1

FLOW IN OPEN CHANNELS

Re = RhV/ν

Usually, R h > 0.1 ft and V > 1 ft/s and νwater=10-5 ft 2/s

Hence, Re = 0.1 X1/10-5 = 104

Re > 750

∴ FLOW IN OPEN CHANNELS IS ALMOST ALWAYS TURBULENT

APPLY ENERGY EQUATION

p1/γ + z1 + V12/2g = p2/γ + z2 + V2

2/2g + hL

But,

p1/γ + z1 = y1 + S0 ∆X AND p2/γ + z2 = y2

y1 + S0 ∆X + V12/2g = y2 + V2

2/2g +hL

If Channel bottom is Horizontal andhL is neglected

z1

y1

z2

V12/2g

y2

V22/2g

∆X

hL

HGL Slope =S

Channel bed Slope = S0

EL Slope = Sf

p1/γp2/γ

S0∆X

2

y1 + V12/2g = y2 + V2

2/2g

y1 + V12/2g = y2 + V2

2/2g = Constant !depth of Channel + Velocity Head is Constant(If frictional losses=0 AND bed of channel is flat.

Lets call this constant: Specific Energy

E = y + V2/2g = y + Q2/2gA2

We like to write flow rate per unit width ofchannel: so define q = Q/b

E = y + q2/2gy2

Or, (E – y)y 2 = q2/2g : Note a cubic equationwith 3 roots. (third one is negative and has nophysical meaning)

(E – y)y2 = q2/2g = constant for a given qSo if we plot E vs y that’s a hyperbola with two values of E for each valueof y

The above diagram is called a “Specific Energy Diagram”

Point C is a dividing point in the flow:

E

y

E = y

y = 0

y

yc Vc2/2g

V2/2gy

V2/2g

C

3

At depth below C large velocities are observed and at depth above C:smaller velocities result.

The velocity at C is called “critical velocity”The depth at C is called “critical depth”

Characteristics for Critical Flow

E = y + Q2/2gA2

dE/dy = 0 and dA = Tdy

0 = 1 + (Q2/gA3)dA/dy

A3/Tc = Q2/g

yc and V c can be obtained as follows:

E = y + V2/2gdifferentiate w.r.t y

or dE/dy = 0 at y=y c

0 = 1 – q2/gyc3 or q 2 = gyc

3

Since Q = VA = V(by) So, q = Q/b = Vy = V cyc

(for rectangular channel)

(Vcyc)2 = gyc

3

or yc = Vc2/g = (q2/g)1/3 ….rectangular channel

Also the energy at the critical point is MINIMUM energy given by

Emin = Ec = yc +Vc2/2g = 1.5yc Also, y c = (2/3)Emin

Criterion for supercrical and subcritical flow

Critical flow at V c = (gyc)1/2

T

y

(For rectangularchannel)

4

Supercritical: V > (gy) 1/2

Subcritical : V < (gy) 1/2

Define a number “Froude Number”

F = V/(gy)1/2

So, F = 1: criticalF > 1: supercritical flowF < 1: subcritical flow

What about slope of channel bed at critical flow?

Recall, mannings equation for flow rate:

2/10

3/249.1SAR

nQ h=For wide and shallow rectangular channel: R h = yCritical slope is given by:

3/1

2

486.1c

cy

gnS

=

CHARACTERISTICS OF CRITICAL, SUPER AND SUBCRITICAL FLOWSUPER CRITICAL SUB

Velocity V>V c V=Vc=(gy)1/2 V<Vc

Depth y<y c y=yc=(q2/g)1/3 y>yc

Slope ofchannel bed

S0>Sc

Steep slopeS0=Sc

3/1

2

486.1c

cy

gnS

=

(in B.G)

S0<Sc

Mild slope

Froude number F>1 F=1 F<1

Since

E = y + V2/2g = y + Q2/2gA2 = y + q2/2gy2

For a fixed specific energy, plot q vs y

At c

EXA

Phy

• S

At a

• FM

y

ritic

MP

sica

luic

crit

reeild

yc

al flow, discharge is m

LES OF OCCURANC

l sign of critical depth

e gate at broad creste

ical gate opening, gat

outfall: effect of gravislope: curvature of ou

S0 < Sc

yc

qmax

q

aximum for a given specific energy.

E OF CRITICAL FLOW

: “Wavy water surface”

d weir

e has no ef

tytfall

EL

3 to 10 yc

0.7

yc

5

fect on flow.

2 yc

6

Steep Slope

• Change in slope (break in grade) (sub to super)

• Change in elevation of channel bed

yc

y1

y2

yc

y2 > y1

Supercritical

yc

y1

ycy2

Subcritical

yc

EL

S0 > Sc

7

• Change in width of channel (changing q)

yc

y1

y2

yc

Supercriticaly2 > y1

yc

y1

ycy2

Subcriticaly2 < y1