1 | % $Id: sgs_models.tex 1526 2015-01-22 12:47:21Z raasch $ |
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2 | \input{header_tmp.tex} |
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3 | %\input{header_lectures.tex} |
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5 | \usepackage[utf8]{inputenc} |
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6 | \usepackage{ngerman} |
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7 | \usepackage{pgf} |
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8 | \usepackage{subfigure} |
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9 | \usepackage{units} |
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11 | \usepackage{hyperref} |
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12 | \newcommand{\event}[1]{\newcommand{\eventname}{#1}} |
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13 | \usepackage{xmpmulti} |
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14 | \usepackage{tikz} |
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15 | \usetikzlibrary{shapes,arrows,positioning} |
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16 | \def\Tiny{\fontsize{4pt}{4pt}\selectfont} |
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17 | |
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18 | %---------- neue Pakete |
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19 | \usepackage{amsmath} |
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20 | \usepackage{amssymb} |
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21 | \usepackage{multicol} |
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22 | \usepackage{pdfcomment} |
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23 | |
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24 | \institute{Institute of Meteorology and Climatology, Leibniz UniversitÀt Hannover} |
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25 | \selectlanguage{english} |
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26 | \date{last update: \today} |
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27 | \event{PALM Seminar} |
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28 | \setbeamertemplate{navigation symbols}{} |
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29 | |
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30 | \setbeamertemplate{footline} |
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31 | {% |
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32 | \begin{beamercolorbox}[rightskip=-0.1cm]& |
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33 | {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}} |
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34 | \end{beamercolorbox} |
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35 | \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,% |
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36 | leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}% |
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37 | {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}% |
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38 | \end{beamercolorbox}% |
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39 | % \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}% |
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40 | % \end{beamercolorbox} |
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41 | }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}} |
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42 | |
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43 | \title[SGS Models]{SGS Models} |
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44 | \author{PALM group} |
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45 | |
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46 | \begin{document} |
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47 | |
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48 | % Folie 1 |
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49 | \begin{frame} |
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50 | \titlepage |
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51 | \end{frame} |
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52 | |
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53 | |
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54 | \section{SGS Models} |
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55 | \subsection{SGS Models} |
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56 | |
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57 | % Folie 2 |
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58 | \begin{frame} |
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59 | \frametitle{SGS Models (I)} |
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60 | \small |
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61 | \begin{itemize} |
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62 | \item<2->The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence). |
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63 | \item<3->Features of small-scale turbulence: local, isotropic, dissipative (inertial subrange) |
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64 | \item<4->SGS stresses should depend on: |
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65 | \begin{itemize} |
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66 | \item local resolved-scale field \hspace{3mm} and / or |
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67 | \item past history of the local fluid (via a PDE) |
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68 | \end{itemize} |
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69 | \item<5->Importance of the model depends on how much energy is contained in the subgrid-scales: |
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70 | \begin{itemize} |
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71 | \item $E_{SGS} / E < 50\%$: results relatively insensitive to the model, (but sensitive to the numerics, e.g. in case of upwind scheme) |
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72 | \item $E_{SGS} / E = 1$: model more important |
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73 | \item<6->\textbf{If the large-scale eddies are not resolved, the SGS model and the LES will fail at all!} |
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74 | \end{itemize} |
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75 | \end{itemize} |
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76 | \end{frame} |
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77 | |
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78 | % Folie 3 |
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79 | \begin{frame} |
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80 | \frametitle{SGS Models (II)} |
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81 | Requirements that a good SGS model must fulfill: |
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82 | \begin{footnotesize} |
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83 | \begin{itemize} |
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84 | \item<2-> Represent interactions with small scales. |
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85 | \item<3-> Provide adequate dissipation\\ (transport of energy from the resolved grid scales to the unresolved grid scales; the rate of dissipation $\varepsilon$ in this context is the flux of energy through the inertial subrange). |
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86 | \item<4-> Dissipation rate must depend on the large scales of the flow rather than being imposed arbitrarily by the model. The SGS model must depend on the large-scale statistics and must be sufficiently flexible to adjust to changes in these statistics. |
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87 | \item<5->In energy conserving codes (ideal for LES) the only way for TKE to leave the resolved modes is by the dissipation provided by the SGS model. |
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88 | \item<6->\underline{The primary goal of an SGS model is to obtain correct statistics of the}\\ |
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89 | \underline{energy containing scales of motion.} |
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90 | \end{itemize} |
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91 | \end{footnotesize} |
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92 | \end{frame} |
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93 | |
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94 | % Folie 4 |
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95 | \begin{frame} |
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96 | \frametitle{SGS Models (III)} |
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97 | \onslide<1-> All the above observations suggest the use of an eddy viscosity type SGS model: |
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98 | \begin{footnotesize} |
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99 | \begin{itemize} |
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100 | \item<2-> Take idea from RANS modeling, introduce eddy viscosity $\nu_T$: |
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101 | \begin{flalign*} |
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102 | &\tau_{ki} = - \nu_T \left( \frac{\partial \overline{u_k}}{\partial x_i}+ \frac{\partial \overline{u_i}}{\partial x_k}\right) = -2 \nu_T \overline{S}_{ki}& \text{with} \hspace{3mm} \overline{S}_{ki} = \frac{1}{2} \left( \frac{\partial \overline{u_k}}{\partial x_i}+ \frac{\partial \overline{u_i}}{\partial x_k}\right)\\ |
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103 | & & \text{filtered strain rate tensor} |
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104 | \end{flalign*} |
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105 | \end{itemize} |
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106 | \end{footnotesize} |
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107 | \vspace{-0.3cm} |
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108 | \onslide<3->Now we need a model for the eddy viscosity: |
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109 | \begin{footnotesize} |
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110 | \begin{itemize} |
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111 | \item<4-> Dimensionality of $\nu_T$ is $l^2/t$ |
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112 | \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characteristic velocity / length scale) |
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113 | \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$ |
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114 | \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element) |
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115 | \begin{flalign*} |
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116 | &q = l \frac{\partial \overline{u}}{\partial x} = l \overline{S}& \text{for 3D: } \overline{S} = \sqrt{2 \overline{S}_{ki}\,\overline{S}_{ki}} \hspace{15mm} \\ |
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117 | & & \text{characteristic filtered rate of strain}\hspace{15mm} |
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118 | \end{flalign*} |
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119 | \end{itemize} |
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120 | \end{footnotesize} |
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121 | \end{frame} |
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122 | |
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123 | |
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124 | \section{Smagorinsky Model} |
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125 | \subsection{The Smagorinsky Model} |
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126 | |
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127 | % Folie 5 |
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128 | \begin{frame} |
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129 | \frametitle{The Smagorinsky Model} |
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130 | \onslide<2->Combine previous expressions to obtain: |
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131 | \begin{equation*} |
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132 | \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S} |
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133 | \end{equation*} |
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134 | \onslide<3-> Model due to Smagorinsky (1963): |
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135 | \begin{itemize} |
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136 | \item<3-> Originally designed at NCAR for global weather modeling. |
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137 | \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theory. |
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138 | \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$ |
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139 | \end{itemize} |
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140 | \end{frame} |
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141 | |
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142 | % Folie 6 |
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143 | \begin{frame} |
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144 | \frametitle{The Smagorinsky Model: Performance} |
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145 | \begin{itemize} |
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146 | \item<2-> Predicts many flows reasonably well |
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147 | \item<3-> Problems: |
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148 | \begin{itemize} |
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149 | \item<3-> Optimum parameter value varies with flow type: |
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150 | \begin{itemize} |
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151 | \item Isotropic turbulence: $C_S \approx 0.2$\\ |
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152 | \item Shear (channel) flows: $C_S \approx 0.065$ |
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153 | \end{itemize} |
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154 | \item<4-> Length scale uncertain with anisotropic filter: |
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155 | \begin{equation*} |
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156 | (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3 |
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157 | \end{equation*} |
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158 | \item<5-> Needs modification to account for: |
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159 | \begin{itemize} |
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160 | \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\ |
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161 | \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall |
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162 | \end{itemize} |
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163 | \end{itemize} |
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164 | \end{itemize} |
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165 | \end{frame} |
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166 | |
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167 | \section{Deardoff Modification} |
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168 | \subsection{Deardoff Modification} |
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169 | |
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170 | % Folie 7 |
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171 | \begin{frame} |
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172 | \frametitle{Deardorff (1980) Modification (Used in PALM) (I)} |
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173 | \footnotesize |
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174 | \onslide<1->{ |
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175 | $ \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}} |
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176 | \small |
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177 | \begin{itemize} |
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178 | \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 |
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179 | local fluid.} |
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180 | \end{itemize} |
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181 | \small |
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182 | \onslide<3->{ |
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183 | $ C_M = const. = 0.1 $ |
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184 | \par\bigskip |
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185 | \scriptsize |
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186 | $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\ |
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187 | 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} |
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188 | \end{cases} $ |
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189 | \small |
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190 | \par\bigskip |
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191 | $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ } |
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192 | \end{frame} |
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193 | |
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194 | % Folie 8 |
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195 | \begin{frame} |
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196 | \frametitle{Deardorff (1980) Modification (Used in PALM) (II)} |
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197 | \begin{itemize} |
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198 | \item{SGS-TKE from prognostic equation} |
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199 | \end{itemize} |
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200 | $ \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 $ |
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201 | \par\bigskip |
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202 | $ \tau_{ki} = -K_{m} \left(\frac{\partial \bar{u_{i}}}{\partial x_{k}} + \frac{\partial \bar{u_{k}}}{\partial x_{i}}\right) + \frac{2}{3}\delta_{ik}\bar{e} \qquad \textnormal{with} \qquad K_{m}=0.1\cdot \Lambda \sqrt{\bar{e}}$ |
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203 | \par\bigskip |
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204 | $ H_{k}=\overline{u_k'\Theta'} = -K_{h}\frac{\partial\bar{\Theta}}{\partial x_{k}} \qquad \textnormal{with} \qquad K_{h}= \left(1+2\frac{\Lambda}{\Delta}\right)$ |
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205 | \par\bigskip |
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206 | $W_{k}=\overline{u_k'q'} = -K_{h}\frac{\partial\bar{q}}{\partial x_{k}}$ |
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207 | \par\bigskip |
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208 | $ \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} $ |
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209 | \par\bigskip |
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210 | $ \nu_e = 2 \nu_T $ |
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211 | \par\bigskip |
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212 | $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $ |
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213 | \end{frame} |
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214 | |
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215 | % Folie 9 |
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216 | \begin{frame} |
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217 | \frametitle{Deardorff (1980) Modification (Used in PALM) (III)} |
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218 | \begin{itemize} |
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219 | \item{There are still problems with this parameterization:} |
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220 | \begin{itemize} |
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221 | \item[-]<2->{The model overestimates the velocity shear near the wall.} |
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222 | \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.} |
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223 | \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.} |
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224 | \end{itemize} |
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225 | \item<5->{Several other SGS models have been developed:} |
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226 | \begin{itemize} |
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227 | \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)} |
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228 | \item[-]<6->{Scale similarity model (Bardina et al.)} |
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229 | \item[-]<7->{Backscatter model (Mason)} |
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230 | \end{itemize} |
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231 | \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent |
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232 | from the different models (Beare et al., 2006, BLM).} |
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233 | \end{itemize} |
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234 | \end{frame} |
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235 | |
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236 | |
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237 | \section{Summary / Important Points for Beginners} |
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238 | \subsection{Summary / Important Points for Beginners} |
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239 | |
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240 | % Folie 10 |
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241 | \begin{frame} |
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242 | \frametitle{Summary / Important Points for Beginners (I)} |
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243 | \begin{columns}[c] |
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244 | \column[T]{0.4\textwidth} |
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245 | \includegraphics<2-7|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png} |
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246 | \includegraphics<8|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png} |
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247 | \includegraphics<9|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png} |
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248 | \includegraphics<10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png} |
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249 | \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}} |
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250 | \column[T]{0.2\textwidth} |
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251 | \vspace{0.9cm} |
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252 | \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png} |
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253 | \par |
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254 | \onslide<8-|handout:1>{\begin{small} fluctuations (\textbf{u},c) \end{small}} |
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255 | \par\bigskip |
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256 | \thicklines |
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257 | \onslide<9-|handout:1>{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}} |
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258 | \vspace{1cm} |
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259 | |
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260 | \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png} |
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261 | \par |
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262 | \onslide<8->{\begin{small} smooth result \end{small}} |
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263 | \column[T]{0.4\textwidth} |
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264 | \includegraphics<1-2|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png} |
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265 | \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png} |
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266 | \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png} |
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267 | \includegraphics<5-10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png} |
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268 | \vspace{1.3cm} |
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269 | \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png} |
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270 | \uncover<7-|handout:1>{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}} |
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271 | \end{columns} |
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272 | \end{frame} |
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273 | |
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274 | % Folie 11 |
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275 | \begin{frame} |
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276 | \frametitle{Summary / Important Points for Beginners (II)} |
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277 | For an LES it always has to be guaranteed that the main energy containing eddies of the respective |
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278 | turbulent flow can really be simulated by the numerical model: |
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279 | \begin{itemize} |
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280 | \item<2->{The grid spacing has to be fine enough.} |
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281 | \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $} |
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282 | \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\ |
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283 | (therefore cyclic boundary conditions are used in most cases).} |
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284 | \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence |
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285 | has to be triggered by imposing random fluctuations.} |
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286 | \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.} |
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287 | \end{itemize} |
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288 | \end{frame} |
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289 | |
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290 | |
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291 | \section{Example Output} |
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292 | \subsection{Example Output} |
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293 | |
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294 | % Folie 12 |
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295 | \begin{frame} |
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296 | \frametitle{Example Output (I)} |
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297 | \begin{itemize} |
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298 | \item{LES of a convective boundary layer} |
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299 | \end{itemize} |
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300 | \includegraphics<1|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png} |
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301 | \includegraphics<2|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png} |
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302 | \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png} |
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303 | \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png} |
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304 | \includegraphics<5|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png} |
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305 | \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png} |
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306 | \includegraphics<7|handout:1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png} |
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307 | \end{frame} |
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308 | |
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309 | % Folie 13 |
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310 | \begin{frame} |
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311 | \frametitle{Example Output (II)} |
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312 | \begin{itemize} |
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313 | \item{LES of a convective boundary layer} |
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314 | \end{itemize} |
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315 | \begin{center} |
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316 | \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png} |
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317 | power spectrum of vertical velocity |
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318 | \end{center} |
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319 | \end{frame} |
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320 | |
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321 | % Folie 14 |
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322 | \begin{frame} |
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323 | \frametitle{Some Symbols} |
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324 | \begin{columns}[c] |
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325 | \column{0.6\textwidth} |
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326 | \begin{tabbing} |
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327 | $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\ |
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328 | $u,v,w$ \\ |
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329 | |
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330 | \\ |
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331 | |
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332 | $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\ |
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333 | $x,y,z$ \\ |
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334 | |
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335 | \\ |
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336 | |
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337 | $\Theta$ \> potential temperature \\ \\ |
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338 | |
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339 | $\Psi$ \> passive scalar \\ \\ |
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340 | |
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341 | $T$ \> actual Temperatur \\ \\ |
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342 | \end{tabbing} |
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343 | \column{0.4\textwidth} |
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344 | \begin{tabbing} |
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345 | $\Phi = gz$ \quad \= geopotential \\ \\ |
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346 | |
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347 | $p$ \> pressure \\ \\ |
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348 | |
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349 | $\rho$ \> density \\ \\ |
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350 | |
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351 | $f_i$ \> Coriolis Parameter \\ \\ |
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352 | |
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353 | $\epsilon_{ijk}$ \> alternating symbol \\ \\ |
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354 | |
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355 | $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\ |
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356 | |
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357 | $Q, Q_\Psi$ \> sources or sinks \\ \\ |
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358 | \end{tabbing} |
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359 | \end{columns} |
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360 | \end{frame} |
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361 | \end{document} |
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