Euler-Euler Two-Fluid Model

Exactly. The Euler-Euler method treats both the gas and liquid phases (or solid and gas phases) as continua and solves independent conservation equations for each phase. It excels in systems with high volume fractions of dispersed phases, such as bubble columns, slurry reactors, and vapor-liquid two-phase flow piping.

The VOF method is an "interface capturing method" that sharply tracks interfaces and is suitable for free-surface flows with large interface structures. On the other hand, the Euler-Euler method is a "dispersed flow model" that handles systems with numerous dispersed bubbles or droplets. It treats them statistically as local volume fractions rather than resolving individual bubbles.

It solves the continuity equation and momentum equation for each phase $k$. The continuity equation is as follows.

Here, $\alpha_k$ is the volume fraction of phase $k$, and $\dot{m}_{lk}$ is the mass transfer rate from phase $l$ to phase $k$. The volume fraction constraint $\sum_k \alpha_k = 1$ holds.

The momentum equation for phase $k$ becomes as follows.

$\mathbf{M}_k$ is the sum of interfacial forces, and this is the core part of the two-fluid model. The pressure $p$ is shared among all phases (shared pressure model), which is the standard approach.

Taking bubble flow as an example, the main forces acting on the dispersed phase (bubbles) are as follows.

For spherical bubbles, Schiller-Naumann is suitable; for deformed bubbles (high Eotvos number), the Ishii-Zuber or Grace model is appropriate. The Ishii-Zuber model automatically switches the drag coefficient according to the bubble regime (spherical, ellipsoidal, cap), making it highly versatile.

Basically, an extended version of the SIMPLE-type algorithm is used. The basic flow is to solve the momentum equations for each phase sequentially and perform pressure correction from the shared pressure.

1. Solve each phase's momentum equation with a tentative velocity field

2. Update the volume fraction equation

3. Solve the pressure correction equation (summing the continuity equations of each phase)

4. Correct the velocity field

5. Update the turbulence equations

6. Repeat until convergence

Yes. However, additional terms may be included in the pressure term for the dispersed phase. For example, in granular flow (Eulerian Granular), a solid pressure $p_s$ is added as a function of volume fraction.

There are three approaches.

In bubble column analysis, additional source terms for Bubble-Induced Turbulence (BIT) are important. The Sato & Sekoguchi model is most commonly used.

Exactly. The wake of bubbles additionally generates turbulent energy. At high void fractions, BIT can become dominant.

The Euler-Euler method has strong nonlinearity and can be difficult to converge.

Steady-state calculations often fail to converge, so it is common to perform transient calculations and take time averages. For systems like bubble columns, run for tens of seconds of physical time before starting to collect statistics.