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1% $Id: exercise_cbl.tex 1534 2015-01-27 09:12:08Z heinze $
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
25\institute{Institute of Meteorology and Climatology, Leibniz UniversitÀt Hannover}
26\selectlanguage{english}
27\date{last update: \today}
28\event{PALM Seminar}
29\setbeamertemplate{navigation symbols}{}
30
31\setbeamertemplate{footline}
32  {
33    \begin{beamercolorbox}[rightskip=-0.1cm]&
34     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
35    \end{beamercolorbox}
36    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,
37      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}
38      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}
39    \end{beamercolorbox}
40    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}
41    \end{beamercolorbox}
42  }
43%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}}
44
45\title[Exercise 1: Convection Between Plates]{Exercise 1: Convection Between Plates}
46\author{PALM group}
47
48\begin{document}
49
50% Folie 1
51\begin{frame}
52\titlepage
53\end{frame}
54
55\section{Exercise}
56\subsection{Exercise}
57
58% Folie 2
59\begin{frame}
60   \frametitle{Exercise 1: Convection Between Plates}
61   
62   Please try to carry out a run with following initial and boundary conditions and create the required output.
63   \begin{itemize}
64      \scriptsize
65           \item<2-> The simulation should represent a stationary convective boundary layer between two uniformly heated/cooled plates with zero mean flow.
66           \item<3-> A free-slip condition for velocity shall be used at the bottom and top boundary.
67           \item<4-> The sensible heat flux at the bottom and top boundary shall be constant throughout the simulation.
68   \end{itemize}   
69   \onslide<5-> Simulation features:
70   \begin{itemize}
71      \scriptsize
72           \item<6-> domain size: about $\unit[2000 \times 2000 \times 1000]{m^3}$ ($x$/$y$/$z$)
73           \item<7-> grid size: $\unit[50]{m}$ equidistant
74           \item<8-> simulated time:    $\unit[3600]{s}$
75           \item<9-> surface heatflux: $\unit[0.1]{K\ m\ s^{-1}}$
76           \item<10-> heatflux at top: $\unit[0.1]{K\ m\ s^{-1}}$
77           \item<11-> initial temperature: $\unit[300]{K}$ everywhere
78           \item<12-> initial velocity: zero everywhere
79   \end{itemize} 
80\end{frame}
81
82% Folie 3
83\begin{frame}
84   \frametitle{Questions to be Answered:}
85   
86   \begin{itemize}
87   \item<1-> How does the flow field look like after 60 minutes of simulated time? (What kind of output do you need to answer this?)
88   \item<2-> How do the horizontally and temporally averaged vertical temperature and heat flux profiles look like?
89   \item<3-> Is it really a large-eddy simulation, i.e., are the subgrid-scale fluxes much smaller than the resolved-scale fluxes? (How long should the averaging time interval be?)
90   \item<4-> How do the total kinetic energy and the maximum velocity components change in time? Has the flow become stationary?
91   \item<5-> Has the domain size and grid size been chosen appropriately?
92   \end{itemize}
93
94\end{frame}
95       
96% Folie 4
97\begin{frame}
98   \frametitle{Hints (I)}
99   \scriptsize
100   
101   PALM parameter names are displayed by courier style, e.g. \textcolor{blue}{\texttt{end\_time}}.\\
102
103   \begin{itemize}
104      \item<2-> Domain size
105      \begin{itemize}
106         \scriptsize
107         \item[-]<2-> Is controlled by grid size (\textcolor{blue}{\texttt{dx}}, \textcolor{blue}{\texttt{dy}}, \textcolor{blue}{\texttt{dz}}) and number of grid points (\textcolor{blue}{\texttt{nx}}, \textcolor{blue}{\texttt{ny}}, \textcolor{blue}{\texttt{nz}}). Since the first grid point along each of the directions has index 0, the total number of grid points used are \textcolor{blue}{\texttt{nx}}+1, \textcolor{blue}{\texttt{ny}}+1, \textcolor{blue}{\texttt{nz}}+1. The total domain size in case of cyclic horizontal boundary conditions is (\textcolor{blue}{\texttt{nx}}+1)*\textcolor{blue}{\texttt{dx}}, (\textcolor{blue}{\texttt{ny}}+1)*\textcolor{blue}{\texttt{dy}}.
108      \end{itemize}
109     
110      \item<3-> Initial profiles
111      \begin{itemize}
112         \scriptsize
113         \item[-]<3-> Constant with height. See parameter \textcolor{blue}{\texttt{initializing\_actions}} for available initialization methods. See \textcolor{blue}{\texttt{ug\_surface}}, \textcolor{blue}{\texttt{vg\_surface}} and \textcolor{blue}{\texttt{pt\_surface}} for initial values of velocity and potential temperature.
114      \end{itemize}
115     
116      \item<4-> Boundary conditions
117       \begin{itemize}
118         \scriptsize
119         \item[-]<4-> For velocity, see \textcolor{blue}{\texttt{bc\_uv\_b}} and \textcolor{blue}{\texttt{bc\_uv\_t}}. See also \textcolor{blue}{\texttt{prandtl\_layer}}, because Neumann conditions don’t allow to use a Prandtl-layer.
120         \item[-]<5-> For temperature / heat flux, see \textcolor{blue}{\texttt{surface\_heatflux}} and \textcolor{blue}{\texttt{top\_heatflux}}. Prescribing of heat flux at the boundary requires a Neumann boundary condition for temperature, see \textcolor{blue}{\texttt{bc\_pt\_b}} and \textcolor{blue}{\texttt{bc\_pt\_t}}.
121         \item[-]<6-> Use a Neumann condition also for the perturbation pressure both at the bottom and the top (\textcolor{blue}{\texttt{bc\_p\_b}}, \textcolor{blue}{\texttt{bc\_p\_t}}).
122      \end{itemize}     
123     
124      \item<7-> Simulation time: See parameter \textcolor{blue}{\texttt{end\_time}}
125     
126   \end{itemize}
127
128\end{frame}
129
130% Folie 5
131\begin{frame}
132   \frametitle{Hints (II)}
133   \footnotesize
134   
135   Hints for data output.
136   
137   \begin{itemize}
138   
139      \item<2-> Variables
140      \begin{itemize}
141         \footnotesize
142         \item[-]<2-> Output variables are chosen with parameters \textcolor{blue}{\texttt{data\_output}} (3d-data or 2d-cross-sections) and \textcolor{blue}{\texttt{data\_output\_pr}} (profiles).
143      \end{itemize}
144
145      \item<3-> Output intervals
146      \begin{itemize}
147         \footnotesize
148         \item[-]<3-> Output intervals are set with parameter \textcolor{blue}{\texttt{dt\_data\_output}}. This parameter affects all output (cross-sections, profiles, etc.). Individual temporal intervals for the different output quantities can be assigned using parameters \textcolor{blue}{\texttt{dt\_do3d}}, \textcolor{blue}{\texttt{dt\_do2d\_xy}}, \textcolor{blue}{\texttt{dt\_do2d\_xz}}, \textcolor{blue}{\texttt{dt\_do2d\_yz}}, \textcolor{blue}{\texttt{dt\_dopr}}, etc.
149      \end{itemize}
150     
151      \item<4-> Time averaging
152      \begin{itemize}
153         \footnotesize
154         \item[-]<4-> Time averaging is controlled with parameters  \textcolor{blue}{\texttt{averaging\_interval}}\textcolor{blue}{\texttt{averaging\_interval\_pr}}\textcolor{blue}{\texttt{dt\_averaging\_input}}\textcolor{blue}{\texttt{dt\_averaging\_input\_pr}}.
155      \end{itemize}
156
157   \end{itemize}
158
159\end{frame}
160
161% Folie 6
162\begin{frame}
163   \frametitle{Further Hints}
164
165   \onslide<2-> You will find some more detailed information to solve this exercise in the PALM-online-documentation under:\\
166   \ \\
167   \small\url{http://palm.muk.uni-hannover.de/wiki/doc/app/examples/cbl}\\
168   \ \\
169   \normalsize (Attention: This documentation is for atmospheric convection with free upper lid.)
170   \ \\
171   \ \\
172   \onslide<3-> \normalsize Please also visit\\
173   \ \\
174   \small\url{http://palm.muk.uni-hannover.de/wiki/doc/app/netcdf}\\
175   \ \\
176   \normalsize where the complete PALM netCDF-data-output and the respective steering parameters are described.
177
178\end{frame}
179
180% Folie 7
181\begin{frame}
182   \frametitle{How to Start?}
183
184   \begin{itemize}
185      \item<2-> Create a data directory for a new run:\\
186           \quad \texttt{cd \~{}/palm/current\_version}\\
187           \quad \texttt{mkdir -p JOBS/uniform\_plates/INPUT}
188           
189           \item<3-> Create the parameter file and set the required parameters in\\
190           \quad \texttt{JOBS/uniform\_plates/INPUT/uniform\_plates\_p3d}
191           
192           \item<4-> Start the run with \texttt{mrun-command}\\
193           \quad \texttt{mrun -d uniform\_plates -h <hi> -K parallel ...}\\
194           and analyze the output files.
195
196   \end{itemize}
197   
198   \ \\
199   
200   \onslide<5-> \huge \centering \textcolor{blue}{Good Luck!}
201
202\end{frame}
203
204% Folie 8
205\section{Results}
206\subsection{Results}
207
208\begin{frame}
209   \frametitle{$xy$-cross sections (instantaneous at $t = \unit[3600]{s}$)}
210   \begin{center}
211      \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_100.eps}
212      \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_500.eps}\\
213      \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_750.eps}
214   \end{center}
215\end{frame}
216
217% Folie 9
218\begin{frame}
219   \frametitle{$xz$-cross sections ($\unit[900]{s}$ average)}
220    \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y250m.eps}
221    \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y500m.eps}\\
222    \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y750m.eps}
223    \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y1000m.eps}
224\end{frame}
225
226% Folie 10
227\begin{frame}
228   \frametitle{Vertical profiles}
229   \begin{center}
230      \includegraphics[angle=90,width=\textwidth]{exercise_cbl_figures/pr_pt.eps}
231   \end{center}
232\end{frame}
233
234% Folie 11
235\begin{frame}
236   \frametitle{LES?}
237   \begin{center}
238      \includegraphics[width=1.0\textwidth]{exercise_cbl_figures/pr_wpt_res_sgs.eps}
239   \end{center}
240\end{frame}
241
242% Folie 12
243\begin{frame}
244   \frametitle{Time series (I)}
245   \begin{center}
246      \includegraphics[angle=90,width=1.0\textwidth]{exercise_cbl_figures/ts.eps}
247   \end{center}
248\end{frame}
249
250% Folie 13
251\begin{frame}
252   \frametitle{Time series (II)}
253   \begin{center}
254      \includegraphics[angle=90,width=1.0\textwidth]{exercise_cbl_figures/ts2.eps}
255   \end{center}
256\end{frame}
257
258
259\subsection{Answers}
260
261% Folie 14
262\begin{frame}
263   \frametitle{Answers to question I}
264   \footnotesize
265   How does the flow field look like after 60 minutes of simulated time?
266   \begin{itemize}
267    \item Useful output: for example instantaneous or time-averaged cross-sections of vertical velocity (frames 8--9).
268    \item Flow field shows narrower updrafts and broader downdrafts, cellular pattern close to the heated/cooled plates in xy-sections of
269          vertical velocity.
270    \item The temporal mean of vertical velocity exhibits a circulation spanning the whole depth of the model domain.
271   \end{itemize}
272\end{frame}
273
274% Folie 15
275\begin{frame}
276   \frametitle{Answers to question II}
277   \footnotesize
278   How do the horizontally and temporally averaged vertical temperature and heat flux profiles look like?
279   \begin{itemize}
280    \item PALM standard profile output contains potential temperature and its vertical flux (shown in frame 10).
281    \item Heating the lower plate and cooling the upper plate induces convection resulting in a well-mixed boundary layer where the
282          potential temperature profile is constant with height.
283          Temperature gradients remain at the domain boundaries since convective turbulence cannot remove them in the vicinity of the walls.
284    \item In case of horizontal homogeneity, the temperature equation reduces  to
285          $\frac{\partial\theta}{\partial t}=-\frac{\partial\overline{w^{\prime}\theta^{\prime}}}{\partial z}$ in the present case. In a
286          stationary state, it follows that $\frac{\partial\theta}{\partial t}= 0 $. Thus, the flux profile
287          $\overline{w^{\prime}\theta^{\prime}}$ has to be constant with height -- as can be seen in frame 10.
288    \item The total vertical heat flux is positive in the whole modeling domain indicating upward transport of warmer air
289          parcels and downward transport of colder air parcels.
290   \end{itemize}
291\end{frame}
292
293% Folie 16
294\begin{frame}
295   \frametitle{Answers to question III}
296   \footnotesize
297   Is it really a large-eddy simulation? Duration of averaging time?
298   \begin{itemize}
299    \item It is a large-eddy simulation because the sub-grid fluxes are negligibly small throughout the bulk of the mixed layer. There, the
300          resolved flux is dominating the total flux indicating a well-resolved turbulent flow (frame 11). Sub-grid fluxes dominate close to
301          the surface where the turbulent-eddies cannot be resolved.
302    \item Typically, the averaging time should contain several large-eddy turnover times. The large-eddy turnover time can be defined as
303          $\tau_{\mathrm{l}}=L/u$ where $L$ is the length-scale of the largest eddies in the flow and $u$ is their typical velocity scale.
304          $\tau_{\mathrm{l}}$ can be interpreted as a typical time a turbulent eddy needs to traverse the modeling domain. In our case,
305          $L$ is proportional to the domain height ($L\approx1000\,\mathrm{m}$) and $u$ is about $5\,\mathrm{ms^{-1}}$ (see time series of
306          wmax on frame 12). Thus, $\tau_{\mathrm{l}}\approx200\,\mathrm{s}$. An averaging time of 600\,s chosen here
307          is, thus, appropriate.
308   \end{itemize}
309\end{frame}
310
311% Folie 17
312\begin{frame}
313   \frametitle{Answers to question IV}
314   \footnotesize
315   Has the flow become stationary?
316   \begin{itemize}
317    \item The time series of total kinetic energy E and the maximum velocities wmax, umax and vmax shown in frames 12-13 exhibit
318          a spin-up phase of the model up to $t\approx2000\,\mathrm{s}$. During this initialization time, turbulence is triggered by
319          random perturbations until turbulence starts to develop.
320    \item A stationary state can be seen by means of an (almost) non-changing E with time. Constant maxima of the velocity
321          components also indicate a stationary flow.
322   \end{itemize}
323\end{frame}
324
325% Folie 18
326\begin{frame}
327   \frametitle{Answers to question V}
328   \footnotesize
329   Has the domain size and grid size been chosen appropriately?
330   \begin{itemize}
331    \item A domain size is generally appropriately chosen in case that several of the dominating flow structures fit into the modeling
332          domain. From the xy-cross sections in frame 8 it becomes apparent that the typical hexagonal flow structures close to the
333          surface can hardly be seen. The xz-cross sections in frame 9 also contain only
334          one circulation. Thus, the domain size in our example seems to be too small to capture several energy-containing flow structures.
335    \item The grid size should be chosen in the way that the dominating flow structures can be represented by at least several
336          grid points (4-5). A grid spacing of 50~m as chosen in this exercise
337          is appropriate since the flow structures exhibit horizontal length scales of about 1~km (see frame 8).
338   \end{itemize}
339\end{frame}
340\end{document}
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