Power-law Index for Velocity Profiles in Open Channel Flows Nian-Sheng Cheng

1. Power-law Index for Velocity Profiles in Open Channel Flows Nian-Sheng Cheng School of Civil and Environmental Engineering Nanyang Technological University, Singapore cnscheng@ntu.edu.sg

2. Consider uniform flows in a wide open channel Power law Log law

3. Power law • Simplest data-driven formula • Often considered empirical • Applicable to larger fraction of the flow domain • Selection of power-law index (1/m) is • subjective/empirical

4. Empirical power-law index • m = 7 (Classical case, 1/7-power law, Prandtl) • m = 6 - 10 (Nikuradze data) • Inferred from flow resistance equation • m = 6 (Manning equation) • m = 8 (Qian and Wan 1999) • m = 4 -12 (Chen 1991) • m = 2 (Smart et al 2002, large-scale roughness)

5. Aim of this study Not to justify if one law is better than the other Rather, it aims to provide an alternative theoretical consideration from which the power law could be derived analytically. The connection of the power law with the log law is explored in light of recently improved understanding of wall turbulence

6. Aim of this study (cont’d) • This study also details • how m varies with flow and boundary conditions • what possible approximations could be • made for practical applications

7. Results to be shown • power law can be derived as a first-order • approximation to log law; • (2) 1/m is a function of • Reynolds number and relative roughness height; • (3) why 1/6-power included in the Manning equation • is of prevalent acceptance; and • (4) smaller m would be required for flows over • very rough boundaries.

8. Log law – a starting point • one of the established theoretical results • for describing velocity profiles • one of classical turbulence theoretical results • in principle applicable for the near-bed • overlap region (< ~20% of flow depth)

9. Log law Why widely accepted? • Justified with theoretical arguments: • Prandtl’s mixing length assumption • von Karman’s dimensional reasoning • Millikan’s asymptotic analysis

10. Log law Scaling considerations (smooth bed): for the inner region matching for the outer region generalized

11. Log law applies to overlap region: /u* << y << h Empirically, 30/u* < y < 0.2h (Nezu and Nakagawa 1993) Approach To derive possible power-form approximations to the log law within the overlap region. The power function so obtained will share the same theoretical grounds as the log law.

13. Log law in terms of inner region parameters yo = hydrodynamic roughness length yir = upper limit of overlap layer / inner region  = ln(yir/yo)

14. From log law  = u/uir  = y/yir ‘L’ denotes the log law ‘P’ denotes the power law From power law or m = 

15. This shows that the power law is a first approximation to the logarithmic law

16. Explanation based on series expansion 0 x < 1 Let x = 1 - 1/

17. 1st order: 2nd order:

19. Determination of power-law index (a) What is hydrodynamic roughness length, yo? yo 0.11/u* hydraulically smooth flows yo 0.033ks hydraulically rough flows

20. Generalized relation • Nikuradze’s experiments: • Both smooth and rough pipes were tested • Rough pipes were prepared with pipe walls • roughened by well-sorted sand particles • r/ks ranges from 15 to 507, • where r is pipe radius and ks is sand diameter.

22. Determination of power-law index (b) Thickness of inner region, yir? 20%h (Open channel flows, Nezu and Nakagawa 1993) 7%r (Smooth pipe flows, Zagarola and Smits 1998) 20% (Turbulent boundary layers, Osterlund et al 2000)

23. Here we assume that yir can be considered as a measure of the extent to which the boundary effect interacts with the flow inertia. Such interactions can be characterized by turbulence statistics, for example, the location of maximum velocity fluctuation or shear stress Example: Location of the maximum (Morrison et al 2004)

24. Location of the maximum (Morrison et al 2004) f = 0.316 Re-0.25 assume

25. For smooth boundary yo = 0.11/u* • Comparing with Barenblatt (1993) yields • = -0.208 and  = 1.661

27. m varies with the two dimensionless parameters, h/ks and u*ks/

28. Noting that u*ks/ = (ks/h)(Uh/)(u*/U) = (ks/h)(Uh/) the variation of m can also be presented in terms of Uh/ flow-depth based Reynolds number and h/ks relative roughness height

29. Variation of m

32. The m-value varies, but within a small range If r/ks increases by 100%, say from r/ks = 31 to 60, m increases only by 9%, i.e. from 5.3 to 5.8, for large Reynolds numbers. If considering the entire data range where r/ks varies from 15 to 507, i.e. an increase by 3280%, m increases from 4.7 to 7.7, i.e. only by 64%.

33. Other indexes might yield the best fit when applying the power law for particular flow measurements, but the power of 1/6 would generally provide acceptable predictions

36. Conclusions • For studying wide open channel flows, the power law is simple but empirical. • The relatively rigorous explanation relates the power law to the classical turbulence theory • The power law can be considered as a first-order approximation to the log law in the overlap region, but is applicable to the majority of the flow domain.

37. Conclusions • m can be associated analytically with Re and h/ks • m appears related to f in a simple fashion • m=1/6 is preferable for many engineering applications • m increases for very rough boundaries

38. The relevant paper published: Cheng, N. S. (2006). “Power-law index for velocity profiles in open channel flows.” Advances in Water Resources, 30(8), 1775-1784.