[915] | 1 | % $Id: basic_equations.tex 945 2012-07-17 15:43:01Z keck $ |
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| 2 | \input{header_tmp.tex} |
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| 3 | %\input{../header_lectures.tex} |
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| 4 | |
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| 5 | \usepackage[utf8]{inputenc} |
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[945] | 6 | \usepackage{ngerman} |
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[915] | 7 | \usepackage{pgf} |
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| 8 | \usetheme{Dresden} |
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| 9 | \usepackage{subfigure} |
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| 10 | \usepackage{units} |
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| 11 | \usepackage{multimedia} |
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| 12 | \usepackage{hyperref} |
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| 13 | \newcommand{\event}[1]{\newcommand{\eventname}{#1}} |
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| 14 | \usepackage{xmpmulti} |
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| 15 | \usepackage{tikz} |
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| 16 | \usetikzlibrary{shapes,arrows,positioning} |
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| 17 | \def\Tiny{\fontsize{4pt}{4pt}\selectfont} |
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| 18 | \usepackage{amsmath} |
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| 19 | \usepackage{amssymb} |
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| 20 | \usepackage{multicol} |
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| 21 | |
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| 22 | |
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| 23 | \institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover} |
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| 24 | \date{last update: \today} |
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| 25 | \event{PALM Seminar} |
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| 26 | \setbeamertemplate{navigation symbols}{} |
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| 27 | |
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| 28 | \setbeamertemplate{footline} |
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| 29 | { |
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| 30 | \begin{beamercolorbox}[rightskip=-0.1cm]& |
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| 31 | {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} |
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| 32 | \end{beamercolorbox} |
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| 33 | \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex, |
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| 34 | leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot} |
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| 35 | {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber} |
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| 36 | \end{beamercolorbox} |
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| 37 | \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot} |
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| 38 | \end{beamercolorbox} |
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| 39 | } |
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| 40 | %\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}} |
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| 41 | |
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| 42 | \title[Basic Equations]{Basic Equations} |
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| 43 | \author{Siegfried Raasch} |
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| 44 | |
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| 45 | \begin{document} |
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| 46 | |
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| 47 | %Folie 1 |
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| 48 | \begin{frame} |
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| 49 | \titlepage |
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| 50 | \end{frame} |
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| 51 | |
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| 52 | |
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| 53 | \section{Basic equations} |
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| 54 | \subsection{Basic equations, Unfiltered} |
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| 55 | |
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| 56 | % Folie 2 |
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| 57 | \begin{frame} |
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| 58 | \frametitle{Basic equations, Unfiltered} |
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| 59 | \begin{itemize} |
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| 60 | \item<2->Navier-Stokes equations |
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| 61 | \begin{equation*} |
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| 62 | \rho \frac{\partial u_i}{\partial t} + \rho u_k |
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| 63 | \frac{\partial u_i}{\partial x_k} = |
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| 64 | - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk} |
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| 65 | f_j u_k - \frac{\partial \phi}{\partial x_i} + \mu |
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| 66 | \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3} |
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| 67 | \frac{\partial}{\partial x_i} \left( |
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| 68 | \frac{\partial u_k}{\partial x_k} \right) \right\} |
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| 69 | \end{equation*} |
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| 70 | \item \onslide<3->First principle |
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| 71 | \begin{equation*} |
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| 72 | \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 |
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| 73 | \end{equation*} |
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| 74 | \item \onslide<4->Equation for passive scalar |
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| 75 | \begin{equation*} |
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| 76 | \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} |
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| 77 | \end{equation*} |
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| 78 | \item \onslide<5->Continuity equation |
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| 79 | \begin{equation*} |
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[945] | 80 | \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} |
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[915] | 81 | \end{equation*} |
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| 82 | \end{itemize} |
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| 83 | \end{frame} |
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| 84 | |
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| 85 | % Folie 3 |
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| 86 | \begin{frame} |
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| 87 | \frametitle{Boussinesq Approximation} |
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[945] | 88 | \footnotesize |
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[915] | 89 | \begin{itemize} |
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| 90 | \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ |
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[945] | 91 | \begin{align*} |
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| 92 | T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\ |
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| 93 | p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\ |
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| 94 | \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& & |
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| 95 | &\psi^{*} << \psi_0& |
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| 96 | \end{align*} |
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[915] | 97 | \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq) |
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| 98 | \begin{equation*} |
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| 99 | \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm} |
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| 100 | \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g}, |
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| 101 | \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g} |
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| 102 | \end{equation*} |
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| 103 | \item \onslide<4->Equation of state |
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| 104 | \begin{equation*} |
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[945] | 105 | 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} |
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[915] | 106 | \end{equation*} |
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| 107 | \begin{equation*} |
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[945] | 108 | \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} + |
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| 109 | \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx |
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[915] | 110 | \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm} |
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| 111 | \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm} |
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| 112 | \end{equation*} |
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| 113 | \end{itemize} |
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| 114 | \end{frame} |
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| 115 | |
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| 116 | % Folie 4 |
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| 117 | \begin{frame} |
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| 118 | \frametitle{Continuity Equation} |
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| 119 | \begin{eqnarray*} |
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| 120 | \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} = |
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| 121 | - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} & |
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| 122 | \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0 |
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| 123 | \hspace{5mm} & \text{anelastic approximation}\\ |
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| 124 | \\ |
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| 125 | \onslide<3-> \rho_0 = const. & \hspace{10mm} |
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| 126 | \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} & |
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| 127 | \text{incompressible flow} |
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| 128 | \end{eqnarray*} |
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| 129 | \end{frame} |
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| 130 | |
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| 131 | % Folie 5 |
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| 132 | \begin{frame} |
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| 133 | \frametitle{Boussinesq Approximated Equations} |
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| 134 | \begin{itemize} |
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| 135 | \item \onslide<2->Navier-Stokes equations |
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| 136 | \begin{equation*} |
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| 137 | \frac{\partial u_i}{\partial t} |
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| 138 | + \frac{\partial u_k u_i}{\partial x_k} = |
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| 139 | - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i} |
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| 140 | - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}} |
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| 141 | + g \frac{T - T_0}{T_0} \delta_{i3} + \nu |
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| 142 | \frac{\partial^2 u_i}{\partial x_k^2} |
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| 143 | \end{equation*} |
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| 144 | \item \onslide<3->First principle |
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| 145 | \begin{equation*} |
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| 146 | \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} = |
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| 147 | \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q |
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| 148 | \end{equation*} |
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| 149 | \item \onslide<4->Equation for passive scalar |
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| 150 | \begin{equation*} |
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| 151 | \frac{\partial \psi}{\partial t} + u_k |
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| 152 | \frac{\partial \psi}{\partial x_k} = \nu_{\psi} |
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| 153 | \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi} |
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| 154 | \end{equation*} |
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| 155 | \item \onslide<5->Continuity equation |
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| 156 | \begin{equation*} |
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| 157 | \frac{\partial u_k}{\partial x_k} = 0 |
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| 158 | \end{equation*} |
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| 159 | \end{itemize} |
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| 160 | \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] |
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| 161 | |
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| 162 | \begin{tikzpicture}[remember picture, overlay] |
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| 163 | \node at (current page.north west){% |
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| 164 | \begin{tikzpicture}[overlay] |
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| 165 | \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!}; |
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| 166 | \end{tikzpicture} |
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| 167 | }; |
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| 168 | \end{tikzpicture} |
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| 169 | \end{frame} |
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| 170 | |
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| 171 | |
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| 172 | \section{Scale Separation} |
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| 173 | \subsection{Scale Separation by Spatial Filtering} |
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| 174 | |
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| 175 | % Folie 6 |
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| 176 | \begin{frame} |
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| 177 | \frametitle{LES - Scale Separation by Spatial Filtering (I)} |
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| 178 | \footnotesize |
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| 179 | \begin{itemize} |
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| 180 | \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}} |
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| 181 | \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).} |
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| 182 | \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).} |
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| 183 | \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.} |
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| 184 | \end{itemize} |
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| 185 | \normalsize |
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| 186 | \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png} |
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| 187 | \end{frame} |
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| 188 | |
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| 189 | |
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| 190 | |
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| 191 | % Folie 7 |
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| 192 | \begin{frame} |
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| 193 | \frametitle{LES - Scale Separation by Spatial Filtering (II)} |
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| 194 | \begin{columns}[T] |
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| 195 | \begin{column}{0.8\textwidth} |
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| 196 | \footnotesize |
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| 197 | \begin{itemize} |
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| 198 | \item<1->The Filter applied is a spatial filter: |
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| 199 | \begin{equation*} |
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| 200 | \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i' |
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| 201 | \end{equation*} |
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| 202 | \begin{equation*} |
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| 203 | \overline{\Psi}'(x_i) = 0 \qquad but \qquad \overline{\overline{\Psi}} \neq \overline{\Psi}(x_i) |
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| 204 | \end{equation*} |
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| 205 | \item<2->Filter applied to the nonlinear advection term: |
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| 206 | \begin{equation*} |
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| 207 | \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}} |
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| 208 | \end{equation*} |
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| 209 | \item<5->Leonard proposes a further decomposition: |
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| 210 | \begin{equation*} |
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| 211 | \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}} |
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| 212 | \end{equation*} |
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| 213 | \begin{equation*} |
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| 214 | \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} |
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| 215 | \end{equation*} |
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| 216 | \end{itemize} |
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| 217 | \end{column} |
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| 218 | \begin{column}{0.32\textwidth} |
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| 219 | \vspace{45mm} |
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| 220 | \begin{footnotesize} |
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| 221 | \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\ |
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| 222 | $C_{ki}$: \textbf{cross-stress} \\ |
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| 223 | $L_{ki}$: \textbf{Leonard-stress} \\ |
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| 224 | $\tau_{ki}$: \textbf{total stress-tensor} |
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| 225 | \end{footnotesize} |
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| 226 | \end{column} |
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| 227 | \end{columns} |
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| 228 | \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] |
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| 229 | \begin{tikzpicture}[remember picture, overlay] |
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| 230 | \node at (current page.north west){ |
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| 231 | \begin{tikzpicture}[overlay] |
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| 232 | \node[plain, draw,anchor=west] at (94mm,-30mm) { |
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| 233 | \begin{footnotesize} |
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| 234 | \noindent \textbf{Ensemble average:} \\ |
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| 235 | \end{footnotesize} |
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| 236 | $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\ |
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| 237 | \vspace{5mm} |
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| 238 | $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$ |
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| 239 | }; |
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| 240 | \end{tikzpicture} |
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| 241 | }; |
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| 242 | \end{tikzpicture} |
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| 243 | \end{frame} |
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| 244 | |
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| 245 | % Folie 8 |
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| 246 | \begin{frame} |
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| 247 | \frametitle{LES - Scale Separation by Spatial Filtering (III)} |
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| 248 | \small |
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| 249 | \begin{itemize} |
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| 250 | \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator |
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| 251 | \begin{flalign*} |
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| 252 | &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\ |
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| 253 | &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\ |
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| 254 | &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] |
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| 255 | \end{flalign*} |
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| 256 | \item<3-> Filter applied to the nonlinear advection term: |
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| 257 | \begin{equation*} |
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| 258 | \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}= |
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| 259 | \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i} |
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| 260 | \end{equation*} |
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| 261 | \end{itemize} |
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| 262 | \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small] |
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| 263 | \begin{tikzpicture}[remember picture, overlay] |
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| 264 | \node at (current page.north west){ |
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| 265 | \begin{tikzpicture}[overlay] |
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| 266 | \node[plain, draw,anchor=west] at (94mm,-30mm) { |
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| 267 | \begin{footnotesize} |
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| 268 | \noindent \textbf{Ensemble average:} \\ |
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| 269 | \end{footnotesize} |
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| 270 | $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\ |
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| 271 | \vspace{5mm} |
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| 272 | $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$ |
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| 273 | }; |
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| 274 | \end{tikzpicture} |
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| 275 | }; |
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| 276 | \end{tikzpicture} |
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| 277 | \end{frame} |
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| 278 | |
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| 279 | |
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| 280 | \section{Filtered equations} |
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| 281 | \subsection{The Filtered Equations} |
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| 282 | |
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| 283 | % Folie 9 |
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| 284 | \begin{frame} |
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| 285 | \frametitle{The Filtered Equations} |
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| 286 | \onslide<2-> |
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| 287 | \begin{equation*} |
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| 288 | \frac{\partial \overline{u_i}}{\partial t} |
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| 289 | + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} = |
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| 290 | - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i} |
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| 291 | - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} |
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| 292 | + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} |
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| 293 | + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} |
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| 294 | - \frac{\partial \tau_{ki}}{\partial x_k} |
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| 295 | \end{equation*} |
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| 296 | |
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| 297 | \begin{footnotesize} |
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| 298 | \begin{itemize} |
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| 299 | \item<3->The previous derivation completely ignores the existance of the computational grid. |
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| 300 | \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$. |
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| 301 | \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$ |
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| 302 | \item<6-> Two possibilities:\\ |
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| 303 | 1. Pre-filtering technique\\ |
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| 304 | ($\Delta x < \Delta$, explicit filtering)\\ |
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| 305 | 2. Linking the analytical filter\\ |
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| 306 | to the computational grid\\ |
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| 307 | ($\Delta x = \Delta$, implicit filtering) |
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| 308 | \end{itemize} |
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| 309 | \end{footnotesize} |
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| 310 | |
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| 311 | \begin{picture}(0.0,0.0) |
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| 312 | \put(140,13){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}} |
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| 313 | \end{picture} |
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| 314 | \end{frame} |
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| 315 | |
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| 316 | %% Folie 10 |
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| 317 | \begin{frame} |
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| 318 | \frametitle{Explicit Versus Implicit Filtering} |
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| 319 | \begin{itemize} |
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| 320 | \item<2-> Explicit filtering: |
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| 321 | \begin{itemize} |
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| 322 | \small |
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| 323 | \item<2-> Requires that the analytical filter is applied explicitly. |
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| 324 | \item<3-> Rarely used in practice, due to additional computational costs. |
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| 325 | \end{itemize} |
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| 326 | \item<4-> Implicit filtering: |
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| 327 | \begin{itemize} |
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| 328 | \small |
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| 329 | \item<4-> The analytical cutoff length is associated with the grid spacing. |
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| 330 | \item<5-> This method does not require the use of an analytical filter. |
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| 331 | \item<6-> The filter characteristic cannot really be controlled. |
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| 332 | \item<7-> Because of its simplicity, this method is used by nearly all LES models. |
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| 333 | \end{itemize} |
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| 334 | \end{itemize} |
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| 335 | |
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| 336 | \onslide<8-> |
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| 337 | \begin{scriptsize} |
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| 338 | \textbf{Literature:}\\ |
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| 339 | \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\ |
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[945] | 340 | \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.\\ |
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[915] | 341 | \end{scriptsize} |
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| 342 | \end{frame} |
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| 343 | |
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| 344 | % Folie 11 |
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| 345 | \begin{frame} |
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| 346 | \frametitle{The Final Set of Equations (PALM)} |
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| 347 | \footnotesize |
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| 348 | \begin{itemize} |
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| 349 | \item<2-> Navier-Stokes equations: |
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| 350 | \onslide<2-> |
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| 351 | \begin{flalign*} |
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| 352 | &\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}& |
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| 353 | \end{flalign*} |
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| 354 | \item<4-> First principle (using potential\\ temperature): |
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| 355 | \onslide<4-> |
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| 356 | \begin{flalign*} |
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| 357 | &\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}& |
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| 358 | \end{flalign*} |
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| 359 | \item<5-> Equation for specific humidity\\ (passive scalar) |
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| 360 | \onslide<5-> |
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| 361 | \begin{flalign*} |
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[945] | 362 | &\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}& |
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[915] | 363 | \end{flalign*} |
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| 364 | \item<6-> |
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| 365 | Continuity equation |
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| 366 | \onslide<6-> |
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| 367 | \begin{flalign*} |
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| 368 | &\frac{\partial \overline{u_k}}{\partial x_k} = 0& |
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| 369 | \end{flalign*} |
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| 370 | \end{itemize} |
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| 371 | |
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| 372 | \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small] |
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| 373 | \begin{tikzpicture}[remember picture, overlay] |
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| 374 | \node at (current page.north west){ |
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| 375 | \begin{tikzpicture}[overlay] |
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| 376 | \node[plain, draw,anchor=west] at (75mm,-45mm) { |
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| 377 | \begin{tiny} |
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| 378 | \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\ |
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| 379 | \end{tiny} |
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| 380 | $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\ |
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| 381 | \vspace{1mm} |
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| 382 | $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$ |
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| 383 | }; |
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| 384 | \end{tikzpicture} |
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| 385 | }; |
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| 386 | \end{tikzpicture} |
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| 387 | |
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| 388 | \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small] |
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| 389 | \begin{tikzpicture}[remember picture, overlay] |
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| 390 | \node at (current page.north west){ |
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| 391 | \begin{tikzpicture}[overlay] |
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| 392 | \node[plain, draw,anchor=west] at (75mm,-70mm) { |
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| 393 | \begin{tiny} |
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| 394 | \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\ |
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| 395 | \end{tiny} |
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| 396 | $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\ |
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[945] | 397 | $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\ |
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| 398 | $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$ |
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[915] | 399 | }; |
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| 400 | \end{tikzpicture} |
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| 401 | }; |
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| 402 | \end{tikzpicture} |
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| 403 | \end{frame} |
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| 404 | |
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| 405 | |
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[945] | 406 | \end{document} |
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