CE 3372 – Lecture 13

OPEN CHANNEL HYDRAULICS

OUTLINE

• Flow Terminology

• Energy Equation

• Critical Depth/Flow

• Flow Profiles for GVF

• Manning’s Equation

OPEN CHANNEL DESIGN CONCEPTS

• Interest to engineers:• Water surface elevation (WSE)

(minimize impact/reduce floods)

• Discharge –Depth relationships

• Channel design

OPEN CHANNELS

• Conduits whose upper boundary of flow is the liquid surface.

TYPES OF FLOW• Steady Flow – flow, depth and velocity may differ from point to point but

remain constant over time

• Unsteady Flow – flow, depth, and velocity is a function of time

• Uniform Flow – occurs in prismatic channels when flow depths are equalno change in velocity within the channel: Q, y, A, S are all

constant

• Non-uniform Flow – velocity is not the same at every point

Temporal

Spatial

OPEN CHANNEL NOMENCLATURE

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

y

OPEN CHANNEL NOMENCLATURE

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

y

Datum

z

OPEN CHANNEL NOMENCLATURE

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

y

Datum

So

1z

OPEN CHANNEL NOMENCLATURE

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

y

Datum

So

1z

1Swse

HGL

OPEN CHANNEL NOMENCLATURE

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

y

Datum

So

1z

1Swse

HGL

EGL

1SfV2/2g

Q=VA

STEADY NON-UNIFORM FLOW

Sketch of steady flow in a channel

Based on cross-sections: Section 1 is upstream Section 2 is downstream

STEADY FLOWThe energy grade line (EGL) is: z_head + P_head + v_head

elevation

pressure (depth)

velocity

Sketch of steady flow in a channel

The hydraulic grade line (HGL) is at the water surface

Hydraulic grade line (HGL)

Energy grade line (EGL) Profile grade line is the channel bottom

The head loss is depicted as the differencebetween a horizontal zero-loss energy grade line and the energy grade line

ENERGY RELATIONSHIPS

• Energy equation for closed conduits

• Energy equation for open conduits

2 21 1 2 2

1 1 2 22 2 L

p V p Vz z h

g ga a

g g+ + = + + +

2 21 2

1 22 2o f

V Vy S x y S x

g g+ D + = + + D

SPECIFIC ENERGY

• The sum of the depth of flow + velocity head (Head relative to the channel bottom)

• For a given discharge, the SE can be calculated for various flow depths including critical depth

g

VyE

2

2

CRITICAL DEPTH

• Depth of flow for a given discharge, where the specific energy is at a minimum

• Occurs when dE/dy = 0 and Fr = 1

• It is important to calculate yc in order to determine if the flow in the channel will be subcritical or supercritical

• Can be found through Specific Energy Diagram

2

2

2gA

QyE

A = By

B

y

Alternate Depths:

Q=qywhere q is the discharge/unit width of channel

Plug & chug. Solve for y

3 roots –1 negative = 2 depths

OPEN CHANNEL FLOWS

• Open channel flow is also classified by the Froude number

• Critical depth, yc occurs at Fr = 1

OPEN CHANNEL FLOWS

Subcritical flow• Low velocities, Fr < 1• Disturbance travels upstream• y > yc

Supercritical flow• High velocities, Fr > 1• Disturbances travel

downstream• y < yc

PA

CRITICAL FLOWT

dy

yHas a minimum at yc

2

2

2gA

QyE 0

dy

dE

dA=

0dEdy

= =

3

2

1c

c

gA

TQ

Arbitrary cross-section

22

FrgA

TV

dA

AD

T=

2

31Q dAgA dy

-

Tdy More general definition of Fr

CRITICAL FLOW – RECTANGULAR CHANNEL

yc

T

Ac

3

2

1c

c

gA

TQ

qTQ TyA cc

3

2

33

32

1cc gy

q

Tgy

Tq

3/12

g

qyc

3cgyq

Only for rectangular channels!

cTT

Given the depth we can find the flow!

qTQ

CRITICAL FLOW: RECTANGULAR CHANNELS3/1

2

g

qyc cc yVq

g

yVy ccc

223

g

Vy cc

2

1gy

V

c

cvelocity head =

g

Vy cc

22

2

2

cc

yyE Eyc

3

2

0.5 (depth)

g

VyE

2

2

g

Vy cc

22

2

1gy

V

c

c

OPEN CHANNEL FLOWS• Similar to pipe flow, open channel flow can be classified into

which is dependent on Reynolds number

OPEN CHANNEL FLOWS

Where V = average velocityRh = hydraulic radius

v = kinematic viscosity

Area represents cross sectional area of the

fluid

Wetted perimeter does not include the

free surface

RECTANGULAR CONDUIT

TRAPEZOIDAL CHANNEL

• common geometry• Engineered (improved)

natural channels are reasonably well approximated by trapezoidal equations• the geometry is important in

drainage engineering

CIRCULAR CONDUIT

• Sweep angle definition matters, book uses 2a.

VARIED FLOW• Gradually varied flow – change in flow depth moving upstream

or downstream is gradual• Rapidly varied flow – change in flow depth occurs over a very

short distance• Ex: waterfall, hydraulic jumps, etc.

• RVF is outside the scope of this course.

GRADUALLY VARIED FLOW

• Equation relating slope of water surface, channel slope, and energy slope:

Variation of Water Surface Elevation

Discharge and Section Geometry

Discharge and Section Geometry

GRADUALLY VARIED FLOW• Procedure to find water surface profile is to integrate the depth taper with distance:

FLOW PROFILESSLOPE DEPTH RELATIONSHIP

Steep yn < yc

Critical yn = yc

Mild yn > yc

Horizontal S0 = 0

Adverse S0 < 0

PROFILE TYPE DEPTH RELATIONSHIP

Type-1 y > yc AND y > yn

Type -2 yc < y < yn OR yn < y < yn

Type -3 y < yc AND y < yn

MANNING EQUATION (1891)

• Depth-Discharge Calculator for any open channel implements Manning's equation

• The equation is the U.S. customary version• A drainage engineer in the US should memorize this

equation!

VALUES OF MANNING N

Lined Canals n Cement plaster 0.011 Untreated gunite 0.016 Wood, planed 0.012 Wood, unplaned 0.013 Concrete, trowled 0.012 Concrete, wood forms, unfinished 0.015 Rubble in cement 0.020 Asphalt, smooth 0.013 Asphalt, rough 0.016

Natural Channels Gravel beds, straight 0.025 Gravel beds plus large boulders 0.040 Earth, straight, with some grass 0.026 Earth, winding, no vegetation 0.030 Earth , winding with vegetation 0.050

d = median size of bed material

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

6/1031.0 dn d in ft

SUMMARY OF OPEN CHANNELS

• All the complications of pipe flow plus ______________________

• 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)

free surface location