lecture14 gradually varied flow3

8/10/2019 Lecture14 Gradually Varied Flow3

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Gradually Varied Flow III

Hydromechani cs VVR090

Spatially Varied Flow

Flow varies with lon gitudinal dist ance.

Examples: sid e-channel spillways, sid e weirs, channels withpermeable boundaries, gutters for c onveying st orm waterrunoff, and drop structures in the bottom of channels.

Two types of flow :

discharge increases with distance

discharge decreases with dis tance

Different principl es govern => different analysisapproach


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Side channel spil lways

Irrigation channel

Hydropower plant

Reservoir contr ol

Side Weirs

Scale model ofside weir (1:22)

Weirs from co mbinedsewer systems


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Collecting Flumes Settling Basins

Increasing Discharge

Example: side-channel spillway

Significant energy loss from mi xing between thewater entering t he channel and the water flowingin the channel.

Hard to quantify the energy loss

=> Use momentum equation in the analysis


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Analysis of Side-Channel Spillway

Simplified analysis: hydrostatic pressure distribution

small S o inflow perpendicular to channel

q * = d Q/d x = Q/x = constant

Apply th e momentum equat ion:

1 2 2 1o oP P Pdx gS Adx M M M =

= 0

The momentum equation may be developed to yield:

2

2

2

2 ( )

1/

o f

Q x dQS S

dy gA dxQdx

gA A T

=

Froude number

S f is compu ted from the Manning formula

Solution proceeds from a control section .


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Decreasing Discharge

Example: sid e weirsNo significant energy loss

=> Water surface profile can be estimated from theenergy principle

Total energy at channel section relative to a datum:

2

22Q

H z ygA

= + +

Differentiating H with respect to x:

2

2 3

1 2 22

, ,o f

dH dz dy Q dQ Q dAdx dx dx g A dx A dx

dz dH dA dA dy dyS S T

dx dx dx dy dx dx

= + +

= = = =

Substitute in relations:

( )( )( )

2

2 2

/ /

1 /o f S S Q gA dQ dxdy

dx Q gA D

=

New term


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Special case:

S o = 0

S f = 0

rectangular channel

( )2 3 2

( ) /

( )

Q x y dQ dxdydx gb y Q x

=

Example: Outflow f rom Channel Bottom

Eleven cubic meters per second are diverted throu gh port s in thebottom of the ch annel between sections 1 and 2. Neglectinghead losses and assuming a hor izontal channel, what depth ofwater is to be expected at section 2? What channel width atsection 2 would be required to prod uce a depth of 2.5 m?


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The specific energy is unchanged from 1 to 2 (noenergy loss es):

2 21 2

1 22 2u u

E y yg g

= + = +

Flow per unit width:

311

1

3212

337.33 m /m s

4.5

22 4.89 m /m s4.5

Qq

b

Qq b

= = =

= = =

Energy equation fr om 1 to 2:

2 2

1 21 2

1 2

22

22

1 12 2

7.33 1 4.89 12.4

2.4 2 2

q q y y

y g y g

yg y g

+ = +

+ = +

Flow condition in s ection 1:

1 1 11

1 1

/ 7.33/ 2.40.63 1

2.4

u q yFr

gy gy g= = = = < Subcritical flow


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Subcritical flow and q decreasing implies that the solutioncorrespondin g to subcritical flow in section 2 must be valid.

y2 = 2.72 m (found by trial-and-error )

Assu me a depth o f y 2 = 2.5 m. What chann el width i srequired?

Employ the energy equation from 1 to 2:

22

2

7.33 1 22 12.4 2.5

2.4 2 2.5 2g b g

+ = +

b 2 = 3.22 m

lecture14 gradually varied flow3

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