%-------------------------------------------------------------------------------
\section{Eddy viscosity Models (\emph{EVM})}
In this section eddy viscosity hypothesis is made which states that the Reynolds stress tensor
is aligned with the rate of strain $\tens{S}$:
%
\begin{equation}
\rho  \tens{R} = \dfrac{2}{3} \rho k \tens{1} - 2 \mu_T \deviator{\tens{S}}
\end{equation} 

%-------------------------------------------------------------------------------
\subsection{Equations for the variables $k$ and $\varepsilon$ (standard $k-\varepsilon$ model)}

\begin{equation}
\left\{
\begin{array}{r c l}
\displaystyle
\rho \der{k }{t} + \grad k \cdot \left( \rho \vect{u} \right) 
- \divs \left[ \left( \mu +\frac{\mu _{t}}{\sigma _{k}} \right)\grad{k}\right] 
&=&\displaystyle
\mathcal{P}+\mathcal{G}-\rho \varepsilon +\Gamma k^{in}+ST_{k}, \\
\displaystyle
\rho \der{ \varepsilon }{t} + \grad \varepsilon \cdot \left(\rho \vect{u} \right)
-\divs \left[ \left( \mu +\frac{\mu _{t}}{\sigma _{\varepsilon }} \right) \grad{\varepsilon}\right] 
&=& \displaystyle
C_{\varepsilon _{1}}\frac{\varepsilon }{k}\left[
\mathcal{P}+(1-C_{\varepsilon _{3}})\mathcal{G}\right] -\rho C_{\varepsilon
_{2}}\frac{\varepsilon ^{2}}{k}+\Gamma \varepsilon ^{in}+ST_{\varepsilon },
\end{array}
\right.
\end{equation}
\nomenclature[rk]{$k$}{turbulent kinetic energy \nomunit{$m^{2}.s^{-2}$}}
\nomenclature[gepslion]{$ \varepsilon $}{turbulent kinetic energy dissipation \nomunit{$m^{2}.s^{-3}$}}
\nomenclature[rproduction]{$\mathcal{P}$}{turbulent kinetic energy production \nomunit{$kg.m^{-1}.s^{-3}$}}
where $\mathcal{P}$ is the production term created by mean shear:
%
\begin{equation}
\begin{array}{rcl}
\mathcal{P} & = & \displaystyle -\rho \tens{R} : \gradt \, \vect{u}
= -\left[-2 \mu_t \deviator{\tens{S}}
+ \frac{2}{3}\rho k \tens{1}\right] : \tens{S}, \\
& = & \displaystyle
2 \mu_t  \deviator{\tens{S}}: \deviator{\tens{S}} 
-\dfrac{2}{3}
\rho k \trace \left( \gradt \,\vect{u} \right),
\end{array}
\end{equation}
\nomenclature[rbuoyancy]{$\mathcal{G}$}{turbulent kinetic energy buoyancy term \nomunit{$kg.m^{-1}.s^{-3}$}}
and
$\mathcal{G}$ is the production term created by gravity effects: 
\begin{equation}
\displaystyle \mathcal{G}= \frac{1}{\rho}\frac{\mu_t}{\sigma_t} \grad \rho \cdot \vect{g}.
\end{equation}

The dynamic turbulent viscosity reads: 
\begin{equation}
\displaystyle \mu_t=\rho C_\mu\frac{k^2}{%
\varepsilon}.
\end{equation}
\nomenclature[rstk]{$ST_{k}$}{additional turbulent kinetic energy source term \nomunit{$kg.m^{-1}.s^{-3}$}}
\nomenclature[rstepsilon]{$ST_{\varepsilon}$}{additional turbulent dissipation source term \nomunit{$kg.m^{-1}.s^{-4}$}}
$ST_{k}$ and $ST_{\varepsilon}$ stand for the additional
source terms prescribed by the user (in rare cases only).

The constants of the model are given in the Table (\ref{tab:k_epsilon_constants}):
\begin{table}[!htp]
\centering
\begin{tabular}{p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}}
$C_\mu$ & $C_{\varepsilon_1}$ & $C_{\varepsilon_2}$ & $\sigma_k$ & $%
\sigma_\varepsilon$ \\ \hline
$0.09$ & $1.44$ & $1.92$ & $1.0$ & $1.3$ 
\end{tabular}%
\caption{Standard $k-\varepsilon$ model constants \cite{Launder:1974}.\label{tab:k_epsilon_constants}}
\end{table}
\nomenclature[rcmu]{$C_\mu$}{eddy viscosity constant}
\nomenclature[rcepsilon1]{$C_{\varepsilon_1}$}{constant of the standard $k-\varepsilon$ model}
\nomenclature[rcepsilon2]{$C_{\varepsilon_2}$}{constant of the standard $k-\varepsilon$ model}
\nomenclature[rcepsilon3]{$C_{\varepsilon_3}$}{constant of the standard $k-\varepsilon$ model depending on the buoyancy term}

$C_{\varepsilon_3}=0$ if $\mathcal{G}\geqslant0$ (unstable stratification)
and $C_{\varepsilon_3}=1$ if $\mathcal{G}\leqslant0$ (stable stratification).

%-------------------------------------------------------------------------------
%\subsection{$k-\varepsilon$ with Linear Production (\emph{LP}) model}

%-------------------------------------------------------------------------------
%\subsection{$k-\omega$ \emph{SST} model}

%-------------------------------------------------------------------------------
\subsection{$k-\epsilon-\overline{v^2}/k$ elliptic blending turbulence model}

The BL-$\overline{v^2}/k$~ \cite{Billard:2012} is a  elliptic-blending based $\overline{v^2}-f$ model. 
It is a low Reynolds number model and as such the wall distance of the first off-wall cell centre must be of order of unity when expressed in viscous unit. 

The following gives details about the model followed by some description of its implementation into \CS. 

\subsubsection{Model description}

This eddy viscosity model solves for $k$ and $\varepsilon$ as turbulence variables, 
representing respectively the turbulent kinetic energy and its dissipation rate, 
as well as two non-dimensional variables, $\varphi=\overline{v^2}/k$ and $\alpha$. 
The first of these latter two represents the ratio of wall normal Reynolds stress 
to turbulent kinetic energy (thus being a measure of the near-wall turbulence anisotropy) 
and the second is a wall proximity sensitive quantity (\emph{i.e.} it takes the value $0$ at a wall and $1$ in the far field). 
The coefficient $\alpha$ is solved for via an elliptic equation 
($L$ representing the turbulence length-scale): 
\begin{equation}
		\alpha - L^2 \Delta \alpha = 1
		\label{eq:turbul:blv2kalpha}
\end{equation}
%
The $\varphi$ transport equation reads: 
%
\begin{equation}\label{eq:turbul:blv2kphi}
\DP{\varphi}  =   
   \left( 1-\alpha^3 \right) f_w
 + \alpha^3 f_h - P \frac{\varphi}{k} 
+  \frac{2}{k} \frac{\nu_t}{\sigma_k} \grad \varphi  \cdot \grad k + \divs \left[ \left( \frac{\nu}{2} + \frac{\nu_t}{\sigma_\varphi}  \right) \grad \varphi \right]
\end{equation}

The aim of the BL-$\overline{v^2}/k$~ model is to stand as a code-friendly version 
of the $\overline{v^2}-f$ model of \cite{Durbin:1991}. In both the wall normal stress 
$\overline{v^2}$ is used in the $\nu_T$ definition to correctly represent the near-wall turbulence 
damping ($T$ is the turbulence time-scale and $C_\mu=0.22$ is calibrated in the logarithmic layer of a channel flow):
%
\begin{equation}
	\nu_T= C_\mu \varphi k ~ T
\end{equation}

The elliptic blending approach mainly allows for an improved robustness. 
Indeed, the original $\overline{v^2}-f$ approach solves for the quantity 
$\overline{v^2}$ and a variable $f$ derived from the wall normal 
pressure term\footnote{the pressure term $\phi_{22}^*$ is not decomposed but modelled as a whole}
 and defined as:
%
\begin{equation}
	f= \dfrac{1}{k} \left[ \underbrace{-\dfrac{2}{\rho}\overline{v \partial_y p}}_{\phi_{22}^*}-2 \nu\overline{\grad v \cdot \grad v}+\varepsilon \dfrac{\overline{v^2}}{k} \right] 
\end{equation}
%
with $f$ being solved using an elliptic equation:
%
\begin{equation}
	f-L^2 \Delta f = f_h
\end{equation}
%
Similarly to the $\alpha$ equation, this elliptic operator allows
 to represent the non-local effects induced by the
 incompressibility of turbulence. The quantity $f_h$ is obtained by
 considering homogeneous modelling (\emph{i.e.} $\Delta f=0$) of $f$ using 
for pressure strain-rate term the model of \cite{Launder:1975}. The correct
 asymptotic behaviour of the variable $\overline{v^2}$ is ensured
 by the following boundary condition:
%
\begin{equation}
   \lim_{y\to 0} f = \lim_{y\to 0} \dfrac{-20 \nu^2 \overline{v^2}}{ \varepsilon y^4}
   \label{eq:turbul:blv2kbc_f_durbin}
\end{equation}
%
This requires a balance between $O(y^4)$ terms which proves to be numerically problematic.
 In the \mbox{BL-$\overline{v^2}/k$~} model the elliptic equation is simply solved
 for a non-dimensional quantity with an homogeneous Dirichlet boundary condition, 
therefore alleviating the stiffness associated to the boundary condition of the 
elliptic variable. The inclusion of $\alpha$ in the definition of $f$ allows a blending 
between the near-wall and the homogeneous form $f=(1-\alpha^3) f_w+\alpha^3 f_h$ in the 
$\varphi$ equation. For the $f_h$ model the proposal of \cite{Speziale:1991} is preferred 
for its better reproduction of the pressure term in a boundary layer. 

The model also solves a $k$-$\varepsilon$ system somewhat modified compared to the one 
generally adopted by $\overline{v^2}-f$ models. The $k$ and $\varepsilon$ equations 
adopted by the BL-$\overline{v^2}/k$~ model read:
%
% K equation
\begin{equation} \label{eq:turbul:blv2kk}
   \DP{ k} = P -  \varepsilon 
           + \divs \left[ \left( \frac{\nu}{2}  + \frac{\nu_t}{\sigma_k}  \right) \grad k \right] 
    - C_{\varepsilon 3} (1-\alpha)^3 \frac{k}{\varepsilon} 2 \nu  \nu_t \left( \partial_k \partial_j U_i \right)\left( \partial_k \partial_j  U_i \right)
 \end{equation}
 % Epsilon equation
\begin{equation}  \label{eq:turbul:blv2kep}
	\DP{\varepsilon} = \frac{C_{\varepsilon 1} P - C_{\varepsilon 2}^* \varepsilon }{T} 
          + \divs \left[ \left( \frac{\nu}{2}  + \frac{\nu_t}{\sigma_{\varepsilon}}  \right) \grad \varepsilon \right]  
\end{equation}

The $k$ equation includes the so-called ``E term'' dependent on the second velocity 
derivatives squared, similar to that introduced into the $\varepsilon$ equation 
by \cite{Jones:1972}, and the homogeneous part of the dissipation rate is 
independently accounted for (following the suggested formulation of \cite{Jakirlic:2002}). 
This implies that the quantity $\varepsilon$ resolved by the model has a different definition 
to that conventionally employed in $k$-$\varepsilon$ schemes (\emph{i.e.} a change of 
variable $\varepsilon \to \varepsilon + (1-\alpha)^3\frac{k}{\varepsilon}E+\frac{1}{2}\nu\partial_{jj}k$). 
This has the beneficial effect of reducing the Reynolds number dependence of the near-wall 
value of the turbulence variables and of the time and length scales, $T$ and $L$ respectively, 
yielding better near-wall prediction of the blending variable and the turbulent viscosity, 
and hence mean flow quantities, for both low and high Reynolds numbers. 
 
A further feature is that the coefficient $C_{\varepsilon_2}^*$ is taken as a function 
of the turbulent transport of $k$ to $\varepsilon$ ratio  (as proposed by \cite{parneix1996second}):

\begin{equation}
  C_{\varepsilon 2}^* = C_{\varepsilon 2} 
- \alpha^3 (0.4-C_{\varepsilon 2}) \tanh \left( \left|  \frac{\divs (\nu_t / \sigma_k \grad k)}{\varepsilon}   \right| ^{3/2}  \right)
\end{equation}

This improves the predictions of the dissipation rate in the defect layer of a channel flow (where the turbulent transport becomes significant) and yields better results in wake flows \cite{parneix1996second}. Full details of the scheme can be found in \cite{Billard:2012}.

%-------------------------------------------
\subsubsection{\emph{in extenso} definition}

\paragraph{Equations:}
Equations for the turbulence kinetic energy $k$, the turbulence dissipation rate $\varepsilon$, 
the non-dimensionnal wall-normal Reynolds stress component $\varphi=\overline{v^2}/k$ and 
the elliptic blending parameter $\alpha$ are given in Eq.~\eqref{eq:turbul:blv2kk}, 
Eq.~\eqref{eq:turbul:blv2kep},  Eq.~\eqref{eq:turbul:blv2kphi} and Eq.~\eqref{eq:turbul:blv2kalpha} respectively. 

%-------------------------------
\paragraph{Scales and constants:}

The definition of the turbulent viscosity is given in Eq.~\eqref{eq:turbul:blv2knut}. 
The near-wall and far field models, $f_w$ and $f_h$ for the $\varphi$ source term, $f$ are 
expressed in Eq.~\eqref{eq:turbul:blv2kfw} and Eq.~\eqref{eq:turbul:blv2kfh}. 
The definition of the variable coefficient $C_{\varepsilon_2}^*$ is given in Eq.~\eqref{eq:turbul:blv2kcep2}

Finally the time and length scales entering the definition of $\nu_t$, the equation of 
$\varepsilon$ and the definition of $f_h$ as well as the equation of $\alpha$ are given in 
Eq.~\eqref{eq:turbul:blv2kscales}. The viscous limiter used as lower bound of the time scale 
has a finite wall value and therefore enables avoid the singularity consecutive to the definition 
of the $\varepsilon$ sink term if the term $-C_{\varepsilon 2}^* \dfrac{\varepsilon^2}{k}$ were 
used in place of $-C_{\varepsilon 2}^* \dfrac{\varepsilon}{T}$. 
Similarly a viscous (Kolmogorov) limiter is used for the length-scale definition $L$ to 
avoid numerical problems which would raise in the $\alpha$ equation numerical resolution 
if the length-scale were to tend to zero at wall. The upper limiter $T_{lim}$ is used to 
enforce the Bradshaw hypothesis (proportionality between shear stress and turbulent 
kinetic energy in a boundary layer $\overline{uv}/k=C (=0.6/\sqrt{3}$ with the present approach)), 
and corrects, in a wider range of cases, the excessive production rate returned by 
the eddy viscosity formulation (\emph{i.e.} allowing a linear rather than a quadratic 
dependance on $S$ for large strain rate).

\begin{equation}
	\nu_t= C_\mu \varphi k ~ \min(T,T_{lim})
\label{eq:turbul:blv2knut}
\end{equation}

\begin{equation}
f_w = - \dfrac{\varepsilon}{2} \dfrac{\varphi}{k}
\label{eq:turbul:blv2kfw}
\end{equation}
\begin{equation}
f_h = - \dfrac{1}{T} \left(  C_1 -1 +C_2 \dfrac{P}{\varepsilon}   \right) \left(  \varphi - \dfrac{2}{3}  \right)
\label{eq:turbul:blv2kfh}
\end{equation}

\begin{equation}
         	C_{\varepsilon 2}^*=C_{\varepsilon 2}+ \alpha^3 \left( C_{\varepsilon 4}-C_{\varepsilon 2} \right) \tanh \left( \left| \frac{\divs \left(\nu_t/\sigma_k \grad k\right)}{\varepsilon} \right|^{3/2}   \right)
	\label{eq:turbul:blv2kcep2}
\end{equation}

\begin{equation}
	\label{eq:turbul:blv2kscales}
		\begin{cases}
			L = \sqrt{C_{L}^{2}\left(\dfrac{k^{3}}{\varepsilon^{2}}+C_{\eta}^{2}\dfrac{\nu^{3/2}}{\varepsilon^{1/2}}\right)} & \\
			T = \sqrt{\dfrac{k^{2}}{\varepsilon^{2}}+C_{T}^{2}\dfrac{\nu}{\varepsilon}} & \\
			T_{lim} = \dfrac{0.6}{\sqrt{6}C_\mu \varphi \sqrt{\tens{S} : \tens{S}}}
		\end{cases}
\end{equation}

%---------------------
\paragraph{Constants:}
\tablename~ \ref{tab:turbul:cstblv2k} gives the value adopted for the constants of the model.

\begin{table}[!htp]
\centering
\begin{tabular}{ccccccc}
	\hline 
	$C_{\varepsilon 1}$ & $C_{\varepsilon 2}$ & $C_{\varepsilon 3} $ & $C_{\varepsilon 4}$ & $\sigma_k$ & $\sigma_{\varepsilon}$\\
	\hline
	$ 1.44 $ & $1.83$ & $2.3$ & $0.4$ & $1$ & $1.5$ \\
	\hline
\end{tabular}

\begin{tabular}{cccccccc}
	\hline 
	 $C_\mu$ & $C_T$ & $C_{L}$ & $C_{\eta}$ & $C_1$ & $C_2$ & $\sigma_\varphi$\\
	\hline
	 $0.22$ & $4$ & $0.164$ & $75$ & $1.7$ & $0.9$  &  $1$ \\
	\hline
\end{tabular}
\caption{\label{tab:turbul:cstblv2k} Constants of the BL-$\overline{v^2}/k$~ model}
\end{table}

%----------------------------------
\subsubsection{Boundary conditions}

The turbulent variables wall boundary conditions are given in Eq.~\eqref{eq:turbul:bcblv2k} 
($y$ being the wall-distance)\footnote{Note that the $\varepsilon$ wall boundary condition 
is halved compared to what is used in the $\varphi-\overline{f}$ model (\texttt{iturb=50}) consequently to the change of variable described above}: 

\begin{equation}
		\begin{cases}
			\displaystyle   \lim_{y\to0} k =  0 & \\
			\displaystyle \lim_{y\to0} \varepsilon =  \lim_{y\to0} \dfrac{\nu k}{y^2} & \\
			\displaystyle \lim_{y\to0} \varphi =  0 & \\
			\displaystyle \lim_{y\to0} \alpha =  0 & \\
		\end{cases}
		\label{eq:turbul:bcblv2k}
\end{equation}

%-------------------------------
\subsubsection{Remaining issues}
\begin{itemize}
  \item The actual source term $\left( \partial_k \partial_j U_i \right)\left( \partial_k \partial_j  U_i \right)$ should be written as 
  $\displaystyle{\sum_{i=1}^3 \sum_{j=1}^3 \sum_{k=1}^3  \left( \partial_k \partial_j  U_i \right)^2}$ and not $\displaystyle{\sum_{i=1}^3 \left(\sum_{j=1}^3 \sum_{k=1}^3  (\partial_k \partial_j  U_i )\right)^2}$ as it is, incorrectly, in the present implementation. 
  \item Issues regarding grid/code dependancy of this term have already been raised \cite{Iaccarino:2001}. Alternatives may be worth investigating, such as $\left( \partial_{jj}  U_i \right)^2$.
\end{itemize}


%-------------------------------------------------------------------------------
\subsection{Spalart-Allmaras model}
The Spalart-Allmaras turbulence model \cite{Spalart:1992} is an \emph{EVM} \emph{RANS} model developed in the $90$'s in aeronautics, and is therefore well suited
for studying a flow around an air-plane wing for instance.

%--------------------------------
\subsubsection{Model description}
It consists of a transport equation of a scalar $\tilde{nu}$ dirctly linked to the turbulent viscosity $\mu_T$. 

More recently, this model has been extended by Aupoix \cite{Aupoix:2003} to rough wall for studying atmospheric flows. It was also successfully applied to flow in turbo-machinery where variants of this model has been developed. 

The transport equation of $\tilde{\nu}$ (pseudo turbulent viscosity, which tends to it far from walls) reads\footnote{
the present formulation is a simplified one presented by Aupoix \cite{Aupoix:2003} where transition terms have been neglected.
}
\begin{equation}\label{eq:turbul:SA_used}
\rho \der{\tilde{\nu}}{ t} + \grad \tilde{\nu} \cdot\left( \rho \vect{u} \right)
= c_{b1} \rho  \tilde{S} \tilde{\nu}
- c_{w1}f_{w} \rho \left( \dfrac{\tilde{\nu}}{d} \right)^2
+ \dfrac{1}{ \sigma} \left[ \divs \left( (\mu + \rho \tilde{\nu}) \grad \tilde{\nu} \right)
+ c_{b2} \rho \left| \grad \tilde{\nu} \right| ^2 
  \right]
  + \Gamma \left( \tilde{\nu}^{in} - \tilde{\nu}^n \right) +ST^{imp} \tilde{\nu} +ST^{exp}
\end{equation}
%
where $\tilde{\nu}^{in}$ is the injection value of $\tilde{\nu}$ in case of any mass source term, and $ST^{imp}_{\tilde{\nu}}$ and $ST^{exp}_{\tilde{\nu}}$ 
are respectively the implicit and explicit additional user source terms and
where
%
\begin{equation}\label{eq:def_fun_SA}
\begin{array}{ r c l}
\mu_T & =& \rho f_{v1} \tilde{\nu} \\
f_{v1} &=& \dfrac{\chi^3 }{\chi^3+ c_{v1}^3} \\
\chi & = & \dfrac{\tilde{\nu}}{ \nu} \\
\tilde{S} &=& \Omega + \dfrac{\tilde{\nu}}{ \kappa ^2 d^2} f_{v2} \\
f_{v2} &=& 1 - \dfrac{ \tilde{\nu}}{\nu + \tilde{\nu} f_{v1}} \\
f_w &=& g \left[ \dfrac{1+ c_{w3}^6 }{g^6+ c_{w3}^6} \right]^\frac{1}{6} \\
g &=& r +c_{w2} \left( r^6 -r \right) \\
r &=& \min \left[ \dfrac{ \tilde{\nu} }{ \tilde{S} \kappa^ 2 d^2 } ; \; 10 \right]
\end{array}
\end{equation}
%
The constants are defined in \tablename~\ref{tab:const_SA}.

\begin{table}[!htbc]
\centering
$
\begin{array}{c | c | c | c | c | c | c | c }
\sigma      & c_{b1} & c_{b2} & \kappa & c_{w2} & c_{w3} & c_{v1} & c_{w1} \\
\hline
\frac{2}{3} & 0.1355 & 0.622  & 0.41   & 0.3    & 2      & 7.1    & \dfrac{c_{b1}}{ \kappa^2}+ \dfrac{1+c_{b2}}{ \sigma}  
\end{array}
$
\caption{Constants of the Spalart Allmaras model.\label{tab:const_SA}}
\end{table}

%----------------------------
\subsubsection{Time stepping}

Equation \eqref{eq:turbul:SA_used} can be rewritten with the $\theta$-scheme presented in \chaptername~\ref{chapter:timstp} as
%
\begin{equation}\label{eq:turbul:SA_time}
\begin{array}{c}
\overbrace{
\left[ \dfrac{\rho}{\Delta t} + \max \left( c_{w1} f_{w} \rho  \dfrac{\tilde{\nu}^n}{d^2} - c_{b1}\rho \tilde{S} , \, 0 \right) - \theta T_s^{imp} + \theta \Gamma^n \right] 
}^{\texttt{implicit term}}
\delta \tilde{\nu}^{n+1}
%
\\
+ \theta
\left(
\underbrace{
 \grad \delta \tilde{\nu}^{n+1} \cdot \left( \rho \vect{u} \right) 
}_{
\texttt{implicit part of the convection}
}
- 
\underbrace{
\divs \left[ \dfrac{\mu + \rho \tilde{\nu}}{\sigma} \grad \delta \tilde{\nu}^{n+1} \right] 
}_{
\texttt{implicit part of the diffusion}
}
\right) 
\\
= 
\\
- \grad \tilde{\nu}^{n} \cdot \left( \rho \vect{u} \right) 
+ 
\divs \left[ \dfrac{\mu + \rho \tilde{\nu}}{\sigma} \grad \tilde{\nu}^{n} \right] 
\\
+
c_{b1} \rho \tilde{S} \tilde{\nu}^n
- c_{w1}f_{w}\rho \left( \dfrac{\tilde{\nu}}{d} \right)^2
+ \dfrac{c_{b2}\rho}{ \sigma} 
 \left| \grad \tilde{\nu} \right| ^2 
 + \Gamma \left( \tilde{\nu}^{in} - \tilde{\nu}^n \right) +ST^{imp} \tilde{\nu}^n +ST^{exp}
\end{array}
\end{equation}
where $\delta \tilde{\nu}^{n+1} \equiv \tilde{\nu}^{n+1} - \tilde{\nu}^{n}$.

\begin{remark}
The term $\left( c_{w1} f_{w} \rho  \dfrac{\tilde{\nu}^n}{d^2} - c_{b1}\rho \tilde{S} \right)$ is implicit 
so that $\tilde{\nu} $ does not require any clipping to remain positive if an \emph{upwind} convective scheme and no flux 
reconstruction are chosen.
\end{remark}


%----------------------------------
\subsubsection{Boundary conditions}

\paragraph{Smooth walls:}
the boundary condition on $\tilde{\nu}$ is a standard zero Dirichlet boundary condition on the walls 
(see \chaptername~\ref{chapter:bndcnd} for the encoding of standard Dirichlet conditions).

Note that the model gives a log law outside of the viscous sub-layer, \emph{i.e.}:
\begin{equation}
\begin{array}{r c l}
\tilde{\nu} & \simeq & \kappa u^\star d \\
\tilde{S} & \simeq & \dfrac{u^\star}{ \kappa d} 
\end{array}
\end{equation}

\paragraph{Rough walls:}
In case of rough walls, let us define:
\begin{equation}
\begin{array}{rcl}
\chi_{rough} & = & \chi + c_{R1} \dfrac{h_s}{d_{rug}} \\
d_{rough} & = & d + d_0 \\
d_0     & = & \exp \left( -8.5 \kappa \right) h_s \simeq 0.03 h_s
\end{array}
\end{equation}
%
where $h_s$ is the roughness size. The Dirichlet boundary conditions 
is replaced by the following Neumann boundary condition:
%
\begin{equation}\label{eq:eq_neumann_nu}
\left. \der{\tilde{ \nu}}{n} \right|_\fib = \dfrac{\left. \tilde{ \nu}\right|_\fib }{ d_0}
\end{equation}
%
A development in series in then written:
\begin{equation}\label{eq:eq_neumann_nu_vf}
 \tilde{\nu}_\fib = \tilde{\nu}_{\centip} -   \grad_\fib \tilde{ \nu} \cdot \vect{\centip \centf} 
\end{equation}
Finally, that is a Robbin type boundary condition formulated as follows in \CS:
which reads as follows in \CS:
\begin{equation}\label{eq:eq_neumann_nu_coef}
 \tilde{\nu}_\fib = A^g_\fib - B^g_\fib \tilde{\nu}_\centip
\end{equation}
with $A^g_\fib=0$ and
%
\begin{equation}\label{eq:eq_neumann_nu_coefb}
 B^g_\fib = \dfrac{d_0}{d_0 + \overline{\centip \centf} } 
\end{equation}

\paragraph{Inlet:} 
the profile of $\tilde{\nu}$ is imposed, the value is deduced from the profiles imposed on $k$ and $\varepsilon$ for a $k-\varepsilon$turbulence 
model assuming $\tilde{\nu } \simeq \nu_T$.



%-------------------------------------------------------------------------------
\section{Differential Reynolds Stress Models (\emph{DRSM})}
In this section, the presented models solve a differential transport equation
on the Reynolds' stresses tensor.
%-------------------------------------------------------------------------------
\subsection{Equations for the Reynolds stress tensor components $R_{ij}$ 
and $\varepsilon$ (\emph{LRR} model)}
%
\nomenclature[rrt2]{$\tens{R}$}{Reynolds stress tensor \nomunit{$m^{2}.s^{-2}$}}
\nomenclature[rrij]{$R_{ij}$}{componant $ij$ of the Reynolds stress tensor \nomunit{$m^{2}.s^{-2}$}}

\begin{equation}
\left\{
\begin{array}{rcll}
\displaystyle
 \rho \der{\tens{R}}{t} 
+ \gradtt \, \tens{R} \cdot \left( \rho \vect{u} \right)
- \divt \left( \mu \gradtt \,\tens{R} \right) 
& = & 
\displaystyle
\tens{\mathit{d}} +
\tens{\mathcal{P}} + \tens{G}+ \tens{\Phi} 
-\rho \tens{\varepsilon} & \displaystyle+\Gamma \tens{R}^{in}+\tens{ST}_{R_{ij}},
\\
\displaystyle
\rho \der{\varepsilon }{t}
+ \grad \varepsilon \cdot \left( \rho \vect{u} \right)
- \dive\left(\mu \grad{\varepsilon}\right) 
& = & \displaystyle 
\mathit{{d}+C_{\varepsilon _{1}}\frac{\varepsilon }{k}\left[ \mathcal{P}%
+G_{\varepsilon }\right] -\rho C_{\varepsilon _{2}}\frac{\varepsilon ^{2}}{k}} 
& \displaystyle+\Gamma \varepsilon ^{in}+ST_{\varepsilon },
\end{array}%
\right.
\end{equation}
\nomenclature[rproductiont2]{$\tens{\mathcal{P}}$}{turbulent production tensor \nomunit{$kg.m^{-1}.s^{-3}$}}
\nomenclature[rbuoyancyt2]{$\tens{\mathcal{G}}$}{turbulent buoyancy production tensor \nomunit{$kg.m^{-1}.s^{-3}$}}
where
$\tens{\mathcal{P}}$ stands for the turbulence production tensor associated
with mean flow strain-rate and $\tens{\mathcal{G}}$ is stands for the
production- tensor associated with buoyancy effects:
\begin{equation}
\begin{array}{r c l}
\displaystyle \tens{ \mathcal{P}} & = & \displaystyle-\rho \left[ \tens{R} \cdot \gradt \, \vect{u} 
+ \gradt \, \vect{u}  \cdot \tens{R}\right], \\
\tens{ \mathcal{G}} & = &
\displaystyle \frac{3}{2}\frac{C_{\mu }}{\sigma _{t}}
\frac{k}{\varepsilon }
\left[\vect{r} \otimes \vect{g} +\vect{g} \otimes \vect{r}  \right].
\end{array}
\end{equation}
where $ \vect{r} \equiv \tens{R} \cdot \grad  \rho$ and 
$G_{\varepsilon }= \Max \left(0, \, \frac{1}{2}\trace \tens{\mathcal{G}}\right)$.
\nomenclature[rrt1]{$\vect{r}$}{vector of the Reynolds stress tensor times the density gradient}
\nomenclature[rbuoyancyepsilon]{$G_{\varepsilon }$}{turbulent buoyancy term for dissipation}

With these definitions the following relations hold:
\begin{equation}
\begin{array}{r c l}
\displaystyle k &=&\frac{1}{2} \trace{\tens{R}}, \\
\mathcal{P} &=&\frac{1}{2} \trace \left( \tens{\mathcal{P}} \right) ,
\end{array}
\end{equation}

$\tens{\Phi}$ is the term representing pressure-velocity correlations:
\nomenclature[gphit2]{$\tens{\Phi}$}{pressure-velocity correlation tensor \nomunit{$kg.s^{-3}$}}
\begin{equation}
\displaystyle \tens{\Phi} = \tens{\phi}_{1}+ \tens{\phi}_{2}+ \tens{\phi}_{3}+ \tens{\phi}_{w},
\end{equation}%
%
\begin{equation}
\begin{array}{r c l}
\tens{\phi}_{1} &=& \displaystyle -\rho \,C_{1}\frac{\varepsilon }{k}%
\deviator{\tens{R}}, \\
\tens{\phi}_{2} &=& -\rho \,C_{2} 
\deviator{\tens{\mathcal{P}}}, \\
\tens{\phi}_{3} &=& -C_{3} \deviator{ \tens{G} }.
\end{array}
\end{equation}

The term $\tens{\phi}_{w}$ is called \emph{wall echo term} (by default, it is not
accounted for: see \doxygenfile{turrij_8f90}{html programmer's documentation of the subroutine} and the appendix \ref{ap:turrij}).

The dissipation term, $\tens{\varepsilon}$ , is considered isotropic:
\nomenclature[gepsilont2]{$\tens{\varepsilon}$}{turbulent kinetic energy dissipation tensor \nomunit{$m^{2}.s^{-3}$}}
\begin{equation}
\displaystyle \tens{\varepsilon}=\frac{2}{3}\ \varepsilon \tens{1}.
\end{equation}

The turbulent diffusion terms are:
\begin{equation}
\begin{array}{r c l}
\tens{d} & = & C_{S} \displaystyle \divt \left( \rho \frac{k}{\varepsilon }%
\tens{R} \cdot \gradtt \, \tens{R} \right), \\
d & = & C_{\varepsilon }\displaystyle \dive \left( \rho \frac{k}{\varepsilon} 
\tens{R} \cdot \grad \varepsilon \right).
\end{array}
\end{equation}

In the rare event of mass sources, $\Gamma R_{ij}^{in}$ and $\Gamma
\varepsilon ^{i}$ are the corresponding injection terms. $ST_{R_{ij}}$ and $%
ST_{\varepsilon }$ are also rarely used additional source terms that can
prescribed by the user.

\begin{table}[!htp]
\begin{center}
\begin{tabular}{p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}|p{0,8cm}}
$C_\mu$ & $C_{\varepsilon}$ & $C_{\varepsilon_1}$ & $C_{\varepsilon_2}$ & $%
C_1$ & $C_2$ & $C_3$ & $C_S$ & $C^{\prime}_1$ & $C^{\prime}_2$ \\ \hline
$0.09$ & $0.18$ & $1.44$ & $1.92$ & $1.8$ & $0.6$ & $0.55$ & $0.22$ & $0.5$
& $0.3$ 
\end{tabular}
\end{center}
\caption{Model constants of the \emph{LRR} $R_{ij}-\varepsilon$ model \cite{Launder:1975}.}
\end{table}

%-------------------------------------------------------------------------------
\section{Large-Eddy Simulation (\emph{LES})}

%-------------------------------------------------------------------------------
\subsection{Standard Smagorinsky model}

\begin{equation}
\mu_{t}=\rho \, \left( C_{s}\,\overline{\Delta } \right)^{2}
\sqrt{2\overline{\tens{S}} \,: \, \overline{\tens{S}}},
\end{equation}%
\nomenclature[odotproductdouble]{$:$}{double dot product}
%
where $\overline{\tens{S}}$ the filtered strain rate tensor components:

\begin{equation}
\overline{\tens{S}}= \symmetric{\overline{\tens{S}}} =
\frac{1}{2} \left[ \gradt \, \vect{\overline{u}} + \transpose{\left( \gradt \, \vect{\overline{u}} \right)}
\right].
\end{equation}
%
\nomenclature[osymmetric]{$\symmetric{ \left(\tens{.}\right)}$}{symmetric part of a tensor}
%
Here, $\overline{u_{i}}$ stands for the $i^{th}$ resolved velocity component
\footnote{%
In the case of implicit filtering, the discretization in space introduces a
spectral low pass filter: only the structures larger that twice the size of
the cells are accounted for. Those structures are called ''the resolved
scales'', whereas the rest, $u_{i}-\overline{u_{i}}$, is referred to as
''unresolved scales'' or ''sub-grid scales''. The influence of the
unresolved scales on the resolved scales have to be modelled.}. 

$C$ is the Smagorinsky constant. Its theoretical value is $0.18$ for
homogeneous isotropic turbulence, but the value $0.065$ is classic for
channel flow. 

$\overline{\Delta }$ is the filter width associated with the finite volume
formulation (implicit filtering which corresponds to the integration over a
cell). The value recommended for hexahedral cells is: $\overline{\Delta }
=2 \norm{\vol{\celli}}^{\frac{1}{3}}$where $\norm{\vol{\celli}}$ is the volume of the cell $\celli$.

%-------------------------------------------------------------------------------
\subsection{Dynamic Smagorinsky model}
A second filter is introduced: it is an explicit filter with a
characteristic width $\widetilde{\Delta }$ superior to that of the implicit
filter ($\overline{\Delta }$). If $\varia$ is a discrete variable defined over
the computational domain, the variable obtained after applying the explicit
filter to $\varia$ is noted $\tilde{\varia}$. Moreover, with:

\begin{equation}
\begin{array}{ r c l}
\tens{L} & = &\widetilde{\overline{\vect{u}} \otimes \overline{\vect{u}}}
-\widetilde{\overline{\vect{u}}} \otimes \widetilde{\overline{ \vect{u}}}, \\
\tens{\tau} & = & \overline{ \vect{u} \otimes \vect{u}}-\overline{\vect{u}} \otimes \overline{ \vect{u}}, \\
\tens{T} &= &\widetilde{\overline{ \vect{u} \otimes \vect{u}}}-\widetilde{\overline{\vect{u}}} \otimes
\widetilde{\overline{ \vect{u}}},
\end{array}
\end{equation}
Germano identity reads:
\begin{equation}
\tens{L} = \tens{T}-\widetilde{\tens{\tau}}.
\end{equation}

Both dynamic models described hereafter rely on the estimation of the tensors
$\tens{T}$ and $\tens{\tau}$ as functions of the filter widths and of the
strain rate tensor (Smagorinsky model). The following modelling is adopted%
\footnote{$\delta_{ij}$ stands for the Kroeneker symbol.}:

\begin{equation}
\begin{array}{ r c l}
T_{ij}-\frac{1}{3}\trace \tens{T} \delta_{ij} &=& -2 C \widetilde{\Delta}^2 |\widetilde{%
\overline{D_{ij}}}| \widetilde{\overline{D_{ij}}}, \\
\tau_{ij}-\frac{1}{3} \tens{\tau } \delta_{ij} &=& -2 C^* \overline{\Delta}^2 |%
\overline{D_{ij}}| \overline{D_{ij}} ,
\end{array}
\end{equation}
where 
$\overline{u}$ stands for the \emph{implicit-filtered} value of a variable $u$
defined at the centres of the cells and $\overline{u}$ represents the
\emph{explicit-filtered} value associated with the variable $u$. It follows that
the numerical computation of $L_{ij}$ is possible, since it requires the
explicit filtering to be applied to implicitly filtered variables only 
(\emph{i.e.} to the variables explicitly computed).

On the following assumption:

\begin{equation}
C = C^*,
\end{equation}
and assuming that $C^*$ is only slightly non-uniform, so that it can be
taken out of the explicit filtering operator, the following equation is
obtained:

\begin{equation}
\deviator{\tens{L}} =  C \left(
\tens{ \alpha}- \tens{\widetilde{\beta}} \right),
\end{equation}
with:
\begin{equation}
\begin{array}{rcl}
\alpha_{ij} &=& -2 \widetilde{\Delta}^2 |\widetilde{\overline{D_{ij}}}|
\widetilde{\overline{D_{ij}}} , \\
\beta_{ij} &=& -2 \overline{\Delta}^2 |\overline{D_{ij}}| \overline{D_{ij}}.
\end{array}
\end{equation}

Since we are left with six equations to determine one single variable, the
least square method is used\footnote{
$\trace \tens{L}$ has no effect for
incompressible flows.}. With:
\begin{equation}
\tens{E} = \tens{L}-C \left( \tens{\alpha} - \tens{\widetilde{\beta}} \right),
\end{equation}
the value for $C$ is obtained by solving the following equation:
\begin{equation}
\frac{\partial \tens{E} : \tens{E}}{\partial C} = 0.
\end{equation}

Finally:
\begin{equation}
C = \frac{ \tens{M} : \tens{L} }{ \tens{M} : \tens{M}},
\end{equation}
with
\begin{equation}
\tens{M} = \tens{\alpha} - \tens{\widetilde{\beta}}.
\end{equation}

This method allows to calculate the Smagorinsky "constant" dynamically at
each time step and at each cell. However, the value obtained for $C$ can be
subjected to strong variations. Hence, this approach is often restricted to
flows presenting one or more homogeneous directions (Homogeneous Isotropic
Turbulence, 2D flows presenting an homogeneous span-wise direction...).
Indeed, in such cases, the model can be (and is) stabilized by replacing $C$
by an average value of $C$ computed over the homogeneous direction(s).

For a general case (without any homogeneous direction), a specific averaging
is introduced to stabilize the model: for any given cell of the mesh, the
averaged Smagorinsky constant is obtained as an average of $C$ over the
"extended neighbouring" of the cell (the set of cells that share at least
one vertex with the cell considered). More precisely, the average value
(also denoted $C$ hereafter) is calculated as indicated below:

\begin{equation}
C = \frac{\widetilde{ \tens{M} : \tens{L}}} {\widetilde{ \tens{M} : \tens{M}}}
\end{equation}


%-------------------------------------------------------------------------------
%\subsection{WALE model}

