10_tutorial_conservative_allen_cahn_two_phase.ipynb 95.3 KB
 Markus Holzer committed Jul 01, 2021 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# The conservative Allen-Cahn model for high Reynolds number, two phase flow with large-density and viscosity constrast" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from pystencils.session import *\n", "from lbmpy.session import *\n", "\n", "from pystencils.simp import sympy_cse\n", "from pystencils.boundaries import BoundaryHandling\n", "\n", "from lbmpy.phasefield_allen_cahn.contact_angle import ContactAngle\n", "from lbmpy.phasefield_allen_cahn.force_model import MultiphaseForceModel, CentralMomentMultiphaseForceModel\n", "from lbmpy.phasefield_allen_cahn.kernel_equations import *\n", "from lbmpy.phasefield_allen_cahn.parameter_calculation import calculate_parameters_rti\n", "\n", "from lbmpy.advanced_streaming import LBMPeriodicityHandling\n", "from lbmpy.boundaries import NoSlip, LatticeBoltzmannBoundaryHandling" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If pycuda is installed the simulation automatically runs on GPU" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {},  Markus Holzer committed Oct 27, 2021 42 43 44 45 46 47 48 49 50  "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "No pycuda installed\n" ] } ],  Markus Holzer committed Jul 01, 2021 51 52 53 54 55 56  "source": [ "try:\n", " import pycuda\n", "except ImportError:\n", " pycuda = None\n", " gpu = False\n",  Markus Holzer committed Oct 27, 2021 57  " target = ps.Target.CPU\n",  Markus Holzer committed Jul 01, 2021 58 59 60 61  " print('No pycuda installed')\n", "\n", "if pycuda:\n", " gpu = True\n",  Markus Holzer committed Oct 27, 2021 62  " target = ps.Target.GPU"  Markus Holzer committed Jul 01, 2021 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The conservative Allen-Cahn model (CACM) for two-phase flow is based on the work of Fakhari et al. (2017) [Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios](http://dx.doi.org/10.1103/PhysRevE.96.053301). The model can be created for two-dimensional problems as well as three-dimensional problems, which have been described by Mitchell et al. (2018) [Development of a three-dimensional\n", "phase-field lattice Boltzmann method for the study of immiscible fluids at high density ratios](http://dx.doi.org/10.1103/PhysRevE.96.053301). Furthermore, cascaded lattice Boltzmann methods can be combined with the model which was described in [A cascaded phase-field lattice Boltzmann model for the simulation of incompressible, immiscible fluids with high density contrast](http://dx.doi.org/10.1016/j.camwa.2019.08.018)\n", "\n", "\n", "The CACM is suitable for simulating highly complex two phase flow problems with high-density ratios and high Reynolds numbers. In this tutorial, an overview is provided on how to derive the model with lbmpy. For this, the model is defined with two LBM populations. One for the interface tracking, which we call the phase-field LB step and one for recovering the hydrodynamic properties. The latter is called the hydrodynamic LB step." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Geometry Setup\n", "\n", "First of all, the stencils for the phase-field LB step as well as the stencil for the hydrodynamic LB step are defined. According to the stencils, the simulation can be performed in either 2D- or 3D-space. For 2D simulations, only the D2Q9 stencil is supported. For 3D simulations, the D3Q15, D3Q19 and the D3Q27 stencil are supported. Note here that the cascaded LBM can not be derived for D3Q15 stencils." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [  Markus Holzer committed Sep 21, 2021 91 92  "stencil_phase = LBStencil(Stencil.D2Q9)\n", "stencil_hydro = LBStencil(Stencil.D2Q9)\n",  Markus Holzer committed Jul 01, 2021 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275  "assert(len(stencil_phase[0]) == len(stencil_hydro[0]))\n", "\n", "dimensions = len(stencil_phase[0])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Definition of the domain size" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# domain \n", "L0 = 256\n", "domain_size = (L0, 4 * L0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Parameter definition\n", "\n", "The next step is to calculate all parameters which are needed for the simulation. In this example, a Rayleigh-Taylor instability test case is set up. The parameter calculation for this setup is already implemented in lbmpy and can be used with the dimensionless parameters which describe the problem." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "# time step\n", "timesteps = 8000\n", "\n", "# reference time\n", "reference_time = 4000\n", "# density of the heavier fluid\n", "rho_H = 1.0\n", "\n", "# calculate the parameters for the RTI\n", "parameters = calculate_parameters_rti(reference_length=L0,\n", " reference_time=reference_time,\n", " density_heavy=rho_H,\n", " capillary_number=0.44,\n", " reynolds_number=3000,\n", " atwood_number=0.998,\n", " peclet_number=1000,\n", " density_ratio=1000,\n", " viscosity_ratio=100)\n", "# get the parameters\n", "rho_L = parameters.get(\"density_light\")\n", "\n", "mu_H = parameters.get(\"dynamic_viscosity_heavy\")\n", "mu_L = parameters.get(\"dynamic_viscosity_light\")\n", "\n", "tau_H = parameters.get(\"relaxation_time_heavy\")\n", "tau_L = parameters.get(\"relaxation_time_light\")\n", "\n", "sigma = parameters.get(\"surface_tension\")\n", "M = parameters.get(\"mobility\")\n", "gravitational_acceleration = parameters.get(\"gravitational_acceleration\")\n", "\n", "\n", "drho3 = (rho_H - rho_L)/3\n", "# interface thickness\n", "W = 5\n", "# coeffcient related to surface tension\n", "beta = 12.0 * (sigma/W)\n", "# coeffcient related to surface tension\n", "kappa = 1.5 * sigma*W\n", "# relaxation rate allen cahn (h)\n", "w_c = 1.0/(0.5 + (3.0 * M))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Fields\n", "\n", "As a next step all fields which are needed get defined. To do so, we create a datahandling object. More details about it can be found in the third tutorial of the [pystencils framework]( http://pycodegen.pages.walberla.net/pystencils/). This object holds all fields and manages the kernel runs." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# create a datahandling object\n", "dh = ps.create_data_handling((domain_size), periodicity=(True, False), parallel=False, default_target=target)\n", "\n", "# pdf fields. g is used for hydrodynamics and h for the interface tracking \n", "g = dh.add_array(\"g\", values_per_cell=len(stencil_hydro))\n", "dh.fill(\"g\", 0.0, ghost_layers=True)\n", "h = dh.add_array(\"h\",values_per_cell=len(stencil_phase))\n", "dh.fill(\"h\", 0.0, ghost_layers=True)\n", "\n", "g_tmp = dh.add_array(\"g_tmp\", values_per_cell=len(stencil_hydro))\n", "dh.fill(\"g_tmp\", 0.0, ghost_layers=True)\n", "h_tmp = dh.add_array(\"h_tmp\",values_per_cell=len(stencil_phase))\n", "dh.fill(\"h_tmp\", 0.0, ghost_layers=True)\n", "\n", "# velocity field\n", "u = dh.add_array(\"u\", values_per_cell=dh.dim)\n", "dh.fill(\"u\", 0.0, ghost_layers=True)\n", "\n", "# phase-field\n", "C = dh.add_array(\"C\")\n", "dh.fill(\"C\", 0.0, ghost_layers=True)\n", "C_tmp = dh.add_array(\"C_tmp\")\n", "dh.fill(\"C_tmp\", 0.0, ghost_layers=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As a next step the relaxation time is stated in a symbolic form. It is calculated via interpolation." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "# relaxation time and rate\n", "tau = 0.5 + tau_L + (C.center) * (tau_H - tau_L)\n", "s8 = 1/(tau)\n", "\n", "# density for the whole domain\n", "rho = rho_L + (C.center) * (rho_H - rho_L)\n", "\n", "# body force\n", "body_force = [0, 0, 0]\n", "body_force[1] = gravitational_acceleration * rho" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Definition of the lattice Boltzmann methods" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For both LB steps, a weighted orthogonal MRT (WMRT) method is used. It is also possible to change the method to a simpler SRT scheme or a more complicated CLBM scheme. The CLBM scheme can be obtained by commenting in the python snippets in the notebook cells below. Note here that the hydrodynamic LB step is formulated as an incompressible velocity-based LBM. Thus, the velocity terms can not be removed from the equilibrium in the central moment space." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", " \n", "
MomentEq. Value Relaxation Rate
$1$$\\rho$$0$
$x$$\\rho u_{0}$$1.8208262344103532$
$y$$\\rho u_{1}$$1.8208262344103532$
$x^{2} - y^{2}$$\\rho u_{0}^{2} - \\rho u_{1}^{2}$$1$
$x y$$\\rho u_{0} u_{1}$$1$
$3 x^{2} + 3 y^{2} - 2$$3 \\rho u_{0}^{2} + 3 \\rho u_{1}^{2}$$1$
$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", " \n", " \n", "\n", " \n", " \n",  Markus Holzer committed Oct 27, 2021 276  " \n",  Markus Holzer committed Jul 01, 2021 277 278 279 280  " \n", "\n", " \n", " \n",  Markus Holzer committed Oct 27, 2021 281  " \n",  Markus Holzer committed Jul 01, 2021 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317  " \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 Oct 27, 2021 318  ""  Markus Holzer committed Jul 01, 2021 319 320 321 322 323 324 325 326  ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [  Markus Holzer committed Sep 21, 2021 327 328  "config_phase = LBMConfig(stencil=stencil_phase, method=Method.MRT, compressible=True,\n", " weighted=True, relaxation_rates=[1, 1, 1, 1, 1])\n",  Markus Holzer committed Jul 01, 2021 329  "\n",  Markus Holzer committed Sep 21, 2021 330  "method_phase = create_lb_method(lbm_config=config_phase)\n",  Markus Holzer committed Jul 01, 2021 331 332  "method_phase.set_first_moment_relaxation_rate(w_c)\n", "\n",  Markus Holzer committed Sep 21, 2021 333 334 335 336  "# config_phase = LBMConfig(stencil=stencil_phase, method=Method.CENTRAL_MOMENT, compressible=True,\n", "# weighted=True, relaxation_rates=[0, w_c, 1, 1, 1], equilibrium_order=4)\n", "\n", "# method_phase = create_lb_method(lbm_config=config_phase)\n",  Markus Holzer committed Jul 01, 2021 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404  "method_phase" ] }, { "cell_type": "code", "execution_count": 9, "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}$$u_{0}^{2} - u_{1}^{2}$$\\frac{1}{0.664004086170318 - 0.147603677553286 {{C}_{(0,0)}}}$
$x y$$u_{0} u_{1}$$\\frac{1}{0.664004086170318 - 0.147603677553286 {{C}_{(0,0)}}}$
$3 x^{2} + 3 y^{2} - 2$$3 u_{0}^{2} + 3 u_{1}^{2}$$1$
$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", " \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 Oct 27, 2021 405  ""  Markus Holzer committed Jul 01, 2021 406 407 408 409 410 411 412 413  ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [  Markus Holzer committed Sep 21, 2021 414 415 416 417 418 419 420 421  "config_hydro = LBMConfig(stencil=stencil_hydro, method=Method.MRT, compressible=False,\n", " weighted=True, relaxation_rates=[s8, 1, 1, 1])\n", "\n", "method_hydro = create_lb_method(lbm_config=config_hydro)\n", "\n", "\n", "# config_hydro = LBMConfig(stencil=stencil_hydro, method=Method.CENTRAL_MOMENT, compressible=False,\n", "# weighted=True, relaxation_rates=[s8, 1, 1], equilibrium_order=4)\n",  Markus Holzer committed Jul 01, 2021 422  "\n",  Markus Holzer committed Sep 21, 2021 423  "# method_hydro = create_lb_method(lbm_config=config_hydro)\n",  Markus Holzer committed Jul 01, 2021 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500  "\n", "method_hydro" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Initialization" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The probability distribution functions (pdfs) are initialised with the equilibrium distribution for the LB methods." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "h_updates = initializer_kernel_phase_field_lb(h, C, u, method_phase, W)\n", "g_updates = initializer_kernel_hydro_lb(g, u, method_hydro)\n", "\n", "h_init = ps.create_kernel(h_updates, target=dh.default_target, cpu_openmp=True).compile()\n", "g_init = ps.create_kernel(g_updates, target=dh.default_target, cpu_openmp=True).compile()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Following this, the phase field is initialised directly in python." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [], "source": [ "# initialize the domain\n", "def Initialize_distributions():\n", " Nx = domain_size[0]\n", " Ny = domain_size[1]\n", " \n", " for block in dh.iterate(ghost_layers=True, inner_ghost_layers=False):\n", " x = np.zeros_like(block.midpoint_arrays[0])\n", " x[:, :] = block.midpoint_arrays[0]\n", " \n", " y = np.zeros_like(block.midpoint_arrays[1])\n", " y[:, :] = block.midpoint_arrays[1]\n", "\n", " y -= 2 * L0\n", " tmp = 0.1 * Nx * np.cos((2 * np.pi * x) / Nx)\n", " init_values = 0.5 + 0.5 * np.tanh((y - tmp) / (W / 2))\n", " block[\"C\"][:, :] = init_values\n", " block[\"C_tmp\"][:, :] = init_values\n", " \n", " if gpu:\n", " dh.all_to_gpu() \n", " \n", " dh.run_kernel(h_init)\n", " dh.run_kernel(g_init)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [  Markus Holzer committed Oct 27, 2021 501  ""  Markus Holzer committed Jul 01, 2021 502 503 504 505 506 507 508 509  ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": {  Markus Holzer committed Oct 27, 2021 510  "image/png": 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\n", 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