Acronym1R
"Acronym1_R” combines Acronym1 scheme with a Manning-Strickler relation for flow resistance. It first computes a value of u∗ from specified values of the water discharge Qw, the channel width B and the bed slope H. It then implements the same code as “Acronym1”.
Model introduction
“Acronym1_R” is used for computing the volume bedload transport rate per unit width and bedload grain size distribution from a specified surface grain size distribution (with sand removed) (Db,i, Ff,i), i = 1..N+1, a specific gravity of the sediment (here equal to R + 1), a water discharge Q, a channel width B and a streamwise bed slope S.
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)
[math]\displaystyle{ \tau_{b} B = \rho u_{*} ^2 B = \rho g S \left ( H B - H ^2 \right ) }[/math] (26)
[math]\displaystyle{ {\frac{U}{u_{*}}} = \alpha _{r} \left ( {\frac{H}{k_{s}}} \right )^ {\frac{1}{6}} }[/math] (27)
[math]\displaystyle{ U = {\frac{Q}{BH}} }[/math] (28)
[math]\displaystyle{ H = \left ({\frac{k_{s} ^ {\frac{1}{3}} Q ^2}{\alpha_{r} ^2 g B^2 S}} \right )^ {\frac{3}{10}} c_{1} }[/math] (29)
[math]\displaystyle{ c_{1}= \left ( 1 - {\frac{H}{B}} \right ) ^ \left ({\frac{-3}{10}}\right ) }[/math] (30)
[math]\displaystyle{ k_{s} = n_{k} D_{s90} }[/math] (31)
| 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, R + 1 | - |
| g | acceleration of gravity | m / s2 |
| τb | boundary shear stress on the bed | kg / (m s) |
| u | shear velocity on the bed | 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 |
| Q | water discharge | m3 |
| B | channel width | m |
| S | a streamwise bed slope | - |
| n | roughness factor | - |
Output
| Symbol | Description | Unit |
|---|---|---|
| qbT | total volume gravel bedload transport rate per unit width summed over all sizes | - |
| τg | shields number | - |
| H | water depth | m |
| u* | shear velocity | m / s |
| Dg | geometric mean | mm |
| 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.
The channel is assumed to be rectangular, with the vertical sidewalls having the same roughness as the bed. The roughness height ks is computed using equation 31. Here nk is a user-specified dimensionless roughness factor. The author suggests a value of 2 for nk.
Depth is computed according to the relation for momentum balance in the bed region using equation 26 applicable to normal (steady, streamwise uniform) flow. Flow resistance on the bed region is computed using a Manning-Strickler resistance relation (equation 27).
Equation 29, 30 is solved iteratively for H in the code. Once H is known, the shear velocity u∗ is computed from equation 26, and the calculation proceeds using the same algorithm as “Acronym1”. It is implicitly assumed in the calculation that all of the boundary shear stress consists of skin friction, with form drag neglected.
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.
