Chapter 10, Problem 10.66P

To determine

(a)

If the water will rise above the bump.

Answer to Problem 10.66P

The water level increases over the bump since it is super critical flow.

Explanation of Solution

Given:

Velocity V₁ = 5.5 m/s

y₁ = 90cm

Concept Used:

$\begin{array}{l}Po\mathrm{int}1isattheflowbeforethebump.\\ Ponit2isabovethebump\\ Fr=\frac{V}{V_c}\end{array}$

Calculation:

$\begin{array}{l}Po\mathrm{int}1isattheflowbeforethebump.\\ Ponit2isabovethebump\\ F{r}_1=\frac{V_1}{V_{1c}}\Rightarrow \frac{5.5}{\sqrt{g{y}_1}}\Rightarrow \frac{5.5}{\sqrt{9.81*0.9}}\\ F{r}_1=1.85\\ SIncetheFroudenumberisgreaterthan1.Theflowis\sup ercritical.\\ Therfore\text{},thewaterincreasesoverthebump\end{array}$

Conclusion:

The water level increases over the bump.

To determine

(b)

The Froude number over the bump if bump height is 30 cm.

Answer to Problem 10.66P

The Froude number over the bump is 1.388

Explanation of Solution

Given:

Velocity V₁ = 5.5 m/s

y₁ = 90 cm

Bump height = 30cm

Concept Used:

$\begin{array}{l}Po\mathrm{int}1isattheflowbeforethebump.\\ Ponit2isabovethebump\\ y_2=initialheight+bumpheight\\ Fr=\frac{V}{V_c}\end{array}$

Calculation:

$\begin{array}{l}Po\mathrm{int}1isattheflowbeforethebump.\\ Ponit2isabovethebump\\ Foruniformflow,\\ q=y_1{V}_1=y_2{V}_2\\ E_2=\frac{V_1{}^2}{2g}+y_1-\Delta h\\ E_2=\frac{{5.5}^2}{2*9.81}+0.9-0.3\\ E_2=2.141m\\ y_2{}^3-E_2{y}_2{}^2+\frac{q^2}{2g}=0\\ y_2{}^3-2.141y_2{}^2+\frac{{(0.9*5.5)}^2}{2*9.81}=0\\ Onsolvingfor{y}_2,weget\\ y_2=1.09m\\ \\ F{r}_{bump}=\frac{V_2}{V_{2c}}\Rightarrow \frac{y_1{V}_1}{y_2\sqrt{g{y}_2}}\Rightarrow \frac{0.9*5.5}{1.09\sqrt{9.81*1.09}}\\ F{r}_{bump}=1.388\end{array}$

Conclusion:

The Froude number over the bump is 1.388.

To determine

(c)

The bump height that will cause critical flow.

Explanation of Solution

Given:

Velocity V₁ = 5.5 m/s

y₁ = 90 cm

Concept Used:

$\begin{array}{l}\Delta h=E_1-E_2\\ \Delta histheheightofthebump\\ E=\frac{V^2}{2g}+y\\ E_c=\frac{3}{2}{y}_c\\ y_c={\left(\frac{q^2}{g}\right)}^{1/3}\end{array}$

Calculation:

$\begin{array}{l}\Delta h=E_1-E_2\\ \Delta histheheightofthebump\\ E_1=\frac{V_1{}^2}{2g}+y_1\\ E_1=\frac{{5.5}^2}{2*9.81}+0.9\\ E_1=2.441\\ Forenergywithcriticalflow,\\ E_2=\frac{3}{2}{y}_c\\ E_2=\frac{3}{2}{\left(\frac{q^2}{g}\right)}^{1/3}\\ E_2=\frac{3}{2}{\left(\frac{{(5.5*0.9)}^2}{9.81}\right)}^{1/3}\\ E_2=2.02m\\ \\ \Delta h=E_1-E_2\\ \Delta h=2.411-2.02\\ \Delta h=0.42m\end{array}$

Conclusion:

The Froude number over the bump is 1.37.