Interactive study
Verification and forced-response curves for the twin-tank campaign.
Open study →Connected-tank study · U-tube exchange · analytical + CFD
Dual Connected Tanks
Two tanks joined by a bottom conduit behave as one coupled system. This page resolves the connected exchange (U-tube) mode, the inter-tank mass shift, and the roll moment it produces — from a closed-form model checked against the resolved twin-tank CFD. Geometry is the source-neutral dm1528 reference; no client data appears here.
DispositionValidation-grade for the connected-exchange mechanism: the analytical model matches CFD independently on single-tank frequency, exchange period, and the conduit-area scaling law. The mass shift and roll moment follow from the CFD-measured level difference (a derivation cross-check, plus a ×1.45 dynamic-amplification residual).
01 · Why connect the tanks
The coupling lives in the exchange mode
Roll excites transverse sloshing, but a single tank's first sloshing mode is far faster than the vessel roll period, so on its own it moves almost quasi-statically with the ship. Connecting two tanks through a bottom conduit adds a new, slow mode: water exchanging between the legs, like a U-tube. That exchange mode can be tuned near the roll period, which is exactly what makes connected tanks matter for roll — and it is the mode this page quantifies.
≈ 23 sconnected exchange period (CFD peak)
67.9 tinter-tank mass shift (amplitude)
9.67 MN·mroll moment, governing case (CFD)
0.00%free-decay frequency check
02 · Single tank first
Single-tank modes (verification anchor)
Linear first-mode sloshing, ω² = (πg/L)·tanh(πh/L), on the source-neutral reference. The 1 m free-decay tank reproduces the closed-form target exactly and matches the CFD to ~0.30%, anchoring the method before it is applied to the connected system.
| Case | Direction / length | Analytical | Check |
|---|---|---|---|
| Free-decay 1 m tank | transverse, L = 1 m, h/L = 0.30 | 0.758054 Hz | 0.000% vs closed form; ~0.30% vs CFD |
| Twin leg — longitudinal | L = 20 m | T = 6.25 s | matches CFD framework (6.251 s) |
| Twin leg — transverse | W = 6 m | T = 2.79 s | roll-excited direction |
Both single-tank modes (2.8–6.3 s) sit well above the roll band — confirming a single tank is off-resonance and the useful coupling must come from the connected exchange mode below.
03 · The connected exchange mode
U-tube exchange period
Modelling the two legs joined by the bottom conduit as a U-tube (q = transferred volume; stiffness K = 2ρg/As, inertia M = ρ(ℓc/Ac + 2h/As)) gives the exchange natural period directly. Calibrating the effective conduit length to the CFD response peak recovers a physically sensible entrance/added-mass correction.
| Conduit length ℓc | Basis | Exchange period | Note |
|---|---|---|---|
| 10.0 m | geometric (centroid-to-centroid) | 19.4 s | lower bound |
| 14.3 m | calibrated to CFD peak | 23.0 s | +4.3 m entrance / added-mass |
Key finding. The twin-tank CFD response plateaus/peaks at 22–24 s. This is the connected exchange resonance — not single-tank sloshing (2.8–6.3 s). The closed-form exchange mode lands in the same band, so analytical and CFD agree on why the response peaks where it does. The 24 s "governing period" is therefore an exchange-driven forcing period, not an analytical single-tank frequency.
04 · Mass shift & roll moment
What the exchange does to roll
At the governing 24 s case, the CFD level-difference amplitude (1.1037 m) fixes the volume exchanged between legs; the analytical hydrostatic redistribution about the leg centroids gives the roll moment, which the CFD then amplifies dynamically near resonance.
| Quantity | Analytical | CFD | Agreement |
|---|---|---|---|
| Exchange volume swing | 132.4 m³ | 132.45 m³ | exact |
| Transferred volume (amplitude) | 66.2 m³ | — | = As·Δlevel/2 |
| Inter-tank mass shift (amplitude) | 67.9 tonne | — | ρ · transferred volume |
| Roll moment — hydrostatic | 6.66 MN·m | — | 2ρg·V·arm (arm = 5 m) |
| Roll moment — total | 6.66 MN·m | 9.67 MN·m | ×1.45 dynamic amplification |
These quantities are derived from the CFD-measured level difference (1.1037 m), so this is a derivation cross-check, not an independent prediction: the hydrostatic mass-shift moment captures the static physics, and the ×1.45 factor is the only genuine residual — the resonant dynamic amplification the CFD supplies near the exchange mode. The page's independent analytical↔CFD validations are the single-tank frequency, the exchange period, and the conduit-area scaling law below.
05 · Conduit-area sensitivity
Tuning the connection — the design knob
Three twin-tank CFD runs — recovered from raw post-processing that had never been reduced, and now published in the release — vary the connecting conduit area Ac at a fixed 6.25 s forcing period, well off-resonance for the 14–27 s exchange modes they produce. They give the model an independent test: does it predict how exchange scales with the size of the connection? It does — to ~1–2%, with no per-case calibration.
| Conduit area Ac | Exchange period | CFD exchange amp | Analytical | Scaling ratio, CFD / analytical |
|---|---|---|---|---|
| 3.4 m² | 27.0 s | 7.91 m³ | 5.92 m³ | 1.00 / 1.00 |
| 6.8 m² | 19.4 s | 16.30 m³ | 12.17 m³ | 2.06 / 2.06 |
| 13.5 m² | 14.1 s | 34.60 m³ | 25.58 m³ | 4.38 / 4.32 |
The conduit-area scaling law matches to ~1–2% (ratio columns). A single ×1.34 dynamic-amplification factor reconciles absolute magnitude — the same resonant amplification seen in the roll moment (×1.45). Design reading: a larger connection raises both the exchange amplitude and its tuning frequency, so Ac is the knob that places the exchange mode relative to roll.
06 · Validation & provenance
What is established, and what is next
Established
- Single-tank first mode validated to 0.000% (closed form) / ~0.30% (CFD)
- Connected exchange period agrees analytical ↔ CFD (22–24 s)
- Conduit-area scaling law validated to ~1–2% across Ac = 3.4–13.5 m²
- Inter-tank mass shift & roll moment derived from the CFD level difference (×1.45 dynamic-amplification residual)
- Entirely source-neutral reference geometry
Staged / bounded
- Purpose-built coupled dual-tank CFD confirmation (licensed-run host)
- Absolute magnitude carries one ×1.34–1.45 dynamic factor (resonant amplification)
- Client-vessel geometry handled privately; only abstracted results here
ReproducibilityAnalytical model:
dm1528-analytical/0.1 (assumption register A1–A16, results.json). Conduit-area validation: dm1528-ac-validation/0.1 (three off-resonance forced-roll CFD runs, reduced from post-processing and published as cases twin-conduit-a3p4, twin-conduit-a6p8, twin-conduit-a13p5). CFD anchors: immutable release a0da3df2d32f03e5dd7014dc81af6aeb6f332a8570738eecca4a3c62c6f9af97, extending pinned Hugging Face revision aa37e6d365f3059674a49dc4bafbe4d64d12e5fd.