% $Id: basic_equations.tex 945 2012-07-17 15:43:01Z fricke $ \input{header_tmp.tex} %\input{../header_lectures.tex} \usepackage[utf8]{inputenc} \usepackage{ngerman} \usepackage{pgf} \usetheme{Dresden} \usepackage{subfigure} \usepackage{units} \usepackage{multimedia} \usepackage{hyperref} \newcommand{\event}[1]{\newcommand{\eventname}{#1}} \usepackage{xmpmulti} \usepackage{tikz} \usetikzlibrary{shapes,arrows,positioning} \def\Tiny{\fontsize{4pt}{4pt}\selectfont} \usepackage{amsmath} \usepackage{amssymb} \usepackage{multicol} \institute{Institut für Meteorologie und Klimatologie, Leibniz Universität Hannover} \date{last update: \today} \event{PALM Seminar} \setbeamertemplate{navigation symbols}{} \setbeamertemplate{footline} { \begin{beamercolorbox}[rightskip=-0.1cm]& {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} \end{beamercolorbox} \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex, leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot} {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber} \end{beamercolorbox} \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot} \end{beamercolorbox} } %\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}} \title[Basic Equations]{Basic Equations} \author{Siegfried Raasch} \begin{document} %Folie 1 \begin{frame} \titlepage \end{frame} \section{Basic equations} \subsection{Basic equations, Unfiltered} % Folie 2 \begin{frame} \frametitle{Basic equations, Unfiltered} \begin{itemize} \item<2->Navier-Stokes equations \begin{equation*} \rho \frac{\partial u_i}{\partial t} + \rho u_k \frac{\partial u_i}{\partial x_k} = - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk} f_j u_k - \frac{\partial \phi}{\partial x_i} + \mu \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3} \frac{\partial}{\partial x_i} \left( \frac{\partial u_k}{\partial x_k} \right) \right\} \end{equation*} \item \onslide<3->First principle \begin{equation*} \rho \frac{\partial T}{\partial t} + \rho u_k \frac{\partial T}{\partial x_k} = \mu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q \end{equation*} \item \onslide<4->Equation for passive scalar \begin{equation*} \rho \frac{\partial \psi}{\partial t} + \rho u_k \frac{\partial \psi}{\partial x_k} = \mu_{\psi} \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi} \end{equation*} \item \onslide<5->Continuity equation \begin{equation*} \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} \end{equation*} \end{itemize} \end{frame} % Folie 3 \begin{frame} \frametitle{Boussinesq Approximation} \footnotesize \begin{itemize} \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ \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*} \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g}, \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g} \end{equation*} \item \onslide<4->Equation of state \begin{equation*} 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} \end{equation*} \end{itemize} \end{frame} % Folie 4 \begin{frame} \frametitle{Continuity Equation} \begin{eqnarray*} \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} = - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} & \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0 \hspace{5mm} & \text{anelastic approximation}\\ \\ \onslide<3-> \rho_0 = const. & \hspace{10mm} \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} & \text{incompressible flow} \end{eqnarray*} \end{frame} % Folie 5 \begin{frame} \frametitle{Boussinesq Approximated Equations} \begin{itemize} \item \onslide<2->Navier-Stokes equations \begin{equation*} \frac{\partial u_i}{\partial t} + \frac{\partial u_k u_i}{\partial x_k} = - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i} - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}} + g \frac{T - T_0}{T_0} \delta_{i3} + \nu \frac{\partial^2 u_i}{\partial x_k^2} \end{equation*} \item \onslide<3->First principle \begin{equation*} \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} = \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q \end{equation*} \item \onslide<4->Equation for passive scalar \begin{equation*} \frac{\partial \psi}{\partial t} + u_k \frac{\partial \psi}{\partial x_k} = \nu_{\psi} \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi} \end{equation*} \item \onslide<5->Continuity equation \begin{equation*} \frac{\partial u_k}{\partial x_k} = 0 \end{equation*} \end{itemize} \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){% \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!}; \end{tikzpicture} }; \end{tikzpicture} \end{frame} \section{Scale Separation} \subsection{Scale Separation by Spatial Filtering} % Folie 6 \begin{frame} \frametitle{LES - Scale Separation by Spatial Filtering (I)} \footnotesize \begin{itemize} \item<1->{LES technique is based on scale separation, in order to reduce the number of degrees of freedom of the solution. \begin{math} \boxed{\Psi(x_i , t) = \overline{\Psi}(x_i , t) + \Psi'(x_i , t)} \end{math}} \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).} \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).} \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.} \end{itemize} \normalsize \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png} \end{frame} % Folie 7 \begin{frame} \frametitle{LES - Scale Separation by Spatial Filtering (II)} \begin{columns}[T] \begin{column}{0.8\textwidth} \footnotesize \begin{itemize} \item<1->The Filter applied is a spatial filter: \begin{equation*} \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i' \end{equation*} \begin{equation*} \overline{\Psi}'(x_i) = 0 \qquad but \qquad \overline{\overline{\Psi}} \neq \overline{\Psi}(x_i) \end{equation*} \item<2->Filter applied to the nonlinear advection term: \begin{equation*} \overline{u_k u_i} = \overline{(\overline{u_k} + u_k')(\overline{u_i} + u_i')} = \overline{\overline{u_k}\,\overline{u_i}} + \underbrace{\overline{\overline{u_k} u_i'} + \overline{u_k' \overline{u_i}}}_{C_{ki}} + \underbrace{\overline{u_k' u_i'}}_{R_{ki}} \end{equation*} \item<5->Leonard proposes a further decomposition: \begin{equation*} \overline{\overline{u_k}\,\overline{u_i}} = \overline{u_k}\,\overline{u_i} + \underbrace{\left( \overline{\overline{u_k}\,\overline{u_i}} - \overline{u_k}\,\overline{u_i} \right)}_{L_{ki}} \end{equation*} \begin{equation*} \overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + L_{ki} + C_{ki} + R_{ki} = \overline{u_k}\,\overline{u_i} + \tau_{ki} \end{equation*} \end{itemize} \end{column} \begin{column}{0.32\textwidth} \vspace{45mm} \begin{footnotesize} \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\ $C_{ki}$: \textbf{cross-stress} \\ $L_{ki}$: \textbf{Leonard-stress} \\ $\tau_{ki}$: \textbf{total stress-tensor} \end{footnotesize} \end{column} \end{columns} \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){ \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (94mm,-30mm) { \begin{footnotesize} \noindent \textbf{Ensemble average:} \\ \end{footnotesize} $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\ \vspace{5mm} $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$ }; \end{tikzpicture} }; \end{tikzpicture} \end{frame} % Folie 8 \begin{frame} \frametitle{LES - Scale Separation by Spatial Filtering (III)} \small \begin{itemize} \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator \begin{flalign*} &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\ &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\ &V=\left[ x - \frac{\Delta x}{2}, x + \frac{\Delta x}{2} \right] \times \left[ y - \frac{\Delta y}{2}, y + \frac{\Delta y}{2} \right] \times \left[ z - \frac{\Delta z}{2}, z + \frac{\Delta z}{2} \right] \end{flalign*} \item<3-> Filter applied to the nonlinear advection term: \begin{equation*} \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}= \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i} \end{equation*} \end{itemize} \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){ \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (94mm,-30mm) { \begin{footnotesize} \noindent \textbf{Ensemble average:} \\ \end{footnotesize} $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\ \vspace{5mm} $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$ }; \end{tikzpicture} }; \end{tikzpicture} \end{frame} \section{Filtered equations} \subsection{The Filtered Equations} % Folie 9 \begin{frame} \frametitle{The Filtered Equations} \onslide<2-> \begin{equation*} \frac{\partial \overline{u_i}}{\partial t} + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} = - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i} - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} - \frac{\partial \tau_{ki}}{\partial x_k} \end{equation*} \begin{footnotesize} \begin{itemize} \item<3->The previous derivation completely ignores the existance of the computational grid. \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$. \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$ \item<6-> Two possibilities:\\ 1. Pre-filtering technique\\ ($\Delta x < \Delta$, explicit filtering)\\ 2. Linking the analytical filter\\ to the computational grid\\ ($\Delta x = \Delta$, implicit filtering) \end{itemize} \end{footnotesize} \begin{picture}(0.0,0.0) \put(140,13){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}} \end{picture} \end{frame} %% Folie 10 \begin{frame} \frametitle{Explicit Versus Implicit Filtering} \begin{itemize} \item<2-> Explicit filtering: \begin{itemize} \small \item<2-> Requires that the analytical filter is applied explicitly. \item<3-> Rarely used in practice, due to additional computational costs. \end{itemize} \item<4-> Implicit filtering: \begin{itemize} \small \item<4-> The analytical cutoff length is associated with the grid spacing. \item<5-> This method does not require the use of an analytical filter. \item<6-> The filter characteristic cannot really be controlled. \item<7-> Because of its simplicity, this method is used by nearly all LES models. \end{itemize} \end{itemize} \onslide<8-> \begin{scriptsize} \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.\\ \end{scriptsize} \end{frame} % Folie 11 \begin{frame} \frametitle{The Final Set of Equations (PALM)} \footnotesize \begin{itemize} \item<2-> Navier-Stokes equations: \onslide<2-> \begin{flalign*} &\frac{\partial \overline{u_i}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} - \frac{1}{\rho_0} \frac{\partial \overline{\pi}^*}{\partial x_i} - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} - \frac{\partial \tau_{ki}^r}{\partial x_k}& \end{flalign*} \item<4-> First principle (using potential\\ temperature): \onslide<4-> \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}& \end{flalign*} \item<5-> Equation for specific humidity\\ (passive scalar) \onslide<5-> \begin{flalign*} &\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-> Continuity equation \onslide<6-> \begin{flalign*} &\frac{\partial \overline{u_k}}{\partial x_k} = 0& \end{flalign*} \end{itemize} \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){ \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (75mm,-45mm) { \begin{tiny} \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\ \end{tiny} $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\ \vspace{1mm} $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$ }; \end{tikzpicture} }; \end{tikzpicture} \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small] \begin{tikzpicture}[remember picture, overlay] \node at (current page.north west){ \begin{tikzpicture}[overlay] \node[plain, draw,anchor=west] at (75mm,-70mm) { \begin{tiny} \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\ \end{tiny} $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\ $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\ $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$ }; \end{tikzpicture} }; \end{tikzpicture} \end{frame} \end{document}