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. 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

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

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 \frac{q_{cb}^{*}}{\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 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 q_{cb}^{*}} is in m^2/s, 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 d_{50}} is the median grain size, 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 s} is the gravity, 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 \theta_{cw,m}} and 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 \theta_{cw}} are the mean and maximum Shields parameters due to waves and currents respectively, 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 \theta_{c}} , 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 \theta_{cr}} is the critical Shields parameter due to currents, 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_c} and 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 b_c} are empirical coefficients.

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

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 \frac{q_s^{*}}{\sqrt{ (s-1) g d^3 }} = U c_R \frac{\varepsilon}{\omega_s} \biggl[ 1 - \exp{ \biggl( - \frac{w_s d}{\epsilon}} \biggr) \biggr] }

(2)

where 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 U} is the depth-averaged current velocity, 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 \omega_s} is the sediment fall velocity, 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 \varepsilon } is the sediment diffusivity, and g

The reference sediment concentration is obtained from

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

(3)

where the coefficient 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_{cR}} is given by

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

(4)

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

The sediment mixing coefficient is calculated as

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 \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).

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 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} }

(6)

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 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} }

(7)

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

The critical velocity is estimated as

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

(7)

where 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 U_{crc}} and 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 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):

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 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} }

(7)

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 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} }

(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

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 q_{t*} = A \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U_c }{\rho g } \biggr] }

(6)

where 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 \tau_{b,max} } is the maximum shear stress, 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 \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

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

(6)

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

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

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

(6)

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

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

The wave friction factor is calculated as (Nielsen 1992) 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 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

$A_{s}=A_{sb}+A_{ss}$

$A_{sb}={\frac {0.005h(d_{50}/h)^{1.2}}{[(s-1)gd_{50}]^{1.2}}}$

$A_{s}={\frac {0.012d_{50}D_{*}^{-0.6}}{[(s-1)gd_{50}]^{1.2}}}$

$C_{d}={\biggl [}{\frac {0.4}{\ln {(h/z_{0})}-1}}{\biggr ]}^{2}$

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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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