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

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

$$\frac{\partial (\varrho \mathbf{\text{v}})}{\partial t}+\mathrm{\nabla }\cdot (\varrho \mathbf{\text{v}}{\mathbf{\text{v}}}^{\text{T}})=\mathrm{\nabla }\cdot ({\mu }_{\text{eff}}(\mathrm{\nabla }\mathbf{\text{v}}+\mathrm{\nabla }{\mathbf{\text{v}}}^{\text{T}}))-\mathrm{\nabla }p+\varrho \mathbf{\text{g}}-\mathbf{\text{f}}$$

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

$${\mu }_{\text{eff}}=\mu +{\mu }_{\text{t}}$$

.

This model, published by Spalart and Allmaras 1992 [82], 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 [95] and Versteeg [92], the correction term which account for the transition or trip, is dropped from the original equations, such that the balance equation simplifies to:

$$\frac{\partial \tilde{\nu }\varrho }{\partial t}+\mathrm{\nabla }\cdot \left(\tilde{\nu }\varrho \mathbf{\text{v}}\right)-c_{\text{b1}}\tilde{S}\tilde{\nu }\varrho -\frac{1}{{\sigma }_{\tilde{\nu }}}\mathrm{\nabla }\cdot \left([\mu +\tilde{\nu }\varrho ]\mathrm{\nabla }\tilde{\nu }\right)-\frac{c_{\text{b2}}}{{\sigma }_{\tilde{\nu }}}\varrho {\left|\mathrm{\nabla }\tilde{\nu }\right|}^2+c_{\text{w1}}{f}_{\text{w}}\varrho \frac{{\tilde{\nu }}^2}{y^2}=0$$

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

$$\tilde{S}=\sqrt{2\mathbf{\Omega }\cdot \mathbf{\Omega }}+\frac{\tilde{\nu }}{{\kappa }^2{y}^2}{f}_{\text{v2}}$$

$$\mathbf{\Omega }=\frac{1}{2}(\mathrm{\nabla }{\mathbf{\text{v}}}_{\text{g}}-\mathrm{\nabla }{\mathbf{\text{v}}}_{\text{g}}^T)$$

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:

$$\chi =\frac{\tilde{\nu }}{\nu }$$

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

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

$$f_{\text{w}}=g_{\text{w}}{\left(\frac{1+c_{\text{w3}}^6}{g_{\text{w}}^6+c_{\text{w3}}^6}\right)}^{\frac{1}{6}}$$

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

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

$${\sigma }_{\tilde{\nu }}=\frac{2}{3}$$

$$c_{\text{b1}}=0.1355$$

$$c_{\text{b2}}=0.622$$

$$c_{\text{v1}}=7.1$$

$$c_{\text{w1}}=\frac{c_{\text{b1}}}{{\kappa }^2}+\frac{1+c_{\text{b2}}}{{\sigma }_{\tilde{\nu }}}$$

$$c_{\text{w2}}=0.3$$

$$c_{\text{w3}}=2$$

$$\kappa =0.41$$

$${\sigma }_{\tilde{\nu }}=2/3$$

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

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