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\section{Sediments}
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\fluidity\ is capable of simulating an unlimited number of sediment concentration classes.
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Each class has a separate grain size, density and settling velocity. The sediment behaves
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as any other tracer field, except it is subject to a settling velocity:
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\fluidity\ is capable of simulating an unlimited number of sediment concentration fields.
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Each sediment field, with concentration, $c_{i}$, behaves as any other tracer field,
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except that it is subject to a settling velocity, $u_{si}$. The equation of conservation
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of suspended sediment mass thus takes the form:
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\begin{equation}\label{eq:sediment_conc}
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\ppt{S_m} + \nabla\cdot(\bmu S_m) = \nabla\cdot(\kaptens\nabla S_m) - \sigma S_m
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The advection in the vertical direction is then modified with a downwards sinking
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velocity $u_{\mathrm{sink}}$.
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\ppt{c_i} + \nabla \cdot c_i ({\bf u} - \delta_{j3}u_{si}) = \nabla \cdot (\kaptens \nabla c_i)
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Source and absorption terms have been removed from the above equation. These will only be
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present on region boundaries.
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Each sediment field represents a discrete sediment type with a specific diameter and
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density. A distribution of sediment types can be achieved by using multiple sediment
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{\bf Notes on model set up}
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Each sediment field must have a sinking velocity. Note that this is not shown as a
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required element in the options tree as it is inherited as a standard option for all
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A sediment density, and sediment bedload field must also be defined. The sediment bedload
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field stores a record of sediment that has exited the modelled region due to settling of
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To use sediment, a linear equation of state must also be enabled
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\option{\ldots/equation\_of\_state/fluids/linear}
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\subsection{Hindered Sinking Velocity}
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The effect of suspended sediment concentration on the fall velocity can be taken into
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account by making the Sinking Velocity field diagnostic. The equation of Richardson and
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Zaki [1954] is then used to calculate the hindered sinking velocity, $u_{si}$, based upon
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the unhindered sinking velocity, $u_{s0}$, and the total concentration of sediment, $c$.
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\begin{equation}\label{eq:hindered_sinking_velocity}
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u_{si} = u_{s0}(1-c)^{2.39}
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\subsection{Deposition and erosion}
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A surface can be defined, the sea-bed, which is a sink for sediment. Once sediment
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fluxes through this surface it is removed from the system and stored in a separate
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field: the SedimentFlux field. Each sediment class has an equivalent Flux field.
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A surface can be defined, the sea-bed, which is a sink for sediment. Once sediment fluxes
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through this surface it is removed from the system and stored in a separate field: the
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Bedload field. Each sediment class has its on bedload field.
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Erosion of this bed can be modelled by applying the sediment\_reentrainment boundary
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condition. There are several options for the re-entrainment algorithm that is used to
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calculate the amount of sediment eroded from the bed.
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1. Garcia's re-entrainment algorithm
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Erosion occurs at a rate based upon the shear velocity of the flow at the bed, $u^*$, the
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distribution of particle classes in the bed, and the particle Reynolds number,
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$R_{p,i}$. The dimensionless entrainment rate for the i$^{th}$ sediment class, $E_i$, is
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given by the following equation:
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E_i = F_i \frac{AZ_i^5}{1-AZ_i^5/0.3}
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Z_i = \lambda_m \frac{u^*}{u_{si}} R_{p,i}^{0.6} \left (\frac{d_i}{d_{50}} \right)^{0.2}
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Where $F_i$ is the volume fraction of the relevant sediment class in the bed, $d_i$ is the
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diameter of the sediment in the i$^{th}$ sediment class and $d_{50}$ is the diameter for
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which 50\% of the sediment in the bed is finer. $A$ is a constant of value $1.3 \times
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$u^*$ and $R_{p,i}$ are defined by the following equations:
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u^* = \sqrt{\tau_b/\rho}
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R_{p,i} = \sqrt{Rgd^{3}}/\nu
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This is given dimension by multiplying by the sinking velocity, $u_{si}$, such that the
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total entrainment flux is:
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2. Generic re-entrainment algorithm
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Erosion occurs when the bed-shear stress is greater than the critical shear stress. Each
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sediment class has a separate shear stress, which can be input or calculated depending
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on the options chosen. Erosion flux, $E_m$ is implemented as a Neumann boundary condition
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on the deposition/erosion surface.
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sediment class has a separate shear stress, which can be input or calculated depending on
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the options chosen. Erosion flux, $E_m$ is implemented as a Neumann boundary condition on
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the bedload/erosion surface.
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\begin{equation}\label{eq:sediment_erosion_rate}
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E_m = E_{0m}\left(1-\varphi\right)\frac{\tau_{sf} - \tau_{cm}}{\tau_{cm}}
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E_m = E_{0m}\left(1-\varphi\right)\frac{\tau_{sf} - \tau_{cm}}{\tau_{cm}}
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where $E_{0m}$ is the bed erodibility constant (kgm$^{-1}$s${-1}$) for sediment class $m$,
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$\tau_{sf}$ is the bed-shear stress, $\varphi$ is the bed porosity (typically 0.3)
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and $\tau_{cm}$ is the critical shear stress
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for sediment class $m$. The critical shear stress can be input by the user or
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automatically calculated using:
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$\tau_{sf}$ is the bed-shear stress, $\varphi$ is the bed porosity (typically 0.3) and
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$\tau_{cm}$ is the critical shear stress for sediment class $m$. The critical shear stress
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can be input by the user or automatically calculated using:
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\begin{equation}\label{eq:critical_shear_stress}
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\tau_{cm} = 0.041\left(s-1\right)\rho gD
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\tau_{cm} = 0.041\left(s-1\right)\rho gD
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where s is the relative density of the sediment, i.e. $\frac{\rho_{S_{m}}}{\rho}$ and $D$ is the sediment
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diameter (mm). The SedimentFlux field effectively mixes the deposited sediment, so order of deposition
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where s is the relative density of the sediment, i.e. $\frac{\rho_{S_{m}}}{\rho}$ and $D$
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is the sediment diameter (mm). The SedimentDepositon field effectively mixes the deposited
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sediment, so order of bedload is not preserved.
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\subsection{Sediment concentration dependent viscosity}
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The viscosity is also affected by the concentration of suspended sediment. This can be
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taken account for by using the sediment\_concentration\_dependent\_viscosity algorithm on
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a diagnostic viscosity field. If using a sub-grid scale parameterisation this must be
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applied to the relevant background viscosity field.
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The equation used is that suggested by Krieger and Dougherty, 1959, and more recently by
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Sequeiros, 2009. Viscosity, $\nu$, is a function of the zero sediment concentration
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viscosity, $\nu_0$, and the total sediment concentration, $c$, as follows.
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\begin{equation}\label{eq:sediment_concentration_dependent_viscosity}
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\nu = \nu_{0}(1-c/0.65)^{-1.625}
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Note: a ZeroSedimentConcentrationViscosity tensor field is required.
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manual/embedded_models.tex
