CMS-Flow:Incipient Motion

Incipient Motion

In the case of the Lund-CIRP (Camenen and Larson 2005, 2007, 2008) and Watanabe (1987) formulas, the incipient motion is based on the critical Shields parameter and estimated using the formula proposed by Soulsby (1997):

$\Theta _{cr}={\frac {0.3}{1+1.2d_{*}}}+0.055\left[1-exp(-0.02d_{*})\right]$

(1)

in which the dimensionless grain size (d_*) is defined

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

(2)

The critical shear stress for incipient motion is given by

${\frac {\tau _{cr}}{g(\rho _{s}-\rho )d}}=\Theta _{cr}$

(3)

The critical depth-averaged velocity for currents alone (U_crc) is calculated using the formula proposed by van Rijn (1984 c):

$U_{crc}=\left\{{\begin{aligned}&0.19\ d_{50}^{0.1}log_{10}\left({\frac {4h}{d_{90}}}\right),\quad \quad for\ 0.1\leq d_{50}\leq 0.5\ mm\\&8.5\ d_{50}^{0.6}log_{10}\left({\frac {4h}{d_{90}}}\right),\quad \quad for\ 0.5\leq d_{50}\leq 2.0\ mm\end{aligned}}\right.$

(4)

where d₅₀ and d₉₀ are the sediment grain size in meters of 50^th and 90^th percentiles, respectively. The above criteria are used in the van Rijn (2007 a,b) and Soulsby-van Rijn (Soulsby 1997) transport formulas.

The critical bottom orbital velocity magnitude for waves alone is calculated using the formulation of Komar and Miller (1975):

$U_{crw}={\begin{cases}0.24[(s-1)g]^{0.66}(d_{50})^{0.33}T_{p}^{0.33},&{\text{for }}0.1\leq d_{50}\leq 0.5\ mm\\0.95[(s-1)g]^{0.57}(d_{50})^{0.43}T_{p}^{0.14},&{\text{for }}0.5\leq d_{50}\leq 2.0\ mm\end{cases}}$

(5)

where T_p is the peak wave period.

References

• Camenen, B., and M. Larson. 2005. A general formula for non-cohesive bed-load sediment transport. Estuarine, Coastal and Shelf Science (63)2:49–260.
• Camenen, B., and M. Larson. 2007. A unified sediment transport formulation for coastal inlet application. ERDC/CHL CR-07-1. Vicksburg, MS: US Army Engineer Research and Development Center.
• Camenen, B., and M. Larson. 2008. A general formula for noncohesive suspended sediment transport. Journal of Coastal Research 24 (3):615–627.
• Komar, P. D., and M. C. Miller. 1975. On the comparison between the threshold of sediment motion under waves and unidirectional currents with a discussion of the practical evaluation of the threshold. Journal of Sedimentary Petrology (45):362–367.
• Soulsby, R. L. 1997. Dynamics of marine sands. London, England: Thomas Telford Publications.
• van Rijn, L. C. 1984c. Sediment transport, Part III: Bed forms and alluvial roughness. Journal of Hydraulic Engineering, ASCE 110(12):1733–1754.
• Watanabe, A. 1987. 3-dimensional numerical model of beach evolution. In Proceedings, Coastal Sediments ’87, 802–817. New Orleans, LA.

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