Changeset 987 for palm/trunk/TUTORIAL/SOURCE
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 Sep 5, 2012 9:27:35 AM (11 years ago)
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palm/trunk/TUTORIAL/SOURCE/sgs_models.tex
r945 r987 164 164 165 165 166 % Folie 7 167 \begin{frame} 168 \frametitle{The Smagorinsky Model: Performance} 169 \begin{itemize} 170 \item<2> Predicts many flows reasonably well 171 \item<3> Problems: 172 \begin{itemize} 173 \item<3> Optimum parameter value varies with flow type: 174 \begin{itemize} 175 \item Isotropic turbulence: $C_S \approx 0.2$\\ 176 \item Shear (channel) flows: $C_S \approx 0.065$ 177 \end{itemize} 178 \item<4> Length scale uncertain with anisotropic filter: 179 \begin{equation*} 180 (\Delta_x \Delta_y \Delta_z)^{1/3} \hspace{5mm} (\Delta_x + \Delta_y + \Delta_z)/3 181 \end{equation*} 182 \item<5> Needs modification to account for: 183 \begin{itemize} 184 \item stratification: $C_S = F(Ri,...)$, $Ri$: Richardson number\\ 185 \item nearwall region: $C_S = F(z+)$, $z+$: distance from wall 186 \end{itemize} 187 \end{itemize} 188 \end{itemize} 189 \end{frame} 190 191 192 \section{Deardoff Modification} 193 \subsection{Deardoff Modification} 194 195 % Folie 8 196 \begin{frame} 197 \frametitle{Deardorff (1980) Modification (Used in PALM) (I)} 198 \footnotesize 199 \onslide<1>{ 200 $ \nu_T = Cql = C_M \Lambda \sqrt{\bar{e}} $ \quad \textbf{with} \quad $ \bar{e} = \frac{\overline{u_i' u_i'}}{2} $ \quad \textbf{SGSturbulent kinetic energy}} 201 \normalsize 202 \begin{itemize} 203 \item<2>{The SGSTKE allows a much better estimation of the velocity scale for the SGS fluctuations and also contains information about the past history of the local fluid.} 204 \end{itemize} 205 \onslide<3>{ 206 $ C_M = const. = 0.1 $ 207 \par\bigskip 208 \scriptsize 209 $ \Lambda = \begin{cases} min\left( 0.7 \cdot z, \Delta \right), & \textbf{unstable or neutral stratification} \\ 210 min\left( 0.7 \cdot z, \Delta, 0.76 \sqrt{\bar{e}} \left[ \frac{g}{\Theta_0} \frac{\partial \bar{\Theta}}{\partial z} \right]^{1/2} \right), & \textbf{stable stratification} 211 \end{cases} $ 212 \normalsize 213 \par\bigskip 214 $ \Delta = \left( \Delta x \Delta y \Delta z \right)^{1/3} $ } 215 \end{frame} 216 217 % Folie 9 218 \begin{frame} 219 \frametitle{Deardorff (1980) Modification (Used in PALM) (II)} 220 \begin{itemize} 221 \item{SGSTKE from prognostic equation} 222 \end{itemize} 223 $ \frac{\partial \bar{e}}{\partial t} = \bar{u_k} \frac{\partial \bar{e}}{\partial x_k}  \tau_{ki} \frac{\partial \bar{u_i}}{\partial x_k} + \frac{g}{\Theta_0} \overline{u_3' \Theta'}  \frac{\partial}{\partial x_k} \left\{ \overline{u_k' \left( e' + \frac{\pi'}{\rho_0} \right)} \right\}  \epsilon $ 224 \par\bigskip 225 $ \frac{\partial}{\partial x_k} \left[ \overline{u_k' \left( e' + \frac{\pi'}{\rho_0} \right)} \right] =  \frac{\partial}{\partial x_k} \nu_e \frac{\partial \bar{e}}{\partial x_k} $ 226 \par\bigskip 227 $ \nu_e = 2 \nu_T $ 228 \par\bigskip 229 $ \epsilon = C_{\epsilon} \frac{\bar{e}^{3/2}}{\Lambda} \qquad \qquad C_{\epsilon} = 0.19 + 0.74\frac{\Lambda}{\Delta} $ 230 \end{frame} 231 232 % Folie 10 233 \begin{frame} 234 \frametitle{Deardorff (1980) Modification (Used in PALM) (III)} 235 \begin{itemize} 236 \item{There are still problems with this parameterization:} 237 \begin{itemize} 238 \item[]<2>{The model overestimates the velocity shear near the wall.} 239 \item[]<3>{$\textrm{C}_\mathrm{M}$ is still a constant but actually varies for different types of flows.} 240 \item[]<4>{Backscatter of energy from the SGSturbulence to the resolvedscale flow can not be considered.} 241 \end{itemize} 242 \item<5>{Several other SGS models have been developed:} 243 \begin{itemize} 244 \item[]<5>{Two part eddy viscosity model (Sullivan, et al.)} 245 \item[]<6>{Scale similarity model (Bardina et al.)} 246 \item[]<7>{Backscatter model (Mason)} 247 \end{itemize} 248 \item<8>{However, for fine grid resolutions ($\textrm{E}_\mathrm{SGS} << \ \textrm{E}$) LES results become almost independent 249 from the different models (Beare et al., 2006, BLM).} 250 \end{itemize} 251 \end{frame} 252 253 254 \section{Summary / Important Points for Beginners} 255 \subsection{Summary / Important Points for Beginners} 256 257 % Folie 11 258 \begin{frame} 259 \frametitle{Summary / Important Points for Beginners (I)} 260 \begin{columns}[c] 261 \column[T]{0.4\textwidth} 262 \includegraphics<27>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_2.png} 263 \includegraphics<8>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_8.png} 264 \includegraphics<9>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_9.png} 265 \includegraphics<10>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_10.png} 266 \onslide<810>{\begin{flushright} \begin{tiny} after Schatzmann and Leitl (2001) \end{tiny} \end{flushright}} 267 \column[T]{0.2\textwidth} 268 \vspace{0.9cm} 269 \includegraphics<810>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png} 270 \par 271 \onslide<8>{\begin{small} fluctuations (\textbf{u},c) \end{small}} 272 \par\bigskip 273 \thicklines 274 \onslide<9>{\mbox{\line(6,0){5} \, \line(1,0){5} \, \line(1,0){5} \quad \begin{small} {critical concentration level} \end{small}}} 275 \vspace{1cm} 276 277 \includegraphics<810>[width=0.7\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_arrow.png} 278 \par 279 \onslide<8>{\begin{small} smooth result \end{small}} 280 \column[T]{0.4\textwidth} 281 \includegraphics<12>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_1_neu.png} 282 \includegraphics<3>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_3_neu.png} 283 \includegraphics<4>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_4.png} 284 \includegraphics<510>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_5.png} 285 \vspace{1.3cm} 286 \includegraphics<6>[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_6_neu.png} 287 \uncover<7>{\includegraphics[width=\textwidth]{sgs_models_figures/Important_Points/Important_Points_1_7_neu.png}} 288 \end{columns} 289 \end{frame} 290 291 % Folie 12 292 \begin{frame} 293 \frametitle{Summary / Important Points for Beginners (II)} 294 For an LES it always has to be guaranteed that the main energy containing eddies of the respective 295 turbulent flow can really be simulated by the numerical model: 296 \begin{itemize} 297 \item<2>{The grid spacing has to be fine enough.} 298 \item<3>{$\textrm{E}_\mathrm{SGS} < (<<) \ \textrm{E} $} 299 \item<4>{The inflow/outflow boundaries must not effect the flow turbulence \\ 300 (therefore cyclic boundary conditions are used in most cases).} 301 \item<5>{In case of homogeneous initial and boundary conditions, the onset of turbulence 302 has to be triggered by imposing random fluctuations.} 303 \item<6>{Simulations have to be run for a long time to reach a stationary state and stable statistics.} 304 \end{itemize} 305 \end{frame} 306 307 308 \section{Example Output} 309 \subsection{Example Output} 310 311 % Folie 13 312 \begin{frame} 313 \frametitle{Example Output (I)} 314 \begin{itemize} 315 \item{LES of a convective boundary layer} 316 \end{itemize} 317 \includegraphics<1>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_1.png} 318 \includegraphics<2>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_2.png} 319 \includegraphics<3>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_3.png} 320 \includegraphics<4>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_4.png} 321 \includegraphics<5>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_5.png} 322 \includegraphics<6>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_6.png} 323 \includegraphics<7>[width=\textwidth]{sgs_models_figures/Example_Output_1/Example_Output_1_7.png} 324 \end{frame} 325 326 % Folie 14 327 \begin{frame} 328 \frametitle{Example Output (II)} 329 \begin{itemize} 330 \item{LES of a convective boundary layer} 331 \end{itemize} 332 \begin{center} 333 \includegraphics[width=0.8\textwidth]{sgs_models_figures/Example_output_2.png} 334 power spectrum of vertical velocity 335 \end{center} 336 \end{frame} 337 338 % Folie 15 339 \begin{frame} 340 \frametitle{Some Symbols} 341 \begin{columns}[c] 342 \column{0.6\textwidth} 343 \begin{tabbing} 344 $u_i \quad (i = 1,2,3)$ \quad \= velocity components \\ 345 $u,v,w$ \\ 346 347 \\ 348 349 $x_i \quad (i = 1,2,3)$ \> spatial coordinates \\ 350 $x,y,z$ \\ 351 352 \\ 353 354 $\Theta$ \> potential temperature \\ \\ 355 356 $\Psi$ \> passive scalar \\ \\ 357 358 $T$ \> actual Temperatur \\ \\ 359 \end{tabbing} 360 \column{0.4\textwidth} 361 \begin{tabbing} 362 $\Phi = gz$ \quad \= geopotential \\ \\ 363 364 $p$ \> pressure \\ \\ 365 366 $\rho$ \> density \\ \\ 367 368 $f_i$ \> Coriolis Parameter \\ \\ 369 370 $\epsilon_{ijk}$ \> alternating symbol \\ \\ 371 372 $\nu, \nu_\Psi$ \> molecular diffusivity \\ \\ 373 374 $Q, Q_\Psi$ \> sources or sinks \\ \\ 375 \end{tabbing} 376 \end{columns} 377 \end{frame} 166 378 \end{document}
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