--- title: "Effect of increasing hump height on water surface" subject: "Hydraulics" chapter: 7 key: "semester_4/hydraulics:hump-height-gradual-increase-specific-energy-diagram" source: https://www.bctengineer.com/q/semester_4/hydraulics/hump-height-gradual-increase-specific-energy-diagram site: BCT Engineering --- # Effect of increasing hump height on water surface Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow - Asked in: 2082 Kartik, 8 marks; 2078 Chaitra, 6 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/hump-height-gradual-increase-specific-energy-diagram - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7 ## Question Explain briefly, by sketching the graph between specific energy and water depth of a rectangular channel, how the depth upstream of a hump changes with the height of hump as it is gradually increased in three stages: (i) less than the critical hump height; (ii) at the critical hump height, and (iii) exceeding the critical hump height. (The same analysis may be asked for a hump together with a constriction of constant width.) ## Short answer For a hump less than critical height, upstream depth is unchanged and depth over the hump falls along the subcritical limb. At critical hump height, upstream depth is still unchanged but the hump acts as a control with critical flow. If the hump exceeds critical height, the flow backs up, upstream depth increases, and critical depth occurs over the hump. ## Answer #### Set-up Consider a rectangular channel with discharge $q$ per unit width, approach depth $y_1$ (subcritical) and specific energy $E_1=y_1+\dfrac{q^2}{2gy_1^2}$. A hump of height $\Delta z$ is placed on the bed. With no energy loss, the specific energy over the hump is $$ E_2=E_1-\Delta z $$ On the specific energy diagram, moving onto the hump shifts the state to the left by $\Delta z$. The critical hump height is $\Delta z_c=E_1-E_{min}=E_1-\tfrac32y_c$. ```text y ^ | . y1 |----------------------* (1) upstream, E1 | . y2 |-------------* (2) over hump, case (i): E2 = E1 - dz | . yc |-----* critical point, E_min (case ii: E2 = E_min) | . | . | . supercritical limb +-----+-------+--------+-----> E E_min E2 E1 |<- dz ->| ``` #### (i) Hump height less than critical ($\Delta z<\Delta z_c$) - $E_2=E_1-\Delta z>E_{min}$, so flow over the hump is possible with the same discharge. - The upstream depth $y_1$ is **unchanged**. - The depth over the hump falls along the subcritical limb (it is lower than $y_1$ but still greater than $y_c$). The water surface drops over the hump. - After the hump the depth rises again to $y_1$. #### (ii) Hump height equal to critical ($\Delta z=\Delta z_c$) - $E_2=E_{min}$, so the depth over the hump is exactly **critical** ($y=y_c$). - The upstream depth is still $y_1$ (unchanged), but this is the limit: the hump now acts as a control (like a broad-crested weir). - The discharge is the maximum that can pass for this upstream energy. #### (iii) Hump height greater than critical ($\Delta z>\Delta z_c$) - $E_1-\Delta zy_c$ | unchanged | | $\Delta z=\Delta z_c$ | critical | unchanged (limit) | | $\Delta z>\Delta z_c$ | critical | increases (backwater) | If the hump is combined with a constriction of constant width, the same logic applies, with $q$ increased in the contracted section: the critical condition is $E_1-\Delta z=\tfrac32y_c$, with $y_c=\sqrt[3]{q_2^2/g}$ and $q_2=Q/b_2$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. On the specific energy diagram, what does passing over a hump of height Δz (no losses) do to the flow state? A. It shifts the state right by Δz, since E2 = E1 + Δz B. It shifts the state left by Δz, since E2 = E1 - Δz C. It moves the state to the alternate depth at the same E D. It leaves E unchanged because the discharge is constant Answer: B. The hump raises the bed, so the energy above the bed falls: E2 = E1 - Δz, a shift to the left on the diagram. (Key point: Effect of hump on specific energy) 2. For a hump lower than the critical height, what happens to the upstream depth and to the depth over the hump? A. Upstream depth rises; depth over hump is critical B. Upstream depth unchanged; depth over hump falls but stays above yc C. Upstream depth falls; depth over hump is supercritical D. Both upstream and hump depths stay unchanged Answer: B. E2 stays above Emin, so the same discharge passes. The depth over the hump drops on the subcritical limb while y1 is unchanged. (Key point: Hump below critical height) 3. The critical hump height Δzc is given by A. E1 - 1.5yc B. E1 + 1.5yc C. 1.5yc - E1 D. y1 - yc Answer: A. Critical flow over the hump needs E2 = Emin = 1.5yc, so Δzc = E1 - Emin = E1 - 1.5yc. (Key point: Critical hump height) 4. When the hump height exceeds the critical height, what happens? A. Depth over the hump drops below yc and the upstream depth is unchanged B. The flow becomes supercritical upstream of the hump C. Critical depth occurs over the hump, the upstream depth rises (backwater) so that E1' = Emin + Δz D. The discharge increases to use the extra height Answer: C. Available energy would be below Emin, so the flow backs up until upstream energy equals Emin + Δz, with critical flow over the hump. 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