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3.1-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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3.1-git
DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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A single-phase, isothermal k-omega 2-Eq. model. More...
#include <dumux/common/properties.hh>#include <dumux/freeflow/properties.hh>#include <dumux/freeflow/rans/model.hh>#include <dumux/freeflow/rans/twoeq/indices.hh>#include <dumux/freeflow/turbulencemodel.hh>#include "problem.hh"#include "fluxvariables.hh"#include "localresidual.hh"#include "volumevariables.hh"#include "iofields.hh"Go to the source code of this file.
A single-phase, isothermal k-omega 2-Eq. model.
Single-phase Reynolds-Averaged Navier-Stokes flow.
This model implements a single-phase, isothermal Reynolds-Averaged Navier-Stokes model, solving the momentum balance equation
\[ \frac{\partial (\varrho \textbf{v})}{\partial t} + \nabla \cdot (\varrho \textbf{v} \textbf{v}^{\textup{T}}) = \nabla \cdot (\mu_\textrm{eff} (\nabla \textbf{v} + \nabla \textbf{v}^{\textup{T}})) - \nabla p + \varrho \textbf{g} - \textbf{f} \]
The effective viscosity is composed of the fluid and the eddy viscosity:
\[ \mu_\textrm{eff} = \mu + \mu_\textrm{t} \]
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Two additional PDEs, one for the turbulentKineticEnergy (k) and a second for the dissipation (omega) are used to calculate the eddy viscosity for this model. The model is taken from Wilcox, 2008 [78].
Turbulent Kinetic Energy balance:
\[ \frac{\partial \left( k \right)}{\partial t} + \nabla \cdot \left( \mathbf{v} k \right) - \nabla \cdot \left[ \left( \nu + \sigma_\textrm{k} \nu_\textrm{t} \right) \nabla k \right] - P + \beta_k^{*} k \omega = 0 \]
with \( P = 2 \nu_\textrm{t} \mathbf{S} \cdot \mathbf{S} \) and \( S_{ij} = \frac{1}{2} \left[ \frac{\partial}{\partial x_i} v_j + \frac{\partial}{\partial x_j} v_i \right] \) based on \( a_{ij} \cdot b_{ij} = \sum_{i,j} a_{ij} b_{ij} \).
Dissipation balance:
\[ \frac{\partial \left( \omega \right)}{\partial t} + \nabla \cdot \left( \mathbf{v} \omega \right) - \nabla \cdot \left[ \left( \nu + \sigma_{\omega} \nu_\textrm{t} \right) \nabla \omega \right] - \alpha \frac{\omega}{k} P + \beta_{\omega} \omega^2 - \frac{\sigma_d}{\omega} \nabla k \nabla \omega = 0 \]
The kinematic eddy viscosity \( \nu_\textrm{t} \) is calculated as follows:
\[ \nu_\textrm{t} = \frac{k}{\tilde{\omega}} \]
With a limited dissipation:
\[ \tilde{\omega} = \textrm{max} \left\{ \omega, 0.875 \sqrt{\frac{P}{\nu_\textrm{t} \beta_\textrm{k}}} \right\} \]
And a cross-diffusion coefficient \( \sigma_\textrm{d} \)
\[ \sigma_\text{d} = \begin{cases} 0 & \mbox{, if } \; \nabla k \cdot \nabla \omega \le 0 \\ 0.125 & \mbox{, if } \; \nabla k \cdot \nabla \omega > 0 \end{cases}. \]
Namespaces | |
| namespace | Dumux |
| make the local view function available whenever we use the grid geometry | |
| namespace | Dumux::Properties |
| namespace | Dumux::Properties::TTag |
| Type tag for numeric models. | |
1.9.3