Two-Layer Model

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[[Image:Fig4.34.png|thumb|400 px|altVelocity profiles in the turbulent boundary layer on a flat plate: (a) velocity distributions at different instances, (b) 17 superimposed profiles, and (c) time-averaged profile  (Cebeci, 2004; Reprinted with permission from Elsevier) |Figure 1: Velocity profiles in the turbulent boundary layer on a flat plate: (a) velocity distributions at different instances, (b) 17 superimposed profiles, and (c) time-averaged profile  (Cebeci, 2004; Reprinted with permission from Elsevier).]]
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[[Image:Fig4.34.png|thumb|400 px|altVelocity profiles in the turbulent boundary layer on a flat plate: (a) velocity distributions at different instances, (b) 17 superimposed profiles, and (c) time-averaged profile  (Cebeci, 2004; Reprinted with permission from Elsevier) | Velocity profiles in the turbulent boundary layer on a flat plate: (a) velocity distributions at different instances, (b) 17 superimposed profiles, and (c) time-averaged profile  (Cebeci, 2004; Reprinted with permission from Elsevier).]]
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The above mixing length theory assumes that the turbulent flow takes place within the entire boundary layer thickness. In reality, the turbulent boundary layer can be further divided into two regions (see Fig. 4.34). The first region is viscous sublayer and it occupies less than 1% of the total turbulent boundary layer thickness. The momentum and heat transfer in this region are dominated by viscous shear and heat conduction, respectively. Outside of the viscous sublayer is a full turbulent region which comprises almost the entirety of the turbulent boundary layer.  Figure 4.34(a) shows the instantaneous velocity profiles in a turbulent boundary layer on a flat plate taken at 17 instances. The 17 superimposed profiles as well as the average profile are shown in Fig. 4.34 (b) and (c), respectively. It can be clearly seen from 4.34(c) that the velocity increases sharply in the sublayer near the wall and its change in the second region is fairly insignificant. While the thickness of the turbulent boundary layer increases with increasing x, the viscous sublayer remains at a fairly constant thickness. As the thickness of the turbulent boundary layer increases, the viscous sublayer inhabits a smaller and smaller portion of the entire turbulent boundary layer. Therefore, the mixing length model based on the assumption that the entire turbulent boundary layer is full turbulent flow should be improved by considering the two-layer structure of the turbulent boundary layer.
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The above mixing length theory assumes that the turbulent flow takes place within the entire boundary layer thickness. In reality, the turbulent boundary layer can be further divided into two regions (see figure to the right). The first region is viscous sublayer and it occupies less than 1% of the total turbulent boundary layer thickness. The momentum and heat transfer in this region are dominated by viscous shear and heat conduction, respectively. Outside of the viscous sublayer is a full turbulent region which comprises almost the entirety of the turbulent boundary layer.  Figure (a) shows the instantaneous velocity profiles in a turbulent boundary layer on a flat plate taken at 17 instances. The 17 superimposed profiles as well as the average profile are shown in Fig. (b) and (c), respectively. It can be clearly seen from figure(c) that the velocity increases sharply in the sublayer near the wall and its change in the second region is fairly insignificant. While the thickness of the turbulent boundary layer increases with increasing ''x'', the viscous sublayer remains at a fairly constant thickness. As the thickness of the turbulent boundary layer increases, the viscous sublayer inhabits a smaller and smaller portion of the entire turbulent boundary layer. Therefore, the mixing length model based on the assumption that the entire turbulent boundary layer is full turbulent flow should be improved by considering the two-layer structure of the turbulent boundary layer.
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In the two-layer structure of the turbulent boundary layer, the velocity is dominated by viscous shear stress in the sublayer and by turbulent mixing in the fully turbulent region. Near the wall, the shear stress is equal to the shear stress at the wall, i.e., <math>\bar{\tau }_{yx}=\tau _{w}</math>. Thus, eq. (4.402) near the wall becomes fre
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In the two-layer structure of the turbulent boundary layer, the velocity is dominated by viscous shear stress in the sublayer and by turbulent mixing in the fully turbulent region. Near the wall, the shear stress is equal to the shear stress at the wall, i.e., <math>\bar{\tau }_{yx}=\tau _{w}</math>. Thus, eq. (4) in [[Algebraic Models for Eddy Diffusivity|Algebraic Models for Eddy Diffusivity]] near the wall becomes
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where the partial derivative <math>\partial \bar{u}/\partial y</math> has been approximated as <math>d\bar{u}/dy</math> because <math>\bar{u}(\partial \bar{u}/\partial x)\approx 0</math> in the region near the wall. For the viscous sublayer, <math>\varepsilon _{M}=0</math> and eq. (4.417) is simplified to:
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where the partial derivative <math>\partial \bar{u}/\partial y</math> has been approximated as <math>d\bar{u}/dy</math> because <math>\bar{u}(\partial \bar{u}/\partial x)\approx 0</math> in the region near the wall. For the viscous sublayer, <math>\varepsilon _{M}=0</math> and eq. (1) is simplified to:
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Equation (4.418) can be integrated from the wall to yield
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Equation (2) can be integrated from the wall to yield
<math>\int_{0}^{{\bar{u}}}{d\bar{u}}=\int_{0}^{y}{\frac{\tau _{w}}{\mu }dy}</math>
<math>\int_{0}^{{\bar{u}}}{d\bar{u}}=\int_{0}^{y}{\frac{\tau _{w}}{\mu }dy}</math>
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is shear velocity or friction velocity, eq. (4.419) becomes
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is shear velocity or friction velocity, eq. (3) becomes
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which is the velocity distribution in the viscous sublayer.  
which is the velocity distribution in the viscous sublayer.  
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For the fully turbulent region, <math>\varepsilon _{M}\gg \nu </math> and eq. (4.417) is simplified to:
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For the fully turbulent region, <math>\varepsilon _{M}\gg \nu </math> and eq. (1) is simplified to:
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Substituting eq. (4.420), eq. (4.424) can be nondimensionalized into  
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Substituting eq. (4), eq. (8) can be nondimensionalized into  
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In order to integrate eq. (4.425), we must know the thickness of the viscous sublayer. For the case without blowing or suction on the wall and zero-pressure gradient, we can choose <math>y^{+}=10.8</math> as the dimensionless thickness of the viscous sublayer. Therefore, eq. (4.425) can be integrated from <math>y^{+}=10.8</math>, i.e.,
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In order to integrate eq. (9), we must know the thickness of the viscous sublayer. For the case without blowing or suction on the wall and zero-pressure gradient, we can choose <math>y^{+}=10.8</math> as the dimensionless thickness of the viscous sublayer. Therefore, eq. (9) can be integrated from <math>y^{+}=10.8</math>, i.e.,
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where the von Kármán’s constant is taken as <math>\kappa =0.4</math>. Equation (4.427) is often referred to as the law of the wall. It can also be approximated into the following power law form:
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where the von Kármán’s constant is taken as <math>\kappa =0.4</math>. Equation (11) is often referred to as the law of the wall. It can also be approximated into the following power law form:
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which fits into eq. (4.427) at least up to <math>y^{+}=1500</math>.
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which fits into eq. (11) at least up to <math>y^{+}=1500</math>.
The two-layer model can then be summarized as
The two-layer model can then be summarized as
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[[Image:Fig4.35.png|thumb|400 px|alt=Comparison of two-layer model and experimental results |Figure 2: Comparison of two-layer model and experimental results.]]
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[[Image:Fig4.35.png|thumb|400 px|alt=Comparison of two-layer model and experimental results | Comparison of two-layer model and experimental results.]]
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Figure 2 shows the comparison between the above two-layer model and experimental results obtained using water and air. The Reynolds number shown in the figure is the momentum thickness Reynolds number defined as <math>\operatorname{Re}_{\delta _{2}}=U_{\infty }\delta _{2}/\nu </math>, where <math>\delta _{2}=0.664\sqrt{\nu x/U_{\infty }}</math> is the momentum thickness. It can be seen that the results from the two-layer model agreed with the experimental results very well except at the outer region of the turbulent boundary layer. In addition, the agreement between eq. (4.429) and the experimental results is also not very good in the region near <math>y^{+}=10.8</math>. Some researchers suggested there is a buffer region (<math>5<y^{+}<30</math>) between the sublayer and the fully turbulent region and the velocity profile in the buffer region is (Bejan and Kraus, 2003):
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Figure to the right shows the comparison between the above two-layer model and experimental results obtained using water and air. The Reynolds number shown in the figure is the momentum thickness Reynolds number defined as <math>\operatorname{Re}_{\delta _{2}}=U_{\infty }\delta _{2}/\nu </math>, where <math>\delta _{2}=0.664\sqrt{\nu x/U_{\infty }}</math> is the momentum thickness. It can be seen that the results from the two-layer model agreed with the experimental results very well except at the outer region of the turbulent boundary layer. In addition, the agreement between eq. (13) and the experimental results is also not very good in the region near <math>y^{+}=10.8</math>. Some researchers suggested there is a buffer region (<math>5<y^{+}<30</math>) between the sublayer and the fully turbulent region and the velocity profile in the buffer region is <ref name="BK2003">Bejan, A., and Kraus, A.D., 2003, Heat Transfer Handbook, John Wiley & Sons, Hoboken, NJ.</ref>:
 
 
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Revision as of 02:17, 21 July 2010

 Related Topics Catalog
External Turbulent Flow/Heat Transfer
  1. Turbulent Boundary Layer Equations
  2. Algebraic Models for Eddy Diffusivity
    1. Mixing Length Model
    2. Two-Layer Model
    3. Van Driest Model
  3. K-ε Model
  4. Momentum and Heat Transfer over a Flat Plate
Velocity profiles in the turbulent boundary layer on a flat plate: (a) velocity distributions at different instances, (b) 17 superimposed profiles, and (c) time-averaged profile (Cebeci, 2004; Reprinted with permission from Elsevier).

The above mixing length theory assumes that the turbulent flow takes place within the entire boundary layer thickness. In reality, the turbulent boundary layer can be further divided into two regions (see figure to the right). The first region is viscous sublayer and it occupies less than 1% of the total turbulent boundary layer thickness. The momentum and heat transfer in this region are dominated by viscous shear and heat conduction, respectively. Outside of the viscous sublayer is a full turbulent region which comprises almost the entirety of the turbulent boundary layer. Figure (a) shows the instantaneous velocity profiles in a turbulent boundary layer on a flat plate taken at 17 instances. The 17 superimposed profiles as well as the average profile are shown in Fig. (b) and (c), respectively. It can be clearly seen from figure(c) that the velocity increases sharply in the sublayer near the wall and its change in the second region is fairly insignificant. While the thickness of the turbulent boundary layer increases with increasing x, the viscous sublayer remains at a fairly constant thickness. As the thickness of the turbulent boundary layer increases, the viscous sublayer inhabits a smaller and smaller portion of the entire turbulent boundary layer. Therefore, the mixing length model based on the assumption that the entire turbulent boundary layer is full turbulent flow should be improved by considering the two-layer structure of the turbulent boundary layer.

In the two-layer structure of the turbulent boundary layer, the velocity is dominated by viscous shear stress in the sublayer and by turbulent mixing in the fully turbulent region. Near the wall, the shear stress is equal to the shear stress at the wall, i.e., \bar{\tau }_{yx}=\tau _{w}. Thus, eq. (4) in Algebraic Models for Eddy Diffusivity near the wall becomes

\frac{\tau _{w}}{\rho }=(\nu +\varepsilon _{M})\frac{d\bar{u}}{dy}

(1)

where the partial derivative \partial \bar{u}/\partial y has been approximated as d\bar{u}/dy because \bar{u}(\partial \bar{u}/\partial x)\approx 0 in the region near the wall. For the viscous sublayer, \varepsilon _{M}=0 and eq. (1) is simplified to:

d\bar{u}=\frac{\tau _{w}}{\mu }dy

(2)

Equation (2) can be integrated from the wall to yield

\int_{0}^{{\bar{u}}}{d\bar{u}}=\int_{0}^{y}{\frac{\tau _{w}}{\mu }dy}

i.e.,

\bar{u}=\frac{\tau _{w}}{\mu }y

(3)

Introducing the wall coordinate

y^{+}=\frac{yu_{\tau }}{\nu },\text{ }u^{+}=\frac{{\bar{u}}}{u_{\tau }}

(4)

where

u_{\tau }=\sqrt{\frac{\tau _{w}}{\rho }}=\sqrt{\frac{c_{f}}{2}}U_{\infty }

(5)

is shear velocity or friction velocity, eq. (3) becomes

\begin{matrix}{}\\\end{matrix}u^{+}=y^{+}

(6)

which is the velocity distribution in the viscous sublayer. For the fully turbulent region, \varepsilon _{M}\gg \nu and eq. (1) is simplified to:

\frac{\tau _{w}}{\rho }=\varepsilon _{M}\frac{d\bar{u}}{dy}

(7)

which can be rewritten to the following format using the Prandtl’s mixing length model

\tau _{w}=\rho \kappa ^{2}y^{2}\left( \frac{d\bar{u}}{dy} \right)^{2}

(8)

Substituting eq. (4), eq. (8) can be nondimensionalized into

\frac{du^{+}}{dy^{+}}=\frac{1}{\kappa y^{+}}

(9)

In order to integrate eq. (9), we must know the thickness of the viscous sublayer. For the case without blowing or suction on the wall and zero-pressure gradient, we can choose y + = 10.8 as the dimensionless thickness of the viscous sublayer. Therefore, eq. (9) can be integrated from y + = 10.8, i.e.,

\int_{10.8}^{u^{+}}{du^{+}}=\int_{10.8}^{y^{+}}{\frac{1}{\kappa y^{+}}dy^{+}}

(10)

which can be integrated to yield

\begin{matrix}{}\\\end{matrix}u^{+}=2.44\ln y^{+}+5.0

(11)

where the von Kármán’s constant is taken as κ = 0.4. Equation (11) is often referred to as the law of the wall. It can also be approximated into the following power law form:

\begin{matrix}{}\\\end{matrix}u^{+}=8.75(y^{+})^{1/7}

(12)

which fits into eq. (11) at least up to y + = 1500.

The two-layer model can then be summarized as

u^{+}=\left\{ \begin{matrix}
   y^{+} & y^{+}<10.8  \\
   2.44\ln y^{+}+5.0 & y^{+}>10.8  \\
\end{matrix} \right.

(13)
Comparison of two-layer model and experimental results
Comparison of two-layer model and experimental results.

Figure to the right shows the comparison between the above two-layer model and experimental results obtained using water and air. The Reynolds number shown in the figure is the momentum thickness Reynolds number defined as \operatorname{Re}_{\delta _{2}}=U_{\infty }\delta _{2}/\nu , where \delta _{2}=0.664\sqrt{\nu x/U_{\infty }} is the momentum thickness. It can be seen that the results from the two-layer model agreed with the experimental results very well except at the outer region of the turbulent boundary layer. In addition, the agreement between eq. (13) and the experimental results is also not very good in the region near y + = 10.8. Some researchers suggested there is a buffer region (5 < y + < 30) between the sublayer and the fully turbulent region and the velocity profile in the buffer region is [1]:

\begin{matrix}{}\\\end{matrix}u^{+}=5+5\ln (y^{+}/5)

(14)

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