# Changeset 987 for palm/trunk/TUTORIAL/SOURCE

Ignore:
Timestamp:
Sep 5, 2012 9:27:35 AM (11 years ago)
Message:

sgs_model.tex update

File:
1 edited

### Legend:

Unmodified
 r945 % Folie 7 \begin{frame} \frametitle{The Smagorinsky Model: Performance} \begin{itemize} \item<2-> Predicts many flows reasonably well \item<3-> Problems: \begin{itemize} \item<3-> Optimum parameter value varies with flow type: \begin{itemize} \item Isotropic turbulence: $C_S \approx 0.2$\\ \item Shear (channel) flows: $C_S \approx 0.065$ \end{itemize} \item<4-> Length scale uncertain with anisotropic filter: \begin{equation*} (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3 \end{equation*} \item<5-> Needs modification to account for: \begin{itemize} \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\ \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall \end{itemize} \end{itemize} \end{itemize} \end{frame} \section{Deardoff Modification} \subsection{Deardoff Modification} % Folie 8 \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 9 \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 10 \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 11 \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_neu.png} \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.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_neu.png} \uncover<7->{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}} \end{columns} \end{frame} % Folie 12 \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 13 \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 14 \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 15 \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}