source: palm/trunk/TUTORIAL/SOURCE/sgs_models.tex @ 1787

Last change on this file since 1787 was 1526, checked in by keck, 10 years ago

several updates in the tutorial

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1% $Id: sgs_models.tex 1526 2015-01-22 12:47:21Z raasch $
2\input{header_tmp.tex}
3%\input{header_lectures.tex}
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 UniversitÀt Hannover}
25\selectlanguage{english}
26\date{last update: \today}
27\event{PALM Seminar}
28\setbeamertemplate{navigation symbols}{}
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,%
36      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}%
37      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}%
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[SGS Models]{SGS Models}
44\author{PALM group}
45
46\begin{document}
47
48% Folie 1
49\begin{frame}
50\titlepage
51\end{frame}
52
53
54\section{SGS Models}
55\subsection{SGS Models}
56
57% Folie 2
58\begin{frame}
59   \frametitle{SGS Models (I)}
60   \small
61   \begin{itemize}
62      \item<2->The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence).
63      \item<3->Features of small-scale turbulence: local, isotropic, dissipative (inertial subrange)
64      \item<4->SGS stresses should depend on:
65      \begin{itemize}
66         \item local resolved-scale field \hspace{3mm} and / or
67         \item past history of the local fluid (via a PDE)
68      \end{itemize}
69      \item<5->Importance of the model depends on how much energy is contained in the subgrid-scales:
70      \begin{itemize}
71         \item $E_{SGS} / E < 50\%$: results relatively insensitive to the model, (but sensitive to the numerics, e.g. in case of upwind scheme)
72         \item $E_{SGS} / E = 1$: model more important
73         \item<6->\textbf{If the large-scale eddies are not resolved, the SGS model and the LES will fail at all!} 
74      \end{itemize}
75   \end{itemize}
76\end{frame}
77
78% Folie 3
79\begin{frame}
80   \frametitle{SGS Models (II)}
81   Requirements that a good SGS model must fulfill:
82   \begin{footnotesize}
83      \begin{itemize}
84         \item<2-> Represent interactions with small scales.
85         \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).
86         \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.
87         \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.
88         \item<6->\underline{The primary goal of an SGS model is to obtain correct statistics of the}\\ 
89         \underline{energy containing scales of motion.}
90      \end{itemize}
91   \end{footnotesize}
92\end{frame}
93
94% Folie 4
95\begin{frame}
96   \frametitle{SGS Models (III)}
97   \onslide<1-> All the above observations suggest the use of an eddy viscosity type SGS model:
98   \begin{footnotesize}
99      \begin{itemize}
100         \item<2-> Take idea from RANS modeling, introduce eddy viscosity $\nu_T$:
101         \begin{flalign*}
102            &\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)\\
103            & & \text{filtered strain rate tensor}
104         \end{flalign*}
105      \end{itemize}
106   \end{footnotesize}
107   \vspace{-0.3cm}
108   \onslide<3->Now we need a model for the eddy viscosity:
109   \begin{footnotesize}
110      \begin{itemize}
111         \item<4-> Dimensionality of $\nu_T$ is $l^2/t$
112         \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characteristic velocity / length scale)
113         \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$
114         \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element)
115         \begin{flalign*}
116            &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} \\
117            & & \text{characteristic filtered rate of strain}\hspace{15mm}
118         \end{flalign*}
119      \end{itemize}
120   \end{footnotesize}
121\end{frame}
122
123
124\section{Smagorinsky Model}
125\subsection{The Smagorinsky Model}
126
127% Folie 5
128\begin{frame}
129   \frametitle{The Smagorinsky Model}
130   \onslide<2->Combine previous expressions to obtain:
131   \begin{equation*}
132      \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S}
133   \end{equation*}
134   \onslide<3-> Model due to Smagorinsky (1963):
135   \begin{itemize}
136      \item<3-> Originally designed at NCAR for global weather modeling.
137      \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theory.
138      \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$
139   \end{itemize}
140\end{frame}
141
142% Folie 6
143\begin{frame}
144   \frametitle{The Smagorinsky Model: Performance}
145   \begin{itemize}
146      \item<2-> Predicts many flows reasonably well
147      \item<3-> Problems:
148      \begin{itemize}
149         \item<3-> Optimum parameter value varies with flow type:
150         \begin{itemize}
151            \item Isotropic turbulence: $C_S \approx 0.2$\\
152            \item Shear (channel) flows: $C_S \approx 0.065$
153         \end{itemize}
154         \item<4-> Length scale uncertain with anisotropic filter:
155         \begin{equation*}
156            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
157         \end{equation*}
158         \item<5-> Needs modification to account for:
159         \begin{itemize}
160            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
161            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
162         \end{itemize}
163      \end{itemize}
164   \end{itemize}
165\end{frame}
166
167\section{Deardoff Modification}
168\subsection{Deardoff Modification}
169
170% Folie 7
171\begin{frame}
172   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
173   \footnotesize
174   \onslide<1->{
175      $ \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}}
176   \small
177   \begin{itemize}
178      \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
179                              local fluid.}
180   \end{itemize} 
181   \small
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      \small
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   $ \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}}$
203   \par\bigskip   
204   $ 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)$
205   \par\bigskip   
206   $W_{k}=\overline{u_k'q'} = -K_{h}\frac{\partial\bar{q}}{\partial x_{k}}$     
207            \par\bigskip 
208   $ \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} $
209   \par\bigskip
210   $ \nu_e = 2 \nu_T $
211   \par\bigskip
212   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
213\end{frame}
214
215% Folie 9
216\begin{frame}
217   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
218   \begin{itemize}
219      \item{There are still problems with this parameterization:}
220      \begin{itemize}
221         \item[-]<2->{The model overestimates the velocity shear near the wall.}
222         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
223         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
224      \end{itemize}
225      \item<5->{Several other SGS models have been developed:}
226      \begin{itemize}
227         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
228         \item[-]<6->{Scale similarity model (Bardina et al.)}
229         \item[-]<7->{Backscatter model (Mason)}
230      \end{itemize}
231      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
232               from the different models (Beare et al., 2006, BLM).}
233   \end{itemize} 
234\end{frame}
235
236
237\section{Summary / Important Points for Beginners}
238\subsection{Summary / Important Points for Beginners}
239
240% Folie 10
241\begin{frame}
242   \frametitle{Summary / Important Points for Beginners (I)}
243   \begin{columns}[c]
244   \column[T]{0.4\textwidth} 
245      \includegraphics<2-7|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
246      \includegraphics<8|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
247      \includegraphics<9|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
248      \includegraphics<10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
249      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}             
250   \column[T]{0.2\textwidth}
251      \vspace{0.9cm}
252      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
253      \par
254      \onslide<8-|handout:1>{\begin{small} fluctuations (\textbf{u},c) \end{small}}
255      \par\bigskip
256      \thicklines
257      \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}}}
258      \vspace{1cm}
259     
260      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
261      \par
262      \onslide<8->{\begin{small} smooth result \end{small}}   
263   \column[T]{0.4\textwidth}     
264      \includegraphics<1-2|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png}
265      \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png}
266      \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
267      \includegraphics<5-10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
268      \vspace{1.3cm}
269      \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png}
270      \uncover<7-|handout:1>{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}}       
271   \end{columns}
272\end{frame}
273
274% Folie 11
275\begin{frame}
276   \frametitle{Summary / Important Points for Beginners (II)}
277    For an LES it always has to be guaranteed that the main energy containing eddies of the respective
278    turbulent flow can really be simulated by the numerical model:     
279    \begin{itemize}
280       \item<2->{The grid spacing has to be fine enough.}
281       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
282       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
283                (therefore cyclic boundary conditions are used in most cases).}
284       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence
285                  has to be triggered by imposing random fluctuations.}
286       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
287    \end{itemize}     
288\end{frame}
289
290
291\section{Example Output}
292\subsection{Example Output}
293
294% Folie 12
295\begin{frame}
296   \frametitle{Example Output (I)}
297   \begin{itemize}
298      \item{LES of a convective boundary layer}
299   \end{itemize}
300   \includegraphics<1|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
301   \includegraphics<2|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
302   \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
303   \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
304   \includegraphics<5|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
305   \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
306   \includegraphics<7|handout:1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
307\end{frame}
308
309% Folie 13
310\begin{frame}
311   \frametitle{Example Output (II)}
312   \begin{itemize}
313      \item{LES of a convective boundary layer}
314   \end{itemize}
315   \begin{center}
316      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
317      power spectrum of vertical velocity
318   \end{center}
319\end{frame}
320
321% Folie 14
322\begin{frame}
323   \frametitle{Some Symbols}
324   \begin{columns}[c]
325      \column{0.6\textwidth}
326      \begin{tabbing}
327      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
328      $u,v,w$ \\ 
329
330      \\
331     
332      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
333      $x,y,z$ \\
334
335      \\
336
337      $\Theta$ \> potential temperature \\ \\
338
339      $\Psi$ \> passive scalar \\ \\
340
341      $T$ \> actual Temperatur \\ \\
342      \end{tabbing}
343   \column{0.4\textwidth}
344      \begin{tabbing}
345      $\Phi = gz$  \quad \= geopotential \\ \\
346
347      $p$ \> pressure \\ \\
348
349      $\rho$ \> density \\ \\
350
351      $f_i$ \> Coriolis Parameter \\ \\
352
353      $\epsilon_{ijk}$ \> alternating symbol \\ \\
354
355      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\
356
357      $Q, Q_\Psi$ \> sources or sinks \\ \\
358      \end{tabbing}
359   \end{columns}
360\end{frame}
361\end{document}
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