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1% $Id: fundamentals_of_les.tex 945 2012-07-17 15:43:01Z maronga$
4
5\usepackage[utf8]{inputenc}
6\usepackage[T1]{fontenc}
7\usepackage{pgf}
8\usetheme{Dresden}
9\usepackage{subfigure}
10\usepackage{units}
11\usepackage{multimedia}
12\usepackage{hyperref}
13\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
14\usepackage{xmpmulti}
15\usepackage{tikz}
16\usepackage{pdfcomment}
17\usetikzlibrary{shapes,arrows,positioning}
18\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
19
20%---------- neue Pakete
21\usepackage{amsmath}
22\usepackage{amssymb}
23\usepackage{multicol}
24
25\institute{Institut fÃŒr Meteorologie und Klimatologie, Leibniz UniversitÃ€t Hannover}
26\date{last update: \today}
27\event{PALM Seminar}
29
30\setbeamertemplate{footline}
31  {%
32    \begin{beamercolorbox}[rightskip=-0.1cm]&
33     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
34    \end{beamercolorbox}
35    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,%
38    \end{beamercolorbox}%
39%    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}%
40%    \end{beamercolorbox}
41  }%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.eps}}
42
43\title[Fundamentals of Large-Eddy Simulation]{Fundamentals of Large-Eddy Simulation}
44\author{Siegfried Raasch}
45
46% Notes:
47% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
48% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
49\begin{document}
50%Folie 1
51\begin{frame}
52\titlepage
53\pdfnote{maronga}{Welcome to the PALM Tutorial!}
54\end{frame}
55
56\section{The Role of Turbulence}
57\subsection{The Role of Turbulence}
58
59% Folie 2
60\begin{frame}
61   \frametitle{The Role of Turbulence (I)}
62   \begin{itemize}
63      \item<1->{\textbf{Most flows in nature \& technical applications are turbulent}}
64      \item<2->{\textbf{Significance of Turbulence}}
65      \begin{itemize}
66         \item<2->{\underline{Meteorology / Oceanography:} Transport processes of momentum, heat, water vapor as well as other scalars}
67         \item<2->{\underline{Health care:} Air pollution}
68         \item<2->{\underline{Aviation, Engineering:} Wind impact on buildings, power output of windfarms}
69      \end{itemize}
70      \item<3->{\textbf{Characteristics of turbulence}}
71      \begin{itemize}
72         \item<3->{non-periodical, 3D stochastic movements}
73         \item<3->{mixes air and its properties on scales between large-scale advection and molecular diffusion}
74         \item<3->{non-linear $\rightarrow$ energy is distributed smoothly with wavelength}
75         \item<3->{wide range of spatial and temporal scales}
76      \end{itemize}
77   \end{itemize}
78\end{frame}
79
80% Folie 3
81\begin{frame}
82   \frametitle{The Role of Turbulence (II)}
83   \begin{columns}[c]
84   \column{0.5\textwidth}
85      \scriptsize
86      \begin{itemize}
87         \item<2->{\textbf{Large eddies:} $\unit[10^3]{m}$ ($L$), $\unit[1]{h}$ \\
88                   \textbf{Small eddies:} $\unit[10^{-3}]{m}$ ($\eta$), \unit[0.1]{s}}
89         \item<3->{\textbf{Energy production and dissipation on different scales}}
90         \begin{itemize}
91            \item<3->{\begin{scriptsize} Large scales: shear and buoyant production \end{scriptsize}}
92            \item<3->{\begin{scriptsize} Small scales: viscous dissipation \end{scriptsize}}
93         \end{itemize}
94         \item<4->{\textbf{Large eddies contain most energy}}
96                   Large eddies are broken up by instabilities and their energy is handled down to smaller scales.}
97      \end{itemize}
98      \normalsize
99   \column{0.5\textwidth}
100      \onslide<3->{
101         \includegraphics[width=\textwidth, height=0.9\textheight]{fundamentals_of_les_figures/Role_of_Turbulence_2.png}}
102   \end{columns}
103\end{frame}
104
105\section{The Reynolds Number}
106\subsection{The Reynolds Number}
107
108% Folie 4
109\begin{frame}
110   \frametitle{The Reynolds Number (Re)}
111   \begin{columns}[c]
112   \column{0.6\textwidth}
113      \onslide<1->{
114         $\frac{L}{\eta} \approx Re^{3/4} \approx 10^6$ \quad \begin{small} (in the atmosphere) \end{small}}
115      \par\bigskip
116      \onslide<2->{
117         $Re = \frac{\left| \textbf{u} \cdot \nabla \textbf{u} \right|}{\left| \nu \nabla^2 \textbf{u} \right|} \hat{=} \frac{LU}{\nu} \qquad \frac{\textnormal{inertia forces}}{\textnormal{viscous forces}}$}
118   \column{0.4\textwidth}
119      \footnotesize
120      \onslide<1->{
121         \textbf{u} 3D wind vector
122
123         $\nu$ kinematic molecular viscosity
124
125         $L$ outer scale of turbulence
126
127         $U$ characteristic velocity scale
128
129         $\eta$ inner scale of turbulence
130            \begin{scriptsize}(Kolmogorov dissipation length) \end{scriptsize} }
131   \end{columns}
132   \normalsize
133   \par\bigskip
134   \par\bigskip
135   \onslide<3->{
136      $\Rightarrow$ \underline{Number of gridpoints for a 3D simulation:}
137   \par\bigskip
138   $\left( \frac{L}{\eta} \right)^3 \approx Re^{9/4} \approx 10^{18}$ (in the atmosphere)}
139\end{frame}
140
141\section{Classes of Turbulence Models}
142\subsection{Classes of Turbulence Models}
143
144% Folie 5
145\begin{frame}
146   \frametitle{Classes of Turbulence Models (I)}
147   \begin{itemize}
148      \item{\textbf{Direct numerical Simulation (DNS)}}
149      \begin{itemize}
150         \item<2->{\textbf{Most straight-forward approach:}}
151         \begin{itemize}
152            \item<2->{Resolve all scales of turbulent flow explicitly.}
153         \end{itemize}
155         \begin{itemize}
156            \item<3->{(In principle) a very accurate turbulence representation.}
157         \end{itemize}
158         \item<4->{\textbf{Problem:}}
159         \begin{itemize}
160            \item<4->{Limited computer resources  (1996: $\sim$ $10^8$, today: $\sim$ $10^{11}$ gridpoints,
161                      but $\sim$ $10^{18}$ gridpoints needed, see prior slide).}
162            \item<4->{$\unit[1]{h}$ simulation of $10^9$ ($2048^3$) gridpoints on $512$ processors of the HLRN supercomputer needs $\unit[10]{h}$ CPU time.}
163         \end{itemize}
164         \item<5->{\textbf{Consequences:}}
165         \begin{itemize}
166            \item<5->{DNS is restricted to moderately turbulent flows (low Reynolds-number flows).}
167            \item<5->{Highly turbulent atmospheric turbulent flows cannot be simulated.}
168         \end{itemize}
169      \end{itemize}
170   \end{itemize}
171\end{frame}
172
173% Folie 6
174\begin{frame}
175   \frametitle{Classes of Turbulence Models (II)}
176   \begin{itemize}
177      \item{\textbf{Reynolds averaged (Navier-Stokes) simulation (RANS)}}
178      \begin{itemize}
179         \item<2->{\textbf{Opposite strategy:}}
180         \begin{itemize}
181            \item<2->{Applications that only require average statistics of the flow (i.e. the mean flow).}
182            \item<2->{Integrate merely the ensemble-averaged equations.}
183            \item<2->{Parameterize turbulence over the whole eddy spectrum.}
184         \end{itemize}
186         \begin{itemize}
187            \item<3->{Computationally inexpensive, fast.}
188         \end{itemize}
189         \item<4->{\textbf{Problem:}}
190         \begin{itemize}
191            \item<4->{Turbulent fluctuations not explicitly captured.}
192            \item<4->{Parameterizations are very sensitive to large-eddy structure that depends on
193                      environmental conditions such as geometry and stratification $\rightarrow$
194                      Parameterizations are not valid for a wide range of different flows.}
195         \end{itemize}
196         \item<5->{\textbf{Consequence:}}
197         \begin{itemize}
198            \item<5->{Not suitable for detailed turbulence studies.}
199         \end{itemize}
200      \end{itemize}
201   \end{itemize}
202\end{frame}
203
204% Folie 7
205\begin{frame}
206   \frametitle{Classes of Turbulence Models (III)}
207   \begin{itemize}
208      \item{\textbf{Large eddy simulation (LES)}}
209      \begin{itemize}
210         \item<2->{Seeks to combine advantages and avoid disadvantages of DNS and RANS by \underline{treating
211                   large scales and small scales separately}, based on Kolmogorov's (1941) similarity theory of turbulence.}
212         \item<3->{Large eddies are explicitly resolved.}
213         \item<4->{The impact of small eddies on the large-scale flow is parameterized.}
215         \begin{itemize}
216            \item<5->{Highly turbulent flows can be simulated.}
217            \item<5->{Local homogeneity and isotropy at large \textit{Re} (Kolmogorov's $1^\mathrm{st}$ hypothesis) leaves
218                      parameterizations uniformly valid for a wide range of different flows.}
219         \end{itemize}
220      \end{itemize}
221   \end{itemize}
222\end{frame}
223
224\section{Concept of LES}
225\subsection{Concept of LES}
226
227 % Folie 8
228 \begin{frame}
229    \frametitle{Concept of Large Eddy Simulation (I)}
230    \begin{columns}
231       \column{0.55\textwidth}
232          \begin{itemize}
233             \item<1->{\textbf{Filtering}}
234             \begin{footnotesize}
235                \begin{itemize}
236                   \item<2->{Spectral cut at wavelength $\Delta x$.}
237                   \item<3->{Structures larger than $\Delta x$ are explicitly calculated (resolved scales).}
238                   \item<4->{Structures smaller than $\Delta x$ must be filtered out (subgrid scales), formally known as low-pass filtering.}
239                   \item<5->{Like for Reynolds averaging: split variables in mean part and fluctuation, spatially average the model equations, e.g.:}
240                \end{itemize}
241             \end{footnotesize}
242             \onslide<6->{\begin{center} $w = \overline{w} + w', \theta = \overline{\theta} + \theta'$ \end{center}}
243          \end{itemize}
244       \column{0.45\textwidth}
245          \includegraphics[width=\textwidth]{fundamentals_of_les_figures/Concept_of_LES.png}
246    \end{columns}
247 \end{frame}
248
249% Folie 9
250\begin{frame}
251   \frametitle{Concept of Large Eddy Simulation (II)}
252   \begin{itemize}
253      \item<1->{\textbf{Parameterization}}
254      \begin{footnotesize}
255      \begin{itemize}
256         \item<2->{The filter procedure removes the small scales from the model equations, but it produces new unknowns, mainly averages of fluctuation products.}
257         \begin{itemize}
258            \item<2->{eg. $\overline{w'\theta'}$}
259         \end{itemize}
260         \item<3->{These unknowns describe the effect of the unresolved, small scales on the resolved, large scales; therefore it is important to include them in the model.}
261         \item<4->{We do not have information about the variables (e.g., vertical wind component and potential temperature) on these small scales of their fluctuations.}
262         \item<5->{Therefore, these unknowns have to be parameterized using information from the resolved scales.}
263         \begin{itemize}
264            \item<5->{A typical example is the flux-gradient relationship, e.g.,}
265         \end{itemize}
266      \end{itemize}
267      \end{footnotesize}
268   \end{itemize}
269   \onslide<5->{
270      \begin{center}
271      $\overline{w'\theta'} = - \nu_\mathrm{h} \cdot \frac{\partial \overline{\theta}}{\partial z}$
272      \end{center}}
273\end{frame}
274
275\end{document}
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