Model help:Acronym1D

Acronym1D

“Acronym1_D” combines the scheme of “Acronym1_R” with a flow duration curve. The bedload transport rate and bedload grain size distribution are computed for each flow of the curve, and then averaged to yield a mean bedload transport rate and a mean bedload grain size distribution.

Model introduction

“Acronym1_D” simply adds a flow duration curve to the algorithm of “Acronym1_R” in order to compute the average volume gravel bedload transport rate per unit width q_bTa, as well as the average bedload grain size distribution (D_b,i, p_af,i), i = 1..N+1. In addition, it computes the values Q_a, H_a, u_∗a and τ_ga∗ corresponding to annual mean values of the water discharge, depth, shear velocity and Shields stress based on surface geometric mean size. The values D_alg, σ_alg, D_al90, D_al70, D_al50 and D_al30 associated with the mean grain size distribution of the bedload are computed along with the corresponding values for the surface material, D_sg, σ_sg, D_s90, D_s70, D_s50 and D_s30. Finally, the program computes the volume gravel bedload transport rate per unit width q_bT, the water discharge Q_w, flow depth H, the shear velocity u_∗ and the Shields stress τ_g∗ associated with each range in the flow duration curve, along with the fraction of time p_Q that the flow is in that range.

The flow duration curve is specified in terms of the pairs (Q_wd,k, p_eQ,k), k = 1..M+1, where Q_wd,k denotes the kth discharge and p_eQ,k denotes the percentage of time this flow is exceeded. Here k = 1 corresponds to the highest flow in the curve, with an exceedance percentage p_eQ of zero, and k = M+1 corresponds to the lowest flow in the curve, with an exceedance percentage p_eQ of 100. The lowest flow on the curve Q_wd,M must exceed zero.

Model parameters

First tab header

Second tab header

Etc. tab header

Parameter	Description	Unit
First parameter	Description parameter	[Units]

Parameter	Description	Unit
First parameter	Description parameter	[Units]

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{ Q_{wr,k}= {/frac{1}{2}} \left ( Q_{wd,k} + Q_{wd, k+1} \right ) }[/math]

(26)

[math]\displaystyle{ P_{Q,k}= {\frac{p_{eQ,k+1} - p_{eQ,k}}{100}} }[/math]

(27)

Nomenclature

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
Qwd,k	the kth discharge	m3 / s
peQ,k	percentage of time that flow is exceeded (here k=1 corresponding to the highest flow in the curve, with an exceedance percentage peQ of zero, and k=M+1 corresponds to the lowest flow in the curve, with an exceedance percentage peQ of 100)	-
S	bed slope	-
B	channel width	m
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
Qwa	mean annual water discharge	m3 / s
Dg	geometric mean	mm
σg	geometric standard deviation	-
Dx	diameter such that x% of the distribution is finer	mm
Qwr	mean water discharge for each range	m3 / s
pQ	fraction of time in that range	-
u*	shear velocity on the range	m / s
τ*	shields number on the range	-

Notes

• Note on equations

In order to implement the equations 10~16, it is necessary to specify a) the surface grain size distribution (D_f,i, F_f,i) and b) the shear velocity u_∗. This results in a predicted values of q_bi.

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 bedload transport rate per unit width q_bT and the fractions pi in the bedload can be calculated as equations 17, 18.

The results are presented in terms of q_bT 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 D_lg and σ_lg, respectively, from the relations of equations 19, 20, 21, 22.

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)

Let D_sx and D_lx 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 D_l50 denote the median sizes of the surface and bedload material, respectively. 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 Dlx where p_f,i ≤ x ≤ p_f,i+1. Then we got equation 23, 24, using equation 25.

The characteristic flow Q_wr,k in each range and fraction of time the flow is in that range p_Q,k are computed with equation 26, 27, here k is ranged from 1 to M.

Let Y_k be any parameter defined for each of the flow ranges k = 1..M. The mean value Ya averaged over the flow duration curve is then given as Y _{= /Sigma Yk pQ,k For example, if the fractions in the bedload in each grain size range within flow range k are given as pk,i then the average fractions of the bedload pai are given as pai = /Sigma pk,i pQ,k}

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)

Gary Parker

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.

Links

• [Model:Acronym1]