1-D Two-Fluid Model (TFM) Overview

• 1-D TFM is a reduced-order multiphase flow model representing two interpenetrating continua with phase-wise conservation laws and customized interfacial closures.
• Various formulations target stratified, packed-bed, suspension, and compressible flows, each characterized by specific assumptions and drag or rheological closures.
• Variational, asymptotic, and analytic techniques underpin the model’s derivation, while numerical methods ensure phase-fraction bounds and energy conservation.

The 1-D Two-Fluid Model (TFM) denotes a class of reduced-order multiphase flow formulations in which two phases are represented as interpenetrating continua in one spatial dimension, with phasic volume fractions, densities, velocities, and closure laws for interfacial exchange. Across the recent literature, the same label covers stratified incompressible flow in channels and pipes, gas–liquid flow through packed-bed reactors, dense suspensions, plasma models, and compressible free-surface flow. The defining feature is therefore not a single universal equation set, but a one-dimensional continuum description in which phase-wise conservation laws, shared constraints, and interfacial closures are specialized to the application of interest (Buist et al., 2021, Nagrani et al., 2024, 0802.3013).

1. Scope and model families

Recent work uses “1-D TFM” for several related but non-identical formulations. Some models evolve separate mass and momentum equations for each phase with a shared pressure field; some impose equal velocity, pressure, and temperature; some reformulate the state in terms of superficial velocity and drift flux; and, in laser–plasma usage, the term can even denote a relativistic electron fluid coupled to a static ion background and Maxwell’s equations. This suggests that the term is best read as a family name for one-dimensional multiphase continuum models rather than as a unique canonical system (Clausse et al., 2021, Verma et al., 2017, Lukáčová-Medvid'ová et al., 2023).

Variant	Distinguishing assumption	Representative context
Stratified incompressible TFM	Cross-sectionally averaged hold-ups, velocities, and pressure	Channels and pipes
Packed-bed 1-D TFM	Steady, fully developed, cross-sectionally uniform, shared pressure gradients	Gas–liquid flow through packed beds
Suspension TFM	Fluid and particle phases as interpenetrating continua with their own mass and momentum equations	Shear-dominated non-Brownian suspensions
Homogeneous compressible TFM	Same velocity, pressure and temperature for both phases	Free-surface compressible air–water flow
Drift-flux/Hamiltonian form	State represented by superficial velocity and drift-flux	Vertical air–water flow, slug formation
Single-temperature two-fluid model	Separate phase masses and velocities, one temperature and entropy	All-Mach compressible two-fluid flow

The application range is correspondingly broad. The model is used for bubble and pulse flow regimes in microgravity packed beds, particle migration in dense suspensions, Kelvin–Helmholtz-dominated stratified flow in pipes, and all-Mach compressible two-fluid dynamics. In each case, the one-dimensional reduction is paired with closures that carry most of the physical specificity (Nagrani et al., 2024, Municchi et al., 2018, López-de-Bertodano et al., 4 Sep 2025).

2. Conservation structure and closure architecture

A common starting point is phase-wise mass conservation,

$$\partial_t(\alpha_k \rho_k) + \partial_x(\alpha_k \rho_k u_k) = 0,\qquad k=1,2,$$

with $\alpha_1+\alpha_2=1$ in two-phase settings. Momentum and energy equations vary more strongly across formulations. In the all-topology model derived through Hamilton’s Stationary Action Principle, the one-dimensional system includes volume fraction transport at the mixture velocity,

$$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$

together with phasic momentum and energy balances containing non-conservative interfacial terms. The variational closure yields

$$u_I = Y_1u_1 + Y_2u_2,\qquad p_I = Y_2p_1 + Y_1p_2,\qquad (pu)_I = Y_1 p_2 u_1 + Y_2 p_1 u_2,$$

where $(pu)_I$ is introduced as an interfacial work term (Haegeman et al., 30 Sep 2025).

In other branches of the literature, closure enters through rheology, drag, or heat transfer rather than through a variational principle. For shear-dominated suspensions, the particle-phase stress includes an anisotropic migration contribution,

$$\Sigma_s = -\mu_f \eta_N(\phi)\dot\gamma_{\mathrm{eff}} Q,$$

and accurate prediction of particle migration hinges on the closure of this anisotropic stress rather than on an ad hoc diffusive flux term (Municchi et al., 2018). In non-isothermal dense suspensions, separate energy equations for particles and fluid are closed by an inter-phase heat transfer coefficient,

$$K_h(\phi,\dot{\gamma}) = K_{h,0}\left[1 + \beta\phi\left(\|\dot{\mathbf{S}}_p\| d_p^2/\alpha_p\right)^m\right],$$

which is calibrated against experiment and coupled to a thermo-rheological migration force (Nagrani et al., 2021).

A recurring structural choice is whether pressure is shared. In the compressible generic two-fluid model, the single pressure $p$ is uniquely determined by the products of volume fractions and densities $\phi_k\rho_k$, and the numerical method is built around that relation (Wu, 2023). In the homogeneous compressible free-surface model, both fluids share the same velocity, pressure and temperature, so the interface is represented by a diffuse transition in the volume fractions rather than by an explicitly tracked surface (0802.3013).

3. Variational, asymptotic, and analytic formulations

One major theoretical line derives 1-D TFM equations from variational principles. A Hamilton-principle formulation replaces the average phase velocities by the superficial velocity

$j=\alpha_1 u_1+\alpha_2 u_2$

and the drift flux

$\alpha_1+\alpha_2=1$0

This generates conservation equations for the global center-of-mass flow and the relative velocity between fluids. Well-posed equations can then be obtained by modelling the storage of kinetic energy in fluctuation structures induced by the interaction between fluids, like wakes and vortexes, so that regularization does not suppress the instabilities responsible for flow-pattern formation and transition. A specific vertical air–water case predicts the formation of the slug flow regime as trains of non-linear waves (Clausse et al., 2021).

A related but distinct development is the all-topology compressible two-fluid model derived through Hamilton’s Stationary Action Principle. It is fully closed, hyperbolic, symmetrizable, admits an entropy conservation law, and provides uniquely defined jump conditions for its non-conservative products. In one dimension, the resulting framework supplies a mathematically explicit treatment of weak solutions while keeping the interfacial closures inside the variational structure (Haegeman et al., 30 Sep 2025).

Another theoretical route is asymptotic reduction. For an isentropic fluid–fluid interaction model with a large symmetric drag force, the simplified two-fluid flow model with a single velocity field is justified as the asymptotic limit as the drag parameter $\alpha_1+\alpha_2=1$1. The same analysis yields an additional relation showing that the density of one fluid species can be resolved from the density and velocity of the other species, termed the testing flow (Liu, 2022).

Exact analytic benchmarks also exist. A one-dimensional coupled Euler–Navier–Stokes two-fluid hydrodynamic model with a common pressure field and adjustable void fraction admits self-similar solutions. Under the self-similar Ansatz, both velocities decay as $\alpha_1+\alpha_2=1$2, pressure decays as $\alpha_1+\alpha_2=1$3, and the explicit solutions involve the error function. These solutions are useful as test cases for numerical solvers and for qualitative analysis of viscosity, density, and void-fraction dependence (Barna et al., 2020).

4. Stratified incompressible flow in channels and pipes

For incompressible, isothermal stratified flow in channels and pipes, the 1-D TFM uses cross-sectionally averaged hold-ups, velocities, and pressure. A central result is that the model satisfies a mechanical energy conservation equation implied by the mass and momentum equations, even in the presence of non-conservative pressure terms and for ducts with an arbitrary cross-sectional shape. The mechanical energy density is

$\alpha_1+\alpha_2=1$4

and the local balance reads

$\alpha_1+\alpha_2=1$5

with pressure work entering through the flux term $\alpha_1+\alpha_2=1$6 (Buist et al., 2021).

This energy structure has been extended to include axial diffusion, friction, and surface tension. In that framework, surface tension can be added in an energy-conserving manner, while diffusion and friction have a strictly dissipative effect on the energy. A semi-discrete spatial discretization can be constructed so that the discrete model mirrors the continuous conservation and dissipation properties, and a flux-limited advective scheme can be energy-conserving in smooth regions while strictly dissipative at sharp gradients. The resulting model is linearly stable to short wavelength perturbations, exhibits nonlinear damping near shocks, and yields smoothly converging numerical solutions even under conditions for which the basic two-fluid model is ill-posed (Buist et al., 2023).

The ill-posedness of the basic model is closely tied to Kelvin–Helmholtz growth. In a recent nonlinear-stability study, runaway Kelvin–Helmholtz instabilities are controlled by a simple turbulent viscosity closure,

$\alpha_1+\alpha_2=1$7

compatible with inertial coupling. The resulting one-dimensional dynamics includes turbulent cascades, chaos, and the formation of churn or slug flow, so stabilization is supplied by a physical dissipation mechanism rather than by a purely aphysical regularization (López-de-Bertodano et al., 4 Sep 2025).

5. Packed-bed reactors and dense suspensions

A highly specific 1-D TFM has been developed for gas–liquid flow through packed-bed reactors under microgravity conditions. Using NASA’s Packed Bed Reactor Experiments conducted on parabolic flights and aboard the International Space Station, interphase drag correlations were obtained for glass (wetting) and Teflon (nonwetting) spheres with $\alpha_1+\alpha_2=1$8 mm and porosity $\alpha_1+\alpha_2=1$9. With an Ergun-type closure for liquid–solid drag, the gas–liquid interphase drag $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$0 becomes the only unknown in the 1-D TFM. A data-driven procedure correlates $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$1 with the liquid and gas Reynolds numbers and the Suratman number, and two-dimensional transient ANSYS Fluent simulations using an Euler–Euler formulation show good agreement with the experimental pressure drops. The reported model errors are typically within 25% of experimental values, with errors up to 40% at the lowest flow rates (Nagrani et al., 2024).

In dense non-Brownian suspensions, the TFM closes the full particle–fluid system rather than reducing it to a suspension balance model. The particle phase is given its own momentum equation and an anisotropic stress analogous to that used in the suspension balance model. This allows the TFM to predict shear-induced particle migration without assuming a steady suspension velocity or a Stokesian fluid, and it can be extended to include buoyancy and kinetic collisional models. Benchmark simulations in OpenFOAM, including curvilinear coordinates and three-dimensional flow, show good agreement with previous experimental and numerical results (Municchi et al., 2018).

A thermal extension adds separate energy equations and calibrated inter-phase heat transfer. In annular Couette flow, both the shear and thermal gradients are responsible for particle migration, and the model identifies a thermo-rheological migration force that is proportional to local shear rate and temperature gradient through the temperature dependence of viscosity. Calibration against experiment gives the best fit $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$2 and $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$3 for the inter-phase heat transfer closure. For $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$4, migration is reduced and can be suppressed at high thermal Peclet number; for $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$5, migration is enhanced. In eccentric Couette cells, the system Nusselt number for $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$6 peaks at $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$7 (Nagrani et al., 2021).

6. Plasma and compressible formulations

In plasma applications, “1-D TFM” can denote a relativistic one-dimensional electron fluid coupled self-consistently to Maxwell’s equations, with ions treated as a static, neutralizing background. Simulations initialized from exact stationary or propagating soliton solutions show that time-dependent, localized, non-propagating structures can form either from soliton collisions or from perturbations of stationary states. These structures survive for several hundreds of plasma periods, and after wave breaking compact coherent remnants with trapped radiation persist in both fluid and PIC simulations (Verma et al., 2017).

A different plasma TFM is derived from plasma kinetic equations by moment model reduction with globally hyperbolic regularization, followed by Maxwellian iteration. The resulting model is formally the same as the five-moment two-fluid model except for the closure relations: the pressure tensor is anisotropic and the heat flux is present. By using the Shakhov collision operator, the model inherits the correct Prandtl number, which supplies the capacity to depict problems with anisotropic pressure tensor and large heat flux (Li et al., 2020).

Compressible two-fluid formulations also appear in free-surface and all-Mach settings. One homogeneous model for compressible air–water flow uses the same velocity, pressure and temperature for both phases, so the free surface becomes a thin three dimensional zone. The method can naturally handle wave breaking and other topological changes, and the governing system is unconditionally hyperbolic for reasonable equations of state (0802.3013). A separate single-temperature model derived in the Symmetric Hyperbolic Thermodynamically Compatible framework retains separate phase masses and velocities but only one temperature and entropy. Its implicit–explicit finite volume scheme is stable for large time steps controlled by the interface transport and is asymptotic preserving for weakly compressible Euler equations with variable volume fraction (Lukáčová-Medvid'ová et al., 2023).

7. Constraints, numerical methods, and recurring issues

Constraint enforcement is a persistent numerical issue because phase fractions are bounded quantities. A phase-bounded finite element method addresses this by formulating the phase-fraction equation as a nonlinear variational inequality with box constraints and solving it with PETSc’s SNES. In this approach, boundedness is coupled directly to the weak form, no artificial diffusion nor ad hoc remapping or limiting is introduced, and violations are reduced to solver tolerances of approximately $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$8 (Treeratanaphitak et al., 2021).

Another line of work removes constraints analytically rather than enforcing them iteratively. The pressure-free two-fluid model simultaneously eliminates the volume constraint and the pressure from the one-dimensional incompressible TFM, producing four evolution equations without additional constraints. The formulation keeps the conservation properties of the original TFM, preserves the correct shock relations, satisfies the volume and volumetric flow constraints exactly, and can be advanced with explicit Runge–Kutta methods on a staggered grid. Benchmark cases report computational cost reductions of approximately 40% (Sanderse et al., 2020).

For compressible equal-pressure models, projection methods remain viable. A recent projection method for a generic compressible two-fluid model uses the fact that the single pressure can be uniquely determined from the partial masses $$\partial_t \alpha_1 + u\,\partial_x \alpha_1 = 0,\qquad u=Y_1u_1+Y_2u_2,$$9, introduces carefully chosen intermediate variables and a stabilizing term, and proves an energy-stable fully discrete scheme (Wu, 2023). Across these developments, the recurring themes are the same: preserving phase-fraction bounds, respecting conservation and shock relations, and regularizing short-wavelength instability without erasing the physically relevant interfacial dynamics. This suggests that, in contemporary 1-D TFM research, the decisive distinctions lie less in one-dimensionality itself than in how closures, constraints, and stability mechanisms are encoded.