Flow Resistance, Channel Gradient, and Hydraulic Geometry

1. 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

2. Flow Resistance Equations • Chezy (1769) • Manning (1889) • Darcy-Weisbach (SI units)

3. Resistance Coefficients • By assuming a roughness coefficient, u can be determined • Use an input parameters for numerical models (Julien, 2002)

4. Resistance Coefficients as a function of Bed Shear Stress (Bed Configuration) (van Rijn, 1993)

5. 3. Longitudinal Profiles Outline • Controls on channel gradient • Downstream variations in discharge, bed slope, and bed texture (downstream fining) • Downstream fining  channel concavity

6. Amazon River Longitudinal Bed Profile Rhine River (Knighton, 1998)

7. River Bollin Nigel Creek River Towy Longitudinal Bed Profile (Knighton, 1998)

8. 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

9. Controls on Gradient (2) • Leopold and Maddock (1953): S 1/Q • Lane (1955): Expanded concept of graded stream • Hack (1957): S D50, 1/AD

10. Longitudinal Variations in Q, S, and Bed Texture, MS River +4° -3° -3°

11. Downstream Fining MS River Allt Dubhaig

12. Downstream Fining • D0 initial grain size, L downstream distance, a 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

13. For Mississippi River Data QB (cfs) SDB (mm) d (m) t (Pa) US 260 0.035 270 0.4 124 DS 2,070,000 0.00008 0.16 13 10  +4° -3° -3° +1° -1° 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

14. 1D Exner Equation Change in bedload with distance with gain/loss to suspended load as modulated by grain settling velocity Change in bed height with time Change in total load with distance • 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

15. 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

16. x Rod Level e1 Rod d1 e2  x1, y1 Water surface Ground surface d2  Water surface slope: (taken positive in the downstream direction) x = x2  x1 y = (e2d2)  (e1d1) slope = y/x x2, y2 Measurement of Stream Channel Gradient Ground surface slope ≠ water surface slope

17. 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”

18. DS Determining hydraulic geometry (Richards, 1982)

19. f = 0.52 At-a-station; Sugar Creek, MD m = 0.30 b = 0.18 (Leopold, Wolman, and Miller, 1964)

20. Downstream Same flow frequency (Morisawa, 1985)

21. 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)

22. Downstream b > f > m; b~0.5, f~0.4, m~0.1 (Knighton, 1998)

23. 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

24. Hydraulic Geometry as a Tool • Used in stream channel design • Identification of unstable stream corridors and unstable stream systems • Concept of channel equilibrium

25. 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

26. Additional Considerations depth velocity width (Richards, 1982)

27. Typical Stream Discharge Determination w0,d0,v0 wn+1,dn+1,vn+1 Tape measure w1 w2 wn,dn,vn w3 T T Q1 Q2 Q3 Qn Qn+1 Right Benchmark (looking downstream) Left Benchmark (looking downstream) d1 d2 d3 v1 Ground surface Current meter For d<0.75 m, located at 0.4d ; For d>0.75 m, average of 0.2d and 0.8d v2 v3 Discharge determination: Discharge = width  depth  velocity Q = w  d  v Q = Q1 + Q2 + Q3 … + Qn+1

28. 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

29. 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