.. include:: /images.rst MMS01: Single-phase, 2D, sinusoidal functions --------------------------------------------- .. _description-3: Description ~~~~~~~~~~~ A sinusoidal divergence-free manufactured solution [12, 13] for the fluid pressure, :math:P_{g}, and :math:x and :math:y velocity components, :math:u_{g} and :math:v_{g}, respectively, is used for the verification of steady-state, single-phase flows on a 2D grid. .. math:: u_{g} = u_{g0}\operatorname{}\left( 2\pi\left( x + y \right) \right)\\ v_{g} = v_{g0}\operatorname{}\left( 2\pi\left( x + y \right) \right)\\ P_{g} = P_{g0}\cos\left( 2\pi\left( x + y \right) \right) :label: mms1eq1 :numref:fig9 shows a color contour of the pressure field and velocity streamlines for the manufactured solution using constants :math:P_{g0} = 100\ \text{Pa}, :math:u_{g0} = 5.0\ \mathrm{m \cdot sec}^{- 1}, and , :math:v_{g0} = 5.0\mathrm{\ m \cdot sec}^{- 1}. .. _fig9: .. figure:: /media/image11.png :align: center Pressure contours and velocity streamlines for 2D, single-phase, simple sinusoidal manufactured solution on a 64x64 cell grid. .. _setup-3: Setup ~~~~~ .. _table13: .. csv-table:: MMS-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 and Central discretization schemes", " ", " " " ", " ", " " "**Geometry**", " ", " " "Coordinate system", "Cartesian", " " "x-length", "1.0", "$$m$$" "y-length", "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)" " ", " ", " " "**Initial Conditions**", " ", " " "Pressure *(gauge)*, :math:P_{g}", "0.0", "(Pa)" "x-velocity, :math:u_{g}", "5.0", "(m·s\ :sup:-1)" "y-velocity, :math:v_{g}", "5.0", "(m·s\ :sup:-1)" " ", " ", " " "**Boundary Conditions** :sup:‡", " ", " " "All boundaries", "Mass inflow", " " **†** Material properties selected to ensure comparable contribution from convection and diffusion terms. :sup:‡ The manufactured solution is imposed on all boundaries (i.e., Dirichlet specification). .. _results-3: Results ~~~~~~~ Numerical solutions were obtained using both Superbee and Central discretization schemes for 8x8, 16x16, 32x32, 64x64, and 128x128 grid meshes. The Superbee scheme order of accuracy tests show a first-order rate of convergence for pressure under the :math:L_{\infty}\ \ norm as illustrated in :numref:fig10 (a), whereas the formal order for this scheme is two. The largest errors in pressure are local to boundary cells along the West (y=0) and South (x=0) edges of the domain as shown in :numref:fig11 (a). This is an artifact of the staggered grid implementation in MFIX where only a single ghost cell layer is present along West and South boundaries, reducing higher-order upwind schemes to first-order. This effect also occurs along the Bottom (z=0) edge of the domain for three-dimensional simulations. Further investigation is needed to determine to what extent the errors introduced at the boundary propagate into the domain interior. .. _fig10: .. figure:: /media/image13.png :align: center Observed orders of accuracy for 2D, single-phase, sinusoidal manufactured solution. (a) Superbee scheme, (b) Central scheme. .. _fig11: .. figure:: /media/image15.png :align: center Errors in pressure for 2D, single-phase, sinusoidal manufactured solution for grid resolution (64x64). (a) Superbee scheme, (b) Central scheme The Central scheme results, depicted in :numref:fig10 (b), show second order accuracy for all variables. The formal order for the Central scheme is recovered because no up-winding is performed, thereby averting solution deterioration at the boundaries. The errors in pressure near the boundaries are consistent with the scheme’s formal order as can be seen from :numref:fig11 (b). Notes ~~~~~ During initial testing, it was discovered that the strain-tensor cross terms for the momentum equations were not calculated within steady-state sub-iterations which lead to large errors (not shown). These errors do not appear in cases with zero shear at the boundaries. Transient simulations recalculate these cross-terms at the start of each time-step making it difficult to determine the effect on the solution. The significance of this simplification (likely done to reduce computational expense) on real-world application problems is unknown and should be investigated. For MMS tests, this issue was circumvented by recalculating the cross-terms of the strain-tensor at each sub-iteration.