1-Eq. Models

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

Description

One-equation turbulence model by Spalart-Allmaras.

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:

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

.

This model, published by Spalart and Allmaras 1992 [64], 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 [78] and Versteeg [75], the correction term which account for the transition or trip, is dropped from the original equations, such that the balance equation simplifies to:

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

Files
file	freeflow/rans/oneeq/fluxvariables.hh
file	freeflow/rans/oneeq/indices.hh
file	dumux/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	dumux/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::staggered >
	The flux variables class for the one-equation model by Spalart-Allmaras using the staggered grid discretization. More...
class	Dumux::OneEqResidualImpl< TypeTag, BaseLocalResidual, DiscretizationMethod::staggered >
	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, GetPropType< TypeTag, Properties::GridGeometry >::discMethod >
	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, GetPropType< TypeTag, Properties::GridGeometry >::discMethod >
	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, GetPropType<TypeTag, Properties::GridGeometry>::discMethod>

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.

Note

Not all specializations are currently implemented

OneEqResidual

template<class TypeTag , class BaseLocalResidual >

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

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.

Note

Not all specializations are currently implemented