From 417f0922ea85c874227948fd64df9a66bc6b13f3 Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Fri, 10 Jul 2026 07:34:03 -0400 Subject: [PATCH 1/6] Tutorials: Add flow around sphere and natural convection --- docs/source_docs/refs.bib | 11 + ...ady-backward-facing-step-richardson_L1.png | Bin ...ady-backward-facing-step-richardson_Lu.png | Bin ...g-step-streamfunction_contours-no-fill.png | Bin ...rd-facing-step-streamfunction_contours.png | Bin ...kward-facing-step-u_velocity_all_cases.png | Bin ...kward-facing-step-v_velocity_all_cases.png | Bin ...facing-step-vorticity_contours-no-fill.png | Bin ...ackward-facing-step-vorticity_contours.png | Bin ...ard-facing-step-wall_pressure_profiles.png | Bin ...steady-flow-around-cylinder-drag-coeff.png | Bin ...steady-flow-around-cylinder-lift-coeff.png | Bin .../images/steady-flow-around-cylinder.png | Bin ...y-flow-around-sphere-midplane_contour.png} | Bin ...teady-flow-around-sphere-force_history.png | Bin 0 -> 504558 bytes ...y-flow-around-sphere-midplane_snapshot.png | Bin 0 -> 481164 bytes ...y-flow-around-sphere-strouhal_spectrum.png | Bin 0 -> 142519 bytes ...eady-flow-around-sphere-windowed_stats.png | Bin 0 -> 653329 bytes docs/source_docs/tutorials/fluid/index.rst | 27 ++ .../steady-backward-facing-step.rst | 2 +- .../steady-flow-around-cylinder.rst | 2 +- .../steady-flow-around-sphere.rst | 4 +- .../fluid/unsteady-flow-around-sphere.rst | 234 ++++++++++++++++++ docs/source_docs/tutorials/index.rst | 64 ++++- 24 files changed, 332 insertions(+), 12 deletions(-) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-richardson_L1.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-richardson_Lu.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-streamfunction_contours-no-fill.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-streamfunction_contours.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-u_velocity_all_cases.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-v_velocity_all_cases.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-vorticity_contours-no-fill.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-vorticity_contours.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-backward-facing-step-wall_pressure_profiles.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-flow-around-cylinder-drag-coeff.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-flow-around-cylinder-lift-coeff.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/images/steady-flow-around-cylinder.png (100%) rename docs/source_docs/tutorials/{incompressible-fluid/images/steady-flow-past-sphere-midplane_contour.png => fluid/images/steady-flow-around-sphere-midplane_contour.png} (100%) create mode 100644 docs/source_docs/tutorials/fluid/images/unsteady-flow-around-sphere-force_history.png create mode 100644 docs/source_docs/tutorials/fluid/images/unsteady-flow-around-sphere-midplane_snapshot.png create mode 100644 docs/source_docs/tutorials/fluid/images/unsteady-flow-around-sphere-strouhal_spectrum.png create mode 100644 docs/source_docs/tutorials/fluid/images/unsteady-flow-around-sphere-windowed_stats.png create mode 100644 docs/source_docs/tutorials/fluid/index.rst rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/steady-backward-facing-step.rst (99%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/steady-flow-around-cylinder.rst (98%) rename docs/source_docs/tutorials/{incompressible-fluid => fluid}/steady-flow-around-sphere.rst (98%) create mode 100644 docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst diff --git a/docs/source_docs/refs.bib b/docs/source_docs/refs.bib index 6fa2b3d..cc77c42 100644 --- a/docs/source_docs/refs.bib +++ b/docs/source_docs/refs.bib @@ -551,6 +551,17 @@ doi = {https://doi.org/10.1016/j.jcp.2022.111305}, pages={385–400} } +@article{sakamoto90, + author = {Sakamoto, H. and Haniu, H.}, + title = {A Study on Vortex Shedding From Spheres in a Uniform Flow}, + journal = {Journal of Fluids Engineering}, + volume = {112}, + number = {4}, + pages = {386-392}, + year = {1990}, + month = {12}, +} + @article{sato81, author = {Sato, Y. and Sadatomi, M.}, title = {Momentum and heat transfer in two-phase bubble flow - I. Theory}, diff --git a/docs/source_docs/tutorials/incompressible-fluid/images/steady-backward-facing-step-richardson_L1.png b/docs/source_docs/tutorials/fluid/images/steady-backward-facing-step-richardson_L1.png similarity index 100% rename from docs/source_docs/tutorials/incompressible-fluid/images/steady-backward-facing-step-richardson_L1.png rename to docs/source_docs/tutorials/fluid/images/steady-backward-facing-step-richardson_L1.png diff --git a/docs/source_docs/tutorials/incompressible-fluid/images/steady-backward-facing-step-richardson_Lu.png b/docs/source_docs/tutorials/fluid/images/steady-backward-facing-step-richardson_Lu.png similarity index 100% rename from docs/source_docs/tutorials/incompressible-fluid/images/steady-backward-facing-step-richardson_Lu.png rename to docs/source_docs/tutorials/fluid/images/steady-backward-facing-step-richardson_Lu.png diff --git 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z?}Y+r^yJ5(snLZ$E3*sFdH%KAl;CgITn9$<-7}qOO#heEnD?N z*|ywu1*;CrL;0v3j(?Z4tl94Nu>a!^_TZV@FN!b(%{{!wz#~-YPqL8X{O?Okck!(d zIoZE$rFKULsD`#baLk{nBssNV*Nn8NN$j)3v?iJ7<#!v-U-JCWjcH3&CTTyi@p3H+ z%?rzyIr6TI`8yQbl2|lhkf3y z$t~Ste5%j(v4TqF(HkOxR}$U9be-ZFa>> zdD4RfnGHJT51++dzZk62|Ki0L6NgtTLuA9=`pRBBn50*DvaXANUmdfYWx*O<-B|Wr zn@^b~B;C%DauHtb>D!nqI7QnpqsVlH@AE%FthS1cb2PYuZ6?k+)RNeHiz(DJ&1kCe zRU3JoRk!om54z9r(rEM#F=ff(+xDa^)KhY1@>(^QSL{|5u6LKLj{K0|`M;szXvd diff --git a/docs/source_docs/tutorials/index.rst b/docs/source_docs/tutorials/index.rst index 2b616fe..37e9329 100644 --- a/docs/source_docs/tutorials/index.rst +++ b/docs/source_docs/tutorials/index.rst @@ -1,12 +1,60 @@ -Incompressible Fluid -==================== +.. _Tutorials: -These tutorials describe complete MFIX-Exa cases and the main input choices -used to build them. +Overview +======== + +These tutorials walk through setting up, running, and analyzing MFIX-Exa +cases, starting from simple, well-characterized flows and building toward +more complex setups. Each tutorial includes the complete a description of +the important input sections, and an analysis of the results. + +Tutorials are grouped by the physical model they use. Within each category, +tutorials vary in which additional physical capabilities they exercise -- for +example, the energy equation, species transport, reactions, or a rotating +reference frame. The table below summarizes this across all tutorials. + +.. list-table:: Tutorial capability overview + :header-rows: 1 + :widths: 28 14 10 10 14 12 12 + + * - Tutorial + - Category + - Steady/Unsteady + - Energy + - Species/Reactions + - Rotating Frame + - Multi-Fluid + * - :ref:`Steady flow around a cylinder ` + - Fluid + - Steady + - + - + - + - + * - :ref:`Steady backward-facing step ` + - Fluid + - Steady + - + - + - + - + * - :ref:`Steady flow around a sphere ` + - Fluid + - Steady + - + - + - + - + * - :ref:`Unsteady flow around a sphere ` + - Fluid + - Unsteady + - + - + - + - .. toctree:: - :maxdepth: 1 + :maxdepth: 2 + :caption: Tutorials - incompressible-fluid/steady-backward-facing-step - incompressible-fluid/steady-flow-around-cylinder - incompressible-fluid/steady-flow-around-sphere + fluid/index -- GitLab From 532713ea54a1f0dd7e3b9dbbcaab788c785fb3eb Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Fri, 10 Jul 2026 08:44:47 -0400 Subject: [PATCH 2/6] Add prelim steady natural convection cavity --- docs/source_docs/refs.bib | 30 ++ ...nvection-cavity-nu_convergence_history.png | Bin 0 -> 132128 bytes ...l-convection-cavity-richardson_Nu_xmax.png | Bin 0 -> 217477 bytes ...l-convection-cavity-richardson_Nu_xmin.png | Bin 0 -> 214420 bytes ...al-convection-cavity-temperature-field.png | Bin 0 -> 288600 bytes docs/source_docs/tutorials/fluid/index.rst | 1 + .../steady-natural-convection-cavity.rst | 331 ++++++++++++++++++ docs/source_docs/tutorials/index.rst | 9 + 8 files changed, 371 insertions(+) create mode 100644 docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-nu_convergence_history.png create mode 100644 docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-richardson_Nu_xmax.png create mode 100644 docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-richardson_Nu_xmin.png create mode 100644 docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-temperature-field.png create mode 100644 docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst diff --git a/docs/source_docs/refs.bib b/docs/source_docs/refs.bib index cc77c42..26558b6 100644 --- a/docs/source_docs/refs.bib +++ b/docs/source_docs/refs.bib @@ -214,6 +214,17 @@ doi="10.1007/978-1-4613-8643-8_2" } +@article{devahldavis83, + author = {de Vahl Davis, G.}, + title = {Natural convection of air in a square cavity: a bench mark numerical solution}, + journal = {International Journal for Numerical Methods in Fluids}, + volume = {3}, + number = {3}, + pages = {249--264}, + year = {1983}, + doi = {10.1002/fld.1650030305} +} + @article{ding90, author = {Ding, Jianmin and Gidaspow, Dimitri}, title = {A bubbling fluidization model using kinetic theory of granular flow}, @@ -364,6 +375,16 @@ doi = {https://doi.org/10.1016/j.jcp.2022.111305}, DOI={10.1017/S0022112094000996}, } +@article{hortmann90, + author = {Hortmann, M. and Peri\'{c}, M. and Scheuerer, G.}, + title = {Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions}, + journal = {International Journal for Numerical Methods in Fluids}, + volume = {11}, + number = {2}, + pages = {189--207}, + year = {1990}, + doi = {10.1002/fld.1650110206} +} @article{Ish79, author = {Ishii, M. and Zuber, N. }, @@ -421,6 +442,15 @@ doi = {https://doi.org/10.1016/j.jcp.2022.111305}, pages = {2297-2304} } +@article{lequere2005, + author = {Le Quere, P. and Weisman, C. and Paillere, H. and Vierendeels, J. and Dick, E. and Becker, R. and Braack, M. and Locke, J.}, + title = {Modelling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part 1. Reference solution}, + journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, + volume = {39}, + pages = {609--616}, + year = {2005}, + doi = {10.1051/m2an:2005027} +} @techreport{Lilly66, author={Lilly, D.K.}, diff --git a/docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-nu_convergence_history.png b/docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-nu_convergence_history.png new file mode 100644 index 0000000000000000000000000000000000000000..7e1bf55d83b9a61397b3b37c05eeadf5b690c1cd GIT binary patch literal 132128 zcmeAS@N?(olHy`uVBq!ia0y~y;L2lQU{&E@V_;yo!C!HSfq}uQ#5JNMI6tkVJh3R1 z!8fs_ASb^hCo@T*EVZaOGe6H*&(J{6Qb!@9q@Z|X3hf)oq$aCKB6h7T(i6bPe=9q`n+!-6VQAjD

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a/docs/source_docs/tutorials/fluid/index.rst +++ b/docs/source_docs/tutorials/fluid/index.rst @@ -15,6 +15,7 @@ fluids by density within the same solver, such as a Rayleigh-Taylor case. steady-backward-facing-step steady-flow-around-sphere unsteady-flow-around-sphere + steady-natural-convection-cavity Planned --------- diff --git a/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst new file mode 100644 index 0000000..ecfa2b6 --- /dev/null +++ b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst @@ -0,0 +1,331 @@ +.. _TutorialSteadyNaturalConvectionCavity: + +Steady natural convection in a differentially heated cavity +============================================================== + +This tutorial sets up buoyancy-driven flow in a closed, differentially +heated square cavity -- the classical de Vahl Davis benchmark +:cite:p:`devahldavis83`. Unlike the earlier :ref:`sphere +` tutorials, this case exercises MFIX-Exa's +energy equation and ideal-gas equation of state: buoyancy comes directly +from a real density variation with temperature, not a linearized +Boussinesq source term (MFIX-Exa has no Boussinesq mode). A grid-refinement +study validates the computed Nusselt number against the benchmark. + +Features +-------- + +* Incompressible-free, buoyancy-driven fluid flow with no particles. +* Ideal-gas equation of state in a sealed (closed) domain -- + ``mfix.constraint = IdealGasClosedSystem``. +* Steady-state pseudo-time advance. +* Vertical hot/cold walls represented as an embedded boundary; adiabatic + top/bottom walls handled as plain domain boundary conditions. +* Quasi-2D domain using a thin, periodic spanwise (:math:`z`) direction. +* Grid-refinement study (:math:`64^2`, :math:`128^2`, :math:`256^2`) with + Richardson extrapolation of the wall Nusselt number, compared against + de Vahl Davis :cite:p:`devahldavis83` and Hortmann et al. + :cite:p:`hortmann90`. + +Case description +----------------- + +The cavity is a unit cube in :math:`x`-:math:`y`, thin and periodic in +:math:`z`. The left (:math:`x=-0.5`) and right (:math:`x=+0.5`) walls are +held at fixed temperatures :math:`T_h=310\text{ K}` and +:math:`T_l=290\text{ K}`; the top and bottom walls are adiabatic, no-slip. +Gravity points in :math:`-y`. + +.. math:: + + Ra = \frac{g\,\beta\,\Delta T\,L^3}{\nu\,\alpha} = 1\times10^6, + \qquad Pr = \frac{\nu}{\alpha} = 0.717, + \qquad \frac{\Delta T}{T_{ref}} = \frac{20}{300} = 6.7\% + +with :math:`\beta=1/T_{ref}` for an ideal gas. The temperature difference +is kept small relative to :math:`T_{ref}` deliberately, so the ideal-gas +solve stays close to the Boussinesq limit that the classical benchmark +assumes -- this is a near-Boussinesq validation case, not a +large-temperature-difference one. (A companion case at +:math:`\Delta T/T_{ref}=120\%`, validating against the compressible +benchmark of Le Quere :cite:p:`lequere2005` instead, is a natural +extension -- see below.) + +To keep the domain and mesh clean for a tutorial (:math:`L=1`, +:math:`n_{cell}=64/128/256`) rather than reproducing a literal small +physical box, the viscosity and thermal conductivity are solved for +directly from the target :math:`Ra`, :math:`Pr`, :math:`\Delta T`, and +:math:`L`, rather than taken as literal air properties: + +.. math:: + + \mu = 8.061\times10^{-4}\text{ Pa}\cdot\text{s}, \qquad + \lambda = 1.1292\text{ W/m/K} + +This is a dimensionless match, not literal air -- the same approach used +for most cavity benchmarks in the literature. The specific heat and +molecular weight *are* real air values (:math:`C_p=1004.5` J/kg/K, +:math:`M=0.028970` kg/mol), and are checked for a physically valid ideal +gas before use: + +.. math:: + + C_v = C_p - \frac{R}{M} = 1004.5 - \frac{8.31446}{0.028970} + = 717.6\text{ J/kg/K} > 0, \qquad \gamma = C_p/C_v \approx 1.400 + +.. note:: + + A negative :math:`C_v` (i.e. :math:`C_p < R/M` for the chosen molecular + weight) is unphysical and has been observed to cause the solver to + either fail to converge or converge to the wrong solution. Always check + this before trusting a new fluid-property set. + +Important input sections +-------------------------- + +The domain is a unit cube, periodic in :math:`z`: + +.. code-block:: text + + geometry.is_periodic = 0 0 1 + geometry.prob_lo = -0.5 -0.5 0.0 + geometry.prob_hi = 0.5 0.5 0.125 + + amr.n_cell = 64 64 8 + +The hot/cold walls are represented as an embedded boundary covering the +:math:`x` faces of the domain; :math:`y` (top/bottom) is left open and +handled by ordinary domain boundary conditions, and :math:`z` is periodic: + +.. code-block:: text + + mfix.geometry = box + + box.Lo = -0.5 -1.0 -1.0 + box.Hi = 0.5 1.0 1.0 + + box.internal_flow = true + box.offset = 1.e-8 + + ... + + bc.left = eb + bc.left.eb.normal = 1. 0. 0. + bc.left.eb.temperature = constant + bc.left.eb.temperature.constant = 310. # Th + + bc.right = eb + bc.right.eb.normal = -1. 0. 0. + bc.right.eb.temperature = constant + bc.right.eb.temperature.constant = 290. # Tl + + bc.top = no-slip # adiabatic (no temperature spec) + bc.bottom = no-slip + +The energy equation and the ideal-gas equation of state are enabled +together in a sealed domain: + +.. code-block:: text + + mfix.advect_density = 1 + mfix.advect_enthalpy = 1 + mfix.constraint = IdealGasClosedSystem + +.. warning:: + + The specific-heat key must be given without the ``.model`` suffix: + ``fluid.specific_heat = constant``, *not* + ``fluid.specific_heat.model = constant``. The latter is the + "corrected" form of several other deprecation-warning fixes + (``fluid.viscosity.molecular.model``, + ``fluid.thermal_conductivity.model``) and looks equally valid, but it + fails to wire up the temperature-to-enthalpy conversion needed when + setting the initial condition, causing the solve to fail at + initialization. ``fluid.thermal_conductivity.model`` does *not* have + this problem -- it appears specific to ``specific_heat``. + +.. code-block:: text + + fluid.specific_heat = constant + fluid.specific_heat.constant = 1004.5 + + fluid.viscosity.molecular.model = constant + fluid.viscosity.molecular.constant = 8.061e-4 + + fluid.thermal_conductivity.model = constant + fluid.thermal_conductivity.constant = 1.1292 + +The case runs to a pseudo steady state: + +.. code-block:: text + + mfix.steady_state = 1 + mfix.steady_state_tol = 1.e-6 + mfix.steady_state_maxiter = 100000 + + mfix.dt_max = 1.0e-2 + +The sharp jump between the uniform :math:`300\text{ K}` initial condition +and the :math:`310\text{ K}`/:math:`290\text{ K}` EB wall temperatures at +:math:`t=0` produces an ill-conditioned solve at severely cut cells with +the default bottom solver; ``hypre`` avoids this: + +.. code-block:: text + + mac_proj.bottom_solver = hypre + nodal_proj.bottom_solver = hypre + +.. note:: + + The complete input file is located in the MFIX-Exa source directory as + ``tutorials/fluid/inputs.natural-convection-cavity`` + +Post-processing +----------------- + +Nusselt number +~~~~~~~~~~~~~~ + +The local Nusselt number at each vertical wall is + +.. math:: + + Nu = \frac{L}{T_h-T_l}\left.\frac{\partial T}{\partial x}\right|_{wall} + +computed with a one-sided finite-difference stencil, +:math:`\partial T/\partial x = (2T_1-3T_2+T_3)/\Delta x`, on the first +three cell-center planes off each wall and averaged over :math:`y`. + +.. _fig_steady_natural_convection_cavity_temperature_field: + +.. figure:: ./images/steady-natural-convection-cavity-temperature-field.png + :width: 60% + :align: center + :alt: Dimensionless temperature field in the differentially heated cavity, no radiation, Ra=1e6. + + Dimensionless temperature field :math:`\tilde{T}=(T-T_l)/(T_h-T_l)`, :math:`128^2` grid. + +The field shows the expected structure for :math:`Ra=10^6`: thin thermal +boundary layers along the hot and cold walls, a stably stratified core, +and a single large recirculating cell -- consistent with de Vahl Davis's +description at this Rayleigh number. + +Grid convergence +~~~~~~~~~~~~~~~~ + +.. list-table:: Wall Nusselt number vs. grid, Ra=1e6 + :header-rows: 1 + + * - Mesh + - :math:`Nu_{xmin}` (hot wall) + - :math:`Nu_{xmax}` (cold wall) + * - :math:`64\times64` + - 9.7986 + - 9.8098 + * - :math:`128\times128` + - 9.0170 + - 9.0240 + * - :math:`256\times256` + - 8.8572 + - 8.8584 + * - de Vahl Davis / Hortmann et al. + - 8.80--8.83 + - 8.80--8.83 + +Richardson extrapolation across the three grids gives an observed order of +accuracy of :math:`p\approx2.29` (hot wall) / :math:`2.25` (cold wall) and +an extrapolated value at :math:`\Delta x\to0` within 0.1% of the benchmark: + +.. list-table:: Richardson extrapolation vs. de Vahl Davis :cite:p:`devahldavis83` / Hortmann et al. :cite:p:`hortmann90` + :header-rows: 1 + + * - Quantity + - Observed order :math:`p` + - Extrapolated (:math:`\Delta x\to0`) + - Reference + - % difference + * - :math:`Nu_{xmin}` + - 2.290 + - 8.8162 + - 8.82 + - -0.04% + * - :math:`Nu_{xmax}` + - 2.246 + - 8.8141 + - 8.82 + - -0.07% + +.. _fig_steady_natural_convection_cavity_richardson: + +.. figure:: ./images/steady-natural-convection-cavity-richardson_Nu_xmin.png + :width: 90% + :align: center + :alt: Richardson extrapolation of the hot-wall Nusselt number across three grid levels. + + Richardson extrapolation, :math:`Nu_{xmin}` (hot wall). + +Both walls agree with each other to within 0.02 in :math:`Nu` at every +grid level, as expected from overall energy balance at steady state, and +both converge cleanly toward the classical benchmark as the mesh is +refined. + +Steady-state convergence check +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +``mfix.steady_state_tol`` being satisfied confirms the residual dropped +below tolerance, not that the converged state is the physically correct +one -- a run can converge to the wrong solution (observed previously with +an unphysical, negative-:math:`C_v` fluid-property set). Tracking +:math:`Nu` and the mean thermodynamic pressure over the last several +plotfiles confirms a genuine plateau rather than a false-positive stop: + +.. figure:: ./images/steady-natural-convection-cavity-nu_convergence_history.png + :width: 80% + :align: center + :alt: Nusselt number history over the run, 128x128 grid, showing a fast initial transient followed by a flat plateau. + + :math:`Nu` convergence history, :math:`128\times128`. + +Excluding the initial transient (first output only), a block-bootstrap +estimate over the last 13 plotfiles gives +:math:`Nu_{xmin}=9.0188\pm0.0015` and :math:`Nu_{xmax}=9.0248\pm0.0009` +(95% CI) -- a tight, flat plateau consistent with the point value used in +the grid-convergence table above, not a still-drifting solution. + +Thermodynamic pressure check +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As a second, independent, mesh-cheap validation check, mass conservation +in the closed ideal-gas domain fixes the domain-mean thermodynamic +pressure relative to the wall temperatures. For a purely conductive +profile, this works out to the ratio of the logarithmic mean wall +temperature to the reference temperature: + +.. math:: + + \frac{P_{mean}}{P_o} = \frac{T_{lm}}{T_{ref}}, \qquad + T_{lm} = \frac{T_h-T_l}{\ln(T_h/T_l)} + +For this case, :math:`T_{lm}/T_{ref}=0.99963`, matching the computed +:math:`P_{mean}/P_o=0.99960` to within 0.004% -- and essentially +unchanged across all three grids (0.999592 / 0.999599 / 0.999600), since +it is a domain-integral quantity dominated by the fixed wall temperatures +rather than a locally-resolved wall gradient. This makes it a useful, +cheap sanity check that is already converged at the coarsest mesh, well +before :math:`Nu` itself finishes converging. + +Extensions +----------- + +* A :math:`512\times512` grid level is a natural addition to the + Richardson study above, if available. +* A companion case at :math:`\Delta T/T_{ref}=120\%` + (:math:`T_h=960\text{ K}`, :math:`T_l=240\text{ K}`, + :math:`T_{ref}=600\text{ K}`, same :math:`Ra`, :math:`Pr`) tests how far + the ideal-gas solve departs from the Boussinesq benchmark as the + temperature difference grows, validated instead against Le Quere + :cite:p:`lequere2005`. The thermodynamic-pressure check above is + expected to show a much larger, and more convection-sensitive, + departure from its pure-conduction prediction at that :math:`\Delta T`. +* Radiative coupling between the walls (not covered here) is a further + extension once the pure-convection case above is finalized. diff --git a/docs/source_docs/tutorials/index.rst b/docs/source_docs/tutorials/index.rst index 37e9329..1e69b66 100644 --- a/docs/source_docs/tutorials/index.rst +++ b/docs/source_docs/tutorials/index.rst @@ -13,6 +13,8 @@ tutorials vary in which additional physical capabilities they exercise -- for example, the energy equation, species transport, reactions, or a rotating reference frame. The table below summarizes this across all tutorials. +.. |check| unicode:: U+2713 + .. list-table:: Tutorial capability overview :header-rows: 1 :widths: 28 14 10 10 14 12 12 @@ -52,6 +54,13 @@ reference frame. The table below summarizes this across all tutorials. - - - + * - :ref:`Steady natural convection ` + - Fluid + - Steady + - |check| + - + - + - .. toctree:: :maxdepth: 2 -- GitLab From 2ad4e578490f480b56f118afda5c6662060760ef Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Fri, 10 Jul 2026 15:14:35 -0400 Subject: [PATCH 3/6] Update with lastest results --- ...nvection-cavity-nu_convergence_history.png | Bin 132128 -> 125161 bytes ...l-convection-cavity-richardson_Nu_xmax.png | Bin 217477 -> 230388 bytes ...l-convection-cavity-richardson_Nu_xmin.png | Bin 214420 -> 225311 bytes ...al-convection-cavity-temperature-field.png | Bin 288600 -> 291252 bytes docs/source_docs/tutorials/index.rst | 38 +++++++++--------- 5 files changed, 18 insertions(+), 20 deletions(-) diff --git a/docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-nu_convergence_history.png b/docs/source_docs/tutorials/fluid/images/steady-natural-convection-cavity-nu_convergence_history.png index 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b/docs/source_docs/tutorials/index.rst @@ -15,52 +15,50 @@ reference frame. The table below summarizes this across all tutorials. .. |check| unicode:: U+2713 + +Fluid +----- + .. list-table:: Tutorial capability overview :header-rows: 1 - :widths: 28 14 10 10 14 12 12 + :widths: 28 14 14 10 12 12 * - Tutorial - - Category - - Steady/Unsteady + - Time advance + - Reference Frame - Energy - - Species/Reactions - - Rotating Frame + - Species - Multi-Fluid - * - :ref:`Steady flow around a cylinder ` - - Fluid + * - :ref:`Flow around a cylinder ` - Steady + - Inertial - - - - - - * - :ref:`Steady backward-facing step ` - - Fluid + * - :ref:`Backward-facing step ` - Steady + - Inertial - - - - - - * - :ref:`Steady flow around a sphere ` - - Fluid + * - :ref:`Flow around a sphere ` - Steady + - Inertial - - - - - - * - :ref:`Unsteady flow around a sphere ` - - Fluid + * - :ref:`Flow around a sphere ` - Unsteady + - Inertial - - - - - - * - :ref:`Steady natural convection ` - - Fluid + * - :ref:`Natural convection in square cavity` - Steady + - Inertial - |check| - - - - .. toctree:: :maxdepth: 2 -- GitLab From 9f784d5eff508a38374fcb3066041469f77f3ede Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Fri, 10 Jul 2026 15:33:06 -0400 Subject: [PATCH 4/6] Updates and missed files --- ...nsteady-flow-around-sphere-384x256x256.mp4 | Bin 0 -> 3670015 bytes ...nvection-cavity-nu_convergence_history.png | Bin 125161 -> 125032 bytes ...l-convection-cavity-richardson_Nu_xmax.png | Bin 230388 -> 230457 bytes ...l-convection-cavity-richardson_Nu_xmin.png | Bin 225311 -> 225199 bytes ...al-convection-cavity-temperature-field.png | Bin 291252 -> 291208 bytes 5 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 docs/source_docs/_static/animations/unsteady-flow-around-sphere-384x256x256.mp4 diff --git a/docs/source_docs/_static/animations/unsteady-flow-around-sphere-384x256x256.mp4 b/docs/source_docs/_static/animations/unsteady-flow-around-sphere-384x256x256.mp4 new file mode 100644 index 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Lari and M. Baneshi and S.A. {Gandjalikhan Nassab} and A. Komiya and S. Maruyama}, +title = {Combined heat transfer of radiation and natural convection in a square cavity containing participating gases}, +journal = {International Journal of Heat and Mass Transfer}, +volume = {54}, +number = {23}, +pages = {5087-5099}, +year = {2011} +} + @article{Lat011, author = {Lathouwers, D. and Bellan, J. }, title = {Modeling of dense gas-solid reative mixtures applied to biomass pyrolysis in a fluidized bed}, diff --git a/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst b/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst index 9bb8e51..3feeafc 100644 --- a/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst +++ b/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst @@ -11,8 +11,8 @@ a sudden expansion and the effect of the pressure-outlet boundary condition. Features -------- -* Incompressible fluid with no particles. -* Steady-state advance. +* Incompressible fluid flow. +* Steady-state pseudo-time advance. * Backward-facing step represented by the inlet velocity profile. * Parabolic velocity inlet, pressure outlet, no-slip walls, and a periodic spanwise direction. diff --git a/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst b/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst index 5836a88..8408715 100644 --- a/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst +++ b/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst @@ -10,8 +10,8 @@ cylinder-flow example that uses MFIX-Exa's embedded-boundary geometry. Features -------- -* Incompressible fluid with no particles. -* Steady-state advance +* Incompressible fluid flow. +* Steady-state pseudo-time advance * Quasi-2D domain using a thin, periodic spanwise direction. * Cylinder represented by a predefined embedded boundary. * Velocity inlet, pressure outlet, no-slip walls, and a periodic spanwise direction. diff --git a/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst index e7f9829..c398eef 100644 --- a/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst +++ b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst @@ -16,8 +16,8 @@ length are computed as part of the analysis. Features -------- -* Incompressible fluid with no particles. -* Steady-state advance. +* Incompressible fluid flow. +* Steady-state pseudo-time advance. * Fully three-dimensional embedded-boundary sphere geometry. * Uniform freestream mass inflow, pressure outlet, and free-slip lateral boundaries approximating an unconfined far field. diff --git a/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst index ecfa2b6..85cc6af 100644 --- a/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst +++ b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst @@ -15,7 +15,7 @@ study validates the computed Nusselt number against the benchmark. Features -------- -* Incompressible-free, buoyancy-driven fluid flow with no particles. +* Buoyancy-driven fluid flow. * Ideal-gas equation of state in a sealed (closed) domain -- ``mfix.constraint = IdealGasClosedSystem``. * Steady-state pseudo-time advance. @@ -30,55 +30,63 @@ Features Case description ----------------- -The cavity is a unit cube in :math:`x`-:math:`y`, thin and periodic in -:math:`z`. The left (:math:`x=-0.5`) and right (:math:`x=+0.5`) walls are -held at fixed temperatures :math:`T_h=310\text{ K}` and -:math:`T_l=290\text{ K}`; the top and bottom walls are adiabatic, no-slip. -Gravity points in :math:`-y`. +The cavity is a unit cube in :math:`x`-:math:`y` with characteristic length +:math:`L=1\text{ m}`, thin and periodic in :math:`z`. The left +(:math:`x=-0.5`) and right (:math:`x=+0.5`) walls are held at fixed +temperatures :math:`T_h=310\text{ K}` and :math:`T_l=290\text{ K}`. The top +and bottom walls are adiabatic and no-slip. Gravity points in the :math:`-y` +direction. This case is characterized by the Rayleigh number (:math:`Ra`) +and Prandtl number (:math:`Pr`): .. math:: Ra = \frac{g\,\beta\,\Delta T\,L^3}{\nu\,\alpha} = 1\times10^6, - \qquad Pr = \frac{\nu}{\alpha} = 0.717, - \qquad \frac{\Delta T}{T_{ref}} = \frac{20}{300} = 6.7\% - -with :math:`\beta=1/T_{ref}` for an ideal gas. The temperature difference -is kept small relative to :math:`T_{ref}` deliberately, so the ideal-gas -solve stays close to the Boussinesq limit that the classical benchmark -assumes -- this is a near-Boussinesq validation case, not a -large-temperature-difference one. (A companion case at + \qquad Pr = \frac{\mu c_p}{\kappa} = 0.717 + +where :math:`\beta=1/T_{ref}` for an ideal gas fluid. + +This setup is a near-Boussinesq validation case; therefore, the temperature +difference is kept small relative to the reference temperature +:math:`T_{ref} = 300\text{ K}`: + +.. math:: + + \frac{\Delta T}{T_{ref}} = \frac{20}{300} = 6.7\% + +This approach keeps the ideal-gas solve close to the Boussinesq limit that +the classical benchmark assumes. (A companion case at :math:`\Delta T/T_{ref}=120\%`, validating against the compressible -benchmark of Le Quere :cite:p:`lequere2005` instead, is a natural +benchmark of Le Quéré :cite:p:`lequere2005` instead, is a natural extension -- see below.) -To keep the domain and mesh clean for a tutorial (:math:`L=1`, -:math:`n_{cell}=64/128/256`) rather than reproducing a literal small -physical box, the viscosity and thermal conductivity are solved for -directly from the target :math:`Ra`, :math:`Pr`, :math:`\Delta T`, and -:math:`L`, rather than taken as literal air properties: +The viscosity (:math:`\mu`) and thermal conductivity (:math:`\kappa`) are +solved for from the target :math:`Ra`, :math:`Pr`, :math:`\Delta T`, and +:math:`L`. .. math:: \mu = 8.061\times10^{-4}\text{ Pa}\cdot\text{s}, \qquad - \lambda = 1.1292\text{ W/m/K} + \kappa = 1.1292\text{ W/(m}\cdot\text{K)} -This is a dimensionless match, not literal air -- the same approach used -for most cavity benchmarks in the literature. The specific heat and -molecular weight *are* real air values (:math:`C_p=1004.5` J/kg/K, -:math:`M=0.028970` kg/mol), and are checked for a physically valid ideal -gas before use: +The specific heat and molecular weight are prescribed constants, +:math:`C_p=1004.5\text{ J/(kg}\cdot\text{K)}` and :math:`M=0.028970\text{ kg/mol}`, +respectively. The values are checked for a physically valid ideal gas before use: .. math:: C_v = C_p - \frac{R}{M} = 1004.5 - \frac{8.31446}{0.028970} - = 717.6\text{ J/kg/K} > 0, \qquad \gamma = C_p/C_v \approx 1.400 + = 717.5\text{ J/(kg}\cdot\text{K)} > 0, \qquad \gamma = C_p/C_v \approx 1.400 -.. note:: - A negative :math:`C_v` (i.e. :math:`C_p < R/M` for the chosen molecular - weight) is unphysical and has been observed to cause the solver to - either fail to converge or converge to the wrong solution. Always check - this before trusting a new fluid-property set. +.. warning:: + + When choosing fluid properties to satisfy specific dimensionless groups + (like :math:`Ra` and :math:`Pr`), always verify that the resulting + thermodynamic properties remain physical. Arbitrarily adjusting + :math:`C_p` or molecular weight without this check can accidentally + yield a negative constant-volume specific heat (:math:`C_v \le 0`), + leading to an unphysical ideal gas state and solver divergence. + Important input sections -------------------------- @@ -94,8 +102,9 @@ The domain is a unit cube, periodic in :math:`z`: amr.n_cell = 64 64 8 The hot/cold walls are represented as an embedded boundary covering the -:math:`x` faces of the domain; :math:`y` (top/bottom) is left open and -handled by ordinary domain boundary conditions, and :math:`z` is periodic: +:math:`x` faces of the domain; :math:`y` (top/bottom) are defined as +no-slip walls and handled by ordinary domain boundary conditions, and +:math:`z` is periodic: .. code-block:: text @@ -122,6 +131,7 @@ handled by ordinary domain boundary conditions, and :math:`z` is periodic: bc.top = no-slip # adiabatic (no temperature spec) bc.bottom = no-slip + The energy equation and the ideal-gas equation of state are enabled together in a sealed domain: @@ -129,31 +139,22 @@ together in a sealed domain: mfix.advect_density = 1 mfix.advect_enthalpy = 1 - mfix.constraint = IdealGasClosedSystem + mfix.constraint = IdealGasClosedSystem -.. warning:: - - The specific-heat key must be given without the ``.model`` suffix: - ``fluid.specific_heat = constant``, *not* - ``fluid.specific_heat.model = constant``. The latter is the - "corrected" form of several other deprecation-warning fixes - (``fluid.viscosity.molecular.model``, - ``fluid.thermal_conductivity.model``) and looks equally valid, but it - fails to wire up the temperature-to-enthalpy conversion needed when - setting the initial condition, causing the solve to fail at - initialization. ``fluid.thermal_conductivity.model`` does *not* have - this problem -- it appears specific to ``specific_heat``. +The calculated viscosity (:math:`\mu`), thermal conductivity (:math:`\kappa`), +and prescribed specific heat (\(:math:`C_p`\)) are set as constant fluid +properties in the configuration deck: .. code-block:: text - fluid.specific_heat = constant - fluid.specific_heat.constant = 1004.5 + fluid.specific_heat.model = constant + fluid.specific_heat.constant = 1004.5 fluid.viscosity.molecular.model = constant fluid.viscosity.molecular.constant = 8.061e-4 - fluid.thermal_conductivity.model = constant - fluid.thermal_conductivity.constant = 1.1292 + fluid.thermal_conductivity.model = constant + fluid.thermal_conductivity.constant = 1.1292 The case runs to a pseudo steady state: @@ -165,15 +166,6 @@ The case runs to a pseudo steady state: mfix.dt_max = 1.0e-2 -The sharp jump between the uniform :math:`300\text{ K}` initial condition -and the :math:`310\text{ K}`/:math:`290\text{ K}` EB wall temperatures at -:math:`t=0` produces an ill-conditioned solve at severely cut cells with -the default bottom solver; ``hypre`` avoids this: - -.. code-block:: text - - mac_proj.bottom_solver = hypre - nodal_proj.bottom_solver = hypre .. note:: @@ -181,7 +173,7 @@ the default bottom solver; ``hypre`` avoids this: ``tutorials/fluid/inputs.natural-convection-cavity`` Post-processing ------------------ +--------------- Nusselt number ~~~~~~~~~~~~~~ @@ -192,9 +184,13 @@ The local Nusselt number at each vertical wall is Nu = \frac{L}{T_h-T_l}\left.\frac{\partial T}{\partial x}\right|_{wall} -computed with a one-sided finite-difference stencil, -:math:`\partial T/\partial x = (2T_1-3T_2+T_3)/\Delta x`, on the first -three cell-center planes off each wall and averaged over :math:`y`. +computed with a second-order, one-sided finite-difference stencil on the first +three cell-center planes off the wall and averaged over :math:`y`. At the left (hot) +wall, this is evaluated as: + +.. math:: + + \left.\frac{\partial T}{\partial x}\right|_{wall} = \frac{-2T_1+3T_2-T_3}{\Delta x} .. _fig_steady_natural_convection_cavity_temperature_field: @@ -207,14 +203,14 @@ three cell-center planes off each wall and averaged over :math:`y`. The field shows the expected structure for :math:`Ra=10^6`: thin thermal boundary layers along the hot and cold walls, a stably stratified core, -and a single large recirculating cell -- consistent with de Vahl Davis's +and a single large recirculating cell—consistent with de Vahl Davis's description at this Rayleigh number. Grid convergence ~~~~~~~~~~~~~~~~ .. list-table:: Wall Nusselt number vs. grid, Ra=1e6 - :header-rows: 1 + :header-rows: 1 * - Mesh - :math:`Nu_{xmin}` (hot wall) @@ -228,32 +224,34 @@ Grid convergence * - :math:`256\times256` - 8.8572 - 8.8584 + * - :math:`512\times512` + - 8.8331 + - 8.8331 * - de Vahl Davis / Hortmann et al. - - 8.80--8.83 - - 8.80--8.83 + - 8.80 - 8.83 + - 8.80 - 8.83 -Richardson extrapolation across the three grids gives an observed order of -accuracy of :math:`p\approx2.29` (hot wall) / :math:`2.25` (cold wall) and -an extrapolated value at :math:`\Delta x\to0` within 0.1% of the benchmark: +Richardson extrapolation across the three finest grids gives an observed order of +accuracy of :math:`p\approx2.74` (hot wall) / :math:`2.72` (cold wall). The +extrapolated value at the continuum limit (:math:`\Delta x\to0`) lands at +the benchmark values from de Vahl Davis :cite:p:`devahldavis83` and +Hortmann et al. :cite:p:`hortmann90`: -.. list-table:: Richardson extrapolation vs. de Vahl Davis :cite:p:`devahldavis83` / Hortmann et al. :cite:p:`hortmann90` - :header-rows: 1 +.. list-table:: Richardson extrapolation vs. benchmark literature + :header-rows: 1 * - Quantity - Observed order :math:`p` - Extrapolated (:math:`\Delta x\to0`) - Reference - - % difference * - :math:`Nu_{xmin}` - - 2.290 - - 8.8162 + - 2.741 + - 8.8289 - 8.82 - - -0.04% * - :math:`Nu_{xmax}` - - 2.246 - - 8.8141 + - 2.717 + - 8.8287 - 8.82 - - -0.07% .. _fig_steady_natural_convection_cavity_richardson: @@ -264,61 +262,60 @@ an extrapolated value at :math:`\Delta x\to0` within 0.1% of the benchmark: Richardson extrapolation, :math:`Nu_{xmin}` (hot wall). -Both walls agree with each other to within 0.02 in :math:`Nu` at every -grid level, as expected from overall energy balance at steady state, and -both converge cleanly toward the classical benchmark as the mesh is -refined. +Both walls agree with each other in :math:`Nu` at every grid level, as expected +from overall energy balance at steady state, and both converge cleanly toward +the classical benchmark as the mesh is refined. Steady-state convergence check -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ``mfix.steady_state_tol`` being satisfied confirms the residual dropped below tolerance, not that the converged state is the physically correct -one -- a run can converge to the wrong solution (observed previously with -an unphysical, negative-:math:`C_v` fluid-property set). Tracking -:math:`Nu` and the mean thermodynamic pressure over the last several -plotfiles confirms a genuine plateau rather than a false-positive stop: +one Tracking :math:`Nu` and the mean thermodynamic pressure over the +last several plotfiles confirms a genuine plateau rather than a +false-positive stop: .. figure:: ./images/steady-natural-convection-cavity-nu_convergence_history.png :width: 80% :align: center - :alt: Nusselt number history over the run, 128x128 grid, showing a fast initial transient followed by a flat plateau. + :alt: Nusselt number history over the run, 512x512 grid, showing a fast initial transient followed by a flat plateau. - :math:`Nu` convergence history, :math:`128\times128`. + :math:`Nu` convergence history, :math:`512\times512` grid. Excluding the initial transient (first output only), a block-bootstrap -estimate over the last 13 plotfiles gives -:math:`Nu_{xmin}=9.0188\pm0.0015` and :math:`Nu_{xmax}=9.0248\pm0.0009` -(95% CI) -- a tight, flat plateau consistent with the point value used in +estimate over the last 28 plotfiles gives +:math:`Nu_{xmin}=8.83289\pm0.00032` and :math:`Nu_{xmax}=8.83290\pm0.00035` +(95% CI)—a tight, flat plateau consistent with the point value used in the grid-convergence table above, not a still-drifting solution. Thermodynamic pressure check -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -As a second, independent, mesh-cheap validation check, mass conservation +As a second, independent and mesh-cheap validation check, mass conservation in the closed ideal-gas domain fixes the domain-mean thermodynamic pressure relative to the wall temperatures. For a purely conductive -profile, this works out to the ratio of the logarithmic mean wall -temperature to the reference temperature: +profile, the pressure ratio evaluates to the ratio of the logarithmic +mean wall temperature to the reference temperature: .. math:: \frac{P_{mean}}{P_o} = \frac{T_{lm}}{T_{ref}}, \qquad T_{lm} = \frac{T_h-T_l}{\ln(T_h/T_l)} +where :math:`P_o` is the initial operating pressure of the closed system. + For this case, :math:`T_{lm}/T_{ref}=0.99963`, matching the computed -:math:`P_{mean}/P_o=0.99960` to within 0.004% -- and essentially -unchanged across all three grids (0.999592 / 0.999599 / 0.999600), since -it is a domain-integral quantity dominated by the fixed wall temperatures -rather than a locally-resolved wall gradient. This makes it a useful, -cheap sanity check that is already converged at the coarsest mesh, well -before :math:`Nu` itself finishes converging. +:math:`P_{mean}/P_o=0.99960` to within 0.004%—and essentially +unchanged across all grids (0.999592 / 0.999599 / 0.999600 / 0.999600). Since +this is a domain-integral quantity dominated by the fixed wall temperatures +rather than a locally resolved wall gradient, it serves as a highly effective, +computationally cheap sanity check that is already converged on the coarsest mesh, +well before :math:`Nu` itself finishes converging. + Extensions ----------- -* A :math:`512\times512` grid level is a natural addition to the - Richardson study above, if available. * A companion case at :math:`\Delta T/T_{ref}=120\%` (:math:`T_h=960\text{ K}`, :math:`T_l=240\text{ K}`, :math:`T_{ref}=600\text{ K}`, same :math:`Ra`, :math:`Pr`) tests how far @@ -328,4 +325,4 @@ Extensions expected to show a much larger, and more convection-sensitive, departure from its pure-conduction prediction at that :math:`\Delta T`. * Radiative coupling between the walls (not covered here) is a further - extension once the pure-convection case above is finalized. + extension of the pure-convection case :cite:p:`lari11`. diff --git a/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst b/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst index 322c1a9..3241a17 100644 --- a/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst +++ b/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst @@ -14,7 +14,7 @@ part of the analysis. Features -------- -* Incompressible fluid with no particles. +* Incompressible fluid flow. * Unsteady (transient) time advance using the Godunov scheme. * The same fully three-dimensional embedded-boundary sphere geometry as the steady tutorial. -- GitLab From da1ff114367c3259a41edab5c94874d01ae1264d Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Wed, 15 Jul 2026 15:53:37 -0400 Subject: [PATCH 6/6] Fix file locations --- docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst | 2 +- .../tutorials/fluid/steady-natural-convection-cavity.rst | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst index c398eef..a48f26b 100644 --- a/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst +++ b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst @@ -130,7 +130,7 @@ time advance: .. note:: The complete input file is located in the MFIX-Exa source directory as - ``tutorials/fluid/inputs.steady-flow-past-sphere`` + ``tutorials/fluid/inputs.steady-flow-around-sphere`` Post-processing ----------------- diff --git a/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst index 85cc6af..57755b9 100644 --- a/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst +++ b/docs/source_docs/tutorials/fluid/steady-natural-convection-cavity.rst @@ -170,7 +170,7 @@ The case runs to a pseudo steady state: .. note:: The complete input file is located in the MFIX-Exa source directory as - ``tutorials/fluid/inputs.natural-convection-cavity`` + ``tutorials/fluid/inputs.steady-natural-convection-cavity`` Post-processing --------------- -- GitLab

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z#Oc$kzwYShc=YIz*W**-wegitr-BATSMLv8?56rmLr3RKMnA|5hTYB3l4^mcmWD>g zg+jrJ6|a7{uZ`ZGH~smVQ*FG`uWJ23DjR%Yb-@b9z`(%0=Oe?y!rm8y3;vWyYJ7DIP*e0_cWIIsB~0nn*ipx~?e@r7TY#Kqk~jn3@r>%fKng9c{MhmdKI;Vst0Cv6o AHvj+t literal 0 HcmV?d00001 diff --git a/docs/source_docs/tutorials/fluid/index.rst b/docs/source_docs/tutorials/fluid/index.rst new file mode 100644 index 0000000..ccd1db7 --- /dev/null +++ b/docs/source_docs/tutorials/fluid/index.rst @@ -0,0 +1,27 @@ +.. _TutorialsFluid: + +Fluid +======= + +Tutorials in this category use MFIX-Exa's fluid solver with no particle +phase present (``dem.solve = none``, ``pic.solve = none``). This includes +ordinary single-fluid flow as well as any case that tracks two immiscible +fluids by density within the same solver, such as a Rayleigh-Taylor case. + +.. toctree:: + :maxdepth: 1 + + steady-flow-around-cylinder + steady-backward-facing-step + steady-flow-around-sphere + unsteady-flow-around-sphere + +Planned +--------- + +The following tutorials are on the roadmap but not yet published: + +- Natural convection cavity -- introduces the energy equation and buoyancy. +- Species mixing -- introduces species transport. +- Taylor-Couette flow -- introduces a rotating reference frame. +- Rayleigh-Taylor instability -- two immiscible fluids tracked by density. diff --git a/docs/source_docs/tutorials/incompressible-fluid/steady-backward-facing-step.rst b/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst similarity index 99% rename from docs/source_docs/tutorials/incompressible-fluid/steady-backward-facing-step.rst rename to docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst index 503571f..9bb8e51 100644 --- a/docs/source_docs/tutorials/incompressible-fluid/steady-backward-facing-step.rst +++ b/docs/source_docs/tutorials/fluid/steady-backward-facing-step.rst @@ -167,7 +167,7 @@ and the maximum iterations is increased to 500,000. .. note:: The complete input file is located in the MFIX-Exa source directory as - ``tutorials/incompressible-fluid/inputs.steady-backward-facing-step`` + ``tutorials/fluid/inputs.steady-backward-facing-step`` Post-Processing diff --git a/docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-cylinder.rst b/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst similarity index 98% rename from docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-cylinder.rst rename to docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst index 389a711..5836a88 100644 --- a/docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-cylinder.rst +++ b/docs/source_docs/tutorials/fluid/steady-flow-around-cylinder.rst @@ -113,7 +113,7 @@ is set to ensure :math:`\Delta t \sim (\Delta x)^2`. .. note:: The complete input file is located in the MFIX-Exa source directory as - ``tutorials/incompressible-fluid/inputs.steady-around-cylinder`` + ``tutorials/fluid/inputs.steady-flow-around-cylinder`` Post-Processing diff --git a/docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-sphere.rst b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst similarity index 98% rename from docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-sphere.rst rename to docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst index 1e766db..e7f9829 100644 --- a/docs/source_docs/tutorials/incompressible-fluid/steady-flow-around-sphere.rst +++ b/docs/source_docs/tutorials/fluid/steady-flow-around-sphere.rst @@ -130,7 +130,7 @@ time advance: .. note:: The complete input file is located in the MFIX-Exa source directory as - ``tutorials/incompressible-fluid/inputs.steady-flow-past-sphere`` + ``tutorials/fluid/inputs.steady-flow-past-sphere`` Post-processing ----------------- @@ -153,7 +153,7 @@ bubble immediately behind the sphere: .. _fig_steady_flow_past_sphere_midplane: -.. figure:: ./images/steady-flow-past-sphere-midplane_contour.png +.. figure:: ./images/steady-flow-around-sphere-midplane_contour.png :width: 80% :align: center :alt: Streamwise velocity and recirculation bubble behind a sphere at Reynolds number 100. diff --git a/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst b/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst new file mode 100644 index 0000000..322c1a9 --- /dev/null +++ b/docs/source_docs/tutorials/fluid/unsteady-flow-around-sphere.rst @@ -0,0 +1,234 @@ +.. _TutorialUnsteadyFlowAroundSphere: + +Unsteady flow around a sphere +================================ + +This tutorial extends the :ref:`steady flow around a sphere +` case to :math:`Re=300`, where the wake +sheds periodically as a chain of hairpin vortices rather than settling into +a fixed recirculation bubble. The geometry, domain, and mesh are unchanged +from the steady case; only the Reynolds number and the time-advance scheme +differ. The drag coefficient and vortex-shedding frequency are computed as +part of the analysis. + +Features +-------- + +* Incompressible fluid with no particles. +* Unsteady (transient) time advance using the Godunov scheme. +* The same fully three-dimensional embedded-boundary sphere geometry as the + steady tutorial. +* A transient, space-and-time-dependent inflow velocity used to seed the + wake instability. +* Surface-force (drag and side-force) reporting on the embedded boundary. +* Statistical (batch-means) analysis of the drag coefficient and Strouhal + number, with comparison against Johnson & Patel (1999) :cite:p:`johnson99`. + +Case description +----------------- + +The sphere, domain, and mesh are identical to the steady tutorial: diameter +:math:`D=1`, centered at the origin, with the domain extending 8 diameters +upstream, 16 diameters downstream, and 8 diameters radially in :math:`y` +and :math:`z`, resolved at 16 cells/diameter (384x256x256). + +The Reynolds number is raised to + +.. math:: + + Re = \frac{\rho\, U_\infty\, D}{\mu} = 300, \qquad \mu = 0.0033333, + +with :math:`\rho=1` and :math:`U_\infty=1` unchanged. At this Reynolds +number the wake sheds periodically as a chain of hairpin vortices with +planar (not axisymmetric) symmetry (Sakamoto & Haniu :cite:p:`sakamoto90`); +the transition from the steady axisymmetric wake used in the :math:`Re=100` +tutorial passes through an intermediate steady-but-non-axisymmetric regime +around :math:`Re\approx210`-:math:`270` before periodic shedding begins +around :math:`Re\approx270`-:math:`300`. + +Seeding the instability +~~~~~~~~~~~~~~~~~~~~~~~~ + +The steady axisymmetric solution remains a mathematically valid, though +unstable, equilibrium at :math:`Re=300`. A solver that preserves the +problem's symmetry exactly can sit near that equilibrium indefinitely +without something to break the symmetry. Rather than rely on numerical +noise to trigger the transition, the inflow velocity is given a small, +transient, spatially localized perturbation, antisymmetric in :math:`y`, +that decays away after roughly the first ten time units: + +.. code-block:: text + + bc.inflow.fluid.velocity = "1.0 + 0.05*exp(-t/2.0)*y*exp(-(y*y+z*z))" + +This has the side effect of fixing the shedding plane to the :math:`x`- +:math:`y` plane, rather than leaving the orientation to arbitrary numerical +noise -- a deliberate choice for a reproducible tutorial result. The +:math:`z`-force on the sphere stays at floating-point roundoff throughout +the run, confirming the wake organizes into the intended plane. + +Important input sections +-------------------------- + +The fluid viscosity sets the Reynolds number: + +.. code-block:: text + + fluid.viscosity.molecular.model = constant + fluid.viscosity.molecular.constant = 0.0033333 + +Genuine unsteady time advance is used in place of the steady tutorial's +pseudo steady-state iteration, with the Godunov advection scheme and a +raised CFL number: + +.. code-block:: text + + mfix.advection_type = Godunov + mfix.cfl = 0.9 + + mfix.dt_min = 1.0e-6 + mfix.dt_max = 0.1 + + mfix.stop_time = 1000.0 + +Explicit, CFL-limited unsteady time advance is more sensitive to severely +cut EB cells than the steady tutorial's pseudo-time advance was -- a very +small cut cell can otherwise force either an unphysically small timestep or +outright instability. State redistribution addresses this by redistributing +the update from small cut cells into their larger neighbors: + +.. code-block:: text + + mfix.redistribution_type = StateRedist + mfix.correction_small_volfrac = 1.e-4 + +The ``eb_drag`` report is sampled far more frequently than in the steady +tutorial, to adequately resolve the shedding frequency: + +.. code-block:: text + + mfix.reports.eb_drag.regions = eb_sphere + mfix.reports.eb_drag.eb_sphere.int = 3 + +.. note:: + + The complete input file is located in the MFIX-Exa source directory as + ``tutorials/fluid/inputs.unsteady-flow-around-sphere`` + +Post-processing +----------------- + +The ``eb_drag`` report gives the drag coefficient :math:`C_d` and the +lateral force coefficients :math:`C_y`, :math:`C_z` over the full run: + +.. _fig_unsteady_flow_around_sphere_force_history: + +.. figure:: ./images/unsteady-flow-around-sphere-force_history.png + :width: 80% + :align: center + :alt: Drag and side-force coefficient history for unsteady flow around a sphere at Reynolds number 300. + + Force coefficient history, :math:`Re=300`. + +:math:`C_z` sits at floating-point roundoff throughout, confirming the wake +sheds in the intended plane. :math:`C_y` carries a persistent negative +mean rather than oscillating about zero -- consistent with the "planar" +wake character described in the literature for this Reynolds-number range, +where the wake settles into a steady deflection with periodic shedding +superimposed, rather than a simple side-to-side oscillation. + +A windowed mean/standard-deviation view of :math:`C_y` shows a slow +amplitude modulation, with a period of roughly 34 time units, superimposed +on the underlying shedding oscillation: + +.. figure:: ./images/unsteady-flow-around-sphere-windowed_stats.png + :width: 80% + :align: center + :alt: Windowed mean and standard deviation of the side-force coefficient, showing a persistent amplitude modulation. + + Windowed mean and standard deviation of :math:`C_y`. + +This modulation does not decay over the roughly 970 time units recorded +after the initial transient, indicating it is a persistent secondary +feature of the flow at this resolution rather than an artifact of +insufficient run time. It is directly visible in the frequency spectrum of +:math:`C_y` as sidebands evenly spaced around the true shedding frequency +by the modulation frequency -- amplitude modulation of a single carrier, +not several independent shedding modes: + +.. figure:: ./images/unsteady-flow-around-sphere-strouhal_spectrum.png + :width: 80% + :align: center + :alt: Frequency spectrum of the side-force coefficient, showing the dominant shedding frequency and its sidebands. + + Spectrum of :math:`C_y`, settled window. + +Statistical analysis +~~~~~~~~~~~~~~~~~~~~~ + +The drag coefficient and shedding (Strouhal) frequency were each estimated +using the batch-means method: the settled portion of the record +(:math:`t\ge250`) was split into seven non-overlapping 100-time-unit +windows, each window's quantity was estimated independently, and a +t-distribution interval was computed on the resulting sample. Because these +windows are consecutive segments of one continuous run rather than +independent repeated experiments, this interval should be read as the +spread of the estimate within this run, not a formal confidence interval +over repeated realizations. + +.. list-table:: Batch-means results vs. Johnson & Patel (1999), :math:`Re=300` + :header-rows: 1 + + * - Quantity + - Mean + - 95% interval + - Reference + - % difference + * - :math:`C_d` + - 0.6765 + - (0.6760, 0.6770) + - 0.656 + - +3.1% + * - :math:`St` + - 0.1191 + - (0.1186, 0.1195) + - 0.137 + - -13.1% + +Both intervals are narrow enough that the reference value falls clearly +outside each one -- these are small but real, well-resolved discrepancies +rather than run-to-run noise. The drag coefficient is close to the +reference; the shedding frequency is not. No dedicated grid-refinement +study was performed for this unsteady case (unlike the steady tutorial), +so mesh resolution is a plausible, though unconfirmed, contributor to both +gaps. + +Flow visualization +~~~~~~~~~~~~~~~~~~~ + +A single mid-plane slice, colored by out-of-plane vorticity, shows the +meandering wake: + +.. figure:: ./images/unsteady-flow-around-sphere-midplane_snapshot.png + :width: 80% + :align: center + :alt: Mid-plane out-of-plane vorticity contour showing the meandering wake behind the sphere at Reynolds number 300. + + Out-of-plane vorticity, mid-plane slice, :math:`Re=300`. + +A 2D slice cannot show the shedding process itself, since the shed +structures are inherently three-dimensional. An animation of Q-criterion +isosurfaces, colored by velocity magnitude and generated in ParaView, +shows the wake organizing into a pair of counter-rotating streamwise +vortex legs that periodically pinch off into discrete hairpin loops and +convect downstream. + +.. raw:: html + +

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