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

QQ

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

SSjjjj

jj

22

111

jjjj QCICICQ 32111

tXK

tXKC

tXK

KXtC

tXK

KXtC

)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, Dt = 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

5927.01)15.01(3.2*2

1)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

tXK

tXKC

tXK

KXtC

tXK

KXtC

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 20

Time (hr)

Dis

cha

rge

(cfs

)

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

dxx

QQ

x

Q

t

Adx

)(

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

dAVddt

d

Momentum Equation

• From Newton’s 2nd Law: • Net force = time rate of change of momentum

....

.scvc

dAVVdVdt

dF

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

dAVVdVdt

dF

Sum of forces on the C.V.

Momentum stored within the C.V

Momentum flow across the C. S.

0)(11 2

fo SSgx

yg

A

Q

xAt

Q

A

0)(

fo SSgx

yg

x

VV

t

V

0)(11 2

fo SSgx

yg

A

Q

xAt

Q

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 SSx

y

x

V

g

V

t

V

g

1

Steady, uniform flow

Steady, non-uniform flow

Unsteady, non-uniform flow

15

Applications of different forms of momentum equation

0)(

fo SSgx

yg

x

VV

t

V

• 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