Channel Flow Routing
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Transcript of Channel Flow Routing
Channel Flow Routing
Reading:Applied Hydrology Sections 8.4, 9.1-9.4, 9.7
Brushy Creek Watershed
Dam 7
Subbasin BUT_060
Reservoir
Routing
Subasin Rainfall -Runoff
Reach SBR_080 Downstream of Dam 7
How do we route the flow through Reach SBR_080?
Hydrologic river routing (Muskingum Method)
Wedge storage in reach
IQ
QI
AdvancingFloodWaveI > Q
II
IQ
I Q
RecedingFloodWaveQ > I
KQS Prism
)(Wedge QIKXS
K = travel time of peak through the reachX = weight on inflow versus outflow (0 ≤ X ≤ 0.5)X = 0 Reservoir, storage depends on outflow, no wedgeX = 0.0 - 0.3 Natural stream
)( QIKXKQS
])1([ QXXIKS
5
Muskingum Method (Cont.)])1([ QXXIKS
]})1([])1({[ 111 jjjjjj QXXIQXXIKSS
tQQ
tII
SS jjjjjj
2211
1
jjjj QCICICQ 32111
tXKtXKC
tXKKXtC
tXKKXtC
)1(2)1(2)1(2
2)1(2
2
3
2
1
Recall:
Combine:
If I(t), K and X are known, Q(t) can be calculated using above equations
6
Muskingum - Example• Given:
– Inflow hydrograph– K = 2.3 hr, X = 0.15, t = 1 hour,
Initial Q = 85 cfs• Find:
– Outflow hydrograph using Muskingum routing method
Period Inflow (hr) (cfs)
1 93 2 137 3 208 4 320 5 442 6 546 7 630 8 678 9 691
10 675 11 634 12 571 13 477 14 390 15 329 16 247 17 184 18 134 19 108 20 90
5927.0
1)15.01(3.2*21)15.01(*3.2*2
)1(2)1(2
3442.01)15.01(3.2*2
15.0*3.2*21)1(2
2
0631.01)15.01(3.2*2
15.0*3.2*21)1(2
2
3
2
1
tXKtXKC
tXKKXtC
tXKKXtC
7
Muskingum – Example (Cont.)
jjjj QCICICQ 32111 Period Inflow C1Ij+1 C2Ij C3Qj Outflow
(hr) (cfs) (cfs) 1 93 0 0 0 85 2 137 9 32 50 91 3 208 13 47 54 114 4 320 20 72 68 159 5 442 28 110 95 233 6 546 34 152 138 324 7 630 40 188 192 420 8 678 43 217 249 509 9 691 44 233 301 578
10 675 43 238 343 623 11 634 40 232 369 642 12 571 36 218 380 635 13 477 30 197 376 603 14 390 25 164 357 546 15 329 21 134 324 479 16 247 16 113 284 413 17 184 12 85 245 341 18 134 8 63 202 274 19 108 7 46 162 215 20 90 6 37 128 170
0
100
200
300
400
500
600
700
800
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Time (hr)
Dis
char
ge (c
fs)
C1 = 0.0631, C2 = 0.3442, C3 = 0.5927
Unsteady Flow Routing in Open Channels
• Flow is one-dimensional• Hydrostatic pressure prevails and vertical
accelerations are negligible• Streamline curvature is small. • Bottom slope of the channel is small.• Manning’s equation is used to describe
resistance effects• The fluid is incompressible
Continuity Equation
dxxQQ
xQ
tAdx
)(
Q = inflow to the control volume
q = lateral inflow
Elevation View
Plan View
Rate of change of flow with distance
Outflow from the C.V.
Change in mass
Reynolds transport theorem
....
.0scvc
dAVddtd
Momentum Equation
• From Newton’s 2nd Law: • Net force = time rate of change of momentum
....
.scvc
dAVVdVdtdF
Sum of forces on the C.V.
Momentum stored within the C.V
Momentum flow across the C. S.
Forces acting on the C.V.
Elevation View
Plan View
• Fg = Gravity force due to weight of water in the C.V.
• Ff = friction force due to shear stress along the bottom and sides of the C.V.
• Fe = contraction/expansion force due to abrupt changes in the channel cross-section
• Fw = wind shear force due to frictional resistance of wind at the water surface
• Fp = unbalanced pressure forces due to hydrostatic forces on the left and right hand side of the C.V. and pressure force exerted by banks
Momentum Equation
....
.scvc
dAVVdVdtdF
Sum of forces on the C.V.
Momentum stored within the C.V
Momentum flow across the C. S.
0)(11 2
fo SSgxyg
AQ
xAtQ
A
0)(
fo SSgxyg
xVV
tV
0)(11 2
fo SSgxyg
AQ
xAtQ
A
Momentum Equation(2)
Local acceleration term
Convective acceleration term
Pressure force term
Gravity force term
Friction force term
Kinematic Wave
Diffusion Wave
Dynamic Wave
Momentum Equation (3)
fo SSxy
xV
gV
tV
g
1
Steady, uniform flow
Steady, non-uniform flow
Unsteady, non-uniform flow
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Applications of different forms of momentum equation
0)(
fo SSgxyg
xVV
tV
• Kinematic wave: when gravity forces and friction forces balance each other (steep slope channels with no back water effects)
• Diffusion wave: when pressure forces are important in addition to gravity and frictional forces
• Dynamic wave: when both inertial and pressure forces are important and backwater effects are not negligible (mild slope channels with downstream control, backwater effects)
Kinematic Wave
• Kinematic wave celerity, ck is the speed of movement of the mass of a flood wave downstream– Approximately, ck = 5v/3 where v = water velocity
Muskingum-Cunge Method
• A variant of the Muskingum method that has a more physical hydraulic basis
• This is what Dean Djokic has used in the Brushy Creek HEC-HMS models
• , where Δx = reach length or an increment of this length
• , where B = surface width, S0 is the bed slope
Reach SBR_080 Downstream of Dam 7
How do we route the flow through Reach SBR_080?
Longitudinal profile for reach SBR_080
0.0008
1
1545 ft
Cross-Section for SBR_080
Station Elevation0 797.6057
118.1 790.0711236.2 781.6702
284 777.0652304 777.0652
323.42 783.5712344.26 789.859
365.1 795.4788
0.00 100.00 200.00 300.00 400.00765.00
770.00
775.00
780.00
785.00
790.00
795.00
800.00
Cross-Section
Distance (ft)
Elev
ation
abo
ve d
atum
(ft)
Routing in stream reach downstream of Dam 7