lecture14 gradually varied flow3
Post on 02-Jun-2018
250 Views
Preview:
TRANSCRIPT
-
8/10/2019 Lecture14 Gradually Varied Flow3
1/9
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
-
8/10/2019 Lecture14 Gradually Varied Flow3
2/9
Side channel spil lways
Irrigation channel
Hydropower plant
Reservoir contr ol
Side Weirs
Scale model ofside weir (1:22)
Weirs from co mbinedsewer systems
-
8/10/2019 Lecture14 Gradually Varied Flow3
3/9
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
-
8/10/2019 Lecture14 Gradually Varied Flow3
4/9
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 .
-
8/10/2019 Lecture14 Gradually Varied Flow3
5/9
-
8/10/2019 Lecture14 Gradually Varied Flow3
6/9
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
-
8/10/2019 Lecture14 Gradually Varied Flow3
7/9
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?
-
8/10/2019 Lecture14 Gradually Varied Flow3
8/9
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
-
8/10/2019 Lecture14 Gradually Varied Flow3
9/9
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
top related