EulerEulermodellen

Euler–Euler models, also called Euler–Euler multiphase flow models, are a class of computational fluid dynamics approaches used to simulate flows containing more than one phase, such as gas–liquid or liquid–liquid systems. In these models each phase is treated as an interpenetrating continuum described in an Eulerian frame of reference, in contrast to particle-based or Lagrangian descriptions of discrete droplets or bubbles.

For each phase k, the model assigns a volume fraction α_k, a density ρ_k, and a velocity u_k. The volume fractions satisfy sum_k α_k = 1. Depending on the formulation, the model may use a one-velocity (single-mixture velocity) approach or a two-fluid approach with separate velocities. The transport equations typically include mass and momentum equations for each phase: Mass conservation for phase k: ∂(α_k ρ_k)/∂t + ∇·(α_k ρ_k u_k) = Γ_k, where Γ_k accounts for phase change. Momentum conservation for phase k: ∂(α_k ρ_k u_k)/∂t + ∇·(α_k ρ_k u_k u_k) = -α_k ∇p + ∇·(α_k τ_k) + α_k ρ_k g + M_k, where M_k represents interphase momentum exchange (drag, lift, virtual mass). Interfacial terms and the mixture pressure couple the phases; turbulence and, if appropriate, surface-tension effects may be included via additional models.

Closure and modelling: The equations require closures for α_k evolution (in some formulations), material properties, viscosities, and interphase coupling coefficients. Drag laws, virtual mass terms, and mass transfer Γ_k are typically calibrated from experiments. Turbulence is usually treated with RANS or LES closures, applied to mixture or phase-specific equations.

Numerical aspects and applications: Euler–Euler models are solved on fixed grids using finite-volume or finite-element discretizations. They are well suited to dense bubbly or droplet-laden flows and to systems where interfaces are numerous but not tracked explicitly. Limitations include uncertainties in interphase closures, numerical diffusion of interfaces, and high computational cost for many phases.