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I. Machine Learning in Propeller Design

Machine Learning: The Concept

Machine learning predicts outputs from inputs without explicit programming.

How does it work?

Training — finding patterns in input–output data pairs.
Prediction — estimating outputs for new, unseen inputs.

Main ML Types

Supervised learning — labeled data.
Unsupervised learning — unlabeled data.
Reinforcement learning — rewards and penalties.

What Requires Special Attention

Challenge	Solution
Small dataset size, low data quality	Synthetic data, preprocessing
Overfitting, underfitting, "black-box" behavior	Regularization, cross-validation
High training cost	GPU/TPU acceleration, algorithm optimization

Key ML Applications in Aircraft and Propeller Design

Accurate aerodynamic computation

Surrogate modeling — fast approximate ML-based models.

Large databases

AircraftVerse: 27,714 aerial vehicle design variants for model training.

Physics-Informed Neural Networks (PINN)

Neural networks that respect physical laws.

Multidisciplinary problems & optimization

Acoustics (noise): evolutionary algorithms + ML to balance range, noise, structural strength, etc.

Proposed Architecture of a Propeller Metamodel

Key idea: the model learns to correct BEMT errors using a small number of CFD simulations performed at strategically selected points.

Task: develop a model that computes propeller thrust and torque at BEMT speed with CFD-level accuracy.

Data Generation

Low-fidelity generator In-house BEMT — 2,000 points

High-fidelity generator CADFlo (CFD) — 50 test points

Inputs Geometry, RPM, V

Outputs T, P, efficiency vs. J (RPM)

Multi-Fidelity Architecture

GP_LF = GaussianProcess() # LF(x) over the entire domain GP_delta = GaussianProcess() # Error: HF(x) − ρ·LF(x) HF(x) = ρ·LF(x) + δ(x)

Example output:

T_prediction = 190.5 N
T_std = 2.3 N

Active Learning (Adaptive Infilling)

New high-fidelity points are added through optimization based on the Expected Improvement (EI) criterion:

EI(x) = (η(x) − η_best) · Φ(Z) + σ(x) · φ(Z) Z = (η(x) − η_best) / σ(x), if σ(x) > 0

This implements a "smart error correction" strategy — CFD effort focuses where information gain is highest.

References — Part I

GitHub – peterdsharpe/NeuralFoil: a practical airfoil aerodynamics analysis tool using physics-informed ML, in pure Python/NumPy.
Airfoil Aerodynamic Performance Prediction Using Machine Learning and Surrogate Modeling. Heliyon 10(8), April 2024. DOI: 10.1016/j.heliyon.2024.e29377
Du, X., He, P., Martins, J. R. R. A. (2021). Rapid airfoil design optimization via neural networks-based parameterization and surrogate modeling. Aerospace Science and Technology, 113, 106701. DOI: 10.1016/j.ast.2021.106701
Cobb, A. D. et al. (2023). AircraftVerse: A Large-Scale Multimodal Dataset of Aerial Vehicle Designs. arXiv. DOI: 10.48550/arxiv.2306.05562
Catalani, G., Agarwal, S., Bertrand, X. et al. (2024). Neural fields for rapid aircraft aerodynamics simulations. Sci Rep 14, 25496. DOI: 10.1038/s41598-024-76983-w
A machine learning approach for propeller design and optimization: Part I and Part II. DOI: 10.1016/j.apor.2022.103178; 10.1016/j.apor.2022.103174

II. Toward Training the Model: Propeller Theories and Their Code Implementation

Prerequisites for Model Training

Input/Output datasets

Mathematical model

Experimental data

Computational resources

Propeller Theories

Design: Lifting Line Theory (minimum induced losses).
Analysis: Momentum Theory + Blade Element Theory + J-ref correction.

Assessing Reliability: Validation and Verification

Verification

Checking code against known analytical solutions.
Testing against standard benchmark problems.
Convergence checks with different numbers of sections.
Energy balance verification using multiple methods.

Validation

Evaluating convergence of computational results against experiment.

Propeller Design: Minimum Induced Losses

Inputs

T — thrust, or P — power
V — flight speed (m/s)
D — propeller diameter (m)
N — number of blades
RPM — rotational speed
α₀ — airfoil angle of attack (rad)
+ airfoil aerodynamics

Outputs

Chord distribution c(r)
Pitch angle distribution β(r)
Propeller characteristics: η, C_T, C_P

Design Algorithm

Initial data setup.
Set the reduced circulation ζ.
For each blade element: flow incidence angle → sectional aerodynamics → angle of attack (max L/D) → pitch and chord → induced velocities.
Compute thrust and torque per element, then for the whole propeller.
Select operating mode (thrust- or power-driven).
Correct ζ, apply constraints, deliver output.

Simplifications

Induced velocities are small.
Wake behind the propeller does not contract.
Tip vortices do not interact.
Steady, low-turbulence flow.

Limitations

Low disk loading.
Small number of blades.
High flight altitude.

 Additional capability: evaluation of propeller "optimality" in CADFlo, with applied physical constraints.
    

Propeller Analysis: Blade Element + Momentum Theory + ML Model

Algorithm

Input data and blade discretization.
For each element: induction coefficients → velocities → induced angle of attack → effective angle of attack → aerodynamic coefficients → thrust coefficient → updated induction coefficients.
Convergence check.
Compute thrust and torque per element, then for the entire propeller.

Simplifications and Limitations

Momentum Theory

Rotating disk model.
Constant radial and circumferential velocity components.
Steady, incompressible, inviscid flow.

Blade Element Theory

Two-dimensional airfoil aerodynamics.

Numerical Limitations

Solution may fail to converge when a > 0.4.
Instabilities near the blade tips.

Mathematical Simplifications

Prandtl tip-loss model.
Discrete blade elements.

2D Airfoil Validation: CADFlo & NeuralFoil vs. Experiment

NeuralFoil — a practical airfoil aerodynamics analysis tool using physics-informed machine learning, available in pure Python/NumPy.

Mesh size: 1.8 × 10⁶ cells
AOA 11° — CFD did not converge

Propeller Validation

The full code was validated against experimental data in both:

Dynamic conditions (forward flight) — multiple test cases.
Static conditions — hover/static thrust testing.

Conclusion: Reducing Conceptual Design Margins via ML

Current Error Budget

~25% BEM C_T error

~28% BEM C_P error

~18% CADFlo C_T error

~22% CADFlo C_P error

Plus additional uncertainty: battery degradation, manufacturing tolerances, atmospheric variability.

Impact on UAV Range Predictions

90% of flights (90th percentile = 36 km) — range no more than 36 km.
95% of flights (95th percentile = 33 km) — range no more than 33 km.
Systematic overestimation of expected range by 20–30%.

Example: Requirement "Range ≥ 40 km"

 BEM would indicate the requirement is met with a margin.
The real probability (with all uncertainties) is less than 60%.
For 90% reliability, the nominal design range must be set to ≥ 55 km.
 

🎯 Result: a trained multi-fidelity ML metamodel correcting BEMT against CFD enables up to 12–15% mass savings of the propulsion system by reducing excessive design margins.

References — Propeller Design and Analysis

Patterson, M. D. (2016). Conceptual Design of High-Lift Propeller Systems for Small Electric Aircraft.
Epps, B. P., Kimball, R. W. (2013). Unified Rotor Lifting Line Theory. Journal of Ship Research, 57, 181–201.
Larrabee, E. (1979). Design of Propellers for Motorsoarers. NASA Publication: The Science and Technology of Low Speed and Motorless Flight.
Adkins, C., Liebeck, R. (1994). Design of Optimum Propellers. Journal of Propulsion and Power.
Betz, A., Prandtl, L. (1919). Schraubenpropeller mit geringstem Energieverlust. Göttinger Nachrichten, pp. 193–217.
Goldstein, S. (1929). On the Vortex Theory of Screw Propellers. Proc. R. Soc. Lond., 123(792), pp. 440–465.
Theodorsen, T. (1948). Theory of Propellers. McGraw-Hill Book Company, 1st ed.

Acknowledgments

The work also references contributions related to a propeller design and analysis tool developed on the Sboard platform:

 A. L. Ivanov, A. P. Kuznetsov — JSC "UEC-Aviadvigatel"
 Description of a software tool for propeller design and analysis on the Sboard platform. 