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Open Channel Flow. Open Channel Flow. free surface. 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

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Open channel flow2 l.jpg
Open Channel Flow

free surface

  • 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

depth


Topics in open channel flow l.jpg
Topics in Open Channel Flow

normal depth

  • 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


Classification of flows l.jpg
Classification of Flows

(Temporal)

  • 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

(Spatial)


Momentum and energy equations l.jpg
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

      • ____________

Contractions

We need an equation for losses

Expansion


Open channel flow discharge depth relationship l.jpg
Open Channel Flow: Discharge/Depth Relationship

  • 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?

A

P

no acceleration

geometry

Force balance


Steady uniform flow force balance l.jpg
Steady-Uniform Flow: Force Balance

toP D x

Shear force=________

Energy grade line

Hydraulic grade line

P

Wetted perimeter = __

b

gA Dx sinq

c

Gravitational force = ________

Dx

a

d

W cos 

Shear force

W

Hydraulic radius

W sin 

Turbulence

Relationship between shear and velocity? ______________


Open conduits dimensional analysis l.jpg
Open Conduits:Dimensional Analysis

  • Geometric parameters

    • ___________________

    • ___________________

    • ___________________

  • Write the functional relationship

  • Does Fr affect shear? _________

Hydraulic radius (Rh)

Channel length (l)

Roughness (e)

No!


Pressure coefficient for open channel flow l.jpg
Pressure Coefficient for Open Channel Flow?

Pressure Coefficient

(Energy Loss Coefficient)

Head loss coefficient

Friction slope

Slope of EGL

Friction slope coefficient


Dimensional analysis l.jpg
Dimensional Analysis

Head loss  length of channel

(like f in Darcy-Weisbach)


Chezy equation 1768 l.jpg
Chezy equation (1768)

  • Introduced by the French engineer Antoine Chezy in 1768 while designing a canal for the water-supply system of Paris

compare

where C = Chezy coefficient

where 60 is for rough and 150 is for smooth

also a function of R (like f in Darcy-Weisbach)


Darcy weisbach equation 1840 l.jpg
Darcy-Weisbach equation (1840)

f = Darcy-Weisbach friction factor

For rock-bedded streams

where d84 = rock size larger than 84% of the rocks in a random sample


Manning equation 1891 l.jpg
Manning Equation (1891)

  • Most popular in U.S. for open channels

(MKS units!)

T /L1/3

Dimensions of n?

NO!

Is n only a function of roughness?

(English system)

Bottom slope

very sensitive to n


Values of manning n l.jpg
Values of Manning n

n = f(surface roughness, channel irregularity, stage...)

d in ft

d = median size of bed material

d in m


Trapezoidal channel l.jpg
Trapezoidal Channel

  • Derive P = f(y) and A = f(y) for a trapezoidal channel

  • How would you obtain y = f(Q)?

1

y

z

b

Use Solver!


Flow in round conduits l.jpg
Flow in Round Conduits

radians

r

A

y

Maximum discharge when y = ______

0.938d

T


Open channel flow energy relations l.jpg
Open Channel Flow: Energy Relations

velocity head

energy grade line

hydraulic grade line

water surface

Bottom slope (So) not necessarily equal to surface slope (Sf)


Energy relationships l.jpg
Energy relationships

Pipe flow

z - measured from horizontal datum

From diagram on previous slide...

Turbulent flow (1)

y - depth of flow

Energy Equation for Open Channel Flow


Specific energy l.jpg

y

Specific Energy

  • The sum of the depth of flow and the velocity head is the specific energy:

y - potential energy

- kinetic energy

If channel bottom is horizontal and no head loss

For a change in bottom elevation


Specific energy20 l.jpg
Specific Energy

In a channel with constant discharge, Q

where A=f(y)

Consider rectangular channel (A=By) and Q=qB

q is the discharge per unit width of channel

A

y

B

3 roots (one is negative)


Specific energy sluice gate l.jpg
Specific Energy: Sluice Gate

sluice gate

q = 5.5 m2/s

y2 = 0.45 m

V2 = 12.2 m/s

EGL

1

E2 = 8 m

2

Given downstream depth and discharge, find upstream depth.

alternate

y1 and y2 are ___________ depths (same specific energy)

Why not use momentum conservation to find y1?


Specific energy raise the sluice gate l.jpg
Specific Energy: Raise the Sluice Gate

sluice gate

EGL

2

1

as sluice gate is raised y1 approaches y2 and E is minimized: Maximum discharge for given energy.


Specific energy step up l.jpg
Specific Energy: Step Up

Short, smooth step with rise Dy in channel

Given upstream depth and discharge find y2

Dy

Increase step height?


Critical flow l.jpg
Critical Flow

yc

Arbitrary cross-section

Find critical depth, yc

T

dy

dA

A

A=f(y)

y

P

T=surface width

Hydraulic Depth


Critical flow rectangular channel l.jpg
Critical Flow:Rectangular channel

T

yc

Ac

Only for rectangular channels!

Given the depth we can find the flow!


Critical flow relationships rectangular channels l.jpg
Critical Flow Relationships:Rectangular Channels

because

inertial

force

Froude number

gravity

force

velocity head =

0.5 (depth)


Critical flow27 l.jpg
Critical 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

    • ___________________________________________

Difficult to measure depth

Unique relationship between depth and discharge


Broad crested weir l.jpg
Broad-crested Weir

E

H

yc

Broad-crested

weir

P

Hard to measure yc

E measured from top of weir

Cd corrects for using H rather than E.


Broad crested weir example l.jpg
Broad-crested Weir: Example

  • Calculate the flow and the depth upstream. The channel is 3 m wide. Is H approximately equal to E?

E

yc

yc=0.3 m

Broad-crested

weir

0.5

How do you find flow?____________________

Critical flow relation

Energy equation

How do you find H?______________________

Solution


Hydraulic jump l.jpg
Hydraulic 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?



Hydraulic jump32 l.jpg
Hydraulic Jump

Conservation of Momentum

hL

EGL

y2

y1

L


Hydraulic jump conjugate depths l.jpg
Hydraulic Jump:Conjugate Depths

For a rectangular channel make the following substitutions

Froude number

Much algebra

valid for slopes < 0.02


Hydraulic jump energy loss and length l.jpg
Hydraulic Jump:Energy Loss and Length

  • Energy Loss

algebra

significant energy loss (to turbulence) in jump

  • Length of jump

No general theoretical solution

Experiments show

for


Gradually varied flow l.jpg
Gradually Varied Flow

Energy equation for non-uniform, steady flow

T

dy

A

y

P


Gradually varied flow36 l.jpg
Gradually Varied Flow

Change in KE

Change in PE

We are holding Q constant!


Gradually varied flow37 l.jpg
Gradually Varied Flow

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

yn is when


Surface profiles l.jpg
Surface 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

Note: These slopes are f(Q)!


Surface profiles39 l.jpg
Surface Profiles

Normal depth

Obstruction

Steep slope (S2)

Sluice gate

Steep slope

Hydraulic Jump

S0 - Sf 1 - Fr2 dy/dx

+ + +

yn

yc

- + -

- - +


More surface profiles l.jpg
More Surface Profiles

S0 - Sf 1 - Fr2 dy/dx

1 + + +

yc

2 + - -

yn

3 - - +


Direct step method l.jpg
Direct Step Method

energy equation

solve for Dx

rectangular channel

prismatic channel


Direct step method friction slope l.jpg
Direct Step MethodFriction Slope

Darcy-Weisbach

Manning

SI units

English units


Direct step l.jpg
Direct 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 Dy is + or -)

    • choose Dy 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 Dx

prismatic


Direct step method44 l.jpg
Direct Step Method

=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)

=(G16-G15)/((F15+F16)/2-So)

A

B

C

D

E

F

G

H

I

J

K

L

M

y

A

P

Rh

V

Sf

E

Dx

x

T

Fr

bottom

surface

0.900

1.799

4.223

0.426

0.139

0.00004

0.901

0

3.799

0.065

0.000

0.900

0.870

1.687

4.089

0.412

0.148

0.00005

0.871

0.498

0.5

3.679

0.070

0.030

0.900


Standard step l.jpg
Standard 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


What curves are available l.jpg
What curves are available?

S1

S3

Is there a curve between yc and yn that decreases in depth in the upstream direction?


Wave celerity l.jpg

Vw

y

y

y+y

y+y

V

V+V

V-Vw

V+V-Vw

unsteady flow

steady flow

F2

V-Vw

V+V-Vw

Wave Celerity

F1


Wave celerity momentum conservation l.jpg

y

y+y

V-Vw

V+V-Vw

steady flow

Wave Celerity:Momentum Conservation

Per unit width


Wave celerity49 l.jpg

y

y+y

V-Vw

V+V-Vw

steady flow

Wave Celerity

Mass conservation

Momentum


Wave propagation l.jpg
Wave 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

upstream


Most efficient hydraulic sections l.jpg
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


Why isn t the most efficient hydraulic section the best design l.jpg
Why isn’t the most efficient hydraulic section the best design?

Minimum area = least excavation only if top of channel is at grade

Cost of liner

Complexity of form work

Erosion constraint - stability of side walls


Open channel flow discharge measurements l.jpg
Open Channel Flow Discharge Measurements design?

  • Discharge

    • Weir

      • broad crested

      • sharp crested

      • triangular

    • Venturi Flume

    • Spillways

    • Sluice gates

  • Velocity-Area-Integration


Discharge measurements l.jpg
Discharge Measurements design?

  • Sharp-Crested Weir

  • Triangular Weir

  • Broad-Crested Weir

  • Sluice Gate

Explain the exponents of H!


Summary l.jpg
Summary design?

  • 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 location


Broad crested weir solution l.jpg
Broad-crested Weir: Solution design?

E

yc

yc=0.3 m

Broad-crested

weir

0.5


Sluice gate l.jpg

Sluice gate design?

reservoir

2 m

10 cm

S = 0.005

Sluice Gate


Summary overview l.jpg
Summary/Overview design?

  • Energy losses

    • Dimensional Analysis

    • Empirical


Energy equation l.jpg
Energy Equation design?

  • 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 step up60 l.jpg
Specific Energy: Step Up design?

Short, smooth step with rise Dy in channel

Given upstream depth and discharge find y2

Dy

Increase step height?


Critical depth l.jpg
Critical Depth design?

  • Minimum energy for q

    • When

    • When kinetic = potential!

    • Fr=1

    • Fr>1 = Supercritical

    • Fr<1 = Subcritical


What next l.jpg
What next? design?

  • 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


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