Programme for weeks 5-8 Tues 1 NovLecture as normal Fri 4 NovOptional clinic – Vel profile exercise Tues 8 NovOptional clinic – Vel profile exercise Fri 11 NovOptional clinic – Vel profile exercise Tues 15 NovDirected reading 1 Fri 18 Nov Directed reading 1 Tues 22 NovDirected reading 1 Fri 25 NovQ&A session on Vel profile exercise and directed reading Tues 29 NovLecture programme resumes

GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary 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 dont flows continue to accelerate?

Condition for uniform flow Forces promoting movement = forces resisting movement F p = F r F p -F r = 0

Uniform flow (flow resistance) formulae 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 F p = F r F p -F r = 0

Definition diagram A

Condition for uniform flow Forces promoting movement = forces resisting movement Fp = Fr A L g sin P L k u 2 u 2 = A L g sin P L k A/P = R sin = = S (m/m)

The Chezy Equation u 2 = g R S k g/k = constant = C

Uniform flow (flow resistance) formulae 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 Chezy Darcy Weisbach Manning n, C, f f Manning, Chezy and Darcy Weisbach roughness coefficients Constants of proportionality

Flow resistance equations Constants of proportionality are n, C, f f, the Manning, Chezy and Darcy Weisbach roughness coefficients

Slope S Slope, m/m Dimensionless number – i.e. no units Mountain rivers S = Lowland rivers S =

Hydraulic radius, R Hydraulic radius R = A/P, m 0.5 m 1 m 0.5 m 100 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 typenf C Artificial channel, shuttered concrete Excavated channel, earth Excavated channel, gravel Natural channel, < 30 m wide, clean, regular Natural channel, < 30 m wide, some weeds and stones Natural channel < 30 m wide, sluggish weedy pools See wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htmwwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm

Grain roughness is a function of bed particle size Colebrook White equation a depends on channel shape (= c. 12) Grain and form roughness Grain Roughness Form Strickler equation n = 0.151D 50 1/ 6 D x = grain size that x% is finer than

Colebrook White Equation

Form roughness Grain Roughness Form

Shear stress in uniform flows Forces promoting movement = forces resisting movement F p = F r leads to = g R S Du Boys equation – shear stress exerted by flowing water

Uniform flow (flow resistance) formulae 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 Mannings n of Width Depth Cross-section area Wetter perimeter Hydraulic radius Slope Mannings n Density of water1000 kg m -3 Accel of gravity9.81 m s -2 Q = =