### Cavitation Analysis using FreSCo+

**Engineering cavitation simulations** using viscous CFD approaches are usually based on VOF-related so called **Euler-Euler**** techniques. VOF-methods are fairly popular for their efficiency. They consider the two-phase flow as a mixture of a liquid and vapour phase which share the kinematic field. The local vapour fraction is obtained from a transport equation conveying the immiscible properties of the involved phases. Cavitation physics is reduced to a**** modelled mass-transfer rate** between water and vapour that occurs as a source in the volume-conservation equation. Different suggestions to mimic the mass-transfer following a reduced Rayleigh-Plesset equation, i.e. **Sauer, Kunz, Singhal, Senocak, Shyy, Zwart**, are at the users choice. All models can be supplemented by an extension to account for more detailes.

Propeller-flow simulations are typically performed using unstructured (strictly hexahedral-cell) meshes featuring local grid refinement as outlined below.

Illustration of typical grid arrangement applied to the PPTC Case of SVA Potsdam.

Simulations can be performed for all types of cavitation and maintain stable even for massively cavitating flows at very high density ratios due to the semi-implict coupling to the continuity equation.

Simulated vapor contour of massively cavitating flow around the PPTC propeller.

Owing to the importance of turbulence in cavitation physics, both approaches can be combined with statistical (**U-RANS**) closure approaches, scale-resolving techniques (**LES**) or non-zonal hybrids (**DES**) of such methods.

Cavitating propeller flows are difficult to simulate since they are governed by the influence of concentrated vortices, e.g. hub and tip vortices. The predictive accuracy particularly hinges on a fair simulation of the downstream evolution of the primary vortices. An **invariant based vortex-confinement** strategy is also available to support the resolution of vortex cavitation. The latter can significantly improve the predictive performance at moderate computational costs as indicated by the below given cases of a cavitationg 3D foil and the PPTC propeller which features cavitation-related thurst reductions (J=1.019).

Influence of invariant based vortex confinement (left: experiments; right: FreSCo^{+} computation).

Predicted vapour volume around the PPTC propeller (Case1 - J=1.019).

VOF approaches do not take into account local (inhomogeneous) water properties and are restricted to simplified vapour-bubble dynamics. The mass-transfer models rely on empirical assumptions casted into the model parameters, e.g. vaporization and condensation coefficients. The respective parameter values are unfortunately not case-independent and interact with the employed computational scheme. Mind that coefficient values found in the literature often span 4-5 orders of magnitude and their respective choice interacts with the applied turbulence closure. The example included below again refers to the PPTC using different coefficents sets reported for the Zwart model. Results reported for the higher coefficient values (left) are in much better agreement with experiments as regards the root cavitation predicition.

Demonstration of the influence of empirical constants using the Zwart mass-transfer model for the PPTC propeller at J=1.269 (Suction side images for case 2 - top: experiments; bottom-left: higher coefficient values; bottom-right: lower coefficient values).

Even from an engineering point of view, a more sophisticated cavitation simulation technique is of interest. Accordingly we developped an improved cavitation-simulation model based upon more detailed bubble dynamics (**Euler-Lagrange** approach). Supplementary to an Eulerian mixture phase, separate equations for the size and the momentum are solved for a large number of cavitation nuclei/bubbles, composing the discrete vapour phase. The Euler-Lagrange model has been implemented into FreSCo^{+} and parallelised using an efficient hybrid MPI-OpenMP algorithm which features above 70% parallel efficiency for 150 MPI processes each running 50 OpenMP threads (50kcells/process; 3000 bubbles/thread).

The novelty of the present approach refers to the application of fully-coupled Euler-Lagrange technique to turbulent engineering flows. The strategy allows to account for inhomogeneous and transient water-quality effects such as nuclei spectra and variable non-condensable gas content.As depicted by comparison of the simulated cavitation extent above a hydrofoil displayed below, the model is able to resolve the influence of the size distribution of the nuclei found in the water.

Predicted vapor-volume contours above a hydrofoil using an Euler-Euler (Zwart) model (left) and an Euler-Lagrange (EL) model (centre/right). The black line indicates experimental observations of the cavitation extent. The right picture refers to EL-results obtained for the introduction of measured nuclei-size spectrum, the centre pircture refers to EL-results for the same nuclei volume using smaller nuclei.

The final example refers to animations of the predicted cavitation pattern for the PPTC case 1 at J=1.019 already shown above. As indicated by the experiments, pronounced tip-vortex cavitation, in line with root cavitation around mid-chord and hub cavitation, is also observed in the detailed Euler-Lagrange approach.