Granular Boundary Conditions

The granular boundary conditions have been implemented in Simcenter STAR-CCM+ such that the two boundary conditions, shear stress specification and granular temperature specification, can be considered independently.

For the shear-stress (of the particle phase), the following condition is enforced when the partial-slip option is used [483]:

Figure 1. EQUATION_DISPLAY

\frac{v_{s l} \cdot τ_{p} \cdot n}{| v_{s l} |} + \frac{\sqrt{3 θ_{p}} π ϕ ρ_{p} α_{p} g_{0} | v_{s l} |}{6 α_{p, m a x}} = 0

(2386)

where:

The first term on the left-hand side of the equation is the component of particle stress at the wall in the direction of the slip velocity, $v_{s l}$ .
$n$ is the unit normal at the boundary.
$ϕ$ is the specularity coefficient. This value varies between 0 for perfectly specular collisions and 1 for perfectly diffuse collisions.

For the granular temperature specification, the following condition is enforced when the Johnson-Jackson option is used [483]:

Figure 2. EQUATION_DISPLAY

κ \nabla θ_{p} \cdot n + \frac{π (1 - {e_{w}}^{2}) ρ_{p} α_{p} θ_{p} \sqrt{3 θ_{p}}}{4 α_{p, m a x} (1 - (\frac{α_{p}}{α_{p, m a x}})^{\frac{1}{3}})} - \frac{\sqrt{3 θ_{p}} π ϕ ρ_{p} α_{p} {| v_{s l} |}^{2}}{6 α_{p, m a x} (1 - (\frac{α_{p}}{α_{p, m a x}})^{\frac{1}{3}})} = 0

(2387)

where:

$κ$ is the granular diffusion coefficient.
$e_{w}$ is the coefficient of restitution for collisions between particles and the wall.

The second and third terms on the left-hand side of the equation appear only when the Partial-Slip option is used for the shear stress specification.

Boundary Conditions for $k$ and $ϵ$

When an appropriate turbulence model is active, the boundary conditions for $k$ and $ϵ$ are implemented in the following form [459]:

Figure 3. EQUATION_DISPLAY

ν_{p 0} \frac{d k_{p}}{d y} + D_{w} k_{p} = 0

(2388)

Figure 4. EQUATION_DISPLAY

ν_{p 0} \frac{d ϵ_{p}}{d y} + D_{w} ϵ_{p} = 0

(2389)

$D w$ is defined as follows:

Figure 5. EQUATION_DISPLAY

D_{w} = \frac{φ π}{6} \sqrt{3 θ_{0}}

(2390)

where $θ_{0}$ is granular temperature at the wall.

The reference kinematic viscosity of the particle phase is given as:

Figure 6. EQUATION_DISPLAY

ν_{p 0} = \frac{1}{2} (1 + C_{c} \frac{τ_{p}}{τ_{c}})^{- 1} τ_{p} θ

(2391)

where:

$C_{c}$ is a model coefficient. The default value is 5.
$τ_{p}$ is the drag timescale.
$τ_{c}$ is the collision timescale.

This formulation is along the same lines as that for the particle momentum Johnson-Jackson boundary condition:

Figure 7. EQUATION_DISPLAY

ν_{p 0} \frac{d v_{p}}{d y} + D_{w} v_{p} = 0

(2392)

Granular Boundary Conditions

Boundary Conditions for k and ϵ

Boundary Conditions for $k$ and $ϵ$