CMS-Flow:Transport Formula: Difference between revisions

From CIRPwiki
Jump to navigation Jump to search
(Created page with "__NOTOC__ ==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. ...")
 
(Created page with __NOTOC__ ==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. ...)
Line 5: Line 5:
=== Bed load===
=== Bed load===
The current-related bed load transport with wave stirring is given by
The current-related bed load transport with wave stirring is given by
{{Equation|<math> \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 )  </math>|2=1}}
{{Equation|math  \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 )  /math|2=1}}


=== Suspended load ===
=== Suspended load ===
The current-related suspended load transport with wave stirring is given by
The current-related suspended load transport with wave stirring is given by
{{Equation|<math> \frac{q_s}{\sqrt{ (s-1) g d^3 }} = U c_R \frac{\epsilon}{w_s} \biggl[ 1  - \exp{ \biggl( - \frac{w_s d}{\epsilon}} \biggr) \biggr]  </math>|2=2}}
{{Equation|math  \frac{q_s}{\sqrt{ (s-1) g d^3 }} = U c_R \frac{\epsilon}{w_s} \biggl[ 1  - \exp{ \biggl( - \frac{w_s d}{\epsilon}} \biggr) \biggr]  /math|2=2}}


The reference sediment concentration is obtained from  
The reference sediment concentration is obtained from  
{{Equation|<math> c_R = A_{cR}  \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}}  \biggr)  </math>|2=3}}
{{Equation|math c_R = A_{cR}  \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}}  \biggr)  /math|2=3}}


where the coefficient <math>A_{cR}</math> is given by
where the coefficient mathA_{cR}/math is given by
{{Equation|<math> A_{cR} = 3.5x10^3 \exp{ \bigl( - 0.3 d_{*} } \bigr)  </math>|2=4}}
{{Equation|math A_{cR} = 3.5x10^3 \exp{ \bigl( - 0.3 d_{*} } \bigr)  /math|2=4}}


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


The sediment mixing coefficient is calculated as  
The sediment mixing coefficient is calculated as  
{{Equation|<math> \epsilon = h \biggl( \frac{k_b^3 D_b + k_c^3 D_c + k_w^3 D_w}{\rho} \biggr)^{1/3}  </math>|2=5}}
{{Equation|math \epsilon = h \biggl( \frac{k_b^3 D_b + k_c^3 D_c + k_w^3 D_w}{\rho} \biggr)^{1/3}  /math|2=5}}


== van Rijn ==
== van Rijn ==
Line 27: Line 27:
== 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 \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U_c }{\rho g } \biggr]  </math>|2=6}}
{{Equation|math q_{t*} = A \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U_c }{\rho g } \biggr]  /math|2=6}}
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.
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.
   
   
The critical shear stress is determined using  
The critical shear stress is determined using  
{{Equation|<math> \tau_{cr} = (\rho_s - \rho) g d \phi_{cr} </math>|2=6}}
{{Equation|math \tau_{cr} = (\rho_s - \rho) g d \phi_{cr} /math|2=6}}


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>.
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.


If waves are present, the maximum bed shear stress <math>\tau_{b,max} </math> is calculated based on Soulsby (1997)
If waves are present, the maximum bed shear stress math\tau_{b,max} /math is calculated based on Soulsby (1997)
{{Equation|<math> \tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2  + (\tau_w \sin{\phi})^2 } </math>|2=6}}
{{Equation|math \tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2  + (\tau_w \sin{\phi})^2 } /math|2=6}}
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  
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> \tau_{m} = \tau_c \biggl[ 1 +  1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr]  </math>|2=6}}
{{Equation|math \tau_{m} = \tau_c \biggl[ 1 +  1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr]  /math|2=6}}


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 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 wave friction factor is calculated as (Nielsen 1992) <math>f_w = \exp{5.5R^{-0.2}-6.3}</math> where
The wave friction factor is calculated as (Nielsen 1992) mathf_w = \exp{5.5R^{-0.2}-6.3}/math where


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>.
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.


== Soulsby-van Rijn ==
== Soulsby-van Rijn ==
The equilibrium sediment concentration is calculated as (Soulsby 1997)
The equilibrium sediment concentration is calculated as (Soulsby 1997)
{{Equation|<math> C_{*} = \frac{A_{sb}+A_{ss}}{h} \biggl[ \biggl( U_c^2 + 0.018  \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - u_{cr} \biggr]^{2.4}  </math>|2=7}}
{{Equation|math  C_{*} = \frac{A_{sb}+A_{ss}}{h} \biggl[ \biggl( U_c^2 + 0.018  \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - u_{cr} \biggr]^{2.4}  /math|2=7}}


----
----
{| border="1"
{| border=1
! Symbol !! Description !! Units
! Symbol !! Description !! Units
|-
|-
|<math> q_{bc} </math> || Bed load transport rate || m<sup>3</sup>/s
|math q_{bc} /math || Bed load transport rate || msup3/sup/s
|-
|-
|<math> s </math> ||  Relative density || -
|math s /math ||  Relative density || -
|-
|-
|<math> \theta_{c}  </math> || Shields parameter due to currents || -
|math \theta_{c}  /math || Shields parameter due to currents || -
|-
|-
|<math> \theta_{cw} </math> ||  Shields parameter due to waves and currents || -
|math \theta_{cw} /math ||  Shields parameter due to waves and currents || -
|-
|-
|<math> \theta_{cw}</math> ||  Critical shields parameter  || -
|math \theta_{cw}/math ||  Critical shields parameter  || -
|-
|-
|<math> a_c </math> || Empirical coefficient || -
|math a_c /math || Empirical coefficient || -
|-
|-
|<math> b_c </math> || Empirical coefficient || -
|math b_c /math || Empirical coefficient || -
|-
|-
|<math> U_c </math> || Current magnitude || m/s
|math U_c /math || Current magnitude || m/s
|}
|}


== References ==
== 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. (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. (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.  
* 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.  
* Soulsby, D.H. (1997). Dynamics of marine sands. A manual for practical applications, Thomas Telford Publications, London, England, 249 p.  
* 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.

Revision as of 01:16, 17 January 2011

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

  math \frac{q_{b ({{{2}}})

{\sqrt{(s-1)gd^3}} = a_c \sqrt{\theta_c} \theta_{cw}\exp{ \biggl ( -b_c \frac{\theta_{cr}}{\theta_{cw}}} \biggr ) /math|2=1}}

Suspended load

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

  math \frac{q_s}{\sqrt{ (s-1) g d^3 ({{{2}}})

= U c_R \frac{\epsilon}{w_s} \biggl[ 1 - \exp{ \biggl( - \frac{w_s d}{\epsilon}} \biggr) \biggr] /math|2=2}}

The reference sediment concentration is obtained from

  {{{1}}} ({{{2}}})

{\theta_{cw}}} \biggr) /math|2=3}}

where the coefficient mathA_{cR}/math is given by

  {{{1}}} (4)

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

The sediment mixing coefficient is calculated as

  {{{1}}} (5)

van Rijn

Watanabe

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

  {{{1}}} (6)

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.

The critical shear stress is determined using

  {{{1}}} (6)

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.

If waves are present, the maximum bed shear stress math\tau_{b,max} /math is calculated based on Soulsby (1997)

  {{{1}}} (6)

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

  {{{1}}} (6)

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 wave friction factor is calculated as (Nielsen 1992) mathf_w = \exp{5.5R^{-0.2}-6.3}/math where

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.

Soulsby-van Rijn

The equilibrium sediment concentration is calculated as (Soulsby 1997)

  {{{1}}} ({{{2}}})

{h} \biggl[ \biggl( U_c^2 + 0.018 \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - u_{cr} \biggr]^{2.4} /math|2=7}}

Symbol Description Units
math q_{bc} /math Bed load transport rate msup3/sup/s
math s /math Relative density -
math \theta_{c} /math Shields parameter due to currents -
math \theta_{cw} /math Shields parameter due to waves and currents -
math \theta_{cw}/math Critical shields parameter -
math a_c /math Empirical coefficient -
math b_c /math Empirical coefficient -
math U_c /math 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.
  • Watanabe, A. (1987). 3-dimensional numerical model of beach evolution, Proceedings Coastal Sediments '87, ASCE, 802-817.
Retrieved from "https://cirpwiki.info/index.php?title=CMS-Flow:Transport_Formula&oldid=7167"