Water surface over hump with contraction

Given data

$y_1=1.6$ m, $V_1=1.10$ m/s, $b_1=4.0$ m; contracted width $b_2=3.5$ m with a hump of $\Delta z=0.35$ m; no losses.

$$Q=b_1y_1V_1=4\times1.6\times1.10=7.04\ \text{m}^3/\text{s}$$ $$E_1=y_1+\frac{V_1^2}{2g}=1.6+\frac{1.21}{19.62}=1.6617\ \text{m},\qquad Fr_1=\frac{1.10}{\sqrt{9.81\times1.6}}=0.278$$

Check for critical flow at the contraction

$$q_2=\frac{7.04}{3.5}=2.0114\ \text{m}^2/\text{s},\quad y_{c2}=\sqrt[3]{\frac{q_2^2}{g}}=0.7444\ \text{m},\quad E_{min2}=1.1165\ \text{m}$$

Energy available over the hump: $E_2=E_1-\Delta z=1.6617-0.35=1.3117$ m $>E_{min2}$. So the flow passes without choking, and the upstream depth is unchanged.

Depth over the hump (subcritical)

$$y_2+\frac{q_2^2}{2gy_2^2}=y_2+\frac{0.2062}{y_2^2}=1.3117\;\Rightarrow\;y_2=1.1579\ \text{m}$$

(here $q_2^2/2g=0.2062$). Velocity $V_2=q_2/y_2=1.737$ m/s.

Water surface elevations (datum: upstream bed)

• At a section upstream: $1.600$ m.
• At the contracted section: $0.35+1.1579=1.5079$ m.

Drop in water surface $=1.600-1.508=0.092$ m.

Specific energy diagram

 y (m)
1.700|                                  .:
1.600|                              1:         (1) upstream, E1 = 1.662 m
1.500|                           .:
1.400|                       .:
1.300|                    .:
1.158|               .2                        (2) over hump, E2 = 1.312 m
1.100|             . :
1.000|          . :
0.900|        . :
0.800|      .  :
0.700|     .   :                               noses: yc1 = 0.681 m, yc2 = 0.744 m
0.600|      .    :
0.500|         .       :
0.400|                   .            :
     +-----+---+------+-------------+---------> E (m)
 ticks, left to right: E_min1 = 1.021, E_min2 = 1.117, E2 = 1.312, E1 = 1.662
 "." = curve for q1 = 1.76 m²/s,   ":" = curve for q2 = 2.011 m²/s

Point (1) is on the upper (subcritical) limb of the curve for $q_1=1.76$ m²/s; point (2) is on the upper limb of the curve for $q_2=2.011$ m²/s, shifted left by $\Delta z=0.35$ m.

Answer: depth over the hump $=1.158$ m, water surface elevation there $=1.508$ m (above the upstream bed); upstream elevation $=1.600$ m (unchanged).