The document discusses gradually varied flow in water resources engineering, detailing governing equations for different flow conditions, including backwater and drawdown curves. It classifies flow regions and profiles based on channel slopes and provides practical examples of various flow profiles related to obstructions and transitions in channels. Key concepts include critical and normal depth lines, along with their implications for flow behavior in mild, steep, and adverse slopes.

1. Chapter-6
Course Code : CE-421
Course Title : Water Resources Engineering-II
Romana Saila
Assistant Professor
Department of Civil Engineering

2. Gradually Varied Flow
Governing equation for Varied flow :
H= Zb+ h + αU2/2g
dh/dx =
𝑺𝒐−𝑺𝒇
𝟏−𝑭𝒓 𝟐
dh/dx= So
𝟏−(
𝒉 𝒏
𝒉
) 𝑵
𝟏−(
𝒉 𝒄
𝒉
) 𝑴
For wide channel, N=3, M=3
For Belanger equation, N=10/3, M=3
If dh/dx is positive it represents a
backwater curve.
If dh/dx is negative , it represents a
drawdown curve.
Datum
Zb
Bottom /Bed
(So)
h
αU2/2g
Energy Line ,Slope
Sf
Horizontal line
x
H
Flow

3. Classification of Bottom Slopes
Sustaining or positive slopes : Non Sustaining Slopes :
1) Mild 1) Horizontal
 So <Sc and hn>hc So = 0
2) Critical 2) Adverse
 So = Sc and hn= hc So < 0
3) Steep
 So >Sc and hn<hc

4. Classification of Flow Regions or Zones
Classification of Channels

5. Classification of Flow Regions or Zones
 Region 1: Space above the topmost line.
 Region 2: Space between top line and the next lower line.
 Region 3: Space between the second line and the bed.
*NDL= Normal Depth Line ; hn= yo= normal depth
* CDL= Critical Depth Line ; hc= yc= critical depth

6. Gradually Varied Flow Profiles

7. Flow Profile in Mild slope
Governing Equation : dh/dx= So
𝟏−(
𝒉 𝒏
𝒉
) 𝑵
𝟏−(
𝒉 𝒄
𝒉
) 𝑴
Zone/Region 1: y > y0 > yc or h> hn> hc
dh/dx > 0 (+ ve) ; Backwater curve; water level increases
at the direction of the flow.
Zone/Region 2: y0 > y > yc or hn > h > hc
dh/dx < 0 (- ve) ; Drawdown curve; water level decreases
at the direction of the flow.
Zone/Region 3: y0 > yc > y or hn > hc > h
dh/dx > 0 (+ ve) ; Backwater curve; Backwater curve; water level increases at the direction of the
flow.

8. Practical Examples of M1, M2, M3 profile
 Obstructions to flow, such as weirs, dams, control structures and natural features, such
as bends, produce M1 backwater curves . These extend to several kilometers upstream
before merging with the normal depth.

9. Practical Examples of M1, M2, M3 profile
 The M2 profiles occur at a sudden drop of the channel, at constriction type of transitions and at
the canal outlet into pools.
 Where a supercritical stream enters a mild slope channel, M3 type of profile occurs. The flow
leading from a spillway or a sluice gate to a mild slope forms a typical example.

10. Reference
 https://web.itu.edu.tr/~bulu/hydraulics_files/lecture_notes_06.pdf