Model help:Acronym1
Acronym1
The Acronym1 programs implement the Parker (1990a) surface-based bedload transport relation in order to compute gravel bedload transport rates.
Model introduction
This program calculates the volume bedload transport per unit width and the Shields number, based on the surface geometric mean diameter and the bedload GSD with its various derivatives (mean, stand. dev., interpolations), given the surface GSD.
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 (D_{b,i}, D_{b,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 F_{i} 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)
[math]\displaystyle{ G \left ( \phi \right )= \left\{\begin{matrix} 5474 \left ( 1 - {\frac{0.853}{\phi}} \right ) ^ \left (4.5 \right ) & \phi \gt 1.59 \\ exp[ 14.2\left ( \phi - 1\right ) - 9.28 \left ( \phi - 1 \right )^2 ] & 1 \lt = \phi \lt = 1.59 \\ \phi ^\left (14.2 \right ) & \phi \lt 1 \end{matrix}\right. }[/math] (12)
[math]\displaystyle{ \omega= 1 + {\frac{\sigma}{\sigma_{O} \left ( \phi_{sgo} \right ) }} [ \omega_{O} \left ( \phi_{sgo} \right ) - 1 ] }[/math] (13)
- 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] (14)
- fraction of gravel bedload in the ith grain size range
[math]\displaystyle{ p_{i}= {\frac{q_{bi}}{q_{bT}}} }[/math] (15)
- Geometric mean of the bedload
[math]\displaystyle{ D_{lg}= 2 ^\left (\bar\psi_{l} \right ) }[/math] (16)
[math]\displaystyle{ \Psi_{l}= \sum\limits_{i=1}^\left (Np \right ) \Psi_{i} p_{i} }[/math] (17)
- Geometric standard deviation of the bedload
[math]\displaystyle{ \delta_{lg}= 2 ^\left ( \delta_{l} \right ) }[/math] (18)
[math]\displaystyle{ \delta_{l} ^2= \sum\limits_{i=1}^\left (Np \right ) \left ( \Psi_{i} - \bar\Psi_{l} \right )^2 p_{i} }[/math] (19)
- Grain sizes in the bedload material
[math]\displaystyle{ D_{lx}= 2 ^\left (\Psi_{lx} \right ) }[/math] (20)
[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] (21)
[math]\displaystyle{ \Psi_{b, i}= Ln_{2} \left ( D_{b, j} \right ) }[/math] (22)
Symbol | Description | Unit |
---|---|---|
D | grain size | L |
D_{i} | characteristic grain size for the ith grain size range (i=1...N) | L |
F_{i} | fraction in surface layer for the ith grain size range(for i =1...N) | - |
τ_{ssrg} ^{*} | equals to 0.0386 | - |
ψ | grain sizes on the base-2 logarithmic ψ scale | |
ρ | density of water | M / L^{3} |
ρ_{s} | density of sediment | M / L^{3} |
R | submerged specific density of sediment, equals to (ρ_{s} /ρ-1) | - |
u | shear velocity of flow | L / T |
g | acceleration of gravity | L / T^{2} |
τ_{b} | boundary shear stress on the bed | M / (L T) |
u_{*} | shear velocity on the bed, equals to sqrt(τ_{b} / ρ ) | L / T |
p_{i} | fraction of gravel bedload in the ith grain size range | L |
ψ_{s} | equals to τ_{bs} / τ_{b} | |
W_{i} ^{*} | dimensionless bedload transport rate for ith grain size | - |
q_{bi} | 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 | |
ω_{0} | function relation in Parker (1990a, b) bedload relation for mixture | - |
Φ | parameter in Parker (1990a, b) bedload relation for mixures | |
Φ_{sgo} | equals to τ_{sg} ^{*} / τ_{ssrg} ^{*} | - |
ψ_{l} | grain sizes on the base-2 logarithmic ψ scale for bedload | - |
ψ_{lx} | grain sizes on the base-2 logarithmic ψ scale for bedload such that x percent of the material is finer | - |
D_{g} | geometric mean | L |
σ_{g} | geometric standard deviation | - |
D_{x} | diameter such that x% of the distribution is finer | L |
Output
Symbol | Description | Unit |
---|---|---|
q_{bT} | total volume gravel bedload transport rate per unit width summed over all sizes | L^{2} / T |
τ_{sg} ^{*} | Shields number based on surface geometric mean size | - |
D_{sg} | geometric mean size of the surface material | L |
σ_{sg} | geometric standard deviations of the surface materials | - |
D_{lg} | geometric mean size of the bedload | L |
σ_{lg} | geometric standard deviation of the bedload | - |
σ_{l} | arithmetic standard deviations of bedload materials | - |
σ | arithmetic standard deviations of the surface materials | - |
D_{sx} | grain size in the surface material, such that x percentage of the material is finer | L |
D_{lx} | grain size in the bedload material, such that x percentage of the material is finer | L |
σ_{lx} | arithmetic standard deviations of bedload materials | L |
Notes
- Note on equations
The gravel is divided into N grain size ranges bounded by N+1 sizes D_{b},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 (D_{b},i, F_{f},i), i = 1..N+1, where F_{f,i} denotes the percent finer in the surface layer. Here D_{b,1} must be the coarsest size, such that F_{f},1 = 100, and D_{b,N+1} must be the finest size, such that F_{f,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 re-normalized, in determining the surface grain size distribution to be input into Acronym1.
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 q_{bi} are known, the total volume bed load transport rate per unit width q_{bT} and the fractions pi in the bedload can be calculated as equations 14.
The results are presented in terms of q_{bT} and the grain size distribution of the bed load, 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, from the relations of equations 16, 17, 18, 19.
The percent finer in the bedload p_{f,i} for the grain size D_{f,i} is obtained from the fractions p_{i} as follows: p_{f,1} = 100 p_{f,i} = p_{f,i-1} - 100 p_{i-1} (i=2~N+1)
Once F_{f,i} is specified (p_{f},i is computed) the value D_{sx} (D_{lx}) can be computed by interpolation. The interpolation should be done using a logarithmic scale for grain size. For example, consider the computation of D_{lx} where p_{f,i} ≤ x ≤ p_{f,i+1}. Then we got equation 20, 21, using equation 22.
In order to carry out the above calculation it is necessary to specify specific gravity of the sediment R + 1, the shear velocity of the flow u, and the grain size distribution of the material in excess of 2 mm in the surface layer (D_{b,i}, F_{f,i}), i = 1 to N+1. The relation then predicts the total volume bed load transport rate per unit width q_{bT} of material in excess of 2 mm, as well as the grain size distribution of this load (D_{b,i}, p_{f,i}).
- Note on running the program
User may enter GSD on either a 0.00-1.00 scale or a 0%-100% scale, and the program will automatically convert it to a 0% -100% scale.
In the case of a uniform diameter, the user should simply enter the diameter in the diameter column, and 0 in the % passing column.
This formulation does not work for sand, so if sand is present and the border value of 2mm is present the program cuts off all sand (<2mm) and re-normalizes the remaining distribution; if sand is present and the border value of 2mm is not present the program alerts the user and exits.
This formulation requires endpoints at 0% and 100%, so if these boundary values are not present the program will automatically interpolate them.
The program automatically organizes the data, so the user may enter the distribution in whatever order they desire.
Note, the Acronym family of functions do not have a GetData function, because there is no time loop, and all the data is being outputted already.
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.