Turbulent Boundary Layer Equations
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Yuwen Zhang (Talk  contribs) 

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  The generalized governing equations for threedimensional turbulent flow have been presented in  +  {{Turbulence Category}} 
+  The generalized governing equations for threedimensional turbulent flow have been presented in [[Governing equations for transport phenomenagoverning equations]]. For twodimensional steadystate turbulent flow, the governing equations can be simplified to:  
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  {{EquationRef(  +  {{EquationRef(2)}} 
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  {{EquationRef(  +  {{EquationRef(3)}} 
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  {{EquationRef(  +  {{EquationRef(5)}} 
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  To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (  +  To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (2) – (5). While the treatments of the timeaveraged 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 <ref name="ON1999">Oosthuizen, P.H., and Naylor, D., 1999, Introduction to Convective Heat Transfer Analysis, WCB/McGrawHill, New York. </ref><ref>Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.</ref>. Therefore, eqs. (2) – (5) can be simplified to: 
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<math>\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}</math>  <math>\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}</math>  
  
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  {{EquationRef(  +  {{EquationRef(6)}} 
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  {{EquationRef(  +  {{EquationRef(7)}} 
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  {{EquationRef(  +  {{EquationRef(8)}} 
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  where eq. (  +  where eq. (3) became <math>\partial \bar{p}/\partial y=0</math> and the partial derivative of timeaveraged pressure in eq. (2) became: 
<math>\partial \bar{p}/\partial x=d\bar{p}/dx</math>  <math>\partial \bar{p}/\partial x=d\bar{p}/dx</math>  
  , which has been reflected in eq. (  +  , 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 timeaveraged 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 timeaveraged shear stress tensor can be expressed as  
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<math>\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}</math>  <math>\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(9)}} 
}  }  
  where  +  where <math>\mathbf{\bar{\tau }}^{\text{m}}</math> is the contribution of the molecular motion, and <math>\mathbf{\bar{\tau }}^{\text{t}}</math> is caused by eddy level activity – referred to as Reynolds stress. For twodimensional flow, the shear stress can be expressed as 
  <math>\mathbf{\bar{\tau }}^{\text{m}}</math> is the contribution of the molecular motion, and <math>\mathbf{\bar{\tau }}^{\text{t}}</math> is caused by eddy level activity – referred to as Reynolds stress. For twodimensional flow, the shear stress can be expressed as  +  
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<math>\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}\overline{\rho {u}'{v}'}</math>  <math>\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}\overline{\rho {u}'{v}'}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(10)}} 
}  }  
Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The timeaveraged 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 timeaveraged heat and mass flux can be respectively expressed as  
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<math>{\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}'}</math>  <math>{\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}'}</math>  
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  {{EquationRef(  +  {{EquationRef(11)}} 
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The mass flux can be obtained by a similar way:  The mass flux can be obtained by a similar way:  
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<math>\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}}</math>  <math>\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}}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(12)}} 
}  }  
+  
+  ==References==  
+  {{Reflist}} 
Current revision as of 02:40, 27 July 2010
External Turbulent Flow/Heat Transfer

The generalized governing equations for threedimensional turbulent flow have been presented in governing equations. For twodimensional steadystate turbulent flow, the governing equations 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 timeaveraged 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 ^{[1]}^{[2]}. Therefore, eqs. (2) – (5) can be simplified to:



where eq. (3) became and the partial derivative of timeaveraged pressure in eq. (2) became: , 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 timeaveraged shear stress tensor can be expressed as

where is the contribution of the molecular motion, and is caused by eddy level activity – referred to as Reynolds stress. For twodimensional flow, the shear stress can be expressed as

Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The timeaveraged heat and mass flux can be respectively expressed as

The mass flux can be obtained by a similar way:
