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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[Basic Equations]{Basic Equations}
\author{Siegfried Raasch}

\begin{document}

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


\section{Basic equations}
\subsection{Basic equations, Unfiltered}

% Folie 2
\begin{frame}
   \frametitle{Basic equations, Unfiltered}
   \begin{itemize}
       \item<2->Navier-Stokes equations
      \begin{equation*}
         \rho \frac{\partial u_i}{\partial t} + \rho u_k 
         \frac{\partial u_i}{\partial x_k} = 
         - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk} 
         f_j u_k - \frac{\partial \phi}{\partial x_i} + \mu 
         \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3} 
         \frac{\partial}{\partial x_i} \left( 
         \frac{\partial u_k}{\partial x_k} \right) \right\}
      \end{equation*}
      \item \onslide<3->First principle
      \begin{equation*}
         \rho \frac{\partial T}{\partial t} + \rho u_k \frac{\partial T}{\partial x_k} = \mu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
      \end{equation*}
      \item \onslide<4->Equation for passive scalar
      \begin{equation*}
         \rho \frac{\partial \psi}{\partial t} + \rho u_k \frac{\partial \psi}{\partial x_k} = \mu_{\psi} \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
      \end{equation*}
      \item \onslide<5->Continuity equation
      \begin{equation*}
         \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} 
      \end{equation*}
   \end{itemize}
\end{frame}

% Folie 3
\begin{frame}
   \frametitle{Boussinesq Approximation}
   \footnotesize
   \begin{itemize}
      \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ 
      \begin{align*}
         T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\
         p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\
         \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& &
         &\psi^{*} << \psi_0&
      \end{align*} 
      \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq)
      \begin{equation*}
         \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm} 
         \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g}, 
         \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g}
      \end{equation*}
      \item \onslide<4->Equation of state
      \begin{equation*}
         p = \rho R T \rightarrow \ln{p} = \ln{\rho} + \ln{R} + \ln{T} \rightarrow \frac{d p}{p} = \frac{d \rho}{\rho} + \frac{d T}{T} 
      \end{equation*}
      \begin{equation*}
         \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} + 
         \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx 
         \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm} 
         \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm}
      \end{equation*}
   \end{itemize}
\end{frame}

% Folie 4
\begin{frame}
   \frametitle{Continuity Equation}
   \begin{eqnarray*}
      \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} = 
      - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} & 
      \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0 
      \hspace{5mm} & \text{anelastic approximation}\\
      \\
      \onslide<3-> \rho_0 = const. & \hspace{10mm} 
      \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} & 
      \text{incompressible flow}
   \end{eqnarray*}
\end{frame}

% Folie 5
\begin{frame}
   \frametitle{Boussinesq Approximated Equations}
   \begin{itemize}
      \item \onslide<2->Navier-Stokes equations
      \begin{equation*}
         \frac{\partial u_i}{\partial t} 
         + \frac{\partial u_k u_i}{\partial x_k} =  
         - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i} 
         - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}} 
         + g \frac{T - T_0}{T_0} \delta_{i3} + \nu 
         \frac{\partial^2 u_i}{\partial x_k^2}
      \end{equation*}
      \item \onslide<3->First principle
      \begin{equation*}
         \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} = 
         \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
      \end{equation*}
      \item \onslide<4->Equation for passive scalar
      \begin{equation*}
         \frac{\partial \psi}{\partial t} + u_k 
         \frac{\partial \psi}{\partial x_k} = \nu_{\psi} 
         \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
      \end{equation*}
      \item \onslide<5->Continuity equation
      \begin{equation*}
         \frac{\partial u_k}{\partial x_k} = 0
      \end{equation*}
   \end{itemize}
   \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]

   \begin{tikzpicture}[remember picture, overlay]
      \node at (current page.north west){%
      \begin{tikzpicture}[overlay]
         \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!};
      \end{tikzpicture}
      };
   \end{tikzpicture}
\end{frame}


\section{Scale Separation}
\subsection{Scale Separation by Spatial Filtering}

% Folie 6
\begin{frame}
   \frametitle{LES - Scale Separation by Spatial Filtering (I)}
   \footnotesize
   \begin{itemize}
      \item<1->{LES technique is based on scale separation, in order to reduce the number of degrees of freedom of the solution. \begin{math} \boxed{\Psi(x_i , t) = \overline{\Psi}(x_i , t) + \Psi'(x_i , t)} \end{math}}
      \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).}
      \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).}
      \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.}
   \end{itemize}
   \normalsize
   \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png}
\end{frame}



% Folie 7
\begin{frame}
   \frametitle{LES - Scale Separation by Spatial Filtering (II)}
   \begin{columns}[T]
      \begin{column}{0.8\textwidth}
      \footnotesize
      \begin{itemize}
         \item<1->The Filter applied is a spatial filter:
         \begin{equation*} 
            \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i' 
         \end{equation*}
         \begin{equation*} 
            \overline{\Psi}'(x_i) = 0 \qquad but \qquad \overline{\overline{\Psi}} \neq \overline{\Psi}(x_i) 
         \end{equation*}  
         \item<2->Filter applied to the nonlinear advection term:
         \begin{equation*} 
            \overline{u_k u_i} = \overline{(\overline{u_k} + u_k')(\overline{u_i} + u_i')} = \overline{\overline{u_k}\,\overline{u_i}} + \underbrace{\overline{\overline{u_k}          u_i'} + \overline{u_k' \overline{u_i}}}_{C_{ki}} + \underbrace{\overline{u_k' u_i'}}_{R_{ki}} 
         \end{equation*}   
         \item<5->Leonard proposes a further decomposition:
         \begin{equation*} 
            \overline{\overline{u_k}\,\overline{u_i}} = \overline{u_k}\,\overline{u_i} + \underbrace{\left( \overline{\overline{u_k}\,\overline{u_i}} - \overline{u_k}\,\overline{u_i} \right)}_{L_{ki}}                         
         \end{equation*}
         \begin{equation*} 
            \overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + L_{ki} + C_{ki} + R_{ki} = \overline{u_k}\,\overline{u_i} + \tau_{ki} 
         \end{equation*}       
      \end{itemize}
      \end{column}
      \begin{column}{0.32\textwidth}
      \vspace{45mm}     
      \begin{footnotesize}
         \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\
         $C_{ki}$: \textbf{cross-stress} \\
         $L_{ki}$: \textbf{Leonard-stress} \\
         $\tau_{ki}$: \textbf{total stress-tensor}
      \end{footnotesize}
      \end{column}
   \end{columns}
   \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
      \begin{tikzpicture}[remember picture, overlay]
      \node at (current page.north west){
      \begin{tikzpicture}[overlay]
         \node[plain, draw,anchor=west] at (94mm,-30mm) {
         \begin{footnotesize}
            \noindent \textbf{Ensemble average:} \\
         \end{footnotesize}
         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
         \vspace{5mm}
         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
         };
      \end{tikzpicture}
      };
   \end{tikzpicture}   
\end{frame}

% Folie 8
\begin{frame}
   \frametitle{LES - Scale Separation by Spatial Filtering (III)}
   \small
   \begin{itemize}
      \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator
      \begin{flalign*}
         &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\
         &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\
         &V=\left[ x - \frac{\Delta x}{2}, x + \frac{\Delta x}{2} \right] \times \left[ y - \frac{\Delta y}{2}, y + \frac{\Delta y}{2} \right] \times \left[ z - \frac{\Delta z}{2}, z + \frac{\Delta z}{2} \right]
      \end{flalign*}
      \item<3-> Filter applied to the nonlinear advection term:
      \begin{equation*}
         \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}=
         \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i}
      \end{equation*}
   \end{itemize}
   \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
      \begin{tikzpicture}[remember picture, overlay]
      \node at (current page.north west){
      \begin{tikzpicture}[overlay]
         \node[plain, draw,anchor=west] at (94mm,-30mm) {
         \begin{footnotesize}
            \noindent \textbf{Ensemble average:} \\
         \end{footnotesize}
         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
         \vspace{5mm}
         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
         };
      \end{tikzpicture}
      };
   \end{tikzpicture}
 \end{frame}


\section{Filtered equations}
\subsection{The Filtered Equations}

% Folie 9
\begin{frame}
   \frametitle{The Filtered Equations}
   \onslide<2->
   \begin{equation*}
      \frac{\partial \overline{u_i}}{\partial t} 
      + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} = 
      - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i} 
      - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} 
      + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} 
      + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} 
      - \frac{\partial \tau_{ki}}{\partial x_k}
   \end{equation*}

   \begin{footnotesize}
      \begin{itemize}
         \item<3->The previous derivation completely ignores the existance of the computational grid.
         \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$.
         \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$
         \item<6-> Two possibilities:\\
         1. Pre-filtering technique\\
         ($\Delta x < \Delta$,  explicit filtering)\\
         2. Linking the analytical filter\\
         to the computational grid\\
         ($\Delta x = \Delta$, implicit filtering)
      \end{itemize}
   \end{footnotesize}

   \begin{picture}(0.0,0.0) 
      \put(140,13){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}}
   \end{picture}
\end{frame}

%% Folie 10
\begin{frame}
  \frametitle{Explicit Versus Implicit Filtering}
   \begin{itemize}
      \item<2-> Explicit filtering:
      \begin{itemize}
         \small
         \item<2-> Requires that the analytical filter is applied explicitly.
         \item<3-> Rarely used in practice, due to additional computational costs.
      \end{itemize}
      \item<4-> Implicit filtering:
      \begin{itemize}
         \small
         \item<4-> The analytical cutoff length is associated with the grid spacing.
         \item<5-> This method does not require the use of an analytical filter.
         \item<6-> The filter characteristic cannot really be controlled.
         \item<7-> Because of its simplicity, this method is used by nearly all LES models.
      \end{itemize}
   \end{itemize}

   \onslide<8->
   \begin{scriptsize}
      \textbf{Literature:}\\
      \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\
      \textbf{Schumann, U., 1975:} Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., \textbf{18}, 376-404.\\
   \end{scriptsize}
\end{frame}

% Folie 11
\begin{frame}
   \frametitle{The Final Set of Equations (PALM)}
   \footnotesize
   \begin{itemize}
      \item<2-> Navier-Stokes equations:
      \onslide<2->
      \begin{flalign*}
         &\frac{\partial \overline{u_i}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} - \frac{1}{\rho_0} \frac{\partial \overline{\pi}^*}{\partial x_i} - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} - \frac{\partial \tau_{ki}^r}{\partial x_k}&
      \end{flalign*}
      \item<4-> First principle (using potential\\ temperature):
      \onslide<4->
      \begin{flalign*}
         &\frac{\partial \overline{\theta}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{\theta}}{\partial x_k} - \frac{\partial H_k}{\partial x_k} + Q_{\theta}&
      \end{flalign*}
      \item<5-> Equation for specific humidity\\ (passive scalar)
      \onslide<5->
      \begin{flalign*}
         &\frac{\partial \overline{q}}{\partial t} = - \frac{\partial \overline{u_k}\,\overline{q}}{\partial x_k} - \frac{\partial W_k}{\partial x_k} + Q_{w}&
      \end{flalign*}
      \item<6-> 
      Continuity equation
      \onslide<6->
      \begin{flalign*}
         &\frac{\partial \overline{u_k}}{\partial x_k} = 0&
      \end{flalign*}
   \end{itemize}

   \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
   \begin{tikzpicture}[remember picture, overlay]
      \node at (current page.north west){
      \begin{tikzpicture}[overlay]
         \node[plain, draw,anchor=west] at (75mm,-45mm) {
         \begin{tiny}
            \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\
         \end{tiny}
         $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\
         \vspace{1mm}
         $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$
         };
      \end{tikzpicture}
      };
   \end{tikzpicture}

   \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
   \begin{tikzpicture}[remember picture, overlay]
      \node at (current page.north west){
      \begin{tikzpicture}[overlay]
         \node[plain, draw,anchor=west] at (75mm,-70mm) {
         \begin{tiny}
            \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\
         \end{tiny}
         $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\
         $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\
         $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$
         };
      \end{tikzpicture}
      };
   \end{tikzpicture}
\end{frame}


\end{document}
