# Model options

The following inputs are defined using the `mfix` prefix.

| Description | Type | Default
--- | --- | --- | ---
gravity | Gravity vector [required] | Reals | 0 0 0
advect_density | Flag to enable time evolution of fluid density. | int | 0
advect_enthalpy | Flag to enable time evolution of fluid temperature and enthalpy. | int | 0
solve_species | Flag to enable the time evolution of fluid species mass fractions. | int | 0
constraint_type | Select low Mach number constraint. `IncompressibleFluid` constraint is appropriate for cold flow systems that do not have chemical reactions. Fluid temperature and species mass fractions act as passive tracers if advected. `IdealGasOpenSystem` constraint is used if the physical domain contains at least one pressure outflow boundary condition. The specified outflow (ambient) pressure is maintained so that the thermodynamic pressure is constant. `IdealGasClosedSystem` constraint is used for systems that are fully periodic or completely closed with no inflow / outflow boundaries. The thermodynamic pressure evolves in time to balance processes like chemical reactions. | string | IncompressibleFluid

Note

The thermodynamic pressure is a scalar value, not a scalar field. This is a consequence of the low Mach number assumption.

## Fluid discretization

The following inputs are defined using the `mfix` prefix.

Key | Description | Type | Default
--- | --- | --- | ---
advection_type | Advection scheme `mol` Predictor-Corrector Method of Lines. MAC velocities and edge states are extrapolated in space. `godunov` Godunov. MAC velocities and edge states are extrapolated in space and time using a second-order Taylor series expansion. | string | Godunov
redistribution_type | Algorithms to address the ‘small cell problem’ associated with explicit cut-cell algorithms. `StateRedist` [Berger_Giulini_2021] [Giulini_etal_2022] `FluxRedist` [Chern_Colella_1987] [Colella_etal_2006] `NoRedist`: Do not redistribute fluxes. | String | StateRedist
redistribute_before_nodal_proj | Redistribute the velocity field before the nodal projection | Bool | True
redistribute_nodal_proj | Redistribute the velocity field after the nodal projection | Bool | False
use_drag_coeff_in_proj_gp | Algebraically consistent p coeff in proj or (default) simplified form | Bool | False
use_drag_in_godunov | Include a drag term in the Godunov flux or (default) not | Bool | False
correction_small_volfrac | Threshold volume fraction for correcting small cell velocity at the end of the predictor and corrector | Real | 1.e-4

[Berger_Giulini_2021] (1,2)

Berger, M., & Giuliani, A. (2021). A state redistribution algorithm for finite volume schemes on cut cell meshes. Journal of Computational Physics, 428, 109820.

[Giulini_etal_2022] (1,2)

Giuliani, A., Almgren, A. S., Bell, J. B., Berger, M. J., de Frahan, M. H., & Rangarajan, D. (2022). A weighted state redistribution algorithm for embedded boundary grids. Journal of Computational Physics, 464, 111305.

[Chern_Colella_1987]

Chern, I.-L., and Colella, P. (1987). A conservative front tracking method for hyperbolic conservation laws,” LLNL Rep. No. UCRL-97200, Lawrence Livermore National Laboratory.

[Colella_etal_2006]

Colella, P., Graves, D. T., Keen, B. J., & Modiano, D. (2006). A Cartesian grid embedded boundary method for hyperbolic conservation laws, J. Comput. Phys., vol. 211, no. 1, pp. 347–366.

## Additional available constraints

Additional constraints may be imposed on problems which are under-determined such as particle settling in a fully periodic domain. Currently, only particle constraints are supported.

The following inputs are defined using the `particles` prefix.

| Description | Type | Default
--- | --- | --- | ---
constraint | Constraint type. Available options include: ‘mean_velocity’ | String | None

For the mean_velocity constraint, the following inputs can be defined.

| Description | Type | Default
--- | --- | --- | ---
mean_velocity_x | mean particle velocity in dir=0 | Real | Undefined
mean_velocity_y | mean particle velocity in dir=1 | Real | Undefined
mean_velocity_z | mean particle velocity in dir=2 | Real | Undefined

Below is an example for zero mean particle velocity in all three directions.

    particles.constraint = "mean_velocity"

    particles.constraint.mean_velocity_x = 0.
    particles.constraint.mean_velocity_y = 0.
    particles.constraint.mean_velocity_z = 0.

In the above example, at the end of each (fluid) time step, the global mean particle velocity is computed in each direction, then subtracted from each particle’s velocity vector so that the mean particle velocity for the system is zero.

## Deposition scheme

The following inputs are defined using the `mfix` prefix.

| Description | Type | Default
--- | --- | --- | ---
deposition_scheme | The algorithm used to transfer particle properties to the Eulerian grid. An overview of the schemes is provided below. `centroid` `trilinear` `true-dpvm` `trilinear-dpvm-square` | String | trilinear
deposition_scale_factor | The deposition scale factor. Applicable only with true-dpvm and trilinear-dpvm-square deposition schemes. The value must be in the intervarl \([0,\Delta x/2]\) where \(\Delta x\) is the mesh spacing. | Real | 1.0
deposition_diffusion_coeff | If a positive value is set, a diffusion equation with this diffusion coefficient is solved to smooth deposited quantities. | Real | -1.0

In the following subsections, the four deposition methods are briefly described and illustrated.

### Centroid

The centroid deposition scheme transfers particle properties, for example volume, to the Eulerian grid cell containing its center. Illustrated in two-dimensions below, the particle center is in the upper-right cell; therefore, the deposition weight for that cell is 1 and all other weights are zero.

Fig. 5 Example of centroid deposition.

### Trilinear

Trilinear deposition transfers particle properties to the eight Eulerian grid cells surrounding its center. First, low and high weights are calculated for each direction, then they are multiplied together to form eight composite weights. The following figure illustrates bilinear interpolation. The left image shows weights computed in the X-direction, \([0.15, 0.85]\), and the center image shows weights computed in the Y-direction, \([0.25,0.75]\). The right image shows the composite weights for the four cells that comprise the 2D neighborhood surrounding the particle.

Fig. 6 Example of trilinear deposition.

### True-DPVM

The “true” divided particle volume method (DPVM) calculates the actual volume of a particle in each cell by computing the intersection between the grid cell faces and the particle. The fractional volume (volume inside the cell divided by total particle volume) is used as the deposition weight. If the `deposition_scale_factor` is defined, it multiplies the particle radius increasing the effective area to transfer properties over. The true-DPVM scheme is illustrated below where the dashed line represents the scaled particle radius. The grown particle intersects the top two and lower-right cells. Since the grown particle does not intersect the lower-left cell, it has a deposition weight of zero.

Fig. 7 Example of true-dpvm deposition.

### Trilinear DPVM square

The trilinear DPVM square method is similar to the trilinear scheme in that low and high weights are computed for each direction, then multiplied together to give a composite weight for each of the eight cells in the neighborhood. The primary distinction is that the fractional volume cut by the grid cell face for each direction is used to calculate the low and high weights. If the `deposition_scale_factor` is defined, it multiplies the particle radius increasing the effective area to transfer properties over. In the below figure, the left image shows the X-direction weights, and the center image shows the Y-direction weights. The right image shows the resulting composite weights.

Fig. 8 Example of trilinear dpvm square deposition.

## Deposition redistribution

The following inputs are defined using the `mfix` prefix.

Key | Description | Type | Default
--- | --- | --- | ---
deposition_redist_type | Algorithm used to redistribute excess solids to adjacent cells. This typically applies only to small cut-cells along the geometry. `MaxPack` If the solids volume fraction exceeds the prescribed `max_solids_volume_fraction`, then the local average solids volume is computed and solids volume exceeding the average is reallocated to adjacent cells. The fraction of redistributed solids is applied to all transferred properties. `StateRedist` Use the state redistribution algorithm of [Berger_Giulini_2021] and [Giulini_etal_2022] to reallocate transferred quantities in small cells. | String | MaxPack
max_solids_volume_fraction | Threshold solids volume fraction for MaxPack redistribution. Redistribution of transferred quantities only occurs when the solids volume fraction of a cell exceeds this value. | Real | 0.6
deposition_redist_vfrac | Threshold geometric volume fraction `StateRedist` redistribution. Redistribution of transferred quantities always occurs for cells with a geometric volume fraction below this value. | Real | 0.1

## Drag models

The following input is defined using the `mfix` prefix.

| Description | Type | Default
--- | --- | --- | ---
drag_type | Fluid-particle drag model. `WenYu` [Wen_Yu_1966] `Gidaspow` [Ding_Gidaspow_1990], [Lathouwers_Bellan_2000] `BVK2` [BVK_2007], [Tenneti_et_2011], [Tang_etal_2015] `UserDrag` | String | None

Note

The `UserDrag` keyword is used to invoke a user defined drag model. This is accomplished by copying `src/usr/usr_drag.cpp` file into the build directory, implementing the desired drag model, and recompiling the code. An example can be found in `tests/DEM06-x`.

[Wen_Yu_1966]

Wen, C.Y. & Yu, Y.H. (1966). Mechanics of Fluidization. The Chemical Engineering Progress Symposium Series, 162, 100-111.

[Ding_Gidaspow_1990]

Ding, J., & Gidaspow, D. (1990). A bubbling fluidization model using kinetic theory of granular flow. AIChE journal, 36(4), 523-538.

[Lathouwers_Bellan_2000]

Lathouwers D. & Bellan J.(2000). Proceedings of the 2000 U.S. DOE Hydrogen Program Review NREL/CP-570-28890. Available from http://www.eere.energy.gov/hydrogenandfuelcells/pdfs/28890k.pdf.

[BVK_2007]

Beetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007). Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice-Boltzmann simulations. Chemical Engineering Science, 62(1-2), 246-255.

[Tenneti_et_2011]

Tenneti, S., Garg, R., & Subramaniam, S. (2011). Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. International journal of multiphase flow, 37(9), 1072-1092.

[Tang_etal_2015]

Tang, Y., Peters, E. A. J. F., Kuipers, J. A. M., Kriebitzsch, S. H. L., & van der Hoef, M. A. (2015). A new drag correlation from fully resolved simulations of flow past monodisperse static arrays of spheres. AIChE journal, 61(2), 688-698.

## Heat transfer coefficients

The following input is defined using the `mfix` prefix.

| Description | Type | Default
--- | --- | --- | ---
convection_type | Fluid-particle heat transfer coefficient model. `RanzMarshall` [RanzMarshall_1952] `Gunn` [Gunn_1978] | String | None

[RanzMarshall_1952]

Ranz, W.E. & Marshall, W.R. (1952). Friction and transfer coefficients for single particles and packed beds. Chemical Engineering Science, 48(5), 247-253.

[Gunn_1978]

Gunn, D. J. (1978). Transfer of heat or mass to particles in fixed and fluidised beds. International Journal of Heat and Mass Transfer, 21(4), 467-476.