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

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

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

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

Q (Chien and Wan 1999 p.320)

 (Chien and Wan 1999 p.314)

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

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.”

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.

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

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

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

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

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

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

High Transport Rates

Assumption

Derivation Integration

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

Comparison for Low Transport Rates

Moderate rate High rate Low rate

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