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HydraulicsProf. Mohammad Saud Afzal
Department of Civil Engineering
Gradually Varied Flow (GVF)
Gradually Varied Flow (GVF)β’ The flow in a channel is termed GRADUALLY VARIED, if the
flow depth changes gradually over a large length of thechannel.
β’ AssumptionsThe channel is prismatic.
The flow in the channel is steady andand non-uniform.
The cross- sectional shape, size and bedslope are constant
The channel bed- slope is small.
The pressure distribution at any section is hydrostatic.
The resistance to flow at any depth is given by thecorresponding uniform flow equation. Example: Manningβs equation
Remember: In the uniform flow equations, energy slope πΊπ is used in place of bed slope πΊπ . When
Manningβs formula is used we get
πΊπ =πππ½π
πΉ ΰ΅π π
Differential Equation of GVF
β’ The total energy H of a GVF can be
expressed as:
π― = π + π +πΆπ½π
ππ
β’ Assuming πΆ = π, we get
π― = π + π +π½π
ππ
Adapted from Subramanya, K. (1986). Flow in Open
Channels. Tata McGraw- Hill Publishing Co. Ltd.
β’ Differentiating both the sides of the
above equation w.r.t xπ π―
π π=
π π
π π+
π π
π π+
π
π π
π½π
ππ(Eq. 1)
Represents energy slopeπ π―
π π= βπΊπ
Represents bottom slopeπ π
π π= βπΊπ
Represents the water surface slope w.r.t the channel bed
β’ Further,π
π π
π½π
ππ=
π
π π
πΈπ
πππ¨ππ π
π π
orπ
π π
π½π
ππ=
βπΈπ
ππ¨ππ π¨
π π
π π
π π
π π¨
π π= π», where T is the top-
width of the channel
β’ So we can rewrite Eq. 1 as
βπΊπ = βπΊπ +π π
π πβ
πΈππ»
ππ¨ππ π
π π
or
π π
π π=
πΊπβπΊπ
πβπΈππ»
ππ¨πDifferential Equation of GVF
NOTE:πΈππ»
ππ¨π= ππ
π, where ππ is Froude Number
Classification of Flow Profilesβ’ If πΈ,π and πΊπ are fixed, then the normal depth ππ and the
critical depth ππ are fixed.
β’ Three possible relationships that may exist between ππ and ππare:ππ > ππ
ππ < ππ
ππ = ππ
Depth obtained from uniform flow equations
β’ Further, ππ does not exist when:
The channel bed is horizontal.
The channel has an adverse slope.
β’ Based on these, the channels are classified into 5
categories as:
πΊπ = π
πΊπ < π
1. Mild Slope (M) - ππ > ππ
2. Steep Slope (S) - ππ < ππ
3. Critical Slope (C) - ππ = ππ
4. Horizontal Bed (H) - πΊπ = π
5. Adverse Slope (A) - πΊπ < π
Subcritical flow at normal depth
Supercritical flow at normal depth
Critical flow at normal depth
Cannot sustain uniform flow
β’ Lines representing the critical depth (CDL) and the normaldepth (NDL), when drawn in the longitudinal section, divide theflow space into the following 3 regions:
Region 1 β Space above the topmost line.
Region 2 β Space between the top line and the next lowerline.
Region 3 β Space between the second lineand the bed.
Adapted from Subramanya, K. (1986). Flow in Open Channels. Tata McGraw-
Hill Publishing Co. Ltd.
Adapted from Subramanya, K. (1986). Flow in Open Channels.
Tata McGraw- Hill Publishing Co. Ltd.
Mild Slope
Adapted from Subramanya, K. (1986). Flow in
Open Channels. Tata McGraw- Hill Publishing
Co. Ltd.
Steep Slope
Adapted from Subramanya, K. (1986). Flow in
Open Channels. Tata McGraw- Hill Publishing
Co. Ltd.
Critical SlopeAdapted from Subramanya, K. (1986). Flow in
Open Channels. Tata McGraw- Hill Publishing
Co. Ltd.
Horizontal Bed
Adapted from Subramanya, K. (1986). Flow in
Open Channels. Tata McGraw- Hill Publishing
Co. Ltd.
Adverse Slope
Adapted from Subramanya, K. (1986). Flow in
Open Channels. Tata McGraw- Hill Publishing
Co. Ltd.
Problem- 1
β’ Find the rate of change of depth of water in a rectangularchannel 10 m wide and 1.5 m deep, when the water isflowing with a velocity of 1 m/s. The flow of water throughthe channel of bed slope 1 in 4000, is regulated in such a waythat energy line is having a slope of 0.00004.
Solution: π = πππ π = π. π π π½ = ππ/π
πΊπ = π/ππππ πΊπ = π. πππππ
π¨ = π Γ π = ππ Γ π. π = ππππ π» = π = πππ πΈ = π¨π½ = ππππ/π
π π
π π=
πΊπβπΊπ
πβπΈππ»
ππ¨π
π π
π π=
π
ππππβ π.πππππ
πβπππΓππ
π.ππΓπππ
π π
π π= π. ππ Γ ππβπ
Problem- 2β’ A rectangular channel with a bottom width of 4 m and a bottom slope of 0.0008 has a
discharge of 1.5 m3/s. In a gradually varied flow in this channel, the depth at a certainlocation is found to be 0.30 m. Assuming Manningβs n = 0.016, determine the type ofGVF profile.
Solution: π = ππ π = π. πππ π = π. π ππ/π
πΊπ = π. ππππ π§ = π. πππ
Now, πΈ
π=
π.π
π= π. ππππ
ππ
π/π ππ =
ππ
π
π/π
=π.ππππ
π.ππ
π/π
= π. ππππ
Now, πΈ =π
ππ¨πΉπ/ππΊπ
π/ππ. π =
π
π.ππππ Γ ππ
πππ
π+πππ
π/π(π. ππππ)π/π
π. π =π
π.πππΓ ππ/π Γ (π. ππππ)π/π
πππππ/π
(π+πππ)π/π
πππ/π
(π+πππ)π/π = π. ππππ
From trial and errorππ = π. ππππ
ππ > ππ (Mild slope)Also ππ > π > ππ
π΄π
Class Question A wide rectangular channel has a Manningβs coefficient of 0.018. For a discharge
intensity of π. πππ
π/π, identify the possible types of gradually varied flow profiles
produced in the following break in the grade of the channel.πΊππ = π. ππππ πππ πΊππ = π. πππ
Solution: Discharge intensity q = 1.5 ππ
π/π
Critical depth ππ = Ξ€ππ ππ/π
= Ξ€π. ππ π. πππ/π
= π. ππππ
Normal depth ππ βΆ For a wide rectangular channel πΉ = ππ
π =π
πππππ
π/ππππ/π
ππ =ππ
πΊπ
π/π
ππ =π. πππ Γ π. π
πΊπ
π/π
Slope ππ
0.0004 1.197
0.016 0.396
ππ(π) πππ(π¦) πππ(π)
0.612 1.197 0.396
Type of grade change : Mild to Steep
The resulting water surface profiles are:
π΄π curve on Mild Slope and πΊπ curve on steep slope
Rapidly varied flow
Hydraulic Jump due to change in bottom elevation
Rapidly varied flow
RVF due to transition
Hydraulic Jump
β’ For Rapidly varied flow (RVF) dy/dx ~1
β’ Flow depth changes occur over a relatively short distance. One such example is hydraulic jump
β’ These changes in depth can be regarded as discontinuity in free surface elevation (dy/dx β)
β’ Hydraulic jump results when there is a conflict between upstream and downstream influences that control particular section of channel
Hydraulic Jump
β’ E.g. Sluice gate requires supercritical flow at upstream portion of channel whereas obstruction require the flow to be subcritical
β’ Hydraulic jump provides the mechanism to make the transition between the two type of flows
Sluice Gate
Hydraulic Jumpβ’ One of the most simple hydraulic jump occurs in a horizontal,
rectangular channel as below
Hydraulic Jump Geometry
Hydraulic Jump Assumptions
β’ The flow within jump is complex but it is reasonable to assume that flow at sections 1 and 2 are nearly uniform, steady and 1D
β’ Neglect any wall shear stress Οw, within relatively short segment between the sections
β’ Pressure force at either section is hydrostatic
Hydraulic Jump Derivation
β’ x-component of momentum equation for control volume is written as
)()( 12111221 VVbyVVVQFF
2
12
111
byApF c
Where
2
11
ypc
2
22
222
byApF c
2
22
ypc
b is the channel width
Hydraulic Jump Derivationβ’ Momentum equation can be written as
β’ Conservation of mass ( continuity ) gives
β’ Energy conservation gives
)(22
12112
21
2
VVg
yVyy Eq. 19
QbVybVy 2211 Eq. 20
hL is the head loss
Lhg
Vy
g
Vy
22
2
22
2
11
Eq. 21
Hydraulic Jump Derivation
β’ Head loss is due to violent turbulent mixing and dissipation that occur during the jump.
β’ One obvious solution is y1=y2 and hL=0 NO JUMP
β’ Another solution : Combine Eq 19 and 20 to eliminate V2
)()(22
21
2
1
2
11
2
111122
12
yygy
yVV
y
yV
g
yVyy Eq. 21b
Hydraulic Jump Derivation
Where Fr1 is upstream Froude number
Question : Obtain Eq. 21c from Eq. 21b
02)()( 2
1
1
22
1
2 Fry
y
y
yEq. 21c
β’Using quadratic formula we get
)81(1(2
1 2
1
1
2 Fry
y
Hydraulic Jump Derivation
β’ Solution with minus sign is neglected ??, Thus
β’ We can also obtain ππ³/ ππ by using Eq. 21
β’ The result is
)81(1(2
1 2
1
1
2 Fry
y Eq. 22
])(1[2
1 2
2
1
2
1
1
2
1 y
yFr
y
y
y
hL Eq. 23
Hydraulic Jump Derivation
Question : Plot of Eq. 22 and corresponding Eq. 23
Hydraulic Jump Derivation
β’ hL cannot be negative since it violates the law of thermodynamics
β’ This means that y2/y1 cannot be less than 1 and Froude number upstream Fr1 is
always greater than 1 for hydraulic jump to take place.
β’ A flow must be supercritical to produce discontinuity called a hydraulic
jump.
Examples of hydraulic jump
Jump caused by a change in channel slope
Examples of hydraulic jump
Submerged hydraulic jumps that can occur just downstream of a sluice gate
Class Question
2F1F
In a flow through rectangular channel for a certain discharge the Froude number corresponding to the two alternative depths are and . Show that
2
1
2
232
122
2
F
FFF
g
Vy
g
Vy
22
2
22
2
11
1y2y
Solution:Let and be the alternative depths.
12 EE The specific energy
2
2
22
1
2
11
21
21
gy
Vy
gy
Vy
numberFroudeFgy
V 2
2
2
1
2
2
2
1
2
2
2
1
2
2
21
21
F
F
F
F
y
y
3
2
2
22
23
1
2
22
1ygB
QFand
ygB
QF
Since
Also
Where Q = discharge in the channel and B = width of the channel, Hence
32
2
1
2
2
2
1
2
1
2
2
3
2
3
1
F
F
y
yor
F
F
y
y2
1
2
2
32
2
1
2
2
2
1
2
2
F
F
F
F
y
y
Class Question
β’ Water on the horizontal apron of the 30 m wide spillway shown in Fig. has a depth of
0.20 m and a velocity of 5.5 m/s. Determine the depth, after the jump, the Froude
numbers before and after the jump.
Class Question
92.32.0*8.9
5.5
1
11
gy
VFr
Conditions across the jump are determined by the upstream Froude number Fr1
Upstream flow is super critical, and therefore it is possible to generate hydraulic jump
Class Question
We obtain depth ratio across the jump as
07.5)92.3*81(1(2
1)81(1(
2
1 22
1
1
2 Fry
y
01.12.0*07.52 y m
Class Question
We obtain Vπ by equating the flow rate
Subcritical Flow
08.101.1
5.5*2.0)(
2
112
y
VyV m/s
343.001.1*8.9
08.1
2
22
gy
VFr
Class Question
Head loss is obtained as
)2
()2
(2
22
2
11
g
Vy
g
VyhL
671.0Lh m
1) Prove that energy loss in a hydraulic jump occurring in a rectangular channel is
21
3
12
4
)(
yy
yyhL
Eq. 24
Class Question
The loss of mechanical energy that takes place in a hydraulic jump is calculated by the application of energy equation (Bernoulliβs equation). If loss of total head in the pump is ππ³, then we can write by Bernoulliβs equation neglecting the slope of the channel.
ππ + Ξ€π½ππ ππ = ππ + Ξ€π½π
π ππ + ππ³
ππ³ = ππ β ππ + Ξ€π½ππ ππ β Ξ€π½π
π ππ
ππ³ = ππ β ππ +ππ
ππ
π
πππ β
π
πππ π = π½πππ = π½πππ
From Eq 21.c we are putting π½π =π
ππ(πππ =
π½π
πππ)
πππππ + ππ
πππ βπππ
π= π
πππππ + ππ
ππππ
=ππ
ππ
ππ³ = ππ β ππ +ππππ
π + πππππ
π
π
πππ β
π
πππ
Which Finally gives
ππ³ =(ππ β ππ)
π
πππππ
If, in a hydraulic jump occurring in a rectangular channel, the Froude number before the jump is 10.0 and the energy loss is 3.20 m. Estimate (i) the sequent depths (ii) the discharge intensity and (iii) the Froude number after the jump.
Class Question
mEandF L 20.30.101
651.130.108112
1811
2
1 22
1
1
2
F
y
y
21
3
12
4 yy
yyEL
Solution:
The sequent depth ratio
Energy loss
12
3
12
1 4
1
yy
yy
y
EL
08.37651.134
1651.1320.33
1
y
m0863.008.37
20.3
1
11
gy
VF
0863.081.90.10 1
VsmV /201.91
msmyVq //7941.00863.0201.9 3
11
1983017818191781
79410
222
22 .
...
.
gyy
q
gy
VF
=depth before the jump =1y
2y =depth after the jump =13.651Γ0.0863= 1.178 m
Discharge intensity
(iii) Froude number after the jump
(ii)
(i)
A rectangular channel has a width of 1.8 m and carries a discharge of 1.8 at a depth of
0.2 m. Calculate (a) the specific energy, (b) depth alternate to the existing depth and
(c) Froude numbers at the alternate depths.
Class Question
2
11 36.020.08.1 mByA
smAQV /0.536.0
80.1 2
11
m
g
VyE 4742.1
81.92
0.520.0
2
22
111
depthExistingmy 20.01
Solution:Let
Area
Velocity
(a) Specific energy
12 EE 4742.12
2
22
g
Vy
22112
2
2
2
2 ,4742.18.181.92
8.1AVAVas
yy
45.12 y
(b) Let1y2y =depth alternate to
Then
By trial and error,
gyVF
57.32.081.9
0.5,2.0 11
Fmy
π¦2 = 1.45π, π2 =π
π΅π¦2=
1.80
1.80 Γ 1.45= 0.69 π/π
1829.045.181.9
69.02
F
(c) Froude number for a rectangular channel is
For
For
Class Question In hydraulic jump occurring in a rectangular horizontal channel, the discharge per unit width is 2.5 ππ/π/m and the depth before the jump is 0.25 m. Estimate (i) the sequent depth and (ii) the energy loss
Y1
V2
Y2
1V
Solution:
smy
qV
myandmsmq
/0.1025.0
5.2
25.0//5.2
1
1
1
3
Initial Froude number 386.625.081.9
0.10
1
11
gy
VF
2
1
1
2 8112
1F
y
y
22 386.68112
1
25.0
y
depthSequentmy 136.22
12 yy(i) The sequent depth ratio is given by
m
yy
yyEL 141.3
250.0136.24
250.0136.2
4
3
21
3
12
LE(ii) The energy loss is given by
Class Question A hydraulic jump occur in a horizontal triangular channel. If the sequent depths in this channel are 0.60 m and 1.20 m respectively, estimate (i) the flow rate, (ii) Froude number at the beginning and end of the jump and (iii) energy loss in the jump.
Solution:
y m=1
(i) Consider a triangular channel of side slope
m horizontal: 1 vertical in fig (in the present
case m=1)
33
32 ymy
myyA
2
22
my
Q
A
Q
P=pressure force=
M=Momentum flux =
2211 MPMP
2
2
23
2
2
1
23
1
33 ym
Qym
ym
Qym
3
1
3
22
2
2
1
2
3
11yy
gm
yym
Q
For a hydraulic jump in horizontal, frictionless channel
1
2
2
1
2
4
1
233
1
22
1
1
3 y
ywhere
y
yym
g
Q
0.26.02.11
2 y
y
24192.012
2126.0
3
12
2352
g
Q
On simplifying
In the present problem m=1,
smQ /541.1 3
TAgA
QF
52
2
62
2
3
22 22
ygm
Q
ygm
myQ
gA
TQF
494.2
222.66.0181.9
541.12
1
5
2
2
1
F
F
,2
52
22
ygm
QF
II. For triangular channel as such
Froude number at the end of the jump:Since
657.560.0
20.12525
1
2
2
1
y
y
F
F
441.0657.5494.22 F
g
Vy
g
VyEEEL
22
2
22
2
1121
smV
mA
smV
mA
/070.144.154.1
44.12.11
/281.436.0541.1
36.06.01
2
22
2
1
22
1
mEL 276.0258.1534.1
81.92
070.12.1
81.92
281.46.0
22
III. Energy loss
Class Question Water flows in a wide channel at q =10 ππ/(s.m) and ππ = 1.25 m. If the flow undergoes a hydraulic jump, compute (a) ππ, (b) π½π, (c) πππ, (d) ππ, (e) the percentage
dissipation, (f) the power dissipated per unit width, and (g) the temperature rise due to dissipation if πͺπ 4200 J/(kg. K).
Solution:π½π =
π
ππ=ππππ/(s.m)
π. ππ π= π. π π/π
The upstream Froude number is therefore
πππ =π½π
ππππ/π
=π. π
π. ππ(π. ππ) π/π= π. πππ
This is a weak jump. The depth ππ is obtained from πππ
ππ= βπ + (π + ππππ
π)π/π
πππππ
= βπ + (π + π(π. πππ)π)ππ = π. ππ
ππ =π
πππ π. ππ = π. πππ
The downstream velocity is π½π =π½πππππ
=π. π(π. ππ)
π. ππ= π. πππ/π
The downstream Froude number is πππ =π½π
ππππ/π
=π. ππ
π. ππ(π. ππ) π/π= π. πππ
As expected, πππ is subcritical, the dissipation loss is ππ =(π. ππ β π. ππ)π
π(π. ππ)(π. ππ)= π. ππππ
The percentage dissipation relates ππ to upstream energy:
π¬π = ππ +π½ππ
ππ= π. ππ +
ππ
π(π. ππ)= π. ππ
Hence percentage loss = πππππ
π¬π=
πππ(π.πππ)
π.ππ= ππ πππππππ
The power dissipated per unit width is
Power= πππππ = πππππ΅
ππ ππππ
π.ππ. ππππ = ππ. π ππΎ/π
Finally, the mass flow rate is αΆπ = ππ (1000 kg/ππ)[10 ππ/(s.m)] 10,000 kg/(s m), and the
temperature rise from the steady flow energy equation is
Power dissipated = αΆπππβπ»
ππππππΎ
π= 10,000 kg/(s m) ππππ
π±
ππ.π² βπ»
from which βπ» = π. πππππ²
The dissipation is large, but the temperature rise is negligible