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1% $Id: sgs_models.tex 987 2012-09-05 09:27:35Z 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{pdfcomment}
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\begin{document}
46
47% Folie 1
48\begin{frame}
49\titlepage
50\end{frame}
51
52
53\section{SGS Models}
54\subsection{SGS Models}
55
56% Folie 2
57\begin{frame}
58   \frametitle{SGS Models (I)}
59   \small
60   \begin{itemize}
61      \item<2->The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence).
62      \item<3->Features of small-scale turbulence: local, isotropic, dissipative (inertial subrange)
63      \item<4->SGS stresses should depend on:
64      \begin{itemize}
65         \item local resolved-scale field \hspace{3mm} and / or
66         \item past history of the local fluid (via a PDE)
67      \end{itemize}
68      \item<5->Importance of the model depends on how much energy is contained in the subgrid-scales:
69      \begin{itemize}
70         \item $E_{SGS} / E < 50\%$: results relatively insensitive to the model, (but sensitive to the numerics, e.g. in case of upwind scheme)
71         \item $E_{SGS} / E = 1$: model more important
72         \item<6->\textbf{If the large-scale eddies are not resolved, the SGS model and the LES will fail at all!} 
73      \end{itemize}
74   \end{itemize}
75\end{frame}
76
77% Folie 3
78\begin{frame}
79   \frametitle{SGS Models (II)}
80   Requirements that a good SGS model must fulfill:
81   \begin{footnotesize}
82      \begin{itemize}
83         \item<2-> Represent interactions with small scales.
84         \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).
85         \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.
86         \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.
87         \item<6->\underline{The primary goal of an SGS model is to obtain correct statistics of the}\\ 
88         \underline{energy containing scales of motion.}
89      \end{itemize}
90   \end{footnotesize}
91\end{frame}
92
93% Folie 4
94\begin{frame}
95   \frametitle{SGS Models (III)}
96   \onslide<1-> All the above observation suggest the use of an eddy viscosity type SGS model:
97   \begin{footnotesize}
98      \begin{itemize}
99         \item<2-> Take idea from RANS modeling, introduce eddy viscosity $\nu_T$:
100         \begin{flalign*}
101            &\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)\\
102            & & \text{filtered strain rate tensor}
103         \end{flalign*}
104      \end{itemize}
105   \end{footnotesize}
106   \onslide<3->Now we need a model for the eddy viscosity:
107   \begin{footnotesize}
108      \begin{itemize}
109         \item<4-> Dimensionality of $\nu_T$ is $l^2/t$
110         \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characterictic velocity / length scale)
111         \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$
112         \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element)
113         \begin{flalign*}
114            &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} \\
115            & & \text{characterictic filtered rate of strain}\hspace{15mm}
116         \end{flalign*}
117      \end{itemize}
118   \end{footnotesize}
119\end{frame}
120
121
122\section{Smagorinsky Model}
123\subsection{The Smagorinsky Model}
124
125% Folie 5
126\begin{frame}
127   \frametitle{The Smagorinsky Model}
128   \onslide<2->Combine previous expressions to obtain:
129   \begin{equation*}
130      \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S}
131   \end{equation*}
132   \onslide<3-> Model due to Smagorinsky (1963):
133   \begin{itemize}
134      \item<3-> Originally designed at NCAR for global weather modeling.
135      \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theories.
136      \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$
137   \end{itemize}
138\end{frame}
139
140% Folie 6
141\begin{frame}
142   \frametitle{The Smagorinsky Model: Performance}
143   \begin{itemize}
144      \item<2-> Predicts many flows reasonably well
145      \item<3-> Problems:
146      \begin{itemize}
147         \item<3-> Optimum parameter value varies with flow type:
148         \begin{itemize}
149            \item Isotropic turbulence: $C_S \approx 0.2$\\
150            \item Shear (channel) flows: $C_S \approx 0.065$
151         \end{itemize}
152         \item<4-> Length scale uncertain with anisotropic filter:
153         \begin{equation*}
154            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
155         \end{equation*}
156         \item<5-> Needs modification to account for:
157         \begin{itemize}
158            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
159            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
160         \end{itemize}
161      \end{itemize}
162   \end{itemize}
163\end{frame}
164
165
166% Folie 7
167\begin{frame}
168   \frametitle{The Smagorinsky Model: Performance}
169   \begin{itemize}
170      \item<2-> Predicts many flows reasonably well
171      \item<3-> Problems:
172      \begin{itemize}
173         \item<3-> Optimum parameter value varies with flow type:
174         \begin{itemize}
175            \item Isotropic turbulence: $C_S \approx 0.2$\\
176            \item Shear (channel) flows: $C_S \approx 0.065$
177         \end{itemize}
178         \item<4-> Length scale uncertain with anisotropic filter:
179         \begin{equation*}
180            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
181         \end{equation*}
182         \item<5-> Needs modification to account for:
183         \begin{itemize}
184            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
185            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
186         \end{itemize}
187      \end{itemize}
188   \end{itemize}
189\end{frame}
190
191
192\section{Deardoff Modification}
193\subsection{Deardoff Modification}
194
195% Folie 8
196\begin{frame}
197   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
198   \footnotesize
199   \onslide<1->{
200      $ \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}}
201   \normalsize
202   \begin{itemize}
203      \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.}
204   \end{itemize} 
205   \onslide<3->{
206      $ C_M = const. = 0.1 $
207      \par\bigskip
208      \scriptsize
209      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
210                          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}
211                  \end{cases} $     
212      \normalsize
213      \par\bigskip
214      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
215\end{frame}
216
217% Folie 9
218\begin{frame}
219   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
220   \begin{itemize}
221      \item{SGS-TKE from prognostic equation}
222   \end{itemize}
223   $ \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 $                                         
224   \par\bigskip                                       
225   $ \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} $
226   \par\bigskip
227   $ \nu_e = 2 \nu_T $
228   \par\bigskip
229   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
230\end{frame}
231
232% Folie 10
233\begin{frame}
234   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
235   \begin{itemize}
236      \item{There are still problems with this parameterization:}
237      \begin{itemize}
238         \item[-]<2->{The model overestimates the velocity shear near the wall.}
239         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
240         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
241      \end{itemize}
242      \item<5->{Several other SGS models have been developed:}
243      \begin{itemize}
244         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
245         \item[-]<6->{Scale similarity model (Bardina et al.)}
246         \item[-]<7->{Backscatter model (Mason)}
247      \end{itemize}
248      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
249               from the different models (Beare et al., 2006, BLM).}
250   \end{itemize} 
251\end{frame}
252
253
254\section{Summary / Important Points for Beginners}
255\subsection{Summary / Important Points for Beginners}
256
257% Folie 11
258\begin{frame}
259   \frametitle{Summary / Important Points for Beginners (I)}
260   \begin{columns}[c]
261   \column[T]{0.4\textwidth} 
262      \includegraphics<2-7>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
263      \includegraphics<8>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
264      \includegraphics<9>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
265      \includegraphics<10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
266      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}             
267   \column[T]{0.2\textwidth}
268      \vspace{0.9cm}
269      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
270      \par
271      \onslide<8->{\begin{small} fluctuations (\textbf{u},c) \end{small}}
272      \par\bigskip
273      \thicklines
274      \onslide<9->{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}}
275      \vspace{1cm}
276     
277      \includegraphics<8-10>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
278      \par
279      \onslide<8->{\begin{small} smooth result \end{small}}   
280   \column[T]{0.4\textwidth}     
281      \includegraphics<1-2>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png}
282      \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png}
283      \includegraphics<4>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
284      \includegraphics<5-10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
285      \vspace{1.3cm}
286      \includegraphics<6>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png}
287      \uncover<7->{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}}       
288   \end{columns}
289\end{frame}
290
291% Folie 12
292\begin{frame}
293   \frametitle{Summary / Important Points for Beginners (II)}
294    For an LES it always has to be guaranteed that the main energy containing eddies of the respective
295    turbulent flow can really be simulated by the numerical model:     
296    \begin{itemize}
297       \item<2->{The grid spacing has to be fine enough.}
298       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
299       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
300                (therefore cyclic boundary conditions are used in most cases).}
301       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence
302                  has to be triggered by imposing random fluctuations.}
303       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
304    \end{itemize}     
305\end{frame}
306
307
308\section{Example Output}
309\subsection{Example Output}
310
311% Folie 13
312\begin{frame}
313   \frametitle{Example Output (I)}
314   \begin{itemize}
315      \item{LES of a convective boundary layer}
316   \end{itemize}
317   \includegraphics<1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
318   \includegraphics<2>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
319   \includegraphics<3>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
320   \includegraphics<4>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
321   \includegraphics<5>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
322   \includegraphics<6>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
323   \includegraphics<7>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
324\end{frame}
325
326% Folie 14
327\begin{frame}
328   \frametitle{Example Output (II)}
329   \begin{itemize}
330      \item{LES of a convective boundary layer}
331   \end{itemize}
332   \begin{center}
333      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
334      power spectrum of vertical velocity
335   \end{center}
336\end{frame}
337
338% Folie 15
339\begin{frame}
340   \frametitle{Some Symbols}
341   \begin{columns}[c]
342      \column{0.6\textwidth}
343      \begin{tabbing}
344      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
345      $u,v,w$ \\ 
346
347      \\
348     
349      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
350      $x,y,z$ \\
351
352      \\
353
354      $\Theta$ \> potential temperature \\ \\
355
356      $\Psi$ \> passive scalar \\ \\
357
358      $T$ \> actual Temperatur \\ \\
359      \end{tabbing}
360   \column{0.4\textwidth}
361      \begin{tabbing}
362      $\Phi = gz$  \quad \= geopotential \\ \\
363
364      $p$ \> pressure \\ \\
365
366      $\rho$ \> density \\ \\
367
368      $f_i$ \> Coriolis Parameter \\ \\
369
370      $\epsilon_{ijk}$ \> alternating symbol \\ \\
371
372      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
373
374      $Q, Q_\Psi$ \> sources or sinks \\ \\
375      \end{tabbing}
376   \end{columns}
377\end{frame}
378\end{document}
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