Solids model

Solids settings

Enabling the solids solver and specifying options common to both DEM and PIC models. The following inputs are defined with the prefix solids:

	Description	Type	Default
types	Specified name(s) of the solids types or None to disable the solids solver. The user-defined names are used to specify DEM and/or PIC model inputs.	Strings	None
molecular_weight	Value of constant solid molecular weight.	Real	0
density	Specify which density model to use for solids. Options (case-insensive): `None` - solids density specified through initial and boundary conditions; `Mixture` - solids density computed as a weighted average of the solid phase species densities. It requires constant species densities to be defined by the user.	String	None
specific_heat	Specify which specific heat model to use for solids. Options: `constant` - constant specific heat	String	constant
specific_heat.constant	Constant solids specific heat value. A value is required for `constant` specific heat model.	Real	0
reference_temperature	Reference temperature used for specific enthalpy.	Real	0
species	Specify which species can constitute the solids phase. Defined species must be a subset of the species defined in `species.solve`	Strings	None
newton_solver.absolute_tol	Define absolute tolerance for damped Newton solver.	Real	1.e-6
newton_solver.relative_tol	Define relative tolerance for damped Newton solver.	Real	1.e-6
newton_solver.max_iterations	Define max number of iterations for damped Newton solver.	Int	100
thermal_conductivity	Specify which thermal conductivity model to use for solids. Options: `constant` - constant thermal conductivity A thermal conductivity model is not required for solids. Defining a thermal conductivity model enables conductive heat transfer for DEM. However, it has no effect for PIC solids heat transfer.	String	constant
thermal_conductivity.constant	Constant solids thermal conductivity value. A value is required for `constant` thermal conductivity model.	Real	0
min_conduction_dist	The surface roughness separating two touching particles. This value is used to remove the singularity that the particle-fluid-particle conduction model develops at the contact interface.	Real	2.e-8
flpc	The fluid lens proportionality constant (flpc) is used to calculate the radius of the fluid lens surrounding the particle for particle-fluid-particle conduction.	Real	0.4

Below is an example for specifying the solids solver model options.

solids.types = my_solid0  my_solid1

solids.reference_temperature = 298.15

solids.specific_heat = mixture

solids.species = Fe2O3  FeO

DEM model settings

Enabling the DEM solver and specifying model options. The following keys must be defined with the prefix dem:

	Description	Type	Default
solve	Specified name(s) of the DEM types or None to disable the DEM solver. The user-defined names are used to specify DEM model inputs.	Strings	None
friction_coeff.pp	Friction coefficient :: particle to particle collisions [required]	Real	0
friction_coeff.pw	Friction coefficient :: particle to wall collisions [required]	Real	0
spring_const.pp	Normal spring constant :: particle to particle collisions [required]	Real	0
spring_const.pw	Normal spring constant :: particle to wall collisions [required]	Real	0
spring_tang_fac.pp	Tangential-to-normal spring constant factor :: particle to particle collisions	Real	0.2857
spring_tang_fac.pw	Tangential-to-normal spring constant factor :: particle to wall collisions	Real	0.2857
damping_tang_fac.pp	Factor relating the tangential damping coefficient to the normal damping coefficient :: particle to particle collisions	Real	0.5
damping_tang_fac.pw	Factor relating the tangential damping coefficient to the normal damping coefficient :: particle to wall collisions	Real	0.5
implicit_drag	Apply fluid-particle drag force by implicit velocity update.	Int	0
tan_history	Include tangential history force in the collision model.	Bool	false
tan_history.max_contacts	Maximum number of contacts per particle tracked at any point when tangential history is enabled.	Int	10
rolling_friction	Rolling friction model to use when tangential history is enabled. [ACRO11, BVB+19, WK12, ZWY+99] Options: `None` `ModelA` \(\boldsymbol{\tau}_{ij}^{(r)} = -\mu_{r}\hat{r}_{ij} \left\| f_{ij}^{(n)} \right\| \frac{\omega_{ij}}{\left\|\omega_{ij}\right\|}\) `ModelB` \(\boldsymbol{\tau}_{ij}^{(r)} = -\mu_{r}\hat{r}_{ij} \left\| f_{ij}^{(n)} \right\| \boldsymbol{u}_{ij}^{\omega}\)	String	None
rolling_friction.coefficient	Rolling friction coefficient when using a rolling friction model	Real	0

The following inputs use the DEM type names specified using the dem.solve input to define restitution coefficients and are proceeded with the prefix dem.restitution_coeff. These must be defined for all DEM solid-solid and solid-wall combinations.

	Description	Type	Default
[dem_solid].[dem_solid2]	Specifies the restitution coefficient between DEM solids. Order is not important and this could be defined as `[dem_solid2].[dem_solid]`. If both are defined they must be equal.	Real	0
[dem_solid].wall	Specifies the restitution coefficient between solid and the wall. Order is not important and this could be defined as `wall.[dem_solid]`. If both are defined they must be equal.	Real	0

Below is an example for specifying the inputs for two DEM solids.

dem.solve = sand  char

dem.friction_coeff.pp     =     0.25
dem.friction_coeff.pw     =     0.15

dem.spring_const.pp       =   100.0
dem.spring_const.pw       =   100.0

dem.spring_tang_fac.pp    =     0.2857
dem.spring_tang_fac.pw    =     0.2857

dem.damping_tang_fac.pp   =     0.5
dem.damping_tang_fac.pw   =     0.5

dem.restitution_coeff.sand.sand =  0.85
dem.restitution_coeff.sand.char =  0.88
dem.restitution_coeff.char.char =  0.90

dem.restitution_coeff.sand.wall =  0.85
dem.restitution_coeff.char.wall =  0.89

Implicit drag

By default, the fluid-particle drag force is applied explicitly to particles,

\[u_p^{n^{\prime}+1} = u_p^{n^\prime} + dt_{\mathrm{DEM}}\left(\beta^n (u_f^{n+1} - u_p^{n}) + \mathcal{F}_p^{n^\prime}\right)/m_p\]

where \(u_p\) is the p-th particle velocity at the indicated time level, \(\beta^n\) is the drag coefficient, \(u_f^{n+1}\) is the fluid velocity, \(\mathcal{F}_p\) are all non-drag forces (e.g., gravitational, buoyancy, collision etc.), and \(m_p\) is particle mass. Terms exist at different time levels due in part to how the fluid and particle models are coupled and partly because particles typically sub-step in time. Specifically, a time step advances the fluid from \(t^n\) to \(t^{n+1}\), then particles take multiple smaller time steps to traverse the same total time. In the above equation, prime markers differentiate quantities updated at each DEM sub-step.

The updated particle velocity, \(u_p^{n^{\prime}+1}\), is substituted into the drag expression when implicit drag is enabled.

\[u_p^{n^{\prime}+1} = \frac{u_p^{n^\prime} + dt_{\mathrm{DEM}}\left( \beta^n u_f^{n+1} + \mathcal{F}_p^{n^\prime} \right)/m_p}{1 + dt_{\mathrm{DEM}} \beta^n / m_p}\]

By using the updated velocity, the drag force computed for the fluid and particles is inconsistent, and interphase momentum is not conserved. However, implicit drag may be needed for numerical stability when the particle density is much less than the fluid (such as bubbles).

PIC model settings

The following inputs define the PIC solids model and its associated frictional stress parameters:

\[\tau_p = \frac{P_s \varepsilon_p^{\beta}}{\max \left[ (\varepsilon_{cp} - \varepsilon_p), \epsilon (1-\varepsilon_p)\right]}\]

The following inputs are defined with the prefix pic:

	Description	Type	Default
solve	Specified name(s) of the PIC types or None to disable the PIC solver.	Strings	None
close_pack	Solids volume fraction at close pack, \(\varepsilon_{cp}\). Below this threshold, the solid stress tensor has minimal impact.	Real	0.55
pressure_coefficient	Pressure linear scale factor in frictional stress model, \(P_s\).	Real	100.0
beta	Volume fraction exponential scale factor in frictional stress model, \(\beta\).	Real	3.0
small_number	Non-singularity term in frictional stress model, \(\epsilon\).	Real	1.0e-7
damping_factor	An empirical dampening factor for the frictional stress model.	Real	0.4
damping_factor_wall_normal	Normal coefficient of restitution for parcel-wall collisions.	Real	0.3
damping_factor_wall_tangent	Tangential coefficient of restitution for parcel-wall collisions.	Real	0.99
parcels_per_cell_at_pack	Specifies the number of particles that can occupy a regular computational cell near close packing. Larger values reduce sensitivity to small changes in parcel positions, which helps stabilize the computed solids volume fraction and, by extension, the frictional stress model. However, increasing this value also results in simulations tracking a greater number of parcels.	Real	32.0
verbose	Verbosity level for PIC-related output.	Int	0

The solids volume fraction is required to evaluate the frictional stress model when advancing parcels from \(t^n\) to \(t^{n+1}\). Ideally, this evaluation would use either the volume fraction at the midpoint of the time step, \(\varepsilon_p^{n+1/2}\), or at the end of the step, \(\varepsilon_p^{n+1}\).

However, since neither of these values is available a priori, an iterative approach is used. An initial approximation of the volume fraction is used to advance parcels to a tentative position at \(t^{n+1}\). These new positions are then used to compute an updated volume fraction field, which is subsequently used in the next iteration to recompute the tentative parcel positions.

The following keywords control the iterative PIC algorithm and are defined with the prefix pic:

	Description	Type	Default
initial_step_type	Specifies the initial solids volume fraction used in PIC iterations to approximate the volume fraction at the end of the time step, :math:t^{n+1}. Options (case-insensitive): `nth_eps` - use the volume fraction from the previous time step, \(t^{n}\) `zero_eps` - set the initial volume fraction to zero `taylor_approx` - use a Taylor series expansion about parcel positions to approximate the volume fraction at \(t^{n+1}\)	String	“nth_eps”
max_iter	Maximum number of iterations used to approximate the end state	Int	3
advance_vel_p	Specifies the weighting factor used to compute the averaged particle velocity field during parcel advancement. The field-averaged velocity is used when applying forces to parcels. This parameter takes values between 0 and 1, blending the particle velocity at the beginning and end of the time step. A value of `0.5` (default) yields a time-centered average.	Real	0.5
velocity_reference_frame	The velocity reference frame scales the field-averaged velocity used to compute the slip velocity between a parcel and the bulk when when the parcels is moving towards an area of higher solids concentration. This parameter takes values between 0 and 1.	Real	0.5