# Turbulent momentum and heat transfer over a flat plate

(Difference between revisions)
 Revision as of 05:11, 9 April 2010 (view source)← Older edit Current revision as of 06:06, 27 July 2010 (view source) (4 intermediate revisions not shown) Line 1: Line 1: {{Turbulence Category}} {{Turbulence Category}} - To obtain the boundary layer thickness for turbulent flow over a flat plate, von Kármán’s momentum integral can be employed. The integral momentum equation (4.167) that was derived for the case of laminar flow is still valid except that the instantaneous velocity should be replaced by the time-averaged velocity. For the flat plate without bellowing or suction ($v_{w}=0$) and without pressure gradient, eq. (4.167) is simplified to + To obtain the boundary layer thickness for turbulent flow over a flat plate, von Kármán’s momentum integral can be employed. The integral momentum equation that was derived for the case of laminar flow is still valid except that the instantaneous velocity should be replaced by the time-averaged velocity. For the flat plate without bellowing or suction ($v_{w}=0$) and without pressure gradient, the equation is simplified to - {| class="wikitable" border="0" + {| class="wikitable" border="0" - |- + |- - | width="100%" |
+ | width="100%" |
- $\frac{\tau _{w}}{\rho }=\frac{d}{dx}\int_{0}^{\delta }{\bar{u}\left( U_{\infty }-\bar{u} \right)dy}$ + $\frac{\tau _{w}}{\rho }=\frac{d}{dx}\int_{0}^{\delta }{\bar{u}\left( U_{\infty }-\bar{u} \right)dy}$ -
+
- |{{EquationRef|(1)}} + |{{EquationRef|(1)}} - |} + |} - While the velocity profile in the laminar boundary layer can be adequately described by a polynomial function, the velocity profile in a turbulent flow is too complicated to be described by a single function for the entire boundary layer. Equation (4.428) states that the velocity profile in the turbulent boundary layer can be approximated as + - $u^{+}=8.75(y^{+})^{1/7}$ + While the velocity profile in the laminar boundary layer can be adequately described by a polynomial function, the velocity profile in a turbulent flow is too complicated to be described by a single function for the entire boundary layer. Equation (12) in [[Two-Layer Model]] states that the velocity profile in the turbulent boundary layer can be approximated as $u^{+}=8.75(y^{+})^{1/7}$, which can be rewritten into the following dimensionless form: - , which can be rewritten into the following dimensionless form: + {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- Line 18: Line 17: |{{EquationRef|(2)}} |{{EquationRef|(2)}} |} |} - Although eq. (4.449) can reasonably represent velocity profile in most parts of the boundary layer, the velocity gradients at both $y=0\text{ and }\delta$ are incorrect: $(\partial \bar{u}/\partial y)_{y=0}\to \infty$ and $(\partial \bar{u}/\partial y)_{y=\delta }\ne 0$. To get the shear stress at the wall, let us substitute eq. (4.420) and (4.421) into eq. (4.428): + Although eq. (2) can reasonably represent velocity profile in most parts of the boundary layer, the velocity gradients at both $y=0\text{ and }\delta$ are incorrect: $(\partial \bar{u}/\partial y)_{y=0}\to \infty$ and $(\partial \bar{u}/\partial y)_{y=\delta }\ne 0$. To get the shear stress at the wall, let us substitute eq. (4) and (5) in [[Two-Layer Model]] into eq. (12) in [[Two-Layer Model]]: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 38: Line 37: |{{EquationRef|(4)}} |{{EquationRef|(4)}} |} |} - which is referred to as the Blasius relation and which is valid for $\operatorname{Re}_{x}<10^{7}$. Substituting eqs. (4.449) and (4.451) into eq. (4.448), one obtains + which is referred to as the Blasius relation and which is valid for $\operatorname{Re}_{x}<10^{7}$. Substituting eqs. (2) and (4) into eq. (1), one obtains {| class="wikitable" border="0" {| class="wikitable" border="0" Line 48: Line 47: |{{EquationRef|(5)}} |{{EquationRef|(5)}} |} |} - Performing the integration and differentiation on the right-hand side of eq. (4.452) yields the following differential equation for the boundary layer thickness: + Performing the integration and differentiation on the right-hand side of eq. (5) yields the following differential equation for the boundary layer thickness: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 68: Line 67: |{{EquationRef|(7)}} |{{EquationRef|(7)}} |} |} - where C is the unspecified integration constant. If we assume that the turbulent boundary layer starts from the edge of the flat plate – which is not a good assumption (see Fig. 4.3) – the integration constant C becomes zero, and eq. (4.454) becomes + where ''C'' is the unspecified integration constant. If we assume that the turbulent boundary layer starts from the edge of the flat plate – which is not a good assumption – the integration constant ''C'' becomes zero, and eq. (7) becomes - + [[Image:Fig4.37.png|thumb|400 px|alt=Comparison of velocity profiles in laminar and turbulent boundary layers | Comparison of velocity profiles in laminar and turbulent boundary layers.]] {| class="wikitable" border="0" {| class="wikitable" border="0" |- |- | width="100%" |
| width="100%" |
$\frac{\delta }{x}=\frac{0.376}{\operatorname{Re}_{x}^{1/5}}$ $\frac{\delta }{x}=\frac{0.376}{\operatorname{Re}_{x}^{1/5}}$ -
|{{EquationRef|(8)}} |{{EquationRef|(8)}} Line 88: Line 86: |{{EquationRef|(9)}} |{{EquationRef|(9)}} |} |} - [[Image:Fig4.37.png|thumb|400 px|alt=Comparison of velocity profiles in laminar and turbulent boundary layers |Figure 1: Comparison of velocity profiles in laminar and turbulent boundary layers.]] - It should be pointed out that eqs. (4.455) and (4.456) are valid for the case that turbulent boundary layer starts from the leading edge of the flat plate and $\operatorname{Re}_{x}<10^{7}$  – beyond which the Blasius relation becomes invalid. + It should be pointed out that eqs. (8) and (9) are valid for the case that turbulent boundary layer starts from the leading edge of the flat plate and $\operatorname{Re}_{x}<10^{7}$  – beyond which the Blasius relation becomes invalid. - Figure 1 shows a comparison of the velocity profiles of laminar and turbulent boundary layers at $\operatorname{Re}_{x}=5\times 10^{6}$ (Welty et al., 2000). It can be seen that the turbulent boundary layer is much thicker than the laminar boundary layer at the same Reynolds number. The mean velocity in the turbulent boundary layer is much higher than that of the laminar boundary layer. The large mean velocity of the turbulent boundary layer allows for a much stronger momentum, heat and mass transfer. Another advantage of the turbulent boundary layer is that it can resist separation better than a laminar boundary layer. Due to its strong ability on momentum, heat and mass transfer and resisting separation, a turbulent boundary layer is desirable in many engineering applications. + + Figure to the right shows a comparison of the velocity profiles of laminar and turbulent boundary layers at $\operatorname{Re}_{x}=5\times 10^{6}$ Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G., 2000, Fundamentals of Momentum, Heat, and Mass Transfer, 4th ed., Wiley, New York.Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.. It can be seen that the turbulent boundary layer is much thicker than the laminar boundary layer at the same Reynolds number. The mean velocity in the turbulent boundary layer is much higher than that of the laminar boundary layer. The large mean velocity of the turbulent boundary layer allows for a much stronger momentum, heat and mass transfer. Another advantage of the turbulent boundary layer is that it can resist separation better than a laminar boundary layer. Due to its strong ability on momentum, heat and mass transfer and resisting separation, a turbulent boundary layer is desirable in many engineering applications. + + ==References== + {{Reflist}}

## Current revision as of 06:06, 27 July 2010

To obtain the boundary layer thickness for turbulent flow over a flat plate, von Kármán’s momentum integral can be employed. The integral momentum equation that was derived for the case of laminar flow is still valid except that the instantaneous velocity should be replaced by the time-averaged velocity. For the flat plate without bellowing or suction (vw = 0) and without pressure gradient, the equation is simplified to $\frac{\tau _{w}}{\rho }=\frac{d}{dx}\int_{0}^{\delta }{\bar{u}\left( U_{\infty }-\bar{u} \right)dy}$ (1)

While the velocity profile in the laminar boundary layer can be adequately described by a polynomial function, the velocity profile in a turbulent flow is too complicated to be described by a single function for the entire boundary layer. Equation (12) in Two-Layer Model states that the velocity profile in the turbulent boundary layer can be approximated as u + = 8.75(y + )1 / 7, which can be rewritten into the following dimensionless form: $\frac{{\bar{u}}}{U_{\infty }}=\left( \frac{y}{\delta } \right)^{1/7}$ (2)

Although eq. (2) can reasonably represent velocity profile in most parts of the boundary layer, the velocity gradients at both y = 0 and δ are incorrect: $(\partial \bar{u}/\partial y)_{y=0}\to \infty$ and $(\partial \bar{u}/\partial y)_{y=\delta }\ne 0$. To get the shear stress at the wall, let us substitute eq. (4) and (5) in Two-Layer Model into eq. (12) in Two-Layer Model: $\frac{U_{\infty }}{\sqrt{\tau _{w}/\rho }}=8.75\left( \frac{\delta \sqrt{\tau _{w}/\rho }}{\nu } \right)^{1/7}$ (3)

Solving for τw yields $\tau _{w}=0.0225\rho U_{\infty }^{2}\left( \frac{\delta U_{\infty }}{\nu } \right)^{-1/4}$ (4)

which is referred to as the Blasius relation and which is valid for $\operatorname{Re}_{x}<10^{7}$. Substituting eqs. (2) and (4) into eq. (1), one obtains $0.0225U_{\infty }^{2}\left( \frac{\delta U_{\infty }}{\nu } \right)^{-1/4}=\frac{d}{dx}\int_{0}^{\delta }{U_{\infty }^{2}\left[ \left( \frac{y}{\delta } \right)^{1/7}-\left( \frac{y}{\delta } \right)^{2/7} \right]dy}$ (5)

Performing the integration and differentiation on the right-hand side of eq. (5) yields the following differential equation for the boundary layer thickness: $0.0225U_{\infty }^{2}\left( \frac{\delta U_{\infty }}{\nu } \right)^{-1/4}=\frac{7}{72}\frac{d\delta }{dx}$ (6)

which can be integrated to obtain: $\left( \frac{\nu }{U_{\infty }} \right)^{1/4}x=3.45\delta ^{5/4}+C$ (7)

where C is the unspecified integration constant. If we assume that the turbulent boundary layer starts from the edge of the flat plate – which is not a good assumption – the integration constant C becomes zero, and eq. (7) becomes $\frac{\delta }{x}=\frac{0.376}{\operatorname{Re}_{x}^{1/5}}$ (8)

The local friction coefficient can be found as $c_{f}=\frac{\tau _{w}}{\rho U_{\infty }^{2}/2}=\frac{0.0576}{\operatorname{Re}_{x}^{1/5}}$ (9)

It should be pointed out that eqs. (8) and (9) are valid for the case that turbulent boundary layer starts from the leading edge of the flat plate and $\operatorname{Re}_{x}<10^{7}$ – beyond which the Blasius relation becomes invalid.

Figure to the right shows a comparison of the velocity profiles of laminar and turbulent boundary layers at $\operatorname{Re}_{x}=5\times 10^{6}$ . It can be seen that the turbulent boundary layer is much thicker than the laminar boundary layer at the same Reynolds number. The mean velocity in the turbulent boundary layer is much higher than that of the laminar boundary layer. The large mean velocity of the turbulent boundary layer allows for a much stronger momentum, heat and mass transfer. Another advantage of the turbulent boundary layer is that it can resist separation better than a laminar boundary layer. Due to its strong ability on momentum, heat and mass transfer and resisting separation, a turbulent boundary layer is desirable in many engineering applications.

## References

1. Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G., 2000, Fundamentals of Momentum, Heat, and Mass Transfer, 4th ed., Wiley, New York.
2. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.