Programme for weeks 5-8

Tues 1 Nov Lecture as normalFri 4 Nov Optional clinic – Vel profile exerciseTues 8 Nov Optional clinic – Vel profile exerciseFri 11 Nov Optional clinic – Vel profile exerciseTues 15 Nov Directed reading 1Fri 18 Nov Directed reading 1Tues 22 Nov Directed reading 1Fri 25 Nov Q&A session on Vel profile exercise

and directed reading Tues 29 Nov Lecture programme resumes

GY2311/GY2312 Lectures 6-7

Fluid Flows

Uniform flowsBoundary layers

DEPARTMENT OF GEOGRAPHY

Sebaskachu River, Labrador – a tortuous meandering river.

Important flow characteristics

velocity, v

shear stress,

shear velocity, u*

discharge, Q

Stream power,

Uniform flow

Steady flow

Steady Unsteady

Why might uniform flows occur?

Why does a fluid flow?

Why don’t flows continue to accelerate?

Condition for uniform flow

Forces promoting movement = forces resisting movement

Fp = Fr

Fp-Fr = 0

Uniform flow (flow resistance) formulae

SRg

SRCu

u = flow velocity, m s-1

C = roughness coefficient

R = hydraulic radius, m

S = bed slope

= shear stress, N m-2

g = accel. due to gravity, m s-2

u is mean downstream flow velocity and is the mean shear stress acting over the

channel boundary (bed and banks)

Condition for uniform flow

Forces promoting movement = forces resisting movement

Fp = Fr

Fp-Fr = 0

Definition diagram

A

Condition for uniform flowForces promoting movement = forces resisting movement

Fp = Fr

A L g sin P L k u2

u2 = A L g sin P L k

A/P = R

sin = = S (m/m)

The Chezy Equation

u2 = g R S

k

kSRg

u

g/k = constant = C

SRCu

Uniform flow (flow resistance) formulae

SRg

SRCu

u = flow velocity, m s-1

C = roughness coefficient

R = hydraulic radius, m

S = bed slope

= shear stress, N m-2

g = accel. due to gravity, m s-1

Flow resistance equations

SRCu Chezy

ffgRS8

u Darcy Weisbach

50670SR

n1

u..

Manning

n, C, ff Manning, Chezy and Darcy Weisbach roughness coefficients

yxSRu

Constants of proportionality

Flow resistance equations

5050SCRu..

50f

50505050

f

SRg8u

.

....

50670SR

n1

u..

yxSRu

Constants of proportionality are n, C, ff , the Manning, Chezy and Darcy Weisbach roughness coefficients

Slope S

yxSRu

Slope, m/m

Dimensionless number – i.e. no units

Mountain rivers S = 0.01-0.1

Lowland rivers S = 0.001-0.0001

Hydraulic radius, R

yxSRu

Hydraulic radius

R = A/P, m

0.5 m

1 m

0.5 m100 m

R=0.5/2 = ¼; 1A = 4P

R=50/101 = c. ½; 1A = 2P

For wide channels, R approximates flow depth

What is flow resistance?

Tabulated values

Channel type n ff C

Artificial channel, shuttered concrete 0.014 0.016 71

Excavated channel, earth 0.022 0.039 45

Excavated channel, gravel 0.025 0.049 40

Natural channel, < 30 m wide, clean, regular

0.03 0.072 33

Natural channel, < 30 m wide, some weeds and stones

0.035 0.093 29

Natural channel < 30 m wide, sluggish weedy pools

0.07 0.4 14

See wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm

50670SR

n1

u..

Grain roughness is a function of bed particle size

84D5.3Ra

log03.2f

1

Colebrook White equation

a depends on channel shape (= c. 12)

Grain and form roughnessGrain

Roughness

Form

Strickler equation

n = 0.151D50

1/6

Dx = grain size that x% is finer than

Colebrook White Equation

Form roughnessGrain

Roughness

Form

Shear stress in uniform flows

Forces promoting movement = forces resisting movement

Fp = Fr

leads to

= g R S

Du Boys equation – shear stress exerted by flowing water

Uniform flow (flow resistance) formulae

SRg

SRCu

u = flow velocity, m s-1

C = roughness coefficient

R = hydraulic radius, m

S = bed slope

= shear stress, N m-2

g = accel. due to gravity, m s-2

Calculate the discharge and shear stress acting on the bed of a 25 m wide channel flowing at a depth of 60 cm. Assume that the channel drops 3 m over a 1000 m reach and that the shape is rectangular with a Manning’s n of 0.035.

Width

Depth

Cross-section area

Wetter perimeter

Hydraulic radius

Slope

Manning’s n

Density of water 1000 kg m-3

Accel of gravity 9.81 m s-2

Q =

=