POROSITY_MODEL

Specifies a porosity model for the flow equation.

Type

AcuSolve Command

Syntax

POROSITY_MODEL("name") {parameters...}

Qualifier

User-given name.

Parameters

type (enumerated) [=none]

Type of the porosity model.

none: No porosity.
constant or const: Darcy-Forchheimer porosity model. Requires permeability, darcy_coefficient, forchheimer_coefficient and permeability_direction.

porosity (real) >0 <=1 [=1]

Porosity value.

permeability_type (enumerated) [=cartesian]

Permeability type.

cartesian: Use Cartesian coordinate system to specify permeabilities and directions.
cylindrical: Use cylindrical coordinate system to specify permeabilities and directions.
spherical: Use spherical coordinate system to specify permeabilities and directions.

direction_1_permeability (real) [=1]

direction_2_permeability (real) [=1]

direction_3_permeability (real) [=1]

Permeability of the porous media in each of the three orthogonal principal axes. Used with constant type. These commands are valid only when permeability_type=cartesian. Also, these commands are equivalent to the previously used permeability array [={1,1,1}], which has been deprecated. AcuSolve will automatically map the permeability_array to the appropriate direction_1_permeability, direction_2_permeability, and direction_3_permeability parameters when reading an input file that uses the deprecated syntax. If both parameters are specified, this mapping will cause the permeability to take precedence over the values that are specified for direction_1_permeability, direction_2_permeability and direction_3_permeability.

cartesian_permeability_directions (array) [={1,0,0;0,1,0;0,0,1}]

Orthogonal principal axes of the permeability values. The vectors are ortho-normalized prior to use. Used with constant type. This command is equivalent to the previously used permeability_direction [={1,0,0;0.1.0;0,0,1}]. If both are specified in the input file, the one that appears last will be used for the computation.

direction_1_permeability_multiplier_function (string) [=none]

direction_2_permeability_multiplier_function (string) [=none]

direction_3_permeability_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. These commands are valid only when permeability_type = cartesian.

radial_permeability (real) [=1]

Radial permeability of the porous media in cylindrical and spherical coordinate systems. This command is valid only when permeability_type = cylindrical or spherical.

axial_permeability (real) [=1]

Axial permeability of the porous media in cylindrical coordinate system. This command is valid only when permeability_type = cylindrical.

tangential_permeability (real) [=1]

Tangential permeability of the porous media in cylindrical and spherical coordinate systems. This command is valid only when permeability_type = cylindrical or spherical.

cylindrical_permeability_axis (array) [={0,0.0;1,0,0}]

Principal axis of the axial permeability value. Other two principal axes in cylindrical coordinate are computed internally and ortho-normalized prior to use.

radial_permeability_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical or spherical.

axial_permeability_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical.

tangential_permeability_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical or spherical.

spherical_permeability_center (array) [={0,0,0}]

Spherical center of permeability values. This command is valid only when permeability_type = spherical.

darcy_coefficient (real) [=0]

Coefficient of Darcy's (linear) term. Used with constant type.

forchheimer_coefficient (real) [=0]

Coefficient of Forchheimer's (quadratic) term. Used with constant type.

darcy_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the Darcy coefficient. If none, no scaling is performed.

forchheimer_multiplier_function (string) [=none]

User-given name of the multiplier function for scaling the Forchheimer coefficient. If none, no scaling is performed.

effectiveness_type (enumerated) [=none]

Type of the effectiveness of the porous model.

constant or const: Requires effectiveness.
piecewise_linear: Piecewise linear curve fit. Requires effectiveness_curve_fit_variable and effectiveness_curve_fit_values.
cubic_spline: Cubic spline curve fit. Requires effectiveness_curve_fit_variable and effectiveness_curve_fit_values.
user_function: User-defined function. Requires effectiveness_user_function, effectiveness_user_values, and effectiveness_user_strings.

effectiveness (real) >0 <=1 [=1]

Effectiveness of the flow resistance exerted by the porous medium.This command is valid only when effectiveness_type = constant.

effectiveness_curve_fit_variable (enumerated) [=temperature]

Independent variable of the curve fit for effectiveness. Used with piecewise_linear and cubic_spline types.

x_coordinate: x coordinate
y_coordinate: y coordinate
z_coordinate: z coordinate
x_reference_coordinate: x reference coordinate
y_reference_coordinate: y reference coordinate
z_reference_coordinate: z reference coordinate
temperature: Temperature
species_1: species 1
species_2: species 2
species_3: species 3
species_4: species 4
species_5: species 5
species_6: species 6
species_7: species 7
species_8: species 8
species_9: species 9

effectiveness_curve_fit_values (array) [={0,0}]

A two-column array of independent-variable/effectiveness data values. Used with piecewise_linear and cubic_spline effectiveness types.

effectiveness_user_function (string) [no default]

Name of the effectiveness user-defined function. Used with effectiveness_type = user_function.

effectiveness_user_values (array) [={}]

Array of values to be passed to the user-defined function. Used with effectiveness_type = user_function.

effectiveness_user_strings (array) [={}]

Array of strings to be passed to the user-defined function. Used with effectiveness_type = user_function.

Description

This command specifies a porosity model for the flow (momentum) equations. This model is only applicable to fluid element sets.

POROSITY_MODEL commands are referenced by MATERIAL_MODEL commands, which in turn are referenced by ELEMENT_SET commands:

POROSITY_MODEL( "my porous media" ) {
   type                                = constant 
   direction_1_permeability            = 1.e-6
   direction_2_permeability            = 1.e-2
   direction_3_permeability            = 1.e-6
   cartesian_permeability_directions   = { 1, 0, 0 ;
                                           0, 1, 0 ;
                                           0, 0, 1 ; }
   darcy_coefficient                   = 0
   forchheimer_coefficient             = 0.5                                       
 }
 MATERIAL_MODEL( "my material model" ) {
   porosity_model                      = "my porous media"
   ...
 }
 ELEMENT_SET( "fluid elements" ) {
   material_model                      = "my material model"
   ...
}

In AcuSolve, a porous media problem can be solved using two methods:

Superficial Velocity
Physical Velocity

In AcuSolve, the porous media is solved at the macroscopic level by averaging the behavior of the flow. The continuity and momentum equation of the flow, in AcuSolve, can be averaged using the superficial averaging method or intrinsic/physical averaging method. AcuSolve, by default, reports the superficially averaged value in the form of superficial velocity. Superficial velocity is calculated based on volumetric flow rate as follows:(1)

_{V_{\sup e r f i c i a l} = \frac{Q}{A}}

where Q is the volumetric flow rate and A is the cross-sectional area of the porous media. The continuity equation in superficial averaged form is divergence free for incompressible flow, which ensures continuity of velocity vector across the porous media boundaries. The superficial averaged method may not be very accurate but is useful when flow and pressure drop needs to be solved at the macro level. In AcuSolve, the below form of superficially averaged continuity and momentum equation has been used:(2)

\frac{\partial ε_{β} ρ_{β}}{\partial t} + \nabla \cdot (ρ_{β} < v_{β} >) = 0

(3)

\frac{ρ_{β}}{ε_{β}} \frac{\partial < v_{β} >}{\partial t} + \frac{1}{ε_{β}^{2}_{}} \nabla \cdot (ρ_{β} < v_{β} > < v_{β} >) = - \nabla < p_{β} >^{β} + ρ_{β} g + \frac{μ_{β}}{ε_{β}} \nabla^{2} < v_{β} > - R f

where

< v_{β} >

is the superficial velocity,

ε_{β}

is the porosity of the porous media, f is the porous media contribution to the momentum equation, p is the pressure, g is the gravity, 𝝆 is the density, and

μ

is the viscosity. R is the rotation tensor to rotate f along the global coordinate system. The term f is defined as:(4)

f = [\frac{C_{D a r c y} μ_{β}}{k} + \frac{C_{F o r c h} ρ_{β}}{\sqrt{k}} < v_{β} >] < v_{β} >

where $C_{D a r c y}$ is the Darcy coefficient, $C_{F o r c h}$ is the Forchheimer coefficient, and $k$ is the permeability.

A more accurate representation of velocity inside the porous media can be attained by solving the continuity and momentum equation inside the porous media using the intrinsic/physical averaging method. AcuSolve solves for physical velocity

< v_{β} >^{β}

inside the porous media using the below form of the flow equation:(5)

\frac{\partial ε_{β} < ρ_{β} >^{β}}{\partial t} + < ρ_{β} v_{β} >^{β} \nabla ε_{β} + ε_{β} \nabla \cdot < (ρ_{β} v_{β}) >^{β} = 0

(6)

\begin{array}{l} ρ_{β} \frac{\partial < v_{β} >^{β}}{\partial t} + ε_{β}^{- 1} \nabla \cdot (ε_{β} ρ_{β} < v_{β} >^{β} < v_{β} >^{β}) = - \nabla < p_{β} >^{β} + ρ_{β} g \\ + μ_{β} [\nabla^{2} < v_{β} >^{β} + ε_{β}^{- 1} \nabla ε_{β} \cdot \nabla < v_{β} >^{β} + ε_{β}^{- 1} \nabla < v_{β} >^{β} \nabla^{2} ε_{β}] - R f \end{array}

where the term f is defined as:(7)

f = ε_{β} [\frac{C_{D a r c y} μ_{β}}{k} + ε \frac{C_{F o r c h} ρ_{β}}{\sqrt{k}} < v_{β} >^{β}] < v_{β} >^{β}

The superficial velocity and the physical velocity are related using the equation below:(8)

< v_{β} > = ε_{β} < v_{β} >^{β}

The darcy_multiplier_function parameter may be used to uniformly scale the Darcy coefficient. The value of this parameter refers to the user-given name of a MULTIPLIER_FUNCTION command in the input file. For example, a Darcy coefficient ramped from zero to 0.5 over the first 10 time steps may be specified by:

POROSITY_MODEL( "ramped Darcy coefficient" ) {
   type                                = constant 
   permeability_type                   = cartesian
   direction_1_permeability            = 1.e-6
   direction_2_permeability            = 1.e-2
   direction_3_permeability            = 1.e-6
   cartesian_permeability_directions   = { 1, 0, 0 ;    0, 1, 0 ;    0, 0, 1 ; }
   darcy_coefficient                   = 1
   forchheimer_coefficient             = 0.5
   darcy_multiplier_function           = "ramped"
   }
   MULTIPLIER_FUNCTION( "ramped" ) {
   type       = piecewise_linear 
   curve_fit_vales                     = { 1, 0 ; 10, 0.5 }
   curve_fit_variable                  = time_step
}

Similarly, forchheimer_multiplier_function may be used to uniformly scale the Forchheimer coefficient. The value of this parameter refers to the user-given name of a MULTIPLIER_FUNCTION command in the input file. For example, using the same MULTIPLIER_FUNCTION command as above, a Forchheimer coefficient ramped from zero to 0.5 over the first 10 time steps may be specified by:

POROSITY_MODEL( "ramped Forchheimer coefficient" ) {
   type                              = constant
   direction_1_permeability          = 1.e-6
   direction_2_permeability          = 1.e-2
   direction_3_permeability          = 1.e-6
   cartesian_permeability_direction  = { 1, 0, 0 ;    0, 1, 0 ;    0, 0, 1 ; }
   darcy_coefficient                 = 0.5
   forchheimer_coefficient           = 1
   forchheimer_multiplier_function   = "ramped"
}

While the default coordinate system is Cartesian, you can also use cylindrical or spherical coordinate systems to specify permeabilities and their directions. The POROSITY_MODEL commands below are an example of cylindrical coordinate system:

POROSITY_MODEL( "my porous media" ) {
    type                          = constant
    permeability_type             = cylindrical 
    radial_permeability           = 1.e-6
    axial_permeability            = 1.e-2
    tangential_permeability       = 1.e-6
    cylindrical_permeability_axis = {0, 0, 0 ;   1, 0, 0 ; }
    darcy_coefficient             = 1
    forchheimer_coefficient       = 0.5
}

and for spherical coordinate system:

POROSITY_MODEL( "my porous media" ) {
    type                        = constant
    permeability_type           = spherical  
    radial_permeability         = 1.e-2
    tangential_permeability     = 1.e-6
    spherical_permeability_axis = {0, 0, 0 ;
    darcy_coefficient           = 1
    forchheimer_coefficient     = 0.5
}

The effectiveness parameter may be used to define a multiplicative constant applied to the force f in equations (Equation 4) and (Equation 7). This parameter controls the effectiveness of the porous medium at providing a resistance to the flow. By setting a value of 1, the porous medium fully exerts its resistance to the flow, while by setting a value of 0, no resistance to the flow is applied.

The porous medium effectiveness can be made temperature dependent to model freezing and melting. The POROSITY_MODEL commands below are an example of a temperature dependent effectiveness going from 1 to 0 at a temperature of 262.15K linearly over 2K:

POROSITY_MODEL( "ModelforMelting" ) {
   type                              = constant
   direction_1_permeability          = 1.0
   direction_2_permeability          = 1.0
   direction_3_permeability          = 1.0
   cartesian_permeability_direction  = { 1, 0, 0 ; 0, 1, 0 ; 0, 0, 1}
   darcy_coefficient                 = 5e+8
   forchheimer_coefficient           = 0
   effectiveness_type                = piecewise_linear
   effectiveness_curve_fit_values    = { 0, 1 ; 261.15, 1; 263.15, 0; 1000, 0 }
   effectiveness_curve_fit_variable  = temperature

An effectiveness of type user_function may be used to model more complex temperature dependencies; see the AcuSolve User-Defined Functions Manual for a detailed description of user-defined functions. An example is provided of a temperature dependent effectiveness going from 1 to 0 at a temperature of 262.15K linearly over 2K as in the previous example. The input command may be given by:

POROSITY_MODEL( "ModelForMelting" ) {
	type                              = constant
	direction_1_permeability          = 1.0
	direction_2_permeability          = 1.0
	direction_3_permeability          = 1.0
	cartesian_permeability_direction  = { 1, 0, 0 ; 0, 1, 0 ; 0, 0, 1 }
	darcy_coefficient                 = 5e+8
	forchheimer_coefficient           = 0
	effectiveness_type                = user_function
	effectiveness_user_function       = usrEffectiveness
	effectiveness_user_values         = {262.15, 2.0}
}

where the user-defined function "usrEffectiveness" may be implemented as follows:

#include "acusim.h"
#include "udf.h"
UDF_PROTOTYPE( usrEffectiveness ) ;				/* function prototype			*/
Void usrEffectiveness (
	UdfHd			udfHd,				/* Opaque handle for accessing data	*/
	Real*			outVec,				/* Output vector			*/
	Integer			nItems,				/* Number of elements			*/
	Integer			vecDim				/* = 1, porosity is a scalar		*/
) {
	Integer			elem ;				/* an element counter			*/
	Real*			temp ;				/* Temperatures of quadrature point	*/
	Real*			usrVals ;			/* user values				*/
	Real			tmelt ;				/* melting temperature			*/
	Real			dmelt ;				/* delta melting		      	*/
	Real			Tp, Tm ;			/* temperature intervals		*/
	Real			phi ;				/* porosity effectiveness		*/
	udfCheckNumUsrVals( udfHd, 2 ) ;			/* check for error			*/
	usrVals= udfGetUsrVals( udfHd ) ;			/* get the user vals			*/
	tmelt	= usrVals[0] ;					/* User Value - Melting temperature	*/
	dmelt	= usrVals[1] ;					/* User Value - Melting delta		*/
	Tp=tmelt+dmelt*0.5 ;
	Tm=tmelt-dmelt*0.5 ;
	// get temperature at integration points
	temp = udfGetElmData( udfHd, UDF_ELM_TEMPERATURE ) ;
	for ( elem = 0 ; elem < nItems ; elem++ ) {
		// compute porosity effectiveness at integration points
		if ( temp[elem] > Tp) {
			phi=0.0 ;
		} else if ( temp[elem] < Tm) {
			phi=1.0 ;
		} else {
			phi=(Tp-temp[elem])/dmelt ;
		}
		outVec[elem] = phi ;
	}
}								/* end of usrEffectiveness */