Propeller CFD Analysis
Theory and Physics
Fundamentals of Propeller Performance
Professor, when predicting ship propeller performance with CFD, what parameters do we look at?
Propeller performance is evaluated by the thrust coefficient $K_T$, torque coefficient $K_Q$, and efficiency $\eta_0$.
Here, $T$ is thrust, $Q$ is torque, $n$ is rotational speed, $D$ is propeller diameter, and $J = V_A/(nD)$ is the advance coefficient. The open water characteristics ($K_T$-$J$ curve and $K_Q$-$J$ curve) represent the fundamental performance of the propeller.
What results are compared against in CFD?
They are compared against towing tank test data for standard propellers (e.g., DTMB 4119, KCS) from ITTC (International Towing Tank Conference) or SMPF (Ship Performance and Propulsion Committee). A difference of within 2% between CFD and experiment for $K_T$ and $K_Q$ is considered good, and within 5% is considered practical.
Numerical Modeling Approaches
What modeling techniques are there for propeller CFD?
There are mainly three. (1) MRF (Moving Reference Frame): Steady-state calculation using a rotating coordinate system around the propeller. Lowest cost. (2) Sliding Mesh (Rigid Body Motion): Unsteady calculation where the propeller is actually rotated. Accurate for hull-propeller interaction. (3) Overset Mesh (Chimera Method): Handles motion by overlapping background and component meshes. Widely used in STAR-CCM+.
Which one should be used?
MRF is sufficient for calculating open water characteristics. Sliding Mesh or Overset is necessary for a propeller in the ship's wake (self-propulsion condition). For cavitation analysis, unsteady Sliding Mesh is standard.
History of Ship Propeller Theory — Rankine-Froude Momentum Theory (1865)
The first theoretical description of propeller thrust was the "Actuator Disk Theory" by William Rankine (1865) and W.J.M. Froude (1878). It idealized the propeller as a virtual, infinitely thin disk imparting momentum to the flow, deriving the simple formula T = ρA(V+va)×2va (where va is induced velocity). This theory captures the essence of thrust generation physics while being simple to calculate, and is still used for initial power and efficiency estimates in design. The theoretical upper limit of propeller efficiency (Betz limit η=1/(1+va/V)) is also derived from this theory and shares the same mathematical structure as the Betz coefficient for wind turbines (η_max≈59.3%). While 130 years later, CFD enables detailed airfoil analysis by fully modeling the blade, Rankine's insightful simplicity is still recounted in educational settings.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine turning on a faucet. At first, water comes out spluttering and unstable, but after a while, the flow becomes steady, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or flow fluctuations each time an engine valve opens/closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled"—meaning this term is set to zero. Since this drastically reduces computational cost, starting with a steady-state solution is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier" air transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is transport by flow, conduction is transfer by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re number flow, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? Right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (less dense) than its surroundings, so buoyancy pushes it up. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated winter room.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Mesh Design
What should I be careful about when generating mesh for a propeller?
The boundary layer mesh near the blade surface is most critical. Place prism layers with $y^+ \approx 1$ to directly resolve the wall with the SST k-ω model. Local refinement is needed at the leading and trailing edges; at least 10 cells should be placed relative to the leading edge radius. The mesh near the tip also needs to be fine to capture tip vortices.
What's a guideline for cell count?
500,000 to 2 million cells per blade is standard. Using periodic boundary conditions to calculate only one blade can reduce cost to 1/$Z$ (where $Z$ is the number of blades). However, for self-propulsion calculations in a ship's wake, the inflow is non-uniform, so a full-blade model is essential. STAR-CCM+'s Trimmed Mesh or Fluent's Polyhedral Mesh offer a good balance of quality and automation.
Cavitation Analysis
What are the CFD models for cavitation?
The Schnerr-Sauer model and Zwart-Gerber-Belamri model are mainstream. They solve gas-liquid two-phase flow using the VOF method, modeling bubble generation (evaporation) in regions where local pressure falls below vapor pressure and bubble collapse (condensation) in regions where pressure recovers.
In Fluent, enable via Multiphase > VOF > Schnerr and Sauer. In STAR-CCM+, set the Cavitation Model in Physics. In OpenFOAM, use interPhaseChangeFoam (which has the Schnerr-Sauer model built-in).
What points require attention in cavitation analysis?
(1) Very small time steps are required ($\Delta t \sim 10^{-5}$ to $10^{-4}$ s). (2) The mesh must be fine enough to resolve the cavity thickness (a few mm). (3) Adding the Reboud correction (density correction) to the SST k-ω turbulence model improves the behavior of sheet and cloud cavitation.
Why is the "Schnerr-Sauer" Cavitation Model Popular?
When analyzing cavitation in propeller CFD, the Schnerr-Sauer model is the most commonly used among OpenFOAM users. While there are several choices like the Zwart-Gerber-Belamri model, Schnerr-Sauer is favored for its ease of use—it requires no tuning of evaporation/condensation coefficients and works once the bubble nucleus density is set. However, for detailed reproduction of actual cavitation collapse (implosion), LES or high-density meshes are needed, and calculations exceeding 1 billion cells are not uncommon. In practice, maintaining mesh quality where "6 uniform prism layers across the entire blade surface" don't collapse is the first major hurdle.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-order accurate, but numerical oscillations occur for Peclet number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy equations drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2–0.3, Velocity: 0.5–0.7 are typical initial values. If diverging, lower the relaxation factors. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this catchball is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream. Although first-order accurate, it is highly stable because it correctly captures flow direction.
Practical Guide
Self-Propulsion Analysis (Self-Propulsion)
What kind of simulation is self-propulsion analysis?
It simulates the state where the propeller propels the ship by placing both the hull and propeller in the same computational domain. By finding the self-propulsion point where thrust and resistance are balanced, the required horsepower of the actual ship is predicted.
What is the calculation procedure?
(1) First, perform resistance calculation for the hull alone (bare hull resistance). (2) Next, separately calculate the propeller's open water characteristics. (3) Perform a Sliding Mesh calculation for hull+propeller, adjusting rotational speed to find the point where thrust = resistance. STAR-CCM+'s Propeller Performance feature automatically searches for the self-propulsion point.
How is the effect of the ship's wake evaluated?
Calculate the wake distribution (wake fraction) at the propeller plane: $w = 1 - V_A / V_S$. Evaluate both nominal wake (without propeller) and effective wake (with propeller). The non-uniformity of the wake causes propeller fluctuating loads, vibration, and cavitation, so detailed analysis of the circumferential distribution is important.
Benchmarks for Verification
Is there data available for verifying propeller CFD?
Let me list some representative benchmark data.
| Propeller | Blades | Features | Data Source |
|---|---|---|---|
| DTMB 4119 | 3 blades | For non-cavitation verification | ITTC |
| PPTC VP1304 | 5 blades | Cavitation/pressure fluctuation | SVA Potsdam |
| KCS |