OPEN-CHANNEL FLOW

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OPEN-CHANNEL FLOW

Open-channel flow is a flow of liquid (basically water) in a conduit with a free surface.

That is a surface on which pressure is equal to local atmospheric pressure.

Free surface

P_atm P_atm

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Classification of Open-Channel Flows

Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface.

p=p_atm Natural flows: rivers,

creeks, floods, etc. Human-made systems:

fresh-water aquaducts, irrigation, sewers,

drainage ditches, etc.

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Open channels

Natural channels Artificial channels

Open cross section Covered cross section

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Total Head at A Cross Section:

The total head at a cross section is:

where H=total head

Z=elevation of the channel bottom

P/g = y = the vertical depth of flow (provided that pressure distribution is hydrostatic)

V²/2g= velocity head g

2 α V γ +

+ P z

= H

2 av

EGL

Datum x

z y

αV²/2g

Q

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S_f:the slope of energy grade line

S_w:the slope of the water surface

Energy Grade Line & Hydraulic Grade Line in Open Channel Flow

S_f :the slope of energy grade line S_w :the slope of the water surface S_o :the slope of the bottom

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z₁

z₂

Datum line Channel bottom

EGL

y₁

y₂

V₁²/2g

V₂²/2g

h_f

1 2

Open-Channel Flow

HGL

EGL HGL

Pipe centerline

Datum line

P₁/γ

z₁ z₂

P₂/γ

V₁²/2g

V₂²/2g

Pipe Flow

1 2

Comparison of Open Channel Flow and Pipe Flow

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Kinds of Open Channel

Canal

Flume

Chute

Drop

Culvert

Open-Flow Tunnel

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Kinds of Open Channel

CANAL is usually a long and mild-sloped channel built in the ground.

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FLUME is a channel usually supported on or above the surface of the ground to carry water across a depression.

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Kinds of Open Channel

CHUTE is a channel having steep slopes.

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DROP is similar to a chute, but the change in elevation is affected in a short distance.

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CULVERT is a covered channel flowing partly full, which is installed to drain water through highway and railroad embankments.

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Kinds of Open Channel

OPEN-FLOW TUNNEL is a comparatively long covered channel used to carry

water through a hill or any obstruction on the ground.

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Types of Open-Channels

Examples of open channels flow are river, streams, flumes, sewers, ditches and lakes.

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Types of Open-Channels

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Classification of Open-Channel Natural Channels: Very irregular in shape.

Rivers, tidal estuaries.

Artificial Channels: Developed by men and usually designed with regular geometric shapes.

Irrigation canals, laboratory flumes.

Prismatic Channels: unvarying cross- section and constant bottom slope.

Artificial channels like Rectangular, Trapezoid

Non-Prismatic Channels varying cross- section and bottom slope.

The natural channels are usually prismatic

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Classification of Open-Channel

Small Slope Channels: having a

bottom slope less than 1 in 10 (10%). Large Slope Channels: having a bottom slope greater than 1 in 10 (10%).

Rigid Boundary Channels: Non- changeable boundaries (bed and sides).

Lined canals, sewers and non-erodible unlined canals

Mobile Boundary Channels: Boundary is composed of loose sedimentary particles moving under the action of flowing water.

An alluvial channel

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THE CHANNEL SECTION is the cross section of a channel taken normal to the direction of the flow.

THE VERTICAL CHANNEL SECTION is the vertical section passing through the lowest or bottom point of the channel section.

d

The channel section (B-B)

The vertical channel section (A-A) y

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Geometric Elements of Channel Section

THE DEPTH OF FLOW, y, is the vertical

distance of the lowest point of a channel section from the free surface.

Datum

z

h θθθθ

θ

y d

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Geometric Elements of Channel Section

THE DEPTH OF FLOW SECTION, d, is the depth of flow normal to the direction of flow.

Datum

θ θθ θ

z h

θ

y d

θ is the channel bottom slope d = ycosθ.

For mild-sloped channels y ≈ d.

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Geometric Elements of Channel Section

THE TOP WIDTH, T,

is the width of the channel section at the free surface.

THE WATER AREA, A,

is the cross-sectional area of the flow normal to the direction of flow.

THE WETTED PERIMETER, P,

is the length of the line of intersection of the channel wetted surface with a cross- sectional plane normal to the direction of flow.

THE HYDRAULIC RADIUS, R = A/P,

is the ratio of the water area to its wetted perimeter.

THE HYDRAULIC DEPTH, D = A/T, is the ratio of the water area to the top width.

d

A = A(d)

T

P

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Geometric Properties of Open Channels

Channel slope

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Open Channel Nomenclature

• Flow depth is the depth of flow at a cross-section measured from the channel bottom.

y

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• Elevation of the channel bottom is the elevation at a cross-section measured from a reference datum (typically MSL).

y

Datum

z

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• Slope of the channel bottom, S_o, is called the topographic slope or channel slope.

y

Datum

S_o

1 z

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Open Channel Nomenclature

• Slope of the water surface is the slope of the HGL, or slope of WSE (water surface elevation).

y

Datum

S_o

1 z

S_wse 1 HGL

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• Slope of the energy grade line (EGL) is called the energy or friction slope.

y

Datum

S_o

1 z

S_wse 1 HGL

EGL S_f 1

V²/2g Q=VA

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Geometric Properties of Open Channels



Top width (T): It is the width of the channel at the free surface as measured perpendicular to the

direction of flow at any given section. It is expressed in meters.



Hydraulic depth (D) It is the ratio of area of cross section (A) to the top with (T).



Section factors (Z): It is the product of the area of cross section (A) to the square root of the

hydraulic depth (D).



 





 T

D A

2 1 3



 







 T

D A A Z

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Geometric Properties of Open Channels



Hydraulic Slope (S): Hydraulic slope of the total energy line is defined as the ratio of drop in total energy line (h

_f

) to the channel length (L).



The stage H is the elevation or vertical distance of

the free surface above the datum. If the lowest

point of the section is chosen as the datum, the

stage is identical with the depth of flow.

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Channel Cross-sections

Cross sections of channels that operate under open

channel flow

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Channel Geometry

The wetted perimeter does not include the free

surface.

Examples of R for common geometries shown in Figure at the left.

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Geometric Properties of Channels

Definitions

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Geometric Properties of Channels

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A

B

b

h

B

b

h m

1

rectangular trapezoidal triangular circular parabolic

B

h m

1

B

θ D h

B

h

flow area

wetted perimeter

P

hydraulic radius

Rh

top width

B

hydraulic depth

Dh

bh (^b⁺^mh)^h mh² ( sin ) ²

8

1 θ − θ D Bh

3 2

h

b+2 b+₂h ₁+m² 2h 1+m² θ ^D 2 1

B B h

2

3

+8 *

*

h b

bh +2

( )

1 2

2h m

b

h mh b

+ +

+

1 2

2 m

mh

+ D



− θ sinθ 4 1

1

2 2

2

8 3

2 h B

h B

+

b b+2mh 2mh

(sinθ/2)^D

2 Ah 3

(^D ^h)

h −

or 2

1 0<ξ ≤

( )

mh b

h mh b

2 +

+ h

2 1

8 2 / sin

sin D



 − θ

θ

θ h

3 2

* ^{Valid for} ^where ^ξ ⁼⁴^h^/^B

1 ξ >

If ^then ^P⁼(^B^/²)[ ¹⁺^ξ² ⁺(¹^/^ξ)^ln(^ξ ⁺ ¹⁺^ξ²)]

h

Geometric elements for different channel cross sections

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Geometric Properties

Channel cross-section notation and formulas

Adapted from USDA-SCS 1972c

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Open Channel Geometric Relationships For

Various Cross Sections

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Effective Forces In Flow Analysis

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EXAMPLE

A rectangular concrete channel is 3 m wide and 2 m high. The water in the channel is 1.5 m deep and is flowing at a rate of 30 m3/s. Determine the flow area, wetted perimeter, and hydraulic radius.

Solution

From the section’s shape (rectangular), we can easily calculate the area as the rectangle’s width multiplied by its depth. Note that the depth used should be the actual depth of flow, not the total height of the cross-section.

The wetted perimeter can also be found easily through simple geometry. A = 3.0 m × 1.5 m = 4.5 m

²

Pw = 3.0 m + 2 × 1.5 m = 6.0 m

R = A / Pw = 4.5 m

²

/ 6.0 m =

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Types of Flow

Criterion: Change in flow depth with respect to time and space

Steady flow (∂y/∂t=0)

Unsteady flow (∂y/∂t≠0)

Uniform Flow

(∂y/∂x=0) Varied Flow (∂y/∂x≠0)

Uniform Flow (∂y/∂x=0)

Varied Flow (∂y/∂x≠0)

GVF RVF GVF RVF

OCF

Time is a criterion

Space is a criterion

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Types of Flow

Criterion: Change in discharge with respect to time and space

OCF

Time is a criterion

Space is a criterion

Steady flow (∂Q/∂t=0)

Unsteady flow (∂Q/∂t≠0)

Continuous Flow

(∂Q/∂x=0)

Spatially- Varied Flow (∂Q/∂x≠0)

Continuous Flow

(∂Q/∂x=0)

Spatially-Varied Flow

(∂Q/∂x≠0)

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Classification of Open-Channel Flows

Obstructions cause the flow depth to vary.

Rapidly varied flow (RVF) occurs over a short distance near the obstacle.

Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF.

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Steady non-uniform flow in a channel.

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Flow classification (Space Criteria – Non Uniform Flow)

• Rapidly varying flows (RVF): flow depths that vary considerably over a short distance

• Gradually varying flow (GVF):

flow depths that vary slowly with distance.

1 2 3 4 5 6 7

8

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State of Flow

Effect of viscosity:

υυ υυ

===

= VR Re

OCF

Laminar OCF, Re < 500

Transitional OCF, 500 < Re < 1000 Turbulent OCF, Re > 1000

Note that R in Reynold number is Hydraulic Radius

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Flow classification (flow particles motion)

Laminar

• Type of fluid flow in which the fluid travels smoothly or in regular paths.

• The flow channel is relatively small, the fluid is moving slowly, and its viscosity is relatively high.

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Flow classification (flow particles motion)

Transitional Turbulent

• The fluid undergoes irregular

fluctuations and mixing. • Most kinds of fluid flow are turbulent except near solid boundaries

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Effective Forces In Flow Analysis (Reynolds number)

Viscous Inertia Gravity

Reynolds number = 𝐼𝑛𝑒𝑡𝑟𝑖𝑎 𝐹𝑜𝑟𝑐𝑒𝑠 𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝐹𝑜𝑟𝑐𝑒𝑠

where 𝑉 = Average velocity of flow, D = pipe diameter, and 𝜐 = Kinematic viscosity of the fluid.

𝑅_𝑒 = 𝑉 × 𝐷 𝜐

𝑅_𝑒 = 𝑉 × 𝑅_ℎ 𝜐

where 𝑉 = Average velocity of flow, 𝑅_ℎ = is hydraulic radius, and 𝜐 = Kinematic viscosity of the fluid.

Laminar flow : Re < 500 (viscous > inertia)

Turbulent flow: Re > 1300 (inertia > viscous) Transitional flow: 500 < Re < 1300

A non-dimensional number

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Effect of Gravity

In open-channel flow the driving force (that is the force causing the motion) is the component of gravity along the channel bottom. Therefore, it is clear that, the effect of gravity is very important in open-channel flow.

In an open-channel flow Froude number is defined as:

In an open-channel flow, there are three types of flow depending on the value of Froude number:

F_r>1 Supercritical Flow F_r=1 Critical Flow

F_r<1 Subcritical Flow

gD

= V F or gD

== V F

and

Force, Gravity

Force Inertia

=

F _r

2 2

r r

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In wave mechanics, the speed of propagation of a small amplitude wave is called the celerity, C.

If we disturb water, which is not moving, a disturbance wave occur, and it propagates in all directions with a celerity, C, as:

For a rectangular channel, the hydraulic depth, D=y.

Therefore, Froude number becomes:

C

= V gy

= V F_r

gy

= C C

C

C C

C

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Now let us consider propagation of a small amplitude wave in a supercritical open channel flow:

Since V > C, it CANNOT propagate upstream it can propagate only towards downstream with a pattern as follows:

This means the flow at upstream will not be affected.

In other words, there is no hydraulic communication between upstream and downstream flow.

F_r> 1, i.e; V > C

C C

Disturbance will be felt only within this region

V

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Now let us consider propagation of a small amplitude wave in a subcritical open channel flow:

Since V < C, it CAN propagate both upstream and downstream with a pattern as follows:

This means the flow at upstream and downstream will both be affected.

In other words, there is hydraulic communication between upstream and downstream flow.

F_r< 1, i.e; V < C C C

V < C

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Now let us consider propagation of a small amplitude wave in a critical open channel flow:

Since V = C, it can propagate only downstream with a pattern as follows:

This means the flow at downstream will be affected.

F_r= 1, i.e; V = C C C

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State of Flow

Effect of gravity:

D in Froude Number is Hydraulic Depth

gD Fr = = = = V

gD

V <<<< V ==== gD V >>>> gD

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Effective Forces In Flow Analysis (Froude number)

Fr > 1 : Flow is supercritical Fr < 1 : Flow is subcritical

Flow is deep, slow with a low energy state

Fr = 1 : Flow is critical

There is a perfect balance between the gravitational and inertial forces.

Viscous Inertia Gravity

Flow is fast flow with a high energy state

𝐹𝑟 = 𝐼𝑛𝑒𝑡𝑟𝑖𝑎 𝐹𝑜𝑟𝑐𝑒𝑠

𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝐹𝑜𝑟𝑐𝑒𝑠 = 𝑉 𝑔𝑅_ℎ

1 2 3

Dimensionless number

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Effective Forces In Flow Analysis (Critical Depth)

Critical depth y_c occurs at Fr = 1

At high flow velocities (Fr > 1), So: y < y_c

At low flow velocities (Fr < 1), So: y > y_c 𝐹𝑟 = 1

𝑅_ℎ = 𝑦

𝑦 = 𝑦_𝑐 = 𝑉² 𝑔 𝐹𝑟² = 𝑉²

𝑔𝑦

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• The wave speed (C) is: 𝐶 = 𝑔𝑦 Critical Flow and Velocity

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• For a wide rectangular channel, the hydraulic depth, R^h=y.

Therefore, Froude number becomes:

𝐹𝑟 = 𝑉

𝑔𝑦 = 𝑉 𝐶

Critical Flow and Velocity

𝐹𝑟 > 1 → 𝑉 > 𝐶

• Since V > C, it CANNOT propagate upstream it can propagate only towards downstream.

𝐹 →

• This means the flow at upstream will not be affected.

• In other words, there is no hydraulic communication between upstream and downstream flow.

Super Critical

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• For a wide rectangular channel:

𝐹𝑟 = 𝑉

𝐹𝑟 < 1 → 𝑉 < 𝐶

• Since V < C, it CAN propagate both upstream and downstream.

𝐹 →

• This means the flow at upstream and downstream will both be affected.

• In other words, there is hydraulic communication between upstream and downstream flow.

Subcritical

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• For a wide rectangular channel:

𝐹𝑟 = 𝑉

𝐹𝑟 = 1 → 𝑉 = 𝐶

• Since V = C, it CAN propagate only downstream.

𝐹 →

• This means the flow at downstream will be affected.

Critical

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Effective Forces In Flow Analysis Example 1-1

10 ft

1 ft

V = 2 ft/s

Determine the type of flow (laminar or turbulent) for the following rectangular channel.

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Example 1-2

Determine the type of flow (laminar or turbulent) for the following trapezoidal channel.

16 ft

4.5 ft V = 6.5 ft/s

1 3

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Example 1-3

Determine the hydraulic radius for the following circular channel.

2.0 m 4.0 m

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Example 1-4

Determine the flow condition (subcritical, critical or supercritical) for a rectangular channel with the flow velocity of 2.32 m/s.

1.06 m

1.83 m

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Example 1-5

Flow in a rectangular channel with the width of 10 ft is critical with the velocity of 11.35 ft/s. Find the depth of flow.

d ft

10 ft

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Example 1-6

Determine the critical depth and flow condition (sub or super critical) in a triangle channel with the side slope of 1:1, depth of 2 ft, and V= 5.28 ft/s.

2.0 ft

1 1

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Q1-1 Determine the flow regime (Laminar, Turbulent or Transitional) in the following triangle channel which the flow velocity is 2.9 ft/s.

Homework 1

2.57 ft 55°

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Q1-2 Calculate the hydraulic radius for the following channel.

3.6 m

1.6 m

2.0 m

4.0 m

Homework 1

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Q1-3 Determine the flow condition (sub or supercritical) in a rectangular flume with the flow velocity of 1.52 m/s and width of 1.36 m, if:

a. Ratio of flume width to flow depth is 4.0 b. Ratio flume width to flow depth is 0.5 c. Flume is very wide ( )

Homework 1

𝑏

𝑦 ≥ 10

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Velocity Profiles

In order to understand the velocity distribution, it is

customary to plot the isovels, which are the equal velocity lines at a cross section.

isovel

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• Velocity is zero on bottom and sides of channel due to no-slip condition the maximum velocity is usually below the free surface.

• It is usually three-dimensional flow.

• However, 1D flow approximation is usually made with good success for many practical problems.

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Velocity Distribution

The velocity distribution in an open-channel flow is quite nonuniform because of :

• Nonuniform shear stress along the wetted perimeter,

• Presence of free surface on which the shear stress is zero.

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• The velocities in channel are not uniformly distributed (usually axisymmetric) in channel section because of presence of a free surface and friction along the channel wall.

• It might be expected to find the maximum velocity at the free surface where the shear force is zero but this is not the case.

Velocity Distribution on Open Channels

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• The maximum velocity is usually found just below the surface.

• The reason is the presence of secondary currents which are circulating from the boundaries towards the section center and resistance at the air/water interface.

Velocity Distribution on Open Channels

• The measured maximum velocity usually appears to occur below the free surface at a distance 0.05 to 0.15 (some references say 0.25) of the depth.

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Kinds of Surface Wave

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Small amplitude Wave Speed

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Conservation of Mass

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Conservation of Momentum

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Momentum Cefficient

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Equation of Motion

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Energy Coefficient

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Values of α and β for typical sections

Typically, beta varies in the range 1.01-1.12 for fairly straight prismatic channels.

Typically, alpha varies in the range 1.03-1.36 for fairly straight prismatic channels.

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Values of α and β for typical sections

Values of alpha and beta can be approximately estimated with the following relations:

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Pressure Distribution

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Horizontal, Parallel Flow

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Parallel Flow in Sloping Channels

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Curvilinear Flow

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Curvilinear Flow