CMS-Flow:Transport Formula

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

math \frac{q_{b

({{{2}}})

{\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

math \frac{q_s}{\sqrt{ (s-1) g d^3

({{{2}}})

= 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

{{{1}}}

({{{2}}})

{\theta_{cw}}} \biggr) /math|2=3}}

where the coefficient mathA_{cR}/math is given by

{{{1}}}

(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

{{{1}}}

(5)

Watanabe

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

{{{1}}}

(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

{{{1}}}

(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)

{{{1}}}

(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

{{{1}}}

(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) mathf_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)

{{{1}}}

({{{2}}})

{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}}

---

Symbol	Description	Units
math q_{bc} /math	Bed load transport rate	msup3/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.