# Changeset 945 for palm/trunk/TUTORIAL/SOURCE

Ignore:
Timestamp:
Jul 17, 2012 3:43:01 PM (11 years ago)
Message:

Location:
palm/trunk/TUTORIAL/SOURCE
Files:
5 edited

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• ## palm/trunk/TUTORIAL/SOURCE/basic_equations.tex

 r915 \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{ngerman} \usepackage{pgf} \usetheme{Dresden} \item \onslide<5->Continuity equation \begin{equation*} \frac{\partial p}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} \end{equation*} \end{itemize} \begin{frame} \frametitle{Boussinesq Approximation} \small \footnotesize \begin{itemize} \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ \begin{equation*} p(x,y,z,t) = p_0(x,y,z,t) + p^{*}(x,y,z,t)\hspace{22mm} \end{equation*} \begin{equation*} \rho(x,y,z,t) = \rho_0(x,y,z,t) + \rho^{*}(x,y,z,t) ; \hspace{5mm} \psi^{*} << \psi_0 \end{equation*} \begin{align*} T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\ p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\ \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& & &\psi^{*} << \psi_0& \end{align*} \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq) \begin{equation*} \item \onslide<4->Equation of state \begin{equation*} p_0 = \rho_0 R T_0 \rightarrow \ln{p_0} = \ln{\rho_0} + \ln{R} + \ln{T_0} \rightarrow \frac{\partial p_0}{p_0} = \frac{\partial \rho_0}{\rho_0} + \frac{\partial T_0}{T_0} \end{equation*} \begin{equation*} \frac{\Delta p_0}{p_0} \approx \frac{\Delta \rho_0}{\rho_0} + \frac{\Delta T_0}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx p = \rho R T \rightarrow \ln{p} = \ln{\rho} + \ln{R} + \ln{T} \rightarrow \frac{d p}{p} = \frac{d \rho}{\rho} + \frac{d T}{T} \end{equation*} \begin{equation*} \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} + \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm} \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm} \textbf{Literature:}\\ \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\ \textbf{Schumann, U., 1975:} Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., \textbf{18}, 376-404. \textbf{Schumann, U., 1975:} Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., \textbf{18}, 376-404.\\ \end{scriptsize} \end{frame} \onslide<5-> \begin{flalign*} &\frac{\partial \overline{\theta}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{\theta}}{\partial x_k} - \frac{\partial H_k}{\partial x_k} + Q_{\theta}& &\frac{\partial \overline{q}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{q}}{\partial x_k} - \frac{\partial W_k}{\partial x_k} + Q_{w}& \end{flalign*} \item<6-> \end{tiny} $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\ $H_{k} = \overline{u_k \theta_k} - \overline{u_k}\,\overline{\theta_i}$\\ $W_{k} = \overline{u_k q_k} - \overline{u_k}\,\overline{q_k}$ $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\ $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$ }; \end{tikzpicture}
• ## palm/trunk/TUTORIAL/SOURCE/fundamentals_of_les.tex

 r915 \usepackage{xmpmulti} \usepackage{tikz} \usepackage{pdfcomment} \usetikzlibrary{shapes,arrows,positioning} \def\Tiny{\fontsize{4pt}{4pt}\selectfont}
• ## palm/trunk/TUTORIAL/SOURCE/non_cyclic_boundary_conditions.tex

 r915 \end{itemize} \end{frame} \section{Motivation} \subsection{Motivation} %Folie 3 \end{frame} \section{How to Create a Turbulent Inflow} \subsection{How to Create a Turbulent Inflow} % Folie 5 \begin{frame} \begin{frame} \frametitle{How to Create a Turbulent Inflow (II)} \small \footnotesize Initial  turbulence is created by a precursor run with cyclic boundary conditions and much smaller domain size than used for the main run. \tikzstyle{line} = [draw, yellow, thick, dashed, -latex'] \tikzstyle{line} = [draw, blue, thick, dashed, -latex'] \begin{tikzpicture} \uncover<1>{\node(picture) {\includegraphics[width=0.4\textwidth]{non_cyclic_figures/create_turbulent_inflow_2/create_turbulent_inflow_1.png}};} \begin{itemize} \item<4->{When the precursor run is finished, data of the last timestep are stored on disc.} \item<5->{These data are then read by the main run and repeatedly mapped to the main run domain, unless it is completely filled.} \item<5->{These data are then read by the main run and repeatedly mapped to the main run domain, until it is completely filled.} \end{itemize}}}; \uncover<6>{\node(picture2) [below=1.8cm of picture.east] {\includegraphics[width=0.9\textwidth]{non_cyclic_figures/create_turbulent_inflow_2/create_turbulent_inflow_4.png}};} \end{frame} \section{Implementation in PALM} \subsection{Implementation in PALM} % Folie 8 \begin{frame} \textbf{Status of availability:} \begin{itemize} \item<2->{Non-cyclic boundary conditions along \textbf{one} of the horizontal directions (x \textbf{or} y).} \item<2->{Non-cyclic boundary conditions along \textbf{one} of the horizontal directions (\textit{x} \textbf{or} \textit{y}).} \begin{itemize} \item<3->{Dirichlet conditions at inflow (stationary vertical profiles, u(z), v(z), pt(z), q(z), w=0).} \item<3->{Dirichlet conditions at inflow (stationary vertical profiles, \textit{u}(\textit{z}), \textit{v}(\textit{z}), \textit{pt}(\textit{z}), \textit{q}(\textit{z}), \textit{w}=0).} \item<4->{Radiation conditions at outflow. Tendencies at the boundary are replaced by e.g.} \end{itemize} \end{frame} \section{Current Applications} \subsection{Current Applications} % Folie 10 \begin{frame} \end{center} \end{frame} \section{How to set up} \subsection{How to set up} % Folie 12 \end{frame} \section{Final remarks} \subsection{Final remarks} % Folie 16 \begin{frame}
• ## palm/trunk/TUTORIAL/SOURCE/program_control.tex

 r915 \usepackage{amssymb} \usepackage{multicol} \usepackage{float} \usepackage{pdfcomment} \institute{Institut fÃŒr Meteorologie und Klimatologie, Leibniz UniversitÃ€t Hannover} \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} \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}