BFC21103 HydraulicsChapter 1. Flow in Open Channel

Updated: October 2021

Comparison between pipe flow and open-channel flow

Datum line

1 2

V1

V2

y1

y2

z2

hf

gV2

22

z1

gV2

21

Datum line

1 2

V1

V2

y1

y2

z2

hf

gV2

22

z1

gV2

21

Pipe flow Open channel flow

Fluid completely fills pipe cross section & no free surface. Liquid flow with upper surface exposed to atmosphere.Fluid flows under pressure. Flow due to gravitational force (slope influenced).Hydraulic grade line does not coincides with water surface.

Hydraulic grade line coincides with water surface.

Maximum velocity occurs at centre of pipe flow. Maximum velocity occurs a little distance below water surface.

Velocity distribution is symmetrical about pipe axis. Velocity distribution depends on channel roughness.

Practical applications of open-channel flow studies:

a. flow depth in rivers, canals and other conveyance conduits,

b. changes in flow depth due to channel controls e.g. weirs, spillways, and gates,

c. changes in river stage during floods,

d. surface runoff from rainfall over land,

e. optimal channel design, and others

1.1 Flow Parameters and Geometric Elementsa. Depth of flow y is the vertical measure of water depth.

Normal depth d is measured normal to the channel bottom.

d = y cos θ

For most applications, d ≈ y since θ ≤ 10% thus cos θ ≈ 1 e.g. cos 1° = 0.9998.

Free surface

Datumθ

θ

So = bottom slope

Sw = water surface slope

b. Flow or discharge Q is the volume of fluid passing a cross-section perpendicular to the direction of flow per unit time.

Mean velocity V is the discharge divided by the cross-sectional area

AQV =

c. Wetted perimeter P is the length of channel perimeter that is wetted or covered by flowing water.

B = bottom width

T = top width

AP

y

A = cross sectional area covered by flowing water

d. Hydraulic radius R is the ratio of the flow area A to wetted perimeter P.

B

T

AP

y

PAR =

e. Hydraulic depth D is the average depth of irregular cross section.

TAD ==

width toparea flow

Public Works Department (JKR) Standard Surface Drains (2014)

JKR Standard Surface Drains (2014)

Channel section Area A

Top width T

Wetted perimeter P

By B B + 2y

Table. Open channel geometries

y

B

T

Rectangular

yz

T

Triangular

1 zy2 2zy 212 zy +

By + zy2 B + 2zy 212 zyB ++yz

T

Trapezoidal

1

B

( )θθ sin228

2

−D DθθsinD

y

Circle2θ

D

T

Water flows in a triangular channel with side slope 3(H) : 2(V). Thedepth of flow is 2.0 m. Find z, A, P, T, R and D.

zy1

Activity 1.1

Activity 1.2

A sewer pipe of 2.0 m diameter is conveying flow as follows.Find A, P, T, R and D.

120°D

yo

For the above compound section, find:

(a) Flow area A

(b) Wetted perimeter P

(c) Hydraulic radius R

Activity 1.3

3 m4 m2 m1 m

2 m

2 m

1 m

A1

A2A3 A4

A5

(a) Flow area A

54321 AAAAAA ++++=

55443322211 2

1 yByByByByzA ++++=

0.11 =z 0.15 =zand

( ) ( ) ( ) ( ) ( )( )112133323421 2 ++++=A

2m5.31=A

3 m4 m2 m1 m

2 m

2 m

1 m

A1

A2A3 A4

A5

3 m4 m2 m1 m

2 m

2 m

1 m

A1

A2A3 A4

A5

(b) Wetted perimeter P

5431 PPPPP +++=

( ) ( )[ ] ( ) ( )25544533

211 112 zyyByyBzyP ++++−+++=

( )[ ] ( )[ ] ( ) ( )22 111331321122 ++++−+++=P

m07.17=P

(a) Flow area A

54321 AAAAAA ++++=

55443322211 2

1 yByByByByzA ++++=

0.11 =z 0.15 =zand

( ) ( ) ( ) ( ) ( )( )112133323421 2 ++++=A

2m5.31=A

(b) Wetted perimeter P

(c) Hydraulic radius R

PAR =

5431 PPPPP +++=

( ) ( )[ ] ( ) ( )25544533

211 112 zyyByyBzyP ++++−+++=

( )[ ] ( )[ ] ( ) ( )22 111331321122 ++++−+++=P

m07.17=P

07.175.31

=R

m845.1=R

1.2 Types of Open Channel

• Prismatic and non-prismatic channels

Prismatic channel is channel which cross-sectional shape, size and bottom slope are constant. Most of man-made (artificial) channels are prismatic channels over long stretches. Examples of man-made channels are irrigation canal, flume, drainage ditches, roadside gutters, drop, chute, culvert and tunnel.

All natural channels generally have varying cross-sections and therefore are non-prismatic. Examples of natural channels are tiny hillside rivulets, through brooks, streams, rivers and tidal estuaries.

• Rigid and mobile boundary channels

Rigid channels are channels with boundaries that is not deformable. Channel geometry and roughness are constant over time. Typical examples are lined canals, sewers and non-erodible unlined canals.

Mobile boundary channels are channels with boundaries that undergo deformation due to continuous process of erosion and deposition due to flow. Examples are unlined man-made channels and natural rivers.

Canalsis usually a long and mild-sloped channel built in the ground, which may be unlined or lined with stoned masonry, concrete, cement, wood or bituminous material.

Griboyedov Canal, St. Petersburg, Russia

Terusan Wan Muhammad Saman, Kedah

This flume diverts water from White River, Washington to generate electricity Bull Run Hydroelectric Project diversion flume

Flumesis a channel of wood, metal, concrete, or masonry, usually supported on or above surface of the ground to carry water across a depression.

Open-channel flume in laboratory

Chuteis a channel having steep slope.

Natural chute (falls) on the left and man-made logging chute on the right on the Coulonge River, Quebec, Canada

Dropis similar to a chute, but the change in elevation is within a short distance.

The spillway of Leasburg Diversion Dam is a vertical hard basin drop structure designed to dissipate energy

Stormwater seweris a drain or drain system designed to drain excess rain from paved streets, parking lots, sidewalks and roofs.

Storm drain receiving urban runoff

Storm sewer

1.3 Types and Classification of Open Channel Flows

Open channel flow

Steady flow Unsteady flow

Uniform flow Non-uniform flow

Gradually-varied flowRapidly-varied flow

Various types of open-channel flow

Open Channel Flow

Classification based on Time

Classification based on Space

Steady Unsteady Uniform Non-Uniform

GVF RVF

Open channel flow conditions can be characterised with respect to space(uniform or non-uniform flows) and time (steady or unsteady flows).

Space - how do flow conditions change along the reach of an open channel system.

a. Uniform flow - depth of flow is the same at every section of the flow dy/dx = 0

b. Non-uniform flow - depth of flow varies along the flow dy/dx ≠ 0

a. Uniform flow

b. Non-uniform flowy1

y2

y y

x

Depth of flow is the same at every section along the channel, 0dd

=xy

Depth of flow varies at different sections along the channel, 0dd

≠xy

Time - how do flow conditions change over time at a specific section in an open channel system.

c. Steady flow - depth of flow does not change/ constant during the time interval under consideration dy/dt = 0

d. Unsteady flow - depth of flow changes with time dy/dt ≠ 0

c. Steady flow

d. Unsteady flow

y1

Time = t1

y2

Time = t2

y1

t3

t2

t1

Depth of flow is the same at every time interval, 0dd

=ty

Depth of flow changes from time to time, 0dd

≠ty

y1 = y2

y1 ≠ y2 ≠ y3

The flow is rapidly varied if the depth changes abruptly over a comparatively short distance. Examples of rapidly varied flow(RVF) are hydraulic jump, hydraulic drop, flow over weir and flow under a sluice gate.

The flow is gradually varied if the depth changes slowly over a comparatively long distance. Examples of gradually varied flow(GVF) are flow over a mild slope and the backing up of flow (backwater).

RVF RVFGVF RVFGVF RVFGVF

Sluice

Hydraulic jump

Flow over weir

Hydraulic drop

Contraction below the sluice

1.4 State of FlowThe state or behaviour of open-channel flow is governed basically by viscosity and gravity effects relative to inertial forces of the flow.

Effect of viscosity - depending on the effect of viscosity relative to inertial forces, the flow may be in laminar, turbulent, or transitional state.

- Reynolds number represents the effect of viscosity relative to inertia,

νVR

=Re

where V is velocity of flow, R is hydraulic radius of conduit and ν is kinematic viscosity (for water at 20°C, ν = 1.004 × 10−6 m2/s, dynamic viscosity µ = 1.002 × 10−3 Ns/m2 and density ρ = 998.2 kg/m3).

Re < 500 → the flow is laminar

500 < Re < 12500 → the flow is transitional

Re > 12500 → the flow is turbulent

The flow is laminar if viscous forces are dominant relative to inertia. Viscosity will determine the flow behaviour. In laminar flow, water particles move in definite smooth paths.

The flow is turbulent if inertial forces are dominant than viscous force. In turbulent flow, water particles move in irregular paths which are not smooth.

νVR

=Re

Effect of gravity - depending on effect of gravity forces relative to inertial forces, the flow may be subcritical, critical and supercritical.

- Froude number represents the ratio of inertial forces to gravity forces,

gDV

=Fr

where V is velocity, D is hydraulic depth of conduit and g is gravity acceleration (g = 9.81 m/s2).

Fr < 1 , flow is in subcritical state

Fr = 1 , flow is in critical state

Fr > 1 , flow is in supercritical state

gDV <→

gDV =→

gDV >→

1.5 Regimes of Flow

A combined effect of viscosity and gravity may produce any one of the following four regimes of flow in an open channel:

a. subcritical - laminar , when Fr < 1 and Re < 500

b. supercritical - laminar , when Fr > 1 and Re < 500

c. supercritical - turbulent , when Fr > 1 and Re > 12500

d. subcritical - turbulent , when Fr < 1 and Re > 12500

Find:

(a) Top surface width T, flow area A, wetted perimeter P, hydraulic radius R, and hydraulic depth D.

(b) If Q = 1.25 m3/s, determine the state of flow. Given kinematic viscosity of water at 20°C = 1.003 × 10−6 m2/s.

(c) If length of channel L = 50 m, find total cost to construct the channel. Given excavation cost = RM 6/m3 and lining cost = RM 80/m2.

Activity 1.4

500 mm

570 mm

30 mm

1

1.5

Activity 1.5If depth of flow y in the following channel is 3.6 m, find top surface flow width T, flow area A, wetted perimeter P, hydraulic radius R, and hydraulic depth D.

Main channel

Right floodplainLeft floodplain

3 m

30 m

113 2

S = 4%S = 1%

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