Propeller Cavitation FSI
Theory and Physics
Overview of Cavitation FSI
Why is FSI necessary for propeller cavitation?
Cavitation (cavitation phenomenon) causes unsteady pressure fluctuations to act on the propeller blade surface, leading to blade vibration, erosion, and noise. Because the elastic deformation of the blade changes the cavitation pattern, fluid-structure interaction is required.
Governing Equations
What is the mathematical model for cavitation like?
The homogeneous mixture flow model is widely used. It solves the Navier-Stokes equations using the mixture density of the liquid and vapor phases. The cavitation number is,
$p_v$ is the vapor pressure. Phase transition is described by a mass transfer model based on the Rayleigh-Plesset equation. The Schnerr-Sauer model and Zwart-Gerber-Belamri model are representative.
$\alpha$ is the vapor volume fraction, $\dot{m}^+, \dot{m}^-$ are the mass transfer rates for evaporation and condensation.
How is the structural side modeled?
Propeller blades are modeled with shell elements or solid elements. For composite propellers (CFRP, etc.), an anisotropic material model considering the laminate configuration is required. Data transfer at the FSI interface is performed on the wet surface.
Cavitation Number σ—A Single Dimensionless Number Determines Fate
The first thing a propeller designer calculates is the cavitation number σ = (p∞ - pv)/(½ρU²). Bubbles start to form when σ falls below around 1, and when it drops below 0.3, the propeller enters "supercavitation," where the entire surface is enveloped in vapor. Interestingly, torpedoes that deliberately utilize supercavitation (like Russia's "Shkval") have achieved speeds of 340 km/h underwater—over four times the speed of conventional torpedoes. A phenomenon that should be absolutely avoided in civilian vessels becomes a weapon's secret technique depending on how it's used—this reversal of thinking is also the appeal of cavitation theory.
Physical Meaning of Each Term
- Structural-Thermal Coupling Term: Thermal expansion due to temperature changes induces structural deformation, and deformation affects the temperature field. $\sigma = D(\varepsilon - \alpha \Delta T)$. 【Everyday Example】Railroad tracks in summer expand and the gaps narrow—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 from temperature differences between hot and cold sections, 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 winds (Vortex-Induced Vibration)—wind forces shake the structure, the shaken structure alters the wind flow, further amplifying 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 changes alter electrical resistance, creating a feedback loop. 【Everyday Example】The nichrome wire in an electric stove heats up (Joule heat) and glows red when current flows—as temperature increases, resistance changes, and current distribution also changes. Eddy current heating in IH cooking heaters and increased sag in power lines due to temperature rise are also examples of this coupling.
- Data Transfer Term: Interpolation resolves mesh mismatches 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 the observation points differ—in CAE coupled analysis, structural 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): Effective 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 with subcycling.
- 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 structural sides. |
| Data transfer error | Dimensionless (%) | Interpolation accuracy depends on mesh density ratio. Below 5% is a guideline. |
Numerical Methods and Implementation
FSI Handling for Rotating Bodies
How is FSI for a rotating propeller handled?
Set up the rotating fluid domain using a sliding mesh / rotating reference frame. For FSI coupling, the propeller blade surface is set as the interface, and structural analysis is performed in the rotating coordinate system.
In Ansys Fluent, it is standard to switch from steady-state calculation using MRF (Moving Reference Frame) to unsteady calculation with sliding mesh. Sliding mesh is essential to capture the unsteady nature of cavitation.
Cavitation Model Settings
How are the parameters for the cavitation model set?
For the Schnerr-Sauer model, the bubble number density $n_b$ is the key parameter. The default is $n_b = 10^{13}$ /m³, but tuning based on comparison with experiments is sometimes necessary.
| Parameter | Schnerr-Sauer | Zwart-Gerber-Belamri |
|---|---|---|
| Nucleation density/radius | $n_b = 10^{13}$ /m³ | $R_b = 10^{-6}$ m |
| Evaporation coefficient | 1.0 | $F_{vap} = 50$ |
| Condensation coefficient | 1.0 | $F_{cond} = 0.01$ |
| Saturation vapor pressure | Temperature dependent | Temperature dependent |
What about time step settings?
360 to 720 steps per propeller revolution (0.5° to 1.0°/step) is recommended. Since the cavitation collapse process is extremely fast, CFL < 1 is desirable locally.
BEM and CFD—The "Dual-Wielding" Strategy for Propeller Analysis
Boundary Element Method (BEM) and CFD have long been used together for propeller fluid analysis. BEM, based on potential flow theory, is extremely fast and suitable for sweeping hundreds of conditions in the initial design stage. On the other hand, LES (Large Eddy Simulation) in CFD can capture bubble generation and collapse in cavitation, but takes hundreds of CPU hours per case. In actual ship design, the standard approach is a three-stage process: "rough exploration with BEM → detailed investigation with CFD → verification with tank testing." Recently, hybrid methods using BEM results as initial conditions for CFD are increasing, with cases showing total analysis time reductions of 30-40% compared to conventional methods.
Monolithic Method
Solves all physical fields simultaneously as a single 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 allows leveraging 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 interface condition consistency. Residual criteria are scaled based on typical values for 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. When coupling iterations oscillate and fail to converge, it is an adaptive method that automatically adjusts the next correction amount based on the previous correction.
Practical Guide
Analysis Procedure
What is the practical procedure for propeller cavitation FSI analysis?
1. Obtain CAD data for propeller blade geometry.
2. Generate fluid mesh (rotating domain + stationary domain).
3. Create structural FE model (blade only. Hub is rigid).
4. Verify FSI operation under wet-non-cavitating conditions.
5. Enable cavitation model and perform main calculation.
6. Evaluate thrust/torque coefficients and cavitation pattern.
How fine does the mesh need to be?
A $y^+$ of 1 or less on the blade surface is desirable. Areas around the blade tip and leading edge, which are origins of cavitation, should be refined in particular.
| Region | Element Size Guideline |
|---|---|
| Blade surface boundary layer | $y^+ < 1$, 15+ layers |
| Tip vortex region | Chord length / 100 |
| Wake region | Chord length / 50 |
| Far field | Chord length / 5 |
What benchmark problems are available for verification?
The PPTC propeller (Potsdam Propeller Test Case) is the standard benchmark by ITTC (International Towing Tank Conference). Measured values for cavitation pattern, thrust coefficient $K_T$, and torque coefficient $K_Q$ are publicly available.
Propeller Replacement on Large Tankers—On-site "Sound" Diagnosis
Veteran marine engineers say, "You can tell propeller cavitation by sound." Normally, there is a low, gentle rotation sound, but when cavitation occurs, abnormal noises like "crackling" or "grinding" can be heard from the stern. In reality, for a 300,000-ton VLCC (Very Large Crude Carrier), the propeller diameter can be over 9 meters, with a rotation speed of only about 90 RPM. Even so, if cavitation occurs, erosion holes over 1 cm deep can form on the blade surface in three months. Predicting erosion amount through simulation and optimizing dry-docking timing could lead to cost savings of tens of millions of yen per year.
Analogy for Analysis Flow
Have you ever blown up 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... FSI analysis repeats this catch-and-throw at each calculation step.
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 alters the flow path, one-way coupling is completely inadequate. A guideline 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 between countries." 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 results.
Software Comparison
Tool Comparison
What software can be used for propeller cavitation FSI?
| Tool | Cavitation Model | FSI | Features |
|---|---|---|---|
| Ansys Fluent | Schnerr-Sauer, Zwart | System Coupling | Extensive track record in rotating body FSI |
| STAR-CCM+ | Schnerr-Sauer | Built-in FSI | Polyhedral Mesh. Strong in maritime industry. |
| OpenFOAM (interPhaseChangeFoam) | Merkle, Kunz, etc. | preCICE coupling | OSS. Free customization. |
| FINE/Marine (Cadence) | Merkle | FSI capable | Formerly NUMECA. Dedicated to marine CFD. |
| HydroStar + NASTRAN | BEM (Potential) | Mode superposition | Simplified FSI. Suitable for initial design. |
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