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DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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freeflow/rans/oneeq/model.hh File Reference

A single-phase, isothermal one-equation turbulence model by Spalart-Allmaras. More...

#include <dumux/common/properties.hh>
#include <dumux/freeflow/properties.hh>
#include <dumux/freeflow/turbulencemodel.hh>
#include <dumux/freeflow/rans/model.hh>
#include <dumux/freeflow/nonisothermal/iofields.hh>
#include "fluxvariables.hh"
#include "indices.hh"
#include "localresidual.hh"
#include "problem.hh"
#include "volumevariables.hh"
#include "iofields.hh"

Go to the source code of this file.

Description

A single-phase, isothermal one-equation turbulence model by Spalart-Allmaras.

Single-phase Reynolds-Averaged Navier-Stokes flow For a detailed model description see freeflow/rans/model.hh.

This model, published by Spalart and Allmaras 1992 [59], uses one additional PDE for a working variable \( \tilde{\nu} \). This variable has the units of a viscosity and can be converted to the eddy viscosity via a model function~( \( f_\text{v1} \)):

\[ \nu_\text{t} = \tilde{\nu} f_\text{v1} \]

Here, as proposed by Wilcox [73] and Versteeg [70], the correction term which account for the transition or trip, is dropped from the original equations, such that the balance equation simplifies to:

\[ \frac{\partial \tilde{\nu}\varrho}{\partial t} + \nabla \cdot \left( \tilde{\nu} \varrho \textbf{v} \right) - c_\text{b1} \tilde{S} \tilde{\nu} \varrho - \frac{1}{\sigma_{\tilde{\nu}}} \nabla \cdot \left( \left[ \mu + \tilde{\nu} \varrho \right] \nabla \tilde{\nu} \right) - \frac{c_\text{b2}}{\sigma_{\tilde{\nu}}} \varrho \left| \nabla \tilde{\nu} \right|^2 + c_\text{w1} f_\text{w} \varrho \frac{\tilde{\nu}^2}{y^2} = 0 \]

Here, a modified mean effective strain rate ( \( \tilde{S} \)) based on the mean rotation rate tensor ( \( \mathbf{\Omega} \)) is used:

\[ \tilde{S} = \sqrt{2 \mathbf{\Omega} \cdot \mathbf{\Omega}} + \frac{\tilde{\nu}}{\kappa^2 y^2} f_\text{v2} \]

\[ \mathbf{\Omega} = \frac{1}{2} \left( \nabla \textbf{v}_\text{g} - \nabla \textbf{v}_\text{g}^{T} \right) \]

This balance equation is linked to the flow geometry by the distance to the closest wall ($y$). Further, the model uses the following functions and expressions:

\[ \chi = \frac{\tilde{\nu}}{\nu} \]

\[ f_\text{v1} = \frac{\chi^3}{\chi^3+c_\text{v1}^3} \]

\[ f_\text{v2} = 1 - \frac{\chi}{1+f_\text{v1}\chi} \]

\[ f_\text{w} = g_\text{w} \left( \frac{1+c_\text{w3}^6}{g^6_\text{w}+c_\text{w3}^6} \right)^\frac{1}{6} \]

\[ g_\text{w} = r_\text{w} + c_\text{w2} (r_\text{w}^6 - r_\text{w}) \]

\[ r_\text{w} = \min \left[ \frac{\tilde{\nu}}{\tilde{S}\kappa^2 y^2},10\right] \]

\[ \sigma_{\tilde{\nu}} = \frac{2}{3} \]

\[ c_\text{b1} = 0.1355 \]

\[ c_\text{b2} = 0.622 \]

\[ c_\text{v1} = 7.1 \]

\[ c_\text{w1} = \frac{c_\text{b1}}{\kappa^2} + \frac{1+c_\text{b2}}{\sigma_{\tilde{\nu}}} \]

\[ c_\text{w2} = 0.3 \]

\[ c_\text{w3} = 2 \]

\[ \kappa = 0.41 \]

\[ \sigma_{\tilde{\nu}} = 2/3 \]

Classes

struct  Dumux::Properties::OneEqModelTraits< dimension >
 Traits for the Spalart-Allmaras model. More...
 
struct  Dumux::Properties::TTag::OneEq
 The type tag for the single-phase, isothermal Spalart-Allmaras model. More...
 
struct  Dumux::Properties::ModelTraits< TypeTag, TTag::OneEq >
 < states some specifics of the isothermal Spalart-Allmaras model More...
 
struct  Dumux::Properties::FluxVariables< TypeTag, TTag::OneEq >
 The flux variables. More...
 
struct  Dumux::Properties::LocalResidual< TypeTag, TTag::OneEq >
 The local residual. More...
 
struct  Dumux::Properties::VolumeVariables< TypeTag, TTag::OneEq >
 Set the volume variables property. More...
 
struct  Dumux::Properties::IOFields< TypeTag, TTag::OneEq >
 The specific I/O fields. More...
 
struct  Dumux::Properties::TTag::OneEqNI
 The type tag for the single-phase, non-isothermal Spalart-Allmaras model. More...
 
struct  Dumux::Properties::ModelTraits< TypeTag, TTag::OneEqNI >
 The model traits of the non-isothermal model. More...
 
struct  Dumux::Properties::VolumeVariables< TypeTag, TTag::OneEqNI >
 Set the volume variables property. More...
 
struct  Dumux::Properties::IOFields< TypeTag, TTag::OneEqNI >
 The specific non-isothermal I/O fields. More...
 

Namespaces

namespace  Dumux
 
namespace  Dumux::Properties
 
namespace  Dumux::Properties::TTag
 Type tag for numeric models.
 
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