Basic Hydraulics: Channels Analysis and design – I.
Transcript of Basic Hydraulics: Channels Analysis and design – I.
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Basic Hydraulics: Channels Analysis and design – I
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Terminology
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Manning’s equation
• Manning’s equation originally developed for open channel flow (by an accountant, no less!)
• Usually written as
v = (1.486/n) R2/3Sf1/2
where v = velocity (ft/sec); n = Manning’s coefficient (also called Manning’s n); R = hydraulic radius (A/P in ft); P = wetted perimeter (ft); Sf = slope of energy gradient line (ft/ft) = hL / L
• Tables of n values available for various surfaces.
• Rearrange Manning's equation to solve for Sf = hL / L (head loss per unit length):
(hL / L) = [Vn / (1.486 R2/3)]2
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Hydraulic radius
• The hydraulic radius (Rh) is the cross sectional area of the flow divided by the wetted perimeter. For a circular pipe flowing full, the hydraulic radius is one-fourth of the diameter. For a wide rectangular channel, the hydraulic radius is approximately equal to the depth.
A = cross–sectional area of the flowing fluid; P = wetted perimeter.
P
ARh
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Slope-area method
• Provides a simple relationship for relating water surface elevation to discharge at a particular channel section.
• Often used for calculating tailwater at culvert outlets and storm drain outlets.
• TxDOT Hydraulic Design Manual suggests using this procedure for small stream crossings or situation for which no unusual flow characteristics are anticipated.
• If crossing is an important one, the Hydraulic Design Manual recommends using a “backwater method.” (More on that later.)
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Slope-area data needs
• Channel cross section: Choose a typical cross section downstream from crossing
• Channel roughness• Channel slope
• Use average bed slope near site• Find from surveys or topographic maps
• Use Manning’s equation to calculate water surface elevation as a function of discharge
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Hydraulic depth
• The Froude Number (Fr) represents the ratio of inertial force to gravitational force and is calculated by:
where dm is the hydraulic mean depth and is defined by dm = A/T where A is the cross-sectional area of the flow and T is the channel top width at the water surface.
m
rgd
vF
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Backwater (Frontwater) Methods• Computing water surface profiles in cases more
complex than slope-area situations uses the energy equation to estimate the water surface elevation at different sections from a known location.
• The plot of the elevations is usually called a back-water curve for M1,M2, and S1 curves and a front-water curve for M3, S2, and S3 curves.
• The variable step method is illustrated by an example to familiarize the participant with the mechanics of the method, however for practical applications use of specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)
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Backwater (Frontwater) Methods• Backwater methods start with the specific
energy at two cross sections
€
E1 = y1 +v12
2g
€
E2 = y2 +v22
2g
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Backwater (Frontwater) Methods• Then the bottom elevations are included as
is the head loss• Therefore the total head at both sections
are equal
€
E1 = y1 +v12
2g
€
E2 = y2 +v22
2g
€
E1 + z1 = E2 + z2 + hL
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Backwater (Frontwater) Methods• Next incorporate the channel bottom slopes
and the energy grade line slope to replace the elevations in terms of these slopes
€
E1 + z1 = E2 + z2 + hL
€
E1 + (z1 − z2) = E2 + hL
€
E1 + S0Δx = E2 + S fΔx
€
S0 − S f =E2 − E1
Δx
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Backwater (Frontwater) Methods• Now use some calculus and section geometry
to convert into discharge, area, and depth
€
S0 − S f = limΔx→0
E2 − E1
Δx=
dE
dx=
dE
dy
dy
dx
€
dE
dy=
d
dy[y +
v 2
2g] =
d
dy[y +
Q2
2gA2]
€
dE
dy=
d
dy[y +
Q2
2gA2] =1−
Q2
2gA3
dA
dy
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Backwater (Frontwater) Methods• Now use some calculus and section geometry
to convert into discharge, area, and depth
€
S0 − S f =dE
dy
dy
dx
€
dE
dy=
d
dy[y +
Q2
2gA2] =1−
Q2
gA3
dA
dy
€
Q2
gA3
dA
dy= Fr2
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Backwater (Frontwater) Methods• Finally insert the substitutions and the
result is an equation that relates depth taper to channel geometry and specific energy
€
S0 − S f = (1− Fr2)dy
dx
€
dy
dx=
S0 − S f
1− Fr2
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Backwater (Frontwater) Methods• “Integrating” the GVF equation from a known
section forward in space or backward in space produces the front- or back-water curve (water surface profile)
€
dy
dx=
S0 − S f
1− Fr2
€
y(x) =S0 − S f
1− Fr2∫ dx
Depth
Position Geometry
GravityFriction
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Backwater (Frontwater) Methods• Recall the original expression of the Froude
Number
• Notice the area and topwidth are incorporated, some algebra and an alternate expression is
€
Fr =V
gA
T
€
Fr2 =V 2
gA
T
=V 2A2T
gA3=
Q2T
gA3
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Backwater (Frontwater) Methods• Now we have the relationships for computing
a water surface profile from some known condition
• Such computation involves:• Select a location where depth is known (or assumed)
• Determine the slope designation, and profile type (M1, S2, etc.)
• Use the designation to decide if integration is downstream or upstream.
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Backwater (Frontwater) Methods• Once these steps are completed, the simplest
method is the constant depth change, variable distance method.
• The energy equation is rearranged to solve for the spatial step as
€
E1 + S0Δx = E2 + S fΔx
€
Δx =E2 − E1
S0 − S f
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Backwater (Frontwater) Methods• This form of the equation suggests the
following algorithm
1.Start at the known section, Q must be specified.
2.Calculate specific energy for the starting section (section 1)
3.Calculate friction slope at the section (Manning’s equation solved for slope is typically used)
€
E1 = y1 +Q2
2gA12
€
S f 1=
Q2n2
1.4921
A12
1
R14 / 3
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Backwater (Frontwater) Methods• This form of the equation suggests the
following algorithm
4.Change the depth slightly, use that value as the depth at section 2
5.Calculate specific energy at section 2
€
y2 = y1 ± Δy
E2 = y2 +Q2
2gA22
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Backwater (Frontwater) Methods• This form of the equation suggests the
following algorithm
6.Calculate friction slope at section 2
7.Compute average friction slope for the reach
€
S f 2=
Q2n2
1.4921
A22
1
R24 / 3
€
S f =S f1 + S f 2
2
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Backwater (Frontwater) Methods• This form of the equation suggests the
following algorithm
8.Solve for the distance to section 2
9.Move to next section and repeat.
€
Δx =E2 − E1
S0 − S f
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Backwater Example• The figure below is a backwater curve in a
rectangular channel with discharge over a dam (somewhere to the right of the figure)
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Backwater Example• The channel is 5 meters wide, bottom slope
is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
• Our goal is to compute the water surface profile and locate the distance upstream where the water flow depth is nearly at normal depth for the channel
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Backwater Example• The channel is 5 meters wide, bottom slope
is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
• Select a location where depth is known (or assumed)
• Use the pool on the right y= 8 meters• We will call this location x=0
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Backwater Example• The channel is 5 meters wide, bottom slope
is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
• Determine the slope designation, and profile type (M1, S2, etc.)
• Compute normal depth in the channel from Manning’s equation, yn = 5 meters
• Compute critical depth in the channel by setting Froude number to unity and solving for depth, yc = 2.3 meters
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Backwater Example• The channel is 5 meters wide, bottom slope
is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
• Determine the slope designation, and profile type
• yn = 5 meters, yc = 2.3 meters, y0=8.
• Using the slope designation the channel is Mild slope.
• Using the type designation the curve will be a Type 1.
• Thus this is an M1 curve.
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Backwater Example• The channel is 5 meters wide, bottom slope
is 0.001, Manning’s n is 0.02 and channel discharge is 55.4 cubic meters per second
• Use the designation to decide if integration is downstream or upstream.• Curve is M1• Downstream control• Integrate upstream (-x direction)
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Backwater Example• Use Excel to build a spreadsheet to
facilitate the computations. A portion of such a sheet is shown
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Backwater Methods• The variable step method was illustrated by
an example to familiarize the participant with the mechanics of the method.
• The example geometry is intentionally simple, and other simplifications are imbedded into the example.
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Backwater Methods• For practical applications use of
specialized software is recommended (HEC-RAS, WSPRO, SWMM, etc.)
• These program are computationally similar to the method presented herein
• These programs have user interfaces to facilitate the data entry
• These programs allow spatial locations to be fixed and depths to be computed, which is far more practical for engineering application.