CMS-Flow:Transport Formula

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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. These are refered to as the Lund-CIRP transport formulas. The general transport formulas can be used for both symmetric and asymmetric waves but for simplicity the waves are assumed to be symmetric in CMS. The bed load transport rate including the stirring effect of waves is given by

  
 \frac{q_{b}}{\sqrt{(s-1) g d_{50}^3}} = a_c \sqrt{\theta_c} \theta_{cw,m}\exp{  \biggl ( -b_c \frac{\theta_{cr}}{\theta_{cw}}} \biggr )
(1)

where q_{b} is in m^2/s, d_{50} is the median grain size, s is the sediment specific gravity or relative density, g is gravitational constant, \theta_{cw,m} and \theta_{cw} are the mean and maximum Shields parameters due to waves and currents respectively, \theta_{c}, \theta_{cr} is the critical Shields parameter due to currents, a_c and b_c are empirical coefficients.

The current-related suspended load transport with wave stirring is given by

  
 \frac{q_s}{\sqrt{ (s-1) g d_{50}^3 }} = U c_R \frac{\varepsilon}{\omega_s} \biggl[ 1  - \exp{ \biggl( - \frac{w_s h}{\varepsilon}} \biggr) \biggr]
(2)

where U is the depth-averaged current velocity, h is the total water depth, \omega_s is the sediment fall velocity, 
\varepsilon is the sediment diffusivity, and c_R is the reference bed concentration. The reference bed concentration is calculated from

  c_R = A_{cR}  \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}}  \biggr) (3)

where the coefficient A_{cR} is given by

  A_{cR} = 3.5 \times 10^3 \exp{ \bigl( - 0.3 D_{*} } \bigr) (4)

where  \nu the kinematic viscosity of water, and D_{*} the dimensionless grain size

  D_{*} = d_{50} \biggl[ \frac{(s-1) g}{ \nu} \biggr] (5)

The sediment fall velocity is calculated using the formula by Soulsby (1997)

 


\omega_s = \frac{\nu}{d} \bigg[ \big( 10.36^2 + 1.049 D_{*}^3 \big)^{1/2} -10.36  \bigg]

(6)

The sediment mixing coefficient is calculated as

  \epsilon = h \biggl( \frac{k_b^3 D_b + k_c^3 D_c + k_w^3 D_w}{\rho} \biggr)^{1/3} (7)

where k_b, k_c, and k_w are coefficients, D_b is the wave breaking dissipation, and D_c and D_w are the bottom friction dissipation due to currents and waves respectively. For more details see Camenen and Larson (2008).

van Rijn

The van Rijn (1984ab) transport equations are used with the recalibrated coefficients of van Rijn (2007ab) are given by

  
q_b = 0.015 \rho_s U h 
  \biggl( \frac{U_e - U_{cr} }{ \sqrt{(s-1) g d_{50}} } \biggr)^{1.5} 
  \biggl( \frac{d_{50}}{h} \biggr)^{1.2} 
(8)
  
  q_s = 0.012 \rho_s U d_{50} 
  \biggl( \frac{U_e - U_{cr} }{ \sqrt{(s-1) g d_{50}}} \biggr)^{2.4} 
  D_{*}^{-0.6} 
(9)

where U_{cr} is the critical depth-averaged velocity for initiation of motion, U_e is the effective depth averaged velocity calculated as U_e = U + 0.4 U_w in which  U_w is the peak orbital velocity based on the significant wave height

The critical velocity is estimated as

  U_{cr} = \beta U_{crc} + (1-\beta) U_{crw} (10)

where U_{crc} and U_{crw} are the critical velocity for currents and waves respectively. As in van Rijn (2007), the critical velocity for currents and waves are calculated based on Komar and Miller (1975):

  
  U_{crc} = 
  \begin{cases} 
0.19 (d_{50})^{0.1} \log{_{10} \big( \frac{4h}{d_{90}} \big) }, & \text{for } 0.1 \le d_{50} \le 0.5 mm \\ 
 8.5 (d_{50})^{0.6} \log{_{10} \big( \frac{4h}{d_{90}} \big) }, & \text{for } 0.5 \le d_{50} \le 2.0 mm
  \end{cases}
(11)
  
  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}
(12)

According to van Rijn (2007) bed load transport formula predicts transport rates with a factor of 2 for velocities higher than 0.6 m/s, but underpredicts transports by a factor of 2-3 for velocities close to initiation of motion.

Watanabe

The equilibrium total load sediment transport rate of Watanabe (1987) is given by

  
q_{t} = A_w \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U }{\rho g } \biggr] 
(13)

where  \tau_{b,max} is the maximum shear stress,  \tau_{cr} is the critical shear stress of incipient motion, and  A is an empirical coefficient typically ranging from 0.1 to 2.

The critical shear stress is determined using

  \tau_{cr} = (\rho_s - \rho) g d \phi_{cr} (14)

In the case of currents only the bed shear stress is determined as  \tau_{c} = \frac{1}{8}\rho g f_c U_c^2 where  f_c is the current friction factor. The friction factor is calculated as  f_c =  0.24log^{-2}(12h/k_{sd}) where  k_{sd}  is the Nikuradse equivalent sand roughness obtained from  k_{sd} = 2.5d_{50} .

If waves are present, the maximum bed shear stress \tau_{b,max} is calculated based on Soulsby (1997)

  \tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2  + (\tau_w \sin{\phi})^2 } (15)

where  \tau_m 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  \phi is the angle between the waves and the current. The mean wave and current bed shear stress is

  
  \tau_{m} = \tau_c \biggl[ 1 +  1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr]
(16)

The wave bed shear stress is given by  \tau_{w} = \frac{1}{2}\rho g f_w U_w^2 where  f_w is the wave friction factor, and  U_w is the wave orbital velocity amplitude based on the significant wave height.

The wave friction factor is calculated as (Nielsen 1992) f_w = \exp{5.5R^{-0.2}-6.3} where  R is the relative roughness defined as  R = A_w/k_{sd} and  A_w  is semi-orbital excursion  A_w = U_w T / (2  \pi) .

Soulsby-van Rijn

Soulsby (1997) proposed the following equation for the total load sediment transport rate under currents and waves

  
  q_t = A_s U \biggl[ \biggl( U^2 + 0.018  \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - U_{cr} \biggr]^{2.4}
(20)

where U_{rms} is the root-mean-squared wave orbital velocity, and C_d is the drag coefficient due to currents alone and the coefficient  A_{s} = A_{sb} + A_{ss} . The coefficients A_{sb} and A_{ss} are related to the bed and suspended transport loads respectively and are given by

  A_{sb} = \frac{ 0.005 h (d_{50}/h)^{1.2} }{ [(s-1)g d_{50} ]^{1.2} } (21)
  A_{s} = \frac{ 0.012 d_{50} D_{*}^{-0.6} }{ [(s-1)g d_{50} ]^{1.2} } (22)

The current drag coefficient is calcualted as

  C_d =  \biggl[ \frac{0.4}{\ln{(h/z_0)}-1 } \biggr]^2 (23)

with a constant bed roughness length z_0 set to 0.006 m.


Symbol Description Units
 q_{bc} Bed load transport rate m3/s
 s Relative density -
 \theta_{c}  Shields parameter due to currents -
 \theta_{cw} Shields parameter due to waves and currents -
 \theta_{cw} Critical shields parameter -
 a_c Empirical coefficient -
 b_c Empirical coefficient -
 U_c 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.
  • van Rijn, L. C. (1984a). "Sediment transport. Part I: Bed load transport", Journal of Hydraulic Engineering, 110(10), 1431–1456.
  • van Rijn, L. C. (1984b). "Sediment transport. Part II: Suspended loadtransport", Journal of Hydraulic Engineering, 110(11), 1613–1641.
  • van Rijn, L.C., (2007a). "Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and Bed-load Transport", Journal of Hydraulic Engineering, 133(6), 649-667.
  • van Rijn, L.C., (2007b). "Unified View of Sediment Transport by Currents and Waves. II: Suspended Transport", Journal of Hydraulic Engineering, 133(6), 668-689.
  • Watanabe, A. (1987). "3-dimensional numerical model of beach evolution," Proceedings Coastal Sediments '87, ASCE, 802-817.

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