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