Model help:RecircFeed
RecircFeed
This provide a calculator for approach to equilibrium in recirculating and feed flumes
Model introduction
This program provides two modules for studying the approach to mobile-bed normal equilibrium in recirculating and sediment-feed flumes containing uniform sediment.
The module "Recirc" implements a calculation for the case of a flume that recirculates water and sediment. The module "Feed" implements a calculation for the case of flume which receives water and sediment feed.
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
- Backwater equation
1) water discharge per unit width
[math]\displaystyle{ q_{w} = U H }[/math] (1)
2)
[math]\displaystyle{ {\frac{\partial H}{\partial x}} = {\frac{S - S_{f}}{1 - Fr^2}} }[/math] (2)
[math]\displaystyle{ S_{f} = C_{f} Fr^2 }[/math] (3)
[math]\displaystyle{ Fr^2 = {\frac{q_{w}^2}{g H^3}} = {\frac{U^2}{g H}} }[/math] (4)
- bed shear stress
[math]\displaystyle{ \tau_{b} = \rho g H S }[/math] (5)
- Friction relations
[math]\displaystyle{ \tau_{b} = \rho C_{f} U^2 }[/math] (6)
[math]\displaystyle{ C_{f}^ \left ({\frac{-1}{2}}\right ) = \alpha_{r} \left ({\frac{H}{k_{c}}}\right ) ^ \left ({\frac{1}{6}}\right ) }[/math] (7)
- Total bed material load
[math]\displaystyle{ {\frac{q_{t}}{\sqrt{R g D} D}} = \alpha _{t} \left ({\frac{\tau _{b}}{\rho R g D} - \tau_{c}^*} \right ) ^ \left (n_{t} \right ) }[/math] (8)
- Sediment transport relation
[math]\displaystyle{ q_{t}^* = {\frac{q_{t}}{\sqrt{R g D}D}} = \left\{\begin{matrix} 0 & if \tau^* \lt \tau_{c}^* \\ \alpha_{t} \left ( \tau^* - \tau_{c}^* \right ) ^ \left (n_{t}\right ) & if \tau^* \gt \tau_{c}^* \end{matrix}\right. }[/math] (9)
[math]\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} }[/math] (10)
- Mobile-bed equilibrium
1)
[math]\displaystyle{ U{\frac{\partial U}{\partial x}} = -g {\frac{\partial H}{\partial x}} - g{\frac{\partial \eta}{\partial x}} - C_{f}{\frac{U^2}{H}} }[/math] (11)
2)
[math]\displaystyle{ {\frac{q_{t}}{\sqrt{R g D}D}} = \left\{\begin{matrix} 0 & if {\frac{C_{f} U^2}{R g D}} \lt \tau_{c}^* \\ \alpha_{L} \left ({\frac{C_{f} U^2}{R g D}} - \tau_{c}^* \right ) ^ \left (N_{L} \right ) & if {\frac{C_{f} U^2}{R g D}} \gt \tau_{c}^*\end{matrix}\right. }[/math] (12)
- Dimensionless parameters
1)
[math]\displaystyle{ H = H_{o} \tilde{H} }[/math] (13)
2)
[math]\displaystyle{ q_{t} = q_{to} \tilde{q} }[/math] (14)
3)
[math]\displaystyle{ x = L \hat{x} }[/math] (15)
4)
[math]\displaystyle{ t = \left ( 1 - \lambda_{p} \right ) {\frac{S_{o} L^2}{q_{to}}} \hat{t} }[/math] (16)
5)
[math]\displaystyle{ \eta = \bar{\eta} + \eta _{d} }[/math] (17)
6)
[math]\displaystyle{ \bar{\eta} = H_{o} \tilde{\eta}_{a} \left ( \hat{t} \right ) }[/math] (18)
7)
[math]\displaystyle{ \eta_{d} = S_{o} L \hat{\eta}_{d} \left ( \hat{x},\hat{t} \right ) }[/math] (19)
8) Backwater relation
[math]\displaystyle{ FI {\frac{\partial \tilde{H}}{\partial \hat{x}}} = {\frac{-{\frac{\partial \hat{\eta}_{d}}{\partial \hat{x}}} - \tilde{H}^ \left (-3\right )}{1 - Fr_{o}^2 \tilde{H}^ \left (-3\right )}} }[/math] (20)
[math]\displaystyle{ Fr_{o} = \left ({\frac{q_{w}^2}{g H_{o}^3}}\right )^ \left ({\frac{1}{2}}\right ) }[/math] (21)
- Relation for sediment conservation
[math]\displaystyle{ FI {\frac{d \tilde{\eta}_{a}}{d \hat{t}}} = \tilde{q}|_{\hat{x} = 0} - \tilde{q}|_{\hat{x} = 1} }[/math] (22)
[math]\displaystyle{ {\frac{\partial \hat{\eta}_{d}}{\partial \hat{t}}} = - {\frac{\partial \tilde{q}}{\partial \hat{x}}} - \left ( \tilde{q}|_{\hat{x} = 0 } - \tilde{q}_{\hat{x} = 1} \right ) }[/math] (23)
- Sediment transport relation
[math]\displaystyle{ \tilde{q} = \left\{\begin{matrix} 0 & \tilde{H}^ \left (-2 \right ) \lt \left (\tau_{r}^*\right )^ \left (-1\right ) \\ \left ({\frac{\tilde{H}^\left (-2\right ) - \left (\tau_{r}^* \right )^ \left (-1\right )}{1 - \left (\tau_{r}^*\right )^\left (-1\right )}} \right )^ \left (N_{L}\right ) & \tilde{H}^\left (-2\right ) \gt \left (\tau_{r}^* \right )^ \left(-1\right )\end{matrix}\right. }[/math] (24)
[math]\displaystyle{ \tau_{r}^* = {\frac{\tau_{o}^*}{\tau_{c}^*}} }[/math] (25)
[math]\displaystyle{ \tau_{o}^* = {\frac{C_{f} U_{o}^2}{R g D}} = {\frac{C_{f}q_{w}^2}{R g D H_{o}^2}} }[/math] (26)
- Boundary condition for a feed flume
[math]\displaystyle{ \tilde{q}|_{\hat{x} = 0 } = 1 }[/math] (27)
[math]\displaystyle{ \tilde{H}|_{\hat{x} = 1} = 1 - \tilde{\eta}_{a} - {\frac{1}{FI}}\left ({\frac{1}{2}} + \hat{\eta}_{d}|_{\hat{x} = 1} \right ) }[/math] (28)
- Initial condition for a feed flume
[math]\displaystyle{ \tilde{\eta}_{a}|_{\hat{t} = 0} = \tilde{\eta}_{al} }[/math] (29)
[math]\displaystyle{ \eta_{d}|_{\hat{t} = 0} = {\frac{S_{l}}{S_{o}}} \left ({\frac{1}{2}} - \hat{x} \right ) }[/math] (30)
- Boundary condition for a recirculating flume
[math]\displaystyle{ \tilde{q}|_{\hat{x} = 0} = \tilde{q}|_{\hat{x} = 1} }[/math] (31)
[math]\displaystyle{ \int _{0}^1 \tilde{H} d \hat{x} = 1 }[/math] (32)
- Initial condition for a recirculating flume
[math]\displaystyle{ \hat{\eta}_{d}|_{\hat{t} = 0} = {\frac{S_{l}}{S_{o}}} \left ({\frac{1}{2}} - \hat{x} \right ) }[/math] (33)
| Symbol | Description | Unit |
|---|---|---|
| Fro | Froude number at normal mobile-bed equilibrium | - |
| ηtilinit | initial value of the dimensionless mean bed elevation (used only in the case of sediment feed) | - |
| τr | geometric standard deviations | - |
| SNI | initial scaled bed slope (ratio of initial bed slope to bed slope at normal flow) | - |
| M | number of spatial intervals in the reach | - |
| dt | dimensionless time step | - |
| Nstep | total number of time steps in a set | - |
| Ntimes | total number of sets in a run | - |
| f | backwater number | - |
| e | initial dimensionless mean bed elevation | m |
| T | ratio of normal to critical Shield stress | - |
| S | bed slope | - |
| t | dimensionless time step | - |
| p | number of prints | - |
| i | number of iterations per print | - |
| x | dimensionless downstream coordinate | - |
| η | bed elevation | L |
| Sl | bed slope | - |
| qbT | dimensionless bedload transport | - |
| H | flow depth | L |
| SNup | upstream end normalized slope | - |
| SNdown | downstream end normalized slope | - |
| U | depth-averaged flow velocity | L / T |
| g | gravitational acceleration | L / T2 |
| qt | volume bed material sediment transport rate per unit width | L2 / T |
| qw | water discharge per unit width | L2 / T |
| τb | boundary shear stress at bed | - |
| L | flume length | L |
| B | flume width | L |
| λp | porosity of bed deposit of sediment | - |
| Sf | down-channel bed slope | - |
| Fr | Froude number | - |
| Cf | bed friction coefficient | - |
| D | grain size | L |
| ρs | sediment density | M / L3 |
| R | sediment submerged specific gravity | - |
| qt * | Einstein number for total bed material load | - |
| τ* | Shields number | - |
| τc * | critical Shields number at the threshold of motion | - |
| αt | coefficient in generic relation for total bed material load | - |
| nt | exponent in the sediment transport relation | - |
| NL | - | |
| Ho | constant channel depth of the unperturbed base flow | - |
| H~ | dimensionless depth | - |
| qto | unperturbed value of qt at base equilibrium | - |
| x^ | dimensionless downstream coordinate | - |
| t^ | dimensionless time | - |
| So | unperturbed value of bed slope S of base equilibrium | - |
| FI | the dimensionless flume number, equals to Ho / So L | - |
| Uo | value of U of unperturbed base flow | - |
| η~a | flume-averaged bed elevation | - |
| η~al | initial value for flume-averaged bed elevation | - |
| η~d | equals to Sl L [0.5 - (x/L)] | - |
| Sl | initial bed slope | - |
| kc | composite roughness height associated with both skin
friction and form drag |
- |
| αL | - | |
| ρ | water density | M / L3 |
| τo * | Shields number of unperturbed flow | - |
| τr * | - |
Notes
This program is meant for use with laboratory flumes, and allows the user to choose either a recirculating flume or a feed flume.
- Constraints on a recirculating flume
a) Water discharge qw is set by the pump.
b) The total amount of water in the flume is conserved. With constant width, constant storage in the return line and negligible storage in the entrance and exit regions. At final equilibrium, when H = Ho, the constraint reduces to HoL = C1, according to which Ho is set by the total amount of water.
c) The total amount of sediment in the flume is conserved.
- Constraints on a feed flume
a) Water discharge qw is set by the pump.
b) The upstream sediment discharge is set by the feeder. Where qtf is the sediment feed rate: qt|_{x = 0} = qtf.
c) Let ξ = η + H denote water surface elevation. The downstream water surface elevation ξd is set by the tailgate.
d) The long-term equilibrium approached in a recirculating flume (without lumps) should be dynamically equivalent to that obtained in a sediment-feed flume.
The inputted Froude number must be less than 1 (i.e. the flow is assumed to be subcritical); if it is not the program will alert the user and automatically quit.
In the outputs file, the normalized slopes upstream and downstream are below the dimensionless eta values, scroll down to find them.
At equilibrium the ratio between the bed slope and the slope at normal flow will both equal 1
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
- Hills, R., 1987, Sediment sorting in meandering rivers, M.S. thesis, University of Minnesota, 73 p.
- Parker, G., 2003, Persistence of sediment lumps in approach to equilibrium in sediment-recirculating flumes, Proceedings, XXX Congress, International Association of Hydraulic Research, Thessaloniki, Greece, August 24-29, downloadable at http://cee.uiuc.edu/people/parkerg/conference_reprints.htm .
- Parker, G. and Wilcock, P., 1993, Sediment feed and recirculating flumes: a fundamental difference, Journal of Hydraulic Engineering, 119(11), 1192‑1204.
