Model help:AgDegNormGravMixPW
AgDegNormGravMixPW
This is the calculator for aggradation and degradation of sediment mixtures in gravel-bed streams.
Model introduction
The program AgDegNormGravMixPW is an extension of AgDegNormal for sediment mixtures in gravel bed rivers where the channel bed material is transported as bedload only.
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
- Characteristic diameter for the ith size range
[math]\displaystyle{ D_{i} = \left ( D_{b,i} \ast D_{b,i+1} \right ) ^ \left ( {\frac{1}{2}} \right ) }[/math] (1)
- The fraction of sample in the ith size range
[math]\displaystyle{ f_{i} = f_{f,i+1} - f_{f,i} }[/math] (2)
- The conservation of sediment in each grain size range (based on Exner equation)
[math]\displaystyle{ \left ( 1 - \lambda _{p} \right ) [L_{a} {\frac{\partial F_{i}}{\partial t}} + \left ( F_{i} - f_{li} \right ) {\frac{\partial L_{a}}{\partial t}}] = -I_{f} {\frac{\partial q_{bT} p_{i}}{\partial x}} + I_{f} f_{li} {\frac{\partial q_{bT}}{\partial x}} }[/math] (3)
- Fraction of sediment in the ith grain size range at the active-layer substrate interface (if ∂η / ∂t > 0)
[math]\displaystyle{ f_{li} = \left\{\begin{matrix} \alpha F_{i} + \left ( 1 - \alpha \right ) p_{i} & {\frac{\partial \eta}{\partial t}} \gt 0 \\ f_{sub,i} & {\frac{\partial \eta}{\partial t}} \lt 0 \end{matrix}\right. }[/math] (4)
- Roughness height due to skin friction
[math]\displaystyle{ k_{s} = n_{k} D_{s90} }[/math] (5)
- Thickness of the active layer
[math]\displaystyle{ L_{a} = n_{a} D_{s90} }[/math] (6)
Symbol | Description | Unit |
---|---|---|
Q | flood discharge | L ^{3} / T |
t | time step | T |
B | river width | L |
D | grain size of the bed sediment | L |
D_{bi} | bound diameter | L |
λ_{p} | bed porosity | - |
α | the parameter that governs the grain size distribution of the sediment at the active layer-substrate interface during bed aggredation | - |
F_{fi} | grain size distribution of the active layer for initial condition | - |
F_{i} | fraction of sediment in the ith grain size range in the active layer | - |
f_{subfi} | fraction of sediment in the ith grain size range in the substrate layer for initial condition | - |
F_{fli} | percent finer than ith grain size range for the bed surface for initial condition | - |
F_{subfli} | percent finer than ith grain size range for the substrate layer for initial condition | - |
f_{li} | fraction of sediment in the ith grain size range in the active-layer substrate interface | - |
p_{i} | fraction of sediment in the ith grain size range in the bedload | - |
F_{sub,i} | fraction of substrate material in the ith size range | - |
D_{s90} | the diameter of the bed surface such that the 90% of the sediment is finer | L |
n_{a} | user specified order-one non dimensional constant | - |
p_{ffi} | the percent that is finer than the ith size range for upstream boundary conditon | - |
η_{d} | fixed bed elevation at the downstream end of the modeled reach | L |
k_{c} | composite roughness height | L |
G | imposed annual sediment transfer rate from upstream | M / L |
G_{tf} | upstream sediment feed rate | - |
ξ_{d} | downstream water surface elevation | L |
L | length of reach under consideration | L |
q_{w} | water discharge per unit width | L^{2} / T |
i | number of time steps per printout | - |
p | number of printouts desired | - |
M | number of spatial intervals | - |
R | submerged specific gravity of sediment | - |
S_{f} | friction slope | - |
F_{r} | Froude number | - |
U | flow velocity | L / T |
g | acceleration of gravity | L/ T^{2} |
α_{r} | coefficient in Manning-Stricker, dimensionless coefficient between 8 and 9 | - |
k_{s} | grain roughness | L |
n_{k} | dimensionless coefficient typically between 2 and 5 | - |
τ^{*} | Shield number | - |
ρ | fluid density | M / L^{3} |
ρ_{s} | sediment density | M / L^{3} |
τ_{c} | critical Shields number for the onset of sediment motion | - |
ψ_{s} | the fraction of bed shear stress | - |
q_{t} ^{*} | Einstein number | - |
I_{f} | flood intermittency | - |
t_{f} | cumulative time the river has been in flood | T |
G_{t} | the annual sediment yield | M / T |
t_{a} | the number of seconds in a year | - |
Q_{f} | sediment transport rate during flood discharge | |
α_{t} | dimensionless coefficient in the sediment transport equation, equals to 8 | - |
n_{t} | exponent in sediment transport relation, equals to 1.5 | - |
τ_{c} ^{*} | reference Shields number in sediment transport relation, equals to 0.047 | |
C_{f} | bed friction coefficient, equals to τ_{b} / (ρ U^{2} ) | - |
C_{Z} | dimensionless Chezy resistance coefficient. | |
S_{l} | initial bed slope of the river | - |
η_{i} | initial bed elevation | L |
D_{sub50} | median size of the substrate layer | L |
D_{subg} | geometric mean size of the substrate layer | L |
L_{a} | thickness of the active layer | L |
x | downstream coordinate | L |
τ | shear stress on bed surface | - |
q_{b} | bed material load | M / T |
Δx | spatial step length, equals to L / M | L |
Q_{w} | flood discharge | L^{3} / T |
Δt | time step | T |
Ntoprint | number of time steps to printout | - |
Nprint | number of printouts | - |
a_{U} | upwinding coefficient (1=full upwind, 0.5=central difference) | - |
α_{s} | coefficient in sediment transport relation | - |
Output
Symbol | Description | Unit |
---|---|---|
η | bed surface elevatioon | L |
H | water depth | L |
τ_{b} | bed shear stress | M / (T^{2} L) |
S | bed slope | - |
q_{t} | total bed material load | L^{2} / T |
Notes
This program computes the time evolution of the long profile of a river of constant width carrying a mixture of gravel sizes, the downstream end of which has a prescribed elevation. In particular, the program computes the time evolution of the spatial profiles of bed elevation, total gravel bedload transport rate and grain size distribution of the surface (active) layer of the bed.
Gravel-bed rivers tend to be poorly-sorted. During floods, bed material load consists almost exclusively of bedload. (Sand is often transported in copious quantities as washload during floods.) The surface material (armor or pavement) tends to be coarser than the substrate. By definition the median size or geometric mean size of the substrate is in the gravel range, but the substrate may contain up to 30% sand in the interstices of an otherwise clast-supported deposit.
The grain size distribution of the bed material is specified in terms of 12 size bounds, Dbi with i = 1 …12, such that f_{fi} denotes the mass fraction of the sample that is finer than D_{bi}. The 12 bound diameters specify 11 grain size ranges defined by (D_{b,i}, D_{b,i+1}) and (f_{f,i}, f_{f,i+1}). For each size range the model computes the characteristic diameter and the fraction of sample in the ith size range.
The flow is assumed normal and the water depth can be computed with either a Manning-Strickler or a Chezy formulation can be used.
The exchange of sediment between the bedload and the bed deposit is modeled with a two-layer model for the channel bed. The bed deposit is divided in two regions, 1) the substrate and 2) the active (or surface or armor) layer.
The grain size distribution of the active layer is assumed to be constant in the vertical and it may vary in the streamwise direction and in time, i.e. the active layer is assumed well-mixed. The grain size distribution of the substrate, in principle, may vary in both streamwise and vertical direction, but it is constant in time. The only way the grain size distribution of the substrate may vary in time is by creating new substrate via bed aggradation. In the present model, the grain size distribution of the substrate is assumed to be constant in space and in time, therefore it works only for the cases of aggradation always and everywhere or degradation always and everywhere. In the case of aggradation followed by degradation, it is necessary to modify the code so that the vertical variation of the grain size distribution of the new substrate created by aggradation is stored in memory.
No attempt is made in this code to decompose the bed resistance into skin friction and form drag. The constant to convert total boundary shear stress to that due to skin friction, ψ_{s}, is set equal to 1 and consequently the composite roughness height for the Manning-Strickler formulation, k_{c}, is equal to the roughness height due to skin friction k_{s}. The roughness height and the thickness of the active layer are computed with the liner functions of the diameter of the bed surface such that the 90% of the sediment is finer.
To compute the bedload transport rate the user can choose from two surface-based bedload transport formulations; those of Parker (1990) and Wilcock and Crowe (2003). In the relation of Parker (1990) the surface grain size distributions need to be renormalized to exclude sand before specification as input to the program. This step is neither necessary nor desirable in the case of the relation of Wilcock and Crowe (2003), where the sand plays an important role in mediating the gravel bedload transport.
The program does not store the vertical and streamwise structure of the new substrate created as the bed aggrades.It is assumed in the model that the grain size distribution of the sediment feed rate does not change in time, the initial grain size distribution of the surface material is the same at every node, the grain size distribution of the substrate is the same at every node and does not vary in the vertical.
The flow is calculated using the normal flow (local equilibrium) approximation.
- Note on model running
In the case of the load relation due to Parker (1990), the grain size distributions are automatically renormalized because the relation is for the transport of gravel only in the case of the load relation due to Wilcock-Crowe (2003), the sand and the fine sediment are retained for the computation
The user will be prompted by the program as to which bedload relation he would like to use.
The input grain size distributions may be on a 0-100% or a 0.00-1.00 scale, and the program will automatically scale.
The input grain size distributions must have bounds at 0% and 100% (1.00) to properly perform the calculation. If the user does not input the bounds the program will automatically interpolate upper and lower bounds D_{bU} and D_{bL} such that f_{fU} = 100 (1.00) and f_{fL} = 0
The water depth is calculated using a Chézy formulation, when only the Chézy coefficient is specified in the input text file. The Manning-Strickler formulation is implemented, when only the coefficients αr and nk are given in the inputfile. When all the three parameters are present, the program will ask the user which formulation they would like to use.
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
Parker, G., 1990, Surface-based bedload transport relation for gravel rivers, Journal of Hydraulic Research, 28(4): 417-436.
Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment, Journal of Hydraulic Engineering, 129(2), 120-128.