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
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 \left ( {\frac{u_{*}}{v_{s}}}, {\frac{H}{k_{c}}}, \zeta_{b} \right ) }[/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)
Symbol | Description | Unit |
---|---|---|
R | sediment specific gravity | - |
B | channel width | L |
D_{50} | median grain size (sand) | L |
D_{90} | grain diameter such that 90% of the distribution is finer | L |
Q | flow discharge | L^{3} / T |
L | reach length | L |
M | number of spatial intervals | |
τ_{s,min} | minimum shear stress due to skin friction | |
Fr_{d} | downstream Froude number | - |
λ_{p} | sediment porosity | - |
I_{f} | flood intermittency | - |
q_{t} | total volume bed material load transport rate per unit width | L^{2} / T |
q_{b} | total volume bedload transport rate per unit width | L^{2} / T |
q_{s} | volume bed material suspended load transport rate per unit width | L^{2} / T |
τ_{s} ^{*} | Shields number due to form drag | - |
τ_{bs} | boundary shear stress due to skin friction | - |
ρ | water density | M / L^{3} |
g | acceleration due to gravity | L / T^{2} |
D_{s50} | median size of surface layer sediment | L |
R | sediment submerged specific gravity | - |
q_{b} ^{*} | Einstein number for bedload transport | - |
τ_{c} ^{*} | 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 | - |
Z_{u} | user-defined variable | - |
A | equals to 5.7 * 10^{-7} | - |
u_{*s} | shear velocity due to skin friction | L / T |
v_{s} | particle terminal fall velocity in quiescent water | L / T |
ν | kinematic viscosity of water | L^{2} / T |
S_{f} | down-channel friction slope | - |
Re_{p} | user-defined variable | - |
u^{*} | shear velocity | L / T |
C_{z} | dimensionless Chezy resistance coefficient | - |
U | depth-averaged flow velocity | L / T |
k_{c} | composite roughness height associated with both skin friction and form drag | L |
κ | Von Karman constant in logarithmic velocity profile | - |
H_{s} | depth associated with skin friction | L |
ζ_{b} | equals to 0.05, in Wright-Parker formulation | - |
S | bed slope | - |
Fr | Froude number | - |
τ_{b} | boundary shear stress | - |
τ_{bf} | boundary shear stress due to dunes | - |
H_{f} | depth associated with dunes | L |
C_{fs} | friction coefficient due to skin friction | - |
D_{s90} | sediment size such that 90 % of the material in the surface layer is finer | - |
τ^{*} | Shields number | - |
τ_{min} ^{*} | minimum Shields number | - |
φ_{s} | ratio of bed shear stress due to skin friction to total bed shear stress | - |
S_{nom} | equals to S_{f} φ_{s}^{-4/3} | - |
τ_{nom} ^{*} | equals to H S_{nom} / R D_{s50} | - |
x | downstream coordinate | L |
ξ_{d} | downstream water surface elevation, must be larger than the beginning point water surface elevation | L |
η | bed surface elevation | L |
H | flow depth | L |
C_{f} | bed friction coefficient | - |
ζ | dimensionless upward normal coordinate | - |
C_{ff} | resistance coefficient due to form drag | - |
q_{w} | water discharge per unit width | L^{2} / T |
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: https://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)
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.