freeflow/rans/twoeq/komega/model.hh File Reference

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.

Description

Single-phase Reynolds-Averaged Navier-Stokes flow.

A single-phase, isothermal Reynolds-Averaged Navier-Stokes model.

This model implements a single-phase, isothermal Reynolds-Averaged Navier-Stokes model, solving the momentum balance equation

\[ \frac{\partial (\varrho \textbf{v})}{\partial t} + \nabla \cdot (\varrho \textbf{v} \textbf{v}^{\text{T}}) = \nabla \cdot (\mu_\textrm{eff} (\nabla \textbf{v} + \nabla \textbf{v}^{\text{T}})) - \nabla p + \varrho \textbf{g} - \textbf{f} \]

The effective viscosity is composed of the fluid and the eddy viscosity:

\[ \mu_\textrm{eff} = \mu + \mu_\textrm{t} \]

.

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 [87].

Turbulent Kinetic Energy balance:

\[ \frac{\partial \left( \varrho k \right)}{\partial t} + \nabla \cdot \left( \mathbf{v} \varrho k \right) - \nabla \cdot \left[ \left( \mu + \sigma_\textrm{k} \mu_\textrm{t} \right) \nabla k \right] - P + \beta_k^{*} k \varrho \omega = 0 \]

with \( P = 2 \mu_\textrm{t} \mathbf{S} \cdot \mathbf{S} \) 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:

\[ \frac{\partial \left( \varrho \omega \right)}{\partial t} + \nabla \cdot \left( \mathbf{v} \varrho \omega \right) - \nabla \cdot \left[ \left( \mu + \sigma_{\omega} \mu_\textrm{t} \right) \nabla \omega \right] - \alpha \frac{\omega}{k} P + \beta_{\omega} \omega^2 - \varrho \frac{\sigma_d}{\omega} \nabla k \nabla \omega = 0 \]

The dynamic eddy viscosity \( \mu_\textrm{t} \) is calculated as follows:

\[ \mu_\textrm{t} = \varrho \frac{k}{\tilde{\omega}} \]

With a limited dissipation:

\[ \tilde{\omega} = \textrm{max} \left\{ \omega, 0.875 \sqrt{\frac{P}{\mu_\textrm{t} \beta_\textrm{k}}} \right\} \]

And a cross-diffusion coefficient \( \sigma_\textrm{d} \)

\[ \sigma_\text{d} = \begin{cases} 0 & \mbox{, if } \; \nabla k \cdot \nabla \omega \le 0 \\ 0.125 & \mbox{, if } \; \nabla k \cdot \nabla \omega > 0 \end{cases}. \]

Classes
struct	Dumux::Properties::KOmegaModelTraits< dimension >
	Traits for the k-omega model. More...
struct	Dumux::Properties::TTag::KOmega
	The type tag for the single-phase, isothermal k-omega model. More...
struct	Dumux::Properties::ModelTraits< TypeTag, TTag::KOmega >
	states some specifics of the isothermal k-omega model More...
struct	Dumux::Properties::FluxVariables< TypeTag, TTag::KOmega >
	The flux variables. More...
struct	Dumux::Properties::LocalResidual< TypeTag, TTag::KOmega >
	The local residual. More...
struct	Dumux::Properties::VolumeVariables< TypeTag, TTag::KOmega >
	Set the volume variables property. More...
struct	Dumux::Properties::IOFields< TypeTag, TTag::KOmega >
	The specific I/O fields. More...
struct	Dumux::Properties::TTag::KOmegaNI
	The type tag for the single-phase, non-isothermal k-omega 2-Eq. model. More...
struct	Dumux::Properties::ModelTraits< TypeTag, TTag::KOmegaNI >
	The model traits of the non-isothermal model. More...
struct	Dumux::Properties::VolumeVariables< TypeTag, TTag::KOmegaNI >
	Set the volume variables property. More...
struct	Dumux::Properties::IOFields< TypeTag, TTag::KOmegaNI >
	The specific non-isothermal I/O fields. More...

Namespaces
namespace	Dumux
namespace	Dumux::Properties
	The energy balance equation for a porous solid.
namespace	Dumux::Properties::TTag
	Type tag for numeric models.

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