flow resistance, channel gradient, and hydraulic geometry
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Flow Resistance, Channel Gradient, and Hydraulic Geometry. 1. Flow Resistance Uniformity and steadiness, turbulence, boundary layers, bed shear stress, velocity 2. Longitudinal Profiles Channel gradient, downstream fining 3. Hydraulic Geometry - PowerPoint PPT PresentationTRANSCRIPT
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Flow Resistance, Channel Gradient, and Hydraulic Geometry
1. Flow Resistance– Uniformity and steadiness, turbulence,
boundary layers, bed shear stress, velocity2. Longitudinal Profiles
– Channel gradient, downstream fining3. Hydraulic Geometry
– General tendencies for exponents, technique for stream gaging
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Flow Resistance Equations• Chezy (1769)
• Manning (1889)
• Darcy-Weisbach(SI units)
RSCu
nSRu
2132
fgRSu 82
channels for wide 2
ddwdwR
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(Julien, 2002)• By assuming a roughness coefficient, u can be determined• Use an input parameters for numerical models
Resistance Coefficients
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Resistance Coefficients as a function of Bed Shear Stress (Bed Configuration)
(van Rijn, 1993)
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3. Longitudinal Profiles
Outline• Controls on channel gradient• Downstream variations in discharge, bed
slope, and bed texture (downstream fining)
• Downstream fining channel concavity
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(Knighton, 1998)
Amazon River
Rhine River
LongitudinalBed Profile
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(Knighton, 1998)
River Bollin Nigel Creek
River Towy LongitudinalBed Profile
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Controls on Gradient (1)• Mackin (1948) - Concept of a graded stream: Over a
period of time, slope is delicately adjusted to provide, with available discharge and channel characteristics, just the velocity required to transport the load supplied
• Rubey (1952): for a constant w/d, S Qs, M (size of bed material load), 1/Q
31
2
2
dQWDQkS ss
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Controls on Gradient (2)• Leopold and Maddock (1953): S 1/Q
• Lane (1955): Expanded concept of graded stream
• Hack (1957): S D50, 1/AD
93.0 to25.0 ; ztQS z
6.0
50006.0
DADS
50DQQS s
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Longitudinal Variations in Q, S, and Bed Texture, MS River
+4° -3° -3°
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Downstream Fining
MS River
Allt Dubhaig
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Downstream Fining12.0 to0006.0;0 LeDD
D0 initial grain size, L downstream distance, sorting or abrasion coefficient
• Sternberg abrasion equation• Abrasion – mechanical breakdown of particles
during transport; rates of DS fining >> rates of abrasion
• Weathering – chemical and mechanical due to long periods of exposure; negligible
• Hydraulic Sorting – size selective deposition
mainly due to a downstream decrease in bed shear stress and turbulence intensity of the river
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For Mississippi River DataQB (cfs) S DB (mm) d (m) t
(Pa)US 260 0.035 270 0.4 124DS 2,070,000 0.00008 0.16 13 10 +4° -3° -3° +1° -1°
d = cQf, f ~ 0.3 to 0.4S = tQz, z ~ -0.65t = gdSt ds, t (Qf)(Qz) t Qn, where n = -0.25 to -0.35Assuming t0 ~ tcmax downstream fining
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1D Exner Equation
ECuxq
xQ
thp bs
bs
1
Change in bed height with time
Change in total load with distance
Change in bedload with distance with gain/loss to suspended load as modulated by grain settling velocity
• Volume transport rates• Can be written for sediment mixtures and multiple
dimensions • Spatial gradients in Qs due to spatial gradients in t• Slope adjustment, and downstream fining, can be
brought on by aggradation and degradation
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DS Fining Profile Concavity?
• Modeling suggests the time-scale for sorting processes to produce downstream fining is shorter than the timescale for bed slope adjustment
• Fluvial systems adjust their bed texture in response to spatial variations in shear stress and sediment supply
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Measurement of Stream Channel Gradient
Ground surface
Water surfacex1, y1
Level
Rode1
e2
d2
d1
x2, y2
x
Water surface slope:(taken positive in the downstream direction)x = x2 x1
y = (e2 d2) (e1 d1) slope = y/x
Rod
Ground surface slope ≠ water surface slope
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Hydraulic Geometry• Q is the dominant independent parameter, and
that dependent parameters are related to Q via simple power functions
• Applied “at-a-station” and “downstream”
baQw fcQd mkQu
mfb kQcQaQudwQ
1 mfb 1 kca
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(Richards, 1982)
DS
Determining hydraulic geometry
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(Leopold, Wolman, and Miller, 1964)
At-a-station; Sugar Creek, MD
f = 0.52
m = 0.30
b = 0.18
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(Morisawa, 1985)
DownstreamSame flow frequency
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(Knighton, 1998)
At-a-station
m > f > band
m > b + fb = 0-0.2
f = 0.3-0.5m = 0.3-0.5
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(Knighton, 1998)
Downstream
b > f > m; b~0.5, f~0.4, m~0.1
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Hydraulic Geometry
• At-a-station: rectangular channels; increase in discharge is “accommodated” by increasing flow depth and flow velocity
• Downstream: increase in discharge is “accommodated” by increasing flow width and depth
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Hydraulic Geometry as a Tool
• Used in stream channel design• Identification of unstable stream corridors
and unstable stream systems• Concept of channel equilibrium
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Additional Considerations• Channel geometry also controlled by
– Grain size and bed composition– Sediment transport rate (bed mobility and roughness)– Bank strength, as assessed by silt-clay content– Vegetation—different exponents depending upon
presence and type• Curved channels and non-linear trends
(compound channels)• Pools & riffles—different exponents
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Additional Considerations
depth
velocity
width
(Richards, 1982)
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Right Benchmark(looking downstream)
Tapemeasure
Left Benchmark(looking downstream)
TT
Ground surface
w0,d0,v0
w1
Q1
v1
w2w3
v2 v3Current meterFor d<0.75 m, located at 0.4d ;For d>0.75 m, average of 0.2d and 0.8d
d1 d2 d3
Q2 Q3 Qn+1
wn+1,dn+1,vn+1
Discharge determination:Discharge = width depth velocityQ = w d v Q = Q1 + Q2 + Q3 … + Qn+1
For example:
22
0101011
vvddwwQ
22
1212122
vvddwwQ
wn,dn,vn
Qn
Width- and depth-averaged flow discharge:
General form:
w
x
d
y
yxvQ0 0
dd
Analytical form:
22
11
1
1
1
11
iin
i
n
i
iiiii
vvddwwQQ
To complete the integration, we will assume
0 ;0 ;
;0 ;0 ;0
111
0000
nnwn vdwwvdw
where n is the number of measurements
Typical Stream Discharge Determination
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Implications for Stream Restoration
• Roughness coefficients (1) enable determination of velocity and (2) are critical input parameters for numerical models
• Exner equation is most commonly used analytic expression to determine bed stability
• Hydraulic geometry is (1) the most widely used analytic framework for stream channel design, and (2) used in the identification of unstable stream corridors and unstable stream systems
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Conclusions• Flow velocity can be determined by assuming
a friction coefficient• Downstream variations in channel gradient,
bed texture, and bed shear stress despite increases in discharge and total sediment load
• Hydraulic geometry assumes discharge is the primary independent parameter
• Hydraulic geometry of river channels shows world-wide tendencies; very powerful “tool”
• A technique for gaging streams is presented