for better understanding of gradually varied flow in fluid mechanics

1. GRADUALLY VARIED FLOW
Prepared by:
Y S G GOVIND BABU
Assistant professor
BVC ENGG COLLEGE-ODALAREVU

2. FLOW CLASSIFICATION
• Uniform (normal) flow: Depth is constant at every section
along length of channel
• Non-uniform (varied) flow: Depth changes along channel
• Rapidly-varied flow: Depth changes suddenly
• Gradually-varied flow: Depth changes gradually

3. • RVF: Rapidly-varied flow
• GVF: Gradually-varied flow
FLOW CLASSIFICATION

4. Figure : Examples for gradually varied flow in open channels.

5. ASSUMPTIONS FOR GRADUALLY-VARIED FLOW
1. The channel is prismatic and the flow is steady.
2. The bed slope, So, is relatively small.
3. The velocity distribution in the vertical section is uniform and the
kinetic energy correction factor is close to unity.
4. Streamlines are parallel and the pressure distribution is hydrostatic.
5. The channel roughness is constant along its length and does not
depend on the depth of flow.

6. ANALYSIS OF GRADUALLY-VARIED FLOW

7. THE EQUATIONS FOR GRADUALLY VARIED FLOW
z+h+
g2
V
=H
2
dx
dz
+
dx
dh
+
g2
V
dx
d
=
dx
dH 2






S-=dx/dz,S-=dx/dH 0
h1
h2
S=Sf: slope of EGL

8. THE EQUATIONS FOR GRADUALLY VARIED FLOW
3
2
3
2
2
2
2
22
Ag
BQ
dx
dh
-=
dh
dA
Ag
Q
dx
dh
-=
Ag2
Q
dh
d
dx
dh
=
dh
dh
Ag2
Q
dx
d
=
g2
V
dx
d
























S-
dx
dh
Ag
BQ
-1=S-
dx
dh
+
Ag
BQ
dx
dh
-=S- 03
2
03
2








Ag/BQ-1
S-S
=
dx
dh
32
0
Fr-1
S-S
=
dx
dh
2
0
If dh/dx is positive the depth is increasing
otherwise decreasing
General governing
Equation for GVF

9. WATER SURFACE PROFILES
Recalling the definitions for the normal depth, yn , and the critical depth,
yc , the following inequalities can be stated

10.  A gradually varied flow profile is classified based on the channel
slope, and the magnitude of flow depth, y in relation to yn and yc .
 The channel slope is classified based on the relative magnitudes of
the normal depth, yn and the critical depth, yc .

11. WATER SURFACE PROFILES CLASSIFICATION

12. WATER SURFACE PROFILES CLASSIFICATION
Flow profiles associated with mild, steep, critical, horizontal,
and adverse slopes are designated as M, S, C, H and A profiles,
respectively.
The space above the channel bed can be divided into three zones
depending upon the inequality defined by equations
Zone 1: Space above the topmost line,
y> yo> yc , y > yc> yo
Zone 2: Space between top line and the
next lower line,
yo > y> yc , yc > y> yo
Zone 3: Space between the second line and
the bed.
yo >yc>y , yc>yo>y

13. PROFILE CLASSIFICATION
The space above both the CDL /
NDL (whichever is upper) is
designated as zone-1.
The space between the CDL and
the NDL is designated as zone-2.
The space between the channel bed
and CDL/NDL (whichever is
lower) is designated as zone-3.

14. Flow profiles are finally classified based on
(i) the channel slope and
(ii) the zone in which they occur.
For example, if the water surface lies in zone-1 in a channel
with mild slope, it is designated as M1 profile. Here, M stands
for a mild channel and 1 stands for zone-1.
DESIGNATION OF FLOW PROFILES

15. Slope
Profile designation Relative
position of y
Type of flow
zone-1 zone-2 zone-3
Adverse
S0 = 0
None
A2 y > yc Subcritical
A3 y < yc Supercritical
Horizontal
S0 = 0
None
H2 y > yc Subcritical
H3 y < yc Supercritical
Mild
0<S0<Sc = 0
M1 y > yn > yc Subcritical
M2 yn > y > yc Subcritical
M3 yn > yc> y Supercritical
Critical
S0 = Sc > 0
C1 y > yc = yn Subcritical
C2 y = yc = yn Uniform - Critical
C3 yc = yn > y Supercritical
Steep
S0 > Sc > 0
S1 y > yc> yn Subcritical
S2 yc > y > yn Supercritical
S3 yc > yn > y Supercritical
Types of Flow Profiles

16. VARIATION OF FLOW DEPTH
Qualitative observations about various types of water surface
profiles can be made and the profile can be sketched without
performing any computations.
This is achieved by considering the signs of the numerator and
the denominator in Equation.
The following analysis helps to know
(i) whether the depth increases or decreases with distance; and
(ii) how the profile approaches the upstream and downstream
limits.

17. GENERAL POINTS TO BE CONSIDERED

18. For a given Q, n, and So at a channel,
yo = Uniform flow depth, yc = Critical flow depth, y = Non-
uniform flow depth.
The depth y is measured vertically from the channel bottom,
the slope of the water surface dy / dx is relative to this channel
bottom.
the prediction of surface
profiles from the
analysis of
Fr-1
S-S
=
dx
dh
2
0
WATER SURFACE PROFILES CLASSIFICATION

19. Consider an S2 profile. In an S2 profile, yc > y > yn ,
yc < y implies that F>1 and
the denominator is negative. yn > y implies that Sf < S0 . Therefore,
This means that flow depth decreases
with distance x. On the downstream
side, as y decreases towards yn it
approaches NDL asymptotically. On
the upstream side, as y increases
toward yc , it approaches CDL almost
vertically. The sketch of an S2 profile
is shown
WATER SURFACE PROFILES CLASSIFICATION

20. Outlining Water Surface Profiles
Proceeding in a similar manner, other water surface profiles can be
sketched. These sketches are shown in Figure

21. Outlining Water Surface Profiles

22. Computation
of
Water Surface Profiles

23. METHODS OF SOLUTIONS OF THE GRADUALLY VARIED FLOW
1. Direct Integration
2. Graphical Integration
3. Numerical Integration
i- The direct step method (distance from depth
for regular channels)
ii- The standard step method, regular channels
(distance from depth for regular channels)
iii- The standard step method, natural channels
(distance from depth for natural channels)

24. Fr-1
S-S
=
dx
d
2
0 fy
GRADUALLY VARIED FLOW
Important Formulas
fo SS =
dx
dE
g2
V
yzH
2
b 
g2
V
yE
2

EzH b 
dx
dE
dx
dz
=
dx
Hd


25. GRADUALLY VARIED FLOW COMPUTATIONS
Analytical solutions to the equations above not available for the
most typically encountered open channel flow situations.
A finite difference approach is applied to the GVF problems.
Channel is divided into short reaches and computations are
carried out from one end of the reach to the other.
Fr-1
S-S
=
dx
d
2
0 fy_
=
dx
fo SS
dE

E: specific energy

26. f
_
o
UD
SS=
x
EE



Sf : average friction
slope in the reach
)SS(
2
1
S fDfuf
_

3/4
u
2
u
2
fu
R
Vn
S  3/4
D
2
D
2
fD
R
Vn
S 
Manning Formula is sufficient to accurately
evaluate the slope of total energy line, Sf
DIRECT STEP METHOD
A nonuniform water surface profile

27. DIRECT STEP METHOD
Subcritical Flow
   
SS
g2/Vyg2/Vy
SS
EE
X
f
_
o
2
UU
2
DD
f
_
o
UD






The condition at the
downstream is known
yD, VD and SfD are known
Chose an appropriate value
for yu
Calculate the corresponding
Vu , Sfu and Sf
Then Calculate X
Supercritical Flow
The condition at the
upstream is known
yu, Vu and Sfu are known
Chose an appropriate value
for yD
Calculate the corresponding
SfD, VD and Sf
Then Calculate X

28. Example
A trapezoidal concrete-lined channel has a constant bed
slope of 0.0015, a bed width of 3 m and side slopes 1:1. A
control gate increased the depth immediately upstream to
4.0m when the discharge is 19 m3/s. Compute WSP to a
depth 5% greater than the uniform flow depth (n=0.017).
Two possibilities exist:
OR

29. Solution
The first task is to calculate the critical and normal depths.
Using Manning formula, the depth of uniform flow:
2/13/2
SAR
n
1
Q  yo = 1.75 m
Using the critical flow condition, the critical depth:
3
2
2
gA
TQ
Fr  yc = 1.36 m
It can be realized that the profile should be M1 since yo > yc
That is to say, the possibility is valid in our problem.

30. y A R E ΔE Sf Sf So-Sf Δx x
4.000 28.00 1.956 4.023 0.000054 0
3.900 26.91 1.918 3.925 0.098 0.000060 0.000057 0.001443 67.98 67.98
3.800 25.84 1.880 3.828 0.098 0.000067 0.000064 0.001436 68.14 136.11
3.700 24.79 1.841 3.730 0.098 0.000075 0.000071 0.001429 68.32 204.44
3.600 23.76 1.802 3.633 0.097 0.000084 0.000080 0.001420 68.54 272.98
3.500 22.75 1.764 3.536 0.097 0.000095 0.000089 0.001411 68.80 341.78
3.400 21.76 1.725 3.439 0.097 0.000107 0.000101 0.001399 69.09 410.87
3.300 20.79 1.686 3.343 0.096 0.000120 0.000113 0.001387 69.44 480.31
3.200 19.84 1.646 3.247 0.096 0.000136 0.000128 0.001372 69.86 550.17
3.100 18.91 1.607 3.151 0.095 0.000155 0.000146 0.001354 70.36 620.53
3.000 18.00 1.567 3.057 0.095 0.000177 0.000166 0.001334 70.96 691.49
1.800 8.64 1.068 2.047 0.066 0.001280 0.001163 0.000337 195.10 1840.24
Solution
   
SS
g2/Vyg2/Vy
SS
EE
X
f
_
o
2
UU
2
DD
f
_
o
UD






yo + (0.05 yo)