.. include:: /images.rst FLD01: Steady, 2D Poiseuille flow --------------------------------- .. _description-9: Description ~~~~~~~~~~~ Plane Poiseuille flow is defined as a steady, laminar flow of a viscous fluid between two horizontal parallel plates separated by a distance, :math:H. Flow is induced by a pressure gradient across the length of the plates, :math:L, and is characterized by a 2D parabolic velocity profile symmetric about the horizontal mid-plane as illustrated in :numref:fld01fig1. .. _fld01fig1: .. figure:: ../media/fld01-setup.png :align: center Plane Poiseuille flow between two flat plates of length L, separated by a distance H. In this problem, the Navier-Stokes equations reduce to a second order, linear, ordinary differential equation (ODE), .. math:: \mu_{g}\frac{d^{2}u_{g}}{dy^{2}} = \frac{dP_{g}}{\text{dx}}, :label: fld01eq1 where :math:\mu_{g} and :math:P_{g} are correspondingly the fluid viscosity and pressure, and :math:u_{g} and :math:v_{g} are respectively the :math:x and :math:y velocity components. Furthermore, it is assumed that gravitational forces are negligible, the pressure gradient is constant, i.e., :math:dP_{g}/dx = C, and all velocity components are zero at the channel walls. The resulting analytical solution to :eq:fld01eq1 is given as .. math:: u_{g}\left( y \right) = - \frac{dP_{g}}{\text{dx}}\frac{1}{2\mu_{g}}y\left( H - y \right). :label: fld01eq2 .. _setup-9: Setup ~~~~~ .. literalinclude:: /subprojects/mfix/tests/fluid/FLD01/mfix.dat :language: fortran .. .. _fld01table1: .. .. csv-table:: FLD-01 Setup, Initial and Boundary Conditions. .. :widths: auto .. :header: "Computational/Physical model", " ", " " .. .. "2D, Steady-state, incompressible", " ", " " .. "Single-phase (no solids)", " ", " " .. "No gravity", " ", " " .. "Thermal energy equation is not solved", " ", " " .. "Turbulence equations are not solved (Laminar)", " ", " " .. "Uniform mesh", " ", " " .. "Superbee discretization scheme", " ", " " .. " ", " ", " " .. "**Geometry**", " ", " " .. "Coordinate system", "Cartesian", " ", "Grid partitions" .. "x-length", "0.2", "$$m$$", "8, 16, 32, 64" .. "y-length", "0.01", "$$m$$", "8, 16, 32, 64" .. " ", " ", " " .. "**Material** :sup:†", " ", " " .. "Fluid density, :math:\rho_{g}", "1.0", "(kg·m\ :sup:-3)" .. "Fluid viscosity, :math:\mu_{g}", "0.001", "(Pa·s)" .. " ", " ", " " .. "**Initial Conditions**", " ", " " .. "Pressure *(gauge)*, :math:P_{g}", "101325", "(Pa)" .. "x-velocity, :math:u_{g}", "10.0", "(m·s\ :sup:-1)" .. "y-velocity, :math:v_{g}", "0.0", "(m·s\ :sup:-1)" .. " ", " ", " " .. "**Boundary Conditions**", " ", " " .. "South boundary", "0.0", "(m·s\ :sup:-1)", "No-Slip wall" .. "North boundary", "0.0", "No-Slip wall" .. "Cyclic West-East boundary with pressure drop", ":math:\Delta P_{g}", "240.0", "$$Pa$$" .. _results-9: Results ~~~~~~~ The analytical and numerical solutions for x-velocity, :math:u_{g}, are shown in :numref:fld01fig2. Only a subset of the numerical solution data points are plotted causing the appearance of a slight shift in presented data points. The observed error demonstrates a second-order rate of convergence with respect to grid size in the y-axial direction. This is attributed to the second-order discretization of the viscous stress term as convection/diffusion terms do not contribute to the solution. .. _fld01fig2: .. figure:: ../media/image33.png :align: center Steady, 2D channel flow x-velocity profile (left), absolute error in x-velocity solution (center), and observed order of accuracy (right) using four grid levels (JMAX = 8, 16, 32, 64). The fluid pressure, :math:P_{g}, varies linearly along the length of the plates as shown in :numref:fld01fig3. The largest observed absolute error is bounded above by :math:10^{- 12} and occurs for the finest mesh. This error is attributed to the convergence criteria of the linear equation system. .. _fld01fig3: .. figure:: ../media/image34.png :align: center :figwidth: 75% Steady, 2D channel flow pressure profile (left) and absolute error in pressure solution (right) using four grid levels (IMAX = 8, 16, 32, 64).