Transient spurious-current spike during sessile droplet relaxation with Huang et al. 2D EBM_VOF sandbox — is this a known artifact?

[Kevin Ripamonti]

Dear all,

I am using the 2D sharp and conservative VOF sandbox from Huang, Han, Zhang, Ni (J. Comput. Phys. 533, 113975, 2025) to run a sessile droplet relaxation benchmark on a flat embedded wall, as part of the fluid-dynamics validation for my Master's thesis on SAW-driven droplets at Politecnico di Milano.

Setup. Water droplet, R = 1.34 mm, σ = 0.0727 N/m, θ_eq = 100°, domain 8 × 8 mm², wall at y = 2 mm aligned with the grid, real water/air properties with viscosity ×100 to accelerate relaxation, T_end = 0.1 s, uniform grids at ML = 6, 7, 8 (no AMR). Include order: myembed.h → navier-stokes/centered.h → embed_contact.h → embed_two-phase.h → embed_tension.h. Field bindings (tmp_c.height, hnew1, oxyi, nc) set in main() before run() as required. The droplet is initialized as a geometric half-circle (θ_init = 90°) and relaxes towards θ_eq = 100°.

Asymptotic plateau — excellent. When the simulation settles between spike events, max|U| decreases monotonically towards remarkably low values:

ML=6: plateau ≈ 1.1 × 10⁻⁴ m/s ML=7: plateau ≈ 2.9 × 10⁻⁵ m/s ML=8: plateau ≈ 4.2 × 10⁻⁶ m/s (reached in the window 7.76e-2 s < t < 9.28e-2 s)

Transient spike events during relaxation. However, during the relaxation phase I observe spikes in max|U|, always localized at the right triple contact point (breaking the left-right symmetry of the droplet). The behaviour is non-monotonic and qualitatively different at different ML:

ML=6: 1 primary spike at t ≈ 3.19e-2 s, peak 2.2 × 10⁻⁴ m/s (ratio ×6) ML=7: 1 primary spike at t ≈ 3.43e-2 s, peak 1.4 × 10⁻³ m/s (ratio ×39) ML=8: alternating cascades of spikes and clean recovery windows global peak 6.7 × 10⁻² m/s at t ≈ 5.0e-2 s (ratio ×470)

At ML=8 the simulation alternates between clean windows of monotonic decay to very low values and cascades of oscillatory spikes that re-excite max|U| up to O(10−2) m/s, followed by recovery. Crucially, after each cascade the solver finds its way back to the low plateau — the ML=8 run reaches 4 × 10⁻⁶ m/s in a 15 ms clean window before a new cascade appears, and ends at 1.1 × 10⁻⁵ m/s at t = 0.1 s, partially recovering from the last cascade.

Additional observations: 

(i) spikes are always localized at the right contact point, independent of ML; 

(ii) V/V₀ remains exactly conserved at ML=6 and ML=7, but develops small correlated oscillations of order 10⁻⁶ at ML=8 (range 0.99999970 – 1.00000067), which is no longer the exact local conservation expected from the method; 

(iii) the recovery time after each individual spike shrinks with refinement — at ML=8 max|U| drops by a factor 10 in just two timesteps — but at ML=8 new cascades are triggered before full recovery is reached; 

(iv) the clean windows at ML=8 reveal that the method is, asymptotically, far more accurate than the spike amplitudes would suggest.

Questions:

1. Has anyone observed these cascades when validating the sandbox at high resolution on flat substrates? Is the spike/stick-slip behaviour a known regime-specific limitation of the parabola-fit HF during quasi-static contact line relaxation?
2. Would initializing the droplet directly at the target contact angle (spherical cap with θ = 100°, zero contact line motion) be expected to eliminate the cascades entirely, exposing the clean ≈ 4 × 10⁻⁶ m/s plateau from t = 0?
3. For a problem where the droplet must first reach a stable equilibrium and then be perturbed by an external body force (a SAW streaming forcing, in my case), would you recommend a different relaxation strategy to avoid exciting the spike cascade?
4. Is there an expected scaling for the plateau vs ML in the framework? My results suggest roughly Δx convergence (factor ~4 from ML to ML+1 between ML=6 and ML=7, factor ~7 between ML=7 and ML=8).

I am happy to share the test case, log files and spurious.dat for all three resolutions. Any insight is very welcome — the asymptotic plateau is the best I have obtained with any method for capillary-dominated contact-line problems, and I would like to make the transient phase reliable as well to move on to the full 3D SAW-driven problem.

Thanks, 

KevinRipamonti M.Sc. student, Politecnico di Milano