.. include:: /images.rst
.. _sec-2p6:
MMS02: Two-phase, 3D, curl-based functions with constant volume fraction
------------------------------------------------------------------------
.. _description-4:
Description
~~~~~~~~~~~
Assuming that gas and solid volume fractions (i.e., :math:`\varepsilon_{g}` and :math:`\varepsilon_{s}`) remain constant, we can see from gas and solid continuity equations that both fluid and solid velocity fields are divergence-free (for constant density of fluid and solids). A manufactured solution for the fluid-phase velocity field is defined using the curl-based approach developed in :cite:`Ani2014`. For the solid-phase velocity field, a set of simple sinusoidal functions is selected (same as those shown in :eq:`mms1eq1`). The manufactured solutions for scalar quantities (pressure, gas temperature, and solid temperature) can be multivariate functions of sines and cosines as defined in :eq:`a1eq1`. The selected functions for all concerned variables are shown over a 3D domain in :numref:`fig12` through :numref:`fig14`.
.. _fig12:
.. figure:: /media/image17.png
:align: center
:figwidth: 75%
Gas phase momentum equation manufactured solutions for 3D, steady-state, two-phase flow verification test case.
.. _fig13:
.. figure:: /media/image18.png
:align: center
Solids phase momentum equation manufactured solutions for 3D, steady-state, two-phase flow verification test case.
.. _fig14:
.. figure:: /media/image19.png
:align: center
Scalar field manufactured solutions for 3D, steady-state, two-phase flow verification test case.
.. _setup-4:
Setup
~~~~~
.. _table14:
.. csv-table:: MMS-02 Setup, Initial and Boundary Conditions.
:widths: auto
:header: "Computational/Physical model", " ", " "
"3D, Steady-state, incompressible", " ", " "
"Two-phase", " ", " "
"No gravity", " ", " "
"Drag model is turned off", " ", " "
"Friction model is turned off", " ", " "
"Thermal energy equations are solved", " ", " "
"Granular energy equation is not solved", " ", " "
"Turbulence equations are not solved (Laminar)", " ", " "
"Central scheme", " ", " "
" ", " ", " "
"**Geometry**", " ", " "
"Coordinate system", "Cartesian", " "
"Domain length, :math:`L` \(x\)", "1.0", "\(m\)"
"Domain height, :math:`H` \(y\)", "1.0", "\(m\)"
"Domain width, :math:`W` \(z\)", "1.0", "\(m\)"
" ", " ", " "
"**Material** :sup:`†`", " ", " "
"Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
"Fluid viscosity, :math:`\mu_{g}`", "1.0", "(Pa·s)"
"Fluid specific heat, :math:`C_{\text{pg}}`", "0.05", "(J·kg\ :sup:`-1`\ ·K\ :sup:`-1`)"
"Fluid thermal conductivity, :math:`k_{g}`", "1.0", "(J·kg\ :sup:`-1`\ ·K\ :sup:`-1`\ ·s\ :sup:`-1`)"
"Solids density, :math:`\rho_{s}`", "2.0", "(kg·m\ :sup:`-3`)"
"Solids viscosity, :math:`\mu_{s}`", "2.0", "(Pa·s)"
"Solids specific heat, :math:`C_{\text{ps}}`", "0.1", "(J·kg\ :sup:`-1`\ ·K\ :sup:`-1`)"
"Solids thermal conductivity, :math:`k_{s}`", "2.0", "(J·kg\ :sup:`-1`\ ·K\ :sup:`-1`\ ·s\ :sup:`-1`)"
" ", " ", " "
"**Initial Conditions**", " ", " "
"Pressure *(gauge)*, :math:`P_{g}`", "0.0", "(Pa)"
"Fluid x-velocity, :math:`u_{g}`", "10.0", "(m·s\ :sup:`-1`)"
"Fluid y-velocity, :math:`v_{g}`", "10.0", "(m·s\ :sup:`-1`)"
"Fluid z-velocity, :math:`w_{g}`", "10.0", "(m·s\ :sup:`-1`)"
"Solids x-velocity, :math:`u_{s}`", "5.0", "(m·s\ :sup:`-1`)"
"Solids y-velocity, :math:`v_{s}`", "5.0", "(m·s\ :sup:`-1`)"
"Solids z-velocity, :math:`w_{s}`", "5.0", "(m·s\ :sup:`-1`)"
"Fluid temperature, :math:`T_{g}`", "350.0", "\(K\)"
"Solids temperature, :math:`T_{s}`", "300.0", "\(K\)"
"Gas volume fraction, :math:`\varepsilon_{g}`", "0.7", "--"
" ", " ", " "
"**Boundary Conditions** :sup:`‡`", " ", " "
"All boundaries", "Mass inflow", " "
**†** Material properties selected to ensure comparable contribution from convection and diffusion terms. Specified values are constant to avoid the introduction of constitutive laws.
:sup:`‡` The manufactured solution is imposed on all boundaries (i.e., Dirichlet specification).
.. _results-4:
Results
~~~~~~~
Numerical solutions were obtained using the Central discretization scheme for 8x8, 16x16, 32x32, 64x64, and 128x128 grid meshes. The observed order approaches second order for both :math:`L_{2}` and :math:`L_{\infty}` norms using the Central scheme, as shown in :numref:`fig15`. This indicates that the numerical discretization terms have been implemented correctly for all derivative terms within the gas momentum equations, solid momentum equations, gas pressure correction equation, gas energy equation, and solid energy equation.
.. _fig15:
.. figure:: /media/image20.png
:align: center
Observed orders of accuracy for 3D, two-phase flows (constant volume fraction) using (a) :math:`\mathbf{L}_{\mathbf{2}}` norms, and (b) :math:`\mathbf{L}_{\mathbf{\infty}}` norms of the discretization error.