Lecture13 Gradually Varied Flow1 2
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Transcript of Lecture13 Gradually Varied Flow1 2
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Gradually Varied Flow I+II
Hydromechanics VVR090
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Gradually Varied Flow
Depth of flow varies with longitudinal distance.
Occurs upstream and downstream control sections.
Governing equation:
21
=
o fS Sdy
dx Fr
(previously Sf= 0 was studied)
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Derivation of Governing Equation
Total energy:
2
2uH z y
g= + +
Differentiating with respect to distance:
( )2 / 2= + +
d u gdH dz dy
dx dx dx dx
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=
=
f
o
dH
Sdx
dzSdx
For a given flow rate:
( )2 2 22
3 3
/ 2d u g Q dA dy Q T dy dyFr
dx gA dy dx gA dx dx
= = =
(slope of energy grade line)
(bottom slope)
21
=
o fS Sdy
dx Fr Resulting equation:
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Definition of Water Surface Slope
Water surface slope dy/dx is defined with respect tothe channel bottom.
Hydrostatic pressure distr ibution is assumed
(streamlines should be reasonably straight and parallel).
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The head loss for a specific reach is equal to the
head loss in the reach for a uniform flow having the
same R and u. Manning equation yields.
The slope of the channel is small
No air entrainment
Fixed velocity distribution
Resistance coefficient constant in the reach under
consideration
2 2
4 / 3f
n uS
R=
Assumptions made when solving the gradually varied flow
equation:
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Classification of Gradually Varied Flow Profiles
The following condit ions prevail:
Ify < yN, then Sf > So
Ify > yN, then Sf < So
IfFr> 1, then y < yc
IfFr< 1, then y > yc
IfSf = So, then y = yN
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Water surface profi les may be classif ied with respect to:
the channel slope
the relationship between y, yN, and yc.
Profile categories:
M (mild) 0 < So < Sc
S (steep) So > Sc > 0
C (critical) So = Sc
A (adverse) So < 0
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Gradually Varied Flow
Profile Classification I
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Gradually Varied Flow Profile Classification II
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Mild Slope (M-Profiles)
Profile types:
1: y > yN > yc => So > Sf and Fr< 1
=> dy/dx > 0
2: yN > y > yc => So < Sf and Fr< 1
=> dy/dx < 0
3: yN > yC > y => So < Sf and Fr> 1=> dy/dx > 0
0 < So < Sc
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Steep Slope (S-Profiles)
Profile types:
1: y > yc > yN => So > Sf and Fr< 1
=> dy/dx > 0
2: yc > y > yN => So > Sf and Fr> 1
=> dy/dx < 0
3: yc > yN > y => So < Sf and Fr> 1=> dy/dx > 0
0 < Sc < So
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Final Form of Water Surface Profile
1. y , Sf 0, Fr 0, and dy/dx So
2. y yN, Sf So, and dy/dx 0
3. y yc, Fr 1, and dy/dx
21
=
o fS Sdy
dx Fr
Asymptotic conditions:
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Transition from Subcrit ical to Supercritical Flow
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Transition from Supercritical to Subcrit ical Flow
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Example: Flow into a Channel from a Reservoir
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Flow Controls
determine the depth in channel either upstream or
downstream such points.
usually feature a change from subcritical to supercrit icalflow
occur at physical barriers, for example, sluice gates,
dams, weirs, drop structures, or changes in channelslope
Locations in the channel where the relationship between the
water depth and flow rate is known (or controllable).
Controls:
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Strategy for Analysis of Open Channel Flow
1. Start at control points2. Proceed upstream or downstream depending on
whether subcritical or supercritical flow occurs,
respectively
Typical approach in the analysis:
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Computation of Gradually Varied Flow
21
=
o fS Sdy
dx Fr Governing equation:
Solutions must begin at a control section and proceedin the direction in which the control operates.
Gradually varied flow may approach uniform flow
asymptotically, but from a practical point of view a
reasonable definition of convergence is applied.
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Uniform Channel
Prismatic channel with constant slope and resistance coefficient.
Apply energy equation over a small distance Dx:
2
2o f
d uy S S
dx g
+ =
Express the equation in difference form:
( )2
2o f
uy S S x
g
+ =
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Over the short distance Dx assume that Manningsequation is suitable to describe the frictional losses (S
f
):
2 2
4 / 3f
n uS
R
=
The equation to be solved may be written:
( )
( )
2
2 2 4 / 3
/ 2
/o mean
y u gx
S n u R
+ =
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Dx i
Reach i
x
y i y i+1
( ) ( )
( )
2 2
1
2 2 4 / 3
1/ 2
/ 2 / 2
/
i ii
o i
y u g y u gx
S n u R
+
+
+ + =
All quantities known at i. Assume yi+1
and computeDx i (u i+1 given by the continuity equation).
u i
u i+1
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Example 6.1
A trapezoidal channel with b = 6.1 m, n = 0.025, z = 2, and So =
0.001 carries a discharge of 28 m3/s. If this channel terminates in
a free overfall, determine the gradually varied flow profile by the
step method.
b = 6.1 m
2
1yN
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Solution:
Compute normal water depth.
( )
( )
2 / 3
2
2
1
2 1
2 1
o
N N
N
N N
N
Q AR S n
A b zy y
P b y z
b zy yR
b y z
=
= += + +
+
= + +
yN = 1.91 m
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Compute crit ical water depth:
( )
1/
2
c c c
c c c
c
u QFr
gD A gA T
A b zy y
T b zy
= = =
= +
= +
yc = 1.14 m
yN > y > yc
Mild slope (yN > yc)M2 profile
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Table for step calculation:
y A P R u u2
/2g Sf Sfav Dx S (Dx)1.14 9.55 11.20 0.85 2.93 0.438 0.00670.0058 3 3
1.24 10.64 11.64 0.91 2.63 0.353 0.0049
0.0044 9.3 12.3
1.32 11.54 12.00 0.96 2.43 0.300 0.0039
and so on( ) ( )2 2
1
, 1/ 2
/ 2 / 2i i
i
o f i
y u g y u gx
S S
+
+
+ + =
( ), 1/ 2 , 1 ,1
2f i f i f i
S S S+ += +
2 2
4 / 3f
n uS
R=
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Other Solution Methods
Problem with the step method is that the water depths isobtained at arbitrary locations (i.e., the water depth is not
calculated at fixed x-locations).
By direct integration of the governing equation this problemcan be circumvented.
Different approaches for direct integration:semi-analytic
trial-and-error
finite difference
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Semi-Analytic Approach
Find solution in terms of closed-form functions (integrals).
Employ suitable approximations to these functions or
some look-up tables.
Approach OK for channels with constant properties.
(for more information, see French)
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Trial-and-Error Approach
Well-suited for computations in non-prismatic channels.
Channel properties (e.g., resistance coefficient and
shape) are a function of longitudinal distance.
Depth is obtained at specific x-locations.
Apply energy equation between two stations locatedD
xapart (z is the elevation of the water surface):
2
2 2
1 2
1 2
2
2 2
f e
f e
uz S x hg
u uz z S x h
g g
+ =
+ = + + +
he: eddy losses
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Equation is solved by trial-and-error (from 2 to 1):
1.Assume y1 u1 (continuity equation)2. Compute S
f
(and he
, if needed)
3. Compute y1 from governing equation. If this value agrees
with the assumed y1, the solution has been found.
Otherwise continue calculations.
Estimate of frictional losses:
( )1 212
f f fS S S= +
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Example 6.4
A trapezoidal channel with b = 20 ft, n = 0.025, z = 2, and So =
0.001 carries a discharge of 1000 ft3/s. If this channel terminates
in a free overfall and there are no eddy losses, determine the
gradually varied flow profile by the trial-and-error step method.
b = 20 ft
2
1yN
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Solution Table
Stn. z y A u u2/2g H1 R Sf Sfav Dx hf H2
0 103.74 3.74 103 9.71 1.46 105.20 2.81 0.00670 105.20
116 104.62 4.50 130 7.69 0.92 105.54 3.24 0.00347 0.00509 116 0.590 105.79
105.02 4.90 146 6.85 0.73 105.75 3.48 0.00251 0.00461 116 0.535 105.73
355 105.56 5.20 158 6.33 0.62 106.18 3.65 0.00201 0.00226 239 0.540 106.27
105.93 5.32 173 5.78 0.52 106.45 3.85 0.00156 0.00204 239 0.724 106.47
745 106.34 5.60 175 5.71 0.51 106.85 3.89 0.00150 0.00153 490 1.14 107.59
106.96 6.21 201 4.98 0.385 107.34 4.21 0.00103 0.00130 490 0.97 107.42
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Finite Difference Approach
Suitable for application on a computer (small length stepsDx might be needed).Can be applied for completely arbitrary channel
configurations and properties.
A range of numerical approaches are available to solve the
governing equations based on finite differences.
The equation is written in difference form and solved in terms
ofy:
( )
2
2 o fu
y S S xg
+ =
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Examples of Gradually Varied Flow
Flow in channel between two reservoirs (lakes):
1. Steep slope, low downstream water level
2. Steep slope, high downstream water level
3. Mild slope, long channel
4. Mild slope, short channel
5. Sluice gate located in the channel
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Steep Slope, Low Downstream Water Level
Critical section at inflow to channel. Normal water depth
occurs some distance downstream in the channel with Fr> 1(yN < ycr). A hydraulic jump develops before water is
discharged to the downstream lake.
Q in the channel depends on H1 and critical section.
Critical
section Hydraulic
jump
Lake
Lake
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Steep Slope, High Downstream Water Level
Downstream water level is high enough to cause dammingeffects to the upstream lake. No critical section occurs in the
inflow section. y > ycr > yN in the channel.
Q depends on H1 and H2.
No critical section
Fr < 1 in the channel,
although it is steep
LakeLake
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Mild Slope, Long Channel
Mild slope and long channel implies that normal water depth
occurs with yN > ycr. Normal water depth is also attained in
the inflow section to the channel. Non-uniform flow develops
in the downstream part of the channel before discharge tothe lake.
Q depends on H1 and yN in the inflow section.
LakeLake
uniform flow non-uniform flow
Normal water depth
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Mild Slope, Short Channel
A short channel implies that normal water depth will not
occur and y > yN > ycr. Non-uniform flow develops in the
entire channel because of the downstream effects of the
lake.
Q depends on H1 and H2.
Lake Lake
Non-uniform flow
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Sluice Gate Located in the Channel
Sluice gate cause damming upstream affecting inflow
from lake. Discharge from sluice gate depends on
upstream water surface elevation over gate opening.
Supercrit ical flow occurs downstream the gate,
followed by a hydraulic jump before the downstream
lake is encountered.
Q depends on H1 and sluice gate properties.
Jump
Sluice gate (Q a
function ofy)Lake
Lake
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Calculation Procedure for Some Gradually
Varied Flows
1. Flow from a reservoir to a long, steeply sloping channel
2. Flow from a reservoir to a long, mildly sloping channel
3. Flow from a reservoir to a short, mildly sloping channel
where a downstream water level affects the flow in the
channel
4. Flow from a reservoir to a short, steeply sloping channel
where a downstream water level affects the flow in the
channel
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Lake
Flow from a Reservoir to a Long, Steeply
Sloping channel
Critical section occurs in inflow section. Employ energy
equation from lake surface to inflow section.
2
12
1
crcr
cr
cr
uH y
g
uFrgy
= +
= =
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Flow from a Reservoir to a Long, Mildly
Sloping Channel
uniform flow non-uniform flow
Lake
Lake
Normal depth occurs in inflow section. Employ energy equation
from lake surface to inf low section.
2
1
2 / 3 1/ 2
2
1
NN
N N o
uH y
g
u R Sn
= +
=
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Flow from Reservoir to Short, Mildly Sloping Channel;
Downstream Water Level Affects Flow in Channel
Downstream lake water level affects inflow from upstream
lake. Non-uniform flow prevails. Q depends on H1 and H2.
Assume Q = Q1. Do a step calculation from downstreamlake water level to inflow section. Employ energy equation
from inflow section to upstrem lake water level. H1 is
regarded as unknown. Calculate for a new flow Q2 which
gives a new upstream lake water level.
Lake Lake
non-uniform flow
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Make a plot ofH1 as a function ofQ.
Determine the correct Q based on the actual upstream lake
water level H1.
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Flow from Reservoir to Short, Steeply Sloping Channel;
Downstream Water Level Affects Flow in Channel
LakeLake
Non-uniform flow Hydraulic
Jump
Non-uniform
flow
Critical section at inflow to channel. Make a step calculation
from upstream lake and downstream lake. The hydraulic jump
occur where the jump equation is satisfied.
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Hydraulic jump is assumed to have negligible spatial
extension.
( )
( )
22
1
1
212
2
1
1 8 12
1 1 8 12
y
Fry
y Fry
= +
= +