Model help:DepDistTotLoadCalc
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DepDistTotLoadCalc
This is an illustration of calculation of depth-discharge relation, bed load transport, suspended load transport and total bed material load for a large, low-slope sand-bed river.
Model introduction
This model is a Depth-Discharge and Total Load calculator, uses:
1. Wright-Parker formulation for flow resistance, 2. Ashida-Michiue formulation for bedload transport, 3. Wright-Parker formulation (without stratification) for suspended load.
Model parameters
Uses ports
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Provides ports
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Main equations
[math]\displaystyle{ \tau_{s} ^* = {\frac{H_{s} S}{R D_{50}}} }[/math] (1)
[math]\displaystyle{ U = 8.32 sqrt \left (g H_{s} S \right ) \left ( {\frac{H_{s}}{k_{s}}} \right ) ^ \left ( {\frac{1}{6}} \right ) }[/math] (2)
[math]\displaystyle{ H = \left ( \Gamma {\frac{R D_{s50}}{S}} \left ( {\frac{sqrt \left ( g \right )}{U}} \right ) ^ \left ( 0.7 \right ) \right ) ^ \left ( {\frac{20}{13}} \right ) }[/math] (3)
[math]\displaystyle{ \Gamma = \left ( {\frac{\tau_{s} ^* - 0.05}{0.7}} \right ) ^ \left ( {\frac{5}{4}} \right ) }[/math] (4)
[math]\displaystyle{ \tau^* = {\frac {H S}{R D_{50}}} }[/math] (5)
[math]\displaystyle{ F_{r}= {\frac{U}{sqrt \left ( g H \right )}} }[/math] (6)
[math]\displaystyle{ u_{*} = sqrt \left ( g H S \right ) }[/math] (7)
[math]\displaystyle{ u_{*s} = sqrt \left ( g H_{s} S \right ) }[/math] (8)
[math]\displaystyle{ q_{b} = sqrt \left ( R g D_{50} \right ) D_{50} \left ( \tau _{s} ^* -0.05 \right ) \left ( sqrt \left ( \tau _{s} ^* \right ) - sqrt \left ( 0.05 \right ) \right ) }[/math] (9)
[math]\displaystyle{ C_{z} = {\frac{U}{u_{*}}} }[/math] (10)
[math]\displaystyle{ k_{c} = {\frac{11H}{e^ \left ( \kappa C_{z} \right )}} }[/math] (11)
[math]\displaystyle{ Z_{u} = {\frac{u_{*s}}{v_{s}}} Re_{p} ^ \left ( 0.6 \right ) S \left ( 0.07 \right ) }[/math] (12)
[math]\displaystyle{ E = {\frac{5.7 * 10^\left ( -7 \right ) Z_{u} ^5}{1 + {\frac{5.7 * 10^\left ( -7 \right )}{0.3}} Z_{u} ^5}} }[/math] (13)
[math]\displaystyle{ q_{s} = {\frac{u_{*} E H}{\kappa}} I }[/math] (14)
[math]\displaystyle{ q_{t} = q_{s} + q_{b} }[/math] (15)
[math]\displaystyle{ I = \Sigma \left ( {\frac{\left ( 1 - \zeta \right ) / \zeta}{\left ( 1 - \zeta _{b} \right ) / \zeta_{b}}} \right ) ^ {\frac{V_{s}}{\kappa u_{*}}} ln \left ( 30 {\frac{H}{k_{c}}} \zeta \right ) d \zeta }[/math] (16)
Nomenclature
| Symbol | Description | Unit |
|---|---|---|
| S | bed slope | - |
| D50 | median sediment size | mm |
| D90 | size such that 90% of the the sediment if finer | mm |
| ks | grain roughness height | mm |
| R | submerged specific gravity of sediment | - |
| ν | kinematic viscosity of water | m2 / s |
| nk | factor such that ks = nk Ds90 | kg/m3 |
| Rep | ||
| Rf | - | |
| vs | fall velocity of size Ds50 | cm / s |
| τb | boundary shear stress on the bed | kg / (m s) |
| u | shear velocity on the bed | m / s |
| qbi | volume gravel bedload transport per unit width of grains in the ith size range | m 2 |
| u* | shear velocity | m / s |
| Dlg | geometric mean of the bedload | mm |
| σlg | geometric standard deviation of the bedload | - |
| Dsx | size in the surface material, such that x percentage of the material is finer | mm |
| Dlx | size in the bedload material, such that x percentage of the material is finer | - |
| Dsx | size in the surface material, such that x percentage of the material is finer | mm |
| Hs | depth associated with skin friction | m |
| U | depth- or cross sectionally-averaged flow velocity or layer-average velocity (turbidity current) | m / s |
| H | cross-sectionally averaged flow depth (river) or flow thickness (turbidity current) | m |
| qw | water discharge per unit width | m2 / s |
| τ* | Shields number | - |
| Fr | Froude number | - |
| u* | shear velocity | m / s |
| u*s | shear velocity due to skin friction | m / s |
| qb | volume bedload transport rate per unit width | m2 / s |
| qb | volume bedload transport rate per unit width | m2 / s |
| Cz | dimensionless Chezy resistance coefficient | - |
| kc | composite roughness height associated with both skin friction and form drag | - |
| E | dimensionless rate of entrainment of bed sediment into suspension | m2 / s |
| I | ||
| qs | volume suspended load transport rate per unit width | m 2 / s |
| qt | volume total bed material transport rate per unit width | m2 / s |
| ζ | dimensionless upward normal coordinate | - |
Notes
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References
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