# Van Driest Model

(Difference between revisions)
 Revision as of 02:18, 21 July 2010 (view source)← Older edit Revision as of 02:26, 21 July 2010 (view source)Newer edit → Line 2: Line 2: [[Image:Fig4.36.png|thumb|400 px|alt=Comparison of Van Driest model (A+=25) with two-layer model and experimental results | Comparison of Van Driest model (A+=25) with two-layer model and experimental results.]] [[Image:Fig4.36.png|thumb|400 px|alt=Comparison of Van Driest model (A+=25) with two-layer model and experimental results | Comparison of Van Driest model (A+=25) with two-layer model and experimental results.]] - + While the above two-layer model works well in the viscous sublayer and the fully turbulent region, a discrepancy between the two-layer model and experimental results in the transition region (near $y^{+}=10.8$) is apparent. The reason for this discrepancy is that the velocities in these two regions were assumed to be dominated by different mechanisms: molecular activities in the viscous sublayer and eddy level activities in the fully turbulent region. The contribution of the eddy diffusion in the viscous sublayer was completely neglected. In fact, relatively large elements of fluid in the viscous sublayer can lift off the wall and immediately be replaced by the other fluid from the fully turbulent region. This phenomenon is referred to as “bursting” Kays, W.M., Crawford, M.E., and Weigand, B., 2005, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY. Therefore, the eddy diffusivity in the viscous sublayer is not really zero and the only point that it becomes zero is at the wall where $y^{+}=0$. - While the above two-layer model works well in the viscous sublayer and the fully turbulent region, a discrepancy between the two-layer model and experimental results in the transition region (near $y^{+}=10.8$) is apparent. The reason for this discrepancy is that the velocities in these two regions were assumed to be dominated by different mechanisms: molecular activities in the viscous sublayer and eddy level activities in the fully turbulent region. The contribution of the eddy diffusion in the viscous sublayer was completely neglected. In fact, relatively large elements of fluid in the viscous sublayer can lift off the wall and immediately be replaced by the other fluid from the fully turbulent region. This phenomenon is referred to as “bursting” (Kays, et al., 2005). Therefore, the eddy diffusivity in the viscous sublayer is not really zero and the only point that it becomes zero is at the wall where $y^{+}=0$. + To account for the effect of eddy diffusivity in the viscous sublayer, the Van Driest hypothesis states that the eddy diffusivity decreases as one nears the wall Van Driest, E.R., 1956, “On Turbulent Flow near a Wall,” J. Aero. Sci., Vol. 23, pp. 1007-1011., instead of two regions where it is “on” or “off.” Therefore, we can use the Prandtl’s mixing length model in the entire turbulent boundary layer by introducing a damping function, i.e., - To account for the effect of eddy diffusivity in the viscous sublayer, the Van Driest hypothesis states that the eddy diffusivity decreases as one nears the wall (Van Driest, 1956), instead of two regions where it is “on” or “off.” Therefore, we can use the Prandtl’s mixing length model in the entire turbulent boundary layer by introducing a damping function, i.e., + {| class="wikitable" border="0" {| class="wikitable" border="0" Line 14: Line 13: |{{EquationRef|(1)}} |{{EquationRef|(1)}} |} |} - where A is an effective sublayer thickness that must be determined empirically. Equation (4.431) can also be expressed in the following dimensionless form: + where ''A'' is an effective sublayer thickness that must be determined empirically. Equation (1) can also be expressed in the following dimensionless form: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 24: Line 23: |{{EquationRef|(2)}} |{{EquationRef|(2)}} |} |} - where $A^{+}=Au_{\tau }/\nu$ is the dimensionless sublayer thickness. The value of A+ must be determined in such a way that the calculated value of u+ outside of the sublayer matches the result obtained by the law of the wall, eq. (4.427). It is found that $A^{+}=25$ can produce the best match. Figure 4.36 shows comparison of the results obtained by the Van Driest model, the two layer model, and experiments. It can be seen that the results produced by the Van Driest model agreed very well with the experimental results in the entire turbulent boundary layer. + where $A^{+}=Au_{\tau }/\nu$ is the dimensionless sublayer thickness. The value of ''A+'' must be determined in such a way that the calculated value of ''u+'' outside of the sublayer matches the result obtained by the law of the wall. It is found that $A^{+}=25$ can produce the best match. The above figure  shows comparison of the results obtained by the Van Driest model, the two layer model, and experiments. It can be seen that the results produced by the Van Driest model agreed very well with the experimental results in the entire turbulent boundary layer. It should be pointed out that $A^{+}=25$ can only produce good results for the cases without blowing or suction and without pressure gradient. When adverse pressure gradient ($d\bar{p}/dx>0$) and wall velocity are present, the following correlation can be used: It should be pointed out that $A^{+}=25$ can only produce good results for the cases without blowing or suction and without pressure gradient. When adverse pressure gradient ($d\bar{p}/dx>0$) and wall velocity are present, the following correlation can be used: Line 45: Line 44: |{{EquationRef|(4)}} |{{EquationRef|(4)}} |} |} - are respectively the blowing velocity and the pressure gradient in the wall coordinate. The empirical constants in eq. (4.433) are $a=7.1,\text{ }b=4.25,$ and $c=10.0$. If $p^{+}>0$, $b=2.9$ and $c=0$. + are respectively the blowing velocity and the pressure gradient in the wall coordinate. The empirical constants in eq. (3) are $a=7.1,\text{ }b=4.25,$ and $c=10.0$. If $p^{+}>0$, $b=2.9$ and $c=0$.

## Revision as of 02:26, 21 July 2010

 External Turbulent Flow/Heat Transfer
Comparison of Van Driest model (A+=25) with two-layer model and experimental results.

While the above two-layer model works well in the viscous sublayer and the fully turbulent region, a discrepancy between the two-layer model and experimental results in the transition region (near y + = 10.8) is apparent. The reason for this discrepancy is that the velocities in these two regions were assumed to be dominated by different mechanisms: molecular activities in the viscous sublayer and eddy level activities in the fully turbulent region. The contribution of the eddy diffusion in the viscous sublayer was completely neglected. In fact, relatively large elements of fluid in the viscous sublayer can lift off the wall and immediately be replaced by the other fluid from the fully turbulent region. This phenomenon is referred to as “bursting” [1]. Therefore, the eddy diffusivity in the viscous sublayer is not really zero and the only point that it becomes zero is at the wall where y + = 0. To account for the effect of eddy diffusivity in the viscous sublayer, the Van Driest hypothesis states that the eddy diffusivity decreases as one nears the wall [2], instead of two regions where it is “on” or “off.” Therefore, we can use the Prandtl’s mixing length model in the entire turbulent boundary layer by introducing a damping function, i.e.,

 $\begin{matrix}{}\\\end{matrix}l=\kappa y(1-e^{y/A})$ (1)

where A is an effective sublayer thickness that must be determined empirically. Equation (1) can also be expressed in the following dimensionless form:

 $l=\kappa y(1-e^{y^{+}/A^{+}})$ (2)

where A + = Auτ / ν is the dimensionless sublayer thickness. The value of A+ must be determined in such a way that the calculated value of u+ outside of the sublayer matches the result obtained by the law of the wall. It is found that A + = 25 can produce the best match. The above figure shows comparison of the results obtained by the Van Driest model, the two layer model, and experiments. It can be seen that the results produced by the Van Driest model agreed very well with the experimental results in the entire turbulent boundary layer. It should be pointed out that A + = 25 can only produce good results for the cases without blowing or suction and without pressure gradient. When adverse pressure gradient ($d\bar{p}/dx>0$) and wall velocity are present, the following correlation can be used:

 $A^{+}=\frac{25}{a\{v_{w}^{+}+b[p^{+}/(1+cv_{w}^{+})]\}+1}$ (3)

where

 $v_{w}^{+}=\frac{v_{w}}{u_{\tau }},\text{ }p^{+}=\frac{\mu d\bar{p}/dx}{\rho ^{1/2}\tau _{w}^{3/2}}$ (4)

are respectively the blowing velocity and the pressure gradient in the wall coordinate. The empirical constants in eq. (3) are a = 7.1, b = 4.25, and c = 10.0. If p + > 0, b = 2.9 and c = 0.

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