Sloshing-Structure Coupling
Theory and Physics
Sloshing Overview
What kind of phenomenon is sloshing?
It is a phenomenon where liquid inside a container violently oscillates due to external excitation (earthquake, waves, vehicle acceleration, etc.). It becomes important in LNG carrier cargo tanks, spent fuel pools at nuclear power plants, rocket propellant tanks, and so on.
Governing Equations
What is the mathematical model for sloshing?
In linear theory, it is described by the velocity potential $\phi$. The natural sloshing frequency of the free surface for a rectangular tank is:
Where $L$ is the tank length, $h$ is the liquid height, and $n$ is the mode number.
For nonlinear sloshing (large amplitude, wave breaking, accompanied by impact pressure), the Navier-Stokes equations are solved using the VOF method.
The impact pressure on the tank wall (sloshing impact) can reach several MPa locally.
How is coupling with the structure handled?
FSI (Fluid-Structure Interaction) becomes necessary when the elastic deformation of the tank wall affects the liquid surface behavior. Particularly in membrane-type LNG tanks (Mark III, NO96), the thin corrugated structure deforms under sloshing impact, which changes the pressure distribution.
Sloshing "Natural Frequency" – The Tank Shape Decides Everything
The most frightening aspect of sloshing is "resonance." The natural frequency of the liquid inside a tank can be calculated for a rectangular tank as f₁ ≈ (1/2π)√(πg/L・tanh(πh/L)) (L is tank length, h is liquid depth). When the tank is about half full and this natural frequency matches the ship's rolling period, the amplitude increases explosively. In the design of early LNG carriers in the 1960s, overlooking this resonance condition led to repeated impact loads on the tank walls and frequent accidents where insulation peeled off. Since then, the industry's operational rule has been to "maintain a full or nearly empty loading condition and avoid passing through the resonance range."
Physical Meaning of Each Term
- Structural-Thermal Coupling Term: Thermal expansion due to temperature change induces structural deformation, and the deformation affects the temperature field. $\sigma = D(\varepsilon - \alpha \Delta T)$. 【Everyday Example】Railroad tracks in summer where the rails expand and the gap narrows – temperature rise → Thermal Expansion → stress generation is a typical example. Warping of electronic circuit boards after soldering is also due to differences in thermal expansion coefficients of different materials. Engine cylinder blocks experience thermal stress due to temperature differences between hot and cold parts, potentially leading to cracks.
- Fluid-Structure Interaction (FSI) Term: Fluid pressure and shear forces deform the structure, and structural deformation changes the fluid domain – a bidirectional interaction. 【Everyday Example】Suspension bridge cables vibrating in strong wind (Vortex-Induced Vibration) – wind force shakes the structure, the shaken structure alters the airflow, further amplifying the vibration. Blood flow in the heart and elastic deformation of blood vessel walls, and aircraft wing flutter (aeroelastic instability) are also typical FSI problems. One-way coupling may suffice in some cases, but bidirectional coupling is essential for large deformations.
- Electromagnetic-Thermal Coupling Term: Joule heating $Q = J^2/\sigma$ causes temperature rise, and temperature change alters electrical resistance, creating a feedback loop. 【Everyday Example】Nichrome wire in an electric heater heats up (Joule heat) and glows red when current flows – as temperature rises, resistance changes, and current distribution also changes. Eddy current heating in IH cooking heaters and increased sag of power lines due to temperature rise are also examples of this coupling.
- Data Transfer Term: Interpolation resolves mesh mismatch between different physical fields. 【Everyday Example】When calculating "feels-like temperature" in weather forecasting by combining "air temperature data" and "wind data," interpolation is needed if the observation points differ – similarly in CAE coupled analysis, structural meshes and CFD meshes generally do not match, so the accuracy of data transfer (interpolation) at the interface directly affects result reliability.
Assumptions and Applicability Limits
- Weak coupling assumption (one-way coupling): Valid when one physical field affects the other but the reverse is negligible.
- Cases requiring strong coupling: Large deformations in FSI, cases with strong temperature dependence in electromagnetic-thermal coupling.
- Separation of time scales: When characteristic times of each physical field differ greatly, efficiency can be improved via sub-cycling.
- Interface condition consistency: Ensure energy and momentum conservation at the coupling interface is satisfied numerically.
- Non-applicable cases: When three or more physical fields are strongly coupled simultaneously, monolithic methods may be necessary.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Thermal expansion coefficient $\alpha$ | 1/K | Steel: ~12×10⁻⁶, Aluminum: ~23×10⁻⁶ |
| Coupled interface force | N/m² (pressure) or N (concentrated force) | Check force balance between fluid and structure sides. |
| Data transfer error | Dimensionless (%) | Interpolation accuracy depends on mesh density ratio. Below 5% is a guideline. |
Numerical Methods and Implementation
Numerical Methods
What methods are used for sloshing FSI?
Is the MPS method a Japanese-origin technique?
It is the Moving Particle Semi-implicit method developed by Professor Seiichi Koshizuka (University of Tokyo). It has been commercialized as Particleworks by Prometech. It can stably track large deformations and splashing of free surfaces and has many achievements in sloshing analysis.
Impact Pressure Evaluation
How is sloshing impact pressure evaluated?
Impact pressure includes air pocket type (compressed air cushion) and flip-through type (direct impact).
- Air pocket type: Air is trapped and compressed between the liquid surface and the wall. Oscillatory pressure waveform. Peak pressure is lower but duration is longer.
- Flip-through type: The liquid surface rises along the wall and strikes it directly. Very high peak pressure but short duration.
When compressibility is considered, two-phase flow analysis including the polytropic process of air is necessary.
SPH and VOF – The "Two Major Schools" of Sloshing Analysis
In sloshing numerical analysis, the grid-based VOF (Volume of Fluid) method and the meshless SPH (Smoothed Particle Hydrodynamics) method have long competed. VOF has a rich track record in OpenFOAM and STAR-CCM+ and its approach to capturing the gas-liquid interface is intuitive. On the other hand, SPH's strength is its natural handling of large liquid deformations (collision with tank walls, splashing). In the shipbuilding industry, IHI and Samsung Heavy Industries actively utilize SPH for sloshing impact pressure analysis. Recently, environments have been established where SPH calculations are parallelized on GPUs, completing simulations with millions of particles in a few hours. The practical answer is not which one is "correct," but to choose based on the analysis objective.
Monolithic Method
Solves all physical fields simultaneously as one system of equations. Stable for strong coupling, but implementation is complex and memory consumption is large.
Partitioned Method (Partitioned Iterative Method)
Solves each physical field independently and exchanges data at the interface. Easy to implement and can utilize existing solvers. Suitable for weak coupling.
Interface Data Transfer
Nearest neighbor (simplest but low accuracy), projection (conservative), RBF interpolation (robust to mesh mismatch). Balance between conservation and accuracy is important.
Sub-iteration
Performs sufficient iterations within each coupling step to ensure consistency of interface conditions. Residual criteria are scaled based on typical values of each physical field.
Aitken Relaxation
Automatically adjusts the relaxation factor for coupling iterations. An adaptive method that prevents divergence from over-relaxation and accelerates convergence.
Stability Condition
Beware of the added mass effect (in fluid-structure coupling when structural density ≈ fluid density). For instability, apply Robin-type interface conditions or the IQN-ILS method.
Analogy for Aitken Relaxation
Aitken relaxation is like "balancing a seesaw." If one side pushes too hard, the other side flies up, and the reaction causes it to push too hard again – Aitken relaxation automatically adjusts the pushing force to suppress this oscillation. It is an adaptive method that automatically adjusts the next correction amount based on the previous correction amount when coupling iterations oscillate and fail to converge.
Practical Guide
LNG Tank Analysis Procedure
How do you proceed with sloshing analysis for an LNG carrier?
1. Calculate hull motion via seakeeping analysis (time history of 6-DOF motion).
2. Set tank geometry and liquid level (partial fill ratio is critical. 20-80% is the dangerous range).
3. Perform CFD (VOF/SPH/MPS) sloshing analysis.
4. Statistical processing of wall pressure (short-term / long-term extreme value distribution).
5. Structural response analysis (input impact pressure into FE model).
Why is a mid-range fill ratio dangerous?
If the liquid level is too low, the liquid mass is small; if too high, the liquid movement is restricted. Sloshing becomes most violent at intermediate fill ratios of 40-70%. The IGC Code (International Code for the Construction and Equipment of Ships Carrying Liquefied Gases in Bulk) restricts navigation within dangerous fill ratio ranges.
Impact Pressure Statistical Processing
How is the variation in impact pressure handled?
Sloshing impact pressure shows extremely large probabilistic variation. Even under identical conditions, peak pressure can vary by more than 10 times between impact events.
Extreme value statistical processing using Gumbel or Weibull distribution is performed to estimate the maximum pressure for the design lifetime. Hundreds of impact events are obtained from a short-term 3-hour analysis to estimate the parameters of the extreme value distribution.
LNG Carrier No.96 Type Tanks – The Crystallization of Sloshing Countermeasures
Among the membrane-type tanks that dominate global LNG transport, GTT's No.96 type is designed with a complex insulation structure to absorb sloshing loads. For large LNG carriers with tank capacities of 140,000 to 170,000 cubic meters, sloshing analysis requires hundreds of simulation cases, exhaustively verifying combinations of wave conditions, loading rates, and sailing speeds. Major Japanese shipyards (Mitsubishi Heavy Industries, Kawasaki Heavy Industries) have dedicated sloshing test equipment (placing a 1/50 scale model on a 6-DOF motion platform) used to validate CFD results. The cost of sloshing analysis for building one LNG carrier reaches tens of millions of yen.
Analysis Flow Analogy
Have you ever inflated a balloon? At that moment, a sophisticated fluid-structure interaction is actually occurring. Internal air pressure (fluid) pushes and expands the rubber wall (structure) → the expanded wall changes the internal pressure distribution → the changed pressure further deforms the wall... Repeating this catch-and-throw at each calculation step is FSI analysis.
Common Pitfalls for Beginners
"One-way coupling should be enough, right?" – This misjudgment is the most dangerous in coupled analysis. If structural deformation is minute, one-way may indeed suffice. However, in cases like heart valve opening/closing where deformation significantly alters the flow path, one-way coupling is completely inadequate. A rule of thumb is "does the deformation exceed 1% of the characteristic length?" If it does, bidirectional coupling is mandatory. If you settle for one-way, the result can be "plausible but actually completely wrong" – this is the scariest pattern.
Thinking About Boundary Conditions
Data exchange at the coupling interface is like "border control." Each country (physical field) has its own laws (governing equations), but if the exchange of people and goods (force, temperature, displacement) at the border (interface) is not managed accurately, the economies (energy balance) of both countries collapse. Interpolation when meshes don't match is like a "translator" – the smaller the mistranslation (interpolation error), the better the result.
Software Comparison
Tool Comparison
What tools are available for sloshing analysis?
| Tool | Method | Features |
|---|---|---|
| Particleworks (Prometech) | MPS Method | Japanese-made. Many achievements in LNG sloshing. |
| FLOW-3D (Flow Science) | VOF (TruVOF) | Specialized in free surface tracking. |
| OpenFOAM (interFoam) | VOF | OSS. Free customization. |
| STAR-CCM+ | VOF/Euler multiphase | Easy setup of tank motion. |
| LS-DYNA | SPH | Originally for impact/crash, also used for sloshing. |
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