Ship Slamming Analysis
Theory and Physics
Physics of Slamming
What is the slamming phenomenon for ships?
It is a phenomenon where an extremely large impact pressure occurs in a very short time when the bottom of a ship navigating in waves strikes the water surface. It is classified into bow flare slamming at the bow, bottom slamming at the bottom, and stern slamming at the stern. The impact pressure reaches the order of MPa, causing local structural damage and whipping vibration.
Governing Equations
Is there a theoretical solution for the impact pressure?
Wagner's (1932) theory is classical. For the water impact of a 2D wedge shape,
Here, $V$ is the impact velocity, $\beta$ is the deadrise angle, and $c(t)$ is the wetted width. As $\beta \to 0$, the pressure diverges to infinity, so a modified model considering the air cushion effect is necessary.
On the fluid side, the incompressible (when impact velocity $V \ll c_{water}$) or compressible Navier-Stokes equations are solved. The free surface is tracked using the VOF method.
On the structural side, it is elastoplastic FEM. The relationship between the time scale of the impact load (order of milliseconds) and the natural period of the structure determines whether the response is dynamic or quasi-static.
Wagner's Wedge Water Entry Theory—Why is the "Water Wall" so Hard?
At the moment a ship's bottom slams against the sea surface, the local pressure can be tens to hundreds of times the hydrostatic pressure. H. Wagner first theorized this in 1932, deriving the relation p_max ≈ ½ρ(πV/2tanβ)² for the maximum pressure when a wedge-shaped cross-section enters water at velocity V (β is the half-apex angle). The key point is that "the smaller the entry angle, the more explosively the pressure increases." For example, with β=5° and an entry speed of 5m/s, the peak pressure reaches several MPa. This is why the bottom panels of small high-speed FRP boats dent with a "bang!" sound after repeated slamming. This theory is still used as the starting point for slamming design today.
Physical Meaning of Each Term
- Structure-Thermal Coupling Term: Thermal expansion due to temperature change induces structural deformation, and deformation affects the temperature field. $\sigma = D(\varepsilon - \alpha \Delta T)$.【Everyday Example】Railroad tracks in summer where the gap narrows as the rails expand—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. Temperature differences between hot and cold parts of an engine cylinder block generate thermal stress, potentially leading to cracks.
- Fluid-Structure Interaction (FSI) Term: Bidirectional interaction where fluid pressure/shear forces deform the structure, and structural deformation changes the fluid domain.【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 vibration. Blood flow in the heart and elastic deformation of vessel walls, 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: Feedback loop where Joule heating $Q = J^2/\sigma$ causes temperature rise, and temperature change alters electrical resistance.【Everyday Example】Nichrome wire in an electric stove heats up (Joule heat) and glows red when current flows—temperature rise changes resistance, altering current distribution. Eddy current heating in IH cooking heaters, 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" by combining "air temperature data" and "wind data" in weather forecasting, interpolation is needed if observation points differ—Similarly in CAE coupled analysis, structural and CFD meshes generally do not match, so data transfer (Interpolation) accuracy 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
- Time Scale Separation: When characteristic times of each physical field differ significantly, efficiency can be improved via subcycling
- Interface Condition Consistency: Ensure energy/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 structural 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 slamming CFD-FSI?
Explicit methods are fundamental due to the impact.
| Method | Fluid | Structure | Features |
|---|---|---|---|
| CFD-VOF + FEM Explicit | OpenFOAM/Fluent | LS-DYNA/Abaqus Explicit | General-purpose. FSI often uses weak coupling |
| SPH + FEM | SPH | FEM | Mesh-free. Strong for spray/splashing |
| ALE (LS-DYNA) | ALE fluid | FEM | *CONSTRAINED_LAGRANGE_IN_SOLID |
| BEM + FEM | Panel method | FEM | Efficient but cannot represent spray |
Is the SPH method suitable for slamming?
SPH is mesh-free and can naturally track large deformations and spray of free surfaces. However, numerical oscillations (noise) in the pressure field are prone to occur, so improved versions like Riemann SPH or δ-SPH are used. LS-DYNA has a built-in SPH solver and can directly couple with FEM structures.
Spatiotemporal Resolution
What resolution is needed to accurately capture impact pressure?
Since the slamming pressure peak disappears in about 0.1~1 ms, a time step below 0.01 ms and a spatial mesh below 1~5 mm near the impact surface are required.
| Parameter | Recommended Value |
|---|---|
| Impact surface mesh size | 1~5 mm |
| Time Step | Below 0.01 ms |
| VOF interface resolution | Minimum 5 cells/water film thickness |
| Pressure sampling | Below 0.001 ms |
The "1000x Wall" Between Tank Tests and CFD—The Reality of Slamming Analysis
Numerical analysis of slamming is a continuous struggle with time step issues. The slamming peak pressure occurs within a few milliseconds, so a time step Δt below 0.01ms is needed to resolve it. On the other hand, an actual ship takes several seconds to pass over one wave. This time scale difference is the "1000x wall." In practice, a one-way coupling approach is mainstream, separating "CFD for slamming alone" and "long-term FEA for the entire hull," and mapping the slamming pressure. In comparison with tank tests, a practical sense is that being within ±20% of the peak pressure is acceptable. The main cause of differences between CFD and experiments is often the "air entrainment effect," which can reduce pressure by 10~30%.
Monolithic Method
Solves all physical fields simultaneously as one system of equations. Stable for strong coupling but complex to implement and memory-intensive.
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 for mesh mismatch). Balance between conservation and accuracy is important.
Sub-iteration
Performs sufficient iterations within each coupling step to ensure interface condition consistency. 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 added mass effect (in fluid-structure coupling when structural density ≈ fluid density). If unstable, apply Robin-type interface conditions or 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 recoil 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
Analysis Procedure
What is the procedure for conducting slamming analysis in practice?
A two-stage approach is common.
Phase 1: Full Ship Seakeeping Analysis
- Calculate ship motions using potential flow BEM or CFD
- Identify slamming occurrence conditions (relative velocity, relative displacement)
Phase 2: Local Slamming Analysis
- Extract impact velocity and angle from Phase 1 results
- Calculate impact pressure and structural response using local CFD-FSI or SPH-FEM
Why separate into two stages?
Because ensuring spatiotemporal resolution for slamming at the full ship scale would lead to astronomical computational costs. Phase 1 identifies "when, where, and at what speed" the impact occurs, and Phase 2 analyzes only that local event with high precision.
Verification Data
Are there benchmark problems available for verification?
The water impact problem for wedge shapes is standard.
| Benchmark | Shape | Verification Target |
|---|---|---|
| Wagner theory | 2D wedge | Impact pressure distribution |
| Zhao & Faltinsen experiment | 2D wedge (β=30°) | Pressure time history |
| Aarsnes elastic wedge | 2D elastic panel | FSI response |
| Luo experiment | 3D bow section | 3D pressure distribution |
Container Ship "Whipping"—The Entire Ship Shakes After Slamming
Immediately after slamming occurs, the entire hull vibrates longitudinally like a plucked guitar string, a phenomenon called "whipping." For large container ships, the natural period is about 2~4 seconds, and bending stress spikes of several hundred MPa repeat 5~10 times at the ship's midsection after slamming. In the 2013 MOL Comfort sinking accident (where a container ship broke in two), it was suggested that this whipping stress may have exceeded design assumptions. Following this accident, the International Association of Classification Societies (IACS) revised regulations to explicitly incorporate whipping stress into container ship design criteria (UR S11A). This was a turning point where slamming-whipping coupled analysis became essential for design.
Analogy for Analysis Flow
Have you ever inflated a balloon? In that moment, a sophisticated fluid-structure interaction is actually happening. 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 catchball at each computational 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微小, one-way may indeed suffice. However, in cases like heart valve opening/closing where deformation significantly changes the flow path, one-way coupling is completely inadequate. A rule of thumb is "whether deformation exceeds 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 software is suitable for slamming analysis?
| Tool | Method | Features |
|---|---|---|
| LS-DYNA |