Model help:Acronym1R

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 Q_w, 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

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

• 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)

• 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)

Nomenclature

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) (D_b,i, F_f,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 q_bT, the Shields stress τ_sg ^∗ based on the surface geometric mean size, the flow depth H, the shear velocity u^∗, the bedload grain size distribution (D_d,i, p_f,i) and the values D_lg, σ_lg, D_l90, D_l70, D_l50 and D_l30 for the bedload, as well as the corresponding values for the surface ,material, D_sg, σ_sg, D_s90, D_s70, D_s50 and D_s30.

The channel is assumed to be rectangular, with the vertical sidewalls having the same roughness as the bed. The roughness height k_s is computed using equation 31. Here n_k is a user-specified dimensionless roughness factor. The author suggests a value of 2 for n_k.

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 (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 16, 17.

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 18, 19, 20, 21.

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 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)

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:Acronym1R]