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 (Ashida and Michiue, 1972)

[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 (Wright and Parker, 2004)

[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 (Wright-Parker formulation)

[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) friction coefficient due to skin friction

[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) Shields number due to form drag

[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) Shields number

[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)

• Calculation of H_s and S_f from known depth H

[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 *	Shields number due to form drag	-
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
R	sediment submerged specific gravity	-
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	equals to 5.7 * 10-7	-
u*s	shear velocity due to skin friction	L / T
vs	particle terminal fall velocity in quiescent water	L / T
ν	kinematic viscosity of water	L2 / T
Sf	down-channel friction slope	-
Rep		-
u*	shear velocity	L / T
Cz	dimensionless Chezy resistance coefficient	-
U	depth-averaged flow velocity	L / T
kc	composite roughness height associated with both skin friction and form drag	L
κ	Von Karman constant in logarithmic velocity profile	-
Hs	depth associated with skin friction	L
ζb	equals to 0.05, in Wright-Parker formulation	-
S	bed slope	-
Fr	Froude number	-
τb	boundary shear stress	-
τbf	boundary shear stress due to dunes	-
Hf	depth associated with dunes	L
Cfs	friction coefficient due to skin friction	-
Ds90	sediment size such that 90 % of the material in the surface layer is finer	-
τ*	Shields number	-
τmin *	minimum Shields number	-
φs	ratio of bed shear stress due to skin friction to total bed shear stress	-
Snom	equals to Sf φs-4/3	-
τnom *	equals to H Snom / R Ds50	-
x	downstream coordinate	L
ξd	downstream water surface elevation, must be larger than the beginning point water surface elevation	L
η	bed surface elevation	L
H	flow depth	m
Hs	flow depth due to skin friction	L

Notes

The program generates a plot of bed and water surface elevations η and ξ versus streamwise distance, as well as a plot of depth H and depth due to skin friction H_s versus streamwise distance.

In the calculation of φ_s, a Newton-Raphson iterative scheme solution is implemented. The solution is initiated with some guess φ_s,1. The calculation is continued until the relative error is under some acceptable limit. There always seems to be a first guess of φ_s for which the Newton-Raphson scheme converges. When τ_nom ^* is only slightly greater than τ_min ^* (in which case φ_s is only slightly less than 1), however, the right initial guess is sometimes hard to find. For example, the scheme may bounce back and forth between two values of φ_s without converging, or may yield at some point a negative value of φ_s. The following technique was adopted to overcome these difficulties:

a) The initial guess for φ_s is set equal to 0.9.

b) Whenever the iterative scheme yields a negative value of φ_s, φ_s is reset to 1.02 and the iterative calculation recommenced.

c) Whenever the calculation does not converge, it is assumed that φ_s is so close to 1 that it can be set equal to 1.

• 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.

Downstream bed elevation is set equal to 0, so that at normal conditions the downstream water surface elevation ξ_d = H_n. The user may then specify a value of ξ_d that differs from H_n (as long as the corresponding downstream Froude number is less than unity), and compute the resulting backwater curve.

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