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.



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