Changing only τ changed the density of a gas at rest — the (τ−½)² stress of Shan–Chen velocity-shift forcing
Apply the same force to a resting D1Q3 gas column and only Shan–Chen forcing gives a τ-dependent density. The shifted equilibrium velocity leaves a fake (τ−½)²F²/ρ stress.
Adding a v Term to a v² Term Got Both Ends Right and Missed the Middle by 42% — Prony's Two-Term Drag Law and Re = 60
Bossut measured drag growing with v², Du Buat with v. Prony's sum a·v²+b·v hands over control at Re = 60 and overpredicts settling speed by up to 42% right there.
A 1 μm Aluminum Particle Heated 4.6× Slower Than the Continuum Formula — Sherman Interpolation in the Knudsen Transition Regime
At 1500 K and 1 atm a 1 μm aluminum particle sits at Kn = 0.67. Nu = 2 shortens heat-up by 4.58; Sherman's harmonic mean puts that time back.
I asked for σ₁₂ = 1.00 and got 0.78 — where an absent phase leaks into the interface in three-component Allen–Cahn
Fixing Σc = 1 with a mean projection in three-component Allen–Cahn lets the absent phase leak into the interface and cuts the realized surface tension by 21.9%. A triple-well penalty closes the gap.
Mobility Up 10×, Interface Width Unchanged — The Anti-Diffusion Term in Conservative Allen–Cahn
Why the anti-diffusion term cancels diffusion exactly, pinning interface width to W rather than to mobility, and where that term enters the LBM equilibrium.
A 9% Smaller Drag Coefficient Left the Droplet 35% Slower — Where the Distortion and Vaporization Corrections Attach
Three drag-coefficient models — rigid sphere, TAB distortion, and Renksizbulut-Yuen vaporization — run on the same droplet and compared by penetration.
Pressure Jumped 21% in a Problem Where Nothing Should Happen — The Abgrall Criterion and the Nonconservative Term
Uniform pressure and velocity stop staying uniform because of the nonconservative term, not the flux. I remeasured Re-Abgrall's H_u operator on one grid.
Raising the time step 5x ran 1.9x faster; 10x ran no faster at all — the window implicit surface tension opens
At density ratio 1000, raising the capillary step 5x ran 1.9x faster. Pushing to 10x did not. Janodet et al. (2025) measured both walls.
Halving the mesh made the blow-up arrive twice as fast — complex eigenvalues in the 6-equation two-fluid model
Slip turns the six-equation two-fluid model complex. The coefficient of interfacial pressure must pass 1.0 to put both slow eigenvalues on the real axis.
[Paper Review] Prying Advection and Pressure Fully Apart — TV Flux Splitting for Baer–Nunziato
Split a Baer–Nunziato flux and sound speed stays in the pressure system. Toro Test 4 beat Rusanov by 1.5×; Noh then sent pressure to −0.12.
[Paper Review] A Droplet Sends Almost Every Sound Wave Back — Acoustic Impedance and a Sharp-Interface Two-Phase Solver
Air-water reflection returns 99.889% of the acoustic energy. A 1% gas fraction drops Wood's sound speed from 1500 m/s to 120.5 m/s.
Why Laundry Dries at 5 °C — Diffusion-Limited Evaporation vs. Heat-Driven Boiling
Laundry dries at 5 °C because the interface vapor fraction is not zero. Boiling waits on heat, and the Lee coefficient r is a penalty, not a property.
Un-smearing the Interface Every Step — Implementing Anti-Diffusion Sharpening (IST)
After 200 steps a helium bubble edge grows from 6 cells to 20. IST post-processing then locks that smeared thickness to the ratio of ε/a.
Why a Droplet at Rest Starts to Flow — Parasitic Currents and Well-Balanced Surface Tension
A 3% curvature error drives the residual to 4.8e5, while exact curvature leaves it at 4.1e-10. That gap is the parasitic current.
Why a Raindrop Is Not Tear-Shaped — The Weber Number and TAB Breakup
The TAB oscillator breaks a drop once the deformation y crosses 1. Undamped, that peak is 2·We/24, so the critical Weber number lands on 12.
Why Droplet Simulations Get Trapped by Δt — Solving Surface Tension Implicitly
From N=64 to N=256, dt_sigma falls by 8. omega*dt_sigma stays 2.221 on every mesh, and explicit integration breaks past about 0.9 dt_sigma.
Giving Each Phase Its Own Equations — The Baer–Nunziato Seven-Equation Model
The printed 324 bar is the 1.2 percent left after two huge stiffened-gas terms cancel. The first relaxation step is what pushes that leftover.
[Paper Review] Second-Order Accuracy in a Single Stage — A GRP Solver for the Kapila Multiphase Model
At N=320 a single-stage GRP reports order 2.03 and an L1 error of 1.55e-5, against 2.31e-2 and order 0.97 for Godunov on one sine wave.
Why Droplets Are Round and Water Climbs the Glass — Surface Tension and Contact Angle
A 5 µm droplet shows a 28800 Pa pressure jump, and a 0.1 mm tube at 30 degrees lifts water 127.1 mm under Jurin's law here.
Why a Shout Above Water Never Reaches Below — Acoustics Across a Diffuse Interface
Near an air–water impedance ratio of 3600, about 99.9% of the sound reflects. The 13.4 degree critical angle is checked in a 1D tube.
[Paper Review] Two-Phase Flow Without a Poisson Solver — Coupling the General Pressure Equation with MSTACS
Raising artificial sound speed from 5 to 15 cuts the stable step by three. OS-MSTACS lists a volume error E_G of 6.57e-8, four orders under OS-CICSAM.
Water Boils at 20°C Too — The Physics of Cavitation
At 20°C vapor pressure is 2.34 kPa. If a venturi throat drives static pressure under that line, water boils and the cavitation number turns negative.
SPH Without Tears — Solving Fluids with Particles, from Kernel to Artificial Viscosity
With γ = 7, Tait turns a 1% density error into a large pressure swing, and a pressure gradient that is not symmetrized leaks pair momentum.
Rhie-Chow Interpolation for Compressible Multiphase Flows
At a water-air density ratio of 1000, the arithmetic mean density is 500.5 and the harmonic mean is 2.0. Filter only the driving pressure.
5-Equation Model Implementation Guide: Building a Multiphase Flow Solver in Python
At volume fraction 0.5 the mixture sound speed is about 24 m/s. The same initial air is near 374 m/s and the water near 1625 m/s.
Interface Capturing Methods: VOF vs Level Set vs Diffuse Interface
A conservative alpha update rings the pressure at a contact. The five-equation model leaves only alpha advection non-conservative, and equilibrium holds.
Compressible Multiphase Flow CFD: Why It's Hard, Why It Matters
Averaging gamma at a gas-liquid interface does not recover the pressure. Near half gas fraction, the mixture sound speed falls to about one hundredth.