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\institute{Institut für Meteorologie und Klimatologie, Leibniz Universität Hannover}
\date{last update: \today}
\event{PALM Seminar}
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\title[Fundamentals of Large-Eddy Simulation]{Fundamentals of Large-Eddy Simulation}
\author{Siegfried Raasch}

% Notes:
% jede subsection bekommt einen punkt im menu (vertikal ausgerichtet.
% jeder frame in einer subsection bekommt einen punkt (horizontal ausgerichtet)
\begin{document}
%Folie 1
\begin{frame}
\titlepage
\pdfnote{maronga}{
   Welcome to the PALM Tutorial!\textCR\textCR 
   We have placed many helpful comments throughout the presentations that will hopefully ease your first steps with PALM.\textCR\textCR 
   In case you find it hard to follow at specific points that have not been (or insufficiently) commented, please let us know! We appreciate feedback that helps improving the tutorial.\textCR\textCR 
   Good luck! - The PALM Group at IMUK
}
\end{frame}

\section{The Role of Turbulence}
\subsection{The Role of Turbulence}

% Folie 2
\begin{frame}
   \frametitle{The Role of Turbulence (I)}
   \begin{itemize}
      \item<1->{\textbf{Most flows in nature \& technical applications are turbulent}}
      \item<2->{\textbf{Significance of Turbulence}}
      \begin{itemize}
         \item<2->{\underline{Meteorology / Oceanography:} Transport processes of momentum, heat, water vapor as well as other scalars}
         \item<2->{\underline{Health care:} Air pollution}
         \item<2->{\underline{Aviation, Engineering:} Wind impact on buildings, power output of windfarms}
      \end{itemize}
      \item<3->{\textbf{Characteristics of turbulence}}
      \begin{itemize}
         \item<3->{non-periodical, 3D stochastic movements}
         \item<3->{mixes air and its properties on scales between large-scale advection and molecular diffusion}
         \item<3->{non-linear $\rightarrow$ energy is distributed smoothly with wavelength}
         \item<3->{wide range of spatial and temporal scales}
      \end{itemize}
   \end{itemize}
\end{frame}

% Folie 3
\begin{frame}
   \frametitle{The Role of Turbulence (II)}
   \begin{columns}[c]
   \column{0.5\textwidth}
      \scriptsize
      \begin{itemize}
         \item<2->{\textbf{Large eddies:} $\unit[10^3]{m}$ ($L$), $\unit[1]{h}$ \\
                   \textbf{Small eddies:} $\unit[10^{-3}]{m}$ ($\eta$), \unit[0.1]{s}}
         \item<3->{\textbf{Energy production and dissipation on different scales}}
         \begin{itemize}
            \item<3->{\begin{scriptsize} Large scales: shear and buoyant production \end{scriptsize}}
            \item<3->{\begin{scriptsize} Small scales: viscous dissipation \end{scriptsize}}
         \end{itemize}
         \item<4->{\textbf{Large eddies contain most energy}}
         \item<5->{\textbf{Energy-cascade} \\ 
                   Large eddies are broken up by instabilities and their energy is handled down to smaller scales.}
      \end{itemize}
      \normalsize
   \column{0.5\textwidth}
      \onslide<3->{
         \includegraphics[width=\textwidth, height=0.9\textheight]{fundamentals_of_les_figures/Role_of_Turbulence_2.png}}
   \end{columns}
\end{frame}

\section{The Reynolds Number}
\subsection{The Reynolds Number}

% Folie 4
\begin{frame}
   \frametitle{The Reynolds Number (Re)}
   \begin{columns}[c]
   \column{0.6\textwidth}
      \onslide<1->{
         $\frac{L}{\eta} \approx Re^{3/4} \approx 10^6$ \quad \begin{small} (in the atmosphere) \end{small}}
      \par\bigskip
      \onslide<2->{
         $Re = \frac{\left| \textbf{u} \cdot \nabla \textbf{u} \right|}{\left| \nu \nabla^2 \textbf{u} \right|} \hat{=} \frac{LU}{\nu} \qquad \frac{\textnormal{inertia forces}}{\textnormal{viscous forces}} $}
   \column{0.4\textwidth}
      \footnotesize
      \onslide<1->{
         \textbf{u} 3D wind vector

         $\nu$ kinematic molecular viscosity 

         $L$ outer scale of turbulence

         $U$ characteristic velocity scale

         $\eta$ inner scale of turbulence
            \begin{scriptsize}(Kolmogorov dissipation length) \end{scriptsize} }             
   \end{columns}
   \normalsize
   \par\bigskip
   \par\bigskip
   \onslide<3->{
      $ \Rightarrow $ \underline{Number of gridpoints for a 3D simulation:}
   \par\bigskip
   $ \left( \frac{L}{\eta} \right)^3 \approx Re^{9/4} \approx 10^{18}$ (in the atmosphere)}
\end{frame}

\section{Classes of Turbulence Models}
\subsection{Classes of Turbulence Models}

% Folie 5
\begin{frame}
   \frametitle{Classes of Turbulence Models (I)}
   \begin{itemize}
      \item{\textbf{Direct numerical Simulation (DNS)}}
      \begin{itemize}
         \item<2->{\textbf{Most straight-forward approach:}}
         \begin{itemize}
            \item<2->{Resolve all scales of turbulent flow explicitly.}
         \end{itemize}
         \item<3->{\textbf{Advantage:}}
         \begin{itemize}
            \item<3->{(In principle) a very accurate turbulence representation.}
         \end{itemize}
         \item<4->{\textbf{Problem:}}
         \begin{itemize}
            \item<4->{Limited computer resources  (1996: $\sim$ $10^8$, today: $\sim$ $10^{11}$ gridpoints, 
                      but $\sim$ $10^{18}$ gridpoints needed, see prior slide).}
            \item<4->{$\unit[1]{h}$ simulation of $10^9$ ($2048^3$) gridpoints on $512$ processors of the HLRN supercomputer needs $\unit[10]{h}$ CPU time.}
         \end{itemize}
         \item<5->{\textbf{Consequences:}}
         \begin{itemize}
            \item<5->{DNS is restricted to moderately turbulent flows (low Reynolds-number flows).}
            \item<5->{Highly turbulent atmospheric turbulent flows cannot be simulated.}
         \end{itemize}
      \end{itemize}
   \end{itemize}
\end{frame}

% Folie 6
\begin{frame}
   \frametitle{Classes of Turbulence Models (II)}
   \begin{itemize}
      \item{\textbf{Reynolds averaged (Navier-Stokes) simulation (RANS)}}
      \begin{itemize}
         \item<2->{\textbf{Opposite strategy:}}
         \begin{itemize}
            \item<2->{Applications that only require average statistics of the flow (i.e. the mean flow).}
            \item<2->{Integrate merely the ensemble-averaged equations.}
            \item<2->{Parameterize turbulence over the whole eddy spectrum.}
         \end{itemize}
         \item<3->{\textbf{Advantage:}}
         \begin{itemize}
            \item<3->{Computationally inexpensive, fast.}
         \end{itemize}
         \item<4->{\textbf{Problem:}}
         \begin{itemize}
            \item<4->{Turbulent fluctuations not explicitly captured.}
            \item<4->{Parameterizations are very sensitive to large-eddy structure that depends on 
                      environmental conditions such as geometry and stratification $\rightarrow$     
                      Parameterizations are not valid for a wide range of different flows.}
         \end{itemize}
         \item<5->{\textbf{Consequence:}}
         \begin{itemize}
            \item<5->{Not suitable for detailed turbulence studies.}
         \end{itemize}
      \end{itemize}
   \end{itemize}
\end{frame}

% Folie 7
\begin{frame}
   \frametitle{Classes of Turbulence Models (III)} 
   \begin{itemize}
      \item{\textbf{Large eddy simulation (LES)}}
      \begin{itemize}
         \item<2->{Seeks to combine advantages and avoid disadvantages of DNS and RANS by \underline{treating 
                   large scales and small scales separately}, based on Kolmogorov's (1941) similarity theory of turbulence.}
         \item<3->{Large eddies are explicitly resolved.}
         \item<4->{The impact of small eddies on the large-scale flow is parameterized.}
         \item<5->{Advantages:}
         \begin{itemize}
            \item<5->{Highly turbulent flows can be simulated.}
            \item<5->{Local homogeneity and isotropy at large \textit{Re} (Kolmogorov's $1^\mathrm{st}$ hypothesis) leaves 
                      parameterizations uniformly valid for a wide range of different flows.}
         \end{itemize}
      \end{itemize}
   \end{itemize}
\end{frame}

\section{Concept of LES}
\subsection{Concept of LES}

 % Folie 8
 \begin{frame}
    \frametitle{Concept of Large Eddy Simulation (I)}
    \begin{columns}
       \column{0.55\textwidth}
          \begin{itemize}
             \item<1->{\textbf{Filtering}}
             \begin{footnotesize}
                \begin{itemize}
                   \item<2->{Spectral cut at wavelength $\Delta x$.}
                   \item<3->{Structures larger than $\Delta x$ are explicitly calculated (resolved scales).}
                   \item<4->{Structures smaller than $\Delta x$ must be filtered out (subgrid scales), formally known as low-pass filtering.}
                   \item<5->{Like for Reynolds averaging: split variables in mean part and fluctuation, spatially average the model equations, e.g.:}
                \end{itemize}
             \end{footnotesize}
             \onslide<6->{\begin{center} $w = \overline{w} + w', \theta = \overline{\theta} + \theta'$ \end{center}}
          \end{itemize}
       \column{0.45\textwidth}
          \includegraphics[width=\textwidth]{fundamentals_of_les_figures/Concept_of_LES.png}
    \end{columns}
 \end{frame}

% Folie 9
\begin{frame}
   \frametitle{Concept of Large Eddy Simulation (II)}
   \begin{itemize}
      \item<1->{\textbf{Parameterization}}   
      \begin{footnotesize}
      \begin{itemize}
         \item<2->{The filter procedure removes the small scales from the model equations, but it produces new unknowns, mainly averages of fluctuation products.}
         \begin{itemize} 
            \item<2->{eg. $\overline{w'\theta'}$}
         \end{itemize}
         \item<3->{These unknowns describe the effect of the unresolved, small scales on the resolved, large scales; therefore it is important to include them in the model.}
         \item<4->{We do not have information about the variables (e.g., vertical wind component and potential temperature) on these small scales of their fluctuations.}
         \item<5->{Therefore, these unknowns have to be parameterized using information from the resolved scales.}
         \begin{itemize}
            \item<5->{A typical example is the flux-gradient relationship, e.g.,}
         \end{itemize}
      \end{itemize}
      \end{footnotesize}
   \end{itemize}
   \onslide<5->{
      \begin{center}
      $ \overline{w'\theta'} = - \nu_\mathrm{h} \cdot \frac{\partial \overline{\theta}}{\partial z} $ 
      \end{center}}
\end{frame}

\end{document}