Wave-Excited Floating Body: Heave-RAO Curve vs Potential-Flow Diffraction

1  Objective

Case 9 (#1302) verified a single point — the long-wave limit where a small floating body is a wave follower (heave RAO → 1). This case turns that one point into a frequency response: the same body and tank, swept across wave period toward resonance. Each CFD point is checked against a potential-flow diffraction prediction for the exact section — the strongest cross-check tier of the CFD validation philosophy (analytical / BEM before experiment).

2  The RAO curve

The blue curve is the potential-flow reference; filled circles are off-resonance CFD solves (green = within band), orange squares are the resonance-region points (see §4), the black diamond is the reused #1302 long-wave point, and faint hollow markers are the coarse-grid points from the resolution study. The reference reproduces the verified #1302 point at T = 3 s from a completely independent method — a genuine cross-method check — then rises toward a resonance peak near T ≈ 0.86 s.

3  Heave response at each period

4  Near resonance: a grid-independent response reduction open finding

The inviscid reference predicts a sharp heave resonance (RAO up to ≈ 2.20 near T ≈ 0.86 s). The CFD does not reproduce it: across the resonance region the body stays close to a wave-follower (RAO ≈ 0.7–1.0), far below the reference. The important question is why — numerical under-resolution of the short wave, or a real reduction in the body's dynamic response. A two-resolution study answers the first half:

The reduction is grid-independent. Doubling the background resolution (18 → 36 cells per wavelength) recovers a larger incident amplitude at the gauges — but the heave amplitude grows in step, so the RAO barely moves and stays far below the inviscid reference. The near-resonance reduction is therefore not a wave-field resolution artifact. What the present forced-wave setup cannot separate is the cause:

• Physical bluff-body damping (expected). The section is a sharp-cornered half-submerged square, not a faired ship section. Flow separation and vortex shedding at its bottom corners add heave damping the inviscid BEM omits, which flattens the resonance peak — the canonical reason CFD and panel methods diverge for bluff sections, and a primary motive for running CFD at all.
• Gauge-to-body incident decay (measurement bias). The RAO is normalised by the incident amplitude at a gauge ~3.5 m upstream of the body; short waves decay measurably over that gap, biasing the measured RAO low. Correcting roughly for the observed decay lifts the resonance-region RAO toward ~1.2 — still below the inviscid peak, so this explains only part of the gap.
• Overset numerical damping (not yet bounded). Interpolation across the overset fringe is mildly dissipative; refining the background grid does not bound it, so it is not ruled out here.

Resolved by the free-decay test (#1332). A still-water free-decay run (report) settles it. Released from a heave displacement, the body rings down with a damped natural frequency within ~0.4% of the reference resonance frequency (its mass, stiffness and added mass are correct — no setup error) and a light, amplitude- independent damping ratio ζ ≈ 0.13 — only ~20% above the potential-flow radiation value (0.10), i.e. linear damping, not the amplitude-dependent separation drag that would flatten a resonance. With that measured damping the body would still resonate to RAO ≈ 1.8 — far above the forced-sweep points here. So the resonance-region reduction is not body damping: it is dominated by the gauge-to-body incident-wave decay above (the short wave is under-delivered to the body over the fetch and the RAO is normalised at a gauge ~3.5 m upstream). The overset body's own dynamics are verified; the off-resonance gate below is unaffected.

5  CFD vs reference — band gate

Reading the gate. The band gate applies off resonance, where the inviscid reference is an accurate ground truth: there CFD and BEM agree within a few percent (PASS). In the resonance region the inviscid reference is only an upper bound (it omits viscous damping), so those points are reported as a measured discrepancy and an open finding (§4) rather than graded pass/fail against a reference that is not ground truth there.

6  The potential-flow reference

The reference is a linear boundary-element (capytaine) solution for the same 0.2 m half-submerged square section, at the CFD's finite water depth (0.4 m). Because the CFD is a 2D slab (an infinitely long cylinder), the BEM models a long box (L = 6 m ≈ 30 beams) whose per-length response converges to the 2D sectional RAO — a box-length sweep (L = 4→8 m) moves the resonance peak by 0.5% and the flank by <1%, so end effects are negligible. Heave RAO = |F₃| / |c₃₃ − (M + a₃₃)ω² + i b₃₃ω|, with the total excitation F₃ the sum of Froude-Krylov and diffraction forces (the diffraction problem alone omits FK and collapses the long-wave limit), and an interior free-surface lid removing the BEM irregular frequencies. The curve is frozen with provenance in docs/api/cfd/cases/wave_excited_body/rao_reference_capytaine.csv; regenerate via generate_rao_reference.py. This is the same potential-flow physics the OrcaWave/AQWA pipeline solves — here computed 2D-consistently for the exact section so the CFD points have a like-for-like check.

7  Numerical model

Every point reuses the verified #1302 geometry (20 m tank, depth 0.4 m, body at x = 13 m, nz = 49 so the still-water line stays on a cell face) and machinery (rigid overset component, heave-only sixDoF, StokesII inlet + active absorption, Goda–Suzuki incident/reflected split for the incident amplitude). Only the wave period, the background nx (the cells-per-wavelength swept in the resolution study), the gauge array (spacing ≈ 0.2λ so the split does not alias) and the solve duration / analysis window (sized from the group velocity) adapt per point — all in digitalmodel.solvers.openfoam.validation.build_sweep_config. The heave amplitude is a single-harmonic least-squares fit over the settled window; RAO = amplitude / incident amplitude.

8  Conclusion

Off-resonance CFD verified against the potential-flow reference; near resonance the CFD response is grid-independent and well below the inviscid peak — a documented open finding (see §4). The overset wave-body machinery, verified at one point in #1302, now traces a heave-RAO curve that matches the potential-flow diffraction reference off resonance (a like-for-like CFD check on the OrcaWave/AQWA pipeline). In the resonance region the CFD response is grid-independent and sits well below the inviscid peak; a free-decay damping test is the recommended follow-up to attribute that reduction and confirm the body's natural frequency.

References

1. Potential-flow reference: capytaine (linear BEM); total heave excitation = Froude-Krylov + diffraction; finite depth 0.4 m; frozen in rao_reference_capytaine.csv.
2. OpenFOAM ESI v2312: multiphase/overInterDyMFoam/floatingBody (overset), interFoam/laminar/waves/stokesII (waves, verified in #1170).
3. Goda, Y. & Suzuki, Y. (1976). Incident/reflected split over a gauge array (N-gauge least squares).
4. digitalmodel issue #1324 (this sweep) under epic #1161; precursor #1302; regression test tests/solvers/openfoam/validation/test_wave_excited_body_rao.py.

Generated from real OpenFOAM v2312 solves and a capytaine BEM reference. All chart data is computed directly from solved fields, gauge records and body-state histories — no values are hand-entered.