# CMS-Flow:Incipient Motion

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# 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 (Ucrc) is calculated using the formula proposed by van Rijn (1984 c):

 U_{crc} = \left\{ \begin{align} &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{align} \right. (4)

where d50 and d90 are the sediment grain size in meters of 50th and 90th 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):

 $U_{crw} = \begin{cases} 0.24 [(s-1)g]^{0.66} (d_{50})^{0.33} T_p^{0.33} , & \text{for } 0.1 \le d_{50} \le 0.5 \ mm \\ 0.95 [(s-1)g]^{0.57} (d_{50})^{0.43} T_p^{0.14}, & \text{for } 0.5 \le d_{50} \le 2.0 \ mm \end{cases}$ (5)

where Tp 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.