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
Flow Resistance Equations• Chezy (1769)
• Manning (1889)
• Darcy-Weisbach(SI units)
RSCu
nSRu
2132
fgRSu 82
channels for wide 2
ddwdwR
(Julien, 2002)• By assuming a roughness coefficient, u can be determined• Use an input parameters for numerical models
Resistance Coefficients
Resistance Coefficients as a function of Bed Shear Stress (Bed Configuration)
(van Rijn, 1993)
3. Longitudinal Profiles
Outline• Controls on channel gradient• Downstream variations in discharge, bed
slope, and bed texture (downstream fining)
• Downstream fining channel concavity
(Knighton, 1998)
Amazon River
Rhine River
LongitudinalBed Profile
(Knighton, 1998)
River Bollin Nigel Creek
River Towy LongitudinalBed Profile
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
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
Longitudinal Variations in Q, S, and Bed Texture, MS River
+4° -3° -3°
Downstream Fining
MS River
Allt Dubhaig
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
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
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
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
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
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
(Richards, 1982)
DS
Determining hydraulic geometry
(Leopold, Wolman, and Miller, 1964)
At-a-station; Sugar Creek, MD
f = 0.52
m = 0.30
b = 0.18
(Morisawa, 1985)
DownstreamSame flow frequency
(Knighton, 1998)
At-a-station
m > f > band
m > b + fb = 0-0.2
f = 0.3-0.5m = 0.3-0.5
(Knighton, 1998)
Downstream
b > f > m; b~0.5, f~0.4, m~0.1
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
Hydraulic Geometry as a Tool
• Used in stream channel design• Identification of unstable stream corridors
and unstable stream systems• Concept of channel equilibrium
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
Additional Considerations
depth
velocity
width
(Richards, 1982)
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
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
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