Sediment Transport: Difference between revisions

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==Lund-CIRP==
Camenen and Larson (2005, 2007, and 2008) developed a general sediment transport formula for bed and suspended load under combined waves and currents.
=== Bed load===
The current-related bed load transport with wave stirring is given by
{{Equation|<math> \frac{q_{b}}{\sqrt{(s-1)gd^3}} = a_c \sqrt{\theta_c} \theta_{cw}\exp{ \biggl ( -b_c \frac{\theta_{cr}}{\theta_{cw}}} \biggr ) </math>|2=1}}
=== Suspended load ===
The current-related suspended load transport with wave stirring is given by
{{Equation|<math> \frac{q_s}{\sqrt{ (s-1) g d^3 }} = U c_R \frac{\epsilon}{w_s} \biggl[ 1 - \exp{ \biggl( - \frac{w_s d}{\epsilon}} \biggr) \biggr] </math>|2=2}}
The reference sediment concentration is obtained from
{{Equation|<math> c_R = A_{cR}  \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}}  \biggr)  </math>|2=3}}
where the coefficient <math>A_{cR}</math> is given by
{{Equation|<math> A_{cR} = 3.5x10^3 \exp{ \bigl( - 0.3 d_{*} } \bigr)  </math>|2=4}}
with <math> d_{*} = d \sqrt{(s-1) g \nu^{-2}} </math> being the dimensionless grain size and <math> \nu </math> the kinematic viscosity of water.
The sediment mixing coefficient is calculated as
{{Equation|<math> \epsilon = h \biggl( \frac{k_b^3 D_b + k_c^3 D_c + k_w^3 D_w}{\rho} \biggr)^{1/3}  </math>|2=5}}
== van Rijn ==
== Watanabe ==
The equilibrium total load sediment transport rate of Watanabe (1987) is given by
{{Equation|<math> q_{t*} = A \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U_c }{\rho g } \biggr]  </math>|2=6}}
where <math> \tau_{b,max} </math> is the maximum shear stress, <math> \tau_{cr} </math> is the critical shear stress of incipient motion, and <math> A </math> is an empirical coefficient typically ranging from 0.1 to 2.
The critical shear stress is determined using
{{Equation|<math> \tau_{cr} = (\rho_s - \rho) g d \phi_{cr} </math>|2=6}}
In the case of currents only the bed shear stress is determined as <math> \tau_{c} = \frac{1}{8}\rho g f_c U_c^2 </math> where <math> f_c </math> is the current friction factor. The friction factor is calculated as <math> f_c = 0.24log^{-2}(12h/k_{sd}) </math> where <math> k_{sd} </math> is the Nikuradse equivalent sand roughness obtained from <math> k_{sd} = 2.5d_{50} </math>.
If waves are present, the maximum bed shear stress <math>\tau_{b,max} </math> is calculated based on Soulsby (1997)
{{Equation|<math> \tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2  + (\tau_w \sin{\phi})^2 } </math>|2=6}}
where <math> \tau_m </math> is the mean shear stress by waves and current over a wave cycle, math> \tau_w </math> is the mean wave bed shear stress, and <math> \phi </math> is the angle between the waves and the current. The mean wave and current bed shear stress is
{{Equation|<math> \tau_{m} = \tau_c \biggl[ 1 + 1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr] </math>|2=6}}
The wave bed shear stress is given by <math> \tau_{w} = \frac{1}{2}\rho g f_w U_w^2 </math> where <math> f_w </math> is the wave friction factor, and <math> U_w </math> is the wave orbital velocity amplitude based on the significant wave height.
The wave friction factor is calculated as (Nielsen 1992) <math>f_w = \exp{5.5R^{-0.2}-6.3}</math> where
where <math> R </math> is the relative roughness defined as <math> R = A_w/k_{sd} </math> and <math> A_w </math> is semi-orbital excursion <math> A_w = U_w T / (2 \pi) </math>.
== Soulsby-van Rijn ==
The equilibrium sediment concentration is calculated as (Soulsby 1997)
{{Equation|<math> C_{*} = \frac{A_{sb}+A_{ss}}{h} \biggl[ \biggl( U_c^2 + 0.018 \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - u_{cr} \biggr]^{2.4}  </math>|2=7}}
----
{| border="1"
! Symbol !! Description !! Units
|-
|<math> q_{bc} </math> || Bed load transport rate || m<sup>3</sup>/s
|-
|<math> s </math> ||  Relative density || -
|-
|<math> \theta_{c}  </math> || Shields parameter due to currents || -
|-
|<math> \theta_{cw} </math> ||  Shields parameter due to waves and currents || -
|-
|<math> \theta_{cw}</math> ||  Critical shields parameter  || -
|-
|<math> a_c </math> || Empirical coefficient || -
|-
|<math> b_c </math> || Empirical coefficient || -
|-
|<math> U_c </math> || Current magnitude || m/s
|}
== References ==
* Camenen, B., and Larson, M. (2005). "A bed load sediment transport formula for the nearshore," Estuarine, Coastal and Shelf Science, 63, 249-260.
* Camenen, B., and Larson, M. (2007). "A unified sediment transport  formulation for coastal inlet applications", ERDC/CHL-TR-06-7, US Army  Engineer Research and Development Center,  Coastal and Hydraulics  Laboratory, Vicksburg, MS.
* Camenen, B., and Larson, M., (2008). "A General Formula for  Non-Cohesive  Suspended Sediment Transport," Journal of Coastal  Research, 24(3), 615-627.
* Soulsby, D.H. (1997). "Dynamics of marine sands. A manual for practical applications," Thomas Telford Publications, London, England, 249 p.
* Watanabe, A. (1987). "3-dimensional numerical model of beach evolution," Proceedings Coastal Sediments '87, ASCE, 802-817.

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