Knowledge / Ship systems

Tank sloshing: from a moving free surface to dynamic structural loads

Explore tank natural periods, fill-level sensitivity, impact pressure, impulse and the limits of scale-model predictions.

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Sloshing is the motion of a liquid free surface relative to its moving tank. It can create distributed dynamic pressure, localized impacts and loads on internal equipment. The important question is not simply how much liquid is aboard, but how tank geometry, fill level and vessel motion combine, and how the resulting load acts on the structure.

Separate equilibrium, inertia and impact

A liquid at rest exerts hydrostatic pressure that increases with depth. When the tank accelerates, the pressure distribution changes to accelerate the liquid. If a free surface develops waves, breaks or strikes a wall or roof, short localized loads can arise on top of the slower pressure variation. These mechanisms require different descriptions.

A static free-surface correction to vessel stability represents an equilibrium effect of liquid redistribution. It does not predict a wall-impact pulse or the load on a pump tower. Conversely, a small high-pressure spot does not automatically mean a large net force over the whole tank. Area and timing are needed to connect pressure with structural loading.

The liquid has its own natural modes

For a rigid rectangular tank with horizontal length L in the direction of oscillation and undisturbed liquid depth h, linear inviscid theory gives the first free-surface mode as ω₁² = gk tanh(kh), where k = π/L. The corresponding period is T₁ = 2π/ω₁. This model assumes small surface displacement and ignores breaking, compressibility and internal obstructions.

The rectangular-tank relationship appears in NASA SP-106, chapter 2. This 1966 mechanics reference remains useful for the idealised equation; it is not a marine structural rule. Changing the direction of motion changes the relevant horizontal dimension, so longitudinal and transverse sloshing generally have different natural periods.

Worked example: lower fill can lengthen the period

Let L = 12.0 m, h = 4.0 m and g = 9.81 m/s². Then k = π/12 = 0.261799 m⁻¹ and kh = 1.04720. With tanh(kh) = 0.780714, ω₁ = √(9.81 × 0.261799 × 0.780714) = 1.41601 rad/s. Therefore f₁ = 0.22536 Hz and T₁ = 4.4373 s.

At h = 1.0 m with the same tank length, tanh(kh) = 0.255978, giving ω₁ = 0.81081 rad/s and T₁ = 7.7493 s. The natural period becomes longer as the liquid becomes shallower in this model. A reduced fill therefore does not automatically move the tank away from the vessel’s dominant motion period.

The calculation estimates frequency, not wave height or pressure. Even an exact match between tank natural period and an excitation period does not determine the final amplitude without damping and forcing magnitude. Nonlinear waves can also shift the response peak away from the small-amplitude prediction.

Tank location and vessel motion matter together

The relevant excitation is acceleration and rotation at the tank, rather than wave height alone. A tank far from the vessel’s pitch or roll reference point can experience substantial translational acceleration from rotational motion. Different headings and speeds change the encountered wave spectrum and the mixture of surge, sway, heave, roll, pitch and yaw.

The ITTC 2024 sloshing procedure connects vessel-motion analysis to generated tank motion and a test matrix. That connection explains why applying an arbitrary sinusoidal tank motion is a mechanism study, not automatically a representative voyage assessment. Liquid motion can also feed back into vessel response when coupling is important.

Worked example: pressure peak is only one descriptor

Consider an idealized triangular pressure pulse above the chosen baseline. Its peak is 150 kPa, total duration 0.060 s, and pressure is assumed uniform and simultaneous over a 0.250 m² patch. Peak force is p_max A = 37 500 N = 37.5 kN. Pressure impulse is 0.5 × 150 000 × 0.060 = 4 500 Pa·s.

Multiplying pressure impulse by patch area gives force impulse 1 125 N·s. A pulse with the same peak but half the duration gives half this impulse. A pulse covering half the area also halves force and force impulse. None of these calculations determines plate stress without stiffness, support conditions and structural dynamics.

The assumption of simultaneous uniform pressure is deliberately strong. A measured point peak cannot simply be applied to an entire bulkhead. DNV’s local-loads explanation places transient sloshing loads in the context of structural response, including approaches that couple fluid and structural behaviour.

Measure the event in space and time

A pressure sensor averages over its active area and responds through its own frequency characteristics. Sparse sensors can miss a small moving impact patch; inadequate sampling can miss the peak or distort duration. Synchronised sensors help establish whether neighbouring loads occur together. Raw records also allow later examination of filtering choices.

The ITTC model-test procedure discusses sensor placement, calibration and impact-data analysis. These details affect the quantity being estimated. The largest observed peak in one short run is a sample maximum, not a known lifetime design load, and an isolated sensor anomaly needs physical corroboration.

A gravity scale does not reproduce every impact effect

For geometrically similar gravity-dominated motion at length scale λ = L_full/L_model, Froude time scaling gives T_full/T_model = √λ. A 1:25 model therefore runs characteristic motions five times faster than the prototype: a 5.0 s full-scale period corresponds to 1.0 s in the model. Density must be accounted for separately in pressure scaling.

Gas entrapment, compressibility, liquid-gas density ratio and local structural flexibility can complicate an impact. Treating all peaks as simple hydrostatic scale multiples can conceal these differences. DNV’s 2014 sloshing note identifies full-scale extrapolation and long-term statistics as assessment issues; that dated overview is not a complete current rule.

Assess the load path and the permitted filling envelope

The loaded system can include tank plating, insulation, internal piping, tower supports and supporting hull structure. Stiffening one part can change load transfer to another. Baffles alter free-surface motion and dissipate energy, but they also attract local loads and affect access, cleaning and piping arrangements. They require design assessment rather than a generic recommendation.

Lloyd’s Register’s public ShipRight index identifies a dedicated sloshing-load and scantling procedure. Actual filling restrictions depend on tank and containment design, cargo and approved operating documentation. No universal “safe percentage” follows from the rectangular-tank example, and changing ballast or cargo distribution must also preserve the vessel’s other loading and stability requirements.

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