Eulerian Multiphase Models

Eulerian multiphase models are used to model interpenetrating fields. The concept of volume fraction is used to model different phases. Nonetheless, the velocities of the phases can differ. Generally, in this type of multiphase flows, there are dispersed phases such as bubbles, droplets, or particles within a carrier field. There are three main approaches for modeling:

Euler-Euler: In this approach, separate but coupled momentum equations, along with individual volume fractions for each phase, are solved to compute the velocities and volume fractions of the phases. This is the most accurate but also the most computationally expensive approach.
Algebraic-Euler (Mixture): In mixture theory, a single mixture momentum equation, together with volume fraction equations for the phase, except one, is solved. The relative velocity of phases is calculated using the Algebraic Slip Model (ASM).
Homogeneous: The homogeneous model is the simplest model and in practice is similar with mixture model, but it assumes that all phases move with the same velocity. Technically, ASM is ignored.

Phasic Equations

As each phase in Eulerian multiphase models moves with different velocities, individual momentum equations govern their motion. The phasic continuity (volume fraction) and momentum equations for each phase k can be written as:

Phasic continuity equation

\frac{\partial}{\partial t} (α_{k} ρ_{k}) + \nabla \cdot (α_{k} ρ_{k} u_{k}) = 0

where $α_{k}$ is the volume fraction of phase $k$ , $ρ_{k}$ is the density of phase $k$ , and $u_{k}$ is the velocity of phase $k$ .

Momentum equation for each phase

\frac{\partial}{\partial t} (α_{k} ρ_{k} u_{k}) + \nabla \cdot (α_{k} ρ_{k} u_{k} u_{k}) = - α_{k} \nabla p + \nabla \cdot α_{k} (τ_{k}) + α_{k} ρ_{k} g + M_{k}

where $p$ is the pressure, $τ_{k}$ is the stress tensor of phase $k$ , $g$ is the gravitational acceleration, and $M_{k}$ represents the momentum exchange between phases.

The last term in the momentum equation represents the momentum exchange between phases. This momentum exchange occurs due to interfacial forces between phases. These forces include drag force, lift force, turbulent dispersion force, wall lubrication force, virtual mass force, surface tension and solid collision force.

In Eulerian multiphase models, individual continuity and momentum equations are solved for the phasic mass (volume) and velocity, while the pressure field is assumed to be shared among all phases. To calculate the shared pressure field, the mean continuity and momentum equations are derived from the phasic equations and solved for the mean velocity of the phases and the shared pressure. The mean continuity and momentum equations are obtained by summing the individual phasic equations as follows:

\frac{\partial}{\partial t} \sum_{k = 1}^{n} (α_{k} ρ_{k}) + \nabla \cdot \sum_{k = 1}^{n} (α_{k} ρ_{k} u_{k}) = 0

\frac{\partial}{\partial t} \sum_{k = 1}^{n} (α_{k} ρ_{k} u_{k}) + \nabla \cdot \sum_{k = 1}^{n} (α_{k} ρ_{k} u_{k} u_{k}) = - \sum_{k = 1}^{n} (α_{k} \nabla p) + \nabla \cdot \sum_{k = 1}^{n} (α_{k}) τ_{k} + \sum_{k = 1}^{n} (α_{k} ρ_{k} g)

The mixture density and mixture velocity of the multiphase system are defined as:

ρ_{m} = \sum_{k = 1}^{n} α_{k} ρ_{k}

u_{m} = \frac{1}{ρ_{m}} \sum_{k = 1}^{n} α_{k} ρ_{k} u_{k}

Note: The phasic velocity differs from the mixture velocity, and the relative velocity between them is referred to as the drift velocity, which is defined as the velocity difference between the velocity of the disperse phase, for example, bubbles, droplets, or particles, and the velocity of the mixture.

The drift velocity accounts for the relative motion between phases, which arises due to differences in density, buoyancy, or interaction forces such as drag.

u_{k} - u_{m} = u_{k}^{d r i f t}

Using the mixture density and velocity, the mixture continuity and momentum equations can be derived through straightforward mathematical manipulations as:

\frac{\partial ρ_{m}}{\partial t} + \nabla \cdot (ρ_{m} u_{m}) = 0

\frac{\partial}{\partial t} (ρ_{m} u_{m}) + \nabla \cdot (ρ_{m} u_{m} u_{m}) = - \nabla p - \nabla \cdot \sum_{k = 1}^{n} (α_{k} ρ_{k} u_{k}^{drift} u_{k}^{drift}) + \nabla \cdot τ_{k} + ρ_{m} g

These two equations are solved in AcuSolve using flow stagger to obtain the mixture velocity and pressure fields.

Eulerian-Eulerian

If the preferred multiphase approach is Eulerian-Eulerian, the mixture continuity and momentum equations are solved in the flow stagger for the mean velocity and pressure. For each dispersed field (k=1,…n-1), individual phasic continuity and momentum equations are solved to obtain the phasic values of volume fraction and velocity.

Algebraic-Eulerian

Similar to Eulerian-Eulerian, the mixture continuity and momentum equations are solved in the flow stagger for the mean velocity and pressure. For each dispersed field (k=1,…n-1), one phasic continuity equation is solved for the phasic values of volume fraction. However, instead of solving for phasic momentum equation, an algebraic equation is solved for the phasic velocity. Algebraic equation is derived following algebra slip model.

Algebraic slip model

The algebraic slip model is used to avoid solving the full phasic momentum equations. To achieve this, the phasic and mean momentum equations are written in non-conservative form (using the mean phasic continuity equations):

α_{k} ρ_{k} \frac{\partial}{\partial t} (u_{k}) + α_{k} ρ_{k} u_{k} \nabla (u_{k}) = - α_{k} \nabla p + \nabla \cdot α_{k} (τ_{k}) + α_{k} ρ_{k} g + M_{k}

\frac{\partial}{\partial t} (ρ_{m} u_{m}) + ρ_{m} u_{m} \nabla (u_{m}) = - \nabla p + \nabla \cdot τ_{m} + ρ_{m} g

These equations can be subtracted to eliminate pressure gradient terms. In addition, the following assumptions are made:

It is assumed that the disperse phase reaches terminal velocity instantaneously, allowing for the transient terms in the phasic equation to be ignored.
Viscous and drift stresses are ignored.
Phasic advection is assumed to be equal to the mean acceleration, as follows:
$u_{k} \nabla \cdot (u_{k}) = u_{m} \nabla \cdot (u_{m})$

Applying these assumptions to the combined equations results in the following:

M_{k} = α_{k} (ρ_{k} - ρ_{m}) (\frac{\partial u_{m}}{\partial t} + u_{m} \nabla \cdot (u_{m}) - g)

As a final assumption, the momentum exchange between phases is considered to be primarily due to drag force, and optionally lift, as follows:

M_{k} = \frac{1}{2} ρ_{c} C_{D} A_{k} |(u_{k} - u_{c})| (u_{k} - u_{c})

where $u_{k}^{s l i p} = u_{k} - u_{c}$ . Therefore, the final form of the algebraic slip model becomes:

|u_{k}^{s l i p}| u_{k}^{s l i p} = \frac{4}{3} \frac{(ρ_{k} - ρ_{m})}{ρ_{k}} (\frac{\partial u_{m}}{\partial t} + u_{m} \nabla \cdot (u_{m}) - g)

The slip velocity and mean velocity are used to obtain the phasic velocity as follows:

u_{k}^{d r i f t} = u_{k}^{s l i p} - \frac{1}{ρ_{m}} \sum_{k = 1}^{n} α_{k} ρ_{k} u_{k}^{s l i p}

u_{k} = u_{m} + u_{k}^{d r i f t}