1. VELOCITY PROFILE AND SHEAR STRESSES CALCULATION IN HIGH VOLUME RELATIVE BED ROUGHNESS FLOW Wojciech Bartnik Andrzej Struzynski Krakow Agriculture University

2. Flow zones – Introduction Laboratory measurements Bed roughness measurements Log-law velocity distribution Calculation of velocity and shear stresses Conclusions Presentation Schedule

3. Bed roughness and water surface acts on the flowing water Flow zones

4. I- laminar flow II- log-law velocity distribution III- wake region IV- free surface region

5. Flow zones flat bed II I III IV

6. Flow zones rough bed I II III IV

7. Flow zones [Williams J.J., 1996]

8. Bed roughness and water surface acts on the shape of flowing water velocity profile. Flow zones

9. The shape of velocity profile depend on: flow depth, av. velocity of flowing water, bed roughness, relative roughness... For hydraulically rough flow conditions I and IV flow zone decreases Fr = 0.074Fr = 1.38 4D Flow zones

10. Laboratory measurements Flume dimensions: l2.0 x 0.5 x 0.6 m (glass walls) Flume rig: micro-propeller flow-meter slope measurements Bed slope, water surface slope Discharge: max 0.13 qm s -1 Artificial grains Ø – 4 to 8 cm

11. Bed roughness measurements homogeneous roughness k s = K (1.926 SF 2 – 0.488 SF + 4.516) Profile-meter AG-1

12. Log-law velocity distribution Maximum velocity moves with relative roughness change flat bed

13. Log-law velocity distribution Maximum velocity moves with relative roughness change rough bed

14. Log-law velocity distribution For the same bed roughness curves are parallel flat bed

15. Log-law velocity distribution For the same bed roughness curves are parallel grains 4M

16. Log-law velocity distribution For the same bed roughness curves are parallel grains 4D

17. Log-law velocity distribution For the same bed roughness curves are parallel grains 6D

18. Log-law velocity distribution For the same bed roughness curves are parallel grains 8D

19. Calculation of velocity and shear stresses Log-law velocity distribution for whole profile is used U/U max = A log (y/Y) + B Modified Prandtl equation B becomes constant -B = 1.12 ± 3%

20. Calculation of velocity and shear stresses U/U max = A log (y/Y) + B A value changes with relative depth Y/K

21. Calculation of velocity and shear stresses U/U max = A log (y/Y) + B Comparison of measured to calculated A constant

22. Calculation of velocity and shear stresses Velocity profile reflects shear stresses Use of logarithmic equation allow calculating  0 for rough flow conditions  0 = 2.303 K U M

23. Calculation of velocity and shear stresses

24. Conclusions Near bed the velocity and velocity profile slope calculations (in logarithmic scale) are correct within the second and third flow zone. The use of equation (4) makes the bed level (zero velocity) estimation error negligible (B=1.12). The use of mentioned method is limited to the rough flow conditions where the maximum velocity lays close to the water surface (the near surface region decreases to 20% of water depth). The measurements of surface velocity, water depth and bed roughness can be used for calculation of water velocity profile and bed shear stresses for rough flow conditions.