Hydrofoil technology is experiencing renewed momentum, driven by high performance sailing developments and increasing demands for transport efficiency. However, one of the most critical and least understood phenomena in hydrofoil design remains ventilation, particularly during take-off conditions when foils operate close to the free surface. 

The paper “Direct Numerical Simulation (DNS) of Flow Around a Hydrofoil: Benchmark Solution” presents the first step in a research series aimed at understanding the fine-scale flow mechanisms that may trigger ventilation. It establishes a high-resolution numerical benchmark based on Direct Numerical Simulation (DNS) for a representative hydrofoil section. 

Background and Motivation 

Ventilation can be initiated by multiple mechanisms, including tip vortex channelling, laminar separation bubbles, cavitation pockets, and subtle geometric imperfections. Some observed behaviours—such as increased ventilation at lower temperatures or smoother surfaces—suggest that transition and laminar flow extent play a larger role than previously assumed. 

Traditional design tools (such as RANS models) are often insufficient to capture the very small-scale structures potentially responsible for these phenomena. DNS, which solves the Navier–Stokes equations without turbulence modelling, allows resolution of all relevant flow scales and is therefore ideal for establishing a benchmark case. 

Methodology

Geometry and Operating Conditions 

The hydrofoil section analysed corresponds to the tip section of an existing hydrofoil, 3D scanned and extruded into a constant-span computational domain  

Simulation parameters: 

• Reynolds number: 500,000 

• Target lift coefficient: 0.5 

• Angle of attack: 0.71° 

The flow was assumed single-phase (no free surface) to isolate hydrodynamic mechanisms. 

Numerical Setup 

The DNS was performed using a finite-volume approach with: 

• Second-order spatial discretisation 

• Fully implicit second-order time integration 

• SIMPLE pressure–velocity coupling 

• No turbulence modelling  

A locally refined Cartesian grid with prism layers near the wall was employed. The finest grid contained approximately 136 million control volumes. 

A systematic four-level grid refinement study was conducted to assess convergence and discretisation error. 

Grid Convergence and Numerical Accuracy 

The study demonstrates strong convergence trends: 

• Lift and drag coefficients approach grid-independent values. 

• Transition locations stabilise between Grid 3 (~50 million cells) and Grid 4 (~136 million cells). 

• Even coarse grids capture macroscopic flow features reasonably well  

However, fine grids are necessary to accurately resolve: 

• Minimum pressure in vortical cores 

• Turbulent fluctuations 

• Wall shear stress variations 

Notably, the average minimum pressure in the domain was approximately 36% lower on the finest grid compared to the coarsest grid, highlighting the importance of resolution for ventilation related mechanisms. 

Key Flow Features Identified 

Laminar Separation Bubble (Suction Side) 

The suction side exhibits: 

• Laminar separation at ~73.5% chord 

• Kelvin–Helmholtz instability development 

• Vortex tube formation 

• Turbulent reattachment  

These vortex tubes contain very low instantaneous pressures, significantly lower than mean values and largely invisible in time-averaged analyses. 

Transition on the Pressure Side 

Transition occurs naturally without laminar separation bubble formation. The flow develops small scale instabilities evolving into turbulence near the trailing edge. 

Unsteady Low-Pressure Structures 

One of the most significant findings is the presence of travelling vortex tubes in the reattachment region that generate: 

• Instantaneous pressures up to three times lower than surface-averaged minima 

• Strong pressure and shear stress fluctuations  

These structures are potential pathways for air ingestion and may contribute directly to ventilation inception—phenomena that lower-fidelity models would fail to detect. 

Engineering Implications 

The study demonstrates that: 

• DNS can provide a reliable benchmark case. 

• Grid refinement to at least ~50 million cells was necessary for accurate transition prediction in our case.  

• Even relatively coarse DNS grids can capture macroscopic behaviour with acceptable engineering accuracy. 

• Fine resolution is critical for capturing low-pressure vortex cores relevant to ventilation and cavitation.  

This benchmark solution can now be used to evaluate: 

• LES and unsteady RANS approaches 

• Panel methods 

• Different foil geometries 

Conclusions

Although a fully resolved DNS at this Reynolds number would require even finer grids, the present solution is sufficiently converged to serve as a reference case for hydrofoil transition and separation modelling. 

Most importantly, the simulation reveals highly unsteady, low pressure vortex structures that could represent previously underestimated ventilation triggers. 

This paper establishes the foundation for a broader research programme investigating hydrofoil ventilation using progressively lower-fidelity tools validated against this DNS benchmark. 

Download the Full Paper 

This article is only a high-level summary. 

To explore the full methodology, detailed figures, pressure and shear stress distributions, grid studies, and complete reference list: 

Subscribe to our newsletter and download the full paper HERE

Stay updated with upcoming publications in this research series, including comparisons with RANS and LES methods, free-surface modelling, and foil shape sensitivity studies.