flow resistant

8/16/2019 flow resistant

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Flow Resistance, Channel

Gradient, and Hydraulic Geometry1. Flow Resistance

– Uniformity and steadiness, turbulence,

boundary layers, bed shear stress, velocity2. Lon itudinal !rofiles

– Channel radient, downstream finin

". Hydraulic Geometry – General tendencies for e#$onents, techni%ue

for stream a in


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Flow Resistance &%uations• Che'y (1)*+

• -annin (1 +

• /arcy0 eisbach( 3 units

RS C u =

nS Ru

2132

=

f gRS u 82 =

channelsfor wide2

d d w

d w R

≈+⋅

=


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(4ulien, 2552• 6y assumin a rou hness coefficient, u can be determined• Use an in$ut $arameters for numerical models

Resistance Coefficients


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". Lon itudinal !rofiles

8utline• Controls on channel radient•

/ownstream variations in dischar e, bedslo$e, and bed te#ture (downstreamfinin

• /ownstream finin ↔ channel concavity


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(9ni hton, 1++

:ma'on River

Rhine River

Lon itudinal6ed !rofile


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(9ni hton, 1++

River 6ollin ;i el Cree<

River =owy Lon itudinal6ed !rofile


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Controls on Gradient (1•

-ac 0 Conce$t of a raded stream? 8ver a$eriod of time, slo$e is delicately ad7usted to $rovide,with available dischar e and channel characteristics, 7ustthe velocity re%uired to trans$ort the load su$$lied

• Rubey (1+@2 ? for a constant wA d , S ∝ Q s , M (si'e of bedmaterial load , 1A Q

31

2

2

= d QW DQ

k S s s


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Controls on Gradient (2•

Leo$old and -addoc< (1+@" ? S ∝ 1AQ

•

Lane (1+@@ ? $anded conce$t of raded stream

• Hac< (1+@) ? S ∝ D@5

, 1A AD

93.0to25.0; −−== z tQS z

6.0

50006.0

=

D A D

S

50 DQQS s∝


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Lon itudinal Bariations in Q, S , and 6ed =e#ture, - River

>D 0"D 0"D


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/ownstream Finin

- River

:llt /ubhai


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/ownstream Finin

12.0to0006.0;0 == − α α Le D DD5 initial rain si'e, L downstream distance, α sortin or abrasion coefficient

• ternber abrasion e%uation• :brasion E mechanical brea


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For -ississi$$i River /ata

Q B (cfs S D B (mm d (m τ (!aU 2*5 5.5"@ 2)5 5.>12>/ 2,5)5,5555.5555 5.1* 1" 15∆

>D 0"D 0"D 1D 01D

d c Q f , f I 5." to 5.>S t Q ' , ' I 05.*@τ ρ gdSτ ∝ ds , τ ∝ (Q f (Q '

τ ∝ J n, where n 05.2@ to 05."@ :ssumin τ 0 I τ cmax → downstream finin


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1/ ner &%uation

( ) ( ) E C u xq

xQ

t h

p b sb s −+∂

∂−=∂∂−=∂

∂−1Chan e in bed

hei ht with timeChan e in totalload with distance

Chan e in bedload with distance

with ainAloss to sus$ended loadas modulated by rain settlinvelocity

• Bolume trans$ort rates• Can be written for sediment mi#tures and multi$le

dimensions• $atial radients in Q s due to s$atial radients in τ • lo$e ad7ustment, and downstream finin , can be

brou ht on by a radation and de radation


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/ Finin ↔ !rofile ConcavityK

• -odelin su ests the time0scale for sortin$rocesses to $roduce downstream finin isshorter than the timescale for bed slo$ead7ustment

• Fluvial systems ad7ust their bed te#ture inres$onse to s$atial variations in shear stressand sediment su$$ly


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Hydraulic Geometry•

Q is the dominant inde$endent $arameter, andthat de$endent $arameters are related to Q viasim$le $ower functions

• :$$lied Mat0a0stationN and MdownstreamN

b

aQw = f

cQd = m

kQu =

( )( )( )m f b kQcQaQud wQ =××=

1=++ m f b 1=×× k ca


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(Richards, 1+ 2

DS

/eterminin hydraulic eometry


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(Leo$old, olman, and -iller, 1+*>

:t0a0station

u ar Cree


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(-orisawa, 1+ @

/ownstreamSame flow frequency


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(9ni hton, 1++

:t0a0station

m f band

m b f b 505.2

f 5."05.@m 5."05.@


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(9ni hton, 1++

/ownstream

b f m bI5.@, fI5.>, mI5.1


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Hydraulic Geometry• :t0a0station? rectan ular channels

increase in dischar e is MaccommodatedNby increasin flow de$th and flow velocity

• /ownstream? increase in dischar e isMaccommodatedN by increasin flow widthand de$th


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Hydraulic Geometry as a =ool• Used in stream channel desi n• 3dentification of unstable stream corridors

and unstable stream systems• Conce$t of channel e%uilibrium


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:dditional Considerations• Channel eometry also controlled by

– Grain si'e and bed com$osition – ediment trans$ort rate (bed mobility and rou hness –

6an< stren th, as assessed by silt0clay content – Be etationOdifferent e#$onents de$endin u$on

$resence and ty$e• Curved channels and non0linear trends

(com$ound channels• !ools P rifflesOdifferent e#$onents


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:dditional Considerations

de$th

velocity

width

(Richards, 1+ 2


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Ri ht 6enchmar<(loo d For d 5.)@ m, avera e of 5.2 d and 5. d

d 1 d 2 d "

Q 2 Q " Q n 1

w n 1 ,d n 1 ,v n 1

/ischar e determination?/ischar e width × de$th × velocityQ w × d × v

Q Q 1 Q 2 Q " Q n+1

w n,d n,v n

Q n

=y$ical tream /ischar e /etermination


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Conclusions• Flow velocity can be determined by assumin

a friction coefficient• /ownstream variations in channel radient,

bed te#ture, and bed shear stress des$iteincreases in dischar e and total sediment load• Hydraulic eometry assumes dischar e is the

$rimary inde$endent $arameter • Hydraulic eometry of river channels shows

world0wide tendencies very $owerful MtoolN• : techni%ue for a in streams is $resented

flow resistant

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