From a2c671bfc9906963c5c18e6172e7fd1c9ebcf391 Mon Sep 17 00:00:00 2001
From: William Fullmer <william.fullmer@netl.doe.gov>
Date: Mon, 11 Nov 2019 11:18:51 -0500
Subject: [PATCH] first pass of QB chapter - Qualitative Benchmarks

---
 docs/source/index.rst                         |   1 +
 docs/source/qb/biseg.rst                      |  18 ++++
 docs/source/qb/figs/boyce_sb_1908_small.png   | Bin 0 -> 352070 bytes
 docs/source/qb/figs/granRT_1908_small.png     | Bin 0 -> 345199 bytes
 docs/source/qb/figs/hcs_ke_1908.png           | Bin 0 -> 5980 bytes
 docs/source/qb/figs/hcs_xy_1908.png           | Bin 0 -> 973089 bytes
 docs/source/qb/figs/mehrdad_1908_small.png    | Bin 0 -> 183844 bytes
 docs/source/qb/figs/netl_biseg_1908_small.png | Bin 0 -> 546542 bytes
 docs/source/qb/granRT.rst                     |  84 ++++++++++++++++++
 docs/source/qb/hcs.rst                        |  84 ++++++++++++++++++
 docs/source/qb/index.rst                      |  28 ++++++
 docs/source/qb/mehrdadsbed.rst                |  54 +++++++++++
 docs/source/qb/refs.rst                       |  82 +++++++++++++++++
 docs/source/qb/single_bubble.rst              |  75 ++++++++++++++++
 14 files changed, 426 insertions(+)
 create mode 100644 docs/source/qb/biseg.rst
 create mode 100644 docs/source/qb/figs/boyce_sb_1908_small.png
 create mode 100644 docs/source/qb/figs/granRT_1908_small.png
 create mode 100644 docs/source/qb/figs/hcs_ke_1908.png
 create mode 100644 docs/source/qb/figs/hcs_xy_1908.png
 create mode 100644 docs/source/qb/figs/mehrdad_1908_small.png
 create mode 100644 docs/source/qb/figs/netl_biseg_1908_small.png
 create mode 100644 docs/source/qb/granRT.rst
 create mode 100644 docs/source/qb/hcs.rst
 create mode 100644 docs/source/qb/index.rst
 create mode 100644 docs/source/qb/mehrdadsbed.rst
 create mode 100644 docs/source/qb/refs.rst
 create mode 100644 docs/source/qb/single_bubble.rst

diff --git a/docs/source/index.rst b/docs/source/index.rst
index f1c57b7..9dc5462 100644
--- a/docs/source/index.rst
+++ b/docs/source/index.rst
@@ -29,6 +29,7 @@ the master branch at the beginning of each month.
    EB
    CITests
    NightlyTests
+   qb/index
    Debugging
 
 Notice
diff --git a/docs/source/qb/biseg.rst b/docs/source/qb/biseg.rst
new file mode 100644
index 0000000..2c01430
--- /dev/null
+++ b/docs/source/qb/biseg.rst
@@ -0,0 +1,18 @@
+.. _Chap:QB:biseg:
+
+Bidisperse Segregation 
+====================== 
+
+wdf TODO ASAP
+
+
+.. figure:: figs/netl_biseg_1908_small.png
+   :width: 16cm
+   :align: center
+   :alt: Sim comparison to bidisperse segregation experiment at NETL 
+
+   Comparison of experiment and MFiX-Exa simulaton for rapid segregation
+   of a bi-disperse particle mixture.   
+
+
+
diff --git a/docs/source/qb/figs/boyce_sb_1908_small.png b/docs/source/qb/figs/boyce_sb_1908_small.png
new file mode 100644
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diff --git a/docs/source/qb/granRT.rst b/docs/source/qb/granRT.rst
new file mode 100644
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--- /dev/null
+++ b/docs/source/qb/granRT.rst
@@ -0,0 +1,84 @@
+.. _Chap:QB:granRT:
+
+Granular Rayleigh-Taylor
+========================
+
+The Rayleigh-Taylor (RT) instability, along with the Kelvin-Helmholtz 
+instability, is among the most well-known and fundamental of all multi-phase 
+flow instabilities [C81]_. The prototypical case involves two (traditional)
+fluids at rest of differing densities with the heavier fluid being accelerated
+into the lighter fluid. The instability manifests itself in a deformation of 
+the interface between the two fluids. The initial deformation grows into an 
+interpenetrating fingering pattern, commonly producing a mushroom cloud 
+pattern, before giving way to micro-scale mixing or bubble-droplet formation 
+if the fluids are immiscible.
+
+
+Vinningland and coworkers [VJFTM07]_ devised a granular analogue of the classic 
+problem in which the heavier "fluid" is an assembly of particles penetrating 
+into a gas, in this case air. The particle assembly is generated by settling 
+under gravity into a random close packed array in a Hele-Shaw cell 
+(rectangular enclosure) which is quickly inverted around a pivot to bring the 
+heavier particle phase above the lighter gas phase. The instability evolution 
+is distinct from the classic fluid-fluid RT, which is described by 
+Vinningland et al. [VJFTM07]_:
+
+ .. line-block::
+
+    The initially flat front defined by the grains subsequently develops 
+    into a pattern of falling granular fingers separated by rising bubbles 
+    of air. A transient coarsening of the front is observed right from the 
+    start by a finger merging process. The coarsening is later stabilized 
+    by new fingers growing from the center of the rising bubbles.
+
+
+The granular-RT is simulated with MFiX-Exa 19.08 with the following setup. 
+The domain is 56 mm wide by 68 mm tall and 1 mm deep which is discretized 
+by a uniform mesh of :math:`224 \times 272 \times 4` CFD cells. No-slip walls 
+are applied at top, bottom, left and right domain extents. 
+The front and back walls are "removed" and treated as periodic in the 
+simulation as the depth is only resolved by four CFD cells.
+No additional geometry definition is required. 
+The particles are assumed monodisperse with a constant diameter of 
+:math:`d_p = 140` microns (:math:`dx^* \approx 1.8`) and density of 
+:math:`\rho_p = 1050` kg/m\ :sup:`3` \. The resitution and sliding friction 
+coefficients are set to 0.9 and 0.25, which are believed to be representative 
+of the polystyrene material. The Wen and Yu drag law [WY66]_ is applied. 
+1.12M particles are initially randomly distributed throughout the domain. 
+The gravity force on the particles is modified by :math:`\tanh 100(t - 2)` 
+so that the particles initially settle upwards, and then the body force is 
+quickly inverted around :math:`t = 2` s, causing the particles to fall 
+downward into the quiescent air. 
+
+
+.. figure:: figs/granRT_1908_small.png
+   :width: 16cm
+   :align: center
+   :alt: Sim comparison to granular Rayleigh-Taylor experiment
+
+   Progression of the granular Rayleigh-Taylor instability in the experiments 
+   of Vinningland et al. [VJFTM07]_ (top row) 
+   compared to the MFiX-Exa result (bottom row). 
+
+
+The figure above shows the progression of an MFiX-Exa 19.08 simulation compared 
+to the experimental results. There is a good match between the experimental and 
+numerical results for roughly the first half of the transien: an initially flat 
+interface gives way to many fingers falling into the gas which merge and form 
+a semi-stable bubble pattern. However, the simulated bubbles appear to be less
+stable than those in the lab, which rise uniformily to the surface. However, in 
+the simulations the center bubble rises slighlty faster than the one on the 
+left, which shifts more weight over the left-hand bubble, which further impedes 
+its rise and eventually "squishes" the left-hand bubble into the center bubble
+as it breaks the surface. Later, the right hand bubble also merges with (what 
+remains of) the center bubble, allowing a path of least resistance for its air 
+to escape without rupturing on the surface on its own. The bubble merging 
+significantly accelerates the second half of the transient (notice the 
+different times between experiment and simulation). Several different 
+variations of this setup were performed: different drag laws, slower inversion 
+time, different combinations of particle restitution and friction coefficients, 
+inclusion of front and back walls. Although all tests were slighlty different, 
+none were able to match the stability of the later time bubble pattern observed 
+experimentally.  
+
+
diff --git a/docs/source/qb/hcs.rst b/docs/source/qb/hcs.rst
new file mode 100644
index 0000000..59332ca
--- /dev/null
+++ b/docs/source/qb/hcs.rst
@@ -0,0 +1,84 @@
+.. _Chap:QB:hcs:
+
+Clustering in the HCS 
+======================
+
+
+The HCS is the simplest non-trivial particulate gas-solid system. The continuum 
+gas-phase is initially at rest. The particles are uniformily distributed in space
+and have zero momentum in all three directions. However, the particle pecular 
+velocity is non-zero, quantified by an initial *granular* temperature, 
+:math:`T_0`. The system is periodic in all direcitons and no external forces act 
+on the system. Under homogeneous conditions, the granular temperature, :math:`T`, 
+is equivalent to two-thirds of the the (massless) mean particle kinetic energy. 
+In the HCS, the Eulerian kinetic theory (KT) model of Garzo et al. [GTSH12]_ 
+reduces to: 
+
+.. math:: 
+   \frac{dT}{dt} = - \frac{2 \gamma}{m} T - \zeta_0 T 
+
+where :math:`m` is the particle mass, :math:`\gamma` is the thermal drag and 
+:math:`\zeta_0` is the zeroth-order cooling rate. The first term on the RHS 
+above represents viscous dissipation due to the interstitial gas while the 
+second term represents collisional dissipation due inelastic particle-particle 
+interactions. The ODE has an analytical solution given by Yin et al. [YZMH13]_ 
+(also see [LBFHHGS16]_ for the exact model used herein which also includes a
+first-order thermal Reynolds number extension to :math:`\gamma`). In the absence 
+of clustering, the granular temperature in the HCS decays according to the 
+analytical solution, known as Haff's law [H83]_ in granular systems:  
+:math:`\gamma = 0`. However, at a critical system size [G05]_, :math:`L^*_c`, 
+(where :math:`L^* = L/d_p`), the initially homogeneous state gives way to the 
+most fundamental of gas-solid instabilities, the clustering instability, which 
+causes :math:`T` (or more accurately :math:`KE`) to deviate significantly from 
+KT solution due to regions of high and low concentration and correlated motion. 
+
+
+To test if MFiX-Exa predicts the expected clustering behavior, a system is set up
+with the following non-dimensional parameters:
+  
+  * initial thermal Reynolds number: :math:`Re_{T_0} = \rho_g d_p \sqrt{T_0} / \mu_g = 20`
+  * density ratio: :math:`\rho^* = \rho_p / \rho_g = 1000`
+  * restitution coefficient: :math:`e = 0.8`
+  * solids concentration: :math:`\phi = \pi N_p / 6 L^*_x L^*_y L^*_z \approx 0.05`
+
+While not specifically studied by Fullmer et al. [FLYH18]_, their results 
+indicate that :math:`L_c^*` may be as large as 50 to 70 at these conditions. 
+In order to avoid the region near critical stability, we use a significnatly
+larger system size: :math:`L^*_x = L^*_y = 256`. The system is thin in the 
+depth dimension, :math:`L^*_z = 8` in order to highlight the clustering 
+phenomena. Therefore, :math:`N_p = 50000`. Because the system is hypothetical, 
+the ideal ``BVK2`` DNS drag law is applied, see [BvK07]_, [TPKKv15]_.
+
+
+.. figure:: figs/hcs_ke_1908.png
+   :width: 8cm
+   :align: center
+   :alt: kinetic energy decay in the HCS
+
+   Decay of the particle mean kinetic energy compared to the KT analytical 
+   soluiton of the GTSH model. 
+
+
+Three replicate systems are simulated with MFiX-Exa 19.08, differing only 
+in initial particle locations and pecular velocities. The particle kinetic 
+energy is averaged in the simulations (red) and  compared to the analytical 
+granular temperature (black) of the HCS as a funciton of time in the figure 
+above. The kinetic energy :math:`KE / KE_0` decays by two to three orders of 
+magnitude in line with the HCS result until clustering and localized mean 
+motion cause a drastic deviation. The final result at :math:`t^* = 1000` 
+for one of the replicates is shown below (at right) compared to the seminal 
+result of Goldhirsch and Zanetti [GZ93]_ (true 2D), the original demonstration 
+of the clustering instability the HCS. 
+
+
+.. figure:: figs/hcs_xy_1908.png
+   :width: 12cm
+   :align: center
+   :alt: clustered state of the HCS
+
+   Clustered state of the HCS observed by Goldhirsch and Zanetti [GZ93]_
+   (left) compared to an MFiX-Exa result (right). 
+
+
+
+
diff --git a/docs/source/qb/index.rst b/docs/source/qb/index.rst
new file mode 100644
index 0000000..22ffd29
--- /dev/null
+++ b/docs/source/qb/index.rst
@@ -0,0 +1,28 @@
+.. _Chap:QB:
+
+Qualitative Benchmarks
+======================
+
+MFiX-Exa uses a three level approach to regression testing spaning from simple
+and/or short smoke tests to validation problems comparing against experimental
+data. Most of the validation benchmarks target the physics of interest to 
+MFiX-Exa's intended audience, i.e., dense bubbling, fast fluidization and 
+pneumatic transport of particulate solids by a light gas. In an effort to widen 
+the phase-space in which MFiX-Exa may be (potentially) applied, a set of 
+qualitative benchmark problems are provided below which focus more on the 
+phenomenology of a given, problem rather than averaged statistical measures. 
+Animations of numerical soluitons to several of the qualitative benchmarks 
+can be seen in the 
+`Gallery <https://amrex-codes.github.io/MFIX-Exa/gallery.html>`_
+
+
+.. toctree::
+   :maxdepth: 1
+
+   hcs
+   granRT
+   mehrdadsbed
+   single_bubble
+   biseg
+   refs
+
diff --git a/docs/source/qb/mehrdadsbed.rst b/docs/source/qb/mehrdadsbed.rst
new file mode 100644
index 0000000..ae35958
--- /dev/null
+++ b/docs/source/qb/mehrdadsbed.rst
@@ -0,0 +1,54 @@
+.. _Chap:QB:mehrdad:
+
+Mehrdad's Bed 
+=============
+
+Typically, bubbling fluidized beds are produced by uniformily--in space and 
+time--driving a gas flow through a particle bed. Slightly different bubbling 
+patterns are observed depending largely on the Geldart classification [G73]_ 
+of the particles and the size of the bed. Like gas bubbles in a liquid bed, 
+the dynamics are almost always highly irregular and chaotic in nature. 
+However, by periodically driving the gas flow, chaotic bubbling can be 
+suppressed yeilding quasi-regular periodic bubbling [PB98]_. Studying bubble 
+control methods, Coppens and coworkers [Cv03]_ realized that periodic bubbling 
+produces starkly regular patterns in thin beds, which may be useful for code 
+validation [WdLC16]_. Shahnam and coworkers took the problem even further and 
+reducing the latteral dimension to produce an oscillating Left-Right single 
+bubble pattern in a setup affectionately referred to as *Mehrdad's bed* [S18]_.  
+
+
+Mehrdad's bed is a rectangular geometry of width :math:`L_x` =  50 mm, 
+height :math:`L_y` = 160 mm, and depth :math:`L_z` =   5 mm. The domain is 
+resolved by a uniform CFD grid of :math:`80 \times 256 \times 8`. No-slip walls 
+are applied on the vertical domain extents with a mass inflow at the bottom and 
+a pressure outflow at the top of the domain. The mass inflow is defined in 
+`usr1.f90` as: 
+
+.. code:: fortran 
+   
+  real(rt),   intent(in ) :: time
+  real(rt) :: usr_pi, usr_umf
+
+  usr_pi  = 4.0d0*ATAN(1.0d0)
+  usr_umf = MIN(0.15d0, 0.1d0*time)
+
+  bc_u_g(1) = usr_umf*(1.3d0 + 0.7d0*DSIN(10.0d0*usr_pi*time))
+
+The bed consists of 188500 particles which are assumed to be monodisperse with 
+constant diameter and density of :math:`d_p = 400` microns and 
+:math:`\rho_p = 2500` kg/m\ :sup:`3` \, respectively. 
+
+
+.. figure:: figs/mehrdad_1908_small.png
+   :width: 16cm
+   :align: center
+   :alt: Sim comparison to Mehrdad's experiment
+
+   Comparison of the experiment (middle two rows) to the MFiX-Exa simulation 
+   (top and bottom rows).    
+   
+The bed is simulated using MFiX-Exa 19.08 for an initial transient period of 
+10 s before an observation window of an additional 5 s. The desired left-right 
+pattern is seen as shown in the figure above. 
+
+
diff --git a/docs/source/qb/refs.rst b/docs/source/qb/refs.rst
new file mode 100644
index 0000000..4631791
--- /dev/null
+++ b/docs/source/qb/refs.rst
@@ -0,0 +1,82 @@
+.. _Chap:QB:refs:
+
+References
+==========
+
+.. [BvK07] R. Beetstra, M.A. van der Hoef, and J.A.M.
+   Kuipers. Drag force of intermediate Reynolds number flow past mono- 
+   and bidisperse arrays of spheres. *AIChE Journal*, **53**, 489--501 (2007).
+
+.. [BPLPM19] C.M. Boyce, A. Penn, M. Lehnert, K.P. Pruessmann, and C.R. Müller. 
+   Magnetic resonance imaging of single bubbles injected into incipiently 
+   fluidized beds. *Chemical Engineering Science*, **200**, 147--166 (2019).
+
+.. [C81] S. Chandrasekhar. *Hydrodynamic and hydromagnetic stability*. 
+   Dover Publications (1981).
+
+.. [Cv03] M.O. Coppens and J.R. van Ommen. Structuring chaotic fluidized beds.
+   *Chemical Engineering Journal*, **96**,  117--124 (2003).
+
+.. [FLYH18] W.D. Fullmer, X. Li, X. Yin and C.M. Hreyna. Notes on clustering 
+   in hte gas-solid HCS. `arxiv:1809.04173 <https://arxiv.org/abs/1809.04173>`_ 
+   (2018).
+
+.. [G73] D. Geldart. Types of gas fluidization. *Powder Technology*, **7** (5), 285--292 (1973) 
+
+.. [G05] V. Garzó. Instabilities in a Free Granular Fluid Described by the 
+   Enskog Equation. *Physical Review E*, **72** (2), 021106 (2005). 
+
+.. [GTSH12] V. Garzó, S. Tenneti, S. Subramaniam, and C.M. Hrenya. Enskog 
+   kinetic theory for monodisperse gas-solid flows. *Journal of Fluid Mechanics*, 
+   **712**, 129--168 (2012).
+
+.. [GZ93] I. Goldhirsch and G. Zanetti. Clustering instability in dissipative 
+   gases. *Physical Review Letters*, **70** (11), 1619--1622 (1993). 
+
+.. [H83] P.K. Haff. Grain Flow as a Fluid-Mechanical Phenomenon. 
+   *Journal of Fluid Mechanics*, **134**, 401--430 (1983). 
+
+.. [LBFHHGS16] P. Liu, T. Brown, W.D. Fullmer, T. Hauser, C.M. Hrenya, 
+   R. Grout and H. Sitaraman. Comprehensive benchmark suite for simulation 
+   of particle laden flows using the discrete element method with performance 
+   profiles from the multiphase flow with interface exchanges (MFiX) code. 
+   Technical report, National Renewable Energy Lab.(NREL), Golden, CO 
+   (United States), (2016). url: https://www.nrel.gov/docs/fy16osti/65637.pdf
+
+.. [PB98] D.V. Pence and D.E. Beasley, Chaos suppression in gas-solid 
+   fluidization, *Chaos*, **8**, 514--519 (1998).
+
+.. [TPKKv15] Y. Tang, E.A.J.F. Peters, J.A.M. Kuipers, 
+   S.H.L. Kriebitzsch, and M.A. van der Hoef. A new drag correlation 
+   from fully resolved simulations of flow past monodisperse static arrays
+   of spheres. *AIChE Journal*, **61** (2), 688--698 (2015).
+
+.. [S18] A. Bakshi, M. Shahnam, A. Gel, T. Li, C. Altantzis, W. Rogers, and A.F. Ghoniem.
+   Comprehensive multivariate sensitivity analysis of CFD-DEM simulations: 
+   Critical model parameters and their impact on fluidization hydrodynamics. 
+   *Powder Technology*, **338**, 519--537 (2018).
+ 
+   J.E. Higham, M. Shahnam, and A. Vaidheeswaran. On the Dynamics of a 
+   Quasi-Two-Dimensional Pulsed-Fludized Bed. 
+   `arxiv:1809.05033 <https://arxiv.org/abs/1809.05033>`_ (2018).  
+
+.. [VJFTM07] J.L. Vinningland, Ø. Johnsen, E.G. Flekkøy, R. Toussaint, 
+   and K.J. Måløy. Granular Rayleigh-Taylor instability: Experiments and 
+   simulations. *Physical Review Letters*, **99** (4), 048001 (2007).
+
+   J.L. Vinningland, Ø. Johnsen, E.G. Flekkøy, R. Toussaint, 
+   and K.J. Måløy. Experiments and simulations of a gravitational granular 
+   flow instability. *Physical Review E*, **76** (5), 051306 (2007).
+
+.. [WY66] C.Y.Wen and Y.H. Yu. Mechanics of fluidization. 
+   *Chemical Engineering Progress Symposium*, **62**, 100--111 (1966).
+
+.. [WdLC16] K. Wu, L. de Martin, Luca, M. and M.-O. Coppens. Pattern formation 
+   in fluidized beds as a tool for model validation: A two-fluid model based study.
+   *Powder Technology*, **295**, 35--42 (2016).
+
+.. [YZMH13] X. Yin, J.R. Zenk, P.P. Mitrano, and C.M. Hrenya. Impact of 
+   Collisional Versus Viscous Dissipation on Flow Instabilities in 
+   Gas-Solid Systems. *Journal of Fluid Mechanics*, **727**, R2 (2013).
+
+
diff --git a/docs/source/qb/single_bubble.rst b/docs/source/qb/single_bubble.rst
new file mode 100644
index 0000000..3f62e5a
--- /dev/null
+++ b/docs/source/qb/single_bubble.rst
@@ -0,0 +1,75 @@
+.. _Chap:QB:singbub:
+
+Single Bubble Injection 
+=======================
+
+Another variation on bubbling typically observed in bubbling fluidized beds 
+is the detailed single-bubble study of Boyce and coworkers [BPLPM19]_. In the 
+experiment, a cylinderical bed is brought to incipient fluidization 
+(just *under* minimum fluidization) by a uniform disributor. Then, an 
+additional volume of gas is abruptly injected from a nozzle located in the 
+center of the bed causing a single bubble to form and rise through the bed. 
+Measurement of the evolution of the bubble formation, rise and 
+is captured with high-speed, high-resolution magnetic resonance imaging (MRI)
+in a 10 mm thick slice through the center of the bed. Two particle types 
+are studied experimentally, only the larger of which is currently used 
+for qualitative benchmarking. 
+
+
+The experimental test section is a cylinderical bed 190 mm in diameter and 
+300 mm tall. The system is modeled in a domain of size 192 mm square by 384 mm 
+tall with a cylinderical EB geometry centered at :math:`(x,z) = (96, 96)` mm. 
+The modeled bed height is larger than the experimentso that it can be resolved 
+by a uniform uniform CFD grid of power 2. Specifically, the applied grid is 
+:math:`32 \times 64 \times 32`, such that :math:`dx^* \approx 2.0`. 
+:math:`N_p = 260`-thousand particles make up a bed of approximately 
+:math:`h_{bed} = 200` static bed height. The particles are of diameter 
+:math:`d_p = 2.93` mm and density 
+:math:`\rho_p = 1040` kg/m\ :sup:`3` \, respectively. Mass inlet and pressure 
+outlet boundary conditions are specified at the bottom and top of the domain, 
+respectively. The nozzle is modeled with a secondary mass inlet covering the 
+center (in x,z) four CFD cells. We note that, as modeled, the area of the 
+injector is roughly three times that of the experimental nozzle, a tube of 
+:math:`7.95` mm diameter. The injection times are adjusted slightly to 
+:math:`\delta t_{inj} = 154.2`, :math:`101.7`, :math:`66.7`, :math:`51.4`, 
+and :math:`25.0` ms so that a uniform jet velocity of :math:`50` m/s can be 
+applied in all cases.
+
+
+A separate defluidization simulation was first carried out first to determine
+:math:`U_{mf} \approx 0.66` m/s using the `WenYu` [WY66]_ drag law, slightly 
+below the experimentally measured value of :math:`U_{mf} = 0.7` m/s. The bed 
+was prepared with two initialization simulations. First, the particle initial
+condition is fluidized above :math:`U_{mf}` at :math:`0.8` m/s for one second 
+using both the uniform distributor and jet sections. Then, the jet section is 
+shut off (velocity in BC set to zero) and the flow in the uniform distributor 
+section is reduced to incipient fluidization at :math:`0.66` m/s for an 
+additional two seconds. Then, beginning at :math:`t = 3` s, the jet region is 
+set to :math:`50` m/s for a specified injection duration given previously. 
+The jet is switched on and off with a step change in `usr1.f90`. 
+
+
+.. figure:: figs/boyce_sb_1908_small.png
+   :width: 16cm
+   :align: center
+   :alt: Sim comparison to single bubble injection experiment of Boyce
+
+   Comparison of experiment and MFiX-Exa simulaton for single bubbles 
+   injected into incipiently fluidized beds for increasing (left to right) 
+   injection times. 
+
+
+The figure above provides a comparison between MFiX-Exa `19.08` simulation 
+results and the experimental measurements. In both cases, the particle data 
+has been averaged onto a :math:`10` mm thick grid of :math:`21 \times 32`
+For the longer injection times, :math:`\delta t_{inj} \approx 100` and 
+:math:`150` ms, the simulated bubbles are larger and more elongated than 
+observed experimentally. However, this trend is not universal, at shorter 
+injection times, the bubble is too small and actually collapses before 
+erupting at the surface. Above, the bubble produced from a :math:`50` ms 
+injection is currently in the process of collapsing. Another interesting 
+feature (perhaps most apparent at :math:`66` ms), is the V-shaped region 
+of particle down flow centered around the bubble centeroid, which appears 
+to be captured rather well by the simulation. 
+
+
-- 
GitLab