Equivalent Roughness Height for CFD Studies on NACA Airfoils

Technical Summary: Equivalent Roughness Height for CFD Studies on NACA Airfoils

Problem Statement
Accurate prediction of aerodynamic performance degradation due to surface roughness is critical for wind turbine blades and other airfoil-based systems. Environmental factors such as dust, ice, insect impact, and erosion alter the leading edge, disrupting the viscous sublayer and increasing skin-friction coefficients. While experimental data, particularly from Abbott and Doenhoff's Theory of Wing Sections, provides a benchmark for "standard roughness" (defined by specific carborundum grain sizes and coverage percentages), translating these physical conditions into Computational Fluid Dynamics (CFD) simulations presents a challenge. CFD turbulence models typically require an equivalent sand-grain roughness height ($k_s$), whereas experimental roughness is often defined by geometric characteristics or surface coverage. There is no direct conversion between the physical grain size used in wind tunnel tests and the uniform numerical roughness parameter required by solvers like ANSYS Fluent.

Methodology
The study proposes a procedure to estimate the equivalent roughness height ($k_s$) based on the percentage of surface area covered by particles. The methodology proceeds as follows:

1. Data Synthesis: The authors bridge the gap between Abbott and Doenhoff's experimental definitions and the theoretical work of Schlichting and Coleman. Schlichting provided tables relating particle geometry and distribution to equivalent roughness, while Coleman et al. later corrected these values, showing that uncorrected estimates could be up to double the actual equivalent roughness.
2. Correlation Development: By analyzing the particle distributions in Schlichting's corrected data (specifically spherical particles), the authors calculated the percentage of surface area covered for each case. A linear relationship was established between the covered surface area percentage and the ratio of equivalent roughness to geometric roughness ($k_{s,eq}/K$). The resulting correlation is:
   $$k_{s,eq} / K = 19.19970 \cdot \text{Covered Surf. [\%]} - 0.12448$$
3. CFD Setup: Simulations were conducted using the $k-\omega$ SST turbulence model with roughness corrections (Hellsten and Laine) for three NACA airfoils: NACA 0012, NACA 2418, and NACA 633-418. The simulations replicated the conditions of Abbott and Doenhoff's wind tunnel tests: a Reynolds number of approximately $6 \times 10^6$ and low turbulence intensity (0.1–0.2%).
4. Calibration: Preliminary simulations were run with three coverage scenarios (5%, 7.5%, and 10%) to determine which equivalent roughness value best matched the experimental lift and drag polars for each specific airfoil geometry.

Key Results

• Coverage Dependence: The study found that the equivalent roughness required to match experimental data varies by airfoil geometry. For the NACA 0012, a 10% coverage area provided the best fit; for the NACA 2418, 5% was optimal; and for the NACA 633-418, 7.5% was the best match. This indicates that the effective roughness is not solely a function of the nominal contamination but also the airfoil's aerodynamic characteristics.
• Accuracy: Within the low-to-moderate angle of attack range (approximately -6° to 5°), the CFD simulations using the derived $k_s$ values reproduced experimental drag coefficients with differences generally below 5% for the NACA 2418 and NACA 633-418. The NACA 0012 showed slightly higher deviations, reaching up to 10% in certain conditions.
• Flow Regime: Preliminary analysis confirmed that all studied configurations operate in the fully rough regime ($k_s^+ > 140$), validating the use of equivalent sand-grain roughness models where viscous sublayer interactions are disrupted.
• Stall Limitations: As expected with steady RANS simulations, discrepancies between numerical and experimental results increased significantly near the stall region. The authors attribute this to the inherent limitations of the $k-\omega$ SST model in capturing large-scale unsteady separation, rather than deficiencies in the roughness estimation procedure.

Significance and Claims
The paper claims to provide a practical framework for translating experimentally defined surface contamination (characterized by particle coverage) into CFD-compatible roughness parameters. By establishing a linear correlation between surface coverage percentage and equivalent roughness height, the methodology allows researchers to bypass the need for detailed geometric descriptors of contamination when modeling aerodynamic degradation.

The authors emphasize that while the correlation was derived from a limited set of experimental data points and assumes regular particle distribution, it offers a viable approach for engineering-level predictions. The study validates that the proposed procedure can reproduce drag polar curves with reasonable accuracy for standard roughness conditions, enabling more realistic numerical simulations of performance loss due to particulate accumulation or surface erosion in wind energy and aerospace applications.

Limitations
The authors explicitly note that the correlation is based on a limited number of data points and assumes regularly distributed spherical particles, which may not reflect the irregular shapes and clustering of real-world contamination. Furthermore, the reliance on steady RANS simulations limits predictive accuracy in the stall region. Future work is suggested to incorporate broader experimental datasets, investigate irregular contamination patterns, and potentially utilize higher-fidelity simulation methods like hybrid RANS-LES.