Drag

With one-way coupling, the continuous phase interacts with the dispersed phase through drag force and the dispersed phase has no influence on the continuous phase. With two-way coupling, the drag force applies in both directions.

The drag force between the continuous and the dispersed phase is computed as a function of the drag coefficient:

Figure 1. EQUATION_DISPLAY

F_{cd}^{D} = \frac{1}{2} C_{D} (\frac{6 α_{d}}{4 D}) ρ_{d} ‖ v_{r} ‖ v_{r}

(2848)

where:

$F_{cd}^{D}$ is the drag force
$C_{D}$ is the drag coefficient
$α_{d}$ is the volume fraction of the dispersed phase
$D$ is the characteristic diameter of the dispersed phase
$ρ_{d}$ is the density of the dispersed phase
$v_{r} = v_{d} - v$ is the relative velocity between the dispersed and the continuous phase

Schiller-Naumann: The drag coefficient according to Schiller-Naumann [543] is calculated as:

Figure 2. EQUATION_DISPLAY

C_{D} = {\begin{matrix} \frac{24}{{Re}_{p}} (1 + \frac{1}{6} {Re}_{p}^{2 / 3}) & , {Re}_{p} \leq 1000 \\ 0.424 & , {Re}_{p} > 1000 \end{matrix}

(2849)

with the particle Reynolds number defined as:

Figure 3. EQUATION_DISPLAY

{Re}_{p} = \frac{\bar{ρ} ‖ v_{r} ‖ D}{\bar{μ}}

(2850)

where:

$\bar{μ}$ is the dynamic viscosity of the multiphase mixture. The mixture dynamic viscosity $\bar{μ}$ can be constant, specified through a field function, or volume-weighted
$\bar{ρ}$ is the density of the multiphase mixture

The mixture density is volume-weighted:

Figure 4. EQUATION_DISPLAY

\bar{ρ} = α_{1} ρ_{1} + α_{2} ρ_{2}

(2851)

Consider a DMP-VOF phase interaction. A cell $c$ is in the VOF-VOF interfacial region with a volume fraction $α_{1}$ of VOF phase $V_{1}$ and a volume fraction $α_{2}$ of VOF phase $V_{2}$ , and $α_{1} + α_{2} = 1$ . The volume fraction of the dispersed phase in the cell is $α_{d}$ . The density, dynamic viscosity, and specific heat of each VOF phase $V_{i}$ are $ρ_{i}$ , $μ_{i}$ , and $C_{p, i}$ , respectively. The dispersed phase properties are $ρ_{d}$ , $μ_{d}$ , and $C_{p, d}$ . The characteristic particle diameter of the dispersed phase is $D$ . The unique velocity field of the background flow is $v$ , and the dispersed phase velocity field is $v_{d}$ . The (relative) phase slip velocity between the dispersed phase and the background flow is $v_{r}$ .

For a DMP-VOF Phase Interaction, the continuous drag force is weighted by the volume fraction:

Figure 5. EQUATION_DISPLAY

F_{di}^{D} = \frac{1}{2} α_{i} C_{D, i} (\frac{6 α_{d}}{4 D}) ρ_{d} ‖ v_{r} ‖ v_{r}

(2852)

Given that $α_{1} + α_{2} = 1$ , when the same constants or field function methods are specified for the drag coefficient in both VOF phases, the same drag force is obtained for both the DMP-VOF Phase Interaction and the DMP-Physics Continuum Interaction.

However, if the Schiller-Naumann correlation Eqn. (2849) is used for the calculation of the drag coefficient, the results are different for the two phase interactions. The phase particle Reynolds number that is used in the calculation of $C_{D, i}$ is defined as:

Figure 6. EQUATION_DISPLAY

R e_{p, i} = \frac{ρ_{i} ‖ v_{r} ‖ D}{μ_{i}}

(2853)

The phase properties $ρ_{i}$ and $μ_{i}$ are used instead of the mixture properties $\bar{ρ}$ and $\bar{μ}$ . $C_{D}$ is continuous across the interface but $C_{D, i}$ is discontinuous across the interface. A linear combination $α_{1} C_{d, 1} + α_{2} C_{d, 2}$ is used for the drag coefficient in the interfacial region, which is different from $C_{D}$ .

By default, the phases are one-way coupled: the background fluid does not feel the reacting force that the particles exert. The drag contributes as $F_{ij}^{D}$ in Eqn. (2846). In two-way coupled cases, the drag force contributes to both the continuous phase and the dispersed phase, but with different signs.