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When configuring ocean problems (or single material/single phase fluids
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problems), only one \option{/material\_phase} is required. Multi-material
722
and multi-phase problems will require one \option{/material\_phase} for each
722
and multiphase problems will require one \option{/material\_phase} for each
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723
phase or material in the problem.
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Note that you must give each of your material phases a name.
1184
1184
\index{conjugate gradient}
1185
1185
\index{multigrid}
1187
As described in Section \ref{ND_Linear_solvers}, for the solution of large sparse linear systems, the so called iterative methods are usually employed. These methods avoid having to explicitly construct the inverse of the matrix, which is generally dense and therefore costly to compute (both in memory and computer time). FLUIDITY is linked to PETSc: a suite of data structures and routines for the scalable (parallel) solution of scientific applications modelled by partial differential equations. It employs the MPI standard for parallelism. FLUIDITY therefore supports any iterative method provided by the PETSc library (http://www-unix.mcs.anl.gov/petsc/petsc-2/snapshots/petsc-dev/docs/manualpages/KSP/KSPType.html --- available methods may depend on the PETSc library installed on your system). Examples include Conjugate Gradient (CG), GMRES and FGMRES (Flexible GMRES). Some options are explicitly listed under \option{solver/iterative\_method}, for example CG: \option{solver/iterative\_method::cg}, whereas others can be selected entering the name of the chosen scheme in \option{solver/iterative\_method}.
1187
As described in Section \ref{ND_Linear_solvers}, for the solution of large sparse linear systems, the so called iterative methods are usually employed. These methods avoid having to explicitly construct the inverse of the matrix, which is generally dense and therefore costly to compute (both in memory and computer time). FLUIDITY is linked to PETSc: a suite of data structures and routines for the scalable (parallel) solution of scientific applications modelled by partial differential equations. It employs the MPI standard for parallelism. FLUIDITY therefore supports any iterative method provided by the PETSc library (http://www-unix.mcs.anl.gov/petsc/petsc-2/snapshots/petsc-dev/docs/manualpages/KSP/KSPType.html --- available methods may depend on the PETSc library installed on your system). Examples include Conjugate Gradient (CG), GMRES and FGMRES (Flexible GMRES). Some options are explicitly listed under \option{solver/iterative\_method}, for example CG: \option{solver/iterative\_method::cg}, whereas
1188
others can be selected entering the name of the chosen scheme in \option{solver/iterative\_method}.
1189
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\subsection{Preconditioner}
1190
1191
\index{linear solvers!preconditioners}
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\subsubsection{Linear fluid EOS}
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\index{equation of state!linear}
1269
The density is a linear function of temperature and salinity:
1270
The density is a linear function of temperature, salinity and any number of generic scalar fields:
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\begin{equation}
1271
\rho=\rho_0 \left(1-\alpha(T-T_0)+\beta (S-S_0)\right),
1272
\rho=\rho_0 \left(1-\alpha(T-T_0)+\beta (S-S_0) - \gamma (F - F_0) \right),
1273
where $\rho_0, \alpha, T_0, \beta$ and $S_0$ are set by the following
1274
where $\rho_0, \alpha, T_0, \beta, S_0, \gamma$ and $F_0$ are set by the following
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\begin{description}
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\item \option{\ldots/linear/reference\_density} sets $\rho_0$
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\item \option{\ldots/linear/temperature\_dependency/reference\_temperature} sets $T_0$
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\item \option{\ldots/linear/salinity\_dependency/salinity\_contraction\_coefficient} sets $\beta$
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\item \option{\ldots/linear/salinity\_dependency/reference\_salinity} sets $S_0$
1282
\item \option{\ldots/linear/generic\_scalar\_field\_dependency/expansion\_coefficient} sets $\gamma$
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\item \option{\ldots/linear/generic\_scalar\_field\_dependency/reference\_value} sets $T_0$
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\end{description}
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Note that for Boussinesq the reference density does not
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influence any of the terms in the
1502
1505
q = (\kaptens\nabla c)\cdot\bmn,\quad \textrm{on}\quad \partial\Omega.
1503
1506
\end{equation*}
1508
\subsubsection{Robin}
1509
\index{boundary conditions!Robin}
1511
A Robin boundary condition sets the diffusive flux term $\vec{n}\cdot\tensor{\kappa}\cdot\grad c$ in weak form as:
1513
- \int_{\partial\Omega} \phi~\vec{n}\cdot\tensor{\kappa}\cdot\grad c =
1514
\int_{\partial\Omega} \phi~h(c-c_{a}),
1516
where $\phi$ is a test function (see section \ref{chap:numerical_discretisation}).
1517
The Robin condition is specified by assigning a value to the order zero $C_{0}$ and order one $C_{1}$ coefficient fields, where
1522
Currently, the Robin boundary condition is only available for Continuous Galerkin and Control Volume
1523
spatial discretisations of a scalar field.
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\subsubsection{Bulk formulae}\label{sec:bulk_formulae}
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These boundary conditions can be used on:
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Models with one prognostic velocity field per \option{material\_phase} are referred to as multiple phase models. The use of these multiple velocity fields permits the inter-penetration and interaction between different phases.
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\subsubsection{Simulation requirements}
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In a multi-phase simulation, each \option{material\_phase} requires:
2317
In a multiphase simulation, each \option{material\_phase} requires:
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\begin{itemize}
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\item an equation of state, and
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\item a PhaseVolumeFraction scalar field.
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\alpha_N = 1 - \sum_{i=1}^{N-1}\alpha_i\textrm{.}
2312
\subsubsection{Inter-phase interactions}
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Inter-phase interactions can be included under \option{../multiphase\_properties} in Diamond. Currently, \fluidity\ only supports fluid-particle drag between the continuous phase and dispersed phase(s), given by:
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\begin{equation}\label{eq:multiphase_drag_term}
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\subsubsection{Inter-phase momentum interaction}
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Currently, \fluidity\ only supports fluid-particle drag between the continuous phase and dispersed phase(s). This can be enabled by switching on the \option{/multiphase\_interaction/fluid\_particle\_drag} option in Diamond and specifying the drag correlation: Stokes, \cite{wen_yu_1966} or \cite{ergun1952}.
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The drag force using the Stokes drag correlation is given by:
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\begin{equation}\label{eq:stokes_drag_force}
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\mathbf{F}_D = \frac{3\ \alpha_p\ C_D\ \alpha_f\ \rho_f\ |\mathbf{u}_f-\mathbf{u}_p|\ (\mathbf{u}_f-\mathbf{u}_p)}{4\ d},
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where $f$ and $p$ denote the fluid (i.e. continuous) and particle (i.e. dispersed) phases respectively, and $d$ is the diameter of a single particle in the dispersed phase. The drag coefficient $C_D$ is defined as:
2318
\begin{equation}\label{eq:drag_coefficient}
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\begin{equation}\label{eq:stokes_drag_coefficient}
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C_D = \frac{24}{\mathrm{Re}},
2325
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Note that $\mu_f$ denotes the isotropic viscosity of the fluid (i.e. continuous) phase.
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Within a dispersed phase, entering a value for $d$ in the \option{../multiphase\_properties/particle\_diameter} option enables fluid-particle interaction between itself and the continuous phase.
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With the drag correlation by \cite{wen_yu_1966}, $\mathbf{F}_D$ and $C_D$ become:
2350
\begin{equation}\label{eq:wen_yu_drag_force}
2351
\mathbf{F}_D = \frac{3\ \alpha_p\ C_D\ \alpha_f\ \rho_f\ |\mathbf{u}_f-\mathbf{u}_p|\ (\mathbf{u}_f-\mathbf{u}_p)}{4\ d\alpha_f^{2.7}},
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\begin{equation}\label{eq:wen_yu_drag_coefficient}
2354
C_D = \frac{24}{\mathrm{Re}}\left(1.0 + 0.15\mathrm{Re}^{0.687}\right).
2356
In contrast to the Stokes drag correlation, the \cite{wen_yu_1966} drag correlation is more suitable for larger particle Reynolds number flows.
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For dense multiphase flows with $\alpha_p > 0.2$, the drag correlation by \cite{ergun1952} is often the most appropriate:
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\begin{equation}\label{eq:ergun_drag_force}
2360
\mathbf{F}_D = \left(150\frac{\alpha_p^2\mu_f}{\alpha_f d_p^2} + 1.75\frac{\alpha_p\rho_f|\mathbf{u}_f-\mathbf{u}_p|}{d_p}\right)\left(\mathbf{u}_f-\mathbf{u}_p\right).
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Note that within each dispersed phase, a value for $d$ must be specified in \option{../multiphase\_properties/particle\_diameter} regardless of which drag correlation is used.
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\subsubsection{Inter-phase energy interaction}
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This subsection considers compressible multiphase flow simulations where each phase has its own InternalEnergy field. Users can choose to include the inter-phase heat transfer term by \cite{gunn1978}, denoted $Q$ in Section \ref{sec:multiphase_equations}, on the RHS of each internal energy equation:
2368
\begin{equation}\label{eq:gunn_heat_transfer_term}
2369
Q = \frac{6 k \alpha_p \mathrm{Nu}_p}{d_p^2}\left(\frac{e_f}{C_f} - \frac{e_p}{C_p}\right),
2373
\mathrm{Nu}_p = \left(7 - 10\alpha_f + 5\alpha_f^2\right)\left(1 + 0.7\mathrm{Re}_p^{0.2}\mathrm{Pr}^{\frac{1}{3}}\right) + \left(1.33 - 2.4\alpha_f + 1.2\alpha_f^2\right)\mathrm{Re}_p^{0.7}\mathrm{Pr}^{\frac{1}{3}},
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\mathrm{Pr} = \frac{C_f \gamma \mu_f}{k},
2380
\mathrm{Re} = \frac{\rho_f d_p |\mathbf{u}_f-\mathbf{u}_p|}{\mu_f},
2382
are the Nusselt, Prandtl and Reynolds number respectively. $C_i$ denotes the specific heat of phase i at constant volume, and $k$ denotes the effective conductivity of the fluid phase; these must be specified in \option{../multiphase\_properties/effective\_conductivity} and \option{../multiphase\_properties/specific\_heat}. Note that we have written the above in terms of internal energy (rather than temperature) by dividing $e_i$ by the specific heat at constant volume.
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To include this term, switch on the \option{/multiphase\_interaction/heat\_transfer} option in Diamond.
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\subsubsection{Current limitations}
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\begin{itemize}
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\item Boussinesq and fully compressible multi-phase simulations are not yet supported.
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\item Boussinesq multiphase simulations are not yet supported.
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\item The momentum equation for each \option{material\_phase} can only be discretised in non-conservative form.
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\item The stress term must be in tensor form, and only works with an isotropic viscosity.
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\item \option{bassi\_rebay} and \option{compact\_discontinuous\_galerkin} are currently the only DG viscosity schemes available for multi-phase flow simulations.
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\item \option{bassi\_rebay} and \option{compact\_discontinuous\_galerkin} are currently the only DG viscosity schemes available for multiphase flow simulations.
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\item Discontinuous \option{PhaseVolumeFraction} fields are not yet supported.
2336
\item The Pressure field only supports \option{continuous\_galerkin} and \option{control\_volume} discretisations.
2337
\item Prescribed velocity fields cannot yet be used in multi-phase simulations.
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\item For compressible multiphase flow simulations, the Pressure field only supports a \option{continuous\_galerkin} discretisation.
2394
\item Prescribed velocity fields cannot yet be used in multiphase simulations.
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\item Fluid-particle drag can currently only support one continuous (i.e. fluid) phase.
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This section describes how to configure \fluidity to simulate single phase incompressible Darcy flow.
2380
First, the \option{Porosity} and \option{Permeability} fields found under the option element \option{/porous\_media} require configuring. Both fields can be either \option{prescribed} or \option{diagnostic}. The \option{Permeability} field can be either a \option{scalar\_field} or a \option{vector\_field}. The \option{Porosity} field will only be used in the transport of scalar fields and the metric tensor. The \option{Permeability} field will only be used in forming the absorption coefficient in the Darcy velocity equation. The use of these two fields should be considered when selecting their associated mesh. If \option{Porosity} is to be included in the transport of scalar fields (and the metric tensor) that are discretised using control volumes, due to the way this method is coded, the mesh associated with the \option{Porosity} must be either element wise (meaning discontinuous order zero) or the same order as field to be transported. It is therefore recommended that the \option{Porosity} field always be associated with a mesh that is discontinuous order zero. It is recommended that the \option{Permeability} field also be associated with a mesh that is discontinuous order zero, due to the recommended Darcy velocity - pressure element pair (being p0p1cv or p0p1 with the continuity tested with the p1 control volume dual space)
2437
First, the \option{Porosity} and \option{Permeability} fields found under the option element \option{/porous\_media} require configuring. Both fields can be either \option{prescribed} or \option{diagnostic}. The \option{Permeability} field can be either a \option{scalar\_field} or a \option{vector\_field}. The \option{Porosity} field will only be used in the transport of scalar fields and the metric tensor. The \option{Permeability} field will only be used in forming the absorption coefficient in the Darcy velocity equation. The use of these two fields should be considered when selecting their associated mesh. If \option{Porosity} is to be included in the transport of scalar fields (and the metric tensor) that are discretised using control volumes, due to the way this method is coded, the mesh associated with the \option{Porosity} must be either element wise (meaning discontinuous order zero) or the same order as field to be transported. It is therefore recommended that the \option{Porosity} field always be
2438
associated with a mesh that is discontinuous order zero. It is recommended that the \option{Permeability} field also be associated with a mesh that is discontinuous order zero, due to the recommended Darcy velocity - pressure element pair (being p0p1cv or p0p1 with the continuity tested with the p1 control volume dual space)
2382
2440
Second, the prognostic \option{Velocity} vector field of the (only) phase is now taken to be the Darcy velocity \ref{sec:porous_media_darcy_flow_equations}. There is no option to select Darcy flow but this can be achieved via a careful selection of the \option{Velocity} field options tree given by:
2383
2441
\begin{itemize}
added
removed
manual/configuring_fluidity.tex
