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1% $Id: basic_equations.tex 1413 2014-05-28 12:21:01Z boeske$
4
5\usepackage[utf8]{inputenc}
6\usepackage{ngerman}
7\usepackage{pgf}
8\usetheme{Dresden}
9\usepackage{subfigure}
10\usepackage{units}
11\usepackage{multimedia}
12\usepackage{hyperref}
13\newcommand{\event}[1]{\newcommand{\eventname}{#1}}
14\usepackage{xmpmulti}
15\usepackage{tikz}
16\usetikzlibrary{shapes,arrows,positioning}
17\def\Tiny{\fontsize{4pt}{4pt}\selectfont}
18\usepackage{amsmath}
19\usepackage{amssymb}
20\usepackage{multicol}
21
22
23\institute{Institut fÃŒr Meteorologie und Klimatologie, Leibniz UniversitÃ€t Hannover}
24\date{last update: \today}
25\event{PALM Seminar}
27
28\setbeamertemplate{footline}
29  {
30    \begin{beamercolorbox}[rightskip=-0.1cm]&
31     {\includegraphics[height=0.65cm]{imuk_logo.pdf}\hfill \includegraphics[height=0.65cm]{luh_logo.pdf}}
32    \end{beamercolorbox}
33    \begin{beamercolorbox}[ht=2.5ex,dp=1.125ex,
36    \end{beamercolorbox}
37    \begin{beamercolorbox}[colsep=1.5pt]{lower separation line foot}
38    \end{beamercolorbox}
39  }
40%\logo{\includegraphics[width=0.3\textwidth]{luhimuk_logo.pdf}}
41
42\title[Basic Equations]{Basic Equations}
43\author{Siegfried Raasch}
44
45\begin{document}
46
47%Folie 1
48\begin{frame}
49   \titlepage
50\end{frame}
51
52
53\section{Basic equations}
54\subsection{Basic equations, Unfiltered}
55
56% Folie 2
57\begin{frame}
58   \frametitle{Basic equations, Unfiltered}
59   \setlength{\leftmargini}{0.3cm}
60   \begin{itemize}
61       \item<2->Navier-Stokes equations
62      \begin{equation*}
63         \rho \frac{\partial u_i}{\partial t} + \rho u_k
64         \frac{\partial u_i}{\partial x_k} =
65         - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk}
66         f_j u_k - \rho \frac{\partial \phi}{\partial x_i} + \mu
67         \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3}
68         \frac{\partial}{\partial x_i} \left(
69         \frac{\partial u_k}{\partial x_k} \right) \right\}
70      \end{equation*}
71      \item \onslide<3->First principle
72      \begin{equation*}
73         \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
74      \end{equation*}
75      \item \onslide<4->Equation for passive scalar
76      \begin{equation*}
77         \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}
78      \end{equation*}
79      \item \onslide<5->Continuity equation
80      \begin{equation*}
81         \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k}
82      \end{equation*}
83   \end{itemize}
84\end{frame}
85
86% Folie 3
87\begin{frame}
88   \frametitle{Boussinesq Approximation}
89   \footnotesize
90   \begin{itemize}
91      \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$
92      \begin{align*}
93         T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\
94         p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\
95         \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& &
96         &\psi^{*} << \psi_0&
97      \end{align*}
98      \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq)
99      \begin{equation*}
100         \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm}
101         \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g},
102         \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g}
103      \end{equation*}
104      \item \onslide<4->Equation of state
105      \begin{equation*}
106         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}
107      \end{equation*}
108      \begin{equation*}
109         \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} +
110         \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx
111         \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm}
112         \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm}
113      \end{equation*}
114   \end{itemize}
115\end{frame}
116
117% Folie 4
118\begin{frame}
119   \frametitle{Continuity Equation}
120   \begin{eqnarray*}
121      \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} =
122      - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} &
123      \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0
124      \hspace{5mm} & \text{anelastic approximation}\\
125      \\
126      \onslide<3-> \rho_0 = const. & \hspace{10mm}
127      \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} &
128      \text{incompressible flow}
129   \end{eqnarray*}
130\end{frame}
131
132% Folie 5
133\begin{frame}
134   \frametitle{Boussinesq Approximated Equations}
135   \begin{itemize}
136      \item \onslide<2->Navier-Stokes equations
137      \begin{equation*}
138         \frac{\partial u_i}{\partial t}
139         + \frac{\partial u_k u_i}{\partial x_k} =
140         - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i}
141         - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}}
142         + g \frac{T - T_0}{T_0} \delta_{i3} + \nu
143         \frac{\partial^2 u_i}{\partial x_k^2}
144      \end{equation*}
145      \item \onslide<3->First principle
146      \begin{equation*}
147         \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} =
148         \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
149      \end{equation*}
150      \item \onslide<4->Equation for passive scalar
151      \begin{equation*}
152         \frac{\partial \psi}{\partial t} + u_k
153         \frac{\partial \psi}{\partial x_k} = \nu_{\psi}
154         \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
155      \end{equation*}
156      \item \onslide<5->Continuity equation
157      \begin{equation*}
158         \frac{\partial u_k}{\partial x_k} = 0
159      \end{equation*}
160   \end{itemize}
161   \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
162
163   \begin{tikzpicture}[remember picture, overlay]
164      \node at (current page.north west){%
165      \begin{tikzpicture}[overlay]
166         \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!};
167      \end{tikzpicture}
168      };
169   \end{tikzpicture}
170\end{frame}
171
172
173\section{Scale Separation}
174\subsection{Scale Separation by Spatial Filtering}
175
176% Folie 6
177\begin{frame}
178   \frametitle{LES - Scale Separation by Spatial Filtering (I)}
179   \footnotesize
180   \begin{itemize}
181      \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}}
182      \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).}
183      \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).}
184      \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.}
185   \end{itemize}
186   \normalsize
187   \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png}
188\end{frame}
189
190
191
192% Folie 7
193\begin{frame}
194   \frametitle{LES - Scale Separation by Spatial Filtering (II)}
195   \begin{columns}[T]
196      \begin{column}{0.8\textwidth}
197      \footnotesize
198      \begin{itemize}
199         \item<1->The Filter applied is a spatial filter:
200         \begin{equation*}
201            \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i'
202         \end{equation*}
203         \begin{equation*}
205         \end{equation*}
206         \item<2->Filter applied to the nonlinear advection term:
207         \begin{equation*}
208            \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}}
209         \end{equation*}
210         \item<5->Leonard proposes a further decomposition:
211         \begin{equation*}
212            \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}}
213         \end{equation*}
214         \begin{equation*}
215            \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}
216         \end{equation*}
217      \end{itemize}
218      \end{column}
219      \begin{column}{0.32\textwidth}
220      \vspace{45mm}
221      \begin{footnotesize}
222         \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\
223         $C_{ki}$: \textbf{cross-stress} \\
224         $L_{ki}$: \textbf{Leonard-stress} \\
225         $\tau_{ki}$: \textbf{total stress-tensor}
226      \end{footnotesize}
227      \end{column}
228   \end{columns}
229   \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
230      \begin{tikzpicture}[remember picture, overlay]
231      \node at (current page.north west){
232      \begin{tikzpicture}[overlay]
233         \node[plain, draw,anchor=west] at (94mm,-30mm) {
234         \begin{footnotesize}
235            \noindent \textbf{Ensemble average:} \\
236         \end{footnotesize}
237         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
238         \vspace{5mm}
239         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
240         };
241      \end{tikzpicture}
242      };
243   \end{tikzpicture}
244\end{frame}
245
246% Folie 8
247\begin{frame}
248   \frametitle{LES - Scale Separation by Spatial Filtering (III)}
249   \small
250   \begin{itemize}
251      \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator
252      \begin{flalign*}
253         &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\
254         &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\
255         &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]
256      \end{flalign*}
257      \item<3-> Filter applied to the nonlinear advection term:
258      \begin{equation*}
259         \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}=
260         \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i}
261      \end{equation*}
262   \end{itemize}
263   \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
264      \begin{tikzpicture}[remember picture, overlay]
265      \node at (current page.north west){
266      \begin{tikzpicture}[overlay]
267         \node[plain, draw,anchor=west] at (94mm,-30mm) {
268         \begin{footnotesize}
269            \noindent \textbf{Ensemble average:} \\
270         \end{footnotesize}
271         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
272         \vspace{5mm}
273         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
274         };
275      \end{tikzpicture}
276      };
277   \end{tikzpicture}
278 \end{frame}
279
280
281\section{Filtered equations}
282\subsection{The Filtered Equations}
283
284% Folie 9
285\begin{frame}
286   \frametitle{The Filtered Equations}
287   \onslide<2->
288   \begin{equation*}
289      \frac{\partial \overline{u_i}}{\partial t}
290      + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} =
291      - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i}
292      - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}}
293      + g \frac{\overline{T}-T_0}{T_0} \delta_{i3}
294      + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2}
295      - \frac{\partial \tau_{ki}}{\partial x_k}
296   \end{equation*}
297
298   \begin{footnotesize}
299      \begin{itemize}
300         \item<3->The previous derivation completely ignores the existance of the computational grid.
301         \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$.
302         \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$
303         \item<6-> Two possibilities:\\
304         1. Pre-filtering technique\\
305         ($\Delta x < \Delta$,  explicit filtering)\\
306         2. Linking the analytical filter\\
307         to the computational grid\\
308         ($\Delta x = \Delta$, implicit filtering)
309      \end{itemize}
310   \end{footnotesize}
311
312   \begin{picture}(0.0,0.0)
313      \put(140,13){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}}
314   \end{picture}
315\end{frame}
316
317%% Folie 10
318\begin{frame}
319  \frametitle{Explicit Versus Implicit Filtering}
320   \begin{itemize}
321      \item<2-> Explicit filtering:
322      \begin{itemize}
323         \small
324         \item<2-> Requires that the analytical filter is applied explicitly.
325         \item<3-> Rarely used in practice, due to additional computational costs.
326      \end{itemize}
327      \item<4-> Implicit filtering:
328      \begin{itemize}
329         \small
330         \item<4-> The analytical cutoff length is associated with the grid spacing.
331         \item<5-> This method does not require the use of an analytical filter.
332         \item<6-> The filter characteristic cannot really be controlled.
333         \item<7-> Because of its simplicity, this method is used by nearly all LES models.
334      \end{itemize}
335   \end{itemize}
336
337   \onslide<8->
338   \begin{scriptsize}
339      \textbf{Literature:}\\
340      \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\
341      \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.\\
342   \end{scriptsize}
343\end{frame}
344
345% Folie 11
346\begin{frame}
347   \frametitle{The Final Set of Equations (PALM)}
348   \footnotesize
349   \begin{itemize}
350      \item<2-> Navier-Stokes equations:
351      \onslide<2->
352      \begin{flalign*}
353         &\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}&
354      \end{flalign*}
355      \item<4-> First principle (using potential\\ temperature):
356      \onslide<4->
357      \begin{flalign*}
358         &\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}&
359      \end{flalign*}
360      \item<5-> Equation for specific humidity\\ (passive scalar)
361      \onslide<5->
362      \begin{flalign*}
363         &\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}&
364      \end{flalign*}
365      \item<6->
366      Continuity equation
367      \onslide<6->
368      \begin{flalign*}
369         &\frac{\partial \overline{u_k}}{\partial x_k} = 0&
370      \end{flalign*}
371   \end{itemize}
372
373   \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
374   \begin{tikzpicture}[remember picture, overlay]
375      \node at (current page.north west){
376      \begin{tikzpicture}[overlay]
377         \node[plain, draw,anchor=west] at (75mm,-45mm) {
378         \begin{tiny}
379            \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\
380         \end{tiny}
381         $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\
382         \vspace{1mm}
383         $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$
384         };
385      \end{tikzpicture}
386      };
387   \end{tikzpicture}
388
389   \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
390   \begin{tikzpicture}[remember picture, overlay]
391      \node at (current page.north west){
392      \begin{tikzpicture}[overlay]
393         \node[plain, draw,anchor=west] at (75mm,-70mm) {
394         \begin{tiny}
395            \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\
396         \end{tiny}
397         $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\
398         $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\
399         $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$
400         };
401      \end{tikzpicture}
402      };
403   \end{tikzpicture}
404\end{frame}
405
406
407\end{document}
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