Changeset 945 for palm/trunk/TUTORIAL/SOURCE/basic_equations.tex

Timestamp:

Jul 17, 2012 3:43:01 PM (12 years ago)

Author:

maronga

Message:

added/updated several tutorial files

File:

• palm/trunk/TUTORIAL/SOURCE/basic_equations.tex (modified) (7 diffs)

palm/trunk/TUTORIAL/SOURCE/basic_equations.tex

@@ -4 +4 @@
 
 \usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
+\usepackage{ngerman}
 \usepackage{pgf}
 \usetheme{Dresden}
@@ -78 +78 @@
       \item \onslide<5->Continuity equation
       \begin{equation*}
-         \frac{\partial p}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} 
+         \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} 
       \end{equation*}
    \end{itemize}
@@ -86 +86 @@
 \begin{frame}
    \frametitle{Boussinesq Approximation}
-   \small
+   \footnotesize
    \begin{itemize}
       \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ 
-      \begin{equation*}
-         p(x,y,z,t) = p_0(x,y,z,t) + p^{*}(x,y,z,t)\hspace{22mm}
-      \end{equation*}
-      \begin{equation*}
-         \rho(x,y,z,t) = \rho_0(x,y,z,t) + \rho^{*}(x,y,z,t) ; 
-         \hspace{5mm} \psi^{*} << \psi_0
-      \end{equation*} 
+      \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*}
@@ -104 +103 @@
       \item \onslide<4->Equation of state
       \begin{equation*}
-         p_0 = \rho_0 R T_0 \rightarrow \ln{p_0} = \ln{\rho_0} + \ln{R} + \ln{T_0} \rightarrow \frac{\partial p_0}{p_0} = \frac{\partial \rho_0}{\rho_0} + \frac{\partial T_0}{T_0} 
-      \end{equation*}
-      \begin{equation*}
-         \frac{\Delta p_0}{p_0} \approx \frac{\Delta \rho_0}{\rho_0} + 
-         \frac{\Delta T_0}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx 
+         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}
@@ -339 +338 @@
       \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.
+      \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}
@@ -361 +360 @@
       \onslide<5->
       \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}&
+         &\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-> 
@@ -396 +395 @@
          \end{tiny}
          $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\
-         $H_{k} = \overline{u_k \theta_k} - \overline{u_k}\,\overline{\theta_i}$\\
-         $W_{k} = \overline{u_k q_k} - \overline{u_k}\,\overline{q_k}$
+         $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\
+         $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$
          };
       \end{tikzpicture}