CMS-Flow:Equilibrium Concentrations and Transport Rates: Difference between revisions

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{{Equation|<math>C_{tk*} = p_{1k}C_{tk}^*</math>|2}}
{{Equation|<math>C_{tk*} = p_{1k}C_{tk}^*</math>|2}}


where p<sub>id</sub>is the fraction of the sediment size (k) in the first (top) bed layer, and C<sub>tk</sub><sup>*</sup> is the potential equilibrium total-load concentration. The potential concentration (C<sub>tk</sub><sup>*</sup>) can be interpreted as the equilibrium concentration for uniform sediment of size d<sub>k</sub>. The above equation is essential for the coupling of sediment transport, bed change, and bed sorting equations.
where p<sub>1k</sub>is the fraction of the sediment size (k) in the first (top) bed layer, and C<sub>tk</sub><sup>*</sup> is the potential equilibrium total-load concentration. The potential concentration (C<sub>tk</sub><sup>*</sup>) can be interpreted as the equilibrium concentration for uniform sediment of size d<sub>k</sub>. The above equation is essential for the coupling of sediment transport, bed change, and bed sorting equations.


=Lund-CIRP=
=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 s given by  
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  


{{Equation|<math> \frac{q_{b}}{\sqrt{(s-1) g d_{50}^3}} = f_b \rho_s \sqrt{\theta_c} \theta_{cw,m}\exp{  \biggl (-4.5 \frac{\theta_{cr}}{\theta_{cw}}} \biggr )</math>|3}}
{{Equation|<math> \frac{q_{b*}}{\sqrt{(s-1) g d_{50}^3}} = f_b \rho_s 12\sqrt{\Theta_c} \Theta_{cw,m}\exp{  \biggl (-4.5 \frac{\Theta_{cr}}{\Theta_{cw}}} \biggr )</math>|3}}


{{Equation|<math> \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]
{{Equation|<math> \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]</math>|4}}
</math>|4}}


where:
where:
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:f<sub>s</sub> = suspended-load scaling factor (default 1.0) [-].
:f<sub>s</sub> = suspended-load scaling factor (default 1.0) [-].


The critical Shields parameter is calculated using Equation (2-100). The mean and maximum Shields parameters are calculated as
The critical Shields parameter is calculated using  


{{Equation|<math>\Theta_{cw,m} = \sqrt{\Theta_c ^2 + \Theta_{w,m}^2 + 2\Theta_c \Theta_{w,m}cos\varphi}</math>|5}}
{{Equation|<math>\Theta_{cr} = \frac{0.3}{1 + 1.2d_*} + 0.055 \left[1 - exp(-0.2d_*)\right]</math>|5}}


{{Equation|<math>\Theta_{cw} = \sqrt{\Theta_c ^2 + \Theta_w ^2 + 2\Theta_c \Theta_w cos\varphi}</math>|6}}
The mean and maximum Shields parameters are calculated as


The mean wave Shields parameter is calculated as <math>\Theta_{w,m} = \Theta_w /2</math>assuming a sinusoidal wave. The Shields parameters for currents and waves are given by
{{Equation|<math>\Theta_{cw,m} = \sqrt{\Theta_c ^2 + \Theta_{w,m}^2 + 2\Theta_c \Theta_{w,m}cos\varphi}</math>|6}}


{{Equation|<math>\Theta_{c|w} = \frac{\tau_{cw}}{g(\rho_s - \rho)d}</math>|7}}
{{Equation|<math>\Theta_{cw} = \sqrt{\Theta_c ^2 + \Theta_w ^2 + 2\Theta_c \Theta_w cos\varphi}</math>|7}}


in which the subscript c|w indicates either the current- (c) or wave-related (w) component. The current-related shear stress (<math>\tau_c</math>) is calculated with Equation (2-11). The wave-related bed shear stress is calculated with Equation (2-12) and the wave friction factor (f<sub>w</sub>) of Swart (1974) given by Equation (2-15).
The mean wave Shields parameter is calculated as <math>\Theta_{w,m} = \Theta_w /2</math> assuming a sinusoidal wave. The Shields parameters for currents and waves are given by
 
{{Equation|<math>\Theta_{c|w} = \frac{\tau_{c|w}}{g(\rho_s - \rho)d}</math>|8}}
 
in which the subscript c|w indicates either the current- (c) or wave-related (w) component. The current-related shear stress (<math>\tau_c</math>) is calculated with  
 
{{Equation|<math>\tau_c = \rho c_b U^2</math>|9}}
 
The wave-related bed shear stress is calculated with  
 
{{Equation|<math>\tau_w = \frac{1}{2}\rho f_w u_w ^2</math>|10}}
 
and the wave friction factor (f<sub>w</sub>) of Swart (1974) is given by  
 
{{Equation|<math>f_w =\left\{\begin{align}
&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{align}
\right.
</math>|11}}


The total bed roughness is assumed to be a linear summation of the grain-related roughness (k<sub>sg</sub>), form-drag (ripple) roughness (k<sub>sr</sub>), and sediment-related roughness (k<sub>ss</sub>):
The total bed roughness is assumed to be a linear summation of the grain-related roughness (k<sub>sg</sub>), form-drag (ripple) roughness (k<sub>sr</sub>), and sediment-related roughness (k<sub>ss</sub>):


{{Equation|<math>k_{s,c|w} = k_{sg} + k_{sr,c|w} + k_{ss,c|w}</math>|8}}
{{Equation|<math>k_{s,c|w} = k_{sg} + k_{sr,c|w} + k_{ss,c|w}</math>|12}}


Here, the grain-related roughness is estimated as k<sub>sg</sub> = 2d<sub>50</sub> The ripple roughness (k<sub>sr</sub>) is calculated as (Soulsby 1997)
Here, the grain-related roughness is estimated as k<sub>sg</sub> = 2d<sub>50</sub> The ripple roughness (k<sub>sr</sub>) is calculated as (Soulsby 1997)


{{Equation|<math>k_{sr,c|w} = 7.5 \frac{H^2 _{r,c|w}}{L_{r,c|w}}</math>|9}}
{{Equation|<math>k_{sr,c|w} = 7.5 \frac{H^2 _{r,c|w}}{L_{r,c|w}}</math>|13}}


where H<sub>r</sub> and L<sub>r</sub> are the ripple height and length, respectively.
where H<sub>r</sub> and L<sub>r</sub> are the ripple height and length, respectively.
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The current- and wave-related sediment roughnesses are estimated as
The current- and wave-related sediment roughnesses are estimated as


{{Equation|<math>k_{ss,c|w} = 5d_{50}\Theta_{c|w}</math>|10}}
{{Equation|<math>k_{ss,c|w} = 5d_{50}\Theta_{c|w}</math>|14}}


The above equation must be solved simultaneously with the expressions for the bottom shear stress because the roughness depends on the stress.
The above equation must be solved simultaneously with the expressions for the bottom shear stress because the roughness depends on the stress.
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The reference concentration is given by
The reference concentration is given by


{{Equation|<math>c_R = A_{cR} \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}}  \biggr)</math>|11}}
{{Equation|<math>c_R = A_{cR}\Theta_{cw,m} \exp{ \biggl(- 4.5 \frac{\Theta_{cr}}{\Theta_{cw}}}  \biggr)</math>|15}}


where the coefficient <math>A_{cR}</math> is determined by the following relationship:
where the coefficient <math>A_{cR}</math> is determined by the following relationship:


{{Equation|<math>A_{cR} = 0.0035 \exp{ \bigl( - 0.3 d_{*} } \bigr) </math>|12}}
{{Equation|<math>A_{cR} = 0.0035 \exp{ \bigl( - 0.3 d_{*} } \bigr) </math>|16}}


The vertical sediment diffusivity is calculated as
The vertical sediment diffusivity is calculated as


{{Equation|<math>\epsilon = h \left(\frac{D_e}{\rho}  \right)^{1/3}</math>|13}}
{{Equation|<math>\epsilon = h \left(\frac{D_e}{\rho}  \right)^{1/3}</math>|17}}


where D<sub>e</sub> is the total effective dissipation given by
where D<sub>e</sub> is the total effective dissipation given by


{{Equation|<math>D_e = k_b ^3 D_{br} + k_c ^3 D_c + k_w ^3 D_w</math>|14}}
{{Equation|<math>D_e = k_b ^3 D_{br} + k_c ^3 D_c + k_w ^3 D_w</math>|18}}


in which k<sub>b</sub>, k<sub>c</sub>, and k<sub>w</sub> are coefficients; D<sub>br</sub> is the wave breaking dissipation (from the wave model); and D<sub>c</sub> and D<sub>w</sub> are the bottom friction dissipation due to currents and waves, respectively. The dissipation from bottom friction due to current (D<sub>c</sub>) and the dissipation from bottom friction due to waves (D<sub>w</sub>) are expressed as
in which k<sub>b</sub>, k<sub>c</sub>, and k<sub>w</sub> are coefficients; D<sub>br</sub> is the wave breaking dissipation (from the wave model); and D<sub>c</sub> and D<sub>w</sub> are the bottom friction dissipation due to currents and waves, respectively. The dissipation from bottom friction due to current (D<sub>c</sub>) and the dissipation from bottom friction due to waves (D<sub>w</sub>) are expressed as


{{Equation|<math>D_{c|w} = \tau_{c|w}u_{*c|w}</math>|15}}
{{Equation|<math>D_{c|w} = \tau_{c|w}u_{*c|w}</math>|19}}


The coefficient k<sub>b</sub>=0.017 (Camenen and Larson 2008), and k<sub>c</sub> and k<sub>w</sub> are a function of the Schmidt number:
The coefficient k<sub>b</sub>=0.017 (Camenen and Larson 2008), and k<sub>c</sub> and k<sub>w</sub> are a function of the Schmidt number:


{{Equation|<math>k_{c|w} = \frac{\kappa}{6} \sigma_{*c|w}</math>|16}}
{{Equation|<math>k_{c|w} = \frac{\kappa}{6} \sigma_{c|w}</math>|20}}


where <math>\sigma_{c|w}</math>is either the current- or wave-related Schmidt number calculated from the following relationships (Camenen and Larson 2008):
where <math>\sigma_{c|w}</math> is either the current- or wave-related Schmidt number calculated from the following relationships (Camenen and Larson 2008):


{{Equation|<math>\sigma_{c|w} =
{{Equation|<math>\sigma_{c|w} =
\left{
\left\{
 
\begin{align}
\begin{align}
&a_{c|w} + b_{c|w} sin^2  
&a_{c|w} + b_{c|w} sin^2  
\left( \frac{\pi}{2} \frac{\omega_s}{u_{*c|w}} \right)  
\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{align}
\right.
</math>|21}}
 
with the coefficients a<sub>c</sub> = 0.4, b<sub>c</sub> = 3.5, a<sub>w</sub> =0.15, and b<sub>w</sub> = 1.5.
 
For multiple-sized (non-uniform) sediments, the fractional equilibrium sediment transport rates are calculated as (Wu and Lin 2011)


for \frac{\omega_s}{u_{*c|w}} \leq 1 \\
{{Equation|<math>\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)</math>|22}}


&1 + (a_{c|w} + b_{c|w} - 1)sin^2
{{Equation|<math>\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]</math>|23}}


\left(\frac{\pi}{2} \frac{u_{*c|w}}{\omega_s}\right)
where:
 
: k = subscript indicating the sediment size class
 
: <math>\xi_k</math> = hiding and exposure function [-]
 
: r<sub>sk</sub> = fraction of suspended load for each size class defined by <math>r_{sk} = \frac{q_{sk}}{q_tk} \simeq \frac{q_{sk*}}{q_{tk*}}</math> where q<sub>sk</sub> and q<sub>tk</sub> are the actual suspended- and total-load transport rates and q<sub>sk*</sub> and q<sub>tk*</sub> are the equilibrium suspended- and total-load transport rates.
 
: p<sub>1k</sub> = fraction of the k<sup>th</sup> sediment size in the first layer [-].
 
The availability of sediment fractions is included through p<sub>1k</sub>, 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
 
{{Equation|<math>q_{b*} = f_b \rho_s 0.015 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} </math>|24}}


for \frac{\omega_s}{u_{*c|w}} > 1
\end{align}


\right}
{{Equation|<math>  q_{s*} = f_s \rho_s 0.012 U d_{50}
  \biggl( \frac{U_e - U_{cr} }{ \sqrt{(s-1) g d_{50}}} \biggr)^{2.4}  d_{*}^{-0.6} </math>|25}}


</math>|17}}
where:


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


<math>U_e</math> = effective depth averaged velocity [m/s]


The effective depth-averaged velocity is calculated as <math>U_e = U + \gamma u_w</math> with <math>\gamma </math>= 0.4 for random waves and <math>\gamma</math> = 0.8 for regular waves. The bottom wave orbital velocity based on linear wave theory is u<sub>w</sub>. For random waves, u<sub>w</sub> = u<sub>ws</sub> where u<sub>ws</sub> is based on the significant wave height and peak wave period


{{Equation|<math>u_{ws} = \frac{\pi H_s}{T_p sinh(kh)}</math>|26}}


The critical depth-averaged velocity is


= van Rijn =
estimated as <math>U_{cr} = \beta_c U_{crc} + (1 - \beta_c )u_{crw} \text{ where } \beta_c = U / (U + u_w) </math> is a blending factor. The critical depth-averaged current velocity (U<sub>crc</sub>) is given by
The van Rijn (1984ab) transport equations are used with the recalibrated coefficients of van Rijn (2007ab) are given by
{{Equation|<math>
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}
</math>|8}}


{{Equation|<math>U_{crc} =
\left\{
\begin{align}
&0.19 d_{50}^{0.1} log_{10} \left(\frac{4h}{d_{90}}\right), \ for \ 0.1 \leq d_{50} \leq 0.5 mm \\
&8.5 d_{50}^{0.6} log_{10} \left(\frac{4h}{d_{90}}\right), \ for \ 0.5 \leq d_{50} \leq 2.0 mm
\end{align}
\right.
</math>|27}}




, and the critical bottom-wave-orbital velocity amplitude (u<sub>crw</sub>) is given


{{Equation|<math>U_{crw} =\left\{
\begin{align}
&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 2.0 mm
\end{align}
\right.
</math>|28}}


{{Equation|<math>
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.
  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}
</math>|9}}


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


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.
{{Equation|<math>q_{bk*} = f_b \rho_s p_{1k} 0.015Uh \left(\frac{U_e - \xi_k ^{1/2}U_{crk}}{\sqrt{(s-1)g d_k}}\right)^{1.5} \left(\frac{d_k}{h}  \right)^{1.2}</math>|29}}


The critical velocity is estimated as
{{Equation|<math>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}</math>|30}}
{{Equation|<math>U_{cr} = \beta U_{crc} + (1-\beta) U_{crw} </math>|10}}


where <math>U_{crc}</math> and <math>U_{crw}</math> 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).
The availability of sediment fractions is included through p<sub>1k</sub>, while hiding and exposure of grain sizes are accounted for by multiplying the critical velocity (U<sub>crk</sub>) by a correction function (<math>\zeta_k^{1/2}</math>).


= Watanabe =
= Watanabe =
The equilibrium total load sediment transport rate of Watanabe (1987) is given by
The equilibrium total load sediment transport rate of Watanabe (1987) is given by
{{Equation|<math>
q_{t} = A_w \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U }{\rho g } \biggr]
</math>|13}}


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.
{{Equation|<math>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)</math>|31}}
 
where:
 
:q<sub>t*</sub> = potential total-load transport rate [kg/m/s]
:r<sub>s</sub> = fraction of suspended load defined by <math>r_{sk} = \frac{q_{sk}}{q_tk} \simeq \frac{q_{sk*}}{q_{tk*}}</math> [-]
:<math>\tau_{bmax}</math> = combined wave-current maximum shear stress [Pa]
:<math>\tau_{cr}</math> = critical shear stress of incipient motion [Pa]
:A<sub>Wat</sub> = empirical coefficient typically ranging from 0.1 to 2.0 [-].
   
   
The critical shear stress is determined using
The critical shear stress is determined from
{{Equation|<math>\tau_{cr} = (\rho_s - \rho) g d \phi_{cr} </math>|14}}


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>.
{{Equation|<math>
\Theta_{cr} = \frac{0.3}{1 + 1.2d_*} + 0.55 \left[1 - exp(-0.02d_*\right] </math>|32}}


If waves are present, the maximum bed shear stress <math>\tau_{b,max} </math> is calculated based on Soulsby (1997)
and
{{Equation|<math>\tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2  + (\tau_w \sin{\phi})^2 }</math>|15}}


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
{{Equation|<math>
{{Equation|<math>
  \tau_{m} = \tau_c \biggl[ 1 +  1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr]
\frac{\tau_{cr}}{g(\rho_s - \rho)d} = \Theta_{cr}
</math>|16}}
</math>|33}}  
 


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 maximum bed shear stress (<math>\tau_{bmax}</math>) is calculated as (Soulsby 1997)


The wave friction factor is calculated as (Nielsen 1992) <math>f_w = \exp{5.5R^{-0.2}-6.3}</math> 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>.
{{Equation|<math>\tau_{bmax} = \sqrt{(\tau_b + \tau_w \cos{\varphi})^2  + (\tau_w \sin{\varphi})^2 }</math>|34}}
 
where <math>\varphi</math> is the angle between the waves and current; <math>\tau_b</math> is the mean shear stress due to waves and currents; and <math>\tau_w</math> is the wave-related bed shear stress which is calculated here using
 
{{Equation|<math>\tau_w = \frac{1}{2}\rho f_w U_w ^2</math>|35}}
 
and
 
{{Equation|<math>f_w = exp(5.5 r^{-0.2} - 6.3)\quad\quad\quad (Nielsen 1992)</math>|36}}
 
The fraction of suspended sediment (r<sub>s</sub>) is estimated using the van Rijn (2007 a,b) transport equations described above. Besides being needed in the total-load transport equation
 
{{Equation|<math>\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} )</math>|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
 
{{Equation|<math>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)</math>|38}}
 
where the subscript ''k'' indicates the sediment size class.


= Soulsby-van Rijn =
= Soulsby-van Rijn =
Soulsby (1997) proposed the following equation for the total load sediment transport rate under currents and waves
{{Equation|<math>
  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}
</math>|20}}


where <math>U_{rms}</math> is the root-mean-squared wave orbital velocity, and <math>C_d</math> is the drag coefficient due to currents alone and the coefficient <math> A_{s} = A_{sb} + A_{ss} </math>. The coefficients <math>A_{sb}</math> and <math>A_{ss}</math> are related to the bed and suspended transport loads respectively and are given by
Soulsby (1997) proposed the following equation for the total load sediment transport rate under currents and waves:
{{Equation|<math>A_{sb} = \frac{ 0.005 h (d_{50}/h)^{1.2} }{ [(s-1)g d_{50} ]^{1.2} } </math>|21}}
 
{{Equation|<math>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}
</math>|39}}
 
{{Equation|<math>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}
</math>|40}}
 
 
where:
 
<math>U_e = \sqrt{U^2 + \frac{0.018}{c_b} u^2 _{rms}}</math> = effective velocity [m/s]
 
u<sub>rms</sub>= root-mean-square bottom wave orbital [m/s]
 
U<sub>crc</sub> = critical depth-averaged velocity for initiation of motion for currents based on Van Rijn (1984c) [m/s].
 
The bed friction coefficient (c<sub>b</sub>) is calculated using
 
{{Equation|<math>c_b = \left[\frac{\kappa}{ln(z_0 /h) + 1} \right]^2</math>|41}}
 
with the bed roughness length (z<sub>0</sub>) set to 0.006 m following Soulsby (1997).


{{Equation|<math>A_{s} = \frac{ 0.012 d_{50} D_{*}^{-0.6} }{ [(s-1)g d_{50} ]^{1.2} } </math>|22}}
The Soulsby-van Rijn formula is modified for multiple-sized sediments similarly to the van Rijn formula in the previous section with the equation


The current drag coefficient is calcualted as
{{Equation|<math>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}</math>|42}}
{{Equation|<math>C_d = \biggl[ \frac{0.4}{\ln{(h/z_0)}-1 } \biggr]^2 </math>|23}}


with a constant bed roughness length <math>z_0 </math> set to 0.006 m.
{{Equation|<math>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}</math>|43}}


As in the case of the van Rijn transport formula, the availability of sediment fractions is included through p<sub>1k</sub>, while hiding and exposure of grain sizes is accounted for by multiplying the critical velocity (U<sub>crk</sub>) by a correction function (<math>\xi_k ^{1/2}</math>). 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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{| border="1"
Line 216: Line 308:
* van Rijn, L. C. (1984a). "Sediment transport. Part I: Bed load transport", Journal of Hydraulic Engineering, 110(10), 1431–1456.
* 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. (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., (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.
* 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.
* 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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Latest revision as of 19:31, 18 February 2015

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 (Ctk*) must be estimated from an empirical formula. The depth-averaged equilibrium concentration is defined as

  (1)

where qtk* is the total-load transport for the kth sediment size class estimated from an empirical formula. For convenience, Ctk* is written in general form as

  (2)

where p1kis the fraction of the sediment size (k) in the first (top) bed layer, and Ctk* is the potential equilibrium total-load concentration. The potential concentration (Ctk*) can be interpreted as the equilibrium concentration for uniform sediment of size dk. 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

  (3)
  (4)

where:

qb* = equilibrium bed-load transport rate [kg/m/s]
qs* = equilibrium suspended-load transport rate [kg/m/s]
= Shields parameters due to currents [-]
= mean Shields parameters due to waves and currents [-]
= maximum Shields parameters due to waves and currents [-]
= critical Shields parameter [-]
= vertical sediment diffusivity [m2/s]
cR = reference bed concentration [kg/m3]
fb= bed-load scaling factor (default 1.0) [-]
fs = suspended-load scaling factor (default 1.0) [-].

The critical Shields parameter is calculated using

  (5)

The mean and maximum Shields parameters are calculated as

  (6)
  (7)

The mean wave Shields parameter is calculated as assuming a sinusoidal wave. The Shields parameters for currents and waves are given by

  (8)

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

  (9)

The wave-related bed shear stress is calculated with

  (10)

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

  (11)

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

  (12)

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

  (13)

where Hr and Lr are the ripple height and length, respectively.

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

  (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

  (15)

where the coefficient is determined by the following relationship:

  (16)

The vertical sediment diffusivity is calculated as

  (17)

where De is the total effective dissipation given by

  (18)

in which kb, kc, and kw are coefficients; Dbr is the wave breaking dissipation (from the wave model); and Dc and Dw are the bottom friction dissipation due to currents and waves, respectively. The dissipation from bottom friction due to current (Dc) and the dissipation from bottom friction due to waves (Dw) are expressed as

  (19)

The coefficient kb=0.017 (Camenen and Larson 2008), and kc and kw are a function of the Schmidt number:

  (20)

where is either the current- or wave-related Schmidt number calculated from the following relationships (Camenen and Larson 2008):

  (21)

with the coefficients ac = 0.4, bc = 3.5, aw =0.15, and bw = 1.5.

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

  (22)
  (23)

where:

k = subscript indicating the sediment size class
= hiding and exposure function [-]
rsk = fraction of suspended load for each size class defined by where qsk and qtk are the actual suspended- and total-load transport rates and qsk* and qtk* are the equilibrium suspended- and total-load transport rates.
p1k = fraction of the kth sediment size in the first layer [-].

The availability of sediment fractions is included through p1k, 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

  (24)


  (25)

where:

= the critical depth-averaged velocity for incipient motion [m/s],

= effective depth averaged velocity [m/s]

The effective depth-averaged velocity is calculated as  with = 0.4 for random waves and = 0.8 for regular waves. The bottom wave orbital velocity based on linear wave theory is uw. For random waves, uw = uws where uws is based on the significant wave height and peak wave period

  (26)

The critical depth-averaged velocity is

estimated as is a blending factor. The critical depth-averaged current velocity (Ucrc) is given by

  (27)


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

  (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

  (29)
  (30)

The availability of sediment fractions is included through p1k, while hiding and exposure of grain sizes are accounted for by multiplying the critical velocity (Ucrk) by a correction function ().

Watanabe

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

  (31)

where:

qt* = potential total-load transport rate [kg/m/s]
rs = fraction of suspended load defined by [-]
= combined wave-current maximum shear stress [Pa]
= critical shear stress of incipient motion [Pa]
AWat = empirical coefficient typically ranging from 0.1 to 2.0 [-].

The critical shear stress is determined from

  (32)

and

  (33)


The maximum bed shear stress () is calculated as (Soulsby 1997)

  (34)

where is the angle between the waves and current; is the mean shear stress due to waves and currents; and is the wave-related bed shear stress which is calculated here using

  (35)

and

  (36)

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

  (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

  (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:

  (39)
  (40)


where:

= effective velocity [m/s]

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

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

The bed friction coefficient (cb) is calculated using

  (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

  (42)
  (43)

As in the case of the van Rijn transport formula, the availability of sediment fractions is included through p1k, while hiding and exposure of grain sizes is accounted for by multiplying the critical velocity (Ucrk) by a correction function (). 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.


Symbol Description Units
Bed load transport rate m3/s
Relative density -
Shields parameter due to currents -
Shields parameter due to waves and currents -
Critical shields parameter -
Empirical coefficient -
Empirical coefficient -
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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