.. include:: /images.rst FLD07: Steady, 2D fully-developed, turbulent channel flow --------------------------------------------------------- .. _description-15: Description ~~~~~~~~~~~ This case uses 2D, fully-developed turbulent channel flow between two horizontal, parallel plates separated by a width, :math:W, to assess the single phase *k-ϵ* model in MFIX. Periodic boundaries with a specified pressure drop are imposed in the y-direction as shown in :numref:fld07fig1. .. _fld07fig1: .. figure:: ../media/fld07-setup.png :align: center Turbulent flow in a 2D channel The pressure drop along the channel is equated to the shear stress at the walls, :math:\tau_{w}. .. math:: W\frac{dP_{g}}{\text{dy}} = {2\tau}_{w} :label: fld07eq1 The shear stress is related to the gas density, :math:\rho_{g}, and friction velocity, :math:v_{*}, .. math:: \tau_{w} = \rho_{g}v_{*}^{2}, :label: fld07eq2 where, the friction velocity, is given by the Reynolds number. .. math:: \text{Re}_{\tau} = \frac{\rho_{g}v_{*}(W/2)}{\mu_{g}} :label: fld07eq3 .. _setup-15: Setup ~~~~~ .. literalinclude:: /subprojects/mfix/tests/fluid/FLD07/mfix.dat :language: bash .. Highlighting with bash (not fortran) because FLD07/mfix.dat uses invalid Fortran syntax: DELP_Y = @(0.0543496*0.0543496) ! (Pa) .. .. _fld07table1: .. .. csv-table:: FLD-07 Setup, Initial and Boundary Conditions. .. :widths: auto .. :header: "Computational/Physical model", " ", " " .. .. "2D, Unsteady, incompressible", " ", " " .. "Single-phase (no solids)", " ", " " .. "No gravity", " ", " " .. "Thermal energy equation is not solved", " ", " " .. "Turbulent, standard *k-ϵ* model", " ", " " .. "Uniform mesh", " ", " " .. "Superbee discretization scheme", " ", " " .. "Time step: 0.02 s (fixed)", " ", " " .. " ", " ", " " .. "**Geometry**", " ", " " .. "Coordinate system", "Cartesian", " ", "Grid partitions" .. "x-length", "2.00", "$$m$$", "6, 12, 18" .. "y-length", "1.00", "$$m$$", "4" .. " ", " ", " " .. "**Material**", " ", " " .. "Fluid density, :math:\rho_{g}", "1.0", "(kg·m\ :sup:-3)" .. "Fluid viscosity, :math:\mu_{g}", "1.0E-04", "(Pa·s)" .. " ", " ", " " .. "**Initial Conditions**", " ", " " .. "x-velocity, :math:u_{g}", "0.00", "$$m·s\ :sup:-1)" .. "y-velocity, :math:v_{g}", "0.00", "\(m·s\ :sup:-1)" .. "Turbulenc kinetic energy, :math:k_{g}", "0.01", "\(m\ :sup:2 \·s\ :sup:-2$$" .. "Turbulenc dissipation rate, :math:ϵ_{g}", "0.01", "$$m\ :sup:2 \·s\ :sup:-3$$" .. " ", " ", " " .. "**Boundary Conditions**", " ", " ", " " .. "West wall", "0.0", $$m·s\ :sup:-1$$, "No-Slip wall" .. "East wall", "0.0", $$m·s\ :sup:-1$$, "No-Slip wall" .. "Cyclic North-South boundary with pressure drop", "2.95E-3", "$$Pa$$", "Specified :math:\Delta P_{g}" .. _results-15: Results ~~~~~~~ The pressure drop in the y-axial direction, domain length and width, and gas density were chosen to reflect the conditions of Lee and Moser :cite:Moser2015 for :math:\text{Re}_{\tau} = 543. The DNS dataset was accessed on November 10, 2016 from http://turbulence.ices.utexas.edu/channel2015/data/LM_Channel_0550\_mean_prof.dat. Transient simulations were performed for better numerical stability. The solution was considered converged when the L\ :sub:2 norms for the gas velocity components, :math:u_{g} and :math:v_{g}, turbulent kinetic energy, :math:k_{g}, and rate of turbulent kinetic energy dissipation, :math:\epsilon_{g}, were all less than 10\ :sup:-10. Simulations were conducted for three mesh levels [6, 12, 18] in the x-axial direction. Mesh levels were selected to ensure that the stream-ways velocity components in computational cells adjacent to the wall were located outside the buffer layer. Specifically, the first stream-ways velocity component should be located at least 30 *wall units* from the wall to be consistent with the :math:k - \epsilon model wall function implementation. .. math:: \frac{\Delta x}{2}\frac{\ v_{*}\rho_{g}}{\mu_{g}} > 30 :label: fld07eq4 The MFIX results are shown in :numref:fld07fig2 along with the direct numerical simulation (DNS) data of Lee and Moser :cite:Moser2015 for :math:\text{Re}_{\tau} = 543. The velocity profiles for the three mesh levels are shown on the left whereas the normalized velocity profiles with respect to wall units are shown on the right. .. _fld07fig2: .. figure:: ../media/image45.jpeg :align: center 2D, fully developed, turbulent channel flow with the DNS data of Lee and Moser :cite:Moser2015 ; (Left) Velocity profile; (Right) Non-dimensionalized channel width and velocity profile.