Hidro
-
Upload
ketut-swandana -
Category
Technology
-
view
740 -
download
1
description
Transcript of Hidro
Monroe L. Weber-Shirk School of Civil and
Environmental Engineering
Open Channel FlowOpen Channel Flow
Open Channel FlowOpen Channel Flow Liquid (water) flow with a ____ ________
(interface between water and air) relevant for
natural channels: rivers, streams engineered channels: canals, sewer
lines or culverts (partially full), storm drains
of interest to hydraulic engineers location of free surface velocity distribution discharge - stage (______) relationships optimal channel design
Liquid (water) flow with a ____ ________ (interface between water and air)
relevant for natural channels: rivers, streams engineered channels: canals, sewer
lines or culverts (partially full), storm drains
of interest to hydraulic engineers location of free surface velocity distribution discharge - stage (______) relationships optimal channel design
free surfacefree surface
depthdepth
Topics in Open Channel FlowTopics in Open Channel Flow
Uniform Flow Discharge-Depth relationships
Channel transitions Control structures (sluice gates, weirs…) Rapid changes in bottom elevation or cross section
Critical, Subcritical and Supercritical Flow Hydraulic Jump Gradually Varied Flow
Classification of flows Surface profiles
Uniform Flow Discharge-Depth relationships
Channel transitions Control structures (sluice gates, weirs…) Rapid changes in bottom elevation or cross section
Critical, Subcritical and Supercritical Flow Hydraulic Jump Gradually Varied Flow
Classification of flows Surface profiles
normal depthnormal depth
Classification of FlowsClassification of Flows Steady and Unsteady
Steady: velocity at a given point does not change with time
Uniform, Gradually Varied, and Nonuniform Uniform: velocity at a given time does not change
within a given length of a channel Gradually varied: gradual changes in velocity with
distance
Laminar and Turbulent Laminar: flow appears to be as a movement of thin
layers on top of each other Turbulent: packets of liquid move in irregular paths
Steady and Unsteady Steady: velocity at a given point does not change with
time
Uniform, Gradually Varied, and Nonuniform Uniform: velocity at a given time does not change
within a given length of a channel Gradually varied: gradual changes in velocity with
distance
Laminar and Turbulent Laminar: flow appears to be as a movement of thin
layers on top of each other Turbulent: packets of liquid move in irregular paths
(Temporal)(Temporal)
(Spatial)(Spatial)
Momentum and Energy Equations
Momentum and Energy Equations
Conservation of Energy “losses” due to conversion of turbulence to heat useful when energy losses are known or small
____________
Must account for losses if applied over long distances _______________________________________________
Conservation of Momentum “losses” due to shear at the boundaries useful when energy losses are unknown
____________
Conservation of Energy “losses” due to conversion of turbulence to heat useful when energy losses are known or small
____________
Must account for losses if applied over long distances _______________________________________________
Conservation of Momentum “losses” due to shear at the boundaries useful when energy losses are unknown
____________
ContractionsContractions
ExpansionExpansion
We need an equation for lossesWe need an equation for losses
Given a long channel of constant slope and cross section find the relationship between discharge and depth
Assume Steady Uniform Flow - ___ _____________ prismatic channel (no change in _________ with distance)
Use Energy and Momentum, Empirical or Dimensional Analysis?
What controls depth given a discharge? Why doesn’t the flow accelerate?
Given a long channel of constant slope and cross section find the relationship between discharge and depth
Assume Steady Uniform Flow - ___ _____________ prismatic channel (no change in _________ with distance)
Use Energy and Momentum, Empirical or Dimensional Analysis?
What controls depth given a discharge? Why doesn’t the flow accelerate?
Open Channel Flow: Discharge/Depth Relationship
Open Channel Flow: Discharge/Depth Relationship
PP
no accelerationno accelerationgeometrygeometry
Force balanceForce balance
AA
Steady-Uniform Flow: Force Balance
Steady-Uniform Flow: Force Balance
W
W sin
x
a
b
c
d
Shear force
Energy grade line
Hydraulic grade line
Shear force =________
0sin xPxA o 0sin xPxA o
sin P
Ao sin
P
Ao
hR = P
AhR =
P
A
sin
cos
sin S
sin
cos
sin S
W cos
g
V
2
2
g
V
2
2
Wetted perimeter = __
Gravitational force = ________
Hydraulic radiusHydraulic radius
Relationship between shear and velocity? ______________
oP x
P
A x sin
TurbulenceTurbulence
Geometric parameters ___________________ ___________________ ___________________
Write the functional relationship
Does Fr affect shear? _________
Geometric parameters ___________________ ___________________ ___________________
Write the functional relationship
Does Fr affect shear? _________
P
ARh
P
ARh Hydraulic radius (Rh)Hydraulic radius (Rh)
Channel length (l)Channel length (l)
Roughness ()Roughness ()
Open Conduits:Dimensional Analysis
Open Conduits:Dimensional Analysis
f , ,Re,ph h
lC r
R Reæ ö
= ç ÷è øF ,M, Wf , ,Re,p
h h
lC r
R Reæ ö
= ç ÷è øF ,M, W
VFr
yg=No!No!
Pressure Coefficient for Open Channel Flow?
Pressure Coefficient for Open Channel Flow?
2
2C
V
pp
2
2C
V
pp
2
2C
V
ghlhl
2
2C
V
ghlhl
2
2C
f
fS
gS l
V= 2
2C
f
fS
gS l
V=
l fh S l=l fh S l=
lhp lhp Pressure CoefficientPressure Coefficient
Head loss coefficientHead loss coefficient
Friction slope coefficientFriction slope coefficient
(Energy Loss Coefficient)(Energy Loss Coefficient)
Friction slopeFriction slope
Slope of EGLSlope of EGL
Dimensional AnalysisDimensional Analysis
f , ,RefS
h h
lC
R Reæ ö
= ç ÷è øf , ,Re
fSh h
lC
R Reæ ö
= ç ÷è ø
f
hS
RC
ll=
f
hS
RC
ll=
2
2 f hgS l RV l
l=2
2 f hgS l RV l
l=2 f hgS R
Vl
=2 f hgS R
Vl
=2
f h
gV S R
l=
2f h
gV S R
l=
f ,RefS
h h
lC
R Reæ ö
= ç ÷è øf ,Re
fSh h
lC
R Reæ ö
= ç ÷è øHead loss length of channelHead loss length of channel
f ,Ref
hS
h
RC
l Re
læ ö
= =ç ÷è øf ,Re
f
hS
h
RC
l Re
læ ö
= =ç ÷è ø
2
2f
fS
gS lC
V= 2
2f
fS
gS lC
V=
(like f in Darcy-Weisbach)
Chezy equation (1768)Chezy equation (1768)
Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the water-supply system of Paris
Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the water-supply system of Paris
h fV C R S= h fV C R S=
150 < C < 60s
m
s
m 150 < C < 60
s
m
s
mwhere C = Chezy coefficientwhere C = Chezy coefficient
where 60 is for rough and 150 is for smoothalso a function of R (like f in Darcy-Weisbach)
where 60 is for rough and 150 is for smoothalso a function of R (like f in Darcy-Weisbach)
2f h
gV S R
l=
2f h
gV S R
l=compare
Darcy-Weisbach equation (1840)Darcy-Weisbach equation (1840)
where d84 = rock size larger than 84% of the rocks in a random sample
where d84 = rock size larger than 84% of the rocks in a random sample
For rock-bedded streamsFor rock-bedded streams
f = Darcy-Weisbach friction factorf = Darcy-Weisbach friction factor
2
84
1
1.2 2.03log h
fRd
=æ öé ù
+ ê úç ÷è øë û
2
84
1
1.2 2.03log h
fRd
=æ öé ù
+ ê úç ÷è øë û
4
4
P
2
d
d
d
ARh
4
4
P
2
d
d
d
ARh
2
2l
l Vh f
d g=
2
2l
l Vh f
d g=
2
4 2lh
l Vh f
R g=
2
4 2lh
l Vh f
R g=
2
4 2fh
l VS l f
R g=
2
4 2fh
l VS l f
R g=
2
8f h
VS R f
g=
2
8f h
VS R f
g=
8f h
gV S R
f=
8f h
gV S R
f=
Manning Equation (1891)Manning Equation (1891)
Most popular in U.S. for open channels Most popular in U.S. for open channels
(English system)(English system)
1/2o
2/3h SR
1
nV 1/2
o2/3h SR
1
nV
1/2o
2/3h SR
49.1
nV 1/2
o2/3h SR
49.1
nV
VAQ VAQ
2/13/21oh SAR
nQ 2/13/21
oh SARn
Q very sensitive to nvery sensitive to n
Dimensions of Dimensions of nn??Dimensions of Dimensions of nn??
Is Is nn only a function of roughness? only a function of roughness?Is Is nn only a function of roughness? only a function of roughness?
(MKS units!)(MKS units!)
NO!NO!
T /L1/3T /L1/3
Bottom slopeBottom slope
Values of Manning nValues of Manning nLined Canals n
Cement plaster 0.011 Untreated gunite 0.016 Wood, planed 0.012 Wood, unplaned 0.013 Concrete, trowled 0.012 Concrete, wood forms, unfinished 0.015 Rubble in cement 0.020 Asphalt, smooth 0.013 Asphalt, rough 0.016
Natural Channels Gravel beds, straight 0.025 Gravel beds plus large boulders 0.040 Earth, straight, with some grass 0.026 Earth, winding, no vegetation 0.030 Earth , winding with vegetation 0.050
Lined Canals n Cement plaster 0.011 Untreated gunite 0.016 Wood, planed 0.012 Wood, unplaned 0.013 Concrete, trowled 0.012 Concrete, wood forms, unfinished 0.015 Rubble in cement 0.020 Asphalt, smooth 0.013 Asphalt, rough 0.016
Natural Channels Gravel beds, straight 0.025 Gravel beds plus large boulders 0.040 Earth, straight, with some grass 0.026 Earth, winding, no vegetation 0.030 Earth , winding with vegetation 0.050
d = median size of bed materiald = median size of bed material
n = f(surface roughness, channel irregularity, stage...)
n = f(surface roughness, channel irregularity, stage...)
6/1038.0 dn 6/1038.0 dn
6/1031.0 dn 6/1031.0 dn d in ftd in ftd in md in m
Trapezoidal ChannelTrapezoidal Channel
Derive P = f(y) and A = f(y) for a trapezoidal channel
How would you obtain y = f(Q)?
Derive P = f(y) and A = f(y) for a trapezoidal channel
How would you obtain y = f(Q)?
z1
b
yzyybA 2 zyybA 2
byzyP 2/1222 byzyP 2/1222
bzyP 2/1212 bzyP 2/1212
2/13/21oh SAR
nQ 2/13/21
oh SARn
Q
Use Solver!Use Solver!
Flow in Round ConduitsFlow in Round Conduits
r
yrarccos
r
yrarccos
cossin2 rA cossin2 rA
sin2rT sin2rT
y
T
A
r
rP 2 rP 2
radiansradians
Maximum discharge when y = ______0.938d0.938d
( ) ( )sin cosr rq q=( ) ( )sin cosr rq q=
Open Channel Flow: Energy Relations
Open Channel Flow: Energy Relations
2g
V21
1
2g
V22
2
xSo
2y1y
x
L fh S x= Denergy grade linehydraulic grade line
velocity head
Bottom slope (So) not necessarily equal to surface slope (Sf)
water surfacewater surface
Energy relationshipsEnergy relationships
2 21 1 2 2
1 1 2 22 2 L
p V p Vz z h
g ga a
g g+ + = + + +
2 21 2
1 22 2o f
V Vy S x y S x
g g+ D + = + + D
Turbulent flow ( 1)
z - measured from horizontal datum
y - depth of flow
Pipe flow
Energy Equation for Open Channel Flow
2 21 2
1 22 2o f
V Vy S x y S x
g g+ + D = + + D
From diagram on previous slide...
Specific EnergySpecific Energy
The sum of the depth of flow and the velocity head is the specific energy:
The sum of the depth of flow and the velocity head is the specific energy:
g
VyE
2
2
If channel bottom is horizontal and no head loss21 EE
y - potential energy
g
V
2
2
- kinetic energy
For a change in bottom elevation1 2E y E- D =
xSExSE o f21
y
Specific EnergySpecific Energy
In a channel with constant discharge, Q
2211 VAVAQ
2
2
2gA
QyE
g
VyE
2
2
where A=f(y)
Consider rectangular channel (A=By) and Q=qB
2
2
2gy
qyE
A
B
y
3 roots (one is negative)
q is the discharge per unit width of channel
0123456789
10
0 1 2 3 4 5 6 7 8 9 10
E
y
Specific Energy: Sluice GateSpecific Energy: Sluice Gate
2
2
2gy
qyE
2
2
2gy
qyE
1
2
21 EE
sluice gatey1
y2
EGL
y1 and y2 are ___________ depths (same specific energy)
Why not use momentum conservation to find y1?
q = 5.5 m2/sy2 = 0.45 mV2 = 12.2 m/s
E2 = 8 m
alternatealternateGiven downstream depth and discharge, find upstream depth.
0
1
2
3
4
0 1 2 3 4
E
ySpecific Energy: Raise the Sluice
GateSpecific Energy: Raise the Sluice
Gate
2
2
2gy
qyE
1 2
E1 E2E1 E2
sluice gate
y1
y2
EGL
as sluice gate is raised y1 approaches y2 and E is minimized: Maximum discharge for given energy.
0
1
2
3
4
0 1 2 3 4
E
y
Specific Energy: Step UpSpecific Energy: Step UpShort, smooth step with rise y in channel
y
1 2E E y= +D
Given upstream depth and discharge find y2
0
1
2
3
4
0 1 2 3 4
E
y
Increase step height?
PPAA
Critical FlowCritical Flow
T
dy
y
T=surface width
Find critical depth, yc
2
2
2gA
QyE
0dy
dE
TdydA 0dEdy
= =
3
2
1c
c
gA
TQ
Arbitrary cross-section
A=f(y)
2
3
2
FrgA
TQ 22
FrgA
TV
dA
AD
T= Hydraulic Depth
2
31Q dAgA dy
-
0
1
2
3
4
0 1 2 3 4
E
y
yc
Critical Flow:Rectangular channel
Critical Flow:Rectangular channel
yc
T
Ac
3
2
1c
c
gA
TQ
qTQ TyA cc
3
2
33
32
1cc gy
q
Tgy
Tq
3/12
g
qyc
3cgyq
Only for rectangular channels!
cTT
Given the depth we can find the flow!
Critical Flow Relationships:Rectangular Channels
Critical Flow Relationships:Rectangular Channels
3/12
g
qyc cc yVq
g
yVy ccc
223
g
Vy cc
2
1gy
V
c
cFroude numberFroude number
velocity head =
because
g
Vy cc
22
2
2
cc
yyE Eyc
3
2
forcegravity force inertial
0.5 (depth)0.5 (depth)
g
VyE
2
2
Kinetic energyPotential energy
Critical FlowCritical Flow
Characteristics Unstable surface Series of standing waves
Occurrence Broad crested weir (and other weirs) Channel Controls (rapid changes in cross-section) Over falls Changes in channel slope from mild to steep
Used for flow measurements ___________________________________________
Characteristics Unstable surface Series of standing waves
Occurrence Broad crested weir (and other weirs) Channel Controls (rapid changes in cross-section) Over falls Changes in channel slope from mild to steep
Used for flow measurements ___________________________________________Unique relationship between depth and discharge
Difficult to measure depth
0
1
2
3
4
0 1 2 3 4
E
y
Broad-crested WeirBroad-crested Weir
H
Pyc
E
3cgyq 3
cQ b gy=
Eyc
3
2
3/ 23/ 22
3Q b g Eæö=
è ø
Cd corrects for using H rather than E.
3/12
g
qyc
Broad-crestedweir
E measured from top of weirE measured from top of weir
3/ 223dQ C b g Hæ ö=
è ø
Hard to measure yc
Broad-crested Weir: ExampleBroad-crested Weir: Example
Calculate the flow and the depth upstream. The channel is 3 m wide. Is H approximately equal to E?
Calculate the flow and the depth upstream. The channel is 3 m wide. Is H approximately equal to E?
0.5yc
E
Broad-crestedweir
yc=0.3 m
Solution
How do you find flow?____________________
How do you find H?______________________
Critical flow relation
Energy equation
Hydraulic JumpHydraulic Jump
Used for energy dissipation Occurs when flow transitions from
supercritical to subcritical base of spillway
We would like to know depth of water downstream from jump as well as the location of the jump
Which equation, Energy or Momentum?
Used for energy dissipation Occurs when flow transitions from
supercritical to subcritical base of spillway
We would like to know depth of water downstream from jump as well as the location of the jump
Which equation, Energy or Momentum?
Hydraulic Jump!Hydraulic Jump!
Hydraulic JumpHydraulic Jump
y1
y2
L
EGLhL
Conservation of Momentum
221121 ApApQVQV
2
gyp
A
QV
222211
2
2
1
2 AgyAgy
A
Q
A
Q
xx ppxx FFMM2121 xx ppxx FFMM
2121
12
11 AVM x 12
11 AVM x
22
22 AVM x 22
22 AVM x
sspp FFFWMM 2121 sspp FFFWMM 2121
Hydraulic Jump:Conjugate DepthsHydraulic Jump:
Conjugate Depths
Much algebra
For a rectangular channel make the following substitutionsByA 11VByQ
11
1
VFr
gy= Froude number
21
12 811
2Fr
yy
valid for slopes < 0.02
Hydraulic Jump:Energy Loss and Length
Hydraulic Jump:Energy Loss and Length
No general theoretical solution
Experiments show
26yL 204 1 Fr
•Length of jump
•Energy Loss
2
2
2gy
qyE
LhEE 21
significant energy loss (to turbulence) in jump
21
312
4 yy
yyhL
algebra
for
Gradually Varied FlowGradually Varied Flow2 2
1 21 22 2o f
V Vy S x y S x
g g+ + D = + + D Energy equation for
non-uniform, steady flow
12 yydy
2
2 f o
Vdy d S dx S dx
g
æ ö+ + =ç ÷è ø
PPAA
T
dy
y
dy
dxS
dy
dxS
g
V
dy
d
dy
dyof
2
2
( )2 2
2 12 1 2 2o f
V VS dx y y S dx
g g
æ ö= - + - +ç ÷è ø
Gradually Varied FlowGradually Varied Flow
2
3
2
3
2
2
22
2
2
22Fr
gA
TQ
dy
dA
gA
Q
gA
Q
dy
d
g
V
dy
d
fo SSFrdx
dy 21
dy
dxS
dy
dxS
g
V
dy
d
dy
dyof
2
2
dy
dxS
dy
dxSFr of 21
Change in KEChange in PEChange in KEChange in PE
We are holding Q constant!
21 Fr
SS
dx
dy fo
Gradually Varied FlowGradually Varied Flow
21 Fr
SS
dx
dy fo
Governing equation
for gradually varied flow
Gives change of water depth with distance along channel Note
So and Sf are positive when sloping down in direction of flow
y is measured from channel bottom dy/dx =0 means water depth is constant
Gives change of water depth with distance along channel Note
So and Sf are positive when sloping down in direction of flow
y is measured from channel bottom dy/dx =0 means water depth is constant
yn is when o fS S=
Surface ProfilesSurface Profiles Mild slope (yn>yc)
in a long channel subcritical flow will occur
Steep slope (yn<yc) in a long channel supercritical flow will occur
Critical slope (yn=yc) in a long channel unstable flow will occur
Horizontal slope (So=0) yn undefined
Adverse slope (So<0) yn undefined
Mild slope (yn>yc) in a long channel subcritical flow will occur
Steep slope (yn<yc) in a long channel supercritical flow will occur
Critical slope (yn=yc) in a long channel unstable flow will occur
Horizontal slope (So=0) yn undefined
Adverse slope (So<0) yn undefined Note: These slopes are f(Q)!
Normal depth
Steep slope (S2)
Hydraulic Jump
Sluice gateSteep slope
Obstruction
Surface ProfilesSurface Profiles
21 Fr
SS
dx
dy fo
S0 - Sf 1 - Fr2 dy/dx
+ + +
- + -
- - + 0
1
2
3
4
0 1 2 3 4
E
y
yn
yc
More Surface ProfilesMore Surface Profiles
S0 - Sf 1 - Fr2 dy/dx
1 + + +
2 + - -
3 - - + 0
1
2
3
4
0 1 2 3 4
E
y
yn
yc
21 Fr
SS
dx
dy fo
Direct Step MethodDirect Step Method
xSg
VyxS
g
Vy fo
22
22
2
21
1
of SS
g
V
g
Vyy
x
22
22
21
21
energy equation
solve for x
1
1
y
qV
2
2
y
qV
2
2
A
QV
1
1
A
QV
rectangular channel
prismatic channel
Direct Step MethodFriction Slope
Direct Step MethodFriction Slope
2 2
4/3fh
n VS
R=
2 2
4/32.22fh
n VS
R=
2
8fh
fVS
gR=
Manning Darcy-Weisbach
SI units
English units
Direct StepDirect Step
Limitation: channel must be _________ (so that velocity is a function of depth only and not a function of x)
Method identify type of profile (determines whether y is + or -) choose y and thus yn+1
calculate hydraulic radius and velocity at yn and yn+1
calculate friction slope yn and yn+1 calculate average friction slope calculate x
Limitation: channel must be _________ (so that velocity is a function of depth only and not a function of x)
Method identify type of profile (determines whether y is + or -) choose y and thus yn+1
calculate hydraulic radius and velocity at yn and yn+1
calculate friction slope yn and yn+1 calculate average friction slope calculate x
prismaticprismatic
Direct Step MethodDirect Step Method
=(G16-G15)/((F15+F16)/2-So)
A B C D E F G H I J K L My A P Rh V Sf E Dx x T Fr bottom surface0.900 1.799 4.223 0.426 0.139 0.00004 0.901 0 3.799 0.065 0.000 0.9000.870 1.687 4.089 0.412 0.148 0.00005 0.871 0.498 0.5 3.679 0.070 0.030 0.900
=y*b+y^2*z
=2*y*(1+z^2)^0.5 +b
=A/P
=Q/A
=(n*V)^2/Rh^(4/3)
=y+(V^2)/(2*g)
of SS
g
V
g
Vyy
x
22
22
21
21
Standard StepStandard Step
Given a depth at one location, determine the depth at a second location
Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate
Can solve in upstream or downstream direction upstream for subcritical downstream for supercritical
Find a depth that satisfies the energy equation
Given a depth at one location, determine the depth at a second location
Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate
Can solve in upstream or downstream direction upstream for subcritical downstream for supercritical
Find a depth that satisfies the energy equation
xSg
VyxS
g
Vy fo
22
22
2
21
1
What curves are available?What curves are available?
S1
S3
Is there a curve between yc and yn that decreases in depth in the upstream direction?
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
05101520
distance upstream (m)
elev
atio
n (m
)
bottom
surface
yc
yn
Wave CelerityWave Celerity
21
2
1gyF 2
2
2
1yygF
2221
2
1yyygFFFr
F1
y+yV+VV
Vw
unsteady flow
y y y+yV+V-VwV-Vw
steady flow
V+V-VwV-Vw
F2
Wave Celerity:Momentum Conservation
Wave Celerity:Momentum Conservation
wwwr VVVVVVVyF
VVVyF wr
VVVyyyyg w 22
2
1
VVVyyyg w 22
1
VVVyg w
y y+yV+V-VwV-Vw
steady flow
12 MMFr yVVM w2
1 Per unit widthPer unit width
Wave CelerityWave Celerity
ww VVVyyVVy
www yVyVVyVyyVyVyVyV
y
yVVV w
VVVyg w
y
yVVyg w
2
2wVVgy wVVc gyc
Mass conservation
y y+yV+V-VwV-Vw
steady flow
Momentum
c
VFr
yg
V
Wave PropagationWave Propagation
Supercritical flow c<V waves only propagate downstream water doesn’t “know” what is happening downstream _________ control
Critical flow c=V
Subcritical flow c>V waves propagate both upstream and downstream
Supercritical flow c<V waves only propagate downstream water doesn’t “know” what is happening downstream _________ control
Critical flow c=V
Subcritical flow c>V waves propagate both upstream and downstream
upstreamupstream
Most Efficient Hydraulic Sections
Most Efficient Hydraulic Sections
A section that gives maximum discharge for a specified flow area Minimum perimeter per area
No frictional losses on the free surface Analogy to pipe flow Best shapes
best best with 2 sides best with 3 sides
A section that gives maximum discharge for a specified flow area Minimum perimeter per area
No frictional losses on the free surface Analogy to pipe flow Best shapes
best best with 2 sides best with 3 sides
Why isn’t the most efficient hydraulic section the best design?
Why isn’t the most efficient hydraulic section the best design?
Minimum area = least excavation only if top of channel is at gradeMinimum area = least excavation only if top of channel is at grade
Cost of linerCost of liner
Complexity of form workComplexity of form work
Erosion constraint - stability of side wallsErosion constraint - stability of side walls
Open Channel Flow Discharge Measurements
Open Channel Flow Discharge Measurements
Discharge Weir
broad crested sharp crested triangular
Venturi Flume Spillways Sluice gates
Velocity-Area-Integration
Discharge Weir
broad crested sharp crested triangular
Venturi Flume Spillways Sluice gates
Velocity-Area-Integration
Discharge MeasurementsDischarge Measurements
Sharp-Crested Weir
Triangular Weir
Broad-Crested Weir
Sluice Gate
Sharp-Crested Weir
Triangular Weir
Broad-Crested Weir
Sluice Gate
5/ 282 tan
15 2dQ C g Hqæ ö=
è ø
3/ 223dQ C b g Hæ ö=
è ø
3/ 222
3 dQ C b gH=
12d gQ C by gy=
Explain the exponents of H!
SummarySummary
All the complications of pipe flow plus additional parameter... _________________
Various descriptions of head loss term Chezy, Manning, Darcy-Weisbach
Importance of Froude Number Fr>1 decrease in E gives increase in y Fr<1 decrease in E gives decrease in y Fr=1 standing waves (also min E given Q)
Methods of calculating location of free surface
All the complications of pipe flow plus additional parameter... _________________
Various descriptions of head loss term Chezy, Manning, Darcy-Weisbach
Importance of Froude Number Fr>1 decrease in E gives increase in y Fr<1 decrease in E gives decrease in y Fr=1 standing waves (also min E given Q)
Methods of calculating location of free surface
free surface locationfree surface location
0
1
2
3
4
0 1 2 3 4
E
y
Broad-crested Weir: SolutionBroad-crested Weir: Solution
0.5yc
E
Broad-crestedweir
yc=0.3 m
3cgyq
32 3.0)/8.9( msmq
smq /5144.0 2
smqLQ /54.1 3Eyc
3
2myE c 45.0
2
32
1 2 0.95E E y m= +D =
21
2
112gy
qyE
435.05.011 myH935.01 y
2
1 1212
qE y
gE- @
Sluice GateSluice Gate
2 m
10 cm
S = 0.005
Sluice gatereservoir
Summary/OverviewSummary/Overview
Energy lossesDimensional AnalysisEmpirical
Energy lossesDimensional AnalysisEmpirical
8f h
gV S R
f=
8f h
gV S R
f=
1/2o
2/3h SR
1
nV 1/2
o2/3h SR
1
nV
Energy EquationEnergy Equation
Specific Energy Two depths with same energy!
How do we know which depth is the right one?
Is the path to the new depth possible?
Specific Energy Two depths with same energy!
How do we know which depth is the right one?
Is the path to the new depth possible?
2 21 2
1 22 2o f
V Vy S x y S x
g g+ + D = + + D
2
22q
ygy
= +g
VyE
2
2
2
22Q
ygA
= +
0
1
2
3
4
0 1 2 3 4
E
y
0
1
2
3
4
0 1 2 3 4
E
y
Specific Energy: Step UpSpecific Energy: Step UpShort, smooth step with rise y in channel
y
1 2E E y= +D
Given upstream depth and discharge find y2
0
1
2
3
4
0 1 2 3 4
E
y
Increase step height?
Critical DepthCritical Depth
Minimum energy for qWhenWhen kinetic = potential!Fr=1
Fr>1 = SupercriticalFr<1 = Subcritical
Minimum energy for qWhenWhen kinetic = potential!Fr=1
Fr>1 = SupercriticalFr<1 = Subcritical
0dy
dE
0
1
2
3
4
0 1 2 3 4
E
y
g
Vy cc
22
2
3
TQ
gA=
3c
q
gy=c
c
VFr
y g=
What next?What next?
Water surface profiles Rapidly varied flow
A way to move from supercritical to subcritical flow (Hydraulic Jump)
Gradually varied flow equations Surface profiles Direct step Standard step
Water surface profiles Rapidly varied flow
A way to move from supercritical to subcritical flow (Hydraulic Jump)
Gradually varied flow equations Surface profiles Direct step Standard step