# Turbulent Boundary Layer Equations

(Difference between revisions)
 Revision as of 03:43, 9 April 2010 (view source)← Older edit Current revision as of 02:40, 27 July 2010 (view source) (5 intermediate revisions not shown) Line 1: Line 1: - The generalized governing equations for three-dimensional turbulent flow have been presented in Chapter 2 (see Section 2.5). For two-dimensional steady-state turbulent flow, the governing equations can be simplified to: + {{Turbulence Category}} + The generalized governing equations for three-dimensional turbulent flow have been presented in [[Governing equations for transport phenomena|governing equations]]. For two-dimensional steady-state turbulent flow, the governing equations can be simplified to: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 16: Line 17: - |{{EquationRef|(1)}} + |{{EquationRef|(2)}} |} |} Line 25: Line 26: - |{{EquationRef|(1)}} + |{{EquationRef|(3)}} |} |} Line 34: Line 35: - |{{EquationRef|(1)}} + |{{EquationRef|(4)}} |} |} Line 43: Line 44: - |{{EquationRef|(1)}} + |{{EquationRef|(5)}} |} |} - To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (4.388) – (4.391). While the treatments of the time-averaged quantities are similar to the cases of laminar flow, special attention must be paid to the time averaging of the products of the fluctuations. It can be shown through a scale analysis that the first terms in the last parentheses on the right hand side of eqs. (4.388) – (4.391) are negligible compared to the second terms in the parentheses (Oosthuizen and Naylor, 1999). Therefore, eqs. (4.388) – (4.391) can be simplified to: + To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (2) – (5). While the treatments of the time-averaged quantities are similar to the cases of laminar flow, special attention must be paid to the time averaging of the products of the fluctuations. It can be shown through a scale analysis that the first terms in the last parentheses on the right hand side of eqs. (2) – (5) are negligible compared to the second terms in the parentheses Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.. Therefore, eqs. (2) – (5) can be simplified to: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 51: Line 52: | width="100%" |
| width="100%" |
$\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\nu \frac{\partial ^{2}\bar{u}}{\partial y^{2}}-\frac{\partial \overline{{v}'{u}'}}{\partial y}$ $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\nu \frac{\partial ^{2}\bar{u}}{\partial y^{2}}-\frac{\partial \overline{{v}'{u}'}}{\partial y}$ -
- |{{EquationRef|(1)}} + |{{EquationRef|(6)}} |} |} Line 62: Line 62: - |{{EquationRef|(1)}} + |{{EquationRef|(7)}} |} |} Line 71: Line 71: - |{{EquationRef|(1)}} + |{{EquationRef|(8)}} |} |} - where eq. (4.389) became $\partial \bar{p}/\partial y=0$ and the partial derivative of time-averaged pressure in eq. (4.388) became: + where eq. (3) became $\partial \bar{p}/\partial y=0$ and the partial derivative of time-averaged pressure in eq. (2) became: $\partial \bar{p}/\partial x=d\bar{p}/dx$ $\partial \bar{p}/\partial x=d\bar{p}/dx$ - , which has been reflected in eq. (4.392). + , which has been reflected in eq. (6). + When molecules or eddies in a turbulent flow cross a control surface, they will carry momentum with them. Thus, the shear stress in a turbulent flow can be caused by both molecular and eddy level activities. While the molecular level activity is the only mechanism of shear stress, transport of momentum by eddies can only be found in turbulent flow. The time-averaged shear stress tensor can be expressed as When molecules or eddies in a turbulent flow cross a control surface, they will carry momentum with them. Thus, the shear stress in a turbulent flow can be caused by both molecular and eddy level activities. While the molecular level activity is the only mechanism of shear stress, transport of momentum by eddies can only be found in turbulent flow. The time-averaged shear stress tensor can be expressed as Line 83: Line 84: $\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}$ $\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(9)}} |} |} - where + where $\mathbf{\bar{\tau }}^{\text{m}}$ is the contribution of the molecular motion, and $\mathbf{\bar{\tau }}^{\text{t}}$ is caused by eddy level activity – referred to as Reynolds stress. For two-dimensional flow, the shear stress can be expressed as - $\mathbf{\bar{\tau }}^{\text{m}}$ is the contribution of the molecular motion, and $\mathbf{\bar{\tau }}^{\text{t}}$ is caused by eddy level activity – referred to as Reynolds stress. For two-dimensional flow, the shear stress can be expressed as + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 93: Line 93: $\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'}$ $\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'}$ - |{{EquationRef|(1)}} + |{{EquationRef|(10)}} |} |} Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The time-averaged heat and mass flux can be respectively expressed as Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The time-averaged heat and mass flux can be respectively expressed as Line 102: Line 102: ${\bar{q}}''_{y}=\mathop{{{\bar{q}}''}}_{y}^{m}+\mathop{{{\bar{q}}''}}_{y}^{t}=-k\frac{\partial \bar{T}}{\partial y}+\rho c_{p}\overline{{v}'{T}'}$ ${\bar{q}}''_{y}=\mathop{{{\bar{q}}''}}_{y}^{m}+\mathop{{{\bar{q}}''}}_{y}^{t}=-k\frac{\partial \bar{T}}{\partial y}+\rho c_{p}\overline{{v}'{T}'}$ - |{{EquationRef|(1)}} + |{{EquationRef|(11)}} |} |} The mass flux can be obtained by a similar way: The mass flux can be obtained by a similar way: Line 111: Line 111: $\bar{{\dot{m}}''}_{y}=\bar{{\dot{m}}''}_{y}^{m}+\bar{{\dot{m}}''}_{y}^{t}=-\rho D\frac{\partial \bar{\omega }_{1}}{\partial y}+\rho \overline{{v}'{\omega }'_{1}}$ $\bar{{\dot{m}}''}_{y}=\bar{{\dot{m}}''}_{y}^{m}+\bar{{\dot{m}}''}_{y}^{t}=-\rho D\frac{\partial \bar{\omega }_{1}}{\partial y}+\rho \overline{{v}'{\omega }'_{1}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(12)}} |} |} + + ==References== + {{Reflist}}

## Current revision as of 02:40, 27 July 2010

 External Turbulent Flow/Heat Transfer Turbulent Boundary Layer Equations Algebraic Models for Eddy Diffusivity K-ε Model Momentum and Heat Transfer over a Flat Plate

The generalized governing equations for three-dimensional turbulent flow have been presented in governing equations. For two-dimensional steady-state turbulent flow, the governing equations can be simplified to: $\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}=0$ (1) $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x}+\nu \left( \frac{\partial ^{2}\bar{u}}{\partial x^{2}}+\frac{\partial ^{2}\bar{u}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'^{2}}}{\partial x}+\frac{\partial \overline{{v}'{u}'}}{\partial y} \right)$ (2) $\bar{u}\frac{\partial \bar{v}}{\partial x}+\bar{v}\frac{\partial \bar{v}}{\partial y}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial y}+\nu \left( \frac{\partial ^{2}\bar{v}}{\partial x^{2}}+\frac{\partial ^{2}\bar{v}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{v}'^{2}}}{\partial x}+\frac{\partial \overline{{u}'{v}'}}{\partial y} \right)$ (3) $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial y}=\alpha \left( \frac{\partial ^{2}\bar{T}}{\partial x^{2}}+\frac{\partial ^{2}\bar{T}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'{T}'}}{\partial x}+\frac{\partial \overline{{v}'{T}'}}{\partial y} \right)$ (4) $\bar{u}\frac{\partial \bar{\omega }}{\partial x}+\bar{v}\frac{\partial \bar{\omega }}{\partial y}=D\left( \frac{\partial ^{2}\bar{\omega }}{\partial x^{2}}+\frac{\partial ^{2}\bar{\omega }}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'{\omega }'}}{\partial x}+\frac{\partial \overline{{v}'{\omega }'}}{\partial y} \right)$ (5)

To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (2) – (5). While the treatments of the time-averaged quantities are similar to the cases of laminar flow, special attention must be paid to the time averaging of the products of the fluctuations. It can be shown through a scale analysis that the first terms in the last parentheses on the right hand side of eqs. (2) – (5) are negligible compared to the second terms in the parentheses . Therefore, eqs. (2) – (5) can be simplified to: $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\nu \frac{\partial ^{2}\bar{u}}{\partial y^{2}}-\frac{\partial \overline{{v}'{u}'}}{\partial y}$ (6) $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial y}=\alpha \frac{\partial ^{2}\bar{T}}{\partial y^{2}}-\frac{\partial \overline{{v}'{T}'}}{\partial y}$ (7) $\bar{u}\frac{\partial \bar{\omega }}{\partial x}+\bar{v}\frac{\partial \bar{\omega }}{\partial y}=D\frac{\partial ^{2}\bar{\omega }}{\partial y^{2}}-\frac{\partial \overline{{v}'{\omega }'}}{\partial y}$ (8)

where eq. (3) became $\partial \bar{p}/\partial y=0$ and the partial derivative of time-averaged pressure in eq. (2) became: $\partial \bar{p}/\partial x=d\bar{p}/dx$ , which has been reflected in eq. (6).

When molecules or eddies in a turbulent flow cross a control surface, they will carry momentum with them. Thus, the shear stress in a turbulent flow can be caused by both molecular and eddy level activities. While the molecular level activity is the only mechanism of shear stress, transport of momentum by eddies can only be found in turbulent flow. The time-averaged shear stress tensor can be expressed as $\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}$ (9)

where $\mathbf{\bar{\tau }}^{\text{m}}$ is the contribution of the molecular motion, and $\mathbf{\bar{\tau }}^{\text{t}}$ is caused by eddy level activity – referred to as Reynolds stress. For two-dimensional flow, the shear stress can be expressed as $\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'}$ (10)

Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The time-averaged heat and mass flux can be respectively expressed as ${\bar{q}}''_{y}=\mathop{{{\bar{q}}''}}_{y}^{m}+\mathop{{{\bar{q}}''}}_{y}^{t}=-k\frac{\partial \bar{T}}{\partial y}+\rho c_{p}\overline{{v}'{T}'}$ (11)

The mass flux can be obtained by a similar way: $\bar{{\dot{m}}''}_{y}=\bar{{\dot{m}}''}_{y}^{m}+\bar{{\dot{m}}''}_{y}^{t}=-\rho D\frac{\partial \bar{\omega }_{1}}{\partial y}+\rho \overline{{v}'{\omega }'_{1}}$ (12)

## References

1. Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGraw-Hill, New York.
2. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.