Phase Slip

For flow of multiphase mixtures with non-homogeneous velocity, the slip between the phases introduces additional terms in the volume fraction transport equation as well as the momentum and energy conservation.

Modeling of the slip velocity is an integral part of the Mixture Multiphase (MMP) model. Although Simcenter STAR-CCM+ solves the momentum transport for the mixture of phases, and the solution variable is the mixture (mass averaged) velocity, the slip velocity model lets you model the effects of the phases moving with different velocities.

In Simcenter STAR-CCM+, for an interaction between phases, you set one fluid to be the primary phase and the other as the secondary phase. In this formulation, the subscript $p$ is used for the primary phase, the subscript $s$ is used for the secondary phase, and the subscript $i$ is used when an expression is valid for any phase.

Denote the relative velocity or slip velocity between two phases $p$ and $s$ as:

Figure 1. EQUATION_DISPLAY

v_{p s} = v_{s} - v_{p}

(2889)

The diffusion velocity is defined as:

Figure 2. EQUATION_DISPLAY

v_{d, i} = v_{i} - v_{m}

(2890)

where $v_{m}$ is the mixture (mass averaged) velocity.

The diffusion and slip velocity are connected by:

Figure 3. EQUATION_DISPLAY

v_{d, p} = - v_{p s} - \sum_{i} Y_{i} v_{s i}

(2891)

with the mass fraction:

Figure 4. EQUATION_DISPLAY

Y_{i} = \frac{α_{i} ρ_{i}}{ρ_{m}}

(2892)

where $α_{i}$ is the volume fraction of phase $i$ , $ρ_{i}$ is the density of phase $i$ , and $ρ_{m}$ is the density of the mixture.

Assuming that all velocities except for $v_{d, p}$ and $v_{d, s}$ are known:

Figure 5. EQUATION_DISPLAY

\begin{matrix} v_{d, p} = \frac{- v_{0} - Y_{s} v_{p s}}{Y_{p} + Y_{s}} \\ v_{d, s} = \frac{- v_{0} + Y_{p} v_{p s}}{Y_{p} + Y_{s}} \end{matrix}

(2893)

and the contribution from diffusion velocities of all other phases is:

Figure 6. EQUATION_DISPLAY

v_{0} = \sum_{i \notin {p, s}} Y_{i} v_{d, i}

(2894)

The slip velocity

v_{p s}

can be user-defined, drag-based or based on Darcy’s law in Porous Media (see Porous Media).

Drag-based slip velocity

Manninen et. al ([639]), define the slip velocity as:

Figure 7. EQUATION_DISPLAY

v_{p s} = \frac{ρ_{s} - ρ_{m}}{ρ_{s}} \frac{24}{{Re}_{p s} c_{d}} τ_{s} b

(2895)

where $ρ_{s}$ is the density of the secondary phase and ${Re}_{p s}$ is the phase pair Reynolds number (also known as the relative Reynolds number). ${Re}_{p s}$ is defined as:

Figure 8. EQUATION_DISPLAY

{Re}_{p s} = \frac{ρ_{p} | v_{p s} | d_{s}}{μ_{p}}

(2896)

The Schiller-Naumann drag coefficient $c_{d}$ is defined as:

Figure 9. EQUATION_DISPLAY

c_{d} = {\begin{matrix} \frac{24}{{Re}_{p s}} (1 + 0.15 {Re}_{p s}^{0.687}) & i f {0 < Re}_{p s} \leq 1000 \\ 0.44 & i f {Re}_{p s} > 1000 \end{matrix}

(2897)

$τ_{s}$ is the particle relaxation time defined as:

Figure 10. EQUATION_DISPLAY

τ_{s} = \frac{ρ_{s} d_{s}^{2}}{18 μ_{p}}

(2898)

where $d_{s}$ is the interaction length scale (particle diameter of the secondary phase) and $μ_{p}$ is the viscosity of the primary phase.

The body forces b are defined as the sum of internal and external body forces. The external forces consist of gravity and rotational forces:

Figure 11. EQUATION_DISPLAY

b_{int} = - \frac{v_{m}^{n} - v_{m}^{n - 1}}{Δ t} - (v_{m} \cdot \nabla) v_{m}

(2899)

Figure 12. EQUATION_DISPLAY

b_{e x t} = g - ω \times v_{m} - ω \times v_{T}

(2900)

where

v_{T}

is the reference frame velocity.

The body forces at time

t_{n}

are then computed with a semi-implicit approach:

Figure 13. EQUATION_DISPLAY

b^{n} = 0.5 b^{n - 1} + 0.5 (b_{e x t} + b_{i n t})

(2901)

Slip Velocity Interaction Length Scale Limiter Option

The interaction length scale limiters for the dispersed phase are used to improve convergence for drag-based slip simulations. This limiter replaces the interaction length scale $d_{s}$ in Eqn. (2896) with the limited interaction length scale, defined as:

Figure 14. EQUATION_DISPLAY

d_{s}^{\lim} = \min (d_{s}, l_{s})

(2902)

where $l_{s}$ is the limiting interaction length scale.

Slip Velocity Limiter Option

Applies a limiter to the maximum slip velocity for either drag-based or Darcy's-law slip simulations. This limiter replaces the slip velocity as computed in Eqn. (2896) with a limited slip velocity, defined as:

Figure 15. EQUATION_DISPLAY

v_{p s}^{\lim} = \frac{v_{p s}}{∥ v_{p s} ∥} \min (v_{p s}, l_{v})

(2903)

where $l_{v}$ is the limiting vector length.