# 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

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
Feed-1 or Recirculate -2
Parameter Description Unit
Equilibrium Froude number (must be < 1) (F) -
Exponent in load relation (n) -
Backwater number (f) -
Initial dimensionless mean bed elevation (e) -
Ratio of normal to critical Shields stress (T) -
Initial normalized slope (S) -
Number of spatial intervals desired (MA) -
Dimensionless time step (t) -
Number of prints desired (p) -
Number of iterations desired -
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

• Backwater equation

1) water discharge per unit width

 $\displaystyle{ q_{w} = U H }$ (1)

2)

 $\displaystyle{ {\frac{\partial H}{\partial x}} = {\frac{S - S_{f}}{1 - Fr^2}} }$ (2)
 $\displaystyle{ S_{f} = C_{f} Fr^2 }$ (3)
 $\displaystyle{ Fr^2 = {\frac{q_{w}^2}{g H^3}} = {\frac{U^2}{g H}} }$ (4)
• bed shear stress
 $\displaystyle{ \tau_{b} = \rho g H S }$ (5)
• Friction relations
 $\displaystyle{ \tau_{b} = \rho C_{f} U^2 }$ (6)
 $\displaystyle{ C_{f}^ \left ({\frac{-1}{2}}\right ) = \alpha_{r} \left ({\frac{H}{k_{c}}}\right ) ^ \left ({\frac{1}{6}}\right ) }$ (7)
 $\displaystyle{ {\frac{q_{t}}{\sqrt{R g D} D}} = \alpha _{t} \left ({\frac{\tau _{b}}{\rho R g D} - \tau_{c}^*} \right ) ^ \left (n_{t} \right ) }$ (8)
• Sediment transport relation
 $\displaystyle{ q_{t}^* = {\frac{q_{t}}{\sqrt{R g D}D}} = \left\{\begin{matrix} 0 & if \tau^* \lt \tau_{c}^* \\ \alpha_{t} \left ( \tau^* - \tau_{c}^* \right ) ^ \left (n_{t}\right ) & if \tau^* \gt \tau_{c}^* \end{matrix}\right. }$ (9)
 $\displaystyle{ \tau^* = {\frac{\tau_{b}}{\rho R g D}} }$ (10)
• Mobile-bed equilibrium

1)

 $\displaystyle{ U{\frac{\partial U}{\partial x}} = -g {\frac{\partial H}{\partial x}} - g{\frac{\partial \eta}{\partial x}} - C_{f}{\frac{U^2}{H}} }$ (11)

2)

 $\displaystyle{ {\frac{q_{t}}{\sqrt{R g D}D}} = \left\{\begin{matrix} 0 & if {\frac{C_{f} U^2}{R g D}} \lt \tau_{c}^* \\ \alpha_{L} \left ({\frac{C_{f} U^2}{R g D}} - \tau_{c}^* \right ) ^ \left (N_{L} \right ) & if {\frac{C_{f} U^2}{R g D}} \gt \tau_{c}^*\end{matrix}\right. }$ (12)
• Dimensionless parameters

1)

 $\displaystyle{ H = H_{o} \tilde{H} }$ (13)

2)

 $\displaystyle{ q_{t} = q_{to} \tilde{q} }$ (14)

3)

 $\displaystyle{ x = L \hat{x} }$ (15)

4)

 $\displaystyle{ t = \left ( 1 - \lambda_{p} \right ) {\frac{S_{o} L^2}{q_{to}}} \hat{t} }$ (16)

5)

 $\displaystyle{ \eta = \bar{\eta} + \eta _{d} }$ (17)

6)

 $\displaystyle{ \bar{\eta} = H_{o} \tilde{\eta}_{a} \left ( \hat{t} \right ) }$ (18)

7)

 $\displaystyle{ \eta_{d} = S_{o} L \hat{\eta}_{d} \left ( \hat{x},\hat{t} \right ) }$ (19)

8) Backwater relation

 $\displaystyle{ FI {\frac{\partial \tilde{H}}{\partial \hat{x}}} = {\frac{-{\frac{\partial \hat{\eta}_{d}}{\partial \hat{x}}} - \tilde{H}^ \left (-3\right )}{1 - Fr_{o}^2 \tilde{H}^ \left (-3\right )}} }$ (20)
 $\displaystyle{ Fr_{o} = \left ({\frac{q_{w}^2}{g H_{o}^3}}\right )^ \left ({\frac{1}{2}}\right ) }$ (21)
• Relation for sediment conservation
 $\displaystyle{ FI {\frac{d \tilde{\eta}_{a}}{d \hat{t}}} = \tilde{q}|_{\hat{x} = 0} - \tilde{q}|_{\hat{x} = 1} }$ (22)
 $\displaystyle{ {\frac{\partial \hat{\eta}_{d}}{\partial \hat{t}}} = - {\frac{\partial \tilde{q}}{\partial \hat{x}}} - \left ( \tilde{q}|_{\hat{x} = 0 } - \tilde{q}_{\hat{x} = 1} \right ) }$ (23)
• Sediment transport relation
 $\displaystyle{ \tilde{q} = \left\{\begin{matrix} 0 & \tilde{H}^ \left (-2 \right ) \lt \left (\tau_{r}^*\right )^ \left (-1\right ) \\ \left ({\frac{\tilde{H}^\left (-2\right ) - \left (\tau_{r}^* \right )^ \left (-1\right )}{1 - \left (\tau_{r}^*\right )^\left (-1\right )}} \right )^ \left (N_{L}\right ) & \tilde{H}^\left (-2\right ) \gt \left (\tau_{r}^* \right )^ \left(-1\right )\end{matrix}\right. }$ (24)
 $\displaystyle{ \tau_{r}^* = {\frac{\tau_{o}^*}{\tau_{c}^*}} }$ (25)
 $\displaystyle{ \tau_{o}^* = {\frac{C_{f} U_{o}^2}{R g D}} = {\frac{C_{f}q_{w}^2}{R g D H_{o}^2}} }$ (26)
• Boundary condition for a feed flume
 $\displaystyle{ \tilde{q}|_{\hat{x} = 0 } = 1 }$ (27)
 $\displaystyle{ \tilde{H}|_{\hat{x} = 1} = 1 - \tilde{\eta}_{a} - {\frac{1}{FI}}\left ({\frac{1}{2}} + \hat{\eta}_{d}|_{\hat{x} = 1} \right ) }$ (28)
• Initial condition for a feed flume
 $\displaystyle{ \tilde{\eta}_{a}|_{\hat{t} = 0} = \tilde{\eta}_{al} }$ (29)
 $\displaystyle{ \eta_{d}|_{\hat{t} = 0} = {\frac{S_{l}}{S_{o}}} \left ({\frac{1}{2}} - \hat{x} \right ) }$ (30)
• Boundary condition for a recirculating flume
 $\displaystyle{ \tilde{q}|_{\hat{x} = 0} = \tilde{q}|_{\hat{x} = 1} }$ (31)
 $\displaystyle{ \int _{0}^1 \tilde{H} d \hat{x} = 1 }$ (32)
• Initial condition for a recirculating flume
 $\displaystyle{ \hat{\eta}_{d}|_{\hat{t} = 0} = {\frac{S_{l}}{S_{o}}} \left ({\frac{1}{2}} - \hat{x} \right ) }$ (33)

## 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: