Reference Books - MD NURUZZAMAN...Course Title: Open Channel Flow Course Code: CE 3261 Reference Books 1. Open Channel Hydraulics by –V.T. Chow 2. Flow in Open channel by - Subramanya

Course Title: Open Channel Flow

Course Code: CE 3261

Reference Books

1. Open Channel Hydraulics by – V.T. Chow

2. Flow in Open channel by - Subramanya

3. Flow Through Open channels by – Ranga Raju

4. Open Channel Flow by M. Hanif Chowdhury

5. Open Channel Flow by Dr. Abdul Halim

Course Content

• Concept of uniform

flow, Chezy and

Manning equations,

estimation of

resistance coefficients

and computation of

uniform flow;

• Hydraulic jump

• Gradually varied

flow; computation of

flow profiles

• Design of channels.

Open Channel Flow (CE 3261)

Chapter: Uniform Flow

Reference Book: Open Channel Flow by Dr. Abdul Halim

Lecture prepared by

Md Nuruzzaman

Lecturer, Department of Civil Engineering

Bangladesh Army University of Engineering and Technology (BAUET)

E-mail: [email protected], [email protected]

Phone: +8801719456829

mailto:[email protected]


Qualifications/Conditions/Criterion for uniform flow

1. The discharge, velocity and depth remain constant along thelength of the channel.

2. The energy line, the water surface and the channel bottom areparallel.

True uniform flow does not occur in natural channels.

Uniform Flow

Question: Explain why a uniform flow cannot occur (a) in africtionless channel and (b) in a horizontal channel (c) in anadverse slope?

Uniform Flow


According to momentum equation,

𝜌𝑄 𝛽2𝑈2 − 𝛽1𝑈1 = 𝐹𝑝1 − 𝐹𝑝2 +𝑊𝑠𝑖𝑛𝜃 − 𝐹𝑓 (1)

where, Q is the flow, U is the velocity, Ff is the friction, W is theweight of water and 𝛽 is the momentum coefficient.

For a uniform flow, 𝑈1 = 𝑈2, 𝛽1 = 𝛽2, 𝐹𝑝1 = 𝐹𝑝2

Hence, for a uniform flow, the momentum equation will be,𝑊𝑠𝑖𝑛𝜃 = 𝐹𝑓

Uniform Flow


The above condition implies that

flow cannot be uniform in a frictionless channel for which 𝐹𝑓 = 0.

flow cannot be uniform in a horizontal channel where 𝜃 = 0.

flow cannot be uniform in an adverse slope channel where both𝑊𝑠𝑖𝑛𝜃 and 𝐹𝑓 act in the same direction.

Uniform Flow

Shear/Tractive/Drag force

When water flows in a channel, the pull of water produces a forcethat acts on the channel bed in the direction of flow. This force isknown as the shear or tractive or drag force.

Shear stress velocity, u∗

= 𝑔𝑅𝑆0

where, R is the hydraulic radius and S0 is the slope

For wide channel, u∗

= 𝑔ℎ𝑆0

Uniform Flow

Laminar or Viscous sublayer

Even in a turbulent flow, there is a very thin layer near theboundary in which the flow is laminar and is known as the laminaror viscous sublayer.

Uniform Flow

Derive an expression for the discharge through a channel byChezy’s formula.

Uniform Flow

Derive an expression for the discharge through a channel byChezy’s formula.

Chezy formula can be found using two assumptions. (1) In steadyuniform flow, the component of gravity force causing the flow isequal to the force of friction or resistance. When the component ofgravity force is small,

the component of gravity force = 𝑊𝑠𝑖𝑛𝜃 = 𝜌𝐴𝐿𝑠𝑖𝑛𝜃 = 𝜌𝐴𝐿𝑡𝑎𝑛𝜃 =𝜌𝐴𝐿𝑆0 = 𝜌𝐴𝐿𝑆𝑓 (1)

Uniform Flow

Question: Derive an expression for the discharge through achannel by Chezy’s formula.

(2) In turbulent flow, the force of resistance per unit wetted areavaries as the square of the mean velocity. The total wetted area isthe product of the wetted perimeter P and the length of the channelL. Hence, the force of resistance is given by

𝐹𝑓 = 𝑘𝑃𝐿𝑈2 (2)

Where, k is a constant of proportionality.

Applying the principle of uniform flow, i.e. 𝑊𝑠𝑖𝑛𝜃 = 𝐹𝑓, we get,

𝜌𝐴𝐿𝑆𝑓 = 𝑘𝑃𝐿𝑈2

𝑈 = 𝐶𝑅1/2𝑆𝑓1/2

Where 𝜌/𝑘 is written as one constant C.

Uniform Flow

Uniform flow formulas

Manning’s formula:

𝑈 =1

𝑛𝑅2/3𝑆𝑓

1/2 (in SI unit)

And 𝑈 =1.486

𝑛𝑅2/3𝑆𝑓

1/2 (in FPS unit)

Where, U is the mean velocity, n is the manning’s coefficient.

Chezy formula:

𝑈 = 𝐶𝑅1/2𝑆𝑓1/2

where, U is the velocity of flow, C is the resistance factor, R is thehydraulic radius and Sf is the energy gradient.

Uniform Flow

Uniform flow formulas

Darcy-Weisbach formula:

ℎ𝑓 = 𝑓𝐿𝑈

𝑑02𝑔

Since, do = 4R and energy gradient, Sf = hf/L, the above formulamay be written as,

𝑈 =8𝑔

𝑓𝑅1/2𝑆𝑓

1/2

where, f is the friction factor.

This formula was basically developed for pipe flow.

Uniform Flow

Factors affecting Manning’s roughness coefficient

(1) Surface roughness

Fine grains in the channel surface result in relatively low ‘n’ valueand coarse grains, in a high value of ‘n’.

(2) Vegetation

Bed surface vegetation will tend to increase Manning’s ‘n’ value.

(3) Channel irregularity

A gradual and uniform channel will not appreciably increase ‘n’value compared to irregular channel.

Uniform Flow

Factors affecting Manning’s roughness coefficient

(4) Channel alignment

Smooth curvature with a large radius will give a relatively low ‘n’value, whereas sharp curvature with severe meandering willincrease ‘n’ value.

(5) Silting and scouring

(6) Obstruction

(8) Stage and discharge

(9) Seasonal change

(10) Suspended material and bed load.

Uniform Flow

Strickler’s formula for computing Manning’s roughnesscoefficient

𝑛 =𝑑50

1/6

21.1= 0.047𝑑50

1/6

Where, d50 is the median diameter of bed materials in meters suchthat 50% of the material by weight is smaller.

Uniform Flow

Problem

An open channel lined with concrete (d50 = 1.50 mm) is laid on aslope of 0.1%. The channel is trapezoidal with b = 6 m and s = 2.Compute the uniform flow discharge in the channel if the depth offlow is 2 m. Also, compute the numerical values of Chezy’s C andfriction factor f.

Uniform Flow

Solution

Sf = S0 = 0.1% = 0.1/100 = 0.001

d50 = 1.50 mm = 0.0015 m

n = 0.047d501/6 = 0.047 x 0.00151/6 = 0.016

A = (b+sh) h = (6 + 2 x 2) x 2 = 20 m2

P = b + 2h 1 + s2 = 6 + 2 x 2 1 + 22 = 14.94 m

R =A

P=

20

14.94= 1.34 m

Q =1

nAR2/3S0

1/2 =1

0.016x 20 x 1.342/3 x 0.0011/2 = 48.04 m3/sec

C =1

nR1/6= 65.6 m1/2/s

f =8g

C2=

8 x 9.81

65.62= 0.02

Uniform Flow

Normal depth

The depth of uniform flow is known as normal depth.

Section factor

When the flow is uniform, the product of flow area and two-thirdspower of hydraulic radius, i.e. AR2/3 is known as the section factorin connection with the Manning formula.

Conveyance

The conveyance for a channel in terms of the Manning formula isgiven by

K =1

𝑛AR2/3

Uniform Flow

Problem: For a trapezoidal channel with b = 6 m, s = 2, n = 0.025and S0 = 0.001, compute the normal depth and velocity when Q =14 m3/s.

Solution

AR2/3 =nQ

S0=0.025 x 14

0.001= 11.068

Uniform Flow

h (m) A (m2) P (m) R (m) 𝐀𝐑𝟐/𝟑

1.00 8.00 10.472 0.764 6.685

2.00 20.00 14.942 1.338 24.285

1.30 11.18 11.814 0.946 10.776

1.32 11.40 11.903 0.958 11.076

Problem: For a trapezoidal channel with b = 6 m, s = 2, n = 0.025and S0 = 0.001, compute the normal depth and velocity when Q =14 m3/s.

So, normal depth = 1.32 m

Normal velocity =𝑄

𝐴=

14

11.4= 1.228 m/s

Uniform Flow

Problem

A channel consists of a main section and two side sections asshown in the following figure. Compute the total discharge and themean velocity of flow for the entire section when, n = 0.025 for themain section, n = 0.035 for the side sections and S0 = 0.0001.

Uniform Flow

Solution

Uniform Flow

Section A (m2) P (m) R (m) Manning’s n

Q

(m3/sec

)

Main 425 56.18 7.56 0.025 654.83

Left 80 24 3.33 0.035 51.00

Right 60 19 3.16 0.035 36.84

565 99.18 742.75

Hence, the total discharge, Q = 742.67 m3/sec

Mean velocity, U =𝑄

𝐴=

742.67

565= 1.31 m/sec

Problem

For a rectangular channel, b = 6 m, n = 0.025 and So = 0.0025.Compute the normal depth and velocity when Q = 20 m3/sec.

Uniform Flow

Estimation of resistance coefficients

Uniform Flow


Uniform Flow


Uniform Flow

Problem

Compute the velocity and discharge in the trapezoidal channelshown, having a bottom width of 20 ft, side slopes 2: 1, and adepth of water 6 ft. Given: Kutter's n = 0.015, and S = 0.005.

Uniform Flow

Solution

Uniform Flow


Chapter: Gradually Varied Flow


Lecture prepared by

Md Nuruzzaman




Phone: +8801719456829



Definition

In gradually varied flow, the depth and mean velocity varygradually along the length of the channel.

Qualifications/Conditions/Criterion for Gradually Varied flow

1. The flow is steady, i.e. hydraulic characteristics of flow remainconstant for the time interval under consideration.

2. The stream lines are practically parallel, i.e. the hydrostaticdistribution of pressure prevails over the channel section.

Gradually Varied Flow

Backwater Curve

The water surface curve or profile represents a backwater curve when the depth of flow increases in the direction of flow (dh/dx >0).

Drawdown Curve

The water surface curve or profile represents a drawdown curvewhen the depth of flow decreases in the direction of flow

(dh/dx <0).

Gradually Varied Flow


Chapter: Design of Channels


Lecture prepared by

Md Nuruzzaman




Phone: +8801719456829



Minimum permissible velocity

The minimum permissible velocity of flow is the lowest meanvelocity of flow that will prevent sedimentation and vegetationgrowth. In general, a velocity of 2 to 3 ft/sec (0.61 to 0.91 m/s) willprevent sedimentation when the silt load of the flow is low. Avelocity of 2.5 ft/sec (0.75 m/s) is sufficient to prevent vegetationgrowth.

Maximum permissible velocity

The maximum permissible velocity is the greatest mean velocitythat will not cause erosion of the channel body. The velocity isgenerally taken as 2 m/s.

Design of Channels

Freeboard

Freeboard is the vertical distance between the top of the channeland the water surface at design condition. It is provided to preventthe water level from overtopping sides of the channel due to itsfluctuation caused by wind, hydraulic jump etc. United StatesBureau of Reclamation (USBR) suggested the following formula tocalculate freeboard.

𝐹𝑏 = 𝑐ℎ

Where Fb is freeboard in ft, h is the depth of flow in ft and c is afactor varying from 1.5 for Q < 20 ft3/sec to 2.5 for Q > 300 ft3/sec.

Design of Channels

Why critical state of flow is undesirable?

When the flow approached the critical state, the water surfacebecomes unstable and wavy and large disturbances are expectedat bends and obstructions. Thus, critical flow is not desirable.

Design of Channels

Advantages of lining channels

The provision of lining a channel

(i) Permits the water to flow at high velocities.

(ii) Decreases seepage and percolation losses.

(iii) Reduces the cost of operation and maintenance.

(iv) Ensures the stability of channel section.

Design of Channels

Best Hydraulic Section

A channel section that conveys the maximum discharge for a givenarea is known as the best hydraulic section. Since Q ∝ AR2/3 and R= A/P, the best hydraulic section gives minimum wetted perimeterP and maximum hydraulic radius R for a given area. It is to benoted that a minimum wetted perimeter also gives minimumamount of lining.

Design of Channels

Show that the best hydraulic section of a rectangular channelis one half of a square.

Design of Channels

Threshold/Incipient/Impending motion

It is the limiting condition at which soil particles begin to move.

Critical Shear Stress

The average shear stress at the boundary of an open channel atwhich the soil particles begin to move is called the critical shearstress.

Shear stress ratio

The ratio of shear stress acting on the soil particle resting on thesloping side of a trapezoidal channel to the shear stress acting onthe soil particle resting on the channel bottom is known as shearstress ratio.

Design of Channels

Shear stress ratio

𝑲 = 𝟏 −𝒔𝒊𝒏𝟐∅

𝒔𝒊𝒏𝟐ψ

Where ψ is the angle of repose and ∅ is the angle of the side slope

Design of Channels

Alluvium channel

An alluvium channel is a channel transporting the same type ofmaterial as that of the channel perimeter.

Regime Channel

A channel is said to be in a regime when it has adjusted its shapeand slope to an equilibrium condition and no silting or scouringoccurs in the channel.

Design of Channels

Kennedy formula of regime channel𝑈 = 0.546ℎ0.64

Where, U is the non-silting and non-scouring velocity and h is thedepth of flow.

Limitations of Kennedy theory

Study yourself

Design of Channels

Problem

Design a stable alluvial channel using the Lacey method. Thechannel is to carry 10 m3/sec of flow through 1 mm sand.

20m3/sec. It is to be excavated in earth containing moderatelyrounded coarse non-cohesive particles with d50= 2 cm,d75=2.5cm and n= 0.025. Determine the section dimensions ofthe channel.

Design of Channels


Chapter: Hydraulic Jump


Lecture prepared by

Md Nuruzzaman




Phone: +8801719456829



Definition

In open channels, when a supercritical flow is made to changeabruptly to subcritical flow, the result is usually an abrupt rise of thewater surface. This feature is known as the hydraulic jump.

Initial depth and sequent or conjugate of hydraulic jump

The depth of supercritical flow before the jump is known as initialdepth.

The depth of subcritical flow after the jump is called sequent orconjugate depth.

Hydraulic Jump

Practical applications of Hydraulic Jump

1. The dissipation of kinetic energy in high-velocity flows over theweirs, spillways, gates, etc.

2. The increase of depth of water in channels for irrigation andwater distribution.

3. The increase of discharge of a sluice gate by repelling thetailwater so that it works under free flow condition.

4. The reduction of uplift pressure under a structure.

5. The mixing of chemicals for water purification.

6. The aeration of flow for city water supplies.

Design of Channels

Types of Hydraulic Jump

Undular jump: The water surface shows undulations and thechange from initial to sequent depth is small and gradual.

Condition: 1 < Fr1 < 1.7.

Weak jump: A series of small roller appears on the jumo surface,but the downstream water surface remains smooth.

Condition: 1.7 < Fr1 < 2.5.

Oscillating jump: The incoming jet oscillates between the bedand the bottom of the surface roller.

Condition: 2.5 < Fr1 < 4.5.

Design of Channels


Steady jump: Appreciable energy dissipation occurs and fairlysmooth water surface downstream is formed.

4.5 < Fr1 < 9.0

Strong jump: The jump surface downstream become very roughand the high-velocity jet generates waves downstream.

Fr1 > 9.0

Design of Channels


Design of Channels

Length of a Jump

The length of a hydraulic jump is the horizontal distance from thefront face or toe of the jump to a point immediately downstreamfrom the roller.

Design of Channels

Efficiency of a Jump

The ratio of the specific energy after the jump to that before thejump, E2/E1 is known as the efficiency of the jump.

Height of a jump

The height of a jump is the difference between the sequent andinitial depths, i.e. hj = h2 – h1.

hj/E1 is called relative height of the jump.

Design of Channels

Problem

Water flows in a horizontal rectangular channel 6 m wide at adepth of 0.52 m and a velocity of 15.2 m/s. If a hydraulic jumpforms in this channel, determine (i) the type of jump (ii) thedownstream depth needed to form the jump (iii) the downstreamFroude number (iv) the efficiency of the jump (v) the relative heightof the jump (vi) the length of the jump (vii) the energy loss orenergy dissipation in the jump (viii) the horse-power dissipation inthe jump.

Design of Channels

Documents

Reference Books - MD NURUZZAMAN...Course Title: Open Channel Flow Course Code: CE 3261 Reference Books 1. Open Channel Hydraulics by –V.T. Chow 2. Flow in Open channel by - Subramanya