Model help:Acronym1R
From CSDMS
Acronym1R
"Acronym1_R” combines Acronym1 scheme with a Manning-Strickler relation for flow resistance.
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
- 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)
- Geometric standard deviations of the surface material
[math]\displaystyle{ \sigma_{sg}= 2 ^\sigma }[/math] (6)
- Arithmetic standard deviations of the surface material
[math]\displaystyle{ \sigma ^2= \sum\limits_{i=1}^N \left (\Psi_{i} - \bar\Psi \right )^2 F_{i} }[/math] (7)
- Bedload 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)
- Roughness height of the channel (including bed and vertical sidewalls)
[math]\displaystyle{ k_{s} = n_{k} D_{s90} }[/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)
Nomenclature
| Symbol | Description | Unit |
|---|---|---|
| D | grain size | L |
| Di | characteristic grain size for the ith grain size range (i=1...N) | L |
| 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 ψ scale | |
| ρ | density of water | M / L3 |
| ρs | density of sediment | M / L3 |
| R | submerged specific density of sediment, equals to (ρs /ρ-1) | - |
| u | shear velocity of flow | L / T |
| g | acceleration of gravity | L / T2 |
| τb | boundary shear stress on the bed | M / (L T) |
| u* | shear velocity on the bed, equals to sqrt(τb / ρ ) | L / T |
| pi | fraction of gravel bedload in the ith grain size range | L |
| Dlg | geometric mean of the bedload | L |
| ψs | equals to τbs / τb | |
| Wi * | dimensionless bedload transport rate for ith grain size | - |
| qbi | volume gravel bedload transport per unit width of grains in the ith size range | L 2 |
| ω | straining relation in Parker (1990a,b) bedload relation for mixtures | |
| G(Φ) | function in Parker (1990a, b) and Wilcock and Crowe (2003) bedload relation for gravel mixture | |
| ω0 | function relation in Parker (1990a, b) bedload relation for mixture | - |
| Φ | parameter in Parker (1990a, b) bedload relation for mixures | |
| Φsgo | equals to τsg * / τssrg * | - |
| ψl | grain sizes on the base-2 logarithmic ψ scale for bedload | - |
| ψlx | grain sizes on the base-2 logarithmic ψ scale for bedload such that x percent of the material is finer | - |
| Dg | geometric mean | L |
| σg | geometric standard deviation | - |
| Dx | diameter such that x% of the distribution is finer | L |
| nx | user-specified dimensionless roughness factor (the author suggests a value of 2) | - |
| ks | roughness height of the channel | - |
| B | channel width | L |
| U | mean velocity | L / T |
| αr | coefficient in Manning-Strickler resistance relation, equals to 8.1 | - |
| Q | water discharge | L3 / T |
| S | channel slope | - |
| c1 | user-specified coefficient | - |
| nk | parameter such that ks = nk Ds90 | - |
| Ds90 | sediment size such that 90 % of the material in the surface layer is finer | - |
Output
| Symbol | Description | Unit |
|---|---|---|
| qbT | total volume gravel bedload transport rate per unit width summed over all sizes | L2 / T |
| τsg * | Shields number based on surface geometric mean size | - |
| Dsg | geometric mean size of the surface material | L |
| σsg | geometric standard deviations of the surface materials | - |
| σ | arithmetic standard deviations of the surface materials | - |
| σlg | geometric standard deviation of the bedload | - |
| Dsx | grain size in the surface material, such that x percentage of the material is finer | L |
| Dlx | grain size in the bedload material, such that x percentage of the material is finer | L |
| σl | arithmetic standard deviations of bedload materials | |
| H | water depth | L |
Notes
- Note on the program
The channel is assumed to be rectangular, with the vertical sidewalls having the same roughness as the bed.
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 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: https://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.(DOI:10.1080/00221689009499058)
- 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.
