Model help:BackwaterWrightParker

BackwaterWrightParker

This is used to calculate backwater curves in sand-bed streams, including the effects of both skin friction and form drag due to skin friction.

Model introduction

This program calculates backwater curves over a sand-bed stream with a specified spatially constant bed slope S. The calculation uses the hydraulic resistance formulation of Wright and Parker (2004) (without the flow stratification correction), as well as calculating the normal depth.

Model parameters

Input files and directories

Run Parameters

About

Parameter	Description	Unit
Input directory	Path to input file
Site prefix	site prefix for Input/Output files	-
Case prefix	Case prefix for Input/Output files	-

Parameter	Description	Unit
bed slope		-
Submerged specific gravity of sediment		-
Median grain size (D50)		-
Grain size such that 90% passes (D90)	grain diameter such that 90% of the distribution is finer	mm
channel width (B)		m
flow discharge (Q)		m3 / s
downstream water surface elevation (k)		m
reach length (L)		m
number of spatial nodes (max of 99)		-

Parameter	Description	Unit
Model name	name of the model	-
Author name	Name of the model author	-
Median grain size (D50)		-

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

• Exner equation of sediment continuity

[math]\displaystyle{ \left ( 1 - \lambda_{p} \right ) = - I_{f} {\frac{\partial q_{t}}{\partial x}} = - I_{f}{\frac{\partial q_{b}}{\partial x}} - I_{f} {\frac{\partial q_{s}}{\partial x}} }[/math]

(1)

• Bedload transport in sand-bed streams

[math]\displaystyle{ \tau_{s}^* = {\frac{\tau_{bs}}{\rho R g D_{s50}}} }[/math]

(2)

[math]\displaystyle{ q_{b}^* = {\frac{q_{b}}{\sqrt{R g D_{s50} D_{s50}}}} = 17 \left ( \tau_{s}^* - \tau_{c}^* \right ) \left ( \sqrt{\tau_{s}^*} - \sqrt{\tau_{c}^*} \right ) }[/math]

(3)

• Entrainment of sand into suspension

[math]\displaystyle{ E = {\frac{A Z_{u}^5}{1 + {\frac{A}{0.3}} Z_{u}^5}} }[/math]

(4)

[math]\displaystyle{ Z_{u} = {\frac{u_{*s}}{v_{s}}} Re_{p}^\left (0.6\right ) S_{f}^ \left (0.07\right ) }[/math]

(5)

[math]\displaystyle{ u_{*s} = \sqrt{{\frac{\tau_{bs}}{\rho}}} }[/math]

(6)

[math]\displaystyle{ Re_{p} = {\frac{\sqrt{R g D_{s50}} D_{s50}}{\nu}} }[/math]

(7)

• Suspended sediment transport rate

[math]\displaystyle{ u_{*} = \left ( g H S_{f} \right )^ \left ({\frac{1}{2}}\right ) }[/math]

(8)

[math]\displaystyle{ u_{*s} = \left ( g H_{s} S_{f} \right )^ \left ({\frac{1}{2}}\right ) }[/math]

(9)

[math]\displaystyle{ C_{z} = {\frac{U}{u_{*}}} }[/math]

(10)

[math]\displaystyle{ k_{c} = 11 {\frac{H}{exp \left ( \kappa C_{z} \right )}} }[/math]

(11)

[math]\displaystyle{ q_{s} = {\frac{E u_{*} H}{\kappa}} I }[/math]

(12)

[math]\displaystyle{ I \left ( {\frac{u_{*}}{v_{s}}}, {\frac{H}{k_{c}}}, \zeta_{b} \right ) = \int_{\zeta_{b}}^1 [{\frac{\left (1 - \zeta \right ) / \zeta}{\left ( 1 - \zeta_{b}\right ) / \zeta_{b}}}]^ \left ({\frac{v_{s}}{\kappa u_{*}}} \right ) ln \left ( 30 {\frac{H}{k_{c}}} \zeta \right) d \zeta }[/math]

(13)

• Gradually varied flow in sand-bed rivers including the effect of bedforms

1) Backwater equation

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

(14)

2) Froude number

[math]\displaystyle{ Fr = {\frac{q_{w}}{g^ \left ({\frac{1}{2}}\right ) H^ \left ( {\frac{3}{2}}\right )}} }[/math]

(15)

3) Friction slope

[math]\displaystyle{ S_{f} = C_{f} {\frac{U^2}{g H}} = {\frac{\tau_{b}}{\rho g H}} = \phi_{s}^ \left ({\frac{-4}{3}}\right ) S_{nom} }[/math]

(16)

4) boundary shear stress in a sand-bed river

[math]\displaystyle{ \tau_{b} = \tau_{bs} + \tau_{bf} = \rho \left ( C_{fs} + C_{ff} \right ) U^2 }[/math]

(17)

5) boundary depth in a sand-bed river

[math]\displaystyle{ H = H_{s} + H_{f} }[/math]

(18)

6)

[math]\displaystyle{ C_{fs}^ \left ({\frac{-1}{2}}\right ) = {\frac{q_{w}}{H \sqrt{g H_{s} S_{f}}}} = 8.32 \left ({\frac{H_{s}}{3D_{s90}}}\right )^ \left ({\frac{1}{6}}\right ) }[/math]

(19)

7)

[math]\displaystyle{ \tau_{s}^* = {\frac{H_{s} S_{f}}{R D_{50}}} = \left\{\begin{matrix} 0.05 + 0.7 \left (\tau^* Fr^ \left (0.7\right ) \right )^ \left (0.8\right ) & \tau^* \gt = \tau_{min}^* \\ \tau^* & \tau^* \lt \tau_{min}^*\end{matrix}\right. }[/math]

(20)

8)

[math]\displaystyle{ \tau^* = {\frac{H S_{f}}{R D_{s50}}} }[/math]

(21)

• Bed shear stress due to skin friction to total bed shear stress

[math]\displaystyle{ \phi = \left\{\begin{matrix} {\frac{0.05 + 0.7 \left ( \tau^* Fr ^ \left (0.7\right ) \right ) ^ \left (0.8\right )}{\tau^*}} & \tau^* \gt =\tau_{min}^* \\ 1 & \tau^* \lt \tau_{min}^* \end{matrix}\right. }[/math]

(22)

• Minimum Shields number

[math]\displaystyle{ \tau_{min}^* = 0.05 + 0.7 \left ( \tau_{min}^* Fr^ \left (0.7\right ) \right ) ^ \left (0.8\right ) }[/math]

(23)

[math]\displaystyle{ F \left (\phi_{s} \right ) = \left\{\begin{matrix} \phi_{s} - [{\frac{\phi_{s}^ \left ({\frac{-1}{3}}\right ) \tau_{nom}^* - 0.05}{0.7 \left ( \tau_{nom}^* \right ) ^ \left ({\frac{4}{5}}\right ) Fr^ \left ({\frac{14}{25}}\right )}}]^ \left ({\frac{-15}{16}}\right ) & \phi_{s} \lt = \left (\tau_{nom}^* / \tau_{min}^* \right )^ \left ({\frac{3}{4}}\right ) \\ \tau_{s} - 1 & \tau_{s} \gt \left ( \tau_{nom}^* / \tau_{min}^* \right) ^ \left ({\frac{3}{4}}\right )\end{matrix}\right. = 0 }[/math]

(24)

• Calculation of the normal flow condition prevailing in the absence of the dredge slot

[math]\displaystyle{ S_{f} = S }[/math]

(25)

[math]\displaystyle{ F_{N} \left (H\right ) = \left\{\begin{matrix} H \phi_{s} \left (H\right ) - {\frac{R D_{50}}{S}}[0.05 + 0.7 \left ({\frac{H S}{R D_{s50}}}\right )^ \left ({\frac{4}{5}}\right )\left ({\frac{q_{w}}{\sqrt{g}H^ \left ({\frac{3}{2}}\right )}}\right )^ \left ({\frac{14}{25}}\right )] & H \gt = {\frac{R D_{50} \tau_{min}^*}{S}} \\ H \phi_{s} \left (H\right ) - H & H \lt {\frac{R D_{50} \tau_{min}^*}{S}} \end{matrix}\right. = 0 }[/math]

(26)

Nomenclature

Symbol	Description	Unit
R	sediment specific gravity, R+1	-
B	channel width	m
D50	median grain size (sand)	mm
D90	grain diameter such that 90% of the distribution is finer	mm
S	bed slope	-
Q	flow discharge	m3 / s
L	reach length	m
M	number of spatial intervals
τs,min	minimum shear stress due to skin friction
Frd	downstream Froude number	m
λp	sediment porosity	-
If	flood intermittency	-
qt	total volume bed material load transport rate per unit width	L2 / T
qb	total volume bedload transport rate per unit width	L2 / T
qs	volume bed material suspended load transport rate per unit width	L2 / T
τs *		-
τbs	boundary shear stress due to skin friction	-
ρ	water density	M / L3
g	acceleration due to gravity	L / T2
Ds50	median size of surface layer sediment	L
qb *		-
τc *	critical Shields number at the threshold of motion, equals to 0.05	-
E	volume rate of entrainment of bed particles into bedload transport per unit bed area per unit time	-
Zu		-
A		-
u*s	shear velocity due to skin friction	L / T
vs	particle terminal fall velocity in quiescent water	L / T

Output

Symbol	Description	Unit
x	downstream coordinate	m
ξd	downstream water surface elevation, must be larger than the beginning point water surface elevation	m
η	bed surface elevation	m
H	flow depth	m
Hs	flow depth due to skin friction	m

Notes

• Note on input parameters:

If the minimum shear stress due to skin friction τ_s,min, calculation bombs at any point the program will end.

If the height due to skin friction H_s, calculation bombs, the program will assign the last value in the calculations to H_s.

If the H_norm calculation bombs, the value for H_norm is not outputted, but this does not affect the other values that are calculated.

This program requires a given downstream water water elevation, ξ_d, such that Fr_d < 1, because the flow is assumed subcritical, and the program will alert the user and quit if the condition is not met.

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

• Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).

• Wright, S. and G. Parker, 2004, Flow resistance and suspended load in sand-bed rivers: simplified stratification model, Journal of Hydraulic Engineering, 130(8), 796-805.

Links

• Model:BackwaterWrightParker