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Yang (1983) – sand

The Yang (1983) formula gives the sediment concentration [ppm]:

\[ \log_{10}c = A + B\ \log_{10}\left( \frac{VI}{w_{s}} - \frac{V_{cr}I}{w_{s}} \right) \]

Equation: Yang formula for the sediment concentration of sand.

The concentration [ppm] is equivalent to [g/m3], see Parker (2004)1.

Where:

  • \(c\) — concentration [ppm]
  • \(V\) — flow velocity [m/s]
  • \(I\) — water level slope
  • \(w_{s}\) — settling velocity [m/s]
  • \(V_{cr}\) — critical flow velocity [m/s]

The water level slope is calculated from:

\[ I = \frac{u_{f}^{2}}{gh} \]

Equation: Water level slope derived from the total friction velocity.

Where:

  • \(u_{f}\) — friction velocity [m/s] equivalent to total shear stress
  • \(h\) — water depth [m]
  • \(g\) — acceleration of gravity [m/s2]

The coefficients \(A\) and \(B\) are functions of the Rouse number and grain Reynolds number:

\[ A = 5.435 - 0.286\ \log_{10}\left( \frac{w_{s}d}{\nu} \right) - 0.457\ \log_{10}\left( \frac{u_{f}}{w_{s}} \right) \]

Equation: Coefficient A in the Yang sand formula.

\[ B = 1.799 - 0.409\ \log_{10}\left( \frac{w_{s}d}{\nu} \right) - 0.314\ \log_{10}\left( \frac{u_{f}}{w_{s}} \right) \]

Equation: Coefficient B in the Yang sand formula.

The critical flow velocity is determined from:

\[ \frac{V_{cr}}{w_{s}} = \begin{cases} 131, & Re < 1.2 \\ \dfrac{2.5}{\log_{10}Re - 0.06} + 0.66, & 1.2 \leq Re \leq 70 \\ 2.05, & Re > 70 \end{cases} \]

Equation: Critical flow velocity as a function of the grain Reynolds number.

Where \(Re\) is a grain Reynolds number:

\[ Re = \frac{u_{f}d}{\nu} \]

Equation: Grain Reynolds number used in the Yang formula.

The (equilibrium) sediment transport is calculated from the concentration by using:

\[ q_{t} = \frac{qc}{s\rho} \]

Equation: Total sediment transport derived from the concentration.

Where:

  • \(q_{t}\) — total sediment transport [m2/s]
  • \(q\) — water flux [m2/s]
  • \(s\) — relative sediment density [-]
  • \(\rho\) — water density [106 g/m3]

The Yang formula is a total load formula, which means the sediment transport has to be divided into bed-load and suspended load:

\[ q_{b} = k_{b}q_{t} \]

Equation: Bed-load part of the Yang total load.

\[ q_{s} = k_{s}q_{t} \]

Equation: Suspended load part of the Yang total load.

Where:

  • \(q_{b}\) — bed-load [m2/s]
  • \(q_{s}\) — suspended load [m2/s]

Where \(k_{b}\) and \(k_{s}\) are bed-load and suspended load factors; there is no requirement that the factors sum to unity.

In a graded sediment model the bed-load for each fraction is modified with the relative weight of each fraction in the bed surface (\(F_{i}\)).

In a graded model the following are specific for each fraction (\(i\)):

  • Grain size \(d_{i}\)
  • Relative density \(s_{i}\)
  • Settling velocity \(w_{si}\)

  1. Parker, G. (2004) "CHAPTER 9: RELATIONS FOR HYDRAULIC RESISTANCE IN RIVERS", downloadable from hydrolab.illinois.edu. ↩