Model help:BackwaterCalculator

BackwaterCalculator

This program is used for backwater calculations in open channel flow.

Model introduction

The program solves the backwater equation for subcritical flow with a predictor – corrector scheme.

Model parameters

Parameter	Description	Unit
Input directory	path to input files
Site prefix	Site prefix for Input/Output files
Case prefix	Case prefix for Input/Output files

Parameter	Description	Unit
Chezy or Manning, Chezy-1 or Manning-2		-

Parameter	Description	Unit
D90 of the surface		mm
hydraulic radius		m
Chezy coefficient		-
initial depth		mm
Mean diameter		mm
alpha (r)		-
roughness coefficient
starting point		m
Slope		m / m
water discharge		m² / s
step size		m

Parameter	Description	Unit
Model name	name of the model	-
Author name	name of the model author	-

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

Water depth calculation

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

(1)

Friction slope

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

(2)

Froude number

[math]\displaystyle{ Fr = {\frac{q_{w}}{\sqrt{g H^3}}} }[/math]

(3)

Non-dimensional friction coefficient

[math]\displaystyle{ C_{f} = {\frac{1}{Cz^2}} }[/math]

(4)

Chezy equation

[math]\displaystyle{ Cz = {\frac{K_{Cz}}{\sqrt{g}}} }[/math]

(5)

Manning-Strickler

[math]\displaystyle{ C_{f} ^ \left ( {\frac{-1}{2}} \right ) = \alpha _{r} \left ( {\frac{H}{k_{s}}} \right ) ^ \left ( {\frac{1}{6}} \right ) }[/math]

(6)

Roughness height due to skin friciton

[math]\displaystyle{ k_{s} = n_{k} D_{90} }[/math]

(7)

Nomenclature

Symbol	Description	Unit
X	Streamwise coordinate	m
ΔX	Spatial step length	m
S	bed slope	-
S_f	friction slope	-
Fr	Froude number	-
C_f	non-dimensional friction coefficient	-
g	acceleration of gravity	m / s²
C_z	non-dimensional Chezy friction coefficient	-
K_cz	dimensional Chezy friction coefficient	-
α_r	user specified parameter	-
k_s	roughness height due to skin friction
n_k	non-dimensional order-one constant	-
D₉₀
H₁	starting water depth at the downstream end of the channel	-
x₁	starting position	m
q_w	water discharge per unit width	m² / s
D₉₀	diameter of the bed surface such that 90% of the distribution is finer	mm

Output

Symbol	Description	Unit
H	depth	m
U	mean flow velocity	m / s
η	bed elevation	m
H_n	water surface at normal flow	m
H_c	critical water depth	m
τ_b	shear stress	-
Fr_n	Froude number at normal flow	-
U_n	mean flow velocity at normal flow	m / s
U_c	critical flow velocity	m / s
τ_bn	bed shear stress at normal flow	N / m²
ξ	water surface elevation	m

Notes

Note on equations

The model uses equation 1 to compute the water depth, H, everywhere in the channel for a given water discharge per unit channel width, q_w, and downstream boundary condition, i.e. a user specified water depth.

The bed slope, S, is assumed constant in the streamwise direction, the friction slope, S_f, and the Froude number, Fr, are defined as equation 2,3.

Note on model running

The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is specified in the input text file. The Manning-Strickler formulation is implemented, when only the coefficients α_r and n_k are given in the input file. When all the three parameters are present, the program will ask the user which formulation they would like to use.

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>]].

Developer(s)

Gary Parker

References

Key papers

Links

Any link, eg. to the model questionnaire, etc.