Changeset 2132 for branches/devmercator2010_1/DOC/TexFiles/Chapters/Chap_ZDF.tex

Timestamp:

2010-09-29T17:31:36+02:00 (14 years ago)

Author:

cbricaud

Message:

add change from DEV_r1784_GLS

File:

• branches/devmercator2010_1/DOC/TexFiles/Chapters/Chap_ZDF.tex (modified) (1 diff)

branches/devmercator2010_1/DOC/TexFiles/Chapters/Chap_ZDF.tex

@@ -233 +233 @@
 
 % -------------------------------------------------------------------------------------------------------------
+%        GLS Generic Length Scale Scheme 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{GLS Generic Length Scale (\key{zdfgls})}
+\label{ZDF_gls}
+
+%--------------------------------------------namgls---------------------------------------------------------
+\namdisplay{namgls}
+%--------------------------------------------------------------------------------------------------------------
+
+The model allows to resolve two prognostic equations for turbulent 
+kinetic energy $\bar {e}$ and a generic length scale \citep{Umlauf_Burchard_2003}. Thanks to the latter, commonly 
+used closures can be retrieved: $k-kl$ \citep{Mellor_Yamada_1982}, $k-{\epsilon }$ \citep{Rodi_1987} and $k-{\omega }$ 
+\citep{Wilcox_1988}. These equations could be written in a generic form with the incorporation 
+of a new variable : ${\psi} = (C^{0}_{\mu})^{p} \ {\bar{e}}^{m} \ l^{n}$.
+
+\begin{equation} \label{Eq_zdfgls_e}
+\frac{\partial \bar{e}}{\partial t} = 
+\frac{A^{vm}}{{\sigma_e} {e_3} }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
+                                                        +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
+-A^{vT}\,N^2
++\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
+                                \;\frac{\partial \bar{e}}{\partial k}} \right]
+- \epsilon \;
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_psi}
+\frac{\partial \psi}{\partial t} = \frac{\psi}{\bar{e}} 
+(\frac{{C_1}A^{vm}}{{\sigma_{\psi}} {e_3} }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
+                                                        +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
+-{C_3}A^{vT}\,N^2- C_2{\epsilon}Fw) 
++\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
+                                \;\frac{\partial \psi}{\partial k}} \right]\;
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_kz}
+   \begin{split}
+         A^{vm} &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
+         A^{vT} &= C_{\mu'}\ \sqrt {\bar{e}} \ l
+   \end{split}
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_eps}
+{\epsilon} = (C^{0}_{\mu}) \frac{\bar {e}^{3/2}}{l} \;
+\end{equation}
+where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}) and $\epsilon$ the dissipation rate.
+In function of the parameters k, m and n, common turbulent closure could be retrieved.
+The constants C1, C2, C3, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function (Fw) depends of the choice of the turbulence model.
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!h]
+\centering
+\includegraphics[scale=0.7]{./TexFiles/Figures/tabgls.png}
+\caption {Values of the parameters in function of the model of turbulence.}
+\label{tabgls}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+About the Mellor-Yamada model, the negativity of n allows to use a wall function to force
+the convergence of the mixing length towards Kzb (K: Kappa and zb: rugosity length) value
+near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ are calculated from stability functions 
+of \citet{Galperin_1988}, \citet{Kantha_Clayson_1994} or \citet{Canuto_2001}.
+$C^{0}_{\mu}$ depends of the choice of the stability function.
+
+The boundary condition at the surface and the bottom could be calculated thanks to Diriclet or Neumann condition.
+The wave effect on the mixing could be also being considered \citep{Craig_Banner_1994}.
+
+-------------------------------------------------------------------------------------------------------------
 %        K Profile Parametrisation (KPP) 
 % -------------------------------------------------------------------------------------------------------------