Changeset 945 for palm/trunk/TUTORIAL/SOURCE/sgs_models.tex

Timestamp:

Jul 17, 2012 3:43:01 PM (12 years ago)

Author:

maronga

Message:

added/updated several tutorial files

File:

• palm/trunk/TUTORIAL/SOURCE/sgs_models.tex (modified) (3 diffs)

palm/trunk/TUTORIAL/SOURCE/sgs_models.tex

@@ -19 +19 @@
 \usepackage{amssymb}
 \usepackage{multicol}
-\usepackage{float}
+\usepackage{pdfcomment}
 
 \institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
@@ -43 +43 @@
 \author{Siegfried Raasch}
 
-% Notes:
-% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
-% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
 \begin{document}
 
@@ -167 +164 @@
 
 
-\section{Deardoff Modification}
-\subsection{Deardoff Modification}
-
-% Folie 7
-\begin{frame}
-   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
-   \footnotesize
-   \onslide<1->{
-      $ \nu_T = Cql = C_M \Lambda \sqrt{\bar{e}} $ \quad \textbf{with} \quad $ \bar{e} = \frac{\overline{u_i' u_i'}}{2} $ \quad \textbf{SGS-turbulent kinetic energy}}
-   \normalsize
-   \begin{itemize}
-      \item<2->{The SGS-TKE allows a much better estimation of the velocity scale for the SGS fluctuations and also contains information about the past history of the local fluid.}
-   \end{itemize} 
-   \onslide<3->{
-      $ C_M = const. = 0.1 $
-      \par\bigskip
-      \scriptsize
-      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
-                          min\left( 0.7 \cdot z, \Delta, 0.76 \sqrt{\bar{e}} \left[ \frac{g}{\Theta_0} \frac{\partial \bar{\Theta}}{\partial z} \right]^{-1/2} \right), & \textbf{stable                             stratification} 
-                  \end{cases} $      
-      \normalsize
-      \par\bigskip
-      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
-\end{frame}
-
-% Folie 8
-\begin{frame}
-   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
-   \begin{itemize}
-      \item{SGS-TKE from prognostic equation}
-   \end{itemize}
-   $ \frac{\partial \bar{e}}{\partial t} = -\bar{u_k} \frac{\partial \bar{e}}{\partial x_k} - \tau_{ki} \frac{\partial \bar{u_i}}{\partial x_k} + \frac{g}{\Theta_0} \overline{u_3'             \Theta'} - \frac{\partial}{\partial x_k} \left\{ \overline{u_k' \left( e' + \frac{\pi'}{\rho_0} \right)} \right\} - \epsilon $                                         
-   \par\bigskip                                        
-   $ \frac{\partial}{\partial x_k} \left[ \overline{u_k' \left( e' + \frac{\pi'}{\rho_0} \right)} \right] = - \frac{\partial}{\partial x_k} \nu_e \frac{\partial \bar{e}}{\partial x_k} $
-   \par\bigskip
-   $ \nu_e = 2 \nu_T $
-   \par\bigskip
-   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
-\end{frame}
-
-% Folie 9
-\begin{frame}
-   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
-   \begin{itemize}
-      \item{There are still problems with this parameterization:}
-      \begin{itemize}
-         \item[-]<2->{The model overestimates the velocity shear near the wall.}
-         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
-         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
-      \end{itemize}
-      \item<5->{Several other SGS models have been developed:}
-      \begin{itemize}
-         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
-         \item[-]<6->{Scale similarity model (Bardina et al.)}
-         \item[-]<7->{Backscatter model (Mason)}
-      \end{itemize}
-      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
-               from the different models (Beare et al., 2006, BLM).}
-   \end{itemize} 
-\end{frame}
-
-
-\section{Summary / Important Points for Beginners}
-\subsection{Summary / Important Points for Beginners}
-
-% Folie 10
-\begin{frame}
-   \frametitle{Summary / Important Points for Beginners (I)}
-   \begin{columns}[c]
-   \column[T]{0.4\textwidth} 
-      \includegraphics<2-7>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
-      \includegraphics<8>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
-      \includegraphics<9>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
-      \includegraphics<10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
-      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}              
-   \column[T]{0.2\textwidth}
-      \vspace{0.9cm}
-      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
-      \par
-      \onslide<8->{\begin{small} fluctuations (\textbf{u},c) \end{small}}
-      \par\bigskip
-      \thicklines
-      \onslide<9->{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}}
-      \vspace{1cm}
-      
-      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
-      \par
-      \onslide<8->{\begin{small} smooth result \end{small}}   
-   \column[T]{0.4\textwidth}     
-      \includegraphics<1-2>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1.png}
-      \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3.png}
-      \includegraphics<4>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
-      \includegraphics<5-10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
-      \vspace{1.3cm}
-      \includegraphics<6>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6.png}
-      \uncover<7->{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7.png}}       
-   \end{columns}
-\end{frame}
-
-% Folie 11
-\begin{frame}
-   \frametitle{Summary / Important Points for Beginners (II)}
-    For an LES it always has to be guaranteed that the main energy containing eddies of the respective 
-    turbulent flow can really be simulated by the numerical model:      
-    \begin{itemize}
-       \item<2->{The grid spacing has to be fine enough.}
-       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
-       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
-                (therefore cyclic boundary conditions are used in most cases).}
-       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence 
-                  has to be triggered by imposing random fluctuations.}
-       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
-    \end{itemize}      
-\end{frame}
-
-
-\section{Example Output}
-\subsection{Example Output}
-
-% Folie 12
-\begin{frame}
-   \frametitle{Example Output (I)}
-   \begin{itemize}
-      \item{LES of a convective boundary layer}
-   \end{itemize}
-   \includegraphics<1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
-   \includegraphics<2>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
-   \includegraphics<3>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
-   \includegraphics<4>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
-   \includegraphics<5>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
-   \includegraphics<6>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
-   \includegraphics<7>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
-\end{frame}
-
-% Folie 13
-\begin{frame}
-   \frametitle{Example Output (II)}
-   \begin{itemize}
-      \item{LES of a convective boundary layer}
-   \end{itemize}
-   \begin{center}
-      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
-      power spectrum of vertical velocity
-   \end{center}
-\end{frame}
-
-% Folie 14
-\begin{frame}
-   \frametitle{Some Symbols}
-   \begin{columns}[c]
-      \column{0.6\textwidth}
-      \begin{tabbing}
-      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
-      $u,v,w$ \\ 
-
-      \\
-      
-      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
-      $x,y,z$ \\
-
-      \\
-
-      $\Theta$ \> potential temperature \\ \\
-
-      $\Psi$ \> passive scalar \\ \\
-
-      $T$ \> actual Temperatur \\ \\
-      \end{tabbing}
-   \column{0.4\textwidth}
-      \begin{tabbing}
-      $\Phi = gz$  \quad \= geopotential \\ \\
-
-      $p$ \> pressure \\ \\
-
-      $\rho$ \> density \\ \\
-
-      $f_i$ \> Coriolis Parameter \\ \\
-
-      $\epsilon_{ijk}$ \> alternating symbol \\ \\
-
-      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
-
-      $Q, Q_\Psi$ \> sources or sinks \\ \\
-      \end{tabbing}
-   \end{columns}
-\end{frame}
 \end{document}