Multi-criteria decision-making for optimal airfoil selection in small wind turbines at low Reynolds numbers

Abstract

This research provides a data-driven methodology to choose optimal NACA airfoils suitable for small HAWTs operating at low Reynolds numbers. 36 airfoils at four different Reynolds numbers (421116–673866) were analyzed based on two important parameters: lift-to-drag ratio (Cl/Cd) and its standard deviation. Objective weighting was ensured before evaluation based on three multi-criteria decision-making methodologies: TOPSIS, AHP, and VIKOR. Borda Score was used to aggregate the rankings, and the consistency of the model was checked using Spearman’s correlation. NACA 4521, 4518, and 4421 emerged as high-performance candidates in all scenarios, with high aerodynamic performance and stability according to the results. The method developed in this research is a replicable method to choose the proper airfoil and will enhance the direction towards further optimization incorporating the structural and manufacturing aspects as well.

Keywords

wind energy airfoil XFoil Lift drag aerodynamic performance small wind turbine MCDM

Introduction

The need for energy has grown dramatically in recent years. Electricity is now essential for human survival (Gopinath and Meher, 2018). Over the last few decades, urbanization and rapid economic growth have been the primary drivers of increased electricity demand. By 2030, worldwide electricity consumption is predicted to reach a total of 30621 TWh in a stated policies scenario, and 31752 TWh in an announced pledges scenario, while in 2050 the annual demand could reach 43672 TWh or 53810 TWh based on either the stated policies scenario or announced pledges scenario, respectively (IEA, 2022). Fossil fuels have been the main energy source used to meet the energy demand of humanity for many years, but they have numerous negative impacts on the environment; hence, the focus of energy generation has shifted towards clean and renewable sources; in recent years, renewable energy has gained global attention for its potential to address both current and future energy needs. Among all the available clean energy sources, wind energy has emerged as one of the most promising renewable energies due to its low cost, sustainability, and low environmental impact (Hsu et al., 2014; Liu, 2016).

Wind energy conversion turbines for small scale are a focus for engineers today because of initiatives aimed at lowering greenhouse gas emissions and addressing climate change. They provide an attractive option for clean energy production in targeted locations. Small wind turbines provide higher flexibility in terms of wind speed conditions and area compared to large turbines (Singh et al., 2012); small wind turbines can be adopted for rural, remote, and residential areas for grid-based or off-grid electricity generation. This enables applications across a wider geographic spread, instead of depending exclusively upon favorable wind conditions (Yang et al., 2019).

Small horizontal-axis wind turbines (SHAWTs) with a rotor swept area of less than 200 m² follow the same aerodynamic principles as larger HAWT systems, per International Electrotechnical Commission regulations (IEC, 2013). The small blade radius in the former applications results in significantly lower Reynolds numbers (Re) across the blade span (Shah et al., 2012). High Reynolds number flows over airfoils result in a faster transition from laminar to turbulent flow within the velocity boundary layer compared to laminar separation; this early transition prevents the unfavorable aerodynamic effects associated with the latter condition (Giguere and Selig, 1997). Also, the design of efficient and effective rotor blades for small horizontal-axis wind turbines is associated with unique and challenging criteria, mainly due to the lower Reynolds number in the small-scale application framework. Operating at low Reynolds numbers introduces aerodynamic challenges, which are unique to the small-scale rotors, including lower lift with high drag and distinctive stalling characteristics (Winslow et al., 2017). In order to handle and address the challenges linked with low Reynolds Numbers and wind speeds operating conditions, multiple research studies have been conducted, and conventional airfoils and bioinspired geometries for SHAWTs have been introduced to achieve better aerodynamic performances (Robles et al., 2023; Singh et al., 2012).

The airfoil is one of the most significant elements of wind turbines, having direct implications for aerodynamic efficiency, energy production, and system stability (Mamadaminov, 2013); for wind energy, particularly in small horizontal-axis wind turbines (HAWTs) at low Reynolds numbers, the choice of optimal airfoil profile is critical to provide high lift-to-drag (Cl/Cd) values, delayed stall onset, and turbulence tolerance (Seifi Davari et al., 2023). According to Giguère et al., conventional airfoils tend to underperform at low Reynolds numbers due to premature boundary layer separation and laminar-turbulent transition deficiencies, reducing the lift and enhancing the drag (Giguère and Selig, 1998); Hansen highlighted the fact that even a 5% rise in Cl/Cd might lead to a 10–15% improvement in turbine efficiency (Hansen, 2008). Moreover, Grasso et al. mentioned the high noise emission, fatigue load, and reducing the working life for small turbines are due to inappropriate airfoil selection (Grasso et al., 2017); Selig and McGranahan conducted extensive low-Re wind tunnel testing of airfoils and found that certain NACA and S-series airfoils exhibited superior aerodynamic behavior due to favorable pressure distributions and smoother stall characteristics (Selig and McGranahan, 2004); moreover, Motta et al. reported that airfoil geometry also affects structural behavior under gust loads and that thicker profiles often yield better mechanical performance at the cost of slight aerodynamic loss (Motta et al., 2015). Also, Yossri et al. studied and investigated the effects of airfoil types and rotor size on the lift-to-drag ratio. In their analysis and study, they found that NACA 4412 achieves and yields the highest lift-to-drag ratio among all the airfoils (Yossri et al., 2021).

Traditionally, airfoil selection has depended significantly on empirical testing and past performance records. With the advancement of computational tools, simulation-based selection methods have gained prominence; some tools are XFOIL, QBlade, and XFLR5. These software model lift, drag, moment, and flow separation behavior for all possible angles of attack and all possible Reynolds numbers. Salgado Fuentes et al. presented a systematic approach to SWT airfoil selection by using XFOIL to calculate the glide ratio and aerodynamics consistency under various conditions (Salgado Fuentes et al., 2016); in addition, both XFLR5 and CFD were employed by Sang et al. to design and optimize a modified airfoil named VAST-EPU-S1010 under extreme low wind speed conditions (Sang et al., 2024). Krishnan and Roy showed that small geometrical changes like trailing thickness and camber have critical implications on low Re performance characteristics, demonstrating the need for geometrical tuning (Krishnan and Roy, 2022); Allen Gardner and Selig (2003) stressed the implementation of Genetic Algorithm (GA) and optimization techniques for inverse airfoil design with minimized human bias in selecting the best design from the pool of solutions (Allen Gardner and Selig, 2003).

With many conflicting criteria, Cl/Cd, structural thickness, stall angle, and cost, Multi-Criteria Decision-Making (MCDM) methods have been indispensable in airfoil selection. AHP, TOPSIS, and VIKOR are the most widely accepted (SARI, 2018). AHP, developed by Saaty in 1980, decomposes complex decisions into hierarchies and uses pairwise comparisons to derive weights; in airfoil selection, AHP allows for systematic prioritization of factors like aerodynamic efficiency versus structural robustness (Kassa et al., 2024). Additionally, entropy-based AHP has been used to reduce subjective bias in decision weights in many applications in the wind energy field (Rehman et al., 2019). TOPSIS, developed by Hwang and Yoon in 1981, ranks alternatives with the intent to minimize distance to the ideal and maximum distance from the worst situation. Batu et al. in 2024 ranked and analyzed airfoils with regards to Cl/Cd and performance variance using entropy-weighted TOPSIS (Batu et al., 2024); Cheng et al. in 2019 have even utilized TOPSIS for evaluating aerospace components, demonstrating its strength in data-heavy environments (Wang and Chang, 2007). VIKOR, proposed by Opricovic (1998), uses compromise ranking to minimize the regret of failing to achieve the optimum performance in all criteria (Opricovic, 1998). It takes both group utility as well as individual regret into consideration, which is very important to balance the trade-offs between lift and structural penalties. Shukla et al. (2020) applied VIKOR to select the material for turbines, indicating it surpasses TOPSIS under high criterion conflict (Raju et al., 2024). Kim and Ahn further confirmed that VIKOR produces consistent results even under incomplete data sets (Kim and Ahn, 2019). AHP-TOPSIS and AHP-VIKOR hybrid or combined approaches have also been popular. Xu et al. employed a hybrid MCDM approach employing both the AHP and VIKOR methodologies in site selection of a wind farm in China (Xu et al., 2020). Furthermore, Konstantinos et al. employed both of TOPSIS and AHP to select a wind farm installation (Konstantinos et al., 2019). Multi-criteria selection methods are increasingly important for small-scale wind turbine blade design. When it comes to airfoil selection criteria, MCDMs have the advantage of simultaneously considering many factors. These criteria may include factors such as lift-to-drag ratio, stall behavior, structural integrity, and manufacturing feasibility. MCDM techniques analyze and rank airfoil designs using mathematical models and decision matrices. This holistic method provides a thorough review of airfoil choices, resulting in better informed and effective decision-making.

The selection of the airfoil is one of the most important steps to conduct carefully while designing or optimizing the blade of a small horizontal-axis wind turbine. Within this in mind, this study emphasizes on evaluating an array of 36 NACA airfoils with different properties and geometries to identify their compatibility with a small horizontal-axis wind turbine under different Reynolds numbers; to conduct this assessment, XFoil was used to simulate the chosen airfoils and collect their characteristics such as the lift and drag rations; then, with an extensive analysis and ranking methodology based on two main criteria which are the lift-to-drag ratio and the aerodynamic stability of these airfoils. Employing the entropy weighting followed by a normalization process, each criterion is used to rank the 36 airfoils based on the three chosen methodologies, TOPSIS, AHP, and VIKOR. Subsequently, the Borda Score approach is employed to aggregate the results obtained from the three techniques into a final ranking of the airfoils. This approach has been tailored specifically to cater to the unique requirements of small horizontal-axis wind turbines operating at low Reynolds number conditions, which ensures a solid, reliable, and methodological selection or ranking process.

Methodology

Airfoil dataset selection and rationale

Before digging into this study, it is important to highlight that the lift, drag, and lift-to-drag ratio data were generated from XFoil, which is a solid and reliable software used to simulate the aerodynamic parameters of airfoils across different angles of attack and Reynolds numbers; nevertheless, operating in low Reynolds numbers is associated with potential limitations and constraints due to its inviscid-viscous coupling and boundary layer transition assumptions.

Although the maximum lift coefficient (Cl) is often a key parameter in aerodynamic assessments, it was deliberately omitted from this study’s multi-criteria analysis. This choice stems from the fact that Cl alone does not reflect the associated drag losses, which are particularly major at low Reynolds numbers where laminar separation and instability are common. Instead, the lift-to-drag ratio (Cl/Cd) is a more meaningful measure of aerodynamic efficiency, an essential factor for wind turbine performance. Also, the variability of Cl/Cd across different angles of attack was examined to evaluate how stable the aerodynamic characteristics are under fluctuating wind conditions.

The aerodynamic performance of a wind turbine is very much a function of airfoil geometry along its blades, particularly for small horizontal-axis wind turbines (HAWTs) at low Reynolds number regimes. A dataset of 36 NACA, as indicated in Table 1, airfoils was compiled for undertaking a comprehensive aerodynamic screening and performance assessment. The selection in this study focuses entirely on the NACA airfoil family to ensure dataset uniformity, public availability, and compatibility with the main simulation tool used, which is XFoil; the exclusion of other airfoil families such as the SG is intentional since many airfoil profiles lack reliable open-source datasets or require specific transition control mechanisms during simulation, which may introduce inconsistencies in comparative analysis. The chosen airfoils comprise different profiles from the 4-digit, 5-digit, and symmetric NACA series with diversity in camber, thickness, and pressure distribution properties. This broad choice was made to cover various aerodynamic characteristics: for instance, cambered profiles NACA 4521, 25028, and 2415 are reputed for higher lift at smaller attack angles, while symmetric airfoils NACA 0012 and 0021 provide structural simplicity and two-way operation, but with inferior lift efficiency. They were selected for their public availability within aerodynamic databases, as well as compatibility with simulation tools like XFOIL, which is widely used for low Reynolds number scenarios; the importance of selecting airfoils carefully has been stressed across various studies. Giguère and Selig found that at Reynolds numbers less than 1,000,000, laminar separation bubbles become prevalent, drastically deteriorating the lift-to-drag ratio (Cl/Cd) and performance as a whole (Giguère and Selig, 1997). Henrik Hansen also mentioned that even a slight improvement in aerodynamic efficiency for flow regimes at these ranges has a major potential for energy conversion improvement within small wind systems (Henrik Hansen, 2017). From a structural perspective, Mouhsine et al. stressed that airfoil thickness not only adds to aerodynamic pressure resistance but also to mechanical stiffness and fatigue life for wind turbine blades (Mouhsine et al., 2018).

Table 1.

List of airfoils selected for the study.

Airfoils
NACA_0024	NACA_25021	NACA_4412	NACA_4521
NACA_25015	NACA_2415	NACA_2424	NACA_4518
NACA_23012	NACA_24021	NACA_2421	NACA_4421
NACA_23015	NACA_23021	NACA_25028	NACA_4524
NACA_24015	NACA_25012	NACA_25026	NACA_4418
NACA_0021	NACA_25018	NACA_25024	NACA_4424
NACA_0018	NACA_23018	NACA_2418	NACA_4515
NACA_0015	NACA_24012	NACA_2412	NACA_4415
NACA_0012	NACA_24018	NACA_23024	NACA_4512

Aerodynamic performance and stability

The initial step of the computation method was the determination of each airfoil’s optimal angle of attack based on its maximum lift-to-drag ratio of Cl/Cd. This is a basic indicator of aerodynamic performance, especially for low-Reynolds-number flow conditions common on small wind turbines. For each airfoil, the Cl/Cd was computed versus AoA, angle of attack, without the use of any interpolation or smoothing function applied. Thus, the result was not affected by artifacts of curve fitting, and the actual peak aerodynamic efficiency from the dataset was the reflected outcome. In addition to identifying peak performance, each airfoil’s performance stability was measured by determining the statistical variance of the Cl/Cd across all attack angles, as indicated in equation (1). A smaller variance indicates steady performance with a wider operation range, which is an advantage when operating within a turbulent wind situation where AoA is constantly changing. This two-part analysis (optimal performance + variance) provided a better aerodynamic assessment.

σ^{2} = \sum_{i = 1}^{N} {(\frac{{C l}_{i}}{{C d}_{i}} - \bar{(\frac{C l}{C d})})}^{2}

(1)

Ranking approach

To ensure objectivity and consistency across all three decision-making methods utilized within this research, TOPSIS, AHP, and VIKOR, an entropy weighting procedure was used to assign relative weights to each evaluation criterion based on their importance. Unlike subjective weighting methods that draw on expert judgment or pairwise comparisons, entropy weighting has its origins within the realm of information theory and takes the intrinsic variability of the data into account to calculate the contribution of each criterion to a decision-making process. Developed originally by Shannon and later generalized for multi-criteria decision-making applications, entropy quantifies the uncertainty or randomness within a dataset (Saraiva, 2023). A criterion that exhibits higher variability across alternatives is treated as carrying higher discriminant power and accordingly is assigned a higher weight, and uniformly distributed criteria are treated as being less informative and are given correspondingly low weights (Saraiva, 2023).

In the study context, the normalized values of two criteria, maximum Cl/Cd ratio and Cl/Cd variance, were subjected to entropy weighting. The decision matrix was initially normalized using a linear min-max method, after which the entropy Ej for each j was calculated using Formula (2) (Saraiva, 2023).

E_{j} = - \frac{1}{\log (n)} \sum_{i = 1}^{n} p_{i j} \log (p_{ij} + ε)

(2)

where p_ij is the proportion of the ith alternative under the jth criterion and n refers to the number of alternatives; to ensure numerical stability during the calculation of the entropy, especially on extreme cases or when logarithmic operations on zero values could lead to undefined behavior such as log(0) a small constant epsilon equal to 10⁻¹²,

ε

was employed to maintain stability in extreme cases. The degree of diversification for each criterion was computed as in equation (3), and the final normalized weight was calculated as indicated in equation (4).

d_{j} = 1 - E_{j}

(3)

w_{j} = \frac{d_{i}}{\sum_{j = 1}^{m} d_{j}}

(4)

where m is the number of criteria. These weights derived from entropy were applied uniformly across the three MCDM methods so that the comparative importance of each criterion was dependent only on the data structure and not on subjective assumptions. This adds to the methodological rigor, reproducibility, and transparency of the airfoil ranking for applications where expert bias or inconsistency has the potential to skew results (Odu, 2019).

A multi-criteria decision-making (MCDM) methodology was used to rank the 36 NACA airfoils. TOPSIS, a technique for ordering preference by similarity to an ideal solution, was the first MCDM technique used to analyze alternatives based on their geometric similarity to an ideal solution (optimal) and a nadir solution (worst). TOPSIS is especially suitable for reconciling trade-offs involving conflicting criteria like aerodynamic efficiency and performance stability (Madanchian and Taherdoost, 2023). In order to remove the scale difference effect across criteria, min-max scaling was applied to all the values. Benefit criteria like the maximum of Cl/Cd were scaled so that they take the highest value as 1, whereas the cost criteria like the variance of Cl/Cd were scaled so that the minimum value takes the value 1. This makes them comparable without distorting the relative sizes of the measures (Vafaei et al., 2016). Ideal best and ideal worst values for every criterion were determined. Euclidean distance from each airfoil with respect to the ideal best and ideal worst was calculated. Next, TOPSIS scores for individual airfoils were calculated using equation (5) (Madanchian and Taherdoost, 2023).

TOPSIS {score}_{i} = \frac{D_{i}^{-}}{D_{i}^{+} + D_{i}^{-}}

(5)

where

D_{i}^{-}

refers to the distance to the ideal worst, and

D_{i}^{+}

indicates the distance to the ideal best; with the highest score linked with the best performance overall (Madanchian and Taherdoost, 2023).

In addition to complementing the TOPSIS analysis for a sound decision-making process, the Analytic Hierarchy Process (AHP) was implemented as a secondary ranking technique for selecting airfoils. Conventional AHP employs pairwise comparisons and expert opinion for determining the relative importance (weights) of criteria. But for the present study, an objective weighting scheme based on the theory of entropy was adopted that eliminated the element of subjectivity and ensured higher levels of transparency and reproducibility. For both criteria to be compared directly, min-max normalization was used:

• For benefit criteria (Max Cl/Cd), scores were normalized so that the best-performing airfoil was given a rating of 1.

• For cost criteria (Cl/Cd variance), the lower the value, the better, so the scale was reversed.

This normalization makes sure that each attribute contributes in proportion to the final ranking, regardless of its raw value range or measurement scale, as suggested by the standard MCDM literature. With both the normalized criteria and entropy-derived weights, the final AHP score for each airfoil was calculated as the weighted sum indicated in equation (6) (Konstantinos et al., 2019).

AHP {score}_{i} = \sum_{i} w_{j} r_{ij}

(6)

where r_ij refers to the normalized value of airfoil i for criterion j and w_j means the weight; this yielded a single performance score per airfoil, which served as the basis for the final ranking.

Before walking into the next and final MCDM method, it is crucial to highlight and mention that while this study or work employs an AHP-style weighted scoring method, it does not follow the traditional or the most adopted AHP framework that involves consistency ratio checks and pairwise comparisons; instead the objective weights derived from entropy are used to ensure reproducibility and eliminate subjectivity. Hence, this approach aligns more and is closer to the entropy-weighted scoring model than conventional or classical AHP.

As a complement to the multi-criteria analysis method, VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) was used to rank the airfoil options using a compromise solution approach. VIKOR, which was formulated by Opricovic and Tzeng, is especially helpful for application in conflicting criteria decision-making situations where not only is the identification of the optimal alternative sought, but a trade-off is also required between collective satisfaction (group utility) and individual regret (Yazdani and Graeml, 2014).

This is especially applicable to airfoil choice, where achieving maximum aerodynamic efficiency (high Cl/Cd) is usually at the cost of aerodynamic stability (low variance of Cl/Cd). Each of the criteria was normalized using the linear min-max technique:

• For benefit criteria, the values were normalized so that the top-performing alternative had a normalized value of 1.

• For costing criteria, smaller values were better, and thus normalization was reversed.

This provided comparability across criteria with different value ranges, consistent with standard VIKOR implementation guidelines.

After preparing the normalized decision matrix and weights, three main indices for each airfoil were calculated:

• S (Group Utility Measure): This is the total performance shortfall for each airfoil from the best solution, taking into account the aggregate of all criteria.

• R (Measures of Regret): This records the largest individual difference from the optimum for any of the criteria—that is, the worst possible for each airfoil.

• Q (Compromise Index): It is the ultimate VIKOR rating calculated by aggregating S and R with a balancing weight v = 0.5, according to conventional tradition, and it is calculated as indicated in equation (7) (Yazdani and Graeml, 2014).

Q_{i} = v \frac{S_{i} - S^{*}}{S^{-} - S^{*}} + (1 - v) \frac{R_{i} - R^{*}}{R^{-} - R^{*}}

(7)

S^* and S⁻ represent the best and worst S values across all alternatives, while R^* and R⁻ are the best and worst R values.

Upon acquiring individual airfoil ranks from the three multi-criteria decision methods, TOPSIS, AHP, and VIKOR, an additional aggregation step was performed to combine the ranks into one decision. This was accomplished with the use of the Borda Count method, an established procedure for aggregating ranks frequently utilized in voting systems and group decision making (Sharif et al., 2015). The objective of this step was to combine the relative ranks of each airfoil into an order that is consensus-driven, basing the order equally upon all three methods. In the Borda Count methodology, each alternative (in this case, each airfoil) is awarded a score that is the total of the ranks given to it from all sources, as shown in equation (8).

Borda {score}_{i} = {Rank}_{TOPSIS} + {Rank}_{AHP} + {Rank}_{VIKOR}

(8)

Lower values are preferable because higher-ranked airfoils (i.e., ranks 1 or 2) make fewer contributions to the total. The Dataset was ranked from bottom to top according to the Borda Score to create the ultimate collective ranking. This implicitly weights all three MCDM methods equally, which is suitable since the same entropy-based weights were utilized in all three.

To compare the consistency and agreement of the three MCDM ranking outcomes, Spearman’s rank correlation coefficient was calculated for each pair of methods (Sedgwick, 2014):

• TOPSIS and AHP

• TOPSIS and VIKOR

• AHP and VIKOR

The non-parametric statistic calculates the strength and direction of the association of two ranked variables. When there is perfect agreement, the value is 1, whereas it is −1 with complete disagreement. The output was printed to verify whether there was statistical agreement among the different methods.

Results and discussion

The lift-to-drag ratio, Cl/Cd, varies across the angles of attack as indicated in Figure 1, which shows the variation of lift/drag ratio (Cl/Cd) with angle of attack (AoA) for the selected set of NACA airfoils tested over four Reynolds numbers: 421116, 505400, 589633, and 673866 (Bougaa and Tenghiri, 2025). The Reynolds numbers used within this study were derived based on the aerodynamic and geometric parameters of an 11 kW small-scale wind turbine. These calculations used the design computations outlined in the author’s thesis. A chord length of 0.25 m was taken at the mid-span of the blade. Besides, the relative wind speed was estimated through the local velocity triangle, considering a tip speed ratio (TSR) ranging from 5 to 8. These computed Reynolds numbers effectively represent the operational conditions typical for small horizontal-axis wind turbines installed at low hub heights in real-world environments. These are the characteristic flow conditions approached by small horizontal-axis wind turbines (HAWTs) at low speeds. The direct method of performance extraction was applied, that is, peak values were extracted directly from the simulation data without smoothing or fitting curves. Throughout all Reynolds numbers, the majority of airfoils behave as expected aerodynamically, with Cl/Cd rising to an apex, usually in the range of approximately 4° to 6°, before declining due to stall; increasing Reynolds numbers produce enhanced performance, which is reflected in more dramatic Cl/Cd peaks and wider optimal ranges of AoA. For example, at Re = 673866, airfoils like the NACA 23021 and the NACA 2412 display both higher peak Cl/Cd values and more gradual decline after the peak, reflecting improved flow attachment and aerodynamic stability. There is an obvious difference seen among symmetric airfoils (e.g., NACA 0012, 0015) and cambered profiles (e.g., NACA 23024, NACA 24012). Symmetrical airfoils have lower values of Cl/Cd and reduced operational ranges because they are symmetric in structure but do not produce much lift at moderate angles of attack. Cambered airfoils show better performance for all flow regimes. These Cl/Cd curves provide the quantitative basis for subsequent entropy-weighted multi-criteria decision-making (MCDM) analyses, with peak Cl/Cd and variability thereof serving as important evaluation indices in TOPSIS, AHP, and VIKOR ranking.

Figure 1.

Visualization of Cl/Cd at each angle of attack at different Re numbers.

Regarding the variance of the Cl/Cd factor or the stability of each airfoil, Figure 2 presents the variation of the Cl/Cd value for a series of NACA airfoils at four Reynolds numbers (421116, 505400, 589633, and 673866). The variation parameter reflects each airfoil’s aerodynamic stability over an interval of angles of attack and indicates performance sensitivity with respect to flow condition variation. Smaller variation means more stable and consistent aerodynamic performance, since small turbines rely upon high aerodynamic stability in turbulent and variable flow conditions. The bar chart shows that there is an obvious distinction among the airfoils depending upon overall Cl/Cd variation; for airfoils like NACA 4521, 25028, and 2424, there is always low variation throughout all Reynolds numbers, which indicates that these airfoils have consistent and stable aerodynamics under variable flow conditions; conversely, airfoils like NACA 0012, 0015, and 0021 have high Cl/Cd variation, especially at increased Reynolds numbers, and indicate higher sensitivity with lesser predictability of performance. Throughout the dataset, increased Reynolds number is systematically correlated with increased Cl/Cd variance, particularly for symmetric airfoils. This indicates that, as higher Reynolds numbers create improved peak performance (Figure 1), they can also add greater aerodynamic response instability, especially where airfoils are not naturally suited to generating lift; this data for variance acts as an input criterion in the ensuing MCDM calculations. This provides support to the maximum Cl/Cd values with an index of stability, allowing for an even-handed choice procedure that takes into consideration both efficacy and dependability.

Figure 2.

Plot of Cl/Cd variance for each airfoil at different Re numbers.

Figures 1 and 2 present two bar charts highlighting the lift-to-drag ratio of each airfoil across multiple angles of attack, and the variance of this ratio for each airfoil at different Reynolds Numbers. For a better illustration and detailed numerical data, Table 2 provides the numerical values of each factor, lift-to-drag ratio, the best or optimal angle of attack, and the variance, at the four adopted Reynolds numbers for the different NACA airfoils.

Table 2.

Airfoil main parameters at each Reynolds number.

Reynolds number	421116			505400			589633			673866
Airfoil	Best angle of attack	Maximum Cl/Cd	Variance Cl/Cd	Best angle of attack	Maximum Cl/Cd	Variance Cl/Cd	Best angle of attack	Maximum Cl/Cd	Variance Cl/Cd	Best angle of attack	Maximum Cl/Cd	Variance Cl/Cd
NACA_0012	6	58.66	1425.34	6	61.97	1558.98	6	64.73	1690.17	7	67.43	1826.34
NACA_0015	7	63.27	1358.08	8	66.04	1526.97	8	69.38	1686.92	8	71.43	1843.66
NACA_0018	9	62.95	1244.41	9	65.72	1423.10	8	108.10	1821.99	10	70.55	1734.74
NACA_0021	10	57.59	1071.87	10	60.85	1227.77	10	63.21	1372.36	10	65.68	1507.47
NACA_0024	10	49.45	845.35	10	52.13	959.11	9	55.40	1078.86	8	58.89	1189.91
NACA_23012	10	72.08	1129.45	11	77.56	1307.84	10	83.03	1462.34	10	87.30	1625.92
NACA_23015	9	67.95	1068.29	8	73.01	1246.09	9	78.30	1399.76	9	82.94	1558.50
NACA_23018	8	65.18	980.11	9	69.48	1095.94	9	73.09	1225.68	9	78.32	1352.95
NACA_23021	9	58.57	858.63	9	61.29	957.68	10	64.26	1061.20	10	67.28	1155.32
NACA_23024	10	48.48	647.40	10	50.51	721.40	10	53.01	802.56	10	54.89	877.95
NACA_24012	9	79.23	1220.43	9	84.65	1391.50	9	89.33	1546.18	9	93.11	1692.16
NACA_24015	10	74.96	1198.94	9	80.72	1385.96	9	85.87	1543.99	9	90.59	1703.62
NACA_24018	8	70.43	1064.51	9	76.19	1211.85	9	81.32	1353.75	9	85.05	1484.12
NACA_24021	9	63.53	897.35	9	68.03	1004.55	10	71.52	1110.57	10	75.24	1214.79
NACA_2412	6	84.69	1132.62	5	89.27	1252.32	5	93.07	1358.14	5	96.24	1455.91
NACA_2415	6	83.81	1147.53	6	88.72	1278.40	6	92.66	1393.62	6	95.76	1502.13
NACA_2418	7	80.09	1046.82	7	85.30	1175.62	7	89.60	1293.08	7	92.82	1400.24
NACA_2421	8	72.56	858.99	8	77.56	972.98	8	81.40	1082.34	8	84.32	1183.25
NACA_2424	8	62.12	660.08	9	65.54	744.40	9	68.35	824.64	9	71.16	903.02
NACA_25012	8	83.59	1266.92	8	88.80	1435.85	8	93.33	1569.52	8	97.16	1717.09
NACA_25015	8	80.41	1274.69	9	86.37	1447.92	8	91.32	1598.93	9	95.50	1739.32
NACA_25018	8	76.32	1128.50	8	81.82	1270.56	9	86.75	1411.91	9	91.29	1548.08
NACA_25021	9	70.55	928.11	9	73.43	1033.55	9	77.18	1145.58	10	80.90	1266.06
NACA_25024	10	59.74	725.70	10	62.19	798.94	10	64.09	874.30	10	67.17	958.23
NACA_25026	10	51.76	597.40	10	53.44	651.68	10	55.48	708.31	8	57.44	766.70
NACA_25028	10	44.59	477.16	10	45.85	516.47	8	47.83	556.12	8	50.72	598.19
NACA_4412	7	102.87	1162.21	6	108.83	1286.50	6	114.47	1401.30	6	118.82	1497.91
NACA_4415	7	96.85	1035.76	7	103.18	1169.32	7	108.55	1285.73	7	113.63	1397.46
NACA_4418	7	88.54	849.36	7	94.29	955.90	7	99.02	1063.67	7	103.57	1162.48
NACA_4421	7	78.92	664.86	7	83.14	737.69	7	87.19	801.93	7	91.32	887.67
NACA_4424	6	69.47	497.63	7	71.79	543.16	7	74.78	589.82	7	77.90	639.37
NACA_4512	5	110.87	1230.52	4	117.25	1352.41	4	122.75	1439.56	4	127.07	1510.33
NACA_4515	5	102.48	1060.73	5	108.53	1163.92	5	113.38	1253.31	5	117.30	1335.90
NACA_4518	5	92.67	833.75	5	97.94	921.23	5	102.42	996.14	5	106.72	1076.36
NACA_4521	6	81.13	616.46	5	85.56	675.65	5	88.90	731.12	5	92.35	782.63
NACA_4524	6	69.25	436.98	6	72.31	471.62	6	74.16	505.29	6	76.60	538.93

While Figure 3 displays the scores from the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) for each airfoil over four Reynolds numbers: 421116, 505400, 589633, and 673866, Table 3 presents a detailed score and ranking of each airfoil across the 4 Reynolds numbers, including all the parameters computed in the TOPSIS approach. The TOPSIS technique combines several criteria here, maximum Cl/Cd and Cl/Cd variation, into one decision score, with increased values denoting improved overall performance; by comparing airfoil performance under different flow conditions, this figure gives an idea of each airfoil’s stability and robustness in terms of aerodynamic efficiency. The outcome indicates that NACA 4521, 4518, and 4421 are consistently the leading ones with scores higher than 0.6 for all Reynolds numbers; these airfoils show excellent tradeoff between good aerodynamic efficiency and robust performance, and they are especially good candidates for low-Reynolds-number applications like small turbines; their leading performance in all four regimes indicates high flexibility and insensitivity to variation in the conditions under which they operate. In contrast, airfoils like NACA 0012, 0015, and 0021 are at the bottom of the scale with TOPSIS values under 0.3, indicating reduced aerodynamic effectiveness and increased instability. This is consistent with their symmetric shape, which performs sub-optimally in applications which are lift-dominated. Interestingly, the modest variation in TOPSIS scores with Reynolds numbers indicates that certain airfoils, such as NACA 23024 and 2412, are quite good at moderate Re but lag in ranking at both ends. This indicates that certain airfoils have good performance averages, but few are excellent under all conditions. These rankings are an initial input into the ultimate airfoil choice, which combines outputs from several MCDM methods.

Figure 3.

Visualization of TOPSIS ranking.

Table 3.

TOPSIS score and ranking.

Re = 421116			Re = 505400			Re = 589633			Re = 673866
Airfoil	TOPSIS score	Rank	Airfoil	TOPSIS score	Rank	Airfoil	TOPSIS score	Rank	Airfoil	TOPSIS score	Rank
NACA_4521	0.67	1	NACA_4521	0.67	1	NACA_4518	0.68	2	NACA_4521	0.67	1
NACA_4518	0.66	2	NACA_4518	0.65	2	NACA_4521	0.67	1	NACA_4518	0.66	2
NACA_4421	0.63	3	NACA_4421	0.63	3	NACA_4421	0.64	3	NACA_4421	0.63	3
NACA_4524	0.63	4	NACA_4524	0.63	4	NACA_4418	0.63	4	NACA_4524	0.62	4
NACA_4418	0.62	5	NACA_4418	0.61	5	NACA_4515	0.63	5	NACA_4515	0.60	5
NACA_4424	0.62	6	NACA_4424	0.61	6	NACA_4524	0.62	6	NACA_4424	0.60	6
NACA_4515	0.60	7	NACA_4515	0.59	7	NACA_4424	0.61	7	NACA_4418	0.60	7
NACA_4415	0.58	8	NACA_4415	0.57	8	NACA_4512	0.59	8	NACA_4512	0.58	8
NACA_4512	0.56	9	NACA_4512	0.56	9	NACA_4415	0.59	9	NACA_4415	0.57	9
NACA_4412	0.55	10	NACA_4412	0.55	10	NACA_4412	0.58	10	NACA_4412	0.56	10
NACA_2424	0.52	11	NACA_2424	0.51	11	NACA_2424	0.51	11	NACA_2424	0.50	11
NACA_2421	0.50	12	NACA_2421	0.49	12	NACA_2421	0.50	12	NACA_25028	0.49	12
NACA_25028	0.49	13	NACA_25028	0.49	13	NACA_25028	0.49	13	NACA_2421	0.47	13
NACA_25026	0.48	14	NACA_25026	0.48	14	NACA_25026	0.48	14	NACA_25026	0.47	14
NACA_25024	0.47	15	NACA_25024	0.47	15	NACA_2418	0.48	15	NACA_25024	0.46	15
NACA_2418	0.46	16	NACA_2418	0.45	16	NACA_2412	0.48	16	NACA_2412	0.45	16
NACA_2412	0.45	17	NACA_2412	0.45	17	NACA_25024	0.47	17	NACA_2418	0.45	17
NACA_23024	0.45	18	NACA_23024	0.45	18	NACA_2415	0.46	18	NACA_2415	0.43	18
NACA_25021	0.45	19	NACA_2415	0.44	19	NACA_25021	0.45	19	NACA_23024	0.43	19
NACA_2415	0.44	20	NACA_25021	0.44	20	NACA_23024	0.45	20	NACA_25021	0.42	20
NACA_24021	0.42	21	NACA_24021	0.41	21	NACA_0018	0.44	21	NACA_24021	0.40	21
NACA_23021	0.41	22	NACA_23021	0.40	22	NACA_24021	0.43	22	NACA_25018	0.39	22
NACA_25012	0.39	23	NACA_25018	0.39	23	NACA_25018	0.42	23	NACA_25012	0.38	23
NACA_25018	0.39	24	NACA_25012	0.39	24	NACA_25012	0.41	24	NACA_23021	0.38	24
NACA_23018	0.38	25	NACA_23018	0.38	25	NACA_23021	0.41	25	NACA_25015	0.37	25
NACA_24012	0.38	26	NACA_24018	0.37	26	NACA_24018	0.40	26	NACA_23018	0.37	26
NACA_24018	0.38	27	NACA_24012	0.37	27	NACA_23018	0.40	27	NACA_24018	0.37	27
NACA_0024	0.37	28	NACA_25015	0.37	28	NACA_24012	0.39	28	NACA_24012	0.36	28
NACA_25015	0.37	29	NACA_0024	0.35	29	NACA_25015	0.39	29	NACA_24015	0.34	29
NACA_23012	0.36	30	NACA_23012	0.35	30	NACA_23012	0.38	30	NACA_23012	0.34	30
NACA_23015	0.36	31	NACA_24015	0.34	31	NACA_24015	0.37	31	NACA_0024	0.33	31
NACA_24015	0.35	32	NACA_23015	0.34	32	NACA_23015	0.36	32	NACA_23015	0.33	32
NACA_0021	0.28	33	NACA_0021	0.26	33	NACA_0024	0.36	33	NACA_0021	0.23	33
NACA_0018	0.23	34	NACA_0018	0.21	34	NACA_0021	0.28	34	NACA_0018	0.19	34
NACA_0015	0.20	35	NACA_0015	0.19	35	NACA_0015	0.21	35	NACA_0015	0.18	35
NACA_0012	0.14	36	NACA_0012	0.15	36	NACA_0012	0.17	36	NACA_0012	0.15	36

The Analytic Hierarchy Process (AHP) scores for all of the chosen NACA airfoils over four Reynolds numbers of 421116, 505400, 589633, and 673866 are shown in Figure 4. While Figure 4 presents a visualization of the final score, Table 4 provides the exact value of each airfoil across the different Reynolds numbers. The AHP scores are calculated through entropy-weighted criteria, which are the maximum Cl/Cd and its variability. Higher values represent higher aerodynamic performance and stability in terms of the weighted contribution from both the criteria. This method offers an open and uniform evaluation method to compare airfoils under different operational conditions. The data show that the highest AHP scores, around or even above 0.7, were obtained consistently by NACA 4521, 4412, and 2414 for all Reynolds numbers. These airfoils have excellent overall performance, with an excellent balance of aerodynamic efficiency and operational stability. These scores are also the least sensitive to Reynolds number variation, indicating robust performance under various flow conditions. Such consistency further supports their candidacy as leading choices for small-scale wind turbine blade design. By comparison, symmetric airfoils like NACA 0012, 0015, and 0021 take the lowest places in the ranking with AHP values under 0.3. These profiles do not just lag behind with regard to Cl/Cd performance, but they have higher variation, which plays a part in their worst overall scores. Certain airfoils, for example, NACA 23009 and 25024, are found to display intermediate performance with moderate AHP values and apparent variability with Reynolds numbers, indicating flow regime sensitivity. Overall, the AHP values are well in accord with the TOPSIS rankings, which supports the identification of a small group of airfoils, particularly NACA 4521, as being optimal under different aerodynamic conditions.

Figure 4.

Airfoil ranking visualization based on AHP.

Table 4.

AHP score and ranking.

Re = 421116			Re = 505400			Re = 589633			Re = 673866
Airfoil	AHP score	Rank	Airfoil	AHP score	Rank	Airfoil	AHP score	Rank	Airfoil	AHP score	Rank
NACA_4524	0.69	1	NACA_4524	0.73	1	NACA_4521	0.69	1	NACA_4524	0.73	1
NACA_4521	0.69	2	NACA_4521	0.70	2	NACA_4524	0.68	2	NACA_4521	0.70	2
NACA_4424	0.66	3	NACA_4424	0.69	3	NACA_4518	0.68	3	NACA_4424	0.69	3
NACA_4518	0.66	4	NACA_4421	0.66	4	NACA_4424	0.65	4	NACA_4421	0.65	4
NACA_4421	0.65	5	NACA_4518	0.65	5	NACA_4421	0.65	5	NACA_4518	0.65	5
NACA_4418	0.62	6	NACA_4418	0.61	6	NACA_4515	0.65	6	NACA_4418	0.59	6
NACA_4515	0.62	7	NACA_4515	0.58	7	NACA_4512	0.64	7	NACA_4515	0.59	7
NACA_4512	0.59	8	NACA_4415	0.55	8	NACA_4418	0.63	8	NACA_25028	0.57	8
NACA_4415	0.59	9	NACA_25028	0.55	9	NACA_4415	0.61	9	NACA_4512	0.56	9
NACA_4412	0.57	10	NACA_2424	0.55	10	NACA_4412	0.60	10	NACA_4415	0.54	10
NACA_2424	0.52	11	NACA_4512	0.54	11	NACA_2424	0.52	11	NACA_2424	0.54	11
NACA_2421	0.50	12	NACA_25026	0.52	12	NACA_2421	0.51	12	NACA_25026	0.53	12
NACA_25028	0.49	13	NACA_4412	0.52	13	NACA_25028	0.49	13	NACA_4412	0.52	13
NACA_23009	0.49	14	NACA_2421	0.50	14	NACA_25026	0.48	14	NACA_25024	0.49	14
NACA_25026	0.48	15	NACA_25024	0.50	15	NACA_2418	0.48	15	NACA_2421	0.48	15
NACA_25024	0.47	16	NACA_23024	0.47	16	NACA_2412	0.48	16	NACA_23024	0.46	16
NACA_2418	0.46	17	NACA_25021	0.44	17	NACA_25024	0.47	17	NACA_2418	0.43	17
NACA_25021	0.45	18	NACA_2418	0.44	18	NACA_2415	0.46	18	NACA_25021	0.42	18
NACA_2412	0.45	19	NACA_24021	0.42	19	NACA_25021	0.45	19	NACA_2412	0.42	19
NACA_2415	0.43	20	NACA_2412	0.42	20	NACA_24021	0.43	20	NACA_24021	0.42	20
NACA_23024	0.43	21	NACA_23021	0.41	21	NACA_23024	0.43	21	NACA_23021	0.40	21
NACA_24021	0.41	22	NACA_2415	0.40	22	NACA_25018	0.41	22	NACA_2415	0.39	22
NACA_23021	0.40	23	NACA_23018	0.39	23	NACA_23021	0.40	23	NACA_23018	0.37	23
NACA_25018	0.39	24	NACA_25018	0.37	24	NACA_24018	0.40	24	NACA_25018	0.35	24
NACA_23018	0.38	25	NACA_24018	0.36	25	NACA_23018	0.40	25	NACA_24018	0.35	25
NACA_24018	0.38	26	NACA_0024	0.35	26	NACA_25012	0.40	26	NACA_0024	0.34	26
NACA_25012	0.37	27	NACA_23015	0.33	27	NACA_0018	0.39	27	NACA_25012	0.30	27
NACA_0024	0.36	28	NACA_25012	0.32	28	NACA_24012	0.38	28	NACA_23015	0.30	28
NACA_25015	0.36	29	NACA_23012	0.32	29	NACA_25015	0.37	29	NACA_24012	0.29	29
NACA_23012	0.36	30	NACA_24012	0.32	30	NACA_23012	0.37	30	NACA_23012	0.29	30
NACA_23015	0.34	31	NACA_25015	0.30	31	NACA_23015	0.36	31	NACA_25015	0.29	31
NACA_24015	0.34	32	NACA_24015	0.30	32	NACA_24015	0.36	32	NACA_24015	0.28	32
NACA_0021	0.33	33	NACA_0021	0.26	33	NACA_0024	0.34	33	NACA_0021	0.23	33
NACA_0018	0.28	34	NACA_0018	0.19	34	NACA_0021	0.27	34	NACA_0018	0.15	34
NACA_0015	0.23	35	NACA_0015	0.14	35	NACA_0015	0.19	35	NACA_0015	0.11	35
NACA_0012	0.17	36	NACA_0012	0.10	36	NACA_0012	0.16	36	NACA_0012	0.10	36

In both TOPSIS and AHP rankings, the NACA 0018 airfoil demonstrates a notably higher score at a Re of 589,633 compared to other Reynolds numbers, as illustrated in Figures 3 and 4. This anomaly diverges from the typical trend observed across the range of tested airfoils and contrasts with its generally limited aerodynamic performance at lower Reynolds numbers. The NACA 0018 profile, being symmetric, is particularly sensitive to flow transition and separation phenomena within the Reynolds number window of 500,000 to 700,000. The unexpectedly enhanced performance at a Re of 589,633 may stem from numerical artifacts fundamental in the XFoil simulation or reflecting a transient stabilization of the flow under these specific conditions; nonetheless, its performance remains among the lowest in other tested regimes. Therefore, this result should be regarded as an outlier rather than a definitive indicator of its aerodynamic suitability. This emphasizes the significance of evaluating multiple flow regimes and stability metrics when selecting airfoils for small wind turbine applications.

Concerning the last MCDM approach employed, Figure 5 shows the values for all selected NACA airfoils’ VIKOR scores (labeled as Q) over four Reynolds numbers: 421116, 505400, 589633, and 67386. Also, Table 5 provides the exact numerical value of the Q score of each airfoil. The VIKOR method, which finds compromise solutions for multi-criteria decision problems, was utilized with entropy-weighted inputs: maximum Cl/Cd as the benefit criterion and Cl/Cd variation as the cost criterion. For the VIKOR model, the best alternatives have lowest Q scores that vary closer to the ideal solution for both group utility and personal regret. The performance clearly displays that NACA 4521, 4418, and 4421 always achieve the lowest Q values, which means they have the best aerodynamic performance and stability under all Reynolds numbers tested; these airfoils do not only have good trade-offs for both efficiency and consistency but also display minimal performance under varied flow conditions; their VIKOR values are normally below 0.1, which supports them being best suited for use in small wind turbines where both robustness and efficiency are needed. On the contrary, symmetric airfoils like the NACA 0012, 0015, and 0021 are ranked highest with VIKOR values nearing or even higher than 0.9. This conforms with their well-documented shortfalls under high lift generation and increased sensitivity to the parameters of Reynolds number and angle of attack. The gradual rise of Q scores from left to right also indicates the wider range of performance among the airfoils under investigation. These results complement the results obtained through TOPSIS and AHP, which affirm that there is a small group of cambered NACA profiles with persistently higher aerodynamic performance under different flow regimes.

Figure 5.

VIKOR ranking.

Table 5.

VIKOR score and ranking.

Re = 421116			Re = 505400			Re = 589633			Re = 673866
Airfoil	VIKOR score	Rank	Airfoil	VIKOR score	Rank	Airfoil	VIKOR score	Rank	Airfoil	VIKOR score	Rank
NACA_4518	0.03	1	NACA_4521	0.05	1	NACA_4518	0.01	1	NACA_4521	0.06	1
NACA_4521	0.04	2	NACA_4518	0.06	2	NACA_4521	0.06	2	NACA_4518	0.07	2
NACA_4418	0.07	3	NACA_4421	0.11	3	NACA_4418	0.10	3	NACA_4421	0.11	3
NACA_4421	0.11	4	NACA_4418	0.12	4	NACA_4421	0.12	4	NACA_4418	0.17	4
NACA_4524	0.19	5	NACA_4524	0.18	5	NACA_4515	0.19	5	NACA_4524	0.21	5
NACA_4424	0.21	6	NACA_4424	0.22	6	NACA_4524	0.23	6	NACA_4424	0.23	6
NACA_4515	0.26	7	NACA_2421	0.30	7	NACA_4424	0.25	7	NACA_4515	0.28	7
NACA_4415	0.26	8	NACA_4515	0.31	8	NACA_4415	0.26	8	NACA_2421	0.32	8
NACA_2421	0.31	10	NACA_4415	0.34	9	NACA_2421	0.32	9	NACA_4415	0.36	9
NACA_2418	0.38	11	NACA_25021	0.40	10	NACA_4512	0.32	10	NACA_25021	0.41	10
NACA_25021	0.38	12	NACA_2424	0.41	11	NACA_4412	0.33	11	NACA_4512	0.42	11
NACA_4412	0.38	13	NACA_2418	0.43	12	NACA_2418	0.38	12	NACA_2424	0.43	12
NACA_4512	0.42	14	NACA_4412	0.45	13	NACA_25021	0.41	13	NACA_4412	0.44	13
NACA_2424	0.42	15	NACA_24021	0.48	14	NACA_2412	0.42	14	NACA_2418	0.45	14
NACA_2412	0.46	16	NACA_25024	0.49	15	NACA_2424	0.44	15	NACA_24021	0.47	15
NACA_24018	0.46	17	NACA_23018	0.49	16	NACA_2415	0.46	16	NACA_23018	0.48	16
NACA_2415	0.49	18	NACA_4512	0.49	17	NACA_24018	0.49	17	NACA_2412	0.49	17
NACA_23015	0.49	19	NACA_2412	0.50	18	NACA_24021	0.49	18	NACA_25024	0.51	18
NACA_25024	0.50	20	NACA_24018	0.52	19	NACA_23018	0.51	19	NACA_2415	0.54	19
NACA_24021	0.50	21	NACA_2415	0.54	20	NACA_25018	0.51	20	NACA_24018	0.57	20
NACA_23018	0.50	22	NACA_25018	0.56	21	NACA_25024	0.53	21	NACA_23021	0.58	21
NACA_25018	0.51	23	NACA_23021	0.57	22	NACA_23015	0.55	22	NACA_25026	0.59	22
NACA_23012	0.54	24	NACA_23015	0.57	23	NACA_23012	0.59	23	NACA_25018	0.61	23
NACA_23021	0.58	25	NACA_25026	0.57	24	NACA_23021	0.60	24	NACA_25028	0.63	24
NACA_25026	0.59	26	NACA_23012	0.63	25	NACA_25026	0.62	25	NACA_23015	0.65	25
NACA_24015	0.61	27	NACA_25028	0.64	26	NACA_25012	0.63	26	NACA_23024	0.67	26
NACA_24012	0.61	28	NACA_23024	0.65	27	NACA_24012	0.63	27	NACA_23012	0.70	27
NACA_25012	0.64	29	NACA_0021	0.69	28	NACA_24015	0.65	28	NACA_0024	0.72	28
NACA_25015	0.67	30	NACA_24012	0.69	29	NACA_25015	0.67	29	NACA_0021	0.73	29
NACA_25028	0.67	31	NACA_24015	0.70	30	NACA_25028	0.69	30	NACA_24012	0.75	30
NACA_23024	0.67	32	NACA_0024	0.72	31	NACA_23024	0.69	31	NACA_25012	0.75	31
NACA_0021	0.69	33	NACA_25012	0.73	32	NACA_0021	0.73	32	NACA_24015	0.77	32
NACA_0018	0.74	34	NACA_25015	0.75	33	NACA_0024	0.75	33	NACA_25015	0.78	33
NACA_0024	0.74	35	NACA_0018	0.82	34	NACA_0018	0.78	34	NACA_0018	0.88	34
NACA_0015	0.88	36	NACA_0015	0.94	35	NACA_0015	0.89	35	NACA_0012	0.99	35
NACA_0012	1.00	37	NACA_0012	1.00	36	NACA_0012	0.92	36	NACA_0015	0.99	36

As a last step, the aggregate results are indicated in Table 6, which shows the Borda Scores for the airfoils ranked according to the considered airfoil evaluations at four different Reynolds numbers (421116, 505400, 589633, and 673866). The Borda Count technique was utilized as an aggregation method, combining individual ranks obtained from the TOPSIS, AHP, and VIKOR models. With this technique, each airfoil receives an accumulated score depending upon its ranking in the individual ranks, the lower the Borda Score, the higher the performance over all decision models. The findings show that NACA 4521, 4518, and 4421 always have the lowest Borda Scores under all Reynolds numbers. This reinforces their dominance with respect to various assessment parameters and methodologies, as earlier suggested through their high TOPSIS and AHP scores and minimal VIKOR score. Their stability and high performance under different flow regimes are evidence of their aerodynamic robustness, positioning them as leading candidates for small turbine blades. On the contrary, symmetric airfoils like NACA 0012, 0015, and 0021 have the highest Borda Scores, an indicator of worse overall ranking under all three models of MCDM. These airfoils not only perform poorly in terms of the lift-to-drag ratio but are also highly variable, which makes their composite ranking very poor.

Table 6.

Borda Scores.

Re = 421116		Re = 505400		Re = 589633		Re = 673866
Airfoil	Borda Score	Airfoil	Borda Score	Airfoil	Borda Score	Airfoil	Borda Score
NACA_4521	5	NACA_4521	4	NACA_4521	5	NACA_4521	4
NACA_4518	7	NACA_4518	9	NACA_4518	5	NACA_4518	9
NACA_4524	10	NACA_4421	10	NACA_4421	12	NACA_4421	10
NACA_4421	12	NACA_4524	10	NACA_4524	14	NACA_4524	10
NACA_4418	14	NACA_4424	15	NACA_4418	15	NACA_4424	15
NACA_4424	15	NACA_4418	15	NACA_4515	16	NACA_4418	17
NACA_4515	21	NACA_4515	22	NACA_4424	18	NACA_4515	19
NACA_4415	25	NACA_4415	25	NACA_4512	25	NACA_4512	28
NACA_4512	31	NACA_2424	32	NACA_4415	26	NACA_4415	28
NACA_4412	33	NACA_2421	33	NACA_4412	31	NACA_2424	34
NACA_2421	34	NACA_4412	36	NACA_2421	33	NACA_2421	36
NACA_2424	37	NACA_4512	37	NACA_2424	37	NACA_4412	36
NACA_2418	45	NACA_25024	45	NACA_2418	42	NACA_25028	44
NACA_25021	50	NACA_2418	46	NACA_2412	46	NACA_25024	47
NACA_25024	52	NACA_25021	47	NACA_25021	51	NACA_25026	48
NACA_2412	53	NACA_25028	48	NACA_2415	52	NACA_2418	48
NACA_25026	56	NACA_25026	50	NACA_25026	53	NACA_25021	48
NACA_25028	57	NACA_24021	54	NACA_25024	55	NACA_2412	52
NACA_2415	59	NACA_2412	55	NACA_25028	56	NACA_24021	56
NACA_24021	65	NACA_23024	61	NACA_24021	60	NACA_2415	59
NACA_23021	71	NACA_2415	61	NACA_25018	65	NACA_23024	61
NACA_24018	71	NACA_23018	64	NACA_24018	67	NACA_23018	65
NACA_23024	72	NACA_23021	65	NACA_23018	71	NACA_23021	66
NACA_25018	72	NACA_25018	68	NACA_23024	72	NACA_25018	69
NACA_23018	73	NACA_24018	70	NACA_23021	72	NACA_24018	72
NACA_25012	80	NACA_23015	82	NACA_25012	76	NACA_25012	81
NACA_23015	80	NACA_25012	84	NACA_0018	82	NACA_0024	85
NACA_24012	83	NACA_23012	84	NACA_24012	83	NACA_23015	85
NACA_23012	85	NACA_0024	86	NACA_23012	83	NACA_24012	87
NACA_25015	91	NACA_24012	86	NACA_23015	85	NACA_23012	87
NACA_24015	92	NACA_25015	92	NACA_25015	87	NACA_25015	89
NACA_0024	97	NACA_24015	93	NACA_24015	91	NACA_24015	93
NACA_0021	101	NACA_0021	94	NACA_0024	99	NACA_0021	95
NACA_0018	104	NACA_0018	102	NACA_0021	100	NACA_0018	102
NACA_0015	108	NACA_0015	105	NACA_0015	105	NACA_0015	106
NACA_0012	111	NACA_0012	108	NACA_0012	108	NACA_0012	107

In order to achieve a solid decision-making process in identifying the most appropriate airfoil, three well-established Multi-Criteria Decision-Making (MCDM) techniques, namely, TOPSIS, AHP, and VIKOR, were used. Since each of the methods has different computational rules and sensitivity to input criteria, a Borda Count-based rank aggregation process was applied to combine the three rankings and determine a consensus solution. The results indicated that there was a very strong conformity among the methods based on Spearman’s rank correlation coefficients; the correlation between TOPSIS and AHP was the most prominent (ρ = 0.9987), reflecting nearly identical prioritization of the airfoils. Likewise, TOPSIS and VIKOR, and AHP and VIKOR, indicated equally robust correlations (ρ = 0.9683 and ρ = 0.9707 respectively), confirming the internal reliability of the decision-making process. These large correlation values imply that the three MCDM techniques provided converging information on the performance of the airfoils, strengthening grounds for confidence in the reliability of the resultant ranking from the Borda Count method. Through integrative aggregation, the role of individual method bias is downplayed, and a stable and balanced choice consequence is achieved.

Conclusion

This work provided an all-around methodologically strict method for choosing the best airfoils for use in small-scale applications for wind turbines. All of the 36 NACA airfoils were evaluated with aero-simulations under four Reynolds numbers of 421116, 505400, 589633, and 673866, which are typical for actual working conditions for small horizontal-axis wind turbines (HAWTs). Each airfoil’s performance was evaluated according to two important aerodynamic parameters: the peak lift-to-drag ratio (Cl/Cd) and their variability with angles of attack. The method applied in Cl/Cd extraction provided objective determination of peak aerodynamic performance without the need for smoothing out curves, whereas variability supplied quantitative data for aerodynamic stability.

To guarantee effective decision-making, three of the best established methods of MCDM, TOPSIS, AHP, and VIKOR, were utilized. All three utilized entropy weighting in order to make an objective assignment of each criterion’s importance, avoiding the subjectiveness frequently linked with expert-weighting. All three models ranked the airfoils independently on a relative performance basis. The individual rankings were aggregated through the Borda Count method, which yielded an overall, composite ranking. Spearman’s rank correlation was also utilized to verify the consistency among the models of the MCDM, which confirmed high agreement and further supported the validity of the overall ranks.

Throughout all Reynolds numbers and models of evaluation, the airfoils NACA 4521, NACA 4518, and NACA 4421 were always among the leading performers. These airfoils provide strong aerodynamic efficiency with high lift-to-drag ratio and low variance, making them suitable choices for small-scale wind turbine applications in turbulent and variable wind environments; their main advantages dwell in their moderate camber and thickness profiles that promote early lift development without excessive drag or instability. These findings have direct implications for wind turbine blade design since choosing and identifying the optimal airfoil reduces the drag and instability factors while it increases the lift, which in turn reduces performance losses due to off-design conditions, enhances energy capture by the blade, and improves the energy efficiency of the whole turbine. Despite the reliability of the results, this study data is based on XFoil simulation results, which could lead to a simplified flow separation phenomena; hence, it is recommended for future work to include high-fidelity CFD simulations and wind tunnel experiments, as well as include structural performance and manufacturing feasibility into the decision framework. Furthermore, the inclusion of other criteria such as acoustic performance could further enhance the comprehensiveness and realism of the airfoil selection process.

Footnotes

ORCID iD

Ilyass Bougaa

Author contributions

Ilyass Bougaa: Conceptualization, Methodology, Investigation, Writing – Original Draft, Visualization.

Nissrine El allami: Writing–Review & Editing, Validation, Investigation.

Lhoussaine Tenghiri: Supervision, Review & Editing, Project Administration, Validation.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be available upon reasonable request.*

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