{ "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, create_generic_mrt\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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\n", 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