multiphase-cfd

name	multiphase-cfd
description	Multiphase CFD — VOF (Volume of Fluid), level set, Euler-Euler, Euler-Lagrange, bubble columns, cavitation (Schnerr-Sauer, Zwart model), droplet breakup (TAB, KH-RT), fluidized beds, drift-flux model, ANSYS Fluent/CFX setup, phase interaction forces (drag, lift, virtual mass), interface resolution requirements.
metadata	{"priority":7,"promptSignals":{"phrases":["multiphase CFD","VOF simulation","two-phase flow CFD","bubble flow simulation","cavitation CFD","volume of fluid"],"minScore":3}}

Multiphase CFD — Complete Skill

Multiphase Flow Regimes and Model Selection

Flow Regime Classification

Based on dispersed phase fraction: Dilute (α_d < 1%): few isolated bubbles/droplets; Euler-Lagrange appropriate Moderate (1–10%): bubble/droplet interactions begin; Euler-Euler or Lagrange with two-way coupling Dense (10–40%): significant phase coupling; Euler-Euler with closure models Packed (> 40%): particle-particle contact dominant; DEM (Discrete Element Method) required

Based on interface character: Sharp interface (distinct boundary): jet breakup, dam break, wave, sloshing → VOF or level set Dispersed phase (bubbles, droplets, particles): Euler-Euler or Euler-Lagrange

VOF (Volume of Fluid) Method

Mathematical Formulation

Phase fraction transport: ∂α_q/∂t + ∇·(u × α_q) = S_αq [α_q = volume fraction of phase q; Σ α_q = 1; S = source term]

Continuity (mixture): ∇·u = 0 [incompressible; or with mass transfer: modified]

Momentum (shared by all phases): ρ(∂u/∂t + u·∇u) = -∇p + ∇·(μ(∇u + ∇uᵀ)) + ρg + F_surface [F_surface = surface tension force]

Mixture properties: ρ = Σ α_q ρ_q; μ = Σ α_q μ_q [volume-weighted average]

Interface Reconstruction

Geometric VOF (PLIC — Piecewise Linear Interface Construction): Reconstruct interface as piecewise linear plane in each cell; high accuracy; second-order

Algebraic VOF (compressive schemes): High Resolution Interface Capturing (HRIC): ANSYS Fluent; bounded; compressive near interface OpenFOAM: interFoam solver uses MULES (Multidimensional Universal Limiter for Explicit Solution)

Surface tension — Continuum Surface Force (CSF, Brackbill et al.): F_st = σ × κ × ∇α [σ = surface tension; κ = curvature = ∇·(∇α/|∇α|)] Parasitic currents: numerical artifact near interface at rest; reduce with smoothed curvature algorithms

Grid Requirements for VOF

Interface resolution: Minimum 5–10 cells across interface thickness (diffuse zone ≈ 1–2 cells in good VOF) Courant number CFL ≤ 0.25 for explicit VOF schemes (stability; interface tracking accuracy) Adaptive mesh refinement (AMR): refine near α = 0.5 contour → reduces cells needed globally

Time step: Δt_surface_tension = √(ρ_min × Δx³ / (2π × σ)) [surface tension time step limit; more restrictive than CFL for small droplets] Both must be satisfied: Δt = min(CFL limit, surface tension limit)

Euler-Euler (Two-Fluid Model)

Governing Equations per Phase

Each phase has own continuity and momentum: ∂(α_q ρ_q)/∂t + ∇·(α_q ρ_q u_q) = Σ ṁ_pq [ṁ_pq = mass transfer from phase p to q] ∂(α_q ρ_q u_q)/∂t + ∇·(α_q ρ_q u_q u_q) = -α_q ∇p + ∇·α_q τ_q + α_q ρ_q g + Σ R_pq + ṁ_pq u_pq

R_pq (interphase momentum exchange): R_pq = -R_qp; includes drag, lift, virtual mass, turbulence dispersion forces

Interfacial Forces

Drag force: F_D = (3/4) × C_D / d_b × α_d × ρ_c × |u_r| × u_r [u_r = u_d - u_c; relative velocity; d_b = bubble/droplet diameter] C_D from drag law: Schiller-Naumann (Re ≤ 1,000); Morsi-Alexander (general) For spherical bubbles: C_D = 24/Re (Stokes); C_D = 24/Re × (1 + 0.15 Re^0.687) (Schiller-Naumann)

Lift force: F_L = C_L × α_d × ρ_c × (u_r × ∇×u_c) [C_L ≈ 0.5 for small bubbles; Tomiyama: C_L depends on Eo number] Direction: perpendicular to relative motion; significant in shear flows (pushes bubbles toward walls or away)

Virtual mass (added mass): F_VM = C_VM × α_d × ρ_c × (Du_c/Dt - Du_d/Dt) [C_VM = 0.5 for spherical bubbles] Important for unsteady bubbly flows; accelerating bubbles

Turbulent dispersion: F_TD = -C_TD × ρ_c × k_c × ∇α_d [C_TD = 1 (Burns model); C_TD = proportional to turbulent diffusivity] Prevents unphysical bubble accumulation; models turbulent mixing

Bubble Column Example

Setup: Domain: 0.5 m × 2 m × 0.5 m; air sparged at bottom; water continuous Boundary: inlet air (u_air, α_air = 1); outlet pressure at top; walls: no-slip Turbulence: k-ε with bubble-induced turbulence (BIT) correction: ε_BIT = 0.6 × (α_d × ρ_c × u_r² × drag) / (ρ_c × k_c) [adds bubble-induced turbulent dissipation]

Key results: gas holdup α_d distribution; circulation patterns; kLa (mass transfer coefficient)

Euler-Lagrange (Dispersed Flow — Particle Tracking)

Lagrangian Particle Equations

Particle (bubble/droplet) equation of motion: m_p × du_p/dt = F_D + F_B + F_VM + F_L + F_p + F_contact [each force on particle p] F_D = (1/2) × C_D × A_p × ρ_c × |u_r| × u_r [drag; A_p = frontal area] F_B = V_p × (ρ_p - ρ_c) × g [buoyancy]

Stokes number: St = τ_p × u_c / L [τ_p = ρ_p d_p² / (18 μ_c); L = characteristic length] St << 1: particles follow flow; St >> 1: particles fly ballistically St ≈ 1: most complex; significant coupling between phases

One-Way vs. Two-Way Coupling

One-way: fluid affects particles; particle back-reaction on fluid neglected; valid for very dilute flow (α_d < 0.1%) Two-way: particle forces feed back to fluid momentum equation; valid for dilute-to-moderate (α_d = 0.1–5%) Four-way: add particle-particle collisions (DEM); for dense flows

Droplet Breakup Models

TAB (Taylor Analogy Breakup): Droplet oscillation → breakup analogy with spring-mass-damper Breakup when: |y| > 1 (y = normalized droplet deformation) y̋ = W_e × ρ_c × u_r² × R / (C_k × σ) - C_d × μ_d × ẏ / (ρ_d × R²) - C_k × y / ... Breakup products: 2 droplets with radius distribution; velocity from oscillation energy

KH-RT (Kelvin-Helmholtz + Rayleigh-Taylor): KH instability → surface wave growth → child droplets shed from parent RT instability → sudden breakup when parent decelerates Used for: diesel injection atomization; two breakup regimes at different Weber numbers

KHRT applicability: We = ρ_c × u_r² × R / σ [Weber number; We > 12: KH breakup; higher → RT breakup]

Cavitation Models

Physics

Cavitation: formation of vapor bubbles in liquid when local pressure drops below vapor pressure P_vap Critical: P_local < P_vap(T) → vapor nucleation → bubble growth → collapse (violent → erosion) Cavitation number: σ_cav = (P_∞ - P_vap) / (0.5 × ρ × U²) [low σ → cavitation likely]

Schnerr-Sauer Model (most common in ANSYS Fluent)

Transport equation for vapor fraction: ∂(α_v ρ_v)/∂t + ∇·(α_v ρ_v u) = ṁ_e + ṁ_c [ṁ_e = evaporation; ṁ_c = condensation]

Source terms: ṁ_e = C_e × (2/3 × ρ_l ρ_v / ρ) × α_nuc × (1-α_v) × √(2/(3ρ_l) × max(P_vap - P, 0)) [evaporation; P < P_vap] ṁ_c = C_c × (2/3 × ρ_l ρ_v / ρ) × α_v × √(2/(3ρ_l) × max(P - P_vap, 0)) [condensation; P > P_vap] C_e = 50 (evaporation); C_c = 0.01 (condensation); α_nuc = 5×10⁻⁴ (nucleation site fraction)

Zwart-Gerber-Belamri model: Similar structure; slightly different coefficients; available in both Fluent and OpenFOAM

Grid requirements for cavitation: Very fine mesh in low-pressure zones (propeller suction, orifice throat, bearing clearances) y⁺ < 5 at solid surfaces for accurate pressure prediction near cavitation inception Time step: CFL ≤ 0.1 in cavitating zones (fast dynamics)

Erosion Prediction

Cavitation erosion correlated to bubble collapse pressure: P_collapse ≈ ρ_l × (P_vap - P_min) → local maximum collapse pressure → erosion index Erosion index: I_e = Σ (collapses × P_collapse²) [integrated over surface; high I_e → erosion-prone]

Fluidized Bed (Dense Granular Flow)

Two approaches: Euler-Euler (TFM — Two-Fluid Model): both gas and solid treated as interpenetrating continua DEM + CFD: explicit particle simulation; more accurate but extremely expensive (N < 10⁶ particles feasible)

Gidaspow drag model (gas-solid): For α_s < 0.2: Wen-Yu drag (dilute) For α_s ≥ 0.2: Ergun equation (dense packed bed) Wen-Yu: f_D = (3/4) × C_D × α_g^(-2.65) × α_s × ρ_g × |u_g - u_s| / d_p Transition between models at α_s = 0.2

OpenFOAM and ANSYS Setup

OpenFOAM Solvers

Solver	Application
interFoam	VOF incompressible two-phase (free surface)
twoPhaseEulerFoam	Euler-Euler two-phase (bubbly flow)
reactingParcelFoam	Euler-Lagrange + reacting sprays
cavitatingFoam	Cavitation (VOF-based)
MPPICFoam	Dense particle flow (MP-PIC = CPFD)

ANSYS Fluent Setup for VOF

Multiphase → Volume of Fluid Phases: primary (water); secondary (air) Interface modeling: sharp interface scheme (geometrical VOF) Surface tension: CSF; σ = 0.072 N/m (water-air at 25°C) Transient: Explicit; Δt from CFL and surface tension limits Body force formulation: enable for buoyancy-driven flows

Post-processing: Interface visualization: iso-surface of α = 0.5 Velocity field in both phases Pressure field: near-interface pressure jump = σ × κ (validate with Young-Laplace)

Standards and References

Source	Scope
Fluent Theory Guide (ANSYS)	Complete model documentation
OpenFOAM User Guide	solver setup, boundary conditions
Brackbill et al. (1992) J. Comp. Phys.	CSF surface tension model
Schnerr & Sauer (2001)	Cavitation transport equation
Gidaspow (1994) Multiphase Flow and Fluidization	Euler-Euler fluidized bed reference
ISO 4126-1	Safety devices for protection against excessive pressure

Output

Provide: flow type (free surface/bubbly/droplet spray/cavitating/fluidized bed), model selected (VOF/Euler-Euler/Euler-Lagrange/DEM) with justification (volume fraction, interface character), governing equations relevant (write key transport equations), interfacial forces included (drag, lift, virtual mass, turbulent dispersion) with models and constants, cavitation model (Schnerr-Sauer/Zwart; P_vap [Pa]; cavitation number σ_cav), grid resolution (cells across interface or CFL limit), time step Δt [s] (CFL and surface tension limits), turbulence model (k-ε or k-ω SST; with BIT correction if bubbly), solver used (Fluent VOF/OpenFOAM interFoam), mesh count and estimated run time, key result (void fraction distribution, jet breakup length, erosion index), and applicable reference (Brackbill 1992, Schnerr 2001, Gidaspow 1994).