{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from lbmpy.session import *\n", "\n", "from pystencils.sympyextensions import prod\n", "from lbmpy.stencils import get_stencil\n", "from lbmpy.moments import get_default_moment_set_for_stencil, moments_up_to_order\n", "from lbmpy.creationfunctions import create_lb_method\n", "from lbmpy.equilibrium import discrete_equilibrium_from_matching_moments\n", "from lbmpy.methods import create_from_equilibrium, DensityVelocityComputation\n", "from lbmpy.quadratic_equilibrium_construction import *\n", "from lbmpy.chapman_enskog import ChapmanEnskogAnalysis\n", "from lbmpy.moments import exponent_to_polynomial_representation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Demo: Theoretical Background - LB Equilibrium Construction using quadratic Ansatz\n", "\n", "According to book by Wolf-Gladrow _\"Lattice-Gas Cellular Automata and Lattice Boltzmann Methods\"_ (2005)\n", "\n", "Through the Chapman Enskog analysis the following necessary conditions can be found in order for a lattice Boltzmann Method to approximate the Navier Stokes mass and momentum conservation equations. In the Chapman Enskog analysis only the moments of the equilibrium distribution functions are used, thus all conditions are formulated with regard to the moments $\\Pi$ of the equilibrium distribution function $f^{(eq)}$\n", "\n", "The conditions are:\n", "- zeroth moment is the density: $\\Pi_0 = \\sum_q f^{(eq)}_q = \\rho$\n", "- first moment is the momentum density, or for incompressible models the velocity:\n", " - compressible: $\\Pi_\\alpha = \\sum_q c_{q\\alpha} f^{(eq)}_q = \\rho u_\\alpha$\n", " - incompressible: $\\Pi_\\alpha = \\sum_q c_{q\\alpha} f^{(eq)}_q = u_\\alpha$\n", "- second moment is related to the pressure tensor and has to be: \n", " $\\Pi_{\\alpha\\beta} = \\sum_q c_{q\\alpha} c_{q\\beta} f^{(eq)}_q = \\rho u_\\alpha u_\\beta + p \\delta_{\\alpha\\beta}$\n", "- third order moments are also used in the Chapman Enskog expansion. The conditions on these moments are harder to formulate and are investigated later. A commonly used, but overly restrictive choice is \n", " $\\Pi_{\\alpha\\beta\\gamma} = p ( \\delta_{\\alpha\\beta} u_\\gamma + \\delta_{\\alpha\\gamma} u_\\beta + \\delta_{\\beta\\gamma} u_\\alpha )$. In Wolf-Gladrows book these conditions on the third order moment are not used but implicitly fulfilled by choosing fixed fractions of the coefficients $\\frac{A_1}{A_2}$ etc.\n", "\n", "Now the following generic quadratic ansatz is used for the equilibrium distribution. \n", "\n", "$$f^{(eq)}_q = A_{|q|} + B_{|q|} (\\mathbf{c}_q \\cdot \\mathbf{u}^2 ) + D_{|q|} \\mathbf{u}^2$$\n", "\n", "\n", "The free parameters $A_{|q|}, B_{|q|}, C_{|q|}$ and $D_{|q|}$ are chosen such that above conditions are fulfilled.\n", "The subscript $|q|$ is an integer and defined as the sum of the absolute values of the corresponding stencil direction. For example: for center $|q|=0$, for direct neighbors like north, east, top $|q|=1$ for 2D diagonals like north-west its 2 and for 3D diagnoals like bottom-north-west its 3.\n", "\n", "_lbmpy_ can create this quadratic ansatz for use for a given stencil:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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