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Timestamp:
Sep 5, 2012 9:27:35 AM (9 years ago)
Author:
maronga
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sgs_model.tex update

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  • palm/trunk/TUTORIAL/SOURCE/sgs_models.tex

    r945 r987  
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    165165
     166% Folie 7
     167\begin{frame}
     168   \frametitle{The Smagorinsky Model: Performance}
     169   \begin{itemize}
     170      \item<2-> Predicts many flows reasonably well
     171      \item<3-> Problems:
     172      \begin{itemize}
     173         \item<3-> Optimum parameter value varies with flow type:
     174         \begin{itemize}
     175            \item Isotropic turbulence: $C_S \approx 0.2$\\
     176            \item Shear (channel) flows: $C_S \approx 0.065$
     177         \end{itemize}
     178         \item<4-> Length scale uncertain with anisotropic filter:
     179         \begin{equation*}
     180            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
     181         \end{equation*}
     182         \item<5-> Needs modification to account for:
     183         \begin{itemize}
     184            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
     185            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
     186         \end{itemize}
     187      \end{itemize}
     188   \end{itemize}
     189\end{frame}
     190
     191
     192\section{Deardoff Modification}
     193\subsection{Deardoff Modification}
     194
     195% Folie 8
     196\begin{frame}
     197   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
     198   \footnotesize
     199   \onslide<1->{
     200      $ \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}}
     201   \normalsize
     202   \begin{itemize}
     203      \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.}
     204   \end{itemize}
     205   \onslide<3->{
     206      $ C_M = const. = 0.1 $
     207      \par\bigskip
     208      \scriptsize
     209      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
     210                          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}
     211                  \end{cases} $     
     212      \normalsize
     213      \par\bigskip
     214      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
     215\end{frame}
     216
     217% Folie 9
     218\begin{frame}
     219   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
     220   \begin{itemize}
     221      \item{SGS-TKE from prognostic equation}
     222   \end{itemize}
     223   $ \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 $                                         
     224   \par\bigskip                                       
     225   $ \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} $
     226   \par\bigskip
     227   $ \nu_e = 2 \nu_T $
     228   \par\bigskip
     229   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
     230\end{frame}
     231
     232% Folie 10
     233\begin{frame}
     234   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
     235   \begin{itemize}
     236      \item{There are still problems with this parameterization:}
     237      \begin{itemize}
     238         \item[-]<2->{The model overestimates the velocity shear near the wall.}
     239         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
     240         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
     241      \end{itemize}
     242      \item<5->{Several other SGS models have been developed:}
     243      \begin{itemize}
     244         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
     245         \item[-]<6->{Scale similarity model (Bardina et al.)}
     246         \item[-]<7->{Backscatter model (Mason)}
     247      \end{itemize}
     248      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
     249               from the different models (Beare et al., 2006, BLM).}
     250   \end{itemize}
     251\end{frame}
     252
     253
     254\section{Summary / Important Points for Beginners}
     255\subsection{Summary / Important Points for Beginners}
     256
     257% Folie 11
     258\begin{frame}
     259   \frametitle{Summary / Important Points for Beginners (I)}
     260   \begin{columns}[c]
     261   \column[T]{0.4\textwidth}
     262      \includegraphics<2-7>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
     263      \includegraphics<8>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
     264      \includegraphics<9>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
     265      \includegraphics<10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
     266      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}             
     267   \column[T]{0.2\textwidth}
     268      \vspace{0.9cm}
     269      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
     270      \par
     271      \onslide<8->{\begin{small} fluctuations (\textbf{u},c) \end{small}}
     272      \par\bigskip
     273      \thicklines
     274      \onslide<9->{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}}
     275      \vspace{1cm}
     276     
     277      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
     278      \par
     279      \onslide<8->{\begin{small} smooth result \end{small}}   
     280   \column[T]{0.4\textwidth}     
     281      \includegraphics<1-2>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png}
     282      \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png}
     283      \includegraphics<4>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
     284      \includegraphics<5-10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
     285      \vspace{1.3cm}
     286      \includegraphics<6>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png}
     287      \uncover<7->{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}}       
     288   \end{columns}
     289\end{frame}
     290
     291% Folie 12
     292\begin{frame}
     293   \frametitle{Summary / Important Points for Beginners (II)}
     294    For an LES it always has to be guaranteed that the main energy containing eddies of the respective
     295    turbulent flow can really be simulated by the numerical model:     
     296    \begin{itemize}
     297       \item<2->{The grid spacing has to be fine enough.}
     298       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
     299       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
     300                (therefore cyclic boundary conditions are used in most cases).}
     301       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence
     302                  has to be triggered by imposing random fluctuations.}
     303       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
     304    \end{itemize}     
     305\end{frame}
     306
     307
     308\section{Example Output}
     309\subsection{Example Output}
     310
     311% Folie 13
     312\begin{frame}
     313   \frametitle{Example Output (I)}
     314   \begin{itemize}
     315      \item{LES of a convective boundary layer}
     316   \end{itemize}
     317   \includegraphics<1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
     318   \includegraphics<2>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
     319   \includegraphics<3>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
     320   \includegraphics<4>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
     321   \includegraphics<5>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
     322   \includegraphics<6>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
     323   \includegraphics<7>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
     324\end{frame}
     325
     326% Folie 14
     327\begin{frame}
     328   \frametitle{Example Output (II)}
     329   \begin{itemize}
     330      \item{LES of a convective boundary layer}
     331   \end{itemize}
     332   \begin{center}
     333      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
     334      power spectrum of vertical velocity
     335   \end{center}
     336\end{frame}
     337
     338% Folie 15
     339\begin{frame}
     340   \frametitle{Some Symbols}
     341   \begin{columns}[c]
     342      \column{0.6\textwidth}
     343      \begin{tabbing}
     344      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
     345      $u,v,w$ \\
     346
     347      \\
     348     
     349      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
     350      $x,y,z$ \\
     351
     352      \\
     353
     354      $\Theta$ \> potential temperature \\ \\
     355
     356      $\Psi$ \> passive scalar \\ \\
     357
     358      $T$ \> actual Temperatur \\ \\
     359      \end{tabbing}
     360   \column{0.4\textwidth}
     361      \begin{tabbing}
     362      $\Phi = gz$  \quad \= geopotential \\ \\
     363
     364      $p$ \> pressure \\ \\
     365
     366      $\rho$ \> density \\ \\
     367
     368      $f_i$ \> Coriolis Parameter \\ \\
     369
     370      $\epsilon_{ijk}$ \> alternating symbol \\ \\
     371
     372      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
     373
     374      $Q, Q_\Psi$ \> sources or sinks \\ \\
     375      \end{tabbing}
     376   \end{columns}
     377\end{frame}
    166378\end{document}
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