00_tutorial_lbmpy_walberla_overview.ipynb 2 MB
MomentEq. Value Relaxation Rate
$1$$\\rho$$1.6$
$x$$u_{0}$$1.6$
$y$$u_{1}$$1.6$
$x^{2}$$\\frac{\\rho}{3} + u_{0}^{2}$$1.6$
$y^{2}$$\\frac{\\rho}{3} + u_{1}^{2}$$1.6$
$x y$$u_{0} u_{1}$$1.6$
$x^{2} y$$\\frac{u_{1}}{3}$$1.6$
$x y^{2}$$\\frac{u_{0}}{3}$$1.6$
$x^{2} y^{2}$$\\frac{\\rho}{9} + \\frac{u_{0}^{2}}{3} + \\frac{u_{1}^{2}}{3}$$1.6$
\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " " ], "text/plain": [ Markus Holzer committed May 12, 2021 250 "" Martin Bauer committed Mar 21, 2019 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ldc.method" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "All moments are relaxed to their equilibrium values with the same rate. We can easily built a more complex method were we choose different rates to separately control viscosity, bulk viscosity and relaxation of higher order moments:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", " \n", "
MomentEq. Value Relaxation Rate
$1$$\\rho$$0$
$x$$u_{0}$$0$
$y$$u_{1}$$0$
$x^{2} - y^{2}$$u_{0}^{2} - u_{1}^{2}$$1.8$
$x y$$u_{0} u_{1}$$1.8$
$3 x^{2} + 3 y^{2} - 2$$3 u_{0}^{2} + 3 u_{1}^{2}$$1.5$
$3 x^{2} y - y$$0$$1$
$3 x y^{2} - x$$0$$1$
$9 x^{2} y^{2} - 3 x^{2} - 3 y^{2} + 1$$0$$1$
\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", Markus Holzer committed Jun 07, 2020 287 " \n", Martin Bauer committed Mar 21, 2019 288 289 290 291 " \n", "\n", " \n", " \n", Markus Holzer committed Jun 07, 2020 292 " \n", Martin Bauer committed Mar 21, 2019 293 294 295 296 " \n", "\n", " \n", " \n", Markus Holzer committed Jun 07, 2020 297 " \n", Martin Bauer committed Mar 21, 2019 298 299 " \n", "\n", Markus Holzer committed Jun 07, 2020 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", " \n", " \n", "\n", " \n", " \n", Martin Bauer committed Mar 21, 2019 317 318 319 " \n", " \n", "\n", Markus Holzer committed Jun 07, 2020 320 321 " \n", " \n", Martin Bauer committed Mar 21, 2019 322 323 324 " \n", " \n", "\n", Markus Holzer committed Jun 07, 2020 325 326 " \n", " \n", Martin Bauer committed Mar 21, 2019 327 328 329 330 331 332 333 " \n", " \n", "\n", " \n", " " ], "text/plain": [ Markus Holzer committed May 12, 2021 334 "" Martin Bauer committed Mar 21, 2019 335 336 337 338 339 340 341 342 ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ Markus Holzer committed Jun 07, 2020 343 344 "mrt_ldc = create_lid_driven_cavity(domain_size=(100, 40), method='mrt',\n", " relaxation_rates=[1.8, 1.5, 1, 1], weighted=True)\n", Martin Bauer committed Mar 21, 2019 345 346 347 348 349 350 351 "mrt_ldc.method" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ Markus Holzer committed Jun 07, 2020 352 "In principle, we can freely choose all entries of the above table (as long as the moments are linear independent) and automatically generate a custom MRT implementation. Since the conserved quantities usually do not affect the collision, they are chosen as zero by default. For special cases, their relaxation rate can be manipulated with set_conserved_moments_relaxation_rate. Furthermore, the above method uses weighted orthogonal moments." Martin Bauer committed Mar 21, 2019 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Modern methods: cumulant and entropic schemes\n", "\n", "Recently more complex lattice Boltzmann methods have been published that improve the stability of standard methods. We demonstrate this by a setting up a shear flow scenario where the upper half is initialized with velocity to the right and the lower half with a velocity to the left. The y-velocity component is initialized with a very small random value." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { Markus Holzer committed May 12, 2021 371 "image/png": 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Martin Bauer committed Mar 21, 2019 372 "text/plain": [ Frederik Hennig committed Feb 02, 2021 373 "