Multiphase flows involve the simultaneous flow of materials with different phases (solid, liquid, or gas) or different chemical properties in a system. In the realm of CFD, these are modeled to understand and predict the interactions between the phases, which are critical in various engineering and scientific processes.
The mathematical modeling of multiphase flows typically involves solving the Navier-Stokes equations for incompressible flow, coupled with additional equations governing the phase interaction and transfer. The key equations include:
$$ \nabla \cdot \mathbf{u} = 0 $$
where $\mathbf{u}$ represents the velocity field.
$$ \rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \nabla \cdot \mu (\nabla \mathbf{u} + (\nabla \mathbf{u})^T) + \mathbf{f} $$
Here, $\rho$ is the density, $p$ is the pressure, $\mu$ is the dynamic viscosity, and $\mathbf{f}$ represents body forces (e.g., gravity).
When dealing with multiphase flows in CFD, additional equations are needed beyond the standard continuity and momentum equations to capture the interactions and transfers between the different phases. These interactions can involve mass, momentum, and energy exchanges, and their accurate representation is crucial for predicting the behavior of multiphase systems. Here's a breakdown of the mathematical formulations often used:
In multiphase systems, mass can be transferred from one phase to another due to processes like evaporation, condensation, or chemical reactions. The equation governing mass transfer between phases $i$ and $j$ can be represented as:
$$ \frac{\partial \rho_i}{\partial t} + \nabla \cdot (\rho_i \mathbf{u}_i) = \dot{m}_{ji} - \dot{m}_{ij}, $$
where:
Momentum transfer in multiphase flows is critical, especially when phases interact dynamically, such as in particle-laden flows or bubbly flows. The momentum equation for each phase might include additional source terms to account for the momentum exchange:
$$ \rho_i \left(\frac{\partial \mathbf{u}_i}{\partial t} + \mathbf{u}_i \cdot \nabla \mathbf{u}_i\right) = -\nabla p_i + \nabla \cdot \mu_i (\nabla \mathbf{u}_i + (\nabla \mathbf{u}_i)^T) + \mathbf{f}_i + \sum_{j \neq i} \mathbf{F}_{ij} $$
where:
Energy transfer becomes significant in multiphase flows when different temperatures or phase changes are involved. The energy equation for each phase might look like:
$$ \rho_i c_{p_i} \left(\frac{\partial T_i}{\partial t} + \mathbf{u}_i \cdot \nabla T_i\right) = \nabla \cdot (k_i \nabla T_i) + Q_i + \sum_{j \neq i} Q_{ij} $$
where:
These additional equations, tailored for specific phase interactions, are crucial for simulating real-world multiphase flow scenarios accurately. They ensure that the behavior of each phase is influenced by and influences the others in a manner that matches physical reality.
There are several types of multiphase flows, commonly categorized based on the phase distribution and interaction dynamics:
In multiphase flow modeling, the terms dispersed-continuous and continuous-continuous phase interactions refer to the ways in which different phases are distributed and interact within a flow field. Understanding these interactions is crucial for selecting the appropriate modeling approaches and for accurately capturing the dynamics of the system.
In dispersed-continuous phase interactions, one phase is dispersed in another continuous phase as small particles, droplets, or bubbles. This type of interaction is characterized by a distinct phase interface with a significant difference in phase volume fractions; typically, the dispersed phase is much less voluminous compared to the continuous phase. Examples include:
Modeling these systems often requires methods that can accurately describe the interactions at the interfaces between the dispersed and continuous phases. Techniques include:
Continuous-continuous phase interactions occur when two or more continuous phases interact with each other, typically without one phase being distinctly dispersed in the other. This can occur in systems where phases have comparable volume fractions or when the interface between them is extensive but not discretely particulate. Examples include:
Modeling these types of interactions often involves:
Both types of interactions are integral to various industrial and natural processes and require different modeling techniques to simulate effectively. Choosing the right approach depends on factors like the scale of the interaction, the properties of the phases, and the desired fidelity of the simulation.