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
This program works the same way as Acronym1, except that it calculates the flow velocity with the user inputted parameters, and it outputs the water depth, H, and shear velocity, u*, in addition to what Acronym1 outputs.
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
- Characteristic grain size for the ith grain size range (spans (Db,i, Db,i+1)) (for i=1...N)
[math]\displaystyle{ D_{i}= \sqrt{ D_{b, i} D_{b, i+1} } }[/math] (1)
- Fraction in the surface layer Fi for the ith grain size range (for i=1...N)
[math]\displaystyle{ F_{i}= \left ( F_{f, i} - F_{f, i+1} \right ) / 100 }[/math] (2)
- Grain Size on the base-2 logarithmic scale:
[math]\displaystyle{ \Psi= LN_{2}\left (D\right) = {\frac{log_{10}\left (D\right)}{log_{10}\left (2\right)}} }[/math] (3)
- Geometric mean size of the surface material
[math]\displaystyle{ D_{sg}=2^\left (\bar\Psi_{s} \right ) }[/math] (4)
[math]\displaystyle{ \bar\Psi_{s}= \sum\limits_{i=1}^N \Psi_{i} F{i} }[/math] (5)
- The geometric standard deviations
[math]\displaystyle{ \sigma_{sg}= 2 ^\sigma }[/math] (6)
- the arithmetic standard deviations
[math]\displaystyle{ \sigma ^2= \sum\limits_{i=1}^N \left (\Psi_{i} - \bar\Psi \right )^2 F_{i} }[/math] (7)
- The transport relation
[math]\displaystyle{ W_{i}^*= {\frac{Rgq_{bi}}{F_{i}u_{*} ^3}}= 0.00218 G \left (\Phi \right ) }[/math] (8)
[math]\displaystyle{ \phi= \omega \Phi_{sgo} \left ( {\frac{D_{i}} {D_{sg}}} \right )^ \left (-0.0951 \right ) }[/math] (9)
[math]\displaystyle{ \Phi_{sgo}= {\frac{\tau_{sg} ^*}{\tau_{ssrg} ^*}} }[/math] (10)
[math]\displaystyle{ \tau_{sg} ^*={\frac{u_{*} ^2}{R g D_{sg}}} }[/math] (11)
- φ> 1.59
[math]\displaystyle{ G \left ( \Phi \right )= 5474 \left ( 1 - {\frac{0.853}{\Phi}} \right ) ^ \left (4.5 \right ) }[/math] (12)
- 1<=φ<=1.59
[math]\displaystyle{ G\left (\Phi \right )= exp[ 14.2\left ( \Phi - 1\right ) - 9.28 \left ( \Phi - 1 \right )^2 ] }[/math] (13)
- φ< 1
[math]\displaystyle{ G\left (\Phi \right )= \Phi ^\left (14.2 \right ) }[/math] (14)
[math]\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \Phi_{sgo} \right ) }} [ \omega_{O} \left ( \Phi_{sgo} \right ) - 1 ] }[/math] (15)
- total volume gravel bedload transport rate per unit width summed over all sizes
[math]\displaystyle{ q_{bT}= \sum\limits_{i=1}^N q_{bi} }[/math] (16)
- fraction of gravel bedload in the ith grain size range
[math]\displaystyle{ p_{i}= {\frac{q_{bi}}{q_{bT}}} }[/math] (17)
- Geometric mean of the bedload
[math]\displaystyle{ D_{lg}= 2 ^\left (\bar\psi_{l} \right ) }[/math] (18)
[math]\displaystyle{ \Psi_{l}= \sum\limits_{i=1}^\left (Np \right ) \Psi_{i} p_{i} }[/math] (19)
- Geometric standard deviation of the bedload
[math]\displaystyle{ \delta_{lg}= 2 ^\left ( \delta_{l} \right ) }[/math] (20)
[math]\displaystyle{ \delta_{l} ^2= \sum\limits_{i=1}^\left (Np \right ) \left ( \Psi_{i} - \bar\Psi_{l} \right )^2 p_{i} }[/math] (21)
- Grain sizes in the bedload material
[math]\displaystyle{ D_{lx}= 2 ^\left (\Psi_{lx} \right ) }[/math] (22)
[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] (23)
[math]\displaystyle{ \Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) }[/math] (24)
[math]\displaystyle{ \tau_{b} B = \rho u_{*} ^2 B = \rho g S \left ( H B - H ^2 \right ) }[/math] (25)
[math]\displaystyle{ {\frac{U}{u_{*}}} = \alpha _{r} \left ( {\frac{H}{k_{s}}} \right )^ {\frac{1}{6}} }[/math] (26)
[math]\displaystyle{ U = {\frac{Q}{BH}} }[/math] (27)
[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] (28)
[math]\displaystyle{ c_{1}= \left ( 1 - {\frac{H}{B}} \right ) ^ \left ({\frac{-3}{10}}\right ) }[/math] (29)
[math]\displaystyle{ k_{s} = n_{k} D_{s90} }[/math] (30)
| Symbol | Description | Unit |
|---|---|---|
| D | grain size | mm |
| Di | characteristic grain size for the ith grain size range (i=1...N) | mm |
| Fi | fraction in surface layer for the ith grain size range(for i =1~N) | - |
| τssrg * | equals to 0.0386 | - |
| ψ | grain sizes on the base-2 logarithmic ψscale | |
| Dsg | geometric mean size of the surface material | mm |
| σsg | geometric standard deviations of the surface materials | - |
| σ | arithmetic standard deviations of the surface materials | - |
| ρ | density of water | kg/m3 |
| ρs | density of sediment | kg/m3 |
| R | submerged specific density of sediment, equals to (ρs /ρ-1) | - |
| u | shear velocity of flow | m / s |
| g | acceleration of gravity | m / s2 |
| τb | boundary shear stress on the bed | kg / (m s) |
| u* | shear velocity on the bed, equals to | m / s |
| qbi | volume gravel bedload transport per unit width of grains in the ith size range | m 2 |
| pi | fraction of gravel bedload in the ith grain size range | mm |
| Dlg | geometric mean of the bedload | mm |
| σlg | geometric standard deviation of the bedload | - |
| Dsx | grain size in the surface material, such that x percentage of the material is finer | mm |
| Dlx | grain size in the bedload material, such that x percentage of the material is finer | - |
| ψs | equals to τbs / τb | |
| Wi * | dimensionless bedload transport rate for ith grain size | - |
| ω | straining relation in Parker (1990a,b) bedload relation for mixtures | |
| τsg * | Shields number based on surface geometric mean size | - |
| G(Φ) | function in Parker (1990a, b) and Wilcock and Crowe (2003) bedload relation for gravel mixture; the function is different in each case | |
| ω0 | function relation in Parker (1990a, b) bedload relation for mixture | - |
| Q | flow discharge | m3 / s |
| S | bed slope | - |
| B | channel width | m |
| n | roughness factor | - |
| Φ | ||
| Φsgo | - | |
| τsgo | ||
| ψl | - | |
| σl | ||
| ψlx | - |
Output
| Symbol | Description | Unit |
|---|---|---|
| qbT | volume bedload transport per unit width | m2 / s |
| Dg | geometric mean | mm |
| τg * | shields number based on surface geometric mean size | kg / (m s) |
| σg | geometric standard deviation | - |
| Dx | diameter such that x% of the distribution is finer | mm |
| H | water depth | m |
| u* | shear velocity | m / s |
Notes
- Note on the program
It 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. The output includes the value of qbT, the Shields stress τsg ∗ based on the surface geometric mean size, the flow depth H, the shear velocity u∗, the bedload grain size distribution (Dd,i, pf,i) and the values Dlg, σlg, Dl90, Dl70, Dl50 and Dl30 for the bedload, as well as the corresponding values for the surface ,material, Dsg, σsg, Ds90, Ds70, Ds50 and Ds30.
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 25 applicable to normal (steady, streamwise uniform) flow. Flow resistance on the bed region is computed using a Manning-Strickler resistance relation (equation 26).
Equation 28, 29 is solved iteratively for H in the code. Once H is known, the shear velocity u∗ is computed from equation 25, 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.
- Note on equations
In order to implement the equations 9~15, 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 16, 17.
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 18, 19, 20, 21.
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 22, 23, using equation 24.
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.
