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

$\frac{q_b}{\sqrt{(s-1)g{d}^3}}=a_c\sqrt{{\theta }_c}{\theta }_{cw}\exp (-b_c\frac{{\theta }_{cr}}{{\theta }_{cw}})$

(1)

Suspended load

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

$\frac{q_s}{\sqrt{(s-1)g{d}^3}}=U{c}_R\frac{ϵ}{w_s}[1-\exp (-\frac{w_{s}d}{ϵ})]$

(2)

The reference sediment concentration is obtained from

$c_R=A_{cR}\exp (-4.5\frac{{\theta }_{cr}}{{\theta }_{cw}})$

(3)

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

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

(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

$ϵ=h(\frac{k_b^3{D}_b+k_c^3{D}_c+k_w^3{D}_w}{\rho }{)}^{1/3}$

(5)

van Rijn

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

$q_b=0.015{\rho }_{s}Uh\left(\frac{U_e-U_{cr}}{\sqrt{(s-1)g{d}_{50}}}{)}^{1.5}\right(\frac{d_{50}}{h}{)}^{1.2}$

(6)

$q_s=0.012{\rho }_{s}U{d}_{50}(\frac{U_e-U_{cr}}{\sqrt{(s-1)g{d}_{50}}}{)}^{2.4}{D}_{*}^{-0.6}$

(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):

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

(7)

$U_{crw}=\{\begin{array}{ll}0.24[(s-1)g{]}^{0.66}(d_{50}{)}^{0.33}{T}_p^{0.33},& \text{for }0.1\le d_{50}\le 0.5mm\\ 0.95[(s-1)g{]}^{0.57}(d_{50}{)}^{0.43}{T}_p^{0.14},& \text{for }0.5\le d_{50}\le 2.0mm\end{array}$

(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

$q_{t*}=A\left[\frac{({\tau }_{b,max}-{\tau }_{cr})U_c}{\rho g}\right]$

(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

${\tau }_{cr}=({\rho }_s-\rho )gd{\varphi }_{cr}$

(6)

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 $f_c=0.24lo{g}^{-2}(12h/k_{sd})$ 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)

${\tau }_{max}=\sqrt{({\tau }_m+{\tau }_w\cos \varphi {)}^2+({\tau }_w\sin \varphi {)}^2}$

(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 $\varphi$ is the angle between the waves and the current. The mean wave and current bed shear stress is

${\tau }_m={\tau }_c[1+1.2(\frac{{\tau }_w}{{\tau }_c+{\tau }_c}{)}^{3.2}]$

(6)

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

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)

Failed to parse (syntax error): {\displaystyle q_t* = \frac{{h} \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 (syntax error): {\displaystyle A_{sb} = \frac{ 0.005 h (d_{50}/h)^{1.2} }{ [(s-1)g d_{50}} ]^{1.2} } } Failed to parse (syntax error): {\displaystyle A_{s} = \frac{ 0.005 h d_{50} D_{*}^{-0.6} }{ [(s-1)g d_{50}} ]^{1.2} } } Failed to parse (syntax error): {\displaystyle C_d = \biggl[ \frac{0.4}{\ln{(h/z_0)}-1 } \biggr]^2 ---- {| border="1" ! Symbol !! Description !! Units |- |<math> 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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