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
– General tendencies for exponents, technique
for stream gaging
Flow Resistance Equations
• Chezy (1769)
u  C RS
23
R S
• Manning (1889) u 
n
• Darcy-Weisbach
(SI units)
R
12
8gRS
u 
f
2
wd
 d for wide channels
w  2d
Resistance Coefficients
• By assuming a roughness coefficient, u can be determined
• Use an input parameters for numerical models
(Julien, 2002)
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
Amazon River
Longitudinal
Bed Profile
Rhine River
(Knighton, 1998)
River Bollin
River Towy
Nigel Creek
Longitudinal
Bed Profile
(Knighton, 1998)
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
 Q DsW 
S  k

2
 Q d 
2
s
13
Controls on Gradient (2)
• Leopold and Maddock (1953): S  1/Q
S  tQ z ; z  0.25 to  0.93
• Lane (1955): Expanded concept of graded stream
QS  Qs D50
• Hack (1957): S  D50, 1/AD
 D50 

S  0.006
 AD 
0 .6
Longitudinal Variations in Q, S, and Bed Texture, MS River
+4°
-3°
-3°
Downstream Fining
MS River
Allt Dubhaig
Downstream Fining
D  D0e L ;  0.0006 to 0.12
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 Data
US
DS

QB (cfs)
260
2,070,000
+4°
S
0.035
0.00008
-3°
DB (mm)
270
0.16
-3°
d = cQf, f ~ 0.3 to 0.4
S = tQz, z ~ -0.65
t = gdS
t  ds, t  (Qf)(Qz)
t  Qn, where n = -0.25 to -0.35
Assuming t0 ~ tcmax  downstream fining
d (m)
0.4
13
+1°
t (Pa)
124
10
-1°
1D Exner Equation
Qs
qb
h
1  p   

 u s Cb  E 
t
x
x
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
x
Rod
Level
e1

Rod
d1
x1, y1
e2
Water surface
Ground surface

d2
Water surface slope:
x2, y2
(taken positive in the downstream direction)
x = x2  x1
y = (e2  d2)  (e1  d1)
Ground surface slope ≠ water surface slope
slope = y/x
Hydraulic Geometry
• Q is the dominant independent parameter, and
that dependent parameters are related to Q via
simple power functions
w  aQ
b
d  cQ
f
u  kQ
  
m
Q  w  d  u  aQb cQ f kQm
b  f  m 1

ac k 1
• Applied “at-a-station” and “downstream”
DS
Determining hydraulic geometry
(Richards, 1982)
f = 0.52
m = 0.30
At-a-station;
Sugar Creek, MD
b = 0.18
(Leopold, Wolman, and Miller, 1964)
Same flow frequency
(Morisawa, 1985)
Downstream
m>f>b
and
m>b+f
b = 0-0.2
f = 0.3-0.5
m = 0.3-0.5
At-a-station
(Knighton, 1998)
Downstream
b > f > m; b~0.5, f~0.4, m~0.1
(Knighton, 1998)
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)
Typical Stream Discharge Determination
w0,d0,v0
Tape
measure
T
wn+1,dn+1,vn+1
w1
w2
wn,dn,vn
w3
T
Q1
Q2
Left Benchmark
(looking downstream)
d1
Q3
d2
Qn Qn+1
Right Benchmark
(looking downstream)
d3
v1
Ground surface
v2
Current meter
Width-v3 and depth-averaged flow discharge:
For d<0.75 m, located at 0.4d ;
w d
For d>0.75 m, average of 0.2d andGeneral
0.8d form: Q 
vdxdy
 
Discharge determination:
Discharge = width  depth  velocity
Q=wdv
Q = Q1 + Q2 + Q3 … + Qn+1
For example:
d d  v v 
Q1  w1  w0   1 0    1 0 
 2   2 
 d  d1   v2  v1 
Q2  w2  w1    2


2
2

 

x 0 y 0
n 1
n 1
i 1
i 1
 d i  d i 1   vi  vi 1 


2

  2 
Analytical form: Q   Qi   wi  wi 1   
To complete the integration, we will assume
w0  0; d 0  00 ; v0  0;
wn 1  ww ; d n 1  0; vn 1  0
where n is the number of measurements
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