source: palm/trunk/TUTORIAL/SOURCE/sgs_models.tex @ 1495

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several updates in the tutorial

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1% $Id: sgs_models.tex 1226 2013-09-18 13:19:19Z 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 observations 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   \vspace{-0.3cm}
107   \onslide<3->Now we need a model for the eddy viscosity:
108   \begin{footnotesize}
109      \begin{itemize}
110         \item<4-> Dimensionality of $\nu_T$ is $l^2/t$
111         \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characteristic velocity / length scale)
112         \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$
113         \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element)
114         \begin{flalign*}
115            &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} \\
116            & & \text{characteristic filtered rate of strain}\hspace{15mm}
117         \end{flalign*}
118      \end{itemize}
119   \end{footnotesize}
120\end{frame}
121
122
123\section{Smagorinsky Model}
124\subsection{The Smagorinsky Model}
125
126% Folie 5
127\begin{frame}
128   \frametitle{The Smagorinsky Model}
129   \onslide<2->Combine previous expressions to obtain:
130   \begin{equation*}
131      \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S}
132   \end{equation*}
133   \onslide<3-> Model due to Smagorinsky (1963):
134   \begin{itemize}
135      \item<3-> Originally designed at NCAR for global weather modeling.
136      \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theories.
137      \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$
138   \end{itemize}
139\end{frame}
140
141% Folie 6
142\begin{frame}
143   \frametitle{The Smagorinsky Model: Performance}
144   \begin{itemize}
145      \item<2-> Predicts many flows reasonably well
146      \item<3-> Problems:
147      \begin{itemize}
148         \item<3-> Optimum parameter value varies with flow type:
149         \begin{itemize}
150            \item Isotropic turbulence: $C_S \approx 0.2$\\
151            \item Shear (channel) flows: $C_S \approx 0.065$
152         \end{itemize}
153         \item<4-> Length scale uncertain with anisotropic filter:
154         \begin{equation*}
155            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
156         \end{equation*}
157         \item<5-> Needs modification to account for:
158         \begin{itemize}
159            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
160            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
161         \end{itemize}
162      \end{itemize}
163   \end{itemize}
164\end{frame}
165
166\section{Deardoff Modification}
167\subsection{Deardoff Modification}
168
169% Folie 7
170\begin{frame}
171   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
172   \footnotesize
173   \onslide<1->{
174      $ \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}}
175   \small
176   \begin{itemize}
177      \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
178                              local fluid.}
179   \end{itemize} 
180   \small
181   \onslide<3->{
182      $ C_M = const. = 0.1 $
183      \par\bigskip
184      \scriptsize
185      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
186                          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}
187                  \end{cases} $     
188      \small
189      \par\bigskip
190      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
191\end{frame}
192
193% Folie 8
194\begin{frame}
195   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
196   \begin{itemize}
197      \item{SGS-TKE from prognostic equation}
198   \end{itemize}
199   $ \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 $                                         
200   \par\bigskip     
201   $ \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}}$
202   \par\bigskip   
203   $ 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)$
204   \par\bigskip   
205   $W_{k}=\overline{u_k'q'} = -K_{h}\frac{\partial\bar{q}}{\partial x_{k}}$     
206            \par\bigskip 
207   $ \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} $
208   \par\bigskip
209   $ \nu_e = 2 \nu_T $
210   \par\bigskip
211   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
212\end{frame}
213
214% Folie 9
215\begin{frame}
216   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
217   \begin{itemize}
218      \item{There are still problems with this parameterization:}
219      \begin{itemize}
220         \item[-]<2->{The model overestimates the velocity shear near the wall.}
221         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
222         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
223      \end{itemize}
224      \item<5->{Several other SGS models have been developed:}
225      \begin{itemize}
226         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
227         \item[-]<6->{Scale similarity model (Bardina et al.)}
228         \item[-]<7->{Backscatter model (Mason)}
229      \end{itemize}
230      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
231               from the different models (Beare et al., 2006, BLM).}
232   \end{itemize} 
233\end{frame}
234
235
236\section{Summary / Important Points for Beginners}
237\subsection{Summary / Important Points for Beginners}
238
239% Folie 10
240\begin{frame}
241   \frametitle{Summary / Important Points for Beginners (I)}
242   \begin{columns}[c]
243   \column[T]{0.4\textwidth} 
244      \includegraphics<2-7|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
245      \includegraphics<8|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
246      \includegraphics<9|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
247      \includegraphics<10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
248      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}             
249   \column[T]{0.2\textwidth}
250      \vspace{0.9cm}
251      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
252      \par
253      \onslide<8-|handout:1>{\begin{small} fluctuations (\textbf{u},c) \end{small}}
254      \par\bigskip
255      \thicklines
256      \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}}}
257      \vspace{1cm}
258     
259      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
260      \par
261      \onslide<8->{\begin{small} smooth result \end{small}}   
262   \column[T]{0.4\textwidth}     
263      \includegraphics<1-2|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png}
264      \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png}
265      \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
266      \includegraphics<5-10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
267      \vspace{1.3cm}
268      \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png}
269      \uncover<7-|handout:1>{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}}       
270   \end{columns}
271\end{frame}
272
273% Folie 11
274\begin{frame}
275   \frametitle{Summary / Important Points for Beginners (II)}
276    For an LES it always has to be guaranteed that the main energy containing eddies of the respective
277    turbulent flow can really be simulated by the numerical model:     
278    \begin{itemize}
279       \item<2->{The grid spacing has to be fine enough.}
280       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
281       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
282                (therefore cyclic boundary conditions are used in most cases).}
283       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence
284                  has to be triggered by imposing random fluctuations.}
285       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
286    \end{itemize}     
287\end{frame}
288
289
290\section{Example Output}
291\subsection{Example Output}
292
293% Folie 12
294\begin{frame}
295   \frametitle{Example Output (I)}
296   \begin{itemize}
297      \item{LES of a convective boundary layer}
298   \end{itemize}
299   \includegraphics<1|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
300   \includegraphics<2|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
301   \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
302   \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
303   \includegraphics<5|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
304   \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
305   \includegraphics<7|handout:1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
306\end{frame}
307
308% Folie 13
309\begin{frame}
310   \frametitle{Example Output (II)}
311   \begin{itemize}
312      \item{LES of a convective boundary layer}
313   \end{itemize}
314   \begin{center}
315      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
316      power spectrum of vertical velocity
317   \end{center}
318\end{frame}
319
320% Folie 14
321\begin{frame}
322   \frametitle{Some Symbols}
323   \begin{columns}[c]
324      \column{0.6\textwidth}
325      \begin{tabbing}
326      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
327      $u,v,w$ \\ 
328
329      \\
330     
331      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
332      $x,y,z$ \\
333
334      \\
335
336      $\Theta$ \> potential temperature \\ \\
337
338      $\Psi$ \> passive scalar \\ \\
339
340      $T$ \> actual Temperatur \\ \\
341      \end{tabbing}
342   \column{0.4\textwidth}
343      \begin{tabbing}
344      $\Phi = gz$  \quad \= geopotential \\ \\
345
346      $p$ \> pressure \\ \\
347
348      $\rho$ \> density \\ \\
349
350      $f_i$ \> Coriolis Parameter \\ \\
351
352      $\epsilon_{ijk}$ \> alternating symbol \\ \\
353
354      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
355
356      $Q, Q_\Psi$ \> sources or sinks \\ \\
357      \end{tabbing}
358   \end{columns}
359\end{frame}
360\end{document}
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