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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!}
\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}