# source:palm/trunk/TUTORIAL/SOURCE/fundamentals_of_les.tex@1649

Last change on this file since 1649 was 1649, checked in by knoop, 8 years ago

Latex lecture update for PALM seminar in Hong Kong

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1% $Id: fundamentals_of_les.tex 1649 2015-09-15 16:34:42Z knoop$
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
7\usepackage{pgf}
8\usepackage{subfigure}
9\usepackage{units}
10\usepackage{multimedia}
11\usepackage{hyperref}
12\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
13\usepackage{xmpmulti}
14\usepackage{tikz}
15\usetikzlibrary{shapes,arrows,positioning}
16\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
17
18%---------- neue Pakete
19\usepackage{amsmath}
20\usepackage{amssymb}
21\usepackage{multicol}
22\usepackage{pdfcomment}
23
24\institute{Institute of Meteorology and Climatology, Leibniz UniversitÃ€t Hannover}
25\selectlanguage{english}
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{PALM group}
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}{
54   Welcome to the PALM Tutorial!\textCR\textCR
55   We have placed many helpful comments throughout the presentations that will hopefully ease your first steps with PALM.\textCR\textCR
56   In case you find it hard to follow at specific points that have not been (or insufficiently) commented, please let us know! We appreciate feedback that helps improving the tutorial.\textCR\textCR
57   Good luck! - The PALM Group at IMUK
58}
59\end{frame}
60
61\section{The Role of Turbulence}
62\subsection{The Role of Turbulence}
63
64% Folie 2
65\begin{frame}
66   \frametitle{The Role of Turbulence (I)}
67   \begin{itemize}
68      \item<1->{\textbf{Most flows in nature \& technical applications are turbulent}}
69      \item<2->{\textbf{Significance of Turbulence}}
70      \begin{itemize}
71         \item<2->{\underline{Meteorology / Oceanography:} Transport processes of momentum, heat, water vapor as well as other scalars}
72         \item<2->{\underline{Health care:} Air pollution}
73         \item<2->{\underline{Aviation, Engineering:} Wind impact on buildings, power output of windfarms}
74      \end{itemize}
75      \item<3->{\textbf{Characteristics of turbulence}}
76      \begin{itemize}
77         \item<3->{non-periodical, 3D stochastic movements}
78         \item<3->{mixes air and its properties on scales between large-scale advection and molecular diffusion}
79         \item<3->{non-linear $\rightarrow$ energy is distributed smoothly with wavelength}
80         \item<3->{wide range of spatial and temporal scales}
81      \end{itemize}
82   \end{itemize}
83\end{frame}
84
85% Folie 3
86\begin{frame}
87   \frametitle{The Role of Turbulence (II)}
88   \begin{columns}[c]
89   \column{0.5\textwidth}
90      \scriptsize
91      \begin{itemize}
92         \item<2->{\textbf{Large eddies:} $\unit[10^3]{m}$ ($L$), $\unit[1]{h}$ \\
93                   \textbf{Small eddies:} $\unit[10^{-3}]{m}$ ($\eta$), \unit[0.1]{s}}
94         \item<3->{\textbf{Energy production and dissipation on different scales}}
95         \begin{itemize}
96            \item<3->{\begin{scriptsize} Large scales: shear and buoyant production \end{scriptsize}}
97            \item<3->{\begin{scriptsize} Small scales: viscous dissipation \end{scriptsize}}
98         \end{itemize}
99         \item<4->{\textbf{Large eddies contain most energy}}
101                   Large eddies are broken up by instabilities and their energy is handled down to smaller scales.}
102      \end{itemize}
103      \normalsize
104   \column{0.5\textwidth}
105      \onslide<3->{
106         \includegraphics[width=\textwidth, height=0.9\textheight]{fundamentals_of_les_figures/Role_of_Turbulence_2.png}}
107   \end{columns}
108\end{frame}
109
110\section{The Reynolds Number}
111\subsection{The Reynolds Number}
112
113% Folie 4
114\begin{frame}
115   \frametitle{The Reynolds Number (Re)}
116   \begin{columns}[c]
117   \column{0.6\textwidth}
118      \onslide<1->{
119         $\frac{L}{\eta} \approx Re^{3/4} \approx 10^6$ \quad \begin{small} (in the atmosphere) \end{small}}
120      \par\bigskip
121      \onslide<2->{
122         $Re = \frac{\left| \textbf{u} \cdot \nabla \textbf{u} \right|}{\left| \nu \nabla^2 \textbf{u} \right|} \hat{=} \frac{LU}{\nu} \qquad \frac{\textnormal{inertia forces}}{\textnormal{viscous forces}}$}
123   \column{0.4\textwidth}
124      \footnotesize
125      \onslide<1->{
126         \textbf{u} 3D wind vector
127
128         $\nu$ kinematic molecular viscosity
129
130         $L$ outer scale of turbulence
131
132         $U$ characteristic velocity scale
133
134         $\eta$ inner scale of turbulence
135            \begin{scriptsize}(Kolmogorov dissipation length) \end{scriptsize} }
136   \end{columns}
137   \normalsize
138   \par\bigskip
139   \par\bigskip
140   \onslide<3->{
141      $\Rightarrow$ \underline{Number of gridpoints for a 3D simulation:}
142   \par\bigskip
143   $\left( \frac{L}{\eta} \right)^3 \approx Re^{9/4} \approx 10^{18}$ (in the atmosphere)}
144\end{frame}
145
146\section{Classes of Turbulence Models}
147\subsection{Classes of Turbulence Models}
148
149% Folie 5
150\begin{frame}
151   \frametitle{Classes of Turbulence Models (I)}
152   \begin{itemize}
153      \item{\textbf{Direct numerical Simulation (DNS)}}
154      \begin{itemize}
155         \item<2->{\textbf{Most straight-forward approach:}}
156         \begin{itemize}
157            \item<2->{Resolve all scales of turbulent flow explicitly.}
158         \end{itemize}
160         \begin{itemize}
161            \item<3->{(In principle) a very accurate turbulence representation.}
162         \end{itemize}
163         \item<4->{\textbf{Problem:}}
164         \begin{itemize}
165            \item<4->{Limited computer resources  (1996: $\sim$ $10^8$, today: $\sim$ $10^{12}$ gridpoints,
166                      but $\sim$ $10^{18}$ gridpoints needed, see prior slide).}
167            \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.}
168         \end{itemize}
169         \item<5->{\textbf{Consequences:}}
170         \begin{itemize}
171            \item<5->{DNS is restricted to moderately turbulent flows (low Reynolds-number flows).}
172            \item<5->{Highly turbulent atmospheric turbulent flows cannot be simulated.}
173         \end{itemize}
174      \end{itemize}
175   \end{itemize}
176\end{frame}
177
178% Folie 6
179\begin{frame}
180   \frametitle{Classes of Turbulence Models (II)}
181   \begin{itemize}
182      \item{\textbf{Reynolds averaged (Navier-Stokes) simulation (RANS)}}
183      \begin{itemize}
184         \item<2->{\textbf{Opposite strategy:}}
185         \begin{itemize}
186            \item<2->{Applications that only require average statistics of the flow (i.e. the mean flow).}
187            \item<2->{Integrate merely the ensemble-averaged equations.}
188            \item<2->{Parameterize turbulence over the whole eddy spectrum.}
189         \end{itemize}
191         \begin{itemize}
192            \item<3->{Computationally inexpensive, fast.}
193         \end{itemize}
194         \item<4->{\textbf{Problem:}}
195         \begin{itemize}
196            \item<4->{Turbulent fluctuations not explicitly captured.}
197            \item<4->{Parameterizations are very sensitive to large-eddy structure that depends on
198                      environmental conditions such as geometry and stratification $\rightarrow$
199                      Parameterizations are not valid for a wide range of different flows.}
200         \end{itemize}
201         \item<5->{\textbf{Consequence:}}
202         \begin{itemize}
203            \item<5->{Not suitable for detailed turbulence studies.}
204         \end{itemize}
205      \end{itemize}
206   \end{itemize}
207\end{frame}
208
209% Folie 7
210\begin{frame}
211   \frametitle{Classes of Turbulence Models (III)}
212   \begin{itemize}
213      \item{\textbf{Large eddy simulation (LES)}}
214      \begin{itemize}
215         \item<2->{Seeks to combine advantages and avoid disadvantages of DNS and RANS by \underline{treating
216                   large scales and small scales separately}, based on Kolmogorov's (1941) similarity theory of turbulence.}
217         \item<3->{Large eddies are explicitly resolved.}
218         \item<4->{The impact of small eddies on the large-scale flow is parameterized.}
220         \begin{itemize}
221            \item<5->{Highly turbulent flows can be simulated.}
222            \item<5->{Local homogeneity and isotropy at large \textit{Re} (Kolmogorov's $1^\mathrm{st}$ hypothesis) leaves
223                      parameterizations uniformly valid for a wide range of different flows.}
224         \end{itemize}
225      \end{itemize}
226   \end{itemize}
227\end{frame}
228
229\section{Concept of LES}
230\subsection{Concept of LES}
231
232 % Folie 8
233 \begin{frame}
234    \frametitle{Concept of Large Eddy Simulation (I)}
235    \begin{columns}
236       \column{0.55\textwidth}
237          \begin{itemize}
238             \item<1->{\textbf{Filtering}}
239             \begin{footnotesize}
240                \begin{itemize}
241                   \item<2->{Spectral cut at wavelength $\Delta x$.}
242                   \item<3->{Structures larger than $\Delta x$ are explicitly calculated (resolved scales).}
243                   \item<4->{Structures smaller than $\Delta x$ must be filtered out (subgrid scales), formally known as low-pass filtering.}
244                   \item<5->{Like for Reynolds averaging: split variables in mean part and fluctuation, spatially average the model equations, e.g.:}
245                \end{itemize}
246             \end{footnotesize}
247             \onslide<6->{\begin{center} $w = \overline{w} + w', \theta = \overline{\theta} + \theta'$ \end{center}}
248          \end{itemize}
249       \column{0.45\textwidth}
250          \includegraphics[width=\textwidth]{fundamentals_of_les_figures/Concept_of_LES.png}
251    \end{columns}
252 \end{frame}
253
254% Folie 9
255\begin{frame}
256   \frametitle{Concept of Large Eddy Simulation (II)}
257   \begin{itemize}
258      \item<1->{\textbf{Parameterization}}
259      \begin{footnotesize}
260      \begin{itemize}
261         \item<2->{The filter procedure removes the small scales from the model equations, but it produces new unknowns, mainly averages of fluctuation products.}
262         \begin{itemize}
263            \item<2->{eg. $\overline{w'\theta'}$}
264         \end{itemize}
265         \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.}
266         \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.}
267         \item<5->{Therefore, these unknowns have to be parameterized using information from the resolved scales.}
268         \begin{itemize}
269            \item<5->{A typical example is the flux-gradient relationship, e.g.,}
270         \end{itemize}
271      \end{itemize}
272      \end{footnotesize}
273   \end{itemize}
274   \onslide<5->{
275      \begin{center}
276      $\overline{w'\theta'} = - \nu_\mathrm{h} \cdot \frac{\partial \overline{\theta}}{\partial z}$
277      \end{center}}
278\end{frame}
279
280\end{document}
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