Hydraulics

Lecture #9

Hydraulic Effiiency

• For the same cross sectional area (A) and the bottom slope (S), the channel delivers a larger discharge with larger hydraulic radius, R=A/P.

2/3 1/20

1Manning equation: hRAQ Sn

• For a given (fixed) cross sectional area, the least perimeter will deliver the largest discharge.

• Such a cross section is called the best hydraulic section.

Example• Find the best hydraulic section for a trapezoid with a constant area. What is Rh?

m

b

1y

A ( )b my y

Find the cross section with a minimum hydraulic radius.

P 2 22 ( )b my y

m

b

1y

22 1b y m

Hydraulic radius hR

2by my

AP

, should be minimuTo have the largest m since is constant. hR P A

2( , , ) --- (1)A b y m by my const

2 --( , , ) 2 -(1 2)P b y m b y m

and have three independent variables ( , , ), but from const, we know that all three cannot be independent.A P b m y

A

We need to eliminate one variable to find P .min

From (1), b mAy

y and substitute this into (2).

2( 2, ) 1mAP y m yy my

2( 2, ) 1mAP y m yy my

occurs whe 0 and 0.n minP PPy m

22 2 1 0 A m

yP my

2

2 01

yP ym m

m

2

2Solve for : 01

P ymymm m

2Since 0 21, 0

1y m

m

1

3m

22

1 Substitue into 2 1 03

Pm m myyA

2

1 2 1 03

13

Ay

Py

23A y

23A y

( 3)1 /m 22 1myAP y my

2 1 12 1333

y y yy

2 3y

23 1232h

yR AP

yy

The best hydraulic section of a trapezoid has 2hyR

Best Hydraulic Section

Gradually Varied Flow

• Uniform flow– The water depth remains a constant  value known as the normal depth. The energy grade line is parallel to the water surface and the channel bottom.

• Rapidly varied flow– Rapid changes in water depth & velocity take place in a short distance. The energy equation cannot be used. The momentum equation is used instead.

• Gradually varied flow– Velocity and water depth changes take place very gradually with distance so the effects of acceleration on the flow between two adjacent sections are negligible. The energy equation can be used.

– Energy is gradually dissipated due to friction, which causes the change of a water surface.

Water Surface Profile in GVF2

2Total energy 2Qg

H y zA

2

3 ( : flow direction)dH dy Q dA dzdx dx gA dx d

xx

Recall ,dA Tdy 2

3

dH dy Q T dy dzdx dx gA dx dx

3

2

1 Q T dy dzgA dx dx

2

31

dH dzdy dx dx

Q TdxgA

3

2VF QAgD

r Tg

(hydraulic depth)AT

D

021fS S

Fdydx r

0 , fS Sdz dHdx dx

In gradually varied flows, the Manning equaion may be appliedto calculate the friction (energy slope) because the water surfacechange from one section to another section is very small.

A wide rectangular channel2/3 1/2Manning equation: 1 h fSR

nQ A0

21fS S

Fdydx r

2 1 2 /hA by yRP b y y b

If (very wide), hb y R y

2/3 1/2( )1fQ by y S

n

2 2

2 10/3fn QS

b y (energy slope in GVF at a specific section)

2 2

0 2 10/3n

n QSb y

(energy slope for a unifrom flow = bottom slope)

This equation can be used to estimate the shape of the water surface profile in gradually varied flows.

02 becomes

1fd S S

Fd ryx

2 2 2 2

02 10/3 2 10/3Using ,fn

n Q n QS Sb y b y

2 2 2 2

2 10/3 2 10/3

2 2(1 / )n

n Q n Qb y b ydy

Q bydxgy

2

3In a rectangular channel, cqyg

10/32 2

2 10/3

2

2 3

1

1

n

n

yn Qb y y

Qgb y

2

32

Qgb

10/3

0

3

1

1

n

c

yydy

dx

y

S

y

10/3

0

2

2 3

1

1

nyy

Qgb

S

y

Channel Classification• Mild channel:   yn >  yc• Steep channel:   yn <  yc• Critical channel:   yn =  yc• Horizontal channel:   S0 = 0• Adverse channel:  S0 < 0

10/3

0

3

1

1

n

c

yydy

dx

y

S

y