source: palm/trunk/TUTORIAL/SOURCE/sgs_models.tex @ 925

Last change on this file since 925 was 915, checked in by maronga, 12 years ago

added first LaTeX source code for the new tutorial

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1% $Id: sgs_models.tex 915 2012-05-30 15:11:11Z maronga $
2\input{header_tmp.tex}
3%\input{header_lectures.tex}
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
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\usetikzlibrary{shapes,arrows,positioning}
17\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
18\usepackage{amsmath}
19\usepackage{amssymb}
20\usepackage{multicol}
21\usepackage{float}
22
23\institute{Institut fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
24\date{last update: \today}
25\event{PALM Seminar}
26\setbeamertemplate{navigation symbols}{}
27
28\setbeamertemplate{footline}
29  {
30    \begin{beamercolorbox}[rightskip=-0.1cm]&
31     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
32    \end{beamercolorbox}
33    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,
34      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}
35      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}
36    \end{beamercolorbox}
37    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}
38    \end{beamercolorbox}
39  }
40%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}}
41
42\title[SGS Models]{SGS Models}
43\author{Siegfried Raasch}
44
45% Notes:
46% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
47% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
48\begin{document}
49
50% Folie 1
51\begin{frame}
52\titlepage
53\end{frame}
54
55
56\section{SGS Models}
57\subsection{SGS Models}
58
59% Folie 2
60\begin{frame}
61   \frametitle{SGS Models (I)}
62   \small
63   \begin{itemize}
64      \item<2->The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence).
65      \item<3->Features of small-scale turbulence: local, isotropic, dissipative (inertial subrange)
66      \item<4->SGS stresses should depend on:
67      \begin{itemize}
68         \item local resolved-scale field \hspace{3mm} and / or
69         \item past history of the local fluid (via a PDE)
70      \end{itemize}
71      \item<5->Importance of the model depends on how much energy is contained in the subgrid-scales:
72      \begin{itemize}
73         \item $E_{SGS} / E < 50\%$: results relatively insensitive to the model, (but sensitive to the numerics, e.g. in case of upwind scheme)
74         \item $E_{SGS} / E = 1$: model more important
75         \item<6->\textbf{If the large-scale eddies are not resolved, the SGS model and the LES will fail at all!} 
76      \end{itemize}
77   \end{itemize}
78\end{frame}
79
80% Folie 3
81\begin{frame}
82   \frametitle{SGS Models (II)}
83   Requirements that a good SGS model must fulfill:
84   \begin{footnotesize}
85      \begin{itemize}
86         \item<2-> Represent interactions with small scales.
87         \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).
88         \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.
89         \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.
90         \item<6->\underline{The primary goal of an SGS model is to obtain correct statistics of the}\\ 
91         \underline{energy containing scales of motion.}
92      \end{itemize}
93   \end{footnotesize}
94\end{frame}
95
96% Folie 4
97\begin{frame}
98   \frametitle{SGS Models (III)}
99   \onslide<1-> All the above observation suggest the use of an eddy viscosity type SGS model:
100   \begin{footnotesize}
101      \begin{itemize}
102         \item<2-> Take idea from RANS modeling, introduce eddy viscosity $\nu_T$:
103         \begin{flalign*}
104            &\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)\\
105            & & \text{filtered strain rate tensor}
106         \end{flalign*}
107      \end{itemize}
108   \end{footnotesize}
109   \onslide<3->Now we need a model for the eddy viscosity:
110   \begin{footnotesize}
111      \begin{itemize}
112         \item<4-> Dimensionality of $\nu_T$ is $l^2/t$
113         \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characterictic velocity / length scale)
114         \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$
115         \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element)
116         \begin{flalign*}
117            &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} \\
118            & & \text{characterictic filtered rate of strain}\hspace{15mm}
119         \end{flalign*}
120      \end{itemize}
121   \end{footnotesize}
122\end{frame}
123
124
125\section{Smagorinsky Model}
126\subsection{The Smagorinsky Model}
127
128% Folie 5
129\begin{frame}
130   \frametitle{The Smagorinsky Model}
131   \onslide<2->Combine previous expressions to obtain:
132   \begin{equation*}
133      \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S}
134   \end{equation*}
135   \onslide<3-> Model due to Smagorinsky (1963):
136   \begin{itemize}
137      \item<3-> Originally designed at NCAR for global weather modeling.
138      \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theories.
139      \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$
140   \end{itemize}
141\end{frame}
142
143% Folie 6
144\begin{frame}
145   \frametitle{The Smagorinsky Model: Performance}
146   \begin{itemize}
147      \item<2-> Predicts many flows reasonably well
148      \item<3-> Problems:
149      \begin{itemize}
150         \item<3-> Optimum parameter value varies with flow type:
151         \begin{itemize}
152            \item Isotropic turbulence: $C_S \approx 0.2$\\
153            \item Shear (channel) flows: $C_S \approx 0.065$
154         \end{itemize}
155         \item<4-> Length scale uncertain with anisotropic filter:
156         \begin{equation*}
157            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
158         \end{equation*}
159         \item<5-> Needs modification to account for:
160         \begin{itemize}
161            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
162            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
163         \end{itemize}
164      \end{itemize}
165   \end{itemize}
166\end{frame}
167
168
169\section{Deardoff Modification}
170\subsection{Deardoff Modification}
171
172% Folie 7
173\begin{frame}
174   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
175   \footnotesize
176   \onslide<1->{
177      $ \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}}
178   \normalsize
179   \begin{itemize}
180      \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 local fluid.}
181   \end{itemize} 
182   \onslide<3->{
183      $ C_M = const. = 0.1 $
184      \par\bigskip
185      \scriptsize
186      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
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}
188                  \end{cases} $     
189      \normalsize
190      \par\bigskip
191      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
192\end{frame}
193
194% Folie 8
195\begin{frame}
196   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
197   \begin{itemize}
198      \item{SGS-TKE from prognostic equation}
199   \end{itemize}
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 $                                         
201   \par\bigskip                                       
202   $ \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} $
203   \par\bigskip
204   $ \nu_e = 2 \nu_T $
205   \par\bigskip
206   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
207\end{frame}
208
209% Folie 9
210\begin{frame}
211   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
212   \begin{itemize}
213      \item{There are still problems with this parameterization:}
214      \begin{itemize}
215         \item[-]<2->{The model overestimates the velocity shear near the wall.}
216         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
217         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
218      \end{itemize}
219      \item<5->{Several other SGS models have been developed:}
220      \begin{itemize}
221         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
222         \item[-]<6->{Scale similarity model (Bardina et al.)}
223         \item[-]<7->{Backscatter model (Mason)}
224      \end{itemize}
225      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
226               from the different models (Beare et al., 2006, BLM).}
227   \end{itemize} 
228\end{frame}
229
230
231\section{Summary / Important Points for Beginners}
232\subsection{Summary / Important Points for Beginners}
233
234% Folie 10
235\begin{frame}
236   \frametitle{Summary / Important Points for Beginners (I)}
237   \begin{columns}[c]
238   \column[T]{0.4\textwidth} 
239      \includegraphics<2-7>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
240      \includegraphics<8>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
241      \includegraphics<9>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
242      \includegraphics<10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
243      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}             
244   \column[T]{0.2\textwidth}
245      \vspace{0.9cm}
246      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
247      \par
248      \onslide<8->{\begin{small} fluctuations (\textbf{u},c) \end{small}}
249      \par\bigskip
250      \thicklines
251      \onslide<9->{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}}
252      \vspace{1cm}
253     
254      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
255      \par
256      \onslide<8->{\begin{small} smooth result \end{small}}   
257   \column[T]{0.4\textwidth}     
258      \includegraphics<1-2>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1.png}
259      \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3.png}
260      \includegraphics<4>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
261      \includegraphics<5-10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
262      \vspace{1.3cm}
263      \includegraphics<6>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6.png}
264      \uncover<7->{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7.png}}       
265   \end{columns}
266\end{frame}
267
268% Folie 11
269\begin{frame}
270   \frametitle{Summary / Important Points for Beginners (II)}
271    For an LES it always has to be guaranteed that the main energy containing eddies of the respective
272    turbulent flow can really be simulated by the numerical model:     
273    \begin{itemize}
274       \item<2->{The grid spacing has to be fine enough.}
275       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
276       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
277                (therefore cyclic boundary conditions are used in most cases).}
278       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence
279                  has to be triggered by imposing random fluctuations.}
280       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
281    \end{itemize}     
282\end{frame}
283
284
285\section{Example Output}
286\subsection{Example Output}
287
288% Folie 12
289\begin{frame}
290   \frametitle{Example Output (I)}
291   \begin{itemize}
292      \item{LES of a convective boundary layer}
293   \end{itemize}
294   \includegraphics<1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
295   \includegraphics<2>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
296   \includegraphics<3>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
297   \includegraphics<4>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
298   \includegraphics<5>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
299   \includegraphics<6>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
300   \includegraphics<7>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
301\end{frame}
302
303% Folie 13
304\begin{frame}
305   \frametitle{Example Output (II)}
306   \begin{itemize}
307      \item{LES of a convective boundary layer}
308   \end{itemize}
309   \begin{center}
310      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
311      power spectrum of vertical velocity
312   \end{center}
313\end{frame}
314
315% Folie 14
316\begin{frame}
317   \frametitle{Some Symbols}
318   \begin{columns}[c]
319      \column{0.6\textwidth}
320      \begin{tabbing}
321      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
322      $u,v,w$ \\ 
323
324      \\
325     
326      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
327      $x,y,z$ \\
328
329      \\
330
331      $\Theta$ \> potential temperature \\ \\
332
333      $\Psi$ \> passive scalar \\ \\
334
335      $T$ \> actual Temperatur \\ \\
336      \end{tabbing}
337   \column{0.4\textwidth}
338      \begin{tabbing}
339      $\Phi = gz$  \quad \= geopotential \\ \\
340
341      $p$ \> pressure \\ \\
342
343      $\rho$ \> density \\ \\
344
345      $f_i$ \> Coriolis Parameter \\ \\
346
347      $\epsilon_{ijk}$ \> alternating symbol \\ \\
348
349      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
350
351      $Q, Q_\Psi$ \> sources or sinks \\ \\
352      \end{tabbing}
353   \end{columns}
354\end{frame}
355\end{document}
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