Physical Velocity Formulation

The porous media model in Simcenter STAR-CCM+ uses the physical velocity formulation. This formulation accounts for the increase in velocity when the flow enters the porous medium.

In the equations below, velocity refers to physical velocity. With regard to energy, the porous media model in Simcenter STAR-CCM+ provides either non-equilibrium or equilibrium modeling. The non-equilibrium model does not assume that fluid and solid phase are at the same temperature. It solves separate energy transport equations for the fluid and the solid phases. The equilibrium model, on the other hand, assumes that the fluid and solid phases are at the same temperature. It solves a single energy transport equation.

This formulation assumes that the step change in porosity only occurs across interfaces. Any step change in porosity in the interior of a region may lead to non-physical results.

Volume Fraction

The volume fraction expresses the volume that is occupied by the

i

th porous phase with respect to the volume occupied by all porous phases, and is related to the porosity and cell volume as follows:

Figure 1. EQUATION_DISPLAY

α_{s i} = \frac{V_{solid, i}}{\sum_{i}^{} V_{solid, i}} = \frac{V_{solid, i}}{V} \cdot \frac{1}{1 - χ}

(1850)

where:

$V_{solid, i}$ is the volume of the $i$ th solid porous phase.
$V$ is the cell volume.
$χ$ is the porosity.

Continuity Equation

Figure 2. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} (ρ χ) d V + \oint_{A} (χ ρ v) \cdot d a = \int_{V} S_{u}^{c} d V

(1851)

where:

$V$ is the volume
$ρ$ is the density
$χ$ is the porosity
$v$ is the physical velocity
$S_{u}^{c}$ is a user-defined source or sink.

Momentum Equation

Figure 3. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} (χ ρ v) d V + \oint_{A} χ ρ v \otimes v \cdot d a = - \oint_{A} χ p I \cdot d a + \oint_{A} χ T \cdot d a + \int_{V} χ f_{b} d V + \int_{V} χ f_{p} d V + \int_{V} S_{u}^{m} d

(1852)

where:

$p$ is pressure.
$I$ is the identity matrix.
$f_{b}$ is the body force vector.
$f_{p}$ is the porous resistance force, where $f_{p} = P_{v} v_{s} + P_{i} | v_{s} | v_{s}$
where:
- $P_{v}$ is the viscous resistance tensor.
- $v_{s}$ is the superficial velocity, $v_{s} = χ v$ .
- $P_{i}$ is the inertial resistance tensor.
$S_{u}^{m}$ is a user-defined momentum source or sink.

Thermal Non-Equilibrium Energy Equations

Figure 4. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} (χ ρ_{fluid} E_{fluid}) d V + \oint_{A} χ ρ_{fluid} H_{fluid} v \cdot d a = - \oint_{A} χ q_{fluid} \cdot d a + \oint_{A} χ T \cdot v \cdot d a + \sum_{solid phases}^{} \int_{V} a_{s i} h_{s i} (T_{fluid} - T_{s i}) d V + \int_{V} χ f_{b} \cdot v d V + \int_{V} S_{u}^{e} d V

(1853)

where:

$E_{fluid}$ is the total energy of the fluid
$H_{fluid}$ is the total enthalpy of the fluid
$a_{s i}$ is the interaction area density of the $i$ th solid phase
$h_{s i}$ is the heat transfer coefficient of the $i$ th solid phase.
$T$ is the stress tensor
$S_{u}^{e}$ is a user-defined energy source or sink for the fluid.
$T_{s i}$ is the solid temperature of the $i$ th solid phase

Figure 5. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} ((1 - χ) α_{s i} ρ_{s i} E_{s i}) d V = - \oint_{A} (1 - χ) α_{s i} {q_{s i}}_{} \cdot d a + \int_{V} a_{s i} h_{s i} (T_{s i} - T_{fluid}) d V + \int_{V} S_{u}^{e} d V

(1854)

where:

$ρ_{s i}$ is the density of the $i$ th solid phase
$E_{s i}$ is the total energy of the $i$ th solid phase
$q_{s i}$ is the conduction heat flux through the $i$ th solid phase

Thermal Equilibrium Energy Equations

Figure 6. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} {(ρE)}_{eff} d V + \oint_{A} χ ρ_{fluid} H_{fluid} v \cdot d a = \oint_{A} k_{eff} \nabla T \cdot d a + \oint_{A} χ T \cdot v \cdot d a + \int_{V} χ f_{b} \cdot v d V + \int_{V} S_{u}^{e} d V

(1855)

Figure 7. EQUATION_DISPLAY

\begin{matrix} {(ρ E)}_{eff} = χ ρ_{fluid} E_{fluid} + (1 - χ) ρ_{solid} C_{p, solid} T_{fluid} \\ ρ_{solid} C_{p, solid} = \sum_{i}^{} α_{s i} ρ_{s i} C_{p, s i} \end{matrix}

(1856)

Figure 8. EQUATION_DISPLAY

\begin{matrix} k_{e ff} = χ k_{fluid} + (1 - χ) k_{solid} \\ k_{solid} = \sum_{i}^{} α_{s i} k_{s i} \end{matrix}

(1857)

where:

$C_{p, solid}$ is the specific heat capacity of the solid
$C_{p, s i}$ is the specific heat capacity of the $i$ th solid phase
$k_{eff}$ is an effective thermal conductivity that is calculated from the thermal conductivities of the fluid $k_{fluid}$ and the solid $k_{solid}$ , where $k_{s i}$ is the effective thermal conductivity of the $i$ th solid phase.

Conditions at the Porous Interface

Across the porous interface, the physical velocity is discontinuous.

The following table lists the porous interface conditions for various variables:


Variable	Porous Interface Condition
Physical velocity	Discontinuous
Superficial velocity	Continous
Static pressure	Discontinuous
Total pressure	Continuous