Model help:AgDegBW

AgDegBW

It is the Calculator for aggradation and degradation of a river reach using a backwater formulation. This program computes 1D bed variation in rivers due to differential sediment transport. The sediment is assumed to be uniform with size D. All sediment transport is assumed to occur in a specified fraction of time during which the river is in flood, specified by an intermittency. A Manning-Strickler relation is used for bed resistance. A generic Meyer-Peter Muller relation is used for sediment transport. The flow is computed using a backwater formulation for gradually varied flow.

Model introduction

The model calculates a) an ambient mobile-bed equilibrium, and b)the response of a river reach to either 1) changed sediment input rate at the upstream end of the reach starting from t = 0 or 2) changed downstream water surface elevation at the downstream end of the reach starting from t = 0, where t is the temporal coordinate. The code is very similar to AgDegNorm. The main difference between the two codes is in the procedure to compute the water depth. In AgDegNorm the flow is assumed normal (i.e. steady and uniform), while in AgDegBW the flow is assumed steady and it is computed solving the backwater equation. The case of Froude-subcritical flow, for which Fr < 1, is considered herein. This implies that integration of the backwater equation must proceed upstream from x = L, with x streamwise coordinate and L length of the modeled reach. Both a Chezy and a Manning-Striclker formulation can be used to compute the flow.

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

• Computation of the flow

The backwater equation

[math]\displaystyle{ {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - F_{r} ^2}} }[/math]

(1)

• Friction slope

[math]\displaystyle{ S_{f} = C_{f} F_{r} ^2 }[/math]

(2)

• Froude number

[math]\displaystyle{ F_{r} ^2 = {\frac{U^2}{g H}} = {\frac{q_{w} ^2}{g H^3}} }[/math]

(3)

• Flow velocity

[math]\displaystyle{ U = {\frac{q_{w}}{H}} }[/math]

(4)

• The bed friction coefficient ( assumed to obey a Manninbg-Strickler resistance )

[math]\displaystyle{ C_f ^ \left ( {\frac{-1}{2}} \right ) = C_{z} = \alpha_{r} \left ( {\frac{H}{k_{c}}} \right ) ^{\frac{1}{6}} }[/math]

(5)

• roughness roughness

[math]\displaystyle{ k_{s} = n_{k} D }[/math]

(6)

• The relation between bed slope S and bed elevation η

[math]\displaystyle{ S = -{\frac{\eta}{x}} }[/math]

(7)

• Water surface elevation

[math]\displaystyle{ \epsilon = \eta + H }[/math]

(8)

• Shields number

[math]\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} = {\frac{C_{f} U^2}{R g D}} = {\frac{C_{f} {\frac{q_{w} ^2}{H^2}}}{R g D}} }[/math]

(9)

• Bed shear stress

[math]\displaystyle{ \tau_{b} = \rho C_{f} U^2 }[/math]

(10)

[math]\displaystyle{ R = {\frac{\rho_{s}}{\rho}} - 1 }[/math]

(11)

• Computation of the sediment transport

equation of the type of Meyer-Peter and Muller

• τ^* > τ_c ^*

[math]\displaystyle{ q_{t} ^* = \alpha_{t} \left ( \phi_{s} \tau ^2 - \tau_{c} ^* \right ) ^ \left ( n_{t} \right ) }[/math]

(12)

• τ^* <= τ_c ^*

[math]\displaystyle{ q_{t} ^* = 0 }[/math]

(13)

• Einstein number

[math]\displaystyle{ q_{t} ^* = {\frac {q\lt sub\gt t\lt /sub\gt }{\sqrt{R g D} D}} }[/math]

(14)

• Equilibrium (graded) states
• Annual sediment yield with a graded state at this slope

[math]\displaystyle{ G_{t} = \rho_{s} q_{t} Bl_{f} t_{a} }[/math]

(15)

[math]\displaystyle{ q_{t} = {\frac{G_{tf}}{\rho_{s}Bl_{f} t_{a}}} }[/math]

(16)

• Computation of bed variation
• Exner equation

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\eta}{t}} = - {\frac{q_{t}}{x}} }[/math]

(17)

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) {\frac{\eta}{t}}= - {\frac{l_{f} q_{t}}{x}} }[/math]

(18)

Nomenclature

Symbol	Description	Unit
Q	flood discharge	m 3 / s
x	streamwise coordinate	m
η	bed elevation	m
t	time step	year
B	channel width	m
D	grain size	mm
l	bed porosity	-
R	submerged specific gravity	-
ξd	downstream water surface elevation	m
qw	water discharge per unit width	m2 / s
kc	roughness height	m
G	imposed annual sediment transfer rate from upstream	tons / annum
Gtf	upstream sediment feed rate	-
ξd	downstream water surface elevation	m
L	length of reach under consideration	m
qw	water discharge per unit width	m2 / s
i	number of time steps per printout	-
p	number of printouts desired	-
M	number of spatial intervals	-
u	upwinding coefficient (1 = full upwind, 0.5 = central difference)	-
r	coefficient in Manning-Strickler Resistance relation	-
a	coefficient in sediment transport relation	-
n	exponent in sediment transport relation	-
P	fraction of bed shear stress that is Skin Friction	-
R	submerged specific gravity of sediment	-
Sf	friction slope	-
Fr	Froude number	-
U	flow velocity	m / s
Cf	bed friction coefficient	-
g	acceleration of gravity	m/ s^2
k	roughness height	mm
αr	coefficient in Manning-Stricker, dimensionless coefficient between 8 and 9	-
ks	grain roughness	m
nk	dimensionless coefficient typically between 2 and 5	-
τ*	Shield number	-
ρ	fluid density	kg / m3
ρs	sediment density	kg / m3
τc	critical Shields number for the onset of sediment motion	-
ψs	the fraction of bed shear stress	-
qt *	Einstein number	-
qt	volume sediment transport rate per unit width	-
If	flood intermittency	-
tf	cumulative time the river has been in flood	s
Gt	the annual sediment yield	tone/yr
ta	the number of seconds in a year	-
λp	the porosity of the bed deposit	-
Qf	sediment transport rate during flood discharge
αt	equals to 8	-
nt	exponent in sediment transport relation, equals to 1.5	-
τc *	reference Shields number in sediment transport relation, equals to 0.047
Cf	bed friction coefficient, equals to τb / (ρ U2 )	-
CZ	dimensionless Chezy resistance coefficient.
Sl	slope of the bed surface	-
x	downstream coordinate	m
τ	shear stress on bed surface	N / m2
qb	bed material load	tons / year
Δx	spatial step length	m
Qw	flood discharge	m3 / s
Δt	time step	year
Ntoprint	number of time steps to printout	-
Nprint	number of printouts	-
aU	upwinding coefficient (1=full upwind, 0.5=central difference)	-
αs	coefficient in sediment transport relation	-

Output

Symbol	Description	Unit
η	bed surface elevatioon	m
H	water depth	m
ξ	water surface elevation	m
τb	bed shear stress	kg / (s^2 m)
S	bed slope	-
qt	total bed material load	m2 / s

Notes

The model computes variation in river bed level η(x, t), where x denotes a streamwise coordinate and t denotes time, in a river with constant width B. The bed sediment is characterized in terms of a single grain size D and submerged specific gravity R. The reach under consideration has length L. Water surface elevation at the downstream end is prescribed. The model is based on a calculation of total bed material load. The model is 1D, assumes a rectangular channel and neglects wall or bank effects.

By modifying the upstream sediment feed rate G_tf and/or the downstream water surface elevation ξ_d, the river can be forced to aggrade or degrade to a new equilibrium. The program computes this evolution.

Actual rivers tend to be morphologically active only during floods. That is, most of the time they are not doing much to modify their morphology. The simplest way to take this into account is to assume an intermittency If such that the river is in flood a fraction I of the time, during which Q = Qf and qt is the sediment transport rate at this discharge (Paola et al., 1992). For the other (1 – If) fraction of time the river is assumed not to be moving sediment.

Output is controlled by the parameters Ntoprint and Nprint. The code will implement Ntoprint time steps. In addition to output pertaining to the initial state, the code implements Nprint outputs, to that the total number of time steps executed is equal to Nprint x Ntoprint.

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

• Paola, C., Heller, P. L. & Angevine, C. L. 1992 The large-scale dynamics of grain-size variation in alluvial basins. I: Theory. Basin Research, 4, 73-90.

• Meyer-Peter, E., and Müller, R. 1948 Formulas for bed-load transport. Proceedings, 2nd Congress International Association for Hydraulic Research, Rotterdam, the Netherlands, 39-64.

Links

• [Model:AgDegBW]