What is Hydraulic jump with classification and Characteristics listed

Hydraulic Jump is formed when super-critical flow meets sub-critical flow.

OR

Sudden turbulent passage of flow from super-critical to sub-critical flow.

Applying linear momentum in the Control Volume
P1 – P2 – Ff + W.sinθ = ρ.Q.(β2u2 – β1u1) …….(1)
Assumption

1. Bed is horizontal i.e W.sinθ ≈ 0
2. Friction losses at boundary is zero i.e Ff ≈ 0
3. Flow is uniform before and after the jump i.e pressure distribution is hydro static in nature.
   P1 = γ.A1.x̅1 and P2 = γ.A2.
   where x̅1,x̅2 → centroid of flow area measured from the free surface of water at section-1 and section-2 respectively.

β1 = β2 = 1.0

Using the all the above assumption and rewriting eq.(1)
⇒ γ.(A1.x̅1 – A2.x̅2) = (γ/g).Q.(u2 – u1) = (Q2/g).(1/A2 – 1/A1)
⇒ Q2/(g.A1) + A1.x̅1 = Q2/(g.A2) + A2.x̅2
Here Q2/(g.A) + A.x̅ = F is known as Specific Force
i.e Specific Force before and after the jump is equal.
Specific Force diagram can be represented as :

For minimum F, dF/dy = 0 ⇒ Q2/g(-1/A2).dA/dy + dA/dy.x̅ = 0
⇒ Q2/(g.A2) = x̅ ⇒ Q2/g = A2.x̅

For a rectangular channel
A1 = b.y1 , x̅1 = y1/2 and
A2 = b.y2 , x̅2 = y2/2
∴ Q2/(g.b.y1) + b.y1.y1/2 = Q2/(g.b.y2) + b.y2.y2/2
⇒ q2/(g.y1) + y12/2 = q2/(g.y2) + y22/2
⇒ (q2/g).(1/y1 – 1/y2) = (y22 – y12)/2
⇒ (q2/g).(y2 – y1)/(y1.y2) = (y2 – y1)(y2 + y1)/2
⇒ q2/g = y1.y2.(y1 + y2)/2

There are two cases possible:

Case I: y1 is known and y2 is variable
⇒ y1.y22 + y12.y2 – 2.q2/g = 0
⇒ y2 = [-y12 ± √(y14 + 8.y1q2/g)]/2.y1 = -y1/2 ± √(y12/4 + 2.q2/g.y1)
⇒ y2/y1 = -1/2 ± √(1/4 + 2.q2/g.y13) = (1/2).[-1 + √(1 + 8.F12)

Case II: y2 is known and y1 is variable
Using the above similarly procedure we get
y1/y2 = (1/2).[-1 + √(1 + 8.F22)

Loss of energy = E1 – E2
ΔE = [(y1 + q2/(2.g.y12)) – y2 + q2/(2.g.y22))]
= (y2 – y1)3/(4.y1.y2)

CLASSIFICATION OF JUMP