# Changeset 1534 for palm/trunk/TUTORIAL

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Timestamp:
Jan 27, 2015 9:12:08 AM (8 years ago)
Message:

Revision of exercise_cbl and exercise_neutral

Location:
palm/trunk/TUTORIAL/SOURCE
Files:
14 deleted
2 edited

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 r1515 \usepackage{ngerman} \usepackage{pgf} \usetheme{Dresden} \usepackage{subfigure} \usepackage{units} \usepackage{tikz} \usetikzlibrary{shapes,arrows,positioning} \usetikzlibrary{decorations.markings}             %neues paket \usetikzlibrary{decorations.pathreplacing}        %neues paket \def\Tiny{\fontsize{4pt}{4pt}\selectfont} %---------- neue Pakete \usepackage{amsmath} \usepackage{amssymb} \usepackage{multicol} \usepackage{pdfcomment} \usepackage{graphicx} \usepackage{listings} \lstset{showspaces=false,language=fortran,basicstyle= \ttfamily,showstringspaces=false,captionpos=b} \institute{Institute of Meteorology and Climatology, Leibniz UniversitÃ€t Hannover} \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?) \item<2-> How do the horizontally and temporally averaged vertical temperature and heat flux profiles look like? \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?) \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?) \item<4-> How do the total kinetic energy and the maximum velocity components change in time? Has the flow become stationary? \item<5-> Has the domain size and grid size been chosen appropriately? \end{itemize} \item<7-> Simulation time \begin{itemize} \scriptsize \item[-]<7-> See parameter \textcolor{blue}{\texttt{end\_time}}. \end{itemize} \item<7-> Simulation time: See parameter \textcolor{blue}{\texttt{end\_time}} \end{itemize} \frametitle{$xy$-cross sections (instantaneous at $t = \unit[3600]{s}$)} \begin{center} \includegraphics[width=0.42\textwidth]{exercise_cbl_figures/xy_w_100.eps} \includegraphics[width=0.42\textwidth]{exercise_cbl_figures/xy_w_500.eps}\\ \includegraphics[width=0.42\textwidth]{exercise_cbl_figures/xy_w_750.eps} \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_100.eps} \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_500.eps}\\ \includegraphics[width=0.4\textwidth]{exercise_cbl_figures/xy_w_750.eps} \end{center} \end{frame} \begin{frame} \frametitle{$xz$-cross sections ($\unit[900]{s}$ average)} \begin{center} \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/xz_w_y250m.eps} \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/xz_w_y500m.eps}\\ \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/xz_w_y750m.eps} \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/xz_w_y1000m.eps} \end{center} \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y250m.eps} \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y500m.eps}\\ \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y750m.eps} \includegraphics[width=0.52\textwidth]{exercise_cbl_figures/xz_w_y1000m.eps} \end{frame} % Folie 10 \begin{frame} \frametitle{Vertical profiles (I)} \frametitle{Vertical profiles} \begin{center} \includegraphics[angle=90,width=\textwidth]{exercise_cbl_figures/pr_pt.eps} \frametitle{LES?} \begin{center} \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/pr_wpt2.eps} \includegraphics[width=0.55\textwidth]{exercise_cbl_figures/pr_wpt_resolved.eps} \includegraphics[width=1.0\textwidth]{exercise_cbl_figures/pr_wpt_res_sgs.eps} \end{center} \end{frame} \end{center} \end{frame} \subsection{Answers} % Folie 14 \begin{frame} \frametitle{Answers to question I} \footnotesize How does the flow field look like after 60 minutes of simulated time? \begin{itemize} \item Useful output: for example instantaneous or time-averaged cross-sections of vertical velocity (frames 8--9). \item Flow field shows narrower updrafts and broader downdrafts, cellular pattern close to the heated/cooled plates in xy-sections of vertical velocity. \item The temporal mean of vertical velocity exhibits a circulation spanning the whole depth of the model domain. \end{itemize} \end{frame} % Folie 15 \begin{frame} \frametitle{Answers to question II} \footnotesize How do the horizontally and temporally averaged vertical temperature and heat flux profiles look like? \begin{itemize} \item PALM standard profile output contains potential temperature and its vertical flux (shown in frame 10). \item Heating the lower plate and cooling the upper plate induces convection resulting in a well-mixed boundary layer where the potential temperature profile is constant with height. Temperature gradients remain at the domain boundaries since convective turbulence cannot remove them in the vicinity of the walls. \item In case of horizontal homogeneity, the temperature equation reduces  to $\frac{\partial\theta}{\partial t}=-\frac{\partial\overline{w^{\prime}\theta^{\prime}}}{\partial z}$ in the present case. In a stationary state, it follows that $\frac{\partial\theta}{\partial t}= 0$. Thus, the flux profile $\overline{w^{\prime}\theta^{\prime}}$ has to be constant with height -- as can be seen in frame 10. \item The total vertical heat flux is positive in the whole modeling domain indicating upward transport of warmer air parcels and downward transport of colder air parcels. \end{itemize} \end{frame} % Folie 16 \begin{frame} \frametitle{Answers to question III} \footnotesize Is it really a large-eddy simulation? Duration of averaging time? \begin{itemize} \item It is a large-eddy simulation because the sub-grid fluxes are negligibly small throughout the bulk of the mixed layer. There, the resolved flux is dominating the total flux indicating a well-resolved turbulent flow (frame 11). Sub-grid fluxes dominate close to the surface where the turbulent-eddies cannot be resolved. \item Typically, the averaging time should contain several large-eddy turnover times. The large-eddy turnover time can be defined as $\tau_{\mathrm{l}}=L/u$ where $L$ is the length-scale of the largest eddies in the flow and $u$ is their typical velocity scale. $\tau_{\mathrm{l}}$ can be interpreted as a typical time a turbulent eddy needs to traverse the modeling domain. In our case, $L$ is proportional to the domain height ($L\approx1000\,\mathrm{m}$) and $u$ is about $5\,\mathrm{ms^{-1}}$ (see time series of wmax on frame 12). Thus, $\tau_{\mathrm{l}}\approx200\,\mathrm{s}$. An averaging time of 600\,s chosen here is, thus, appropriate. \end{itemize} \end{frame} % Folie 17 \begin{frame} \frametitle{Answers to question IV} \footnotesize Has the flow become stationary? \begin{itemize} \item The time series of total kinetic energy E and the maximum velocities wmax, umax and vmax shown in frames 12-13 exhibit a spin-up phase of the model up to $t\approx2000\,\mathrm{s}$. During this initialization time, turbulence is triggered by random perturbations until turbulence starts to develop. \item A stationary state can be seen by means of an (almost) non-changing E with time. Constant maxima of the velocity components also indicate a stationary flow. \end{itemize} \end{frame} % Folie 18 \begin{frame} \frametitle{Answers to question V} \footnotesize Has the domain size and grid size been chosen appropriately? \begin{itemize} \item A domain size is generally appropriately chosen in case that several of the dominating flow structures fit into the modeling domain. From the xy-cross sections in frame 8 it becomes apparent that the typical hexagonal flow structures close to the surface can hardly be seen. The xz-cross sections in frame 9 also contain only one circulation. Thus, the domain size in our example seems to be too small to capture several energy-containing flow structures. \item The grid size should be chosen in the way that the dominating flow structures can be represented by at least several grid points (4-5). A grid spacing of 50~m as chosen in this exercise is appropriate since the flow structures exhibit horizontal length scales of about 1~km (see frame 8). \end{itemize} \end{frame} \end{document}
 r1515 \begin{itemize} \item A neutrally stratified atmospheric boundary layer shall be simulated. \item<2-> The flow shall be driven by a constant large-scale pressure gradient, i.e. a geostrophic wind. \item<2-> The flow shall be driven by a constant large-scale pressure gradient, i.e., a geostrophic wind. \item<3-> At the end of the simulation, turbulence as well as the mean flow should be in a stationary state. \end{itemize} \item<2-> How do the horizontally and temporally averaged vertical velocity and momentum flux profiles look like? \vspace{1em} \item<3-> Is it really a large-eddy simulation, i.e. are the subgrid-scale fluxes much smaller than the resolved-scale fluxes? \item<3-> Is it really a large-eddy simulation, i.e., are the subgrid-scale fluxes much smaller than the resolved-scale fluxes? \vspace{1em} \item<4-> How do the turbulence spectra of $u$, $v$, $w$ along $x$ and along $y$ look like?\\ \item<3-> Output of vertical profile data generated by the 1D-model is controlled by parameter \texttt{\textcolor{blue}{dt\_pr\_1d}}. It is in ASCII-format and it is written into a separate file. You can include the profiles of the 1D-model, which are used to initialize the 3D-model, in the standard profile data output of the 3D-model (which is controlled by parameter \texttt{\textcolor{blue}{data\_output\_pr}}) by adding a \texttt{'\#'} sign to the respective output quantity, e.g. \texttt{\textcolor{blue}{data\_output\_pr} = '\#u'}. \vspace{0.5em} \item<3-> For the 1D-model, please set \texttt{\textcolor{blue}{mixing\_length\_1d} = 'blackadar'} and \texttt{\textcolor{blue}{dissipation\_1d} = 'detering'} in order to get a correct mean boundary layer wind profile. The default settings of these parameters would switch the turbulence parameterization of the 1D-model to the SGS-parameterization of the 3D-LES-model, which represents only the SGS-parts of turbulence. However, for this exercise the 1D-model has to parameterize all scales of turbulence (i.e. it should be used as a RANS-model). \item<3-> For the 1D-model, please set \texttt{\textcolor{blue}{mixing\_length\_1d} = 'blackadar'} and \texttt{\textcolor{blue}{dissipation\_1d} = 'detering'} in order to get a correct mean boundary layer wind profile. The default settings of these parameters would switch the turbulence parameterization of the 1D-model to the SGS-parameterization of the 3D-LES-model, which represents only the SGS-parts of turbulence. However, for this exercise the 1D-model has to parameterize all scales of turbulence (i.e., it should be used as a RANS-model). \end{itemize} % Folie 7 \begin{frame} \frametitle{Time series of TKE} \begin{center} \includegraphics[width=1.0\textwidth]{exercise_neutral_figures/ts.eps} \frametitle{Time series of TKE, umax and wmax} \begin{center} \includegraphics[width=0.62\textwidth]{exercise_neutral_figures/ts_tke_umax_wmax.eps} \end{center} \end{frame} % Folie 8 \begin{frame} \frametitle{Vertical profiles of $\overline{wu}$, $\overline{wv}$} \begin{center} \includegraphics[width=\textwidth]{exercise_neutral_figures/pr_wu.eps}\\ \frametitle{Vertical profiles of $\overline{w}$, $\overline{wu}$, $\overline{wv}$} \begin{center} \includegraphics[width=1.0\textwidth]{exercise_neutral_figures/pr_w_wu_wv.eps} \end{center} \end{frame} $\overline{wu''}$ and $\overline{wv''}$} \begin{center} \includegraphics[width=0.55\textwidth]{exercise_neutral_figures/pr_wu_sgs.eps}\\ \includegraphics[width=0.55\textwidth]{exercise_neutral_figures/pr_wu_resolved.eps}\\ \includegraphics[width=0.55\textwidth]{exercise_neutral_figures/pr_wu_wv_sgs_res.eps} \end{center} \end{frame} % Folie 10 \begin{frame} \frametitle{Spectra of $u$ and $v$} \includegraphics[angle=90,width=1.0\textwidth]{exercise_neutral_figures/sp_u.eps} \end{frame} \frametitle{Spectra of $u$, $v$ and $w$} \begin{center} \includegraphics[angle=90,width=0.7\textwidth]{exercise_neutral_figures/sp_u_v_w.eps} \end{center} \end{frame} \subsection{Answers} % Folie 11 \begin{frame} \frametitle{Spectra of $w$} \begin{center} \includegraphics[angle=90,width=0.6\textwidth]{exercise_neutral_figures/sp_w.eps} \end{center} \frametitle{Answers to question I} \footnotesize How long do you have to simulate until turbulence / mean flow become stationary? \begin{itemize} \item As can be seen in frame 6, a simulation time of about 48~h should at least be taken to obtain a roughly constant kinetic energy. \item The time series of E shows an oscillation with a period of roughly 14~h. This can be attributed to the inertial oscillation affecting the air parcels due to the Coriolis force. This oscillation is damped with time. \item umax and wmax do not change much in time after the spin-up time of roughly 6~h. \end{itemize} \end{frame} % Folie 12 \begin{frame} \frametitle{Answers to question II} \footnotesize How do the horizontally and temporally averaged vertical velocity and momentum flux profiles look like? \begin{itemize} \item The profiles are shown in frame 7. The horizontally averaged vertical velocity is practically zero as the usage of incompressible equations together with cyclic boundary conditions (horizontal homogeneity) suggest. \item wu an wv decrease with height since friction decelerates the flow at the surface. Due to the turning of the wind vector with height (Ekman spiral), the meridional velocity component is non-zero evoking also a non-zero vertical momentum flux of the v-velocity component. \item The non-convergence of the single profiles can be attributed to the inertial oscillation. \end{itemize} \end{frame} % Folie 13 \begin{frame} \frametitle{Answers to question III} \footnotesize Is it really a large-eddy simulation? \begin{itemize} \item Frame 8 shows sub-grid and resolved momentum flux profiles. \item The simulation is an LES since resolved momentum fluxes are the dominant components to the total flux except for the near vicinity of the surface where the unresolved, sub-grid fluxes dominate. \end{itemize} \end{frame} % Folie 14 \begin{frame} \frametitle{Answers to question IV} \footnotesize Can you identify the inertial subrange? \begin{itemize} \item In PALM, the spectral density is normalized by means of the variance and additionally multiplied by the wave number. Thus, the spectral density appearing on the ordinate of the plots in frame 9 is dimensionless. \item The spectra show a maximum spectral density for small wave numbers. Thus, the largest eddies contain the highest variance (or turbulence kinetic energy, TKE). For higher wave numbers the inertial subrange follows where the spectra follow a -2/3 slope in the plot (indicated by a black line). There, the variance follows the energy cascade where larger eddies break-up into smaller eddies. For the highest wave numbers, the spectra depart from the -2/3 slope indicating that dissipation takes place. \item The spectra also show that the production range is not well developed (very flat maxima). This suggests that the modeling domain might be too small to capture relevant larger scales. \end{itemize} \end{frame}