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TRANSCRIPT
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Copyright @ 2009 Dr. Jennifer Duan CEEM
BIA Summer WorkshopHydraulics and Engineering Application
by Jennifer DuanAssociate Professor, CEEM
I. Hydraulic Basics
Copyright @ 2009 Dr. Jennifer Duan CEEM
Copyright @ 2009 Dr. Jennifer Duan CEEM
Water FlowPipe Flow: water flowing in a closed pipe is called pipe flow. Water in a pipe flows from high to low pressure locations. Pipe flows are often pressurized.
Examples: flow in household pipes, in garden hose, in a tube, in tunnels.
Open Channel Flow: Water flows in rivers in which the flowing fluid forms a free surface and is driven by gravity. Water flows from high to low elevations.
Examples water in natural rivers, in sewer channel.
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Characteristics1) Free surface adjusts itself to
accommodate different flow conditions.
• Smooth / waving surface• Flood waves
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Characteristics2) extremely variable cross section
shape and roughness, such as• a wide reach of shallow water• a narrow reach of falls, cascade• concrete or vegetated channels• natural rivers with structures, bridges
Flow Depth (h) Flow depth is the vertical distance from water surface
to channel bottom. Flow depth varies spatially and temporally, for
instance, most rivers are non-wadable in flood seasons.
In natural rivers, flow depth is usually less than 10 ft. To maintain navigation in Mississippi River, the minimum flow depth is 9 ft. Certain fish species need a minimum flow depth of 4 ft to survive.
Copyright @ 2009 Dr. Jennifer Duan CEEM
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Flow Velocity (V) Velocity is the speed that flow travels from one
location to another. Flow velocity also varies spatially and temporally,
for instance, water flows faster in flood seasons. In natural rivers, flow velocity is usually less than 10
ft/s. If flow velocity is too high, soil (also called sediment) on banks and beds will be eroded away. Otherwise, soil (sediment) will deposit to form sand bars and beaches.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Flow Discharge (Q) Flow discharge is the volume of water passing through a cross
section per unit time. It is usually measured by cubic feet per second, or gallon per hour.
Flow discharge can be calculated as the product of flow velocity and cross sectional area. For rectangular cross section, flow discharge is calculated as
Q=VA=VhBwhere Q=discharge, V=velocity, A= Cross section area,h= flow depth, B= channel width
Copyright @ 2009 Dr. Jennifer Duan CEEM
River Bed : Flat or Waving? Although we can not see river
bottom through turbid water, we know sand are at river bottom.
Are river bed flat as water surface?
No. Most river beds are not flat but with sand dunes, just like the dunes in desert. Those dunes are formed by flowing water, and can be miles long in Mississippi River.
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Copyright @ 2009 Dr. Jennifer Duan CEEM
Dimensionless NumbersFroude number is the ratio of
inertia to gravity force.
1) The larger the Froude number, the higher the velocity
2) Froude number for flood flow is often nearly or greater than 1.0; while for mild flow, Froude number is less than 1.0.
ghVFr
Gallatin River, by G.E. Urroz
Logan River below first dam, Utah by G.E. Urroz
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Subcritical, critical, and supercritical flow
Flow is subcritical if Froude number is less than 1.0; critical if Froude number is 1.0; supercritical if Froude number is greater than 1.0.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Dimensionless NumbersReynolds number is the ratio of inertia to viscous force
Vh
Re
Gallatin River, by G.E. Urroz
Logan River below first dam, Utah by G.E. Urroz
1) Larger Reynolds number means more turbulent flow.
2) Reynolds number in natural river is often greater than 2400, which means turbulent.3) For slow moving flow, Reynolds number is less than 2400, called laminar.
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Steady vs Unsteady Flow Unsteady flow means velocity and depth vary with time,
otherwise, it’s steady flow.
steady unsteady
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Uniform vs Non-uniform FlowNonuniform flow means velocity and depth vary spatially in the flow direction, otherwise, it’s uniform flow. Uniform flow only exists in prismatic channel.
Non-uniform uniform
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Basic Concepts Gradually varied flow is nonuniform flow, but the curvature
of the free surface and the accompanying streamlines is so small that the transverse pressure distribution at any station along the flow can be approximated as hydrostatic.
This flow can be treated as one-dimensional flow only considering variation of flow variables (depth and velocity) inthe flow direction.
Most flow in natural rivers are gradually varied as cross sections change shapes and bed slopes vary along the river.
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Copyright @ 2009 Dr. Jennifer Duan CEEM
Basic Concepts Rapidly varied flow is highly nonuniform flow that varies
in the flow direction and/or laterally. If flow varies in the flow direction (e.g. hydraulic jump, spillway flow), momentum equation or 2D model are needed to solve flow depth and velocity.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Basic Concepts Spatially varied flow is a class of non-uniform flow but
owes its non-uniformity to variation in the flow discharge as well as to the imbalance of gravity and resistance force.
II. Flow Principles
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Mass Conservation Since flow density is a constant, flow discharge passing
through one cross section is the same as that through another section when there is no lateral inflow or outflow.
Q1= Q2 = … = Qn
Copyright @ 2009 Dr. Jennifer Duan CEEM
If discharge is calculated as the product of flow velocity and area:
V1A1=V2A2 Flow velocity is smaller at
wider cross sections and higher at narrow cross sections.
Q1
Q2
Q3
Energy Conservation Flow energy consists of internal energy, potential energy,
and kinetic energy. The internal energy relates to flow pressure, potential energy
relates to elevation, and kinetic energy relates to velocity. Pipe flow always flow from high to low pressure while flow
velocity usually is a constant due to constant pipe diameter. Internal energy is consumed by friction from pipe surfaces.
Open channel flow always flow from high to low elevations due to gravity force. In the meantime, potential energy changes to kinetic energy.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Energy Equation The change of flow energy from one cross section to another
is caused by friction forces on boundaries and extra drag forces due to obstacles in rivers, such as bridges, weirs.
The energy conservation equation can be written as
Copyright @ 2009 Dr. Jennifer Duan CEEM
losshgVZP
gVZP
22
22
22
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P = pressure, Z = bed elevation, V = velocity, g = gravity acceleration, γ = specific gravity, hloss = energy loss.
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Energy Equation
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If channel bed elevation remains the same, the energy conservation equation can be written as
losshg
Vhg
Vh 22
22
2
21
1
This equation is called the specific energy equation.
The combination of the 1st two terms is the water surface elevation:
222
111 WSZPWSZP
Momentum Principle The change of flow momentum equals to the total external
forces. The most commonly used momentum equation is the empirical Manning’s equation, written as
n = manning’s roughness coefficient, R = hydraulic radius,S = channel slope.
Hydraulic radius is calculated as P = wetted parameter
Manning’s equation applies to uniform and gradually varied flows. Copyright @ 2009 Dr. Jennifer Duan CEEM
)unitSI(1 2/13/2 SRn
V
PAR
Manning’s Equation For shallow open channel flow, Manning’s equation can be
simplified as
Manning’s equation is dimensional. If using English unit,flowvelocity is ft/s, flow depth is ft. The equation should be
The Manning’s roughness coefficient ranges from 0.008 ( for glass surface) to 0.4 (mature trees on floodplain). The roughness for natural sandy rivers is usually 0.03-0.04, while for vegetated floodplain is 0.06-0.08.
Copyright @ 2009 Dr. Jennifer Duan CEEM
2/13/21 Shn
V
2/13/249.1 Shn
V
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Copyright @ 2009 Dr. Jennifer Duan CEEM
Hydraulic JumpHydraulic Jump is formed when flow changes from supercritical to subcritical.
Example #1: A common example of a hydraulic jump is the roughly circular stationary wave that forms around the central stream of water. The jump is at the transition between the point where the circle appears still and where the turbulence is visible.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Hydraulic Jump
Hydraulic Jump after St Anthony Falls Dam
Example #2: Hydraulic jump is formed when flow passes through the spillway.
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Hydraulic Jump Hydraulic jump is an abrupt change in depth from supercritical to subcritical flow that is always accompanied by a significant energy loss.
Application of the momentum equation to a hydraulic jump in a rectangular channel
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III. Flow Measurement
Copyright @ 2009 Dr. Jennifer Duan CEEM
Flow Measurement Devices Flow discharge in open channel flow can be measured by
weirs, sluice gate, and long-throated flume. There are three types of weirs: sharp-crested, triangular, and
broad crested. Those weirs will be discussed in detail later. Sluice gate measures flow discharge based on the differences
of water surface upstream and downstream of the gate. Long-neck flume creates supercritical flow to measure flow
discharge. All those measurement devices are based on energy
principles.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Sluice Gate
Sluice Gate Geometry
Considering the vena contractaeffect, z2=Cca; for 0
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Rectangular Sharp Crested Weir
2/3232 bHCgQ d
wd P
HC 08.0611.0
in which b = width of weir, Pw = height of weir, Cd = constant coefficient, and can be approximately calculated by
Triangular Weir
2/5
2tan2
158 HgCQ d
where θ = angle of opening, Cd = coefficient ranging from 0.57 to 0.59.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Broad-Crested Weir
2/32/1
32
32 LHgCCQ dv
Use the energy equation at the weir where flow is critical,
3/12
2
2 )/(23
2
gLQ
gAQyH
cce
Fig. Definition sketch of broad-crested weir
Then, the discharge relation,
in which, Cd = coefficient (=0.85) ,Cv = (He/H)3/2 ≈ 1.10, is approaching
velocity coefficient.
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Copyright @ 2009 Dr. Jennifer Duan CEEM
Long-Throated FlumeThe most common long-throated flume is the Parshall Flume with a contraction section. The cross section can be rectangular, trapezoidal, or others.
Copyright @ 2009 Dr. Jennifer Duan CEEM
Long-Throated FlumeThe critical flow depth occurred at the contraction, so the flow discharge is calculated as
)(2 cecd yHgACQ
Cd=coefficient;Ac=cross section area at contraction;He = the total approach energy head including flow velocity head;yc = critical flow depth. The discharge coefficient can be calculated as
7.0/1.0for)07.0( 018.0 lHl
HC eed
IV. Hydraulic Structure
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Copyright @ 2009 Dr. Jennifer Duan CEEM
IntroductionHydraulic structures include spillways, culverts and bridges.
Spillways are used on both large and small dams to pass flood flows, thereby preventing overtopping and failure of the dam.
Pabco Grade Control Structure
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Spillway
Blue Diamond Detention Basin Las Vegas Q=145000cfs
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T. R. Roosevelt Dam, Arizona
www.usbr.gov
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Three Gorges Reservoir Spillways
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CulvertCulverts are designed to carry peak flood discharges under road ways or other embankments to prevent embankments overflows.
Inlet and Outlet of Culvert in Las Vegas
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BridgesBridges convey vehicles over waterways, but they must accommodate through flows of floodwaters without failure due to overtopping or foundation failure by scour.
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Culvert Culverts are designed to carry peak
flood discharges under roadways or other embankments to prevent embankment overflows.
Flow passing through culverts could be open channel or pressurized depending on the head at the inlet.
The culvert performance varies with the head of approaching flow, and the head and discharge relation is called the performance curve.
Culvert Performance Curve
Inlet Control Culvert The USGS classified culvert flow into six types depending primarily on the headwater and tailwater levels and whether the slope is mild or steep. For inlet control culverts, flow discharge is determined by head water depth and critical flow depth in the barrel. The formula is below:
Inlet control culvert flows)(2 ccd yHWgACQ
Where HW=head water depth, yc = critical flow depth, Ac is critical flow area. For pipe culverts with a square edge in a vertical head wall, Cd=0.93 for Hw/d
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Copyright @ 2009 Dr. Jennifer Duan CEEM
Outlet Control – OC-1
RLfK
LSTWHWgAQ
e 41
)(2 0
3/42
224 RK
LgnRLf
n
For OC-1 type of flow, the discharge is calculated as
In which Ke = entrance loss coefficient in Table 6.5; TW=tailwater depth relative to the outlet invert; A = pipe cross section area. The head loss can be written in terms of Manning’s n,
Copyright @ 2009 Dr. Jennifer Duan CEEM
Outlet ControlOC-2: unsubmerged outlet
Fig.(d) has the unsubmerged outlet with critical flow depth occurring there (USGS Type 7)
Fig.(e) has unsubmergedinlet and outlet, and flow is subcritical on the mild slope (USGS Type 2 or 3).
Flow discharge is both cases need to be calculated based on out flow properties.