1. Bedload Transport Rates for Low, Moderate and High Shear Stresses
by Nian-Sheng Cheng
School of Civil & Environmental Engineering
Nanyang Technological University
Singapore

2. Given properties of fluid and particle, D,  and s,
In the dimensionless form,
where,

3. Bedload Formulas with Excess Stress
Authors
 = f()
Meyer-Peter & Muller (1948)
 = 8( - C)1.5
Bagnold (1973)
Englund & Fredsoe (1976)
Yalin (1977)

4. Most of the bed-load functions apply only for  > C
If  < C
then how to predict the sediment transport rate?
What does C or C mean?
What are the approaches for evaluating C?
‘weak movement’ – Subjective
‘small transport rate’ – How small is small?
Shields Diagram
How Good

5. Q
(Chien and Wan 1999 p.320)

6. 
(Chien and Wan 1999 p.314)

7. Sometimes, C works as an adjustable constant
for predictions or data fitting.
1/Q F
(Raudkivi 1998 p.148)

8. Laursen (1999) stated
“When c is used in a sediment transport formula,
the notion becomes fuzzy, but c is then basically a
coefficient in the particular formula that is important
for low rates of transport.
The same sediment particle has as many values
of c as there are formulas.”

9. The fact is that
there is no shear stress below which
no single grain moves
(Paintal 1970; Lavelle and Mofjeld 1987)
The method of determining c
by extrapolating qb to zero is flawed
because most stress-transport relations are
power functions (Buffington 1999)
if qb = 0, then c must be zero.

10. Dimensionless Shear Stress as Function of Bedload Transport Rate
(Buffington 1999)

11. Dimensionless Shear Stress as Function of Bedload Transport Rate
(Buffington 1999)

12. Dimensionless Shear Stress as Function of Bedload Transport Rate
(Buffington 1999)

13. It’s possible, for example, …
Without C or its kind, can we go further?
It’s possible, for example, …

14. A ‘new’ variable: Stream power
Dimensionless stream power:
Just for convenience

15. Simplified Bedload Formulas for High Shear Conditions
Authors
Simplified formulas
Meyer-Peter & Muller (1948)
 = 8
Bagnold (1973)
 = (13.2~19.3)
Englund & Fredsoe (1976)
 = 11.6
Yalin (1977)
 = 13.5

16. High Transport Rates

17. Assumption

18. Derivation
Integration

21. For low transport rates,
only empirical formulas are available.
Examples includes
(Paintal 1971)
(Einstein 1942)

22. Comparison for Low Transport Rates

23. Moderate rate
High rate
Low rate

24. Conclusions: - a good data-fitting exercise;
- no mechanisms included.
Further ‘Exercises’ (much harder)
Viscous effects which may be significant for fine particles
at low transport rates
Role of near-bed turbulence in weak sediment transport
Intensive two-phase interaction at high transport rates,
the shear stress depending on a few factors