Chapter 13, Problem 100P

To determine

The change in water level.

Whether the flow over the bump is sub or supercritical.

Answer to Problem 100P

The rise in water level over the bump is $0.322\text{m}$.

The flow over the bump is supercritical.

Explanation of Solution

Given information:

Velocity of water flow is $10\text{m}/\text{s}$, the flow depth is $0.65\text{m}$, the bump height is $30\text{cm}$.

Write the expression for the Froude number.

  $F_{r_1}=\frac{V_1}{\sqrt{g{y}_1}}$  ...... (I)

Here, the velocity of fluid is $V_1$ the acceleration due to gravity is $g$ and the fluid depth before the bump is $y_1$.

The figure below shows the rise over the bump.

  

  Figure-(1)

Write the expression for the critical flow depth.

  $y_c={\left(\frac{V_1^2{y}_1^2}{g}\right)}^{1/3}$....... (II)

Write the expression for the specific energy before the bump.

  $E_{s1}=y_1+\frac{V^2}{2g}$  ...... (III)

Write the expression for the specific energy over the bump.

  $E_{s2}=E_{s1}-\Delta z_b$  ...... (IV)

Here, the height of the bump is $\Delta z_b$.

Write the expression for the critical specific energy.

  $E_c=\frac{3}{2}{y}_c$  ...... (V)

Here the critical flow depth is $y_c$

Write the expression to calculate the flow over the bump.

  $y_2^3-\left(E_{s1}-\Delta z_b\right)y_2^2+\frac{V_1^2}{2g}{y}_1^2=0$  ...... (VI)

Here, the specific energy before the bump is $E_{s1}$, the specific energy over the bump is $E_{s2}$, the flow depth before the bump is $y_1$ and the flow depth over the bump is $y_2$.

Write the expression for the rise over the bump.

  $B_s=y_2-y_1+\Delta z_b$  ...... (VII)

Calculation:

Substitute $10\text{m}/\text{s}$ for $V_1$

  $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$ and $0.65\text{m}$ for $y_1$ in Equation (I).

  $\begin{array}{c}{F}_{r1}=\frac{10\text{m}/\text{s}}{\sqrt{\left(9.81\text{m}/{\text{s}}^{\text{2}}\right)\times \left(0.65\text{m}\right)}}\\ =\frac{10\text{m}/\text{s}}{\sqrt{\left(6.3765{\text{m}}^2/{\text{s}}^{\text{2}}\right)}}\\ =\frac{10\text{m}/\text{s}}{2.525\text{m}/\text{s}}\\ =3.960\end{array}$

The Froude number is greater than $1$ hence the flow is supercritical before the bump.

Substitute $10\text{m}/\text{s}$ for $V_1$, $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$

  $0.65\text{m}$ for $y_1$ in Equation (II).

  $\begin{array}{c}{y}_c={\left(\frac{{\left(10\text{m}/\text{s}\right)}^2\times {\left(0.65\text{m}\right)}^2}{9.81\text{m}/{\text{s}}^{\text{2}}}\right)}^{1/3}\\ ={\left(\frac{\left(100{\text{m}}^{\text{2}}/{\text{s}}^{\text{2}}\right)\times \left(0.4225{\text{m}}^2\right)}{9.81\text{m}/{\text{s}}^{\text{2}}}\right)}^{1/3}\\ ={\left(\frac{\left(42.25{\text{m}}^4/{\text{s}}^{\text{2}}\right)}{9.81\text{m}/{\text{s}}^{\text{2}}}\right)}^{1/3}\\ ={\left(4.3068{\text{m}}^{\text{3}}\right)}^{1/3}\end{array}$

  $y_c=1.62699\text{m}$

Substitute $10{\text{m}}^{\text{3}}/\text{s}$ for $V$, $0.65\text{m}$ for $y_1$ and $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$ in Equation (III).

  $\begin{array}{c}{E}_{s1}=0.65\text{m}+\frac{{\left(10\text{m}/\text{s}\right)}^2}{2\left(9.81\text{m}/{\text{s}}^{\text{2}}\right)}\\ =0.65\text{m}+5.096\text{m}\\ =\text{5}\text{.74}\text{m}\end{array}$

Substitute $5.74\text{m}$ for $E_{s1}$ and $0.3\text{m}$ for $\Delta z_b$ in Equation (IV).

  $\begin{array}{c}{E}_{s2}=5.74\text{m}-0.3\text{m}\\ =\text{5}\text{.44}\text{m}\end{array}$

Substitute $1.62994\text{m}$ for $y_c$ in Equation (V).

  $\begin{array}{c}{E}_c=\frac{3}{2}\left(1.62994\text{m}\right)\\ =2.44\text{m}\end{array}$

Substitute $5.74\text{m}$ for $E_{s1}$, $0.3\text{m}$ for $\Delta z_b$, $10\text{m}/\text{s}$ for $V_1$

  $0.65\text{m}$ for $y_1$ and $9.81\text{m}/{\text{s}}^{\text{2}}$ in Equation (VI).

  $\begin{array}{l}{y}_2^3-\left(5.74\text{m}-0.3\text{m}\right)y_2^2+\frac{{\left(10\text{m}/\text{s}\right)}^2}{2\left(9.81\text{m}/{\text{s}}^{\text{2}}\right)}{\left(10\text{m}\right)}^2=0\\ y_2^3-5.44y_2^2-2.153=0\end{array}$

After solving by iteration method three roots of the $y_2$ are $5.372\text{m}$, $-0.597\text{m}$ and $0.672\text{m}$.

A physical meaningful root of the equation is $0.672\text{m}$.

Substitute $0.3\text{m}$ for $\Delta z_b$, $0.65\text{m}$ for $y_1$, and $0.672\text{m}$ for $y_2$ in Equation (VII).

  $\begin{array}{c}{B}_s=0.672\text{m}-0.65\text{m}+0.30\text{m}\\ =\text{0}\text{.022}\text{m}-0.30\text{m}\\ =\text{0}\text{.322}\text{m}\end{array}$

Substitute $10\text{m}/\text{s}$ for $V_1$

  $9.81\text{m}/{\text{s}}^{\text{2}}$ for $g$ and $0.322\text{m}$ for $y_1$ in Equation (I).

  $\begin{array}{c}{F}_{r1}=\frac{10\text{m}/\text{s}}{\sqrt{\left(9.81\text{m}/{\text{s}}^{\text{2}}\right)\times \left(0.322\text{m}\right)}}\\ =\frac{10\text{m}/\text{s}}{\sqrt{\left(3.15882{\text{m}}^2/{\text{s}}^{\text{2}}\right)}}\\ =\frac{10\text{m}/\text{s}}{1.777\text{m}/\text{s}}\\ =5.626\end{array}$

The Froude number is greater than $1$ hence the flow is supercritical over the bump.

Conclusion:

The rise in water level over the bump is $0.322\text{m}$.

The flow over the bump is supercritical.