SOLUTION We are to estimate \delta, \delta^{*}, \text { and } \theta based on Eqs. 1 and 2.
Assumptions 1 The flow is turbulent, but steady in the mean. 2 The plate is thin and is aligned parallel to the free stream, so that U(x) = V = constant.
Analysis First we substitute Eq. 2 into Eq. 10–80 and integrate to find momentum thickness,
\theta=\int_{0}^{\infty} \frac{u}{U}\left(1-\frac{u}{U}\right) d y=\int_{0}^{\delta}\left(\frac{y}{\delta}\right)^{1 / 7}\left(1-\left(\frac{y}{\delta}\right)^{1 / 7}\right) d y=\frac{7}{72} \delta (3)
Similarly, we find displacement thickness by integrating Eq. 10–72,
Displacement thickness: \delta^{*}=\int_{0}^{\infty}\left(1-\frac{u}{U}\right) d y (Eq. 10-72)
\delta^{*}=\int_{0}^{\infty}\left(1-\frac{u}{U}\right) d y=\int_{0}^{\delta}\left(1-\left(\frac{y}{\delta}\right)^{1 / 7}\right) d y=\frac{1}{8} \delta (4)
The Kármán integral equation reduces to Eq. 10–97 for a flat plate boundary layer. We substitute Eq. 3 into Eq. 10–97 and rearrange to get
Shape factor: H=\frac{\delta^{*}}{\theta} (Eq. 10-97)
C_{f, x}=2 \frac{d \theta}{d x}=\frac{14}{72} \frac{d \delta}{d x}
from which
\frac{d \delta}{d x}=\frac{72}{14} C_{f, x}=\frac{72}{14} 0.027\left( Re _{x}\right)^{-1 / 7} (5)
where we have substituted Eq. 1 for the local skin friction coefficient. Equation 5 can be integrated directly, yielding
Boundary layer thickness: \frac{\delta}{x} \cong \frac{0.16}{\left(\operatorname{Re}_{x}\right)^{1 / 7}} (6)
Finally, substitution of Eqs. 3 and 4 into Eq. 6 gives approximations for 𝛿* and 𝜃,
Displacement thickness: \frac{\delta^{*}}{x} \cong \frac{0.020}{\left( Re _{x}\right)^{1 / 7}} (7)
and
Momentum thickness: \frac{\theta}{x} \cong \frac{0.016}{\left(\operatorname{Re}_{x}\right)^{1 / 7}} (8)
Discussion The results agree with the expressions given in column (a) of Table 10–4 to two significant digits. Indeed, many of the expressions in Table 10–4 were generated with the help of the Kármán integral equation.
| TABLE 10–4 |
| Summary of expressions for laminar and turbulent boundary layers on a smooth flat plate aligned parallel to a uniform stream* |
| Property |
Laminar |
(a) Turbulent(†) |
(b) Turbulent(‡) |
| Boundary layer thickness |
\frac{\delta}{x}=\frac{4.91}{\sqrt{\operatorname{Re}_{x}}} |
\frac{\delta}{x}\cong\frac{0.16}{\left(\operatorname{Re}_{x}\right)^{1/7}} |
\frac{\delta}{x}\cong\frac{0.38}{\left(\operatorname{Re}_{x}\right)^{1/5}} |
| Displacement thickness |
\frac{\delta*}{x}=\frac{1.72}{\sqrt{\operatorname{Re}_{x}}} |
\frac{\delta^{*}}{x}\cong\frac{0.020}{\left(\operatorname{Re}_{x}\right)^{1/7}} |
\frac{\delta^{*}}{x}\cong\frac{0.048}{\left(\operatorname{Re}_{x}\right)^{1/5}} |
| Momentum thickness |
\frac{\theta}{x}=\frac{0.664}{\sqrt{\operatorname{Re}_{x}}} |
\frac{\theta}{x}\cong\frac{0.016}{\left(\operatorname{Re}_{x}\right)^{1/7}} |
\frac{\theta}{x}\cong\frac{0.037}{\left(\operatorname{Re}_{x}\right)^{1/5}} |
| Local skin friction coefficient |
C_{f, x}=\frac{0.664}{\sqrt{\operatorname{Re}_{x}}} |
C_{f,x}\cong\frac{0.027}{\left(\operatorname{Re}_{x}\right)^{1/7}} |
C_{f, x} \cong \frac{0.059}{\left( Re _{x}\right)^{1 /5}} |
| * Laminar values are exact and are listed to three significant digits, but turbulent values are listed to only two significant digits due to the large uncertainty affiliated with all turbulent flow fields. |
| † Obtained from one-seventh-power law. |
| ‡ Obtained from one-seventh-power law combined with empirical data for turbulent flow through smooth pipes. |
