# 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)