app}}=\rho (\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial y}$ ${{\tau }_{app}}=\rho (\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial y}$ - |{{EquationRef|(1)}} + |{{EquationRef|(3)}} |} |} - is the apparent or total shear stress and y is the distance measured from the wall ($y={{r}_{o}}-r$). The apparent shear stress is equal to ${{\tau }_{w}}$ 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 ($y={{r}_{o}}-r$). The apparent shear stress is equal to ${{\tau }_{w}}$ at the wall and zero at the centerline. Integrating eq. (2) from the centerline to the wall yields, - +
$-\frac{d\bar{p}}{dx}=2\frac{{{\tau }_{w}}}{{{r}_{o}}} - [itex]-\frac{d\bar{p}}{dx}=2\frac{{{\tau }_{w}}}{{{r}_{o}}}$ + - which can be substituted into eq. (5.272) to yield + which can be substituted into eq. (2) to yield {| class="wikitable" border="0" {| class="wikitable" border="0" Line 37: Line 36: $-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}$ $-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}$ - |{{EquationRef|(1)}} + |{{EquationRef|(4)}} |} |} - Integrating eq. (5.274) in the interval of (0, r), one obtains: + Integrating eq. (4) in the interval of (0, ''r''), one obtains: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 46: Line 45: ${{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)$ ${{\tau }_{app}}={{\tau }_{w}}\cdot \left( \frac{r}{{{r}_{o}}} \right)={{\tau }_{w}}\cdot \left( 1-\frac{y}{{{r}_{o}}} \right)$ - |{{EquationRef|(1)}} + |{{EquationRef|(5)}} |} |} - where y is measured from the tube wall ($y={{r}_{0}}-r$). 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: + where ''y'' is measured from the tube wall ($y={{r}_{0}}-r$). Equation (5) 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 [[Algebraic Models for Eddy Diffusivity]]) can be applied near the wall to yield: {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
| width="100%" |
- ${{u}^{+}}=2.5\ln {{y}^{+}}+5.5$ + ${{u}^{+}}=2.5\ln {{y}^{+}}+5.5 - |{{EquationRef|(1)}} + |{{EquationRef|(6)}} |} |} - which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in Section 4.11.2 and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as + which is referred to as the Nikuradse equation. The constants 2.5 and 5.5 are different from those in [[Algebraic Models for Eddy Diffusivity]] and are obtained by curve-fitting to the experimental results. The dimensionless velocity and coordinate are defined as {| class="wikitable" border="0" {| class="wikitable" border="0" Line 64: Line 63: [itex]{{u}^{+}}=\frac{{\bar{u}}}{{{u}_{\tau }}},\text{ }{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }$ ${{u}^{+}}=\frac{{\bar{u}}}{{{u}_{\tau }}},\text{ }{{y}^{+}}=\frac{y{{u}_{\tau }}}{\nu }$ - |{{EquationRef|(1)}} + |{{EquationRef|(7)}} |} |} where where Line 73: Line 72: ${{u}_{\tau }}=\sqrt{\frac{{{\tau }_{w}}}{\rho }}$ ${{u}_{\tau }}=\sqrt{\frac{{{\tau }_{w}}}{\rho }}$ - |{{EquationRef|(1)}} + |{{EquationRef|(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: + It should be pointed out that the Nikuradse equation (6) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (6) is a finite value, not zero as it should be.  In addition, eq. (6) 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 Reichardt, H., 1951, “Die Grundlagen des turbulenten Wärmeüberganges,” Arch. Gesamte Waermetech, Vol. 2, pp. 129-142. suggested the following empirical correlation for the eddy diffusivity: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 82: Line 81: $\frac{{{\varepsilon }_{M}}}{\nu }=\frac{\kappa {{y}^{+}}}{6}\left( 1+\frac{r}{{{r}_{o}}} \right)\left[ 1+2{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]$ $\frac{{{\varepsilon }_{M}}}{\nu }=\frac{\kappa {{y}^{+}}}{6}\left( 1+\frac{r}{{{r}_{o}}} \right)\left[ 1+2{{\left( \frac{r}{{{r}_{o}}} \right)}^{2}} \right]$ - |{{EquationRef|(1)}} + |{{EquationRef|(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 + which becomes ${{\varepsilon }_{M}}/\nu =\kappa {{y}^{+}}$ near the wall – a result that coincides with the mixing length theory. Equation (9) produces finite eddy diffusivity at the centerline. Assuming ${{\varepsilon }_{M}}\gg \nu$, eq. (3) becomes {| class="wikitable" border="0" {| class="wikitable" border="0" Line 91: Line 90: ${{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}$ ${{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}$ - |{{EquationRef|(1)}} + |{{EquationRef|(10)}} |} |} - Substituting eqs. (5.275) and (5.279), and integrating the resultant equation, the following velocity profile is obtained + Substituting eqs. (5) and (9), and integrating the resultant equation, the following velocity profile is obtained {| class="wikitable" border="0" {| class="wikitable" border="0" Line 100: Line 99: ${{u}^{+}}=2.5\ln \left\{ \frac{3(1+r/{{r}_{o}})}{2[1+2{{(r/{{r}_{o}})}^{2}}]}{{y}^{+}} \right\}+5.5$ ${{u}^{+}}=2.5\ln \left\{ \frac{3(1+r/{{r}_{o}})}{2[1+2{{(r/{{r}_{o}})}^{2}}]}{{y}^{+}} \right\}+5.5$ - |{{EquationRef|(1)}} + |{{EquationRef|(11)}} |} |} - which becomes identical to eq. (5.276) near the wall and produces zero slope at the centerline. + which becomes identical to eq. (6) near the wall and produces zero slope at the centerline. The friction factor for internal turbulent flow is defined as The friction factor for internal turbulent flow is defined as Line 110: Line 109: ${{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}$ ${{c}_{f}}=\frac{{{\tau }_{w}}}{\rho \bar{u}_{m}^{2}/2}$ - |{{EquationRef|(1)}} + |{{EquationRef|(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 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 Line 119: Line 118: ${{\bar{u}}_{m}}=\frac{2}{r_{o}^{2}}\int_{0}^{{{r}_{o}}}{\bar{u}rdr}$ ${{\bar{u}}_{m}}=\frac{2}{r_{o}^{2}}\int_{0}^{{{r}_{o}}}{\bar{u}rdr}$ - |{{EquationRef|(1)}} + |{{EquationRef|(13)}} |} |} - The definition of the friction factor, eq. (5.282), can be rewritten as + The definition of the friction factor, eq. (12), can be rewritten as {| class="wikitable" border="0" {| class="wikitable" border="0" Line 128: Line 127: ${{\left( \frac{{{\tau }_{w}}}{\rho } \right)}^{1/2}}={{\bar{u}}_{m}}{{\left( \frac{{{c}_{f}}}{2} \right)}^{1/2}}$ ${{\left( \frac{{{\tau }_{w}}}{\rho } \right)}^{1/2}}={{\bar{u}}_{m}}{{\left( \frac{{{c}_{f}}}{2} \right)}^{1/2}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(14)}} |} |} - 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, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NYFaghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO. {| class="wikitable" border="0" {| class="wikitable" border="0" Line 137: Line 136: ${{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}$ ${{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(15)}} |} |} At the centerline where $\bar{u}={{\bar{u}}_{c}}\text{ and }y={{r}_{0}}$, the centerline velocity satisfies: At the centerline where $\bar{u}={{\bar{u}}_{c}}\text{ and }y={{r}_{0}}$, the centerline velocity satisfies: Line 146: Line 145: $\frac{{{{\bar{u}}}_{c}}}{\sqrt{{{\tau }_{w}}/\rho }}=8.6{{\left( \frac{{{r}_{o}}\sqrt{{{\tau }_{w}}/\rho }}{\nu } \right)}^{1/7}}$ $\frac{{{{\bar{u}}}_{c}}}{\sqrt{{{\tau }_{w}}/\rho }}=8.6{{\left( \frac{{{r}_{o}}\sqrt{{{\tau }_{w}}/\rho }}{\nu } \right)}^{1/7}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(16)}} |} |} The velocity at any radius is related to the centerline velocity by The velocity at any radius is related to the centerline velocity by Line 155: Line 154: $\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$ $\frac{{\bar{u}}}{{{{\bar{u}}}_{c}}}={{\left( \frac{y}{{{r}_{o}}} \right)}^{1/7}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(17)}} |} |} - Substituting eq. (5.287) into eq. (5.283), a relationship between the mean velocity and centerline velocity is obtained: + Substituting eq. (17) into eq. (13), a relationship between the mean velocity and centerline velocity is obtained: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 164: Line 163: ${{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}$ ${{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(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 - [[Image:Fig5.32.png|thumb|400 px|alt=Friction factor for duct flow |Friction factor for duct flow.]] + [[Image:Fig5.32.png|thumb|600 px|alt=Friction factor for duct flow |Friction factor for duct flow.]] - + Substituting eq. (14) and (18) into eq. (16) and considering the definition of Reynolds number ${{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu$, the friction coefficient can be obtained as {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 175: Line 173: ${{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}$ ${{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}$ - |{{EquationRef|(1)}} + |{{EquationRef|(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): + 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 figure to the right): {| class="wikitable" border="0" {| class="wikitable" border="0" Line 184: Line 182: ${{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}$ ${{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(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: + Instead of the one-seventh law, eq. (15), the law of the wall, eq. (6), can be used to obtain the following correlation: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 193: Line 191: $c_{f}^{-1/2}=1.737\ln (c_{f}^{1/2}{{\operatorname{Re}}_{D}})-0.396$ $c_{f}^{-1/2}=1.737\ln (c_{f}^{1/2}{{\operatorname{Re}}_{D}})-0.396$ - |{{EquationRef|(1)}} + |{{EquationRef|(21)}} |} |} - 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 (21) is valid up to Re''D'' = 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: + For fully-developed turbulent flow in a non-circular tube, eq. (21) 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 figure to the right). The effect of the surface roughness can be measured by the roughness Reynolds number defined as: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 205: Line 204: ${{\operatorname{Re}}_{k}}=\frac{{{k}_{s}}{{u}_{\tau }}}{\nu }=\frac{{{k}_{s}}{{({{\tau }_{w}}/\rho )}^{1/2}}}{\nu }$ ${{\operatorname{Re}}_{k}}=\frac{{{k}_{s}}{{u}_{\tau }}}{\nu }=\frac{{{k}_{s}}{{({{\tau }_{w}}/\rho )}^{1/2}}}{\nu }$ - |{{EquationRef|(1)}} + |{{EquationRef|(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: + 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: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 214: Line 213: ${{c}_{f}}\simeq {{\left( 1.74\ln \frac{D}{{{k}_{s}}}+2.28 \right)}^{-2}}$ ${{c}_{f}}\simeq {{\left( 1.74\ln \frac{D}{{{k}_{s}}}+2.28 \right)}^{-2}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(23)}} |} |} + + ==References== + {{Reflist}}

## Current revision as of 09:15, 27 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 equation 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. (2) from the centerline to the wall yields,

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

which can be substituted into eq. (2) to yield

 $-2\frac{{{\tau }_{w}}}{{{r}_{0}}}=\frac{1}{r}\frac{\partial (r{{\tau }_{app}})}{\partial r}$ (4)

Integrating eq. (4) 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) 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 Algebraic Models for Eddy Diffusivity) 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 Algebraic Models for Eddy Diffusivity 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 (6) is invalid near the centerline because the slope of the velocity at the centerline obtained from eq. (6) is a finite value, not zero as it should be. In addition, eq. (6) 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 [1] 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 (9) produces finite eddy diffusivity at the centerline. Assuming ${{\varepsilon }_{M}}\gg \nu$, eq. (3) becomes

 ${{\tau }_{app}}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial y}$ (10)

Substituting eqs. (5) and (9), 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. (6) 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. (12), 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 [2][3]

 ${{u}^{+}}=8.6{{({{y}^{+}})}^{1/7}}$ (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. (17) into eq. (13), a relationship between the mean velocity and centerline velocity is obtained:

 ${{\bar{u}}_{m}}=0.817{{\bar{u}}_{c}}$ (18)
Friction factor for duct flow.

Substituting eq. (14) and (18) into eq. (16) and considering the definition of Reynolds number ${{\operatorname{Re}}_{D}}={{\bar{u}}_{m}}D/\nu$, the friction coefficient can be obtained as

 ${{c}_{f}}=0.078\operatorname{Re}_{D}^{-1/4}$ (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 figure to the right):

 ${{c}_{f}}=0.046\operatorname{Re}_{D}^{-1/5},\text{ for }3\times {{10}^{4}}<\operatorname{Re}<{{10}^{6}}$ (20)

Instead of the one-seventh law, eq. (15), the law of the wall, eq. (6), 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 (21) is valid up to ReD = 106.

For fully-developed turbulent flow in a non-circular tube, eq. (21) 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 figure to the right). 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)

## References

1. Reichardt, H., 1951, “Die Grundlagen des turbulenten Wärmeüberganges,” Arch. Gesamte Waermetech, Vol. 2, pp. 129-142.
2. Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY
3. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.