CMS-Flow:Equilibrium Concentrations and Transport Rates

Equilibrium Concentrations and Transport Rates

In order to close the system of equations describing the sediment transport, bed change, and bed sorting equations, the fractional equilibrium depth-averaged total-load concentration (C_tk*) must be estimated from an empirical formula. The depth-averaged equilibrium concentration is defined as

$C_{tk*}={\frac {q_{tk*}}{Uh}}$

(1)

where q_tk* is the total-load transport for the k^th sediment size class estimated from an empirical formula. For convenience, C_tk* is written in general form as

$C_{tk*}=p_{1k}C_{tk}^{*}$

(2)

where p_1kis the fraction of the sediment size (k) in the first (top) bed layer, and C_tk^* is the potential equilibrium total-load concentration. The potential concentration (C_tk^*) can be interpreted as the equilibrium concentration for uniform sediment of size d_k. The above equation is essential for the coupling of sediment transport, bed change, and bed sorting equations.

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. The current-related bed- and suspended transport rate wave stirring is given by

${\frac {q_{b*}}{\sqrt {(s-1)gd_{50}^{3}}}}=f_{b}\rho _{s}12{\sqrt {\Theta _{c}}}\Theta _{cw,m}\exp {{\biggl (}-4.5{\frac {\Theta _{cr}}{\Theta _{cw}}}}{\biggr )}$

(3)

${\frac {q_{s*}}{\sqrt {(s-1)gd_{50}^{3}}}}=f_{s}\rho _{s}c_{R}U{\frac {\epsilon }{\omega _{s}}}\left[1-exp\left(-{\frac {\omega _{s}h}{\epsilon }}\right)\right]$

(4)

where:

q_b* = equilibrium bed-load transport rate [kg/m/s]

q_s* = equilibrium suspended-load transport rate [kg/m/s]

$\Theta _{c}$ = Shields parameters due to currents [-]

$\Theta _{cw,m}$ = mean Shields parameters due to waves and currents [-]

$\Theta _{cw}$= maximum Shields parameters due to waves and currents [-]

$\Theta _{cr}$ = critical Shields parameter [-]

$\epsilon$ = vertical sediment diffusivity [m²/s]

c_R = reference bed concentration [kg/m³]

f_b= bed-load scaling factor (default 1.0) [-]

f_s = suspended-load scaling factor (default 1.0) [-].

The critical Shields parameter is calculated using

$\Theta _{cr}={\frac {0.3}{1+1.2d_{*}}}+0.055\left[1-exp(-0.2d_{*})\right]$

(5)

The mean and maximum Shields parameters are calculated as

$\Theta _{cw,m}={\sqrt {\Theta _{c}^{2}+\Theta _{w,m}^{2}+2\Theta _{c}\Theta _{w,m}cos\varphi }}$

(6)

$\Theta _{cw}={\sqrt {\Theta _{c}^{2}+\Theta _{w}^{2}+2\Theta _{c}\Theta _{w}cos\varphi }}$

(7)

The mean wave Shields parameter is calculated as $\Theta _{w,m}=\Theta _{w}/2$ assuming a sinusoidal wave. The Shields parameters for currents and waves are given by

$\Theta _{c|w}={\frac {\tau _{c|w}}{g(\rho _{s}-\rho )d}}$

(8)

in which the subscript c|w indicates either the current- (c) or wave-related (w) component. The current-related shear stress ($\tau _{c}$) is calculated with

$\tau _{c}=\rho c_{b}U^{2}$

(9)

The wave-related bed shear stress is calculated with

$\tau _{w}={\frac {1}{2}}\rho f_{w}u_{w}^{2}$

(10)

and the wave friction factor (f_w) of Swart (1974) is given by

$f_{w}=\left\{{\begin{aligned}&exp(5.21r^{-0.19}-6.0)\quad for\ r>1.57\\&0.3\quad \quad \quad \quad \quad \quad \quad \quad \quad for\ r\leq 1.57\end{aligned}}\right.$

(11)

The total bed roughness is assumed to be a linear summation of the grain-related roughness (k_sg), form-drag (ripple) roughness (k_sr), and sediment-related roughness (k_ss):

$k_{s,c|w}=k_{sg}+k_{sr,c|w}+k_{ss,c|w}$

(12)

Here, the grain-related roughness is estimated as k_sg = 2d₅₀ The ripple roughness (k_sr) is calculated as (Soulsby 1997)

$k_{sr,c|w}=7.5{\frac {H_{r,c|w}^{2}}{L_{r,c|w}}}$

(13)

where H_r and L_r are the ripple height and length, respectively.

The current- and wave-related sediment roughnesses are estimated as

$k_{ss,c|w}=5d_{50}\Theta _{c|w}$

(14)

The above equation must be solved simultaneously with the expressions for the bottom shear stress because the roughness depends on the stress.

The reference concentration is given by

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

(15)

where the coefficient $A_{cR}$ is determined by the following relationship:

$A_{cR}=0.0035\exp {{\bigl (}-0.3d_{*}}{\bigr )}$

(16)

The vertical sediment diffusivity is calculated as

$\epsilon =h\left({\frac {D_{e}}{\rho }}\right)^{1/3}$

(17)

where D_e is the total effective dissipation given by

$D_{e}=k_{b}^{3}D_{br}+k_{c}^{3}D_{c}+k_{w}^{3}D_{w}$

(18)

in which k_b, k_c, and k_w are coefficients; D_br is the wave breaking dissipation (from the wave model); and D_c and D_w are the bottom friction dissipation due to currents and waves, respectively. The dissipation from bottom friction due to current (D_c) and the dissipation from bottom friction due to waves (D_w) are expressed as

$D_{c|w}=\tau _{c|w}u_{*c|w}$

(19)

The coefficient k_b=0.017 (Camenen and Larson 2008), and k_c and k_w are a function of the Schmidt number:

$k_{c|w}={\frac {\kappa }{6}}\sigma _{c|w}$

(20)

where $\sigma _{c|w}$ is either the current- or wave-related Schmidt number calculated from the following relationships (Camenen and Larson 2008):

$\sigma _{c|w}=\left\{{\begin{aligned}&a_{c|w}+b_{c|w}sin^{2}\left({\frac {\pi }{2}}{\frac {\omega _{s}}{u_{*c|w}}}\right)\ for{\frac {\omega _{s}}{u_{*c|w}}}\leq 1\\&1+(a_{c|w}+b_{c|w}-1)sin^{2}\left({\frac {\pi }{2}}{\frac {u_{*c|w}}{\omega _{s}}}\right)\ for{\frac {\omega _{s}}{u_{*c|w}}}>1\end{aligned}}\right.$

(21)

with the coefficients a_c = 0.4, b_c = 3.5, a_w =0.15, and b_w = 1.5.

For multiple-sized (non-uniform) sediments, the fractional equilibrium sediment transport rates are calculated as (Wu and Lin 2011)

${\frac {q_{bk*}}{\sqrt {(s-1)gd_{k}^{3}}}}=f_{b}\xi _{k}^{-1}p_{1k}\rho _{s}12{\sqrt {\Theta _{c}}}\Theta _{cw,m}\ exp\left(-4.5{\frac {\Theta _{crk}}{\Theta _{cw}}}\right)$

(22)

${\frac {q_{sk*}}{\sqrt {(s-1)gd_{k}^{3}}}}=f_{s}\xi _{k}^{-1}p_{1k}\rho _{s}c_{Rk}U{\frac {\epsilon _{k}}{\omega _{sk}}}\left[1-exp\left(-{\frac {\omega _{sk}h}{\epsilon _{k}}}\right)\right]$

(23)

where:

k = subscript indicating the sediment size class

$\xi _{k}$ = hiding and exposure function [-]

r_sk = fraction of suspended load for each size class defined by $r_{sk}={\frac {q_{sk}}{q_{t}k}}\simeq {\frac {q_{sk*}}{q_{tk*}}}$ where q_sk and q_tk are the actual suspended- and total-load transport rates and q_sk* and q_tk* are the equilibrium suspended- and total-load transport rates.

p_1k = fraction of the k^th sediment size in the first layer [-].

The availability of sediment fractions is included through p_1k, while hiding and exposure of grain sizes are accounted for by directly multiplying the transport rates.

van Rijn

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

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

(24)

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

(25)

where:

$U_{cr}$ = the critical depth-averaged velocity for incipient motion [m/s],

$U_{e}$ = effective depth averaged velocity [m/s]

The effective depth-averaged velocity is calculated as $U_{e}=U+\gamma u_{w}$ with $\gamma$= 0.4 for random waves and $\gamma$ = 0.8 for regular waves. The bottom wave orbital velocity based on linear wave theory is u_w. For random waves, u_w = u_ws where u_ws is based on the significant wave height and peak wave period

$u_{ws}={\frac {\pi H_{s}}{T_{p}sinh(kh)}}$

(26)

The critical depth-averaged velocity is

estimated as $U_{cr}=\beta _{c}U_{crc}+(1-\beta _{c})u_{crw}{\text{ where }}\beta _{c}=U/(U+u_{w})$ is a blending factor. The critical depth-averaged current velocity (U_crc) is given by

$U_{crc}=\left\{{\begin{aligned}&0.19d_{50}^{0.1}log_{10}\left({\frac {4h}{d_{90}}}\right),\ for\ 0.1\leq d_{50}\leq 0.5mm\\&8.5d_{50}^{0.6}log_{10}\left({\frac {4h}{d_{90}}}\right),\ for\ 0.5\leq d_{50}\leq 2.0mm\end{aligned}}\right.$

(27)

, and the critical bottom-wave-orbital velocity amplitude (u_crw) is given

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

(28)

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

The van Rijn formulas (1984 a,b; 2007 a,b) were originally proposed for well-sorted sediments. The sediment availability is included by multiplication of transport rates with the fraction of the sediment size class in the upper bed layer. The hiding and exposure are considered by a correction factor which multiples to the critical velocity. When applied to multiple-sized sediments, the fractional equilibrium transport rates are calculated as

$q_{bk*}=f_{b}\rho _{s}p_{1k}0.015Uh\left({\frac {U_{e}-\xi _{k}^{1/2}U_{crk}}{\sqrt {(s-1)gd_{k}}}}\right)^{1.5}\left({\frac {d_{k}}{h}}\right)^{1.2}$

(29)

$q_{sk*}=f_{s}\rho _{s}p_{1k}0.012Uh\left({\frac {U_{e}-\xi _{k}^{1/2}U_{crk}}{\sqrt {(s-1)gd_{k}}}}\right)^{2.4}\left({\frac {d_{k}}{h}}\right)d_{*k}^{-0.6}$

(30)

The availability of sediment fractions is included through p_1k, while hiding and exposure of grain sizes are accounted for by multiplying the critical velocity (U_crk) by a correction function ($\zeta _{k}^{1/2}$).

Watanabe

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

$q_{t*}=\left[f_{s}r_{s}+f_{b}(1-r_{s})\right]\rho _{s}A_{Wat}U\left({\frac {\tau _{bmax}-\tau _{cr}}{\rho g}}\right)$

(31)

where:

q_t* = potential total-load transport rate [kg/m/s]

r_s = fraction of suspended load defined by $r_{sk}={\frac {q_{sk}}{q_{t}k}}\simeq {\frac {q_{sk*}}{q_{tk*}}}$ [-]

$\tau _{bmax}$ = combined wave-current maximum shear stress [Pa]

$\tau _{cr}$ = critical shear stress of incipient motion [Pa]

A_Wat = empirical coefficient typically ranging from 0.1 to 2.0 [-].

The critical shear stress is determined from

$\Theta _{cr}={\frac {0.3}{1+1.2d_{*}}}+0.55\left[1-exp(-0.02d_{*})\right]$

(32)

and

${\frac {\tau _{cr}}{g(\rho _{s}-\rho )d}}=\Theta _{cr}$

(33)

The maximum bed shear stress ($\tau _{bmax}$) is calculated as (Soulsby 1997)

$\tau _{bmax}={\sqrt {(\tau _{b}+\tau _{w}\cos {\varphi })^{2}+(\tau _{w}\sin {\varphi })^{2}}}$

(34)

where $\varphi$ is the angle between the waves and current; $\tau _{b}$ is the mean shear stress due to waves and currents; and $\tau _{w}$ is the wave-related bed shear stress which is calculated here using

$\tau _{w}={\frac {1}{2}}\rho f_{w}U_{w}^{2}$

(35)

and

$f_{w}=exp(5.5r^{-0.2}-6.3)\quad \quad \quad (Nielsen1992)$

(36)

The fraction of suspended sediment (r_s) is estimated using the van Rijn (2007 a,b) transport equations described above. Besides being needed in the total-load transport equation

${\frac {\partial }{\partial t}}\left({\frac {hC_{tk}}{\beta _{tk}}}\right)+{\frac {\partial (hU_{j}C_{tk})}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}\left[v_{s}h{\frac {\partial (r_{sk}C_{tk})}{\partial x_{j}}}\right]+\alpha _{t}\omega _{sk}(C_{tk*}-C_{tk})$

(37)

, it also allows the application of the bed- and suspended-load scaling factors in a way similar to all other transport formula.

The Watanabe (1987) transport formula is modified for multiple-sized sediments as

$q_{tk*}=\left[f_{s}r_{sk}+f_{b}(1-r_{sk})\right]\rho _{s}p_{1k}A_{Wat}U\left({\frac {\tau _{bmax}-\xi _{k}\tau _{crk}}{\rho g}}\right)$

(38)

where the subscript k indicates the sediment size class.

Soulsby-van Rijn

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

$q_{b*}=f_{b}\rho _{s}0.005Uh\left({\frac {U_{e}-U_{crc}}{\sqrt {(s-1)gd_{50}}}}\right)^{2.4}\left({\frac {d_{50}}{h}}\right)^{1.2}$

(39)

$q_{s*}=f_{s}\rho _{s}0.012Uh\left({\frac {U_{e}-U_{crc}}{\sqrt {(s-1)gd_{50}}}}\right)^{2.4}\left({\frac {d_{50}}{h}}\right)d_{*}^{-0.6}$

(40)

where:

$U_{e}={\sqrt {U^{2}+{\frac {0.018}{c_{b}}}u_{rms}^{2}}}$ = effective velocity [m/s]

u_rms= root-mean-square bottom wave orbital [m/s]

U_crc = critical depth-averaged velocity for initiation of motion for currents based on Van Rijn (1984c) [m/s].

The bed friction coefficient (c_b) is calculated using

$c_{b}=\left[{\frac {\kappa }{ln(z_{0}/h)+1}}\right]^{2}$

(41)

with the bed roughness length (z₀) set to 0.006 m following Soulsby (1997).

The Soulsby-van Rijn formula is modified for multiple-sized sediments similarly to the van Rijn formula in the previous section with the equation

$q_{bk*}=f_{b}\rho _{s}p_{1k}0.005Uh\left({\frac {U_{e}-\xi _{k}^{1/2}U_{crk}}{\sqrt {(s-1)gd_{k}}}}\right)^{2.4}\left({\frac {d_{k}}{h}}\right)^{1.2}$

(42)

$q_{sk*}=f_{s}\rho _{s}p_{1k}0.012Uh\left({\frac {U_{e}-\xi _{k}^{1/2}U_{crk}}{\sqrt {(s-1)gd_{k}}}}\right)^{2.4}\left({\frac {d_{k}}{h}}\right)d_{*k}^{-0.6}$

(43)

As in the case of the van Rijn transport formula, the availability of sediment fractions is included through p_1k, while hiding and exposure of grain sizes is accounted for by multiplying the critical velocity (U_crk) by a correction function ($\xi _{k}^{1/2}$). It is noted that the Soulsby-van Rijn (Soulsby 1997) formulas are very similar to the van Rijn (1984a,b; 2007a,b) except for the definition of the effective velocity and the recalibration of the bed-load formula coefficients in van Rijn (2007a). The proposed changes for multiple-sized sediments should be verified with measurements or numerical simulations for non-uniformly-sized sediment transport.

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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. 1984c. Sediment transport, Part III: Bed forms and alluvial roughness. Journal of Hydraulic Engineering, ASCE 110(12):1733–1754.
• 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.
• Wu, W., and Q. Lin. 2011. Extension of the Lund-CIRP formula for multiple-sized sediment transport under currents and waves. Oxford, MS: The University of Mississippi, National Center for Computational Hydroscience and Engineering.

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