Basic Hydraulics: Open Channel Flow – I

Open Channel Definitions

• Open channels are conduits whose upper boundary of flow is the liquid surface

• Canals, rivers, streams, bayous, drainage ditches are common examples of open channels.

• Storm and sanitary sewers are are also open channels unless they become surcharged (and thus behave like pressurized systems).

Open Channel Nomenclature

• Elevation (profile) of some open channel

Open Channel Nomenclature

• Flow profile related variables

• Flow depth (pressure head)• Velocity head• Elevation head• Channel slope• Water slope (Hydraulic grade line)• Energy (Friction) slope (Energy grade line)

Open Channel Nomenclature

• Cross Sections

Natural Cross Section Engineered Cross Section

Open Channel Nomenclature

• Cross Section Geometry and Measures

• Flow area (all the “blue”)• Wetted perimeter• Topwidth• Flow depth

• Thalweg (path along bottom of channel)

Open Channel Steady Flow

• For any discharge (Q) the flow at any section is described by:

• Flow depth • Mean section velocity• Flow area (from the cross section geometry)

• Depth-area, depth-topwidth, depth-perimeter are used to estimate changes in depth (or flow) as one moves from section to section

Open Channel Steady Flow Types

• The flow-depth, depth-area, etc. relationships are non-unique, flow “type” is relevant

• Uniform (normal)• Sub-critical• Critical• Super-critical

Cross Section Geometry

• Normal, Critical, Sub-, Super-Critical flow all depend on channel geometry.

• Engineered cross sections almost exclusively use just a handful of convenient geometry (rectangular, trapezoidal, triangular, and circular).

• Natural cross sections are handled in similar fashion as engineered, except numerical integration is used for the depth-area, topwidth-area, and perimeter-area computations.

Cross Section Geometry

• Rectangular Channel

• Depth-Area

• Depth-Topwidth

• Depth-Perimeter€

A(y) = By

€

T(y) = B

€

Pw (y) = B + 2y

• Trapezoidal Channel

• Depth-Area

• Depth-Topwidth

• Depth-Perimeter

Cross Section Geometry

€

A(y) = y(B + my)

€

T(y) = B + 2my

€

Pw (y) = B + 2y 1+ m2

• Triangular Channels

• Special cases of trapezoidal channel

• V-shape; set B=0• J-shape; set B=0, use ½

area, topwidth, and perimeter

Cross Section Geometry

• Circular Channel (Conduit with Free-Surface)

• Contact Angle:

• Depth-Area:

• Depth-Topwidth:

• Depth-Perimeter:

Cross Section Geometry

€

A(y) =D2

4

α

2− sin(

α

2)cos(

α

2)

⎛

⎝ ⎜

⎞

⎠ ⎟

€

T(y) = Dsin(α

2)

€

Pw (y) =Dα

2

€

α(y) = 2cos−1(1−2y

D)

• Irregular Cross Section• Use tabulations for the

hydraulic calculations

Cross Section Geometry

• Irregular Cross Section – Depth-Area

Cross Section Geometry

A1

A2

A3A4

Depth

AreaA1

A1+A2

A1+A2+A3

• Irregular Cross Section – Depth-Area

Cross Section Geometry

T1

T2T3

T4

Depth

TopwidthT1

T1+T2

T1+T2+T3

• Irregular Cross Section – Depth-Perimeter

Cross Section Geometry

P1

P2

P3P4

Depth

PerimeterP1

P1+P2

P1+P2+P3

• Convention is to express station along a section with respect to “looking downstream”• Left bank is left side of

stream looking downstream (into the diagram)

• Right bank is right side of stream looking downstream (into the diagram)

Flow Direction/ Cross Section Geometry

Left Bank Right Bank

Flow Direction

Energy Equation in Open Channel Flow

• Energy equation:

α = velocity head correction factor

€

y1 + z1 +α 1v1

2

2g= y2 + z2 +

α 2v22

2g+ hL

Energy Equation in Open Channel Flow

• When velocity is nearly uniform across the channel the correction factor is usually treated as unity (α = 1)

• Hence, the energy equation is typically written as

€

y1 + z1 +v1

2

2g= y2 + z2 +

v22

2g+ hL

Potential Energy• In pressurized systems the potential energy is the

sum of the pressure and elevation head.

• In open channels, elevation is taken at the bottom of the channel, the analog to pressure is the flow depth.

• Thus the potential energy is the sum of elevation and flow depth

€

Static Head (Potential) = y + z

Kinetic Energy• Kinetic energy is the energy of motion; in

pressurized as well as open channel systems, this energy is represented by the velocity head

• The sum of these two “energy” components is the total dynamic head (usually just “total head”)

€

Kinetic (Velocity) Head =v 2

2g

€

Total Head = y + z +v 2

2g

Hydraulic Grade Line• The hydraulic grade line is coincident with the water

surface.• It represents the static head at any point along the

channel.

Specific Energy

• Total energy is the sum of potential and kinetic components:

• Energy relative to the bottom of the channel is called the specific energy (at a section)

€

E total = z + y +v 2

2g

€

E specific = y +v 2

2g

Specific Energy

• Relationship of Total and Specific Energy (at two different sections)

€

E specific = y +v 2

2g

€

E total = z + y +v 2

2g y1

y2

Specific Energy Calculations• For example • Rectangular channel,

• Q=100cfs; B=10 ft

• So

• Table shows values. Plot on next page = specific energy diagram€

E specific = y +v 2

2g= y +

(10 / y)2

2g= y +

100

2gy 2€

V =Q

A=

100 cfs

(10 ft)(y ft)=

10

y

Specific Energy Diagram

Critical Depth

• Specific energy relationship has a minimum point

• Flow at specified discharge cannot exist below minimum specific energy value

• Depth associated with minimum energy is called “critical depth”

• Critical depth (if it occurs) is a “control section” in a channel

• What is the value of critical depth for the case shown in the previous diagram (and table)?

Flow Classification by Critical Depth

• Subcritical flow –Water depth is above critical depth (velocity is less than the velocity at critical depth)

• Supercritical flow – Water depth is below critical depth (velocity is greater than the velocity at critical depth)

• Critical flow – Water depth is equal to critical depth. 1 to 1 depth-discharge at critical (dashed line)

Q1

Q2

Q3

Flow Classification by Critical Depth

• Classification important in water surface profile (HGL) estimation and discharge measurement.

• Water can exist at two depths except at critical depth• Critical depth important in measuring discharge

• Sub- and Super-Critical classification determine if the controlling section is upstream or downstream.

• Sub- and Super-Critical classification determines if computed HGL will be a front-water or back-water curve.

Conveyance

• The cross sectional properties can be grouped into a single term called conveyance

• Manning’s equation becomes

• Units of conveyance are CFS

€

Q =KS01/ 2€

K =1.49

nAR2 / 3

Normal depth

• Normal depth is another flow condition where the slope of the energy grade line, channel bottom, and the slope of the hydraulic grade line are all the same

• Manning’s equation assuming normal depth is

€

Q =1.49

nAR2 / 3S0

1/ 2

€

A

Pw

Normal depth

• To use need depth-area, depth-perimeter information from channel geometry.

• Then can rearrange (if desired) to express normal depth in terms of discharge, and geometry.

• Computationally more convenient to use a root-finding tool (i.e. Excel Goal Seek/Solver) than to work the algebra because of the exponentation of the geometric variables.

Normal depth

• For example, TxDOT HDM Eq 10-1 is one such Manning’s equation, rearranged to return normal depth in a triangular section (J-shape)

where Q = design flow (cfs); n = Manning’s roughness coefficient ; Sx = pavement cross slope; S = friction slope; d = normal depth (ft).

8/3

2/124.1

S

QnSd x

€

Sx

€

d