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Ackers & White (1973)

The Ackers & White sediment transport calculation1 starts with the non-dimensional grain diameter \(D_{*}\):

\[ D_{*} = d\ \left( \frac{(s - 1)g}{\nu^{2}} \right)^{1/3} \]

Equation: Non-dimensional grain diameter.

Three ranges of \(D_{*}\) are considered in the formula:

  • \(D_{*} < 1\): Fine sediment
  • \(D_{*} = 1 - 60\): Transitional
  • \(D_{*} > 60\): Gravel

Four coefficients derive from the non-dimensional grain diameter: \(n\), \(A\), \(C\), \(m\):

\[ n = \begin{cases} 1, & D_{*} < 1 \\ 1 - 0.56\log_{10}(D_{*}), & 1 \leq D_{*} \leq 60 \\ 0, & 60 < D_{*} \end{cases} \]

Equation: Coefficient n in the Ackers & White formula.

\[ A = \begin{cases} 0.37, & D_{*} < 1 \\ \dfrac{0.23}{\sqrt{D_{*}}} + 0.14, & 1 \leq D_{*} \leq 60 \\ 0.17, & 60 < D_{*} \end{cases} \]

Equation: Coefficient A in the Ackers & White formula.

\[ C = \begin{cases} 2.95 \times 10^{-4}, & D_{*} < 1 \\ 10^{2.86\log_{10}D_{*} - \log_{10}^{2}D_{*} - 3.53}, & 1 \leq D_{*} \leq 60 \\ 0.025, & 60 < D_{*} \end{cases} \]

Equation: Coefficient C in the Ackers & White formula.

\[ m = \begin{cases} 11, & D_{*} < 1 \\ \dfrac{9.66}{D_{*}} + 1.34, & 1 \leq D_{*} \leq 60 \\ 1.5, & 60 < D_{*} \end{cases} \]

Equation: Coefficient m in the Ackers & White formula.

Ackers & White calculates a general sediment mobility number \(F_{gr}\):

\[ F_{gr} = \frac{u_{f}^{n}}{\sqrt{(s - 1)gd}}\ \left( \frac{V}{\sqrt{32}\ \log_{10}\frac{10h}{d}} \right)^{1 - n} \]

Equation: General sediment mobility number.

Where \(V\) is the flow speed.

From the general sediment mobility number, the general transport parameter \(G_{gr}\):

\[ G_{gr} = C\left( \frac{F_{gr}}{A} - 1 \right)^{m} \]

Equation: General transport parameter.

The general sediment mobility number relates to the sediment transport via:

\[ C = G_{gr}\left( \frac{d}{h} \right)\ \left( \frac{V}{u_{f}} \right)^{n} \]

Equation: Volumetric sediment concentration from the general transport parameter.

Where \(C\) is the volumetric sediment concentration. The volumetric sediment transport is hence:

\[ q_{t} = Vh\ G_{gr}\left( \frac{d}{h} \right)\ \left( \frac{V}{u_{f}} \right)^{n} \]

Equation: Volumetric total sediment transport.

Notes regarding Ackers & White

Ackers & White is a total load formula, which can be distributed to bed-load and suspended load by using constant factors. There is no requirement that the formula be used for both bed-load and suspended load, so it is allowed to e.g. use Ackers & White for suspended load and e.g. Meyer-Peter for bed-load.

Ackers & White was developed for single fraction sediment but can be adopted to graded models in which case the load is modified with the local relative weight function (\(F_{i}\)). No other modifications apply to Ackers & White for multi-fraction models.

  1. Ackers, P., and White, W. R. (1973). "Sediment transport: new approach and analysis," Journal of Hydraulics Division, American Society of Civil Engineers, Vol. 99, No. HY11, pp. 2040-2060. ↩