Tank sloshing — natural-period & resonance explorer

The fundamental sloshing natural period of a partially filled tank, from validated analytical theory across tank shapes. The normalized master curve collapses every size onto one dimensionless family; pick a shape to read real periods; and see how partial fill sweeps a tank into resonance with vessel roll — the coupling external diffraction (AQWA/OrcaWave) does not capture. Every value is a live hydrodynamics.sloshing evaluation.

Normalized master curve

Dimensionless sloshing frequency Ω₁ = ω₁√(L_c/g) vs fill / slenderness ratio h/L_c

Any tank of a given shape — any size — lies on its curve. Read Ω₁ off the curve at your fill ratio, then the real period is T = 2π√(L_c/g) / Ω₁. Deep-liquid asymptotes: rectangular √π = 1.77, upright cylinder √1.841 = 1.36 (×√2 on a diameter basis).

Horizontal-cylinder and elliptical curves use the equivalent-rectangle method (good to ~10–15% at mid fill; near-full and violent cases are where a VOF free-surface CFD run takes over).

CFD vs literature — which shape is validated

Which tank shape was run in CFD? The RECTANGULAR tank — measured CFD vs literature theory

The master curve above is literature theory (linear potential flow) for every shape. So far only the rectangular tank has been validated with free-surface CFD (OpenFOAM VOF). This chart isolates it — CFD-measured points against the literature curve, with the error at each fill — and the strip below shows which shapes are CFD-validated versus literature / analytical only.

Literature basis: linear potential theory (Faltinsen & Timokha 2009; Abramson 1966), corroborated by forced-roll and multi-fill CFD / experiments (Delorme / SPHERIC; Rognebakke & Faltinsen; Chen & Xue). Full method, force histories and the adversarially-verified survey: the 2D CFD validation study →

Forced-roll resonance — with vs without

How forced roll reveals the sloshing resonance

Without forced roll, a free-decay CFD run gives the tank's intrinsic first-mode natural period (the point on the master curve). With forced roll, driving the tank at a range of roll periods and measuring the wall run-up traces a resonance curve that peaks at that same natural period — roll excites the mode the free-decay rings down. Response is normalised to each fill's own peak and plotted against the drive period over the natural period, T_drive/T₁.

Anti-roll tuning — CFD-reduced tuned damper

Tune the tank to your vessel: fill for a target roll period, and the damping it delivers

The forced-roll CFD reduces the tank to a single-DOF tuned damper — an equivalent damping ratio ζ (rising with fill) plus added roll damping/inertia. Tuning is a curve, not a single “best fill”: set the tank length L and fill so the sloshing period f₁(fill) matches the vessel roll period; the fill then sets the damping.

Tank length L (m) Target vessel roll period (s)

Tuning fill h/L—

Fill depth h—

Damping there (CFD ζ)—

CFD-reduced tuned-damper parameters (rectangular, 4° roll)

Fill h/L	Natural period T₁ (s)	Equivalent damping ζ	Added roll damping B₄₄ (N·m·s/rad)

ζ and B₄₄ from the forced-roll CFD (in-phase / quadrature reduction; SloshingCouplingModel). Higher fill → more damping. This is a screening-grade linearization, valid as a design anchor, not a full model: ζ is a config-specific equivalent (not an invariant law), fixed at 4° roll (larger amplitude bends the backbone — softening at these fills, h/L ≥ 0.30; shallow fills harden instead — and raises breaking-driven damping); it is 2D, so it cannot capture 3D rotary/swirl modes near resonance; there are no baffles/screens (damping here is wave-breaking only — real anti-roll tanks tune damping with internals); and roll–sway coupling / roll-axis position are not modelled. Only three fills, so ζ/B₄₄ are coarsely interpolated. See the CFD study for the backbone and full caveats.

Shape lookup

Real natural period for a specific tank

Tank shape Size (m)

Fundamental period (½ full)—

Characteristic length L_c—

Aspect / fill ratio—

Fill fraction	Fundamental period T₁ (s)

Resonance screen — marine coupling

Partial-fill sloshing vs vessel roll (the ballast-tank / anti-roll case)

Fill level

Cylindrical T₁—

Vessel roll—

tank

4 s8121620 s

Where CFD takes over. Analytical periods and the class-simplified screen above close most of a scope. Violent / high-fill / impact cases — and the coupled roll response of partially filled ballast tanks used as tuned/anti-roll tanks — are resolved with VOF free-surface CFD (OpenFOAM), feeding a reduced response back into the vessel roll time-domain model. This is the tiering behind the ballast-tank sloshing study. The red rings on the master curve above are the first rung of that ladder: measured 2D interFoam points that pin the rectangular curve to real free-surface CFD (see the CFD verification page).

Literature confidence & way forward

What the literature says — and the CFD that should follow

Supports our approach

Caveats & contrasts

Further CFD analysis (prioritised)

Synthesised from an adversarially-verified deep literature survey (8 agents; fabricated draft figures were removed by a critique pass). Full method & references on the CFD study page →

Worldwide relationship basis

References

Built by scripts/capabilities/build_sloshing_explorer.py from src/digitalmodel/hydrodynamics/sloshing.py. — engine source →