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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[SGS Models]{SGS Models}
\author{Siegfried Raasch}

\begin{document}

% Folie 1
\begin{frame}
\titlepage
\end{frame}


\section{SGS Models}
\subsection{SGS Models}

% Folie 2
\begin{frame}
   \frametitle{SGS Models (I)}
   \small
   \begin{itemize}
      \item<2->The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence).
      \item<3->Features of small-scale turbulence: local, isotropic, dissipative (inertial subrange)
      \item<4->SGS stresses should depend on:
      \begin{itemize}
         \item local resolved-scale field \hspace{3mm} and / or
         \item past history of the local fluid (via a PDE)
      \end{itemize}
      \item<5->Importance of the model depends on how much energy is contained in the subgrid-scales:
      \begin{itemize}
         \item $E_{SGS} / E < 50\%$: results relatively insensitive to the model, (but sensitive to the numerics, e.g. in case of upwind scheme)
         \item $E_{SGS} / E = 1$: model more important 
         \item<6->\textbf{If the large-scale eddies are not resolved, the SGS model and the LES will fail at all!} 
      \end{itemize}
   \end{itemize}
\end{frame}

% Folie 3
\begin{frame}
   \frametitle{SGS Models (II)}
   Requirements that a good SGS model must fulfill:
   \begin{footnotesize}
      \begin{itemize}
         \item<2-> Represent interactions with small scales.
         \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).
         \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.
         \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.
         \item<6->\underline{The primary goal of an SGS model is to obtain correct statistics of the}\\ 
         \underline{energy containing scales of motion.}
      \end{itemize}
   \end{footnotesize}
\end{frame}

% Folie 4
\begin{frame}
   \frametitle{SGS Models (III)}
   \onslide<1-> All the above observations suggest the use of an eddy viscosity type SGS model:
   \begin{footnotesize}
      \begin{itemize}
         \item<2-> Take idea from RANS modeling, introduce eddy viscosity $\nu_T$:
         \begin{flalign*}
            &\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)\\
            & & \text{filtered strain rate tensor}
         \end{flalign*}
      \end{itemize}
   \end{footnotesize}
   \vspace{-0.3cm}
   \onslide<3->Now we need a model for the eddy viscosity:
   \begin{footnotesize}
      \begin{itemize}
         \item<4-> Dimensionality of $\nu_T$ is $l^2/t$
         \item<5-> Obvious choice: $\nu_T = Cql$ \hspace{5mm} (q, l: characteristic velocity / length scale)
         \item<6-> Turbulence length scale is easy to define: largest size of the unresolved scales is $\Delta$ \hspace{10mm} $l = \Delta$
         \item<7-> Velocity scale not obvious (smallest resolved scales, their size is of the order of the variation of velocity over one grid element)
         \begin{flalign*}
            &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} \\
            & & \text{characteristic filtered rate of strain}\hspace{15mm}
         \end{flalign*}
      \end{itemize}
   \end{footnotesize}
\end{frame}


\section{Smagorinsky Model}
\subsection{The Smagorinsky Model}

% Folie 5
\begin{frame}
   \frametitle{The Smagorinsky Model}
   \onslide<2->Combine previous expressions to obtain:
   \begin{equation*}
      \nu_T = C \Delta^2 \overline{S} = (C_S \Delta)^2 \overline{S}
   \end{equation*}
   \onslide<3-> Model due to Smagorinsky (1963):
   \begin{itemize}
      \item<3-> Originally designed at NCAR for global weather modeling.
      \item<4-> Can be derived in several ways: heuristically (above), from inertial range arguments (Lilly), from turbulence theories.
      \item<5-> Constant predicted by all methods (based on theory, decay of isotropic turbulence): $C_S = \sqrt{C} \approx 0.2$
   \end{itemize}
\end{frame}

% Folie 6
\begin{frame}
   \frametitle{The Smagorinsky Model: Performance}
   \begin{itemize}
      \item<2-> Predicts many flows reasonably well
      \item<3-> Problems:
      \begin{itemize}
         \item<3-> Optimum parameter value varies with flow type:
         \begin{itemize}
            \item Isotropic turbulence: $C_S \approx 0.2$\\
            \item Shear (channel) flows: $C_S \approx 0.065$
         \end{itemize}
         \item<4-> Length scale uncertain with anisotropic filter:
         \begin{equation*}
            (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3
         \end{equation*}
         \item<5-> Needs modification to account for:
         \begin{itemize}
            \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\
            \item near-wall region: $C_S = F(z+)$, $z+$: distance from wall
         \end{itemize}
      \end{itemize}
   \end{itemize}
\end{frame}

\section{Deardoff Modification}
\subsection{Deardoff Modification}

% Folie 7
\begin{frame}
   \frametitle{Deardorff (1980) Modification (Used in PALM) (I)}
   \footnotesize
   \onslide<1->{
      $ \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}}
   \small
   \begin{itemize}
      \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
                              local fluid.}
   \end{itemize} 
   \small
   \onslide<3->{
      $ C_M = const. = 0.1 $
      \par\bigskip
      \scriptsize
      $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\
                          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} 
                  \end{cases} $      
      \small
      \par\bigskip
      $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ }
\end{frame}

% Folie 8
\begin{frame}
   \frametitle{Deardorff (1980) Modification (Used in PALM) (II)}
   \begin{itemize}
      \item{SGS-TKE from prognostic equation}
   \end{itemize}
   $ \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 $                                         
   \par\bigskip     
   $ \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}}$
   \par\bigskip   
   $ 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)$
   \par\bigskip    
   $W_{k}=\overline{u_k'q'} = -K_{h}\frac{\partial\bar{q}}{\partial x_{k}}$     
            \par\bigskip 
   $ \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} $
   \par\bigskip
   $ \nu_e = 2 \nu_T $
   \par\bigskip
   $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $
\end{frame}

% Folie 9
\begin{frame}
   \frametitle{Deardorff (1980) Modification (Used in PALM) (III)}
   \begin{itemize}
      \item{There are still problems with this parameterization:}
      \begin{itemize}
         \item[-]<2->{The model overestimates the velocity shear near the wall.}
         \item[-]<3->{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.}
         \item[-]<4->{Backscatter of energy from the SGS-turbulence to the resolved-scale flow can not be considered.}
      \end{itemize}
      \item<5->{Several other SGS models have been developed:}
      \begin{itemize}
         \item[-]<5->{Two part eddy viscosity model (Sullivan, et al.)}
         \item[-]<6->{Scale similarity model (Bardina et al.)}
         \item[-]<7->{Backscatter model (Mason)}
      \end{itemize}
      \item<8->{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent
               from the different models (Beare et al., 2006, BLM).}
   \end{itemize} 
\end{frame}


\section{Summary / Important Points for Beginners}
\subsection{Summary / Important Points for Beginners}

% Folie 10
\begin{frame}
   \frametitle{Summary / Important Points for Beginners (I)}
   \begin{columns}[c]
   \column[T]{0.4\textwidth} 
      \includegraphics<2-7|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png}   
      \includegraphics<8|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png}
      \includegraphics<9|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png}
      \includegraphics<10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png}
      \onslide<8-10>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}}              
   \column[T]{0.2\textwidth}
      \vspace{0.9cm}
      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
      \par
      \onslide<8-|handout:1>{\begin{small} fluctuations (\textbf{u},c) \end{small}}
      \par\bigskip
      \thicklines
      \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}}}
      \vspace{1cm}
      
      \includegraphics<8-10|handout:1>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png}
      \par
      \onslide<8->{\begin{small} smooth result \end{small}}   
   \column[T]{0.4\textwidth}     
      \includegraphics<1-2|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png}
      \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png}
      \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png}
      \includegraphics<5-10|handout:1>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png}
      \vspace{1.3cm}
      \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png}
      \uncover<7-|handout:1>{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}}       
   \end{columns}
\end{frame}

% Folie 11
\begin{frame}
   \frametitle{Summary / Important Points for Beginners (II)}
    For an LES it always has to be guaranteed that the main energy containing eddies of the respective 
    turbulent flow can really be simulated by the numerical model:      
    \begin{itemize}
       \item<2->{The grid spacing has to be fine enough.}
       \item<3->{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $}
       \item<4->{The inflow/outflow boundaries must not effect the flow turbulence \\
                (therefore cyclic boundary conditions are used in most cases).}
       \item<5->{In case of homogeneous initial and boundary conditions, the onset of turbulence 
                  has to be triggered by imposing random fluctuations.}
       \item<6->{Simulations have to be run for a long time to reach a stationary state and stable statistics.}
    \end{itemize}      
\end{frame}


\section{Example Output}
\subsection{Example Output}

% Folie 12
\begin{frame}
   \frametitle{Example Output (I)}
   \begin{itemize}
      \item{LES of a convective boundary layer}
   \end{itemize}
   \includegraphics<1|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png}
   \includegraphics<2|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png}
   \includegraphics<3|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png}
   \includegraphics<4|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png}
   \includegraphics<5|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png}
   \includegraphics<6|handout:0>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png}
   \includegraphics<7|handout:1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png}
\end{frame}

% Folie 13
\begin{frame}
   \frametitle{Example Output (II)}
   \begin{itemize}
      \item{LES of a convective boundary layer}
   \end{itemize}
   \begin{center}
      \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png}
      power spectrum of vertical velocity
   \end{center}
\end{frame}

% Folie 14
\begin{frame}
   \frametitle{Some Symbols}
   \begin{columns}[c]
      \column{0.6\textwidth}
      \begin{tabbing}
      $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\
      $u,v,w$ \\ 

      \\
      
      $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\
      $x,y,z$ \\

      \\

      $\Theta$ \> potential temperature \\ \\

      $\Psi$ \> passive scalar \\ \\

      $T$ \> actual Temperatur \\ \\
      \end{tabbing}
   \column{0.4\textwidth}
      \begin{tabbing}
      $\Phi = gz$  \quad \= geopotential \\ \\

      $p$ \> pressure \\ \\

      $\rho$ \> density \\ \\

      $f_i$ \> Coriolis Parameter \\ \\

      $\epsilon_{ijk}$ \> alternating symbol \\ \\

      $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\

      $Q, Q_\Psi$ \> sources or sinks \\ \\
      \end{tabbing}
   \end{columns}
\end{frame}
\end{document}