source: palm/trunk/TUTORIAL/SOURCE/basic_equations.tex @ 1408

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missing rho added to basic equations in tutorial

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1% $Id: basic_equations.tex 1408 2014-05-19 06:26:39Z raasch $
2\input{header_tmp.tex}
3%\input{../header_lectures.tex}
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 fÌr Meteorologie und Klimatologie, Leibniz UniversitÀt Hannover}
24\date{last update: \today}
25\event{PALM Seminar}
26\setbeamertemplate{navigation symbols}{}
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,
34      leftskip=.3cm,rightskip=0.3cm plus1fil]{title in head/foot}
35      {\leavevmode{\usebeamerfont{author in head/foot}\insertshortauthor} \hfill \eventname \hfill \insertframenumber \; / \inserttotalframenumber}
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   \begin{itemize}
60       \item<2->Navier-Stokes equations
61      \begin{equation*}
62         \rho \frac{\partial u_i}{\partial t} + \rho u_k
63         \frac{\partial u_i}{\partial x_k} =
64         - \frac{\partial p}{\partial x_i} - \rho \varepsilon_{ijk} 
65         f_j u_k - \rho \frac{\partial \phi}{\partial x_i} + \mu 
66         \left\{ \frac{\partial^2 u_i}{\partial x_k^2} + \frac{1}{3} 
67         \frac{\partial}{\partial x_i} \left(
68         \frac{\partial u_k}{\partial x_k} \right) \right\}
69      \end{equation*}
70      \item \onslide<3->First principle
71      \begin{equation*}
72         \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
73      \end{equation*}
74      \item \onslide<4->Equation for passive scalar
75      \begin{equation*}
76         \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}
77      \end{equation*}
78      \item \onslide<5->Continuity equation
79      \begin{equation*}
80         \frac{\partial \rho}{\partial t} = - \frac{\partial \rho u_k}{\partial x_k} 
81      \end{equation*}
82   \end{itemize}
83\end{frame}
84
85% Folie 3
86\begin{frame}
87   \frametitle{Boussinesq Approximation}
88   \footnotesize
89   \begin{itemize}
90      \item \onslide<2->Splitting thermodynamic variables into a basic state $\psi_0$ and a variation $\psi^{*}$ 
91      \begin{align*}
92         T(x,y,z,t) &= T_0(x,y,z) &+& T^{*}(x,y,z,t)&&\\
93         p(x,y,z,t) &= p_0(x,y,z) &+& p^{*}(x,y,z,t)&&\\
94         \rho(x,y,z,t) &= \rho_0(z) &+& \rho^{*}(x,y,z,t);& &
95         &\psi^{*} << \psi_0&
96      \end{align*} 
97      \item \onslide<3->Hydrostatic equilibrium, geostrophic wind (not included in Boussinesq)
98      \begin{equation*}
99         \frac{\partial p_0}{\partial z} = -g \rho_0 \hspace{10mm} 
100         \frac{1}{\rho_0} \frac{\partial p_0}{\partial x} = -f v_\mathrm{g},
101         \hspace{5mm} \frac{1}{\rho_0} \frac{\partial p_0}{\partial y} = f u_\mathrm{g}
102      \end{equation*}
103      \item \onslide<4->Equation of state
104      \begin{equation*}
105         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} 
106      \end{equation*}
107      \begin{equation*}
108         \frac{\Delta p}{p_0} \approx \frac{\Delta \rho}{\rho_0} +
109         \frac{\Delta T}{T_0} \rightarrow \frac{p^{*}}{p_0} \approx 
110         \frac{\rho^{*}}{\rho_0} + \frac{T^{*}}{T_0} \hspace{10mm} 
111         \frac{\rho^{*}}{\rho_0} \approx - \frac{T^{*}}{T_0} \hspace{10mm}
112      \end{equation*}
113   \end{itemize}
114\end{frame}
115
116% Folie 4
117\begin{frame}
118   \frametitle{Continuity Equation}
119   \begin{eqnarray*}
120      \onslide<2-> \dfrac{\partial \rho_0(z)}{\partial t} =
121      - \dfrac{\partial \rho_0(z) u_k}{\partial x_k} & 
122      \hspace{10mm} \dfrac{\partial \rho_0 u_k}{\partial x_k} = 0
123      \hspace{5mm} & \text{anelastic approximation}\\
124      \\
125      \onslide<3-> \rho_0 = const. & \hspace{10mm} 
126      \dfrac{\partial u_k}{\partial x_k} = 0 \hspace{5mm} & 
127      \text{incompressible flow}
128   \end{eqnarray*}
129\end{frame}
130
131% Folie 5
132\begin{frame}
133   \frametitle{Boussinesq Approximated Equations}
134   \begin{itemize}
135      \item \onslide<2->Navier-Stokes equations
136      \begin{equation*}
137         \frac{\partial u_i}{\partial t} 
138         + \frac{\partial u_k u_i}{\partial x_k} = 
139         - \frac{1}{\rho_0}\frac{\partial p^{*}}{\partial x_i} 
140         - \varepsilon_{ijk} f_j u_k + \varepsilon_{i3k} f_3 u_{k_\mathrm{g}} 
141         + g \frac{T - T_0}{T_0} \delta_{i3} + \nu 
142         \frac{\partial^2 u_i}{\partial x_k^2}
143      \end{equation*}
144      \item \onslide<3->First principle
145      \begin{equation*}
146         \frac{\partial T}{\partial t} + u_k \frac{\partial T}{\partial x_k} =
147         \nu_\mathrm{h} \frac{\partial^2 T}{\partial x_k^2} + Q
148      \end{equation*}
149      \item \onslide<4->Equation for passive scalar
150      \begin{equation*}
151         \frac{\partial \psi}{\partial t} + u_k
152         \frac{\partial \psi}{\partial x_k} = \nu_{\psi} 
153         \frac{\partial^2 \psi}{\partial x_k^2} + Q_{\psi}
154      \end{equation*}
155      \item \onslide<5->Continuity equation
156      \begin{equation*}
157         \frac{\partial u_k}{\partial x_k} = 0
158      \end{equation*}
159   \end{itemize}
160   \onslide<6-> \tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
161
162   \begin{tikzpicture}[remember picture, overlay]
163      \node at (current page.north west){%
164      \begin{tikzpicture}[overlay]
165         \node[plain, draw,anchor=west] at (94mm,-55mm) {\noindent This set of equations is valid for almost all kind of CFD models!};
166      \end{tikzpicture}
167      };
168   \end{tikzpicture}
169\end{frame}
170
171
172\section{Scale Separation}
173\subsection{Scale Separation by Spatial Filtering}
174
175% Folie 6
176\begin{frame}
177   \frametitle{LES - Scale Separation by Spatial Filtering (I)}
178   \footnotesize
179   \begin{itemize}
180      \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}}
181      \item<2->{Large / low-frequency modes $\Psi$ are calculated directly (resolved scales).}
182      \item<3->{Small / high-frequency modes $\Psi'$ are parameterized using a statistical model (subgrid / subfilter scales, SGS model).}
183      \item<4->{These two categories of scales are seperated by defining a cutoff length $\Delta$.}
184   \end{itemize}
185   \normalsize
186   \includegraphics[width=\textwidth]{basic_equations_figures/Spatial_Filtering_I.png}
187\end{frame}
188
189
190
191% Folie 7
192\begin{frame}
193   \frametitle{LES - Scale Separation by Spatial Filtering (II)}
194   \begin{columns}[T]
195      \begin{column}{0.8\textwidth}
196      \footnotesize
197      \begin{itemize}
198         \item<1->The Filter applied is a spatial filter:
199         \begin{equation*} 
200            \overline{\Psi}(x_i) = \int_D G(x_i - x_i') \Psi(x_i')dx_i'
201         \end{equation*}
202         \begin{equation*} 
203            \overline{\Psi}'(x_i) = 0 \qquad but \qquad \overline{\overline{\Psi}} \neq \overline{\Psi}(x_i)
204         \end{equation*} 
205         \item<2->Filter applied to the nonlinear advection term:
206         \begin{equation*} 
207            \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}} 
208         \end{equation*}   
209         \item<5->Leonard proposes a further decomposition:
210         \begin{equation*} 
211            \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}}                         
212         \end{equation*}
213         \begin{equation*} 
214            \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} 
215         \end{equation*}       
216      \end{itemize}
217      \end{column}
218      \begin{column}{0.32\textwidth}
219      \vspace{45mm}     
220      \begin{footnotesize}
221         \onslide<3->$R_{ki}$: \textbf{Reynolds-stress} \\
222         $C_{ki}$: \textbf{cross-stress} \\
223         $L_{ki}$: \textbf{Leonard-stress} \\
224         $\tau_{ki}$: \textbf{total stress-tensor}
225      \end{footnotesize}
226      \end{column}
227   \end{columns}
228   \onslide<4->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
229      \begin{tikzpicture}[remember picture, overlay]
230      \node at (current page.north west){
231      \begin{tikzpicture}[overlay]
232         \node[plain, draw,anchor=west] at (94mm,-30mm) {
233         \begin{footnotesize}
234            \noindent \textbf{Ensemble average:} \\
235         \end{footnotesize}
236         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
237         \vspace{5mm}
238         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
239         };
240      \end{tikzpicture}
241      };
242   \end{tikzpicture}   
243\end{frame}
244
245% Folie 8
246\begin{frame}
247   \frametitle{LES - Scale Separation by Spatial Filtering (III)}
248   \small
249   \begin{itemize}
250      \item<2-> Volume-balance approach (Schumann, 1975)\\ advantage: numerical discretization acts as a\\ Reynolds operator
251      \begin{flalign*}
252         &\Psi(V,t)=\frac{1}{\Delta x \cdot \Delta y \cdot \Delta z} = \int \int \int_V \Psi(V',t) dV'&\\
253         &\overline{\Psi'}(x_i)=0 \hspace{5mm} \text{and} \hspace{5mm} \overline{\overline{\Psi}} = \overline{\Psi}\\
254         &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]
255      \end{flalign*}
256      \item<3-> Filter applied to the nonlinear advection term:
257      \begin{equation*}
258         \overline{u_k u_i} = \overline{(\overline{u_k}+u'_k)(\overline{u_i}+u'_i)}=
259         \overline{u_k}\,\overline{u_i}+\overline{u'_k u'_i}
260      \end{equation*}
261   \end{itemize}
262   \onslide<1->\tikzstyle{plain} = [rectangle, draw, text width=0.27\textwidth, font=\small]
263      \begin{tikzpicture}[remember picture, overlay]
264      \node at (current page.north west){
265      \begin{tikzpicture}[overlay]
266         \node[plain, draw,anchor=west] at (94mm,-30mm) {
267         \begin{footnotesize}
268            \noindent \textbf{Ensemble average:} \\
269         \end{footnotesize}
270         $\overline{\overline{\Psi}}(x_i) = \overline{\Psi}(x_i)$\\
271         \vspace{5mm}
272         $\overline{u_k u_i} = \overline{u_k}\,\overline{u_i} + \overline{u_k' u_i'}$
273         };
274      \end{tikzpicture}
275      };
276   \end{tikzpicture}
277 \end{frame}
278
279
280\section{Filtered equations}
281\subsection{The Filtered Equations}
282
283% Folie 9
284\begin{frame}
285   \frametitle{The Filtered Equations}
286   \onslide<2->
287   \begin{equation*}
288      \frac{\partial \overline{u_i}}{\partial t} 
289      + \frac{\partial \overline{u_k}\,\overline{u_i}}{\partial x_k} =
290      - \frac{1}{\rho_0} \frac{\partial \overline{p}^*}{\partial x_i} 
291      - \varepsilon_{ijk}f_j \overline{u_k} + \varepsilon_{i3k} f_3 \overline{u}_{k_\mathrm{g}} 
292      + g \frac{\overline{T}-T_0}{T_0} \delta_{i3} 
293      + \nu \frac{\partial^2 \overline{u_i}}{\partial x_k^2} 
294      - \frac{\partial \tau_{ki}}{\partial x_k}
295   \end{equation*}
296
297   \begin{footnotesize}
298      \begin{itemize}
299         \item<3->The previous derivation completely ignores the existance of the computational grid.
300         \item<4->The computational grid introduces another space scale: the discretization step $\Delta x_i$.
301         \item<5->$\Delta x_i$ has to be small enough to be able to apply the filtering process correctly: $\Delta x_i \le \Delta$
302         \item<6-> Two possibilities:\\
303         1. Pre-filtering technique\\
304         ($\Delta x < \Delta$,  explicit filtering)\\
305         2. Linking the analytical filter\\
306         to the computational grid\\
307         ($\Delta x = \Delta$, implicit filtering)
308      \end{itemize}
309   \end{footnotesize}
310
311   \begin{picture}(0.0,0.0)
312      \put(140,13){\uncover<6->{\includegraphics[width=0.6\textwidth]{basic_equations_figures/explicit_implicit.png}}}
313   \end{picture}
314\end{frame}
315
316%% Folie 10
317\begin{frame}
318  \frametitle{Explicit Versus Implicit Filtering}
319   \begin{itemize}
320      \item<2-> Explicit filtering:
321      \begin{itemize}
322         \small
323         \item<2-> Requires that the analytical filter is applied explicitly.
324         \item<3-> Rarely used in practice, due to additional computational costs.
325      \end{itemize}
326      \item<4-> Implicit filtering:
327      \begin{itemize}
328         \small
329         \item<4-> The analytical cutoff length is associated with the grid spacing.
330         \item<5-> This method does not require the use of an analytical filter.
331         \item<6-> The filter characteristic cannot really be controlled.
332         \item<7-> Because of its simplicity, this method is used by nearly all LES models.
333      \end{itemize}
334   \end{itemize}
335
336   \onslide<8->
337   \begin{scriptsize}
338      \textbf{Literature:}\\
339      \textbf{Sagaut, P., 2001:} Large eddy simulation for incompressible flows: An introduction. Springer Verlag, Berlin/Heidelberg/New York, 319 pp.\\
340      \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.\\
341   \end{scriptsize}
342\end{frame}
343
344% Folie 11
345\begin{frame}
346   \frametitle{The Final Set of Equations (PALM)}
347   \footnotesize
348   \begin{itemize}
349      \item<2-> Navier-Stokes equations:
350      \onslide<2->
351      \begin{flalign*}
352         &\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}&
353      \end{flalign*}
354      \item<4-> First principle (using potential\\ temperature):
355      \onslide<4->
356      \begin{flalign*}
357         &\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}&
358      \end{flalign*}
359      \item<5-> Equation for specific humidity\\ (passive scalar)
360      \onslide<5->
361      \begin{flalign*}
362         &\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}&
363      \end{flalign*}
364      \item<6-> 
365      Continuity equation
366      \onslide<6->
367      \begin{flalign*}
368         &\frac{\partial \overline{u_k}}{\partial x_k} = 0&
369      \end{flalign*}
370   \end{itemize}
371
372   \onslide<3->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
373   \begin{tikzpicture}[remember picture, overlay]
374      \node at (current page.north west){
375      \begin{tikzpicture}[overlay]
376         \node[plain, draw,anchor=west] at (75mm,-45mm) {
377         \begin{tiny}
378            \noindent normal stresses included in the stress tensor are now included in a modified dynamic pressure:\\
379         \end{tiny}
380         $\tau_{ki}^r = \tau_{ki} - \frac{1}{3} \tau_{jj} \delta_{ki}$\\
381         \vspace{1mm}
382         $\overline{\pi}^* = \overline{p}^* + \frac{1}{3} \tau_{jj} \delta_{ki}$
383         };
384      \end{tikzpicture}
385      };
386   \end{tikzpicture}
387
388   \onslide<7->\tikzstyle{plain} = [rectangle, draw, text width=0.40\textwidth, font=\small]
389   \begin{tikzpicture}[remember picture, overlay]
390      \node at (current page.north west){
391      \begin{tikzpicture}[overlay]
392         \node[plain, draw,anchor=west] at (75mm,-70mm) {
393         \begin{tiny}
394            \noindent subgrid-scale stresses (fluxes) to be parameterized in the SGS model:\\
395         \end{tiny}
396         $\tau_{ki} = \overline{u_k u_i} - \overline{u_k}\,\overline{u_i}$\\
397         $H_{k} = \overline{u_k \theta} - \overline{u_k}\,\overline{\theta}$\\
398         $W_{k} = \overline{u_k q} - \overline{u_k}\,\overline{q}$
399         };
400      \end{tikzpicture}
401      };
402   \end{tikzpicture}
403\end{frame}
404
405
406\end{document}
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