Composite materials can be used to fabricate flexible marine propellers, which can improve efficiency and reduce underwater radiated noise. Since the hydrodynamic performance of flexible propellers is determined by their deformation under fluid loading, similarity laws for flexible propeller scaling should account for the deformation in model experiments. This study introduces a non-dimensional parameter to characterise the deformation of flexible propellers and evaluates it through time-domain fluid–structure interaction simulations. A coupled solver combines unsteady Reynolds-Averaged Navier–Stokes equations with a finite-element structural solver. The study focuses on the Wageningen C4-40 propeller geometry under uniform inflow and is limited to isotropic materials.

A set of non-dimensional relations is derived through dimensional analysis, with a form of the Cauchy number expressing deformation amplitude. Validation using a Reynolds–Cauchy similarity approach on two geometrically similar propellers of different diameters confirms consistent deformation and less than 5\% difference in thrust and torque coefficients between model and full-scale propeller. The disparities in performance results are attributed to numerical artefacts in the fluid solver, as the results indicate that the $k-\omega$ SST turbulence model is sensitive to near-wall resolution.

Achieving full-scale Reynolds numbers in propeller test facilities is not feasible, yet simulations demonstrate that flexible propellers are sensitive to viscous forces. The study observes disparities in deformation extent across Reynolds numbers. The deformation of flexible propellers improves flow attachment over the blades. The overall Reynolds-number trends remain similar to the rigid results: thrust coefficients increase and torque coefficients decrease as Reynolds numbers increase. The open-water efficiency depends on both coefficients, and larger Reynolds numbers result in higher efficiencies.

The Froude–Cauchy scaling approach proves suitable for model experiments; however, material availability limits practical implementation. This study indicates that the steady-state deformation is primarily governed by stiffness, with negligible impact from the structural-to-fluid density ratio. In contrast, in unsteady conditions, the structural-to-fluid density ratio affects the modal frequencies, which describe the dynamic behaviour of propeller blades. Particularly, propellers with high skew, rake, or with anisotropic material properties are affected by structural-to-fluid density. Furthermore, this analysis observes that the first blade mode of a zero-skew angle propeller is pure bending, and a 30\% variation in structural density does not alter the natural frequency of this blade. However, coupled bend-twist modes are sensitive to the structural density, which therefore affects blade deformation in unsteady conditions.

In conclusion, the extent of propeller deformation can be controlled by a non-dimensional parameter expressing the ratio of elastic to hydrodynamic forces. For steady open-water conditions, deformation is additionally a function of Reynolds number. In unsteady conditions, the structural-to-fluid density ratio becomes relevant, as it alters coupled bend-twist modal frequencies. Thus, the deformation extent in this regime is, in addition to the Reynolds number, a function of fluid damping and the ratio of natural frequency to revolution rate. This study offers a basis for accurate scaling of flexible propellers in both experimental and computational studies.