Multiphase Chapter 2: Mixture Model

Multiphase systems often involve several materials moving together but interacting in subtle ways through drag, buoyancy, and heat exchange. The Mixture Model captures this behavior by treating the flow as a single, averaged medium with embedded differences between its constituents. This chapter introduces the physical foundations of the model, explains the meaning behind its governing equations, and describes how drift, slip, and interfacial area connect microscopic phase interactions with macroscopic flow behavior. The accompanying application section illustrates how the model is implemented in ANSYS Fluent, including built-in options for cavitation, condensation, and wet-steam modeling, and concludes with industrial examples from propeller and turbine design. The result is a complete, intuitive understanding of the Mixture Model’s role as a bridge between single-phase and fully resolved multiphase simulations.

 

The Mixture Model - Physical Interpretation and Context

Multiphase flows often involve several materials coexisting within the same domain - gas within liquid, liquid within gas, or solid particles suspended in a fluid. In many engineering situations, these phases move more or less together, exchanging mass, momentum, and heat continuously. The mixture model represents such systems in an economical way: it treats the entire blend as one continuous medium while still accounting for how the individual components distribute and drift within it.

Rather than solving a separate set of conservation equations for each phase, the model introduces an averaged “mixture” that possesses effective properties derived from the local proportions of its constituents. Around this single framework, additional relationships describe how the dispersed or secondary phases deviate slightly from the average motion. This approach captures the key physics of coupled multiphase motion without the heavy computational cost of a full Eulerian–Eulerian simulation.


Advantages and Limitations

Because the mixture model only solves one momentum and energy field, its numerical cost remains modest. It still resolves multiphase effects such as buoyancy, slip, and interphase transfer, but within a simpler and more stable structure. It is therefore suitable for flows where the phases travel largely in the same direction and remain in partial equilibrium - for example, aerated flows, bubble columns, light particle suspensions, or cavitating liquids.

Its main limitation appears when the phases interact too strongly or behave independently. If the particles vary widely in size or if detailed interfacial physics such as film formation or breakup must be resolved explicitly, the model’s assumptions no longer hold. In those cases, Eulerian or Volume of Fluid formulations are preferred. The mixture approach is thus a middle ground - less detailed than a full two-fluid model, but far more informative than treating the medium as a single-phase continuum.


Continuity and Transport of Phases

Mass conservation in a mixture flow expresses the idea that the combined density of all materials, multiplied by their local averaged velocity, must remain balanced over time. The mixture density is obtained by weighting each phase’s density according to its local volume fraction. In addition, each secondary phase has its own transport equation for its volume fraction. This equation describes how the dispersed component spreads or accumulates, carried mostly by the average motion of the mixture but corrected by its small drift relative to that motion.

If one phase transforms into another - for instance, through evaporation, condensation, or cavitation - extra source terms account for the mass transfer between them. These coupled balances ensure that the total amount of material remains physically consistent across all phases.


Momentum and Forces Within the Mixture

The momentum balance of the mixture describes how the averaged velocity field evolves under pressure gradients, viscous stresses, body forces such as gravity, and the collective effects of interphase drag. The overall viscosity of the mixture is computed as a weighted blend of the component viscosities, giving the equivalent “thickness” or resistance of the combined fluid.

Because the secondary phases may not move exactly at the same speed as the average flow, their relative motions introduce additional stress-like terms. These drift contributions account for the transfer of momentum between phases and for effects such as particle inertia or buoyant slip. In essence, the momentum equation represents both the large-scale motion of the entire blend and the internal readjustments among its constituents.


Energy and Thermal Coupling

Energy conservation within the mixture encompasses both internal energy and kinetic energy of all phases. It includes convective transport of enthalpy by the moving mixture and conductive or turbulent heat diffusion. The effective thermal conductivity combines molecular and turbulent contributions from each phase.

In systems involving phase change, the energy balance automatically captures latent heat effects through source terms. For example, when vapor bubbles collapse within a liquid, the released energy raises the local temperature, while boiling or cavitation withdraws energy from the surrounding fluid. The equation therefore provides a complete description of how heat and motion interact across the mixture.


Relative Velocities: Slip and Drift

A distinctive feature of multiphase flows is the slight difference in motion between the phases. The slip velocity expresses this difference directly as the velocity of the dispersed phase relative to the continuous one. The drift velocity expresses it relative to the average mixture motion. These quantities link microscopic particle dynamics with the macroscopic averaged flow.

The drift is determined primarily by the balance between drag and inertia. A small, light bubble quickly adapts to the surrounding liquid velocity; a heavy particle or large droplet responds more slowly. The characteristic time required for a dispersed particle to adjust - the relaxation time - depends on its diameter, its density, and the viscosity of the carrier phase. The mixture model incorporates these ideas by estimating the relative velocity from simple force-balance relationships involving drag, gravity, and local acceleration of the surrounding flow.

When the flow is turbulent, eddies cause random fluctuations that mix the phases more strongly. To represent this additional spread, the model introduces a turbulent diffusion term based on the local turbulence intensity, ensuring that dispersed materials do not cluster unrealistically in stagnant regions.


Interfacial Area and Its Role

Wherever different phases coexist, there are interfaces - surfaces separating liquid and gas, or fluid and solid. The interfacial area concentration measures how much of this contact area exists per unit of mixture volume. It governs how effectively mass, momentum, and heat are exchanged between the phases. A greater total interfacial area enhances drag, heat transfer, and chemical reactions.

Because bubbles and droplets continuously merge and split, the total interfacial area is not constant. Models therefore include terms for coalescence (which reduces area) and breakup (which increases it). These terms are commonly correlated with turbulent intensity and flow conditions. Advanced formulations, such as the Population Balance or MUSIG models, treat the bubble or droplet size distribution statistically; simpler approaches use single transport equations with empirical source and sink terms. Correct prediction of interfacial area is crucial, since it directly controls how the phases interact in simulations.


Application Chapter - Practical Implementation in ANSYS Fluent

Within Fluent, the mixture model is activated through the multiphase model panel. The user defines which materials belong to which phases, assigns their properties, and specifies whether slip between the phases should be considered. The solver then computes the mixture’s overall velocity and pressure fields while tracking the volume fraction of each secondary phase.

Because the formulation assumes a pressure-based coupling of density and momentum, it is available only with the pressure-based solver. It cannot be used for inviscid flows, solidification, or melting, and only one phase may be treated as an ideal compressible gas. Parallel execution requires message passing rather than shared memory for correct particle tracking compatibility.

Internally, Fluent discretizes the volume-fraction equations implicitly to enhance stability. Additional numerical options help convergence: the implicit body force correction accounts for the balance between pressure gradients and gravity in buoyancy-driven flows, while the interface modeling options allow different treatments for smooth dispersed mixtures or for regions that include both dispersed and sharp interfaces.


Built-in Sub-Models for Phase Change

Cavitation

Cavitation occurs when a liquid’s local pressure drops below its vapor pressure, allowing vapor bubbles to form and collapse violently. These events create noise, vibration, and surface erosion in pumps, propellers, and injectors. Fluent provides several formulations for representing this process within the mixture framework.

  • Singhal et al. (Full Cavitation Model) captures the key phenomena: phase change, bubble dynamics, turbulence-induced pressure fluctuations, and the presence of dissolved gases. It supports multiple phases and can include thermal and compressibility effects. For numerical stability, it is often used without explicit slip between phases.

  • Zwart–Gerber–Belamri model simplifies the physics by assuming all bubbles share the same radius. It introduces coefficients controlling how rapidly vapor is created or collapsed as pressure deviates from saturation. These coefficients are usually adjusted empirically to match experimental data.

  • Schnerr–Sauer model follows a similar concept but uses a different formulation based on bubble number density. It is maintained mainly for compatibility with older studies and is less commonly recommended today.

In all cases, the correct prediction of vapor fraction and collapse intensity depends on fine spatial resolution near pressure minima and careful control of time-stepping.

Cavitation models can also be combined with scale-resolving turbulence approaches such as SAS or DES to capture the unsteady vortex structures where cavitation originates, for instance near propeller tips.

Wet Steam Condensation

During the rapid expansion of steam, the flow path crosses the saturation line, leading to condensation. The mixture then consists of vapor mixed with tiny liquid droplets - the so-called wet steam region. Predicting this process accurately is critical in steam turbine design, where droplet impingement causes blade erosion and efficiency losses.

Fluent implements a non-equilibrium condensation model based on classical homogeneous nucleation theory. It solves two additional transport equations: one for the liquid mass fraction and another for the droplet number density. The model describes how droplets form, grow, or vanish as the vapor expands and cools.

Because this formulation depends on compressible flow behavior, it is available only for density-based solvers, and boundary conditions are restricted to pressure or mass-flow inlets and pressure outlets. Material properties are automatically linked to steam saturation data.

Evaporation and Condensation at Surfaces (Lee Model)

Surface evaporation and condensation are treated through the Lee model, which relates the rate of mass transfer to how much the local temperature deviates from the saturation temperature. A proportionality constant - the inverse of a characteristic relaxation time - determines how quickly the phase change occurs. The model operates within the mixture, VOF, or Eulerian frameworks, depending on how the phases are defined.

The latent heat of vaporization is specified as the difference in standard-state enthalpies of the vapor and liquid phases. The correct tuning of the evaporation and condensation frequencies is essential: overly large values lead to numerical instability, while too small values underpredict the rate of phase change. Comparing simulation results with experimental data is often necessary to calibrate these coefficients.


Guidelines for Stable Simulation

Cavitating and condensing flows exhibit steep gradients and strong coupling between pressure and volume fraction. For numerical stability, the mixture model benefits from small Courant numbers, reduced relaxation factors for pressure and momentum, and careful boundary treatment to prevent unphysical reverse flow. Using the coupled solver further improves convergence by solving pressure and velocity simultaneously.


Industrial Applications

The mixture model has proven reliable in a range of engineering analyses:

  • Marine propeller design: Rolls-Royce employed cavitation simulations in Fluent to refine the Kamewa CP-A propeller, achieving measurable improvements in hydrodynamic efficiency confirmed by experiments.

  • Hydrofoil cavitation: Numerical predictions of vapor pockets along a NACA-type hydrofoil reproduced experimental pressure distributions and cavity shapes with high accuracy.

  • Horseshoe cloud cavitation: Simulations captured the characteristic vapor horseshoe structure that drives erosion in submerged foils, validating the turbulence and cavitation coupling within the model.

These cases illustrate how the mixture model balances accuracy and practicality, providing physically meaningful predictions without the computational burden of fully resolved multiphase models.


Summary

The mixture model portrays a complex physical reality through a unified lens: a single continuum that carries within it traces of multiple materials. It ensures that the essential interactions - drag, buoyancy, slip, heat transfer, and phase change - are all represented, while simplifying the mathematics to a manageable level. The price of that simplicity is the assumption of near-equilibrium behavior: the phases must share much of their motion. When that condition is satisfied, the mixture model serves as an elegant and efficient window into the dynamics of multiphase flows, from the smallest vapor bubbles in a pump to the condensation mist inside a turbine

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