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SLIDE 1

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❲❡✐rs ❛♥❞ ❋❧✉♠❡s

❙❖❊✷✶✺✻✿ ❋❧✉✐❞s ▲❡❝t✉r❡ ✼

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SLIDE 2

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❘❡❝❛♣

▲❛st ❧❡❝t✉r❡ ④ ❛ss✉♠❡❞ t❤❡ ❝❤❛♥♥❡❧ ✇❛s ✉♥✐❢♦r♠ ✭❝♦♥st❛♥t ❞❡♣t❤✱ s❤❛♣❡✱ s❧♦♣❡ ❡t❝✳✮ ④ st❡❛❞② ✉♥✐❢♦r♠ ✌♦✇ ❋♦✉♥❞ t❤❛t ✿

✎ ❧♦❝❛t✐♦♥ ♦❢ ❢r❡❡ s✉r❢❛❝❡ t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ✎ ✷ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s ④ s✉❜❝r✐t✐❝❛❧ ❛♥❞ s✉♣❡r❝r✐t✐❝❛❧✳

❙♦❧✉t✐♦♥ ❝❛♥ ❥✉♠♣ ❜❡t✇❡❡♥ t❤❡s❡ ✭❤②❞r❛✉❧✐❝ ❥✉♠♣✮

✎ ❋r♦✉❞❡ ♥✉♠❜❡r ✐♠♣♦rt❛♥t ✎ ✌♦✇ ❝❛♥ ❛❧s♦ ❜❡ ❝r✐t✐❝❛❧ ④ ❋r ❂ ✶ ✎ ❙✐♥❣❧❡ s♦❧✉t✐♦♥ ❢♦r t❤✐s ❝❛s❡ ✿ ♠❛①✐♠✉♠ ✌♦✇ ❢♦r ❛ ❣✐✈❡♥

s♣❡❝✐☞❝ ❡♥❡r❣② ❲❤❛t ✐❢ t❤❡ ❝❤❛♥♥❡❧ ✐s ♥♦♥✲✉♥✐❢♦r♠❄ ④ st❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇❄

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SLIDE 3

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇

❋r♦✉❞❡ ♥✉♠❜❡r st✐❧❧ ✐♥❞✐❝❛t❡s ✇❤❡t❤❡r t❤❡ ✌♦✇ ✐s s✉❜✲ ♦r s✉♣❡r✲❝r✐t✐❝❛❧✳ ❆s t❤❡ ❝r♦ss✲s❡❝t✐♦♥ ❝❤❛♥❣❡s✱ t❤❡ ✌♦✇ ❝♦♥❞✐t✐♦♥s ✇✐❧❧ t♦♦✳ ❚❤❡ ❢r❡❡ s✉r❢❛❝❡ ♠❛② ♣❛ss t❤r♦✉❣❤ t❤❡ ❝r✐t✐❝❛❧ ❞❡♣t❤✳ P♦✐♥ts ✇❤❡r❡ t❤✐s ❤❛♣♣❡♥s ❛r❡ ❛ ❧✐♠✐t✐♥❣ ❢❛❝t♦r ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ t❤❡ ❝❤❛♥♥❡❧✱ ❛♥❞ ❛r❡ ❦♥♦✇♥ ❛s ❝♦♥tr♦❧ s❡❝t✐♦♥s✳

Critical depth Hydraulic jump s<sc s>sc d2 dc d1 s<sc

❊①❛♠♣❧❡s ♦❢ t❤✐s ❜❡❤❛✈✐♦✉r ✐♥❝❧✉❞❡ ✇❡✐rs ❛♥❞ ✌✉♠❡s✱ ✉s❡❞ ✭♣r✐♥❝✐♣❛❧❧②✮ t♦ ♠❡❛s✉r❡ ✌♦✇ r❛t❡s ♦❢ ✇❛t❡r✳ ❲❡✐rs ❛r❡ ✈❡rt✐❝❛❧ ♦❜str✉❝t✐♦♥s ④ ❝❛♥ ❜❡ t❤✐♥ ♦r ❜r♦❛❞✲❝r❡st❡❞✳ ❋❧✉♠❡s ❛r❡ ♥❛rr♦✇✐♥❣s ✐♥ t❤❡ ❝❤❛♥♥❡❧✳

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SLIDE 4

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs

❚②♣✐❝❛❧ ❛r❛♥❣❡♠❡♥t ✿

h E dc Total energy

✎ ❙✉✍❝✐❡♥t❧② ❜r♦❛❞ t❤❛t t❤❡ ❢r❡❡ s✉r❢❛❝❡ ✐s ♣❛r❛❧❧❡❧ t♦ t❤❡

❝r❡st

✎ ❋❧♦✇ ✉♣str❡❛♠ tr❛♥q✉✐❧ ④ ❋r ❁ ✶ ✎ ❋❧♦✇ ❞♦✇♥str❡❛♠ ✉♥✐♠♣❡❞❡❞ ④ ❢r❡❡ ❥❡t ④ ❞✐s❝❤❛r❣❡ ♦✈❡r

t❤❡ ✇❡✐r ✇✐❧❧ ❜❡ t❤❡ ♠❛①✐♠✉♠ ♣♦ss✐❜❧❡

✎ ❚❤✉s ✌♦✇ ♦✈❡r ✇❡✐r ❛t ❝r✐t✐❝❛❧ ❞❡♣t❤

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SLIDE 5

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❋❧♦✇ r❛t❡ ♦✈❡r ✇❡✐r ❣✐✈❡♥ ❜② ◗ ❂ ❈❜❊ ✸❂✷ ✐♥ t❡r♠s ♦❢ t❤❡ ✉♣str❡❛♠ s♣❡❝✐☞❝ ❡♥❡r❣② ❊✳ ❯s✉❛❧❧② ❛ss✉♠❡ ❱ ❂ ✵ ✉♣str❡❛♠✱ s♦ ❊ ✬ ❍✱ ❛♥❞ t❤✉s ◗ ❂ ❈❜❍✸❂✷ ④ ❣❡♥❡r❛❧ ✇❡✐r ❡q✉❛t✐♦♥✳ ❆ s✐♥❣❧❡ ♠❡❛s✉r❡♠❡♥t ♦❢ t❤✐s ✈❛❧✉❡ ❣✐✈❡s t❤❡ ❞✐s❝❤❛r❣❡✳ ❋r♦♠ t❤❡♦r❡t✐❝❛❧ ❛♥❛❧②s✐s ❈ ❂ ✶✿✼✵✺ ❜✉t t❤✐s ✐s ✉s✉❛❧❧② ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❡①♣❡r✐♠❡♥t✳ ◆✳❇✳ ❚❤❡ ❞❡♣t❤ ♦✈❡r t❤❡ ❝r❡st ♦❢ t❤❡ ✇❡✐r ✐s ☞①❡❞✳ ❘❛✐s✐♥❣ t❤❡ ✇❡✐r ❝r❡st ✇✐❧❧ ♥♦t ❛❧t❡r t❤✐s✱ ❜✉t ✇✐❧❧ ❛❧t❡r t❤❡ ♦✈❡r❛❧❧ ❞❡♣t❤ ✉♣str❡❛♠✳

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SLIDE 6

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❚❤✐♥ ❲❡✐rs

❆❧t❡r♥❛t✐✈❡ ❛r❛♥❣❡♠❡♥t ④ ✇❡✐r ✐s t❤✐♥ ✭✌❛t ♣❧❛t❡✮ ✇✐t❤ s♣❡❝✐☞❡❞ ♦♣❡♥✐♥❣✳ ❆ss✉♠♣t✐♦♥s ✿

✎ ❋❧♦✇ ❜❡❤✐♥❞ ✇❡✐r ❱ ❂ ✵ ✎ Pr❡ss✉r❡ ✐♥ ♥❛♣♣❡ ❛t♠♦s♣❤❡r✐❝ ✎ ◆♦ ❡♥❡r❣② ❧♦ss❡s ✎ ❙tr❡❛♠❧✐♥❡s ❤♦r✐③♦♥t❛❧ ♦✈❡r ❝r❡st

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SLIDE 7

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r

H δh ∆ H1

❙t❛rt ❢r♦♠ ◗ ❂ ❆❱ ❈♦♥s✐❞❡r t❤❡ t❤✐♥ str✐♣ ♠❛r❦❡❞ ❆ ❂ ❜✍❤

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SLIDE 8

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❱ ❂ ♣ ✷❣❤ s♦ ✍◗ ❂ ❜✍❤ ♣ ✷❣❤ ❛♥❞ t❤✉s ◗ ❂ ❩ ❍

❞◗ ❂ ❩ ❍

❜ ♣ ✷❣❤ ❞❤ ❂ ❜ ♣ ✷❣ ✷ ✸❍✸❂✷

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SLIDE 9

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❱ ❂ ♣ ✷❣❤ s♦ ✍◗ ❂ ❜✍❤ ♣ ✷❣❤ ❛♥❞ t❤✉s ◗ ❂ ❩ ❍

❞◗ ❂ ❩ ❍

❜ ♣ ✷❣❤ ❞❤ ❂ ❜ ♣ ✷❣ ✷ ✸❍✸❂✷

slide-10
SLIDE 10

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❆♣♣❧②✐♥❣ ❇❡r♥♦✉❧❧✐ ❣✐✈❡s ❱ ❂ ♣ ✷❣❤ s♦ ✍◗ ❂ ❜✍❤ ♣ ✷❣❤ ❛♥❞ t❤✉s ◗ ❂ ❩ ❍

❞◗ ❂ ❩ ❍

❜ ♣ ✷❣❤ ❞❤ ❂ ❜ ♣ ✷❣ ✷ ✸❍✸❂✷

slide-11
SLIDE 11

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

◆✳❇✳ ❚❤✐s ✐s ✇r♦♥❣ ✿ ✇❡ ✉s✉❛❧❧② ♠❡❛s✉r❡ ❍✶✱ ♥♦t ❍ ❆ ♠♦r❡ ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ❝❛♥ ❜❡ ❝❛rr✐❡❞ ♦✉t✱ ✇✐t❤ ✁ ✬ ❍✶❂✸✱ ❣✐✈✐♥❣ ◗ ❂ ✵✿✽✶ ✒✷ ✸ ✓ ♣ ✷❣ ❜❍✸❂✷

❚❤✐s ✐s st✐❧❧ ✐♥❛❝❝✉r❛t❡ ✿ ✇❡ ❤❛✈❡ ♥❡❣❧❡❝t❡❞ t❤❡ ✈❡❧♦❝✐t② ❤❡❛❞ ✉♣str❡❛♠ ✭✐♠♣♦rt❛♥t ❢♦r s♠❛❧❧ ❝❤❛♥♥❡❧s✮✳ ■♥ r❡❛❧✐t②✱ ❥✉st ✉s❡ t❤❡ ❢♦r♠✉❧❛ ◗ ❂ ✒✷ ✸ ✓ ❈❞ ♣ ✷❣ ❜❍✸❂✷ ✇❤❡r❡ ✇❡ ❤❛✈❡ ❛ss✉♠❡❞ ❍✶ ✬ ❍ ❈❞ ✐s ❛ ❝♦❡✍❝✐❡♥t t♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❢♦r ❡❛❝❤ ✇❡✐r✳ ●❡♥❡r❛❧❧② ❈❞ ✬ ✵✿✻✷

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SLIDE 12

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❱❡❡ ♥♦t❝❤ ✇✐❡r

❆ s✐♠✐❧❛r ❛♥❛❧②s✐s ❢♦r ❛ ✇❡✐r ♦❢ ❛♥❣❧❡ ✒ ✐s ♣♦ss✐❜❧❡✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ str✐♣ ❜ ❂ ✷✭❍ ❤✮ t❛♥ ✒❂✷ ❛♥❞ t❤❡ s✐♠♣❧❡ ❛♥❛❧②s✐s ❣✐✈❡s ◗ ❂ ✽ ✶✺ ♣ ✷❣ t❛♥ ✒ ✷❍✺❂✷

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SLIDE 13

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

❋❧✉♠❡s

V 2 b2 V 1 b1

✿ ✿ ✿ ♣❧❛♥ ✈✐❡✇ ❲✐❞t❤ ♦❢ ❝❤❛♥♥❡❧ r❡❞✉❝❡❞ ✮ ✌♦✇ s♣❡❡❞s ✉♣ ✮ ❞✐s❝❤❛r❣❡ ♣❡r ✉♥✐t ✇✐❞t❤ ✐♥❝r❡❛s❡s✳ ❙♣❡❝✐☞❝ ❡♥❡r❣② ❝♦♥st❛♥t ✮ ❢♦r s✉❜❝r✐t✐❝❛❧ ✌♦✇✱ ❞❡♣t❤ ❞❡❝r❡❛s❡s ✮ ❢♦r s✉♣❡r❝r✐t✐❝❛❧ ✌♦✇✱ ❞❡♣t❤ ✐♥❝r❡❛s❡s ❚❤✐s ❛rr❛♥❣❡♠❡♥t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r ✌♦✇ ♠❡❛s✉r❡♠❡♥t✳

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SLIDE 14

❲❡✐rs ❛♥❞ ❋❧✉♠❡s ❘❡❝❛♣ ❙t❡❛❞② ♥♦♥✲✉♥✐❢♦r♠ ✌♦✇ ❇r♦❛❞ ❝r❡st❡❞ ✇❡✐rs ❚❤✐♥ ✇❡✐rs ❘❡❝t❛♥❣✉❧❛r ♥♦t❝❤ ✇❡✐r ❱❡❡ ♥♦t❝❤ ✇❡✐r ❋❧✉♠❡s

■❢ t❤❡ ❢r❡❡ s✉r❢❛❝❡ ❞♦❡s ♥♦t ♣❛ss t❤r♦✉❣❤ t❤❡ ❝r✐t✐❝❛❧ ❞❡♣t❤✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ✈❡♥t✉r✐ ✌✉♠❡

h Total energy Critical depth d1 d2

■❢ t❤❡ ❢r❡❡ s✉r❢❛❝❡ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❝r✐t✐❝❛❧ ❞❡♣t❤ ✐♥ t❤❡ t❤r♦❛t✱ t❤❡♥ ✇❡ ❤❛✈❡ ❛ st❛♥❞✐♥❣ ✇❛✈❡ ✌✉♠❡

Total energy h d1 dc hydraulic jump