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

${\frac {q_{b}}{\sqrt {(s-1)gd^{3}}}}=a_{c}{\sqrt {\theta _{c}}}\theta _{cw}\exp {{\biggl (}-b_{c}{\frac {\theta _{cr}}{\theta _{cw}}}}{\biggr )}$

(1)

Suspended load

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

${\frac {q_{s}}{\sqrt {(s-1)gd^{3}}}}=Uc_{R}{\frac {\epsilon }{w_{s}}}{\biggl [}1-\exp {{\biggl (}-{\frac {w_{s}d}{\epsilon }}}{\biggr )}{\biggr ]}$

(2)

The reference sediment concentration is obtained from

$c_{R}=A_{cR}\exp {{\biggl (}-4.5{\frac {\theta _{cr}}{\theta _{cw}}}}{\biggr )}$

(3)

where the coefficient $A_{cR}$ is given by

$A_{cR}=3.5x10^{3}\exp {{\bigl (}-0.3d_{*}}{\bigr )}$

(4)

with $d_{*}=d{\sqrt {(s-1)g\nu ^{-2}}}$ being the dimensionless grain size and $\nu$ the kinematic viscosity of water.

The sediment mixing coefficient is calculated as

$\epsilon =h{\biggl (}{\frac {k_{b}^{3}D_{b}+k_{c}^{3}D_{c}+k_{w}^{3}D_{w}}{\rho }}{\biggr )}^{1/3}$

(5)

van Rijn

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

$q_{b}=0.015\rho _{s}Uh{\biggl (}{\frac {U_{e}-U_{cr}}{\sqrt {(s-1)gd_{50}}}}{\biggr )}^{1.5}{\biggl (}{\frac {d_{50}}{h}}{\biggr )}^{1.2}$

(6)

$q_{s}=0.012\rho _{s}Ud_{50}{\biggl (}{\frac {U_{e}-U_{cr}}{\sqrt {(s-1)gd_{50}}}}{\biggr )}^{2.4}D_{*}^{-0.6}$

(7)

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

The critical velocity is estimated as

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

(7)

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):

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

(7)

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

(7)

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

$q_{t*}=A{\biggl [}{\frac {(\tau _{b,max}-\tau _{cr})U_{c}}{\rho g}}{\biggr ]}$

(6)

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

$\tau _{cr}=(\rho _{s}-\rho )gd\phi _{cr}$

(6)

In the case of currents only the bed shear stress is determined as $\tau _{c}={\frac {1}{8}}\rho gf_{c}U_{c}^{2}$ 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)

$\tau _{max}={\sqrt {(\tau _{m}+\tau _{w}\cos {\phi })^{2}+(\tau _{w}\sin {\phi })^{2}}}$

(6)

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

$\tau _{m}=\tau _{c}{\biggl [}1+1.2{\biggl (}{\frac {\tau _{w}}{\tau _{c}+\tau _{c}}}{\biggr )}^{3.2}{\biggr ]}$

(6)

The wave bed shear stress is given by $\tau _{w}={\frac {1}{2}}\rho gf_{w}U_{w}^{2}$ 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

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

The equilibrium sediment concentration is calculated as (Soulsby 1997)

$q_{t}*=A_{s}U{\biggl [}{\biggl (}U^{2}+0.018{\frac {U_{rms}^{2}}{C_{d}}}{\biggr )}^{0.5}-U_{cr}{\biggr ]}^{2.4}$

(7)

where $A_{s}=A_{sb}+A_{ss}$ Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{sb} = \frac{ 0.005 h (d_{50}/h)^{1.2} }{ [(s-1)g d_{50}} ]^{1.2} } } Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{s} = \frac{ 0.005 h d_{50} D_{*}^{-0.6} }{ [(s-1)g d_{50}} ]^{1.2} } } $C_{d}={\biggl [}{\frac {0.4}{\ln {(h/z_{0})}-1}}{\biggr ]}^{2}$

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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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