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. Two claims here are superseded: the exchange period is 14.31 s (23 s is the amplitude peak), and the ×1.45 residual is resolved in closed form rather than being dynamic amplification — see the note below.
Superseded in part · 2026-07-28Two statements on this page have since been corrected by a measured (period × roll amplitude) grid. The 23 s figure below is the amplitude peak, not the exchange natural period — phase puts the natural period at 14.31 s, and the amplitude peak moves with roll amplitude while the natural period does not. Consequently the effective conduit length calibrated to that peak absorbs a damping-induced shift into an inertia parameter. The ×1.45 residual is also now resolved in closed form: it is the weight moment of the whole fluid mass about the roll axis plus the within-leg free surface, terms the redistribution estimate omits. See Damping & sizing.
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.
14.31 sexchange natural period (phase); 23 s is the amplitude 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 was the original approach here; it is superseded. The response peak is loss-controlled and sits well above the natural period, so anchoring an inertia parameter to it is not valid — the calibration is now anchored to the measured natural period instead.
| Conduit length ℓc | Basis | Exchange period | Note |
|---|---|---|---|
| 10.0 m | geometric (centroid-to-centroid) | 19.4 s | uncorrected geometric-path comparison |
| 5.20 m | calibrated to the measured natural period | 14.31 s | 0.520× geometric; 90° phase crossing |
Key finding. The connected exchange natural period is 14.31 s, identified by the 90° phase crossing. The 22–24 s plateau in the 5° forced-roll sweep is the loss-controlled amplitude peak, not the exchange resonance or tuning period; it shifts from 20 s to 40 s across the measured 2.5°–10° roll-amplitude range.
04 · Mass shift & roll moment
What the exchange does to roll
For the published 24 s, 5° forcing case, the CFD level-difference amplitude (1.1037 m) fixes the volume exchanged between legs. The 6.66 MN·m redistribution estimate omits two static contributions to the total roll moment: the weight moment of the whole fluid mass about the roll axis and the within-leg free surface.
| 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 redistribution-only estimate | 9.67 MN·m | ×1.45 static-term residual; resolved in closed form |
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 was previously read as resonant dynamic amplification. It is not: it is present at quasi-static forcing, where inertia cannot contribute, and is now resolved in closed form as two omitted static terms. 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). The previously applied ×1.34 absolute-magnitude factor must not be interpreted as resonant dynamic amplification; the basis for reconciling absolute exchange magnitude remains unverified — needs owner. Design reading: a larger connection raises both the exchange amplitude and its natural 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 natural period is 14.31 s from the 90° phase crossing
- Conduit-area scaling law validated to ~1–2% across Ac = 3.4–13.5 m²
- Inter-tank mass shift derived from the CFD level difference; the former ×1.45 roll-moment residual is resolved as two omitted static terms
- Entirely source-neutral reference geometry
Staged / bounded
- Purpose-built coupled dual-tank CFD confirmation (licensed-run host)
- The ×1.34 reconciliation of absolute exchange magnitude is unverified and needs owner review; it is not established as dynamic 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 1034bb4efc5d9b390d51d4d546800c5a25bf6673af824e07ed5265da038ec30b, extending pinned Hugging Face revision 51ba5ddbca8dcfa8faeb6de5c592f342f32cfe2d.