Model help:DredgeSlotBW
DredgeSlotBW
This model is a calculator for aggradation and degradation of sediment mixtures in gravel-bed streams subject to cyclic hydrographs.
Model introduction
This program calculates the 1D bed evolution of a sand-bed river after installation of a dredge slot.
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> \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> \tau_{s}^* = {\frac{\tau_{bs}}{\rho R g D_{s50}}} </math> (2)
<math> 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> E = {\frac{A Z_{u}^5}{1 + {\frac{A}{0.3}} Z_{u}^5}} </math> (4)
<math> Z_{u} = {\frac{u_{*s}}{v_{s}}} Re_{p}^\left (0.6\right ) S_{f}^ \left (0.07\right ) </math> (5)
<math> u_{*s} = \sqrt{{\frac{\tau_{bs}}{\rho}}} </math> (6)
<math> Re_{p} = {\frac{\sqrt{R g D_{s50}} D_{s50}}{\nu}} </math> (7)
- Suspended sediment transport rate (Wright-Parker formulation)
<math> u_{*} = \left ( g H S_{f} \right )^ \left ({\frac{1}{2}}\right ) </math> (8)
<math> u_{*s} = \left ( g H_{s} S_{f} \right )^ \left ({\frac{1}{2}}\right ) </math> (9)
<math> C_{z} = {\frac{U}{u_{*}}} </math> (10)
<math> k_{c} = 11 {\frac{H}{exp \left ( \kappa C_{z} \right )}} </math> (11)
<math> q_{s} = {\frac{E u_{*} H}{\kappa}} I </math> (12)
<math> 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> {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - Fr^2}} </math> (14)
2) Froude number
<math> Fr = {\frac{q_{w}}{g^ \left ({\frac{1}{2}}\right ) H^ \left ( {\frac{3}{2}}\right )}} </math> (15)
3) Friction slope
<math> 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> \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> H = H_{s} + H_{f} </math> (18)
6) friction coefficient due to skin friction
<math> 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> \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^* >= \tau_{min}^* \\ \tau^* & \tau^* < \tau_{min}^*\end{matrix}\right. </math> (20)
8) Shields number
<math> \tau^* = {\frac{H S_{f}}{R D_{s50}}} </math> (21)
- Bed shear stress due to skin friction to total bed shear stress
<math> \phi = \left\{\begin{matrix} {\frac{0.05 + 0.7 \left ( \tau^* Fr ^ \left (0.7\right ) \right ) ^ \left (0.8\right )}{\tau^*}} & \tau^* >=\tau_{min}^* \\ 1 & \tau^* < \tau_{min}^* \end{matrix}\right. </math> (22)
- Minimum Shields number
<math> \tau_{min}^* = 0.05 + 0.7 \left ( \tau_{min}^* Fr^ \left (0.7\right ) \right ) ^ \left (0.8\right ) </math> (23)
- Calculation of Hs and Sf from known depth H
<math> 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} <= \left (\tau_{nom}^* / \tau_{min}^* \right )^ \left ({\frac{3}{4}}\right ) \\ \tau_{s} - 1 & \tau_{s} > \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> S_{f} = S </math> (25)
<math> 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 >= {\frac{R D_{50} \tau_{min}^*}{S}} \\ H \phi_{s} \left (H\right ) - H & H < {\frac{R D_{50} \tau_{min}^*}{S}} \end{matrix}\right. = 0 </math> (26)
- Trapping of wash load
1) Concentration of wash load in the ith grain size range
<math> C_{wi} = C_{uwi} exp [- {\frac{V_{swi}}{q_{w}}} \left (x - L_{su} \right )] </math> (27)
2) Exner equation
<math> \left ( 1 - \lambda_{p} \right ) {\frac{\partial \eta}{\partial t}} = - I_{f} {\frac{\partial \left ( q_{b} + q_{s} \right )}{\partial x}} + I_{f} \sum V_{swi} C_{wi} </math> (28)
3) Relaxation distance for suspended sediment profile
<math> C = {\frac{C}{r_{o}}} [1 - exp \left ( - {\frac{r_{o} v_{s}}{q_{w}}} x \right )] </math> (29)
4) Characteristic relaxation distance
<math> L_{sr} = {\frac{q_{w}}{r_{o} v_{s}}} </math> (30)
| Symbol | Description | Unit |
|---|---|---|
| B | channel width | L |
| D50 | median grain size (sand) | L |
| D90 | grain diameter such that 90% of the distribution is finer | L |
| Q | flow discharge | L3 / T |
| L | reach length | L |
| M | number of spatial intervals | |
| τs,min | minimum shear stress due to skin friction | |
| Frd | downstream Froude number | L |
| 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 | L |
| Hslot | depth of dredge slot | L |
| ru | fraction of reach length defining upstream end of dredge slot | - |
| rd | fraction of reach length defining downstream end of dredge slot | - |
| αu | upwinding coefficient, 0.5 < αu <= 1 (1 corresponds to full upwinding) | - |
| Cwi | concentration of wash load in the ith grain size range | - |
| vswi | characteristic settling velocity for that range | L / T |
| Cuwi | value of Cwi at the upstream end of the slot | - |
| Lsu | streamwise position of the upstream end of the slot | - |
| ro | equals to E / C | - |
| Lsr | characteristic relaxation distance for adjustment of the suspended sediment profile | - |
| I | user-defined parameters | - |
| qw | water discharge per unit width | L2 / L |
| Cff | friction coefficient due to form drag | - |
| φ | transverse angle of inclination of bed | - |
| C | depth-flux averaged (river) or layer-flux averaged (turbidity current) volume concentration of suspended sediment | - |
Notes
The model implements BackwaterWrightParker for the case of filling of a dredge slot. It first computes the equilibrium normal flow values of depth H, depth due to skin friction Hs and volume bed load and bed material suspended load transport rates per unit width qb and qs for given values of flood water discharge Qw, flood intermittency If, channel width B, bed sediment sizes D50 and D90 (both assumed constant), sediment submerged specific gravity R and (constant) bed slope S.
A dredge slot is then excavated at time t = 0. The hole has depth Hslot, width B and length (rd - ru)L, where L is reach length, ruL is the upstream end of the dredge slot. and rdL is the downstream end of the dredge slot. Once the slot is excavated, it is allowed to fill without further excavation. Specification of the bed porosity λp, the number of spatial intervals M, the time step Δt, the number of steps to printout Mtoprint, the number of printout after the one corresponding to the initial bed Mprint and the upwinding coefficient au completes the input.
A sufficiently deep dredge slot can capture wash load (e.g. material finer than 62.5 μm) as well as bed material load. As long as the dredge slot is sufficiently deep to prevent re-entrainment of wash load, the rate at which wash load fills the slot can be computed by means of a simple settling model.
- Note on model running
The model is allowed up to 2000 spatial steps, any more than that will cause a memory overwrite and the data cannot be trusted
The Normal flow data is included in the initial output, at the bottom of the file.
The sediment is assumed to be sufficiently uniform such that D50 and D90 do not change in space or time.
The bedload transport calculations use an Ashida-Michiue formulation (1972), the entrainment of suspended sediment uses that of Wright and Parker (2004), as does the formulation for flow resistance.
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.
