1. 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

2. GY2311/GY2312 Lectures 6-7 Fluid Flows Uniform flows Boundary layers DEPARTMENT OF GEOGRAPHY

3. Sebaskachu River, Labrador – a tortuous meandering river. Important flow characteristics velocity, v shear stress, shear velocity, u* discharge, Q Stream power,

4. Uniform flow

5. Steady flow Steady Unsteady

6. Why might uniform flows occur? Why does a fluid flow? Why dont flows continue to accelerate?

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

8. 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)

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

10. Definition diagram A

11. 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)

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

13. 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

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

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

16. Slope S Slope, m/m Dimensionless number – i.e. no units Mountain rivers S = 0.01-0.1 Lowland rivers S = 0.001-0.0001

17. 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

18. What is flow resistance?

19. Tabulated values Channel typenf C Artificial channel, shuttered concrete0.0140.01671 Excavated channel, earth0.0220.03945 Excavated channel, gravel0.0250.04940 Natural channel, < 30 m wide, clean, regular 0.030.07233 Natural channel, < 30 m wide, some weeds and stones 0.0350.09329 Natural channel < 30 m wide, sluggish weedy pools 0.070.414 See wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htmwwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm

20. 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

21. Colebrook White Equation

22. Form roughness Grain Roughness Form

23. 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

24. 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

25. 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 0.035. 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 = =