Model help:Acronym1: Difference between revisions
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| shield | | shield number | ||
| kg / (m s) | | kg / (m s) | ||
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| σ<sub>g | |||
| geometric standard deviation | |||
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| diameter such that x% of the distribution is finer | |||
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Revision as of 15:36, 15 April 2011
Acronym1
The Acronym1 programs implement the Parker (1990a) surface-based bedload transport relation in order to compute gravel bedload transport rates.
Model introduction
The gravel is divided into N grain size ranges bounded by N+1 sizes Db,i, i = 1 to N+1. The grain size distribution of the surface (active) layer of the bed is specified in terms of the N+1 pairs (Db,i, Ff,i), i = 1..N+1, where Ff,i denotes the percent finer in the surface layer. Here Db,1 must be the coarsest size, such that Ff,1 = 100, and Db,N+1 must be the finest size, such that Ff,N+1 = 0.
The finest size must equal or exceed 2 mm. That is, the sand must be removed from the surface size distribution, and the fractions appropriately renormalized, in determining the surface grain size distribution to be input into Acronym1.
Model parameters
Uses ports
This will be something that the CSDMS facility will add
Provides ports
This will be something that the CSDMS facility will add
Main equations
- Grain Size:
[math]\displaystyle{ \Psi= LN_{2}\left (D\right) = {\frac{log_{10}\left (D\right)}{log_{10}\left (2\right)}} }[/math] (1)
[math]\displaystyle{ \Psi_{i}= LN_{2}\left ( D_{i}\right) = {\frac{log_{10}\left (D_{i}\right)}{log_{10}\left (2\right)}} }[/math] (2)
[math]\displaystyle{ D_{i}= Sqrt \left (D_{b, i} D_{b, i+1} \right ) }[/math] (3)
[math]\displaystyle{ F_{i}= \left ( F_{f, i} - F_{f, i+1} \right ) / 100 }[/math] (4)
[math]\displaystyle{ D_{sg}=2^\Psi_{s} }[/math] (5)
[math]\displaystyle{ \Psi_{s}= \Sigma \Psi_{i} F{i} }[/math] (6)
[math]\displaystyle{ \sigma_{sg}= 2 ^\sigma }[/math] (7)
[math]\displaystyle{ \sigma ^2= \Sigma \left (\Psi_{i} - \Psi \right )^2 F_{i} }[/math] (8)
[math]\displaystyle{ W_{i}^*= {\frac{Rgq_{bi}}{F_{i}u_{*} ^3}}= 0.00218 G \left (\Phi \right ) }[/math] (9)
[math]\displaystyle{ \phi= \omega \phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) }[/math] (10)
[math]\displaystyle{ \Phi= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} }[/math] (11)
[math]\displaystyle{ \tau_{sg} ^*={\frac{u_{*} ^2}{Rg D_{sg}}} }[/math] (12)
- φ> 1.59
[math]\displaystyle{ G \left ( \Phi \right )= 5474 \left ( 1 - {\frac{0.853}{\Phi}} \right ) ^ \left (4.5 \right ) }[/math] (13)
- 1<=φ<=1.59
[math]\displaystyle{ G\left (\Phi \right )= exp\left ( 14.2\left ( \Phi - 1\right ) - 9.28 \left ( \Phi - 1 \right )^2 \right ) }[/math] (14)
- φ< 1
[math]\displaystyle{ G\left (\Phi \right )= \Phi ^\left (14.2 \right ) }[/math] (15)
[math]\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \Phi_{sgo} \right ) }} \left ( \omega_{O} \left ( \Phi_{sgo} \right ) - 1 \right ) }[/math] (16)
[math]\displaystyle{ q_{bT}= \Sigma q_{bi} }[/math] (17)
[math]\displaystyle{ p_{i}= {\frac{q_{bi}}{q_{bT}}} }[/math] (18)
[math]\displaystyle{ D_{lg}= 2 ^\left (\psi_{l} \right ) }[/math] (19)
[math]\displaystyle{ \Psi_{l}= \Sigma \Psi_{i} p_{i} }[/math] (20)
[math]\displaystyle{ \delta_{lg}= 2 ^\left ( \delta_{l} \right ) }[/math] (21)
[math]\displaystyle{ \delta_{l} ^2= \Sigma \left ( \Psi_{i} - \Psi_{l} \right )^2 p_{i} }[/math] (22)
[math]\displaystyle{ D_{lx}= 2 ^\left (\Psi_{lx} \right ) }[/math] (23)
[math]\displaystyle{ \Psi_{lx}= \Psi_{b, i+1} + {\frac{\Psi_{b, j} - \Psi_{b, i+1}}{p_{f, i} - p_{f, i+1}}}\left ( x - p_{f, i+1} \right ) }[/math] (24)
[math]\displaystyle{ \Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) }[/math] (25)
Symbol | Description | Unit |
---|---|---|
Di | characteristic grain size (for i =1~N) | mm |
Fi | surface layer (for i =1~N) | - |
ψ | grain sizes | mm |
Dsg | geometric mean size of the surface material | mm |
σsg | geometric standard deviations | - |
σ | arithmetic standard deviations | - |
ρ | density of water | kg/m3 |
ρs | density of sediment | kg/m3 |
R | submerged specific gravity | - |
g | acceleration of gravity | m / s2 |
τb | boundary shear stress on the bed | kg / (m s) |
u | shear velocity of flow | m / s |
qbi | volume gravel bedload transport per unit width of grains in the ith size range | m 2 |
u* | shear velocity | m / s |
Dlg | geometric mean of the bedload | mm |
σlg | geometric standard deviation of the bedload | - |
Dsx | size in the surface material, such that x percentage of the material is finer | mm |
Dlx | size in the bedload material, such that x percentage of the material is finer | - |
Dsx | size in the surface material, such that x percentage of the material is finer | mm |
Output
Symbol | Description | Unit |
---|---|---|
qbT | total volume gravel bedload transport rate per unit width summed over all sizes | - |
Dg | geometric mean | mm |
τg | shield number | kg / (m s) |
σg | geometric standard deviation | - |
Dx | diameter such that x% of the distribution is finer | mm |
Notes
- Note on equations
In order to implement the equations 10~16, it is necessary to specify a) the surface grain size distribution (Df,i, Ff,i) and b) the shear velocity u∗. This results in a predicted values of qbi.
If the boundary shear stress at the bed includes a component of form drag, the component must be removed before computing u∗.
Once the parameters qbi are known the total volume bedload transport rate per unit width qbT and the fractions pi in the bedload can be calculated as equations 17, 18.
The results are presented in terms of qbT and the grain size distribution of the bedload, which is computed from the values of pi. These same fractions pi are used to compute the geometric mean and geometric standard deviation of the bedload Dlg and σlg, respectively, from the relations of equations 19, 20, 21, 22.
The percent finer in the bedload pf,i for the grain size Df,i is obtained from the fractions pi as follows: pf,1 = 100 pf,i = pf,i-1 - 100 pi-1 (i=2~N+1)
Let Dsx and Dlx denote sizes in the surface and bedload material, respectively, such that x percent of the material is finer. For example, if x = 50 then Ds50 and Dl50 denote the median sizes of the surface and bedload material, respectively. Once Ff,i is specified (pf,i is computed) the value Dsx (Dlx) can be computed by interpolation. The interpolation should be done using a logarithmic scale for grain size. For example, consider the computation of Dlx where pf,i ≤ x ≤ pf,i+1. Then we got equation 23, 24, using equation 25.
Examples
An example run with input parameters, BLD files, as well as a figure / movie of the output
Follow the next steps to include images / movies of simulations:
- Upload file: http://csdms.colorado.edu/wiki/Special:Upload
- Create link to the file on your page: [[Image:<file name>]].
See also: Help:Images or Help:Movies
Developer(s)
References
- Parker, G. 1990a Surface based bedload transport relation for gravel rivers. Journal of Hydraulic Research, 28(4), 417 436.
- Parker, G. 1990b The "ACRONYM" series of Pascal programs for computing bedload transport in gravel rivers. External Memorandum M 220, St. Anthony Falls Hydraulic Laboratory, University of Minnesota.