Turbulent momentum and heat transfer over a flat plate
From ThermalFluidsPedia
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{{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  +  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 timeaveraged velocity. For the flat plate without bellowing or suction (<math>v_{w}=0</math>) and without pressure gradient, the equation is simplified to 
  +  { class="wikitable" border="0"  
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  +   width="100%" <center>  
  +  <math>\frac{\tau _{w}}{\rho }=\frac{d}{dx}\int_{0}^{\delta }{\bar{u}\left( U_{\infty }\bar{u} \right)dy}</math>  
  +  </center>  
  +  {{EquationRef(1)}}  
  +  }  
  +  
  <math>u^{+}=8.75(y^{+})^{1/7}</math>  +  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 [[TwoLayer Model]] states that the velocity profile in the turbulent boundary layer can be approximated as <math>u^{+}=8.75(y^{+})^{1/7}</math>, which can be rewritten into the following dimensionless form: 
  , which can be rewritten into the following dimensionless form:  +  
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{{EquationRef(2)}}  {{EquationRef(2)}}  
}  }  
  Although eq. (  +  Although eq. (2) can reasonably represent velocity profile in most parts of the boundary layer, the velocity gradients at both <math>y=0\text{ and }\delta </math> are incorrect: <math>(\partial \bar{u}/\partial y)_{y=0}\to \infty </math> and <math>(\partial \bar{u}/\partial y)_{y=\delta }\ne 0</math>. To get the shear stress at the wall, let us substitute eq. (4) and (5) in [[TwoLayer Model]] into eq. (12) in [[TwoLayer Model]]: 
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{{EquationRef(4)}}  {{EquationRef(4)}}  
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  which is referred to as the Blasius relation and which is valid for <math>\operatorname{Re}_{x}<10^{7}</math>. Substituting eqs. (  +  which is referred to as the Blasius relation and which is valid for <math>\operatorname{Re}_{x}<10^{7}</math>. Substituting eqs. (2) and (4) into eq. (1), one obtains 
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{{EquationRef(5)}}  {{EquationRef(5)}}  
}  }  
  Performing the integration and differentiation on the righthand side of eq. (  +  Performing the integration and differentiation on the righthand side of eq. (5) yields the following differential equation for the boundary layer thickness: 
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{{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  +  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 
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{{EquationRef(9)}}  {{EquationRef(9)}}  
}  }  
  [[Image:Fig4.37.pngthumb400 pxalt=Comparison of velocity profiles in laminar and turbulent boundary layers   +  [[Image:Fig4.37.pngthumb400 pxalt=Comparison of velocity profiles in laminar and turbulent boundary layers  Comparison of velocity profiles in laminar and turbulent boundary layers.]] 
+  
+  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 <math>\operatorname{Re}_{x}<10^{7}</math> – 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 <math>\operatorname{Re}_{x}=5\times 10^{6}</math> <ref name="W2000">Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G., 2000, Fundamentals of Momentum, Heat, and Mass Transfer, 4th ed., Wiley, New York.</ref>. 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.  
  + 
Revision as of 03:08, 21 July 2010
External Turbulent Flow/Heat Transfer 
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 timeaveraged velocity. For the flat plate without bellowing or suction (v_{w} = 0) and without pressure gradient, the equation is simplified to

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 TwoLayer 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:

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: and . To get the shear stress at the wall, let us substitute eq. (4) and (5) in TwoLayer Model into eq. (12) in TwoLayer Model:

Solving for τ_{w} yields

which is referred to as the Blasius relation and which is valid for . Substituting eqs. (2) and (4) into eq. (1), one obtains

Performing the integration and differentiation on the righthand side of eq. (5) yields the following differential equation for the boundary layer thickness:

which can be integrated to obtain:

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

The local friction coefficient can be found as

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 – 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 ^{[1]}. 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.
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