Model help:DredgeSlotBW: Difference between revisions

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==Main equations==
==Main equations==
* bedload calculation using Ashida-Michiue formula
* Exner equation of sediment continuity
::::{|
::::{|
|width=500px|<math>q_{Bj} = 17 \tau _{*j} ^ \left ( {\frac{3}{2}} \right ) \left ( {\frac{1 - \tau _{*cj}}{\tau _{*j}}} \right ) \left ( 1 - \sqrt{ {\frac{\tau _{*cj}}{\tau _{*j}}}} \right ) \ast p_{j} </math>
|width=800px|<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>
|width=50px align="right"|(1)
|width=50p=x align="right"|(1)
|}
|}
* Entrainment of suspended sediment
* Bedload transport in sand-bed streams (Ashida and Michiue, 1972)
::::{|
::::{|
|width=500px|<math> E_{si} = {\frac{B \left ( \lambda X_{i} \right ) ^5 }{1 + {\frac{B}{0.3}} \left ( \lambda X_{i} \right ) ^ 5}} </math>
|width=800px|<math> \tau_{s}^* = {\frac{\tau_{bs}}{\rho R g D_{s50}}} </math>
|width=50px align="right"|(2)
|width=50p=x align="right"|(2)
|}
|}
::::{|
::::{|
|width=500px|<math> X_{i} = \left ( {\frac{u_{*sk}}{v_{si}}} R_{pi} ^ \left (0.6\right ) \right ) S_{0} ^ \left (0.08\right ) \left ( {\frac{D_{i}}{D_{50}}}\right ) ^ \left ( 0.2 \right ) </math>
|width=800px|<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>
|width=50px align="right"|(3)
|width=50p=x align="right"|(3)
|}
|}
* Overall suppression of entrainment due to mixture effects
* Entrainment of sand into suspension (Wright and Parker, 2004)
::::{|
::::{|
|width=500px|<math> \lambda = 1 - 0.28 \delta _{\Phi} </math>
|width=800px|<math> E = {\frac{A Z_{u}^5}{1 + {\frac{A}{0.3}} Z_{u}^5}} </math>
|width=50px align="right"|(4)
|width=50p=x align="right"|(4)
|}
|}
* Particle Reynolds number
::::{|
::::{|
|width=500px|<math> R_{pi} = {\frac{\sqrt{ R g D_{i} } D_{i}}{\nu}} </math>
|width=800px|<math> Z_{u} = {\frac{u_{*s}}{v_{s}}} Re_{p}^\left (0.6\right ) S_{f}^ \left (0.07\right ) </math>
|width=50px align="right"|(5)
|width=50p=x align="right"|(5)
|}
::::{|
|width=800px|<math> u_{*s} = \sqrt{{\frac{\tau_{bs}}{\rho}}} </math>
|width=50p=x align="right"|(6)
|}
::::{|
|width=800px|<math> Re_{p} = {\frac{\sqrt{R g D_{s50}} D_{s50}}{\nu}} </math>
|width=50p=x align="right"|(7)
|}
* Suspended sediment transport rate (Wright-Parker formulation)
::::{|
|width=800px|<math> u_{*} = \left ( g H S_{f} \right )^ \left ({\frac{1}{2}}\right ) </math>
|width=50p=x align="right"|(8)
|}
::::{|
|width=800px|<math> u_{*s} = \left ( g H_{s} S_{f} \right )^ \left ({\frac{1}{2}}\right ) </math>
|width=50p=x align="right"|(9)
|}
::::{|
|width=800px|<math> C_{z} = {\frac{U}{u_{*}}} </math>
|width=50p=x align="right"|(10)
|}
::::{|
|width=800px|<math> k_{c} = 11 {\frac{H}{exp \left ( \kappa C_{z} \right )}} </math>
|width=50p=x align="right"|(11)
|}
::::{|
|width=800px|<math> q_{s} = {\frac{E u_{*} H}{\kappa}} I </math>
|width=50p=x align="right"|(12)
|}
::::{|
|width=800px|<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>
|width=50p=x align="right"|(13)
|}
* Gradually varied flow in sand-bed rivers including the effect of bedforms
1) Backwater equation
::::{|
|width=800px|<math> {\frac{dH}{dx}} = {\frac{S - S_{f}}{1 - Fr^2}} </math>
|width=50p=x align="right"|(14)
|}
2) Froude number
::::{|
|width=800px|<math> Fr = {\frac{q_{w}}{g^ \left ({\frac{1}{2}}\right ) H^ \left ( {\frac{3}{2}}\right )}} </math>
|width=50p=x align="right"|(15)
|}
3) Friction slope
::::{|
|width=800px|<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>
|width=50p=x align="right"|(16)
|}
4) boundary shear stress in a sand-bed river
::::{|
|width=800px|<math> \tau_{b} = \tau_{bs} + \tau_{bf} = \rho \left ( C_{fs} + C_{ff} \right ) U^2 </math>
|width=50p=x align="right"|(17)
|}
5) boundary depth in a sand-bed river
::::{|
|width=800px|<math> H = H_{s} + H_{f} </math>
|width=50p=x align="right"|(18)
|}
6) friction coefficient due to skin friction
::::{|
|width=800px|<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>
|width=50p=x align="right"|(19)
|}
7) Shields number due to form drag
::::{|
|width=800px|<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>
|width=50p=x align="right"|(20)
|}
8) Shields number
::::{|
|width=800px|<math> \tau^* = {\frac{H S_{f}}{R D_{s50}}} </math>
|width=50p=x align="right"|(21)
|}
* Bed shear stress due to skin friction to total bed shear stress
::::{|
|width=800px|<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>
|width=50p=x align="right"|(22)
|}
* Minimum Shields number
::::{|
|width=800px|<math> \tau_{min}^* = 0.05 + 0.7 \left ( \tau_{min}^* Fr^ \left (0.7\right ) \right ) ^ \left (0.8\right ) </math>
|width=50p=x align="right"|(23)
|}
* Calculation of H<sub>s</sub> and S<sub>f</sub> from known depth H
::::{|
|width=800px|<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>
|width=50p=x align="right"|(24)
|}
* Calculation of the normal flow condition prevailing in the absence of the dredge slot
::::{|
|width=800px|<math> S_{f} = S </math>
|width=50p=x align="right"|(25)
|}
::::{|
|width=800px|<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>
|width=50p=x align="right"|(26)
|}
* Trapping of wash load
1) Concentration of wash load in the ith grain size range
::::{|
|width=800px|<math> C_{wi} = C_{uwi} exp [- {\frac{V_{swi}}{q_{w}}} \left (x - L_{su} \right )]  </math>
|width=50p=x align="right"|(27)
|}
2) Exner equation
::::{|
|width=800px|<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>
|width=50p=x align="right"|(28)
|}
3) Relaxation distance for suspended sediment profile
::::{|
|width=800px|<math> C = {\frac{C}{r_{o}}} [1 - exp \left ( - {\frac{r_{o} v_{s}}{q_{w}}} x \right )] </math>
|width=50p=x align="right"|(29)
|}
4) Characteristic relaxation distance
::::{|
|width=800px|<math> L_{sr} = {\frac{q_{w}}{r_{o} v_{s}}} </math>
|width=50p=x align="right"|(30)
|}
|}


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!Symbol!!Description!!Unit
!Symbol!!Description!!Unit
|-
|-
| E<sub>si</sub>
| R
| dimensionless entrainment rate of sediment in the ith grain-size range (volume entrainment per bed area per unit fractional bed content)
| sediment specific gravity, R+1
| -
| -
|-
|-
| λ
| B
| overall suppression of entrainment due to mixture effects
| channel width
| L
|-
| D<sub>50</sub>
| median grain size (sand)
| L
|-
| D<sub>90</sub>
| grain diameter such that 90% of the distribution is finer
| L
|-
| S
| bed slope
| -
| -
|-
|-
| σ<sub>Φ</sub>
| Q
| standard deviation of the bed sediment on the sedimentological Φ scale
| flow discharge
| L<sup>3</sup> / T
|-
| L
| reach length
| L
|-
| M
| number of spatial intervals
|-
|-
| τ<sub>s,min</sub>
| minimum shear stress due to skin friction
|-
|--
| Fr<sub>d</sub>
| downstream Froude number
| L
|-
| λ<sub>p</sub>
| sediment porosity
| -
| -
|-
|-
| u<sub>*sk</sub>
| I<sub>f</sub>
| shear velocity due to sin friction
| flood intermittency
| -
| -
|-
|-
| R<sub>pi</sub>
| q<sub>t</sub>
| particle Reynolds number
| total volume bed material load transport rate per unit width
| L<sup>2</sup> / T
|-
| q<sub>b</sub>
| total volume bedload transport rate per unit width
| L<sup>2</sup> / T
|-
| q<sub>s</sub>
| volume bed material suspended load transport rate per unit width
| L<sup>2</sup> / T
|-
| τ<sub>s</sub> <sup>*</sup>
| Shields number due to form drag
| -
| -
|-
|-
| B
| τ<sub>bs</sub>
| constant in newly proposed entrainment relation, equals to 7.8*10<sup>-7</sup>
| boundary shear stress due to skin friction
| -
| -
|-
|-
| S<sub>0</sub>
| ρ
| the ratio of h/D<sub>50</sub>
| water density
| M / L<sup>3</sup>
|-
| g
| acceleration due to gravity
| L / T<sup>2</sup>
|-
| D<sub>s50</sub>
| median size of surface layer sediment
| L
|-
| R
| sediment submerged specific gravity
| -
| -
|-
|-
| ν
| q<sub>b</sub> <sup>*</sup>
| kinematic viscosity
|  
| -
| -
|-
|-
| D<sub>i</sub>
| τ<sub>c</sub> <sup>*</sup>
| characteristic diameter of the ith grain-size range
| critical Shields number at the threshold of motion, equals to 0.05
| -
| -
|-
|-
| D<sub>50</sub>
| E
| median grain diameter of the bed material
| volume rate of entrainment of bed particles into bedload transport per unit bed area per unit time
| -
| -
|-
|-
| τ<sub>*j</sub>
| Z<sub>u</sub>
| dimensionless tractive force for the j diameter
|
| -
| -
|-
|-
| q<sub>Bj</sub>
| A
| the amount of bed load
| equals to 5.7 * 10<sup>-7</sup>
| -
| -
|-
|-
| τ<sub>*cj</sub>
| u<sub>*s</sub>
| critical dimensionless tractive force for the j diameter
| shear velocity due to skin friction
| L / T
|-
| v<sub>s</sub>
| particle terminal fall velocity in quiescent water
| L / T
|-
| ν
| kinematic viscosity of water
| L<sup>2</sup> / T
|-
| S<sub>f</sub>
| down-channel friction slope
| -
| -
|-
|-
| p<sub>j</sub>
| Re<sub>p</sub>
| abundance ratio of the sediment of j diameter
|  
| -
| -
|-
|-
| x
| u<sup>*</sup>
| downstream coordinate
| shear velocity
| m
| L / T
|-
|-
| η
| C<sub>z</sub>
| bed elevation
| dimensionless Chezy resistance coefficient
| m
| -
|-
|-
| q<sub>b</sub>
| U
| volume bedload transport per unit width
| depth-averaged flow velocity
| tons/annum
| L / T
|-
|-
| q<sub>s</sub>
| k<sub>c</sub>
| volume bedload transport per unit width due to skin friction
| composite roughness height associated with both skin friction and form drag
| tons/annum
| L
|-
|-
| H
| κ
| water depth
| Von Karman constant in logarithmic velocity profile
| m
| -
|-
|-
| H<sub>s</sub>
| H<sub>s</sub>
| water depth due to skin friction
| depth associated with skin friction  
| m
| L
|-
|-
| k<sub>si</sub>
| ζ<sub>b</sub>
| water surface elevation
| equals to 0.05, in Wright-Parker formulation
| m
| -
|-
|-
| Q<sub>w</sub>
| S
| flow discharge
| bed slope
| m<sup>3</sup> / s
| -
|-
| Fr
| Froude number
| -
|-
| τ<sub>b</sub>
| boundary shear stress
| -
|-
| τ<sub>bf</sub>
| boundary shear stress due to dunes
| -
|-
| H<sub>f</sub>
| depth associated with dunes
| L
|-
| C<sub>fs</sub>
| friction coefficient due to skin friction
| -
|-
| D<sub>s90</sub>
| sediment size such that 90 % of the material in the surface layer is finer
| -
|-
| τ<sup>*</sup>
| Shields number
| -
|-
| τ<sub>min</sub> <sup>*</sup>
| minimum Shields number
| -
|-
|-
| I<sub>f</sub>
| φ<sup>s</sup>
| flood intermittency
| ratio of bed shear stress due to skin friction to total bed shear stress
| -
| -
|-
|-
| R<sub>r</sub>
| S<sub>nom</sub>
| submerged specific gravity
| equals to S<sub>f</sub> φ<sub>s</sub><sup>-4/3</sup>
|  
| -
|-
|-
| S
| τ<sub>nom</sub> <sup>*</sup>
| bed slope
| equals to H S<sub>nom</sub> / R D<sub>s50</sub>
| -
| -
|-
|-
| x
| downstream coordinate
| L
|-
| ξ<sub>d</sub>
| downstream water surface elevation, must be larger than the beginning point water surface elevation
| L
|-
| η
| bed surface elevation
| L
|-
| H
| flow depth
| L
|-
| H<sub>s</sub>
| flow depth due to skin friction
| L
| L
| reach length
| m
|-
|-
| H<sub>slot</sub>
| H<sub>slot</sub>
| depth of dredge slot
| depth of dredge slot
| m
| L
|-
|-
| r<sub>U</sub>
| r<sub>u</sub>
| fraction of reach length for upstream end of dredge slot
| fraction of reach length defining upstream end of dredge slot
| -
| -
|-
|-
| r<sub>d</sub>
| r<sub>d</sub>L
| fraction of reach length for downstream end of dredge slot
| fraction of reach length defining downstream end of dredge slot
| -
|- 
| λ<sub>p</sub>
| bed porosity
| -
| -
|-
|-
| a<sub>U</sub>
| α<sub>u</sub>
| upwinding coefficient: a<sub>u</sub> = 1 corresponds to full upwinding, 0.5 for central difference
| upwinding coefficient, 0.5 < α<sub>u</sub> <= 1 (1 corresponds to full upwinding)
| -
| -
|-
|-
| M
| C<sub>wi</sub>
| number of spatial steps (<=2000)
| concentration of wash load in the ith grain size range
| -
| -
|-
|-
| t
| v<sub>swi</sub>
| time step
| characteristic settling velocity for that range
| yr
| L / T
|-
|-
| Δt
| C<sub>uwi</sub>
| time step
| value of C<sub>wi</sub> at the upstream end of the slot
| -
| -
|-
|-
| Δx
| L<sub>su</sub>
| spatial step length
| streamwise position of the upstream end of the slot
| -
| -
|-
|-
| Mtoprint
| r<sub>o</sub>
| number of time steps to printout
| equals to E / C
| -
| -
|-
|-
| Mprint
| L<sub>sr</sub>
| number of printouts
| characteristic relaxation distance for adjustment of the suspended sediment profile
| -
| -
|-
|-
|}
|}


   </div>
   </div>
</div>
</div>
==Notes==
==Notes==
In the calculation of River Bed Elevation Variation with a Dredge Slot: the river is assumed to be sand-bed. The calculation proceeds using a backwater formulation. Flow resistance is computed using the Wright-Parker (2004) formulation. The bedload transport rate is computed using the Ashida-Michiue (1972) formulation. The rate of entrainment into suspension is computed using the Wright-Parker formulation without the stratification correction.
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 H<sub>s</sub> and volume bed load and bed material suspended load transport rates per unit width q<sub>b</sub> and q<sub>s</sub> for given values of flood water discharge Q<sub>w</sub>, flood intermittency I<sub>f</sub>, channel width B, bed sediment sizes D<sub>50</sub> and D<sub>90</sub> (both assumed constant), sediment submerged specific gravity R and (constant) bed slope S.


The calculation begins with the assumption of a prevailing mobile-bed normal flow equilibrium before installation of the dredge slot. The flow depth H, volume bedload transport rate per unit width q<sub>b</sub> and volume suspended transport rate per unit width q<sub>s</sub> at normal flow are computed based on input values of discharge Q<sub>ww</sub>, channel width B, bed material sizes D<sub>50</sub> and D<sub>90</sub>, sediment submerged specific gravity R<sub>r</sub> and bed slope S.
A dredge slot is then excavated at time t = 0. The hole has depth H<sub>slot</sub>, width B and length (r<sub>d</sub> - r<sub>u</sub>)L, where L is reach length, r<sub>u</sub>L is the upstream end of the dredge slot. and r<sub>d</sub>L 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 λ<sub>p</sub>, 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 a<sub>u</sub> completes the input.


The sediment is assumed to be sufficiently uniform so that D<sub>50</sub> and D<sub>90</sub> are unchanging in space and time. The input parameter Inter specifies the fraction of any year for which flood flow prevails. At other times of the year the river is assumed to be morphologically dormant.
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.


The reach is assumed to have length L. The dredge slot is excavated at time t = 0, and then allowed to fill in time with no subsequent excavation. The depth of initial excavation below the bottom of the bed prevailing at normal equilibrium is an input variable with the name Hslot. The dredge slot extends from an upstream point equal to r<sub>u*L</sub> to a downstream point rd*Hslot, where ru and rd are user-input values.
The porosity lamp of the sediment deposit is a user-input parameter.
The bedload transport relation used in the calculation is that of Ashida and Michiue (1972). The formulation for entrainment of sediment into suspension is that of Wright and Parker (2004). The formulation for flow resistance is that of Wright and Parker (2004). The flow stratification correction of Wright-Parker is not implemented here for simplicity. A quasi-equilibrium formulation is used to computed the transport rate of suspended sediment from the entrainment rate.
A backwater calculation is used to compute the flow. The water surface elevation at the downstream end of the reach is held constant at the value associated with normal flow equilibrium.
Iteration is required to compute: a) the flow depth prevailing at normal flow; b) the friction slope and depth prevailing at normal flow, b) the friction slope and depth associated with skin friction associated with skin friction from any given value of depth, and b) the minimum Shields number below which form drag is taken to vanish.


* Note on model running
* Note on model running
Line 367: Line 587:
==References==
==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)
* 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.
* 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.


Revision as of 12:14, 26 May 2011

The CSDMS Help System

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

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
Flow discharge (Q) m3 / s
Flood Intermittency (I) flood intermittency -
Channel width (B) m
Median grain size (sand) (d) median grain diameter of sand mm
Grain size such that 90% is finer (sand) (D) mm
Submerged specific gravity of sediment (R) -
Bed slope (S) -
Reach slope
Reach length (L) m
Depth of dredge slot (H) m
Fraction of reach length defining upstream end of dredge slot (u) -
Fraction of reach length defining downstream end of dredge slot (r) -
Bed porosity (l) -
Upwinding coefficient (1 = full upwinding, 0.5 for central difference) -
Number of spatial steps desired (no more than 2000) (M) -
Time step (t) days
Number of iterations per print statement (i) -
Iterations of prints desired (p) -
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

  • 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 }[/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 Hs and Sf 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)
  • Trapping of wash load

1) Concentration of wash load in the ith grain size range

[math]\displaystyle{ C_{wi} = C_{uwi} exp [- {\frac{V_{swi}}{q_{w}}} \left (x - L_{su} \right )] }[/math] (27)

2) Exner equation

[math]\displaystyle{ \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]\displaystyle{ C = {\frac{C}{r_{o}}} [1 - exp \left ( - {\frac{r_{o} v_{s}}{q_{w}}} x \right )] }[/math] (29)

4) Characteristic relaxation distance

[math]\displaystyle{ L_{sr} = {\frac{q_{w}}{r_{o} v_{s}}} }[/math] (30)

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:

See also: Help:Images or Help:Movies

Developer(s)

Gary Parker

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.

Links

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