1-Eq. Models

One-equation turbulence model by Spalart-Allmaras. More...

Description

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

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\kappa =0.41$$

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

Files
file	freeflow/rans/oneeq/fluxvariables.hh
file	freeflow/rans/oneeq/indices.hh
file	freeflow/rans/oneeq/iofields.hh
file	freeflow/rans/oneeq/localresidual.hh
file	freeflow/rans/oneeq/model.hh
	A single-phase, isothermal one-equation turbulence model by Spalart-Allmaras.
file	freeflow/rans/oneeq/problem.hh
	One-equation turbulence problem base class.
file	freeflow/rans/oneeq/staggered/fluxvariables.hh
file	freeflow/rans/oneeq/staggered/localresidual.hh
file	freeflow/rans/oneeq/volumevariables.hh

Classes
struct	Dumux::OneEqIndices< dimension, numComponents >
	The common indices for the isothermal one-equation turbulence model by Spalart-Allmaras. More...
struct	Dumux::OneEqIOFields
	Adds I/O fields for the one-equation turbulence model by Spalart-Allmaras. More...
struct	Dumux::Properties::OneEqModelTraits< dimension >
	Traits for the Spalart-Allmaras model. More...
class	Dumux::RANSProblemImpl< TypeTag, TurbulenceModel::oneeq >
	One-equation turbulence problem base class. More...
class	Dumux::OneEqFluxVariablesImpl< TypeTag, BaseFluxVariables, DiscretizationMethod >
	The flux variables class for the one-equation model by Spalart-Allmaras using the staggered grid discretization. More...
class	Dumux::OneEqResidualImpl< TypeTag, BaseLocalResidual, DiscretizationMethod >
	Element-wise calculation of the residual for one-equation turbulence models using the staggered discretization. More...
class	Dumux::OneEqVolumeVariables< Traits, NSVolumeVariables >
	Volume variables for the isothermal single-phase one-equation turbulence model by Spalart-Allmaras. More...

Typedefs
template<class TypeTag , class BaseFluxVariables >
using	Dumux::OneEqFluxVariables = OneEqFluxVariablesImpl< TypeTag, BaseFluxVariables, typename GetPropType< TypeTag, Properties::GridGeometry >::DiscretizationMethod >
	The flux variables class for the one-equation turbulence model by Spalart-Allmaras. This is a convenience alias for that actual, discretization-specific flux variables. More...
template<class TypeTag , class BaseLocalResidual >
using	Dumux::OneEqResidual = OneEqResidualImpl< TypeTag, BaseLocalResidual, typename GetPropType< TypeTag, Properties::GridGeometry >::DiscretizationMethod >
	The local residual class for the one-equation turbulence model by Spalart-Allmaras This is a convenience alias for the actual, discretization-specific local residual. More...

Typedef Documentation

OneEqFluxVariables

template<class TypeTag , class BaseFluxVariables >

using Dumux::OneEqFluxVariables = typedef OneEqFluxVariablesImpl<TypeTag, BaseFluxVariables, typename GetPropType<TypeTag, Properties::GridGeometry>::DiscretizationMethod>

Note

Not all specializations are currently implemented

OneEqResidual

template<class TypeTag , class BaseLocalResidual >

using Dumux::OneEqResidual = typedef OneEqResidualImpl<TypeTag, BaseLocalResidual, typename GetPropType<TypeTag, Properties::GridGeometry>::DiscretizationMethod>

Note

Not all specializations are currently implemented