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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 observation 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}
   \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: characterictic 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{characterictic 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}


\end{document}