Fluid model¶

The following inputs are defined using the prefix fluid:

	Description	Type	Default
solve	Specify the name of the fluid or set to empty string or `None` to disable the fluid solver. The name assigned to the fluid solver is used to specify fluid properties and initial and boundary conditions. It is common to use the name `fluid`.	String
species	Specify the species that constitute the fluid. All listed species must be properly defined. See the species definition documentation for additional details.	Strings	None

Description

Type

Default

solve

Specify the name of the fluid or set to empty string or None to disable the fluid solver. The name assigned to the fluid solver is used to specify fluid properties and initial and boundary conditions. It is common to use the name fluid.

String

species

Specify the species that constitute the fluid.

All listed species must be properly defined. See the species definition documentation for additional details.

Strings

None

The root prefix [fluid_name] for the following inputs is the name defined using the fluid.solve keyword. For example, if the fluid solver is named myfluid, then the following keywords are preceded with myfluid and a period. Currently, MFIX-Exa only supports a single fluid; therefore, it is common to name the fluid fluid. This is illustrated later in example input snippets.

Viscosity¶

Molecular viscosity¶

The following inputs are defined using the prefix [fluid_name].viscosity.molecular:

	Description	Type	Default
model	Specify which molecular viscosity model to use. Options: `constant` - constant viscosity `Sutherland` [Sut93] - \(\displaystyle\mu(T) = \mu_{ref}\left(\frac{T}{T_{ref}}\right)^{3/2}\times\frac{(T_{ref}+S)}{(T+S)}\) `Reid` [RPP87] - \(\displaystyle\mu(T) = Ae^{\left(\frac{B}{T} + CT + DT^2 \right)}\) `mixture` - a mixture viscosity is computed from species viscosities and local species mass fractions [BSL06] \(\displaystyle\mu_{mix} = \sum_{\alpha=1}^{N} \frac{X_{\alpha} \mu_{\alpha}}{\sum_{\beta} X_{\beta} \phi_{\alpha \beta}}\) A viscosity model is required if the fluid solver is enabled.	String	constant
constant	Constant fluid viscosity. A value is required for `constant` viscosity model.	Real	None
Sutherland.T_ref	Sutherland model reference temperature. A value is required for `Sutherland` viscosity model.	Real	None
Sutherland.mu_ref	Sutherland model reference viscosity at T_ref. A value is required for `Sutherland` viscosity model.	Real	None
Sutherland.S	Sutherland model temperature. A value is required for `Sutherland` viscosity model.	Real	None
Reid.A Reid.B Reid.C Reid.D	Reid model constants. Values are required for `Reid` viscosity model.	Real	None

Eddy viscosity¶

The following inputs are defined using the prefix fluid.viscosity.eddy:

	Description	Type	Default
model	Specify eddy viscosity model. Options: `None` - No eddy viscosity model `Smagorinsky-Lilly` [Lil66, Sma63] `WALE` [DNP98] `usr`	String	None
Smagorinsky-Lilly.constant	Smagorinsky-Lilly constant which usually has values between 0.1 and 0.2. A value is required when using the `Smagorinsky-Lilly` eddy viscosity model.	Real	None
WALE.constant	WALE eddy viscosity model constant.	Real	0.325

Description

Type

Default

model

Specify eddy viscosity model.

Options:

None - No eddy viscosity model
Smagorinsky-Lilly [Lil66, Sma63]
WALE [DNP98]
usr

String

None

Smagorinsky-Lilly.constant

Smagorinsky-Lilly constant which usually has values between 0.1 and 0.2.

A value is required when using the Smagorinsky-Lilly eddy viscosity model.

Real

None

WALE.constant

WALE eddy viscosity model constant.

Real

0.325

Suspension viscosity¶

The following inputs are defined using the prefix fluid.viscosity.suspension:

	Description	Type	Default
model	Specify suspension viscosity model of the form \(\mu_{susp} = \mu_{mol}(\mu^* - 1)\) Options: `None` - No eddy suspension model \(\mu^* = 1\) `Einstein` [Ein11] \(\mu^* = 1 + 2.5\varepsilon_s\) `Brinkman` [Bri52, GLGP07] \(\mu^* = (1-\varepsilon_s)^{-c}\) `Roscoe` [KD59, MP56, Ros52] \(\mu^* = (1-\varepsilon_s/c_1)^{-c_2}\) `ChengLaw` [CL03] \(\mu^* = e^{2.5(1/(1-\varepsilon_s)^c-1)/c}\) `Sato` [SS81] \(\displaystyle\mu_{pit} = C d_s \rho \left\|\boldsymbol{u} - \boldsymbol{u_s}\right\|\) `Subramaniam` [MTGS15] \(\mu_{pit} = C_\mu k_f^2 / \epsilon_f\)	String	None
Brinkman.constant	Constant for exponent in Brinkman suspension expression. A value is required when using the `Brinkman` model.	Real	None
Roscoe.c1	Constant for max packing in Roscoe suspension expression. A value is required when using the `Roscoe` model.	Real	None
Roscoe.c2	Constant for exponent in Roscoe suspension expression. A value is required when using the `Roscoe` model.	Real	None
ChengLaw.constant	Constant for exponent in ChengLaw suspension expression. A value is required when using the `ChengLaw` model.	Real	None
Sato.constant	Constant for Sato suspension expression.	Real	0.65
Subramaniam.constant	Constant for Subramaniam suspension expression.	Real	0.09

Maximum viscosity¶

The following inputs are defined using the prefix fluid.viscosity:

	Description	Type	Default
max_effective_factor	Max limit of the effective viscosity as a factor of the molecular viscosity \(\mu_{eff} < fac*\mu_{mol}\), where \(\displaystyle\mu_{eff} = \mu_{mol} + \mu_{eddy} + \mu_{susp} + \mu_{pit}\)	Real	1.e6

Description

Type

Default

max_effective_factor

Max limit of the effective viscosity as a factor of the molecular viscosity

\(\mu_{eff} < fac*\mu_{mol}\), where \(\displaystyle\mu_{eff} = \mu_{mol} + \mu_{eddy} + \mu_{susp} + \mu_{pit}\)

Real

1.e6

Molecular weight¶

The following inputs are defined using the prefix [fluid_name]:

	Description	Type	Default
molecular_weight	Constant fluid molecular weight. A molecular weight should only be defined when using one of the `IdealGas` constraints and the species equations are not solved.	Real	0

Description

Type

Default

molecular_weight

Constant fluid molecular weight.

A molecular weight should only be defined when using one of the IdealGas constraints and the species equations are not solved.

Real

Specific heat¶

The following inputs are defined using the prefix [fluid_name].specific_heat:

	Description	Type	Default
model	Specify which specific heat model to use for the fluid. A specific heat model is required if advecting enthalpy. Furthermore, the model must be `mixture` if species equations are solved. Options: `constant` - the fluid has a constant specific heat `NASA7-poly` the fluid specific heat is defined by NASA-7 polynomials. NASA7 polynomial format: \(c_p(T)/R = \sum_{i=0}^4 a_{i}T^{i}\) Additional information on NASA-7 polynomials is provided in species specific heats. `mixture` - a mixture specific heat is computed from species specific heats and local species mass fractions. \(c_{p,\mathrm{mixture}} = \sum_n X_n c_{p,n}\)	String	None
constant	Constant fluid specific heat. A value is required for `constant` specific heat model.	Real	0
NASA7.a[i]	Specific heat polynomial coefficients. Polynomial coefficients are required if the fluid specific heat model is `NASA7-poly`. Additional information on NASA-7 polynomials is provided in species specific heats.	Reals	None
NASA7.Tsplit	Defines the transition temperature between NASA-7 polynomials. If there are `N` sets of coefficients for `N` temperature ranges, then `N-1` transition temperatures must be specified. In the standard case of two temperature ranges, a transition temperature of 1000K is assumed if `Tsplit` is not provided.	Reals	1000
ignore_discontinuities	Ignore discontinuities in NASA-7 polynomials at transition temperatures (`Tsplit`). More information is provided in species specific heats.	Int	0

The following inputs are defined using the prefix [fluid_name].

	Description	Type	Default
enthalpy_of_formation	Enthalpy of formation of fluid. Only used when the fluid specific heat model is `constant`.	Real	0
reference_temperature	Reference temperature used to compute fluid specific enthalpy when using `constant` fluid specific heat model or `mixture` model with `constant` model for species.	Real	298.0

Thermal conductivity¶

The following inputs are defined using the prefix [fluid_name].thermal_conductivity.

	Description	Type	Default
model	Specify which thermal conductivity model to use for the fluid. Options: `constant` - constant thermal conductivity `Sutherland` [Sut93] - \(\displaystyle\kappa(T) = \kappa_{ref}\left(\frac{T}{T_{ref}}\right)^{3/2}\times\frac{(T_{ref}+S)}{(T+S)}\) `power_law` : Power law formulation \(\displaystyle\kappa(T) = \left(\frac{T}{T_{ref}}\right)^n\) A thermal conductivity model is required if advecting enthalpy.	String	None
constant	Value of constant fluid thermal conductivity.	Real	None
Sutherland.T_ref	Sutherland model reference temperature.	Real	None
Sutherland.k_ref	Sutherland model reference conductivity at T_ref.	Real	None
Sutherland.S	Sutherland model temperature.	Real	None
power_law.T_ref	Power law model reference temperature.	Real	None
power_law.exponent	Power law model exponent \(n\)	Real	None

Radiation¶

The following inputs are defined using the prefix [fluid_name].radiation.

	Description	Type	Default
scattering_coeff	Scattering coefficient of the fluid.	Real	None
absorption.model	Specify which radiation absorption coefficient model to use for the fluid. Options: `gray-poly` - The absorption coefficient is defined by a temperature-dependent polynomial (0-5th order). Piecewise polynomials are defined by specifying temperature ranges. `gray-wsgg` - Weighted sum of gray-gases model. For each gray gas n, a gas‑specific absorption coefficient \(k_n\) is provided along with a set of polynomial coefficients \(b_{n,i}\) (0th-5th order). \(\displaystyle a(T) = \sum_n \left( [1-e^{-k_i PS}] \sum_i b_{n,i} T^i \right)\) `mixture` - Mixture-based model combining species contributions. An absorption model is required if radiation is enabled.	String	None

Description

Type

Default

scattering_coeff

Scattering coefficient of the fluid.

Real

None

absorption.model

Specify which radiation absorption coefficient model to use for the fluid. Options:

gray-poly - The absorption coefficient is defined by a temperature-dependent polynomial (0-5th order). Piecewise polynomials are defined by specifying temperature ranges.
gray-wsgg - Weighted sum of gray-gases model. For each gray gas n, a gas‑specific absorption coefficient \(k_n\) is provided along with a set of polynomial coefficients \(b_{n,i}\) (0th-5th order).

\(\displaystyle a(T) = \sum_n \left( [1-e^{-k_i PS}] \sum_i b_{n,i} T^i \right)\)
mixture - Mixture-based model combining species contributions.

An absorption model is required if radiation is enabled.

String

None

Gray polynomial¶

The following inputs are defined using the prefix [fluid_name].radiation.absorption.

	Description	Type	Default
form	Specifies the polynomial form used to represent the absorption coefficient as a function of temperature. Options: `0` - Absorption coefficient is defined as a polynomial in temperature. \(\displaystyle a(T) = a_0 + a_1 T + a_2 T^2 + \ldots\) `1` - Absorption coefficient is defined as a polynomial in the inverse of temperature. \(\displaystyle a(T) = a_0 + a_1 / T + a_2 / T^2 + \ldots\)	Int	0
T_range	Valid temperature range for the polynomial. If no temperature is provided, then the polynomial is assumed valid for all temperatures. Piecewise polynomials are defined by specify multiple ranges. For example, two piecewise polynomials can be specified by providing three temperatures: low, mid and high.	Reals	None
polynomial_coeffs	Polynomial coefficient. Polynomials can be up to 5-th order, and a set of polynomial coefficients must be defined for each temperature range.	Reals	None

Description

Type

Default

form

Specifies the polynomial form used to represent the absorption coefficient as a function of temperature. Options:

0 - Absorption coefficient is defined as a polynomial in temperature.

\(\displaystyle a(T) = a_0 + a_1 T + a_2 T^2 + \ldots\)
1 - Absorption coefficient is defined as a polynomial in the inverse of temperature.

\(\displaystyle a(T) = a_0 + a_1 / T + a_2 / T^2 + \ldots\)

Int

T_range

Valid temperature range for the polynomial. If no temperature is provided, then the polynomial is assumed valid for all temperatures. Piecewise polynomials are defined by specify multiple ranges. For example, two piecewise polynomials can be specified by providing three temperatures: low, mid and high.

Reals

None

polynomial_coeffs

Polynomial coefficient. Polynomials can be up to 5-th order, and a set of polynomial coefficients must be defined for each temperature range.

Reals

None

Example: A single temperature-dependent polynomial

A single polynomial is used over the entire valid temperature range. Here, the absorption coefficient is represented by one set of polynomial coefficients (e.g., 0th–2nd order).:

fluid0.radiation.absorption.model = gray-poly
fluid0.radiation.absorption.T_range = 300. 2400.
fluid0.radiation.absorption.polynomial_coeffs = 32.2454,  4.057e-3, -3.89e-6

Example: Three piecewise constants

The absorption coefficient is defined using multiple temperature segments. Providing three temperatures defines two piecewise regions. Because each region is specified with a single coefficient, the absorption coefficient is constant within each segment.:

fluid0.radiation.absorption.model = gray-poly
fluid0.radiation.absorption.T_range = 300.  600. 1200.
fluid0.radiation.absorption.polynomial_coeffs = 1.
fluid0.radiation.absorption.polynomial_coeffs = 0.1
fluid0.radiation.absorption.polynomial_coeffs = 0.01

Gray WSGG (Weighted Sum of Gray Gases)¶

The following inputs are defined using the prefix [fluid_name].radiation.absorption.

	Description	Type	Default
pressure-path_length	The pressure path length product `PS`. This term includes partial pressure of the absorbing gases `P` and the path length of the gas `S`. A value is required when the species equations are not solved.	Real	None
path_length	The path length of the gas, `S`. A value is required when the species are solved. It is combed with the total partial pressure of the real `gases` based on the local composition to compute the pressure path length product.	Real	None
gases	List of absorbing gas species. The total partial pressure based on the local composition `P` is computed from the listed species. One or more species must be listed if species equations are solved.	Strings	None
T_range	Valid temperature range for the polynomial. If no temperature is provided, then the polynomial is assumed valid for all temperatures. Piecewise polynomials are defined by specify multiple ranges. For example, two piecewise polynomials can be specified by providing three temperatures: low, mid and high.	Reals	None
coeffs	Weighted sum of gray gas coefficients.	Reals	None
polynomial_coeffs	Polynomial coefficient. Polynomials can be up to 5-th order, and a set of polynomial coefficients must be defined for each temperature range.	Reals	None

Example: WSGG model with five gray gases

Example inputs defining a WSGG model. Here, five gray gases are included in the absorption‑coefficient model. This is not a complete input file, nor does it represent a real WSGG model.

Dummy values are provided for illustration only.:

fluid0.radiation.absorption.model = gray-wsgg
fluid0.radiation.absorption.T_range = 600. 2400.
fluid0.radiation.absorption.pressure-path_length = 2.0

fluid0.radiation.absorption.coeffs = 100.  10.  1.  0.1  0.01
fluid0.radiation.absorption.polynomial_coeffs = 0.4068  -1.83E-04  8.68E-09
fluid0.radiation.absorption.polynomial_coeffs = 0.3390  -1.53E-04  7.24E-09
fluid0.radiation.absorption.polynomial_coeffs = 0.2712  -1.22E-04  5.79E-09
fluid0.radiation.absorption.polynomial_coeffs = 0.2034  -9.16E-05  4.34E-09
fluid0.radiation.absorption.polynomial_coeffs = 0.1356  -6.11E-05  2.89E-09

Each polynomial_coeffs line corresponds to one gray gas and provides the polynomial coefficients \(b_{n,i}\) in the temperature‑dependent term \(\sum_i b_{n,i}T^{i-1}\).

Example: WSGG model with partial‑pressure path‑length

Example inputs defining a weighted‑sum‑of‑gray‑gases model using three gray gases. In this case, the pressure path‑length product PS is computed from the partial pressures of real gases CO2 and H2O. As before, this is not a complete input file and does not represent a real WSGG dataset.:

species.solve = O2 H2O CO2 coal ash
fluid.species = CO2 H2O O2

fluid0.radiation.absorption.T_range = 600. 2400.
fluid0.radiation.absorption.path_length = 1.0
fluid0.radiation.absorption.gases = CO2 H2O
fluid0.radiation.absorption.coeffs = 75., 5., 1.
fluid0.radiation.absorption.polynomial_coeffs = 0.6280, -1.69E-04,  -8.78E-08,  2.22E-11
fluid0.radiation.absorption.polynomial_coeffs = 0.3768, -1.01E-04,  -5.27E-08,  1.33E-11
fluid0.radiation.absorption.polynomial_coeffs = 0.2512, -6.75E-05,  -3.51E-08,  8.90E-12

coeffs supplies the gray‑gas absorption coefficients \(k_n\), and each polynomial_coeffs line provides the \(b_{n,i}\) for the corresponding gray gas. The specified gases (CO2, H2O) are used to compute the local partial‑pressure path‑length PS.

Newton solver¶

The following inputs are defined using the prefix fluid.newton_solver and control the convergence criteria of the damped Newton solver used to compute temperature from enthalpy and specific heat.

	Description	Type	Default
absolute_tol	Define absolute tolerance for the Newton solver	Real	1.e-8
relative_tol	Define relative tolerance for the Newton solver	Real	1.e-8
max_iterations	Define max number of iterations for the Newton solver	Int	500

Example inputs¶

Incompressible fluid¶

The following setup shows how to define the most basic fluid model where the only required physical property is fluid viscosity. We are using the IncompressibleFluid constraint; therefore, the constant fluid density is defined by the initial and boundary conditions. The fluid energy and species equations are not solved.

Listing 6 Snippet of inputs defining a simple incompressible fluid. This is not a complete input file.¶

mfix.constraint = IncompressibleFluid

mfix.advect_density  = 0
mfix.advect_enthalpy = 0
mfix.solve_species   = 0


# Fluid model settings
# -----------------------------------------------------------------------
fluid.solve = fluid0

fluid0.viscosity.molecular.model = constant
fluid0.viscosity.molecular.constant = 1.8e-5


# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain

ic.full-domain.fluid0.volfrac   =  1.0
ic.full-domain.fluid0.density   =  1.0

ic.full-domain.fluid0.velocity  =  0.0  0.0  0.0


# Boundary Conditions
# -----------------------------------------------------------------------
bc.regions = inlet  outlet

bc.inlet = mi
bc.inlet.fluid0.density  =  1.0

bc.inlet.fluid0.inflow_type = velocity
bc.inlet.fluid0.velocity =  1.0e-8

bc.outlet = po
bc.outlet.fluid0.pressure =  0.

Simple ideal gas¶

In the following example, we use the IdealGasOpenSystem so that the fluid density is calculated from the ideal gas equation of state. In addition to viscosity, we must also provide the fluid molecular weight. The fluid energy and species equations are not solved, but the constant fluid temperature is required in the initial and boundary conditions to fully specify the equation of state. Lastly, the outflow boundary pressure is taken as the thermodynamic pressure when evaluating the equation of state.

Listing 7 Snippet of inputs defining a simple ideal gas. This is not a complete input file.¶

mfix.constraint = IdealGasOpenSystem

mfix.advect_density  = 1
mfix.advect_enthalpy = 0
mfix.solve_species   = 0


# Fluid model settings
# -----------------------------------------------------------------------
fluid.solve = fluid0

fluid0.viscosity.molecular.model = constant
fluid0.viscosity.molecular.constant = 1.8e-5

fluid0.molecular_weight = 29.0e-3


# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain

ic.full-domain.fluid0.volfrac   =  1.0

ic.full-domain.fluid0.velocity  =  0.0  0.0  0.0

ic.full-domain.fluid0.temperature = 300.0


# Boundary Conditions
# -----------------------------------------------------------------------
bc.regions = inlet  outlet

bc.inlet = mi

bc.inlet.fluid0.inflow_type = velocity
bc.inlet.fluid0.velocity =  1.0e-8

bc.inlet.fluid0.temperature = 300.0

bc.outlet = po
bc.outlet.fluid0.pressure =  101325.

Ideal gas with energy¶

In the last example, we use the IdealGasOpenSystem, however unlike the previous demonstration, the energy equation is solved. Therefore, we must define the fluid viscosity, molecular weight, thermal conductivity, and specific heat. Again, the fluid temperature is required in the initial and boundary conditions, and the outflow boundary pressure is taken as the thermodynamic pressure for the system.

Listing 8 Snippet of inputs defining an idea gas with energy. This is not a complete input file.¶

mfix.constraint = IdealGasOpenSystem

mfix.advect_density  = 1
mfix.advect_enthalpy = 1
mfix.solve_species   = 0


# Fluid model settings
# -----------------------------------------------------------------------
fluid.solve = fluid0

fluid0.viscosity.molecular.model = constant
fluid0.viscosity.molecular.constant = 2.0e-5

fluid0.thermal_conductivity.model = constant
fluid0.thermal_conductivity.constant = 0.026

fluid0.molecular_weight = 28.96518e-3

fluid0.specific_heat = NASA7-poly

fluid0.specific_heat.NASA7.a0 =  5.95960930E+00   3.08792717E+00
fluid0.specific_heat.NASA7.a1 =  3.56839620E+00   1.24597184E-03
fluid0.specific_heat.NASA7.a2 = -6.78729429E-04  -4.23718945E-07
fluid0.specific_heat.NASA7.a3 =  1.55371476E-06   6.74774789E-11
fluid0.specific_heat.NASA7.a4 = -3.29937060E-12  -3.97076972E-15
fluid0.specific_heat.NASA7.a5 = -4.66395387E-13  -9.95262755E+02


# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain

ic.full-domain.fluid0.volfrac   =  1.0

ic.full-domain.fluid0.velocity  =  0.0  0.0  0.0

ic.full-domain.fluid0.temperature = 300.0

# Boundary Conditions
# -----------------------------------------------------------------------
bc.regions = inlet  outlet

bc.inlet = mi

bc.inlet.fluid0.inflow_type = velocity
bc.inlet.fluid0.velocity =  1.0e-8

bc.inlet.fluid0.temperature = 300.0

bc.outlet = po

bc.outlet.fluid0.pressure =  101325.