Airfoil Performance of Small-Scale Vertical Axis Wind Turbines Under Urban Low Wind Speeds Using DMST and LLFVW Models

Abstract

This work presents a comparative analysis of six airfoil profiles for small-scale vertical axis wind turbines (VAWTs) operating under low wind speeds (2–8 m/s) typical of urban environments. Aerodynamic performance during startup and nominal operation is investigated using two widely adopted modeling approaches, the Double Multiple Streamtube (DMST) and the Lifting Line Free Vortex Wake (LLFVW) methods, implemented in the open-source QBlade framework. The objective of the study is to evaluate relative airfoil performance and the consistency of observed trends across aerodynamic models commonly used in early-stage VAWT design. The results demonstrate a fundamental trade-off between self-starting capability at low tip-speed ratios ($\lambda <2$) and power efficiency at nominal operating conditions ($2\le \lambda \le 4$). Low-Reynolds-number and VAWT-oriented airfoils (S1210, E387, and DU 06-W-200) show enhanced startup torque under weak inflow conditions, whereas symmetric NACA airfoils (NACA 0015 and NACA 0018) deliver higher power coefficients once operational tip-speed ratios are achieved. Comparison with experimental benchmark data indicates that the transient LLFVW model yields improved agreement relative to the stationary DMST approach, which tends to overestimate performance at moderate and high tip-speed ratios. Overall, the study provides practical guidance for airfoil selection in micro-scale VAWTs intended for urban applications, where reliable self-starting and efficient operation must be carefully balanced.

1. Introduction

Global urbanization is accelerating, and projections indicate that by 2050 nearly 68% of the world’s population will reside in cities, underscoring the urgency of developing sustainable and resilient urban energy systems. Urban growth intensifies the demand for water, energy, and materials, increasing pressure on non-renewable resources and complicating the transition toward low-carbon development. In Mexico, rapid metropolitan expansion and population densification make it essential to rethink energy production and distribution frameworks. Demographic forecasts estimate that the national population will reach 138 million by 2030 and 146 million by 2050 [1], while nearly 90% of municipalities will be predominantly urban by mid-century [2].

This context highlights a growing need for distributed renewable energy technologies capable of operating efficiently within complex urban environments. Small-scale wind systems offer an opportunity to exploit localized wind resources on rooftops or integrated architectural surfaces [3]. By decentralizing generation, such systems can reduce grid congestion, lower local emissions, and enhance energy resilience during peak-demand periods or supply disruptions. As emphasized by the United Nations, access to clean and efficient energy systems is a cornerstone of safe, resilient, and sustainable cities, aligning with Sustainable Development Goals (SDGs) 7, 11, and 13 [4]. Accordingly, integrating decentralized wind energy into urban planning strategies represents a viable pathway for climate adaptation and improved energy security.

Urbanization trends further reinforce this motivation. Mexico’s major metropolitan regions, including the Valle de México, Monterrey, and Guadalajara, which are projected to grow by approximately 5 million inhabitants by 2030, while state capitals and medium-sized cities will incorporate an additional 8.3 million residents [5]. This increasing concentration of population and infrastructure strengthens the case for compact, low-impact energy technologies that can operate within existing built environments without extensive land-use requirements.

Among the available technologies, micro wind turbines particularly vertical-axis wind turbines (VAWTs) stand out as promising candidates for urban deployment [6]. The highly turbulent and multidirectional wind fields typical of cities often favor VAWTs over horizontal-axis turbines. Their key advantages include omnidirectional operation, low cut-in wind speeds, reduced acoustic emissions, and suitability for low-Reynolds-number flow regimes, all of which are critical for dense urban settings. Accordingly, the present study focuses on lift-driven, Darrieus-type VAWTs, for which aerodynamic performance is governed primarily by airfoil characteristics and can be consistently assessed using blade-element-based models.

A Darrieus-type VAWT consists of straight or curved blades mounted vertically around a central axis. As the rotor rotates, each blade experiences periodic variations in angle of attack, generating lift and torque [7]. The resulting power output depends on blade geometry, chord length, rotor radius, and operating tip-speed ratio [8]. Despite relatively high efficiency at optimal tip-speed ratios ($\lambda \approx 2-5$), conventional Darrieus turbines exhibit poor self-starting performance due to low initial torque [9], which remains a critical limitation for autonomous operation in low-wind urban environments.

Urban wind conditions differ markedly from those of conventional wind energy sites. Mean wind speeds in cities are typically reduced due to surface roughness and flow obstruction [10,11], with reported average velocities between 2 m/s and 6 m/s and gusts reaching 6 m/s to 9 m/s [12,13]. In addition, urban flows exhibit strong wind direction variability and elevated turbulence intensity, frequently exceeding 20–30%, driven by building-induced flow distortion and shear [14]. These characteristics pose significant aerodynamic challenges for micro-scale wind turbines and accentuate the importance of airfoil behavior under unsteady, low-Reynolds-number conditions.

As a representative example of a large, densely built metropolitan area, wind conditions in the Zona Metropolitana del Valle de México (ZMVM) are considered in this study. Data from the Global Wind Atlas indicate mean wind speeds of approximately 1.75 m/s at 10 m height and 3.14 m/s at 50 m, with observed values ranging from 0.19 to 7.32 m/s (Figure 1). While local wind characteristics vary among cities worldwide, these values are broadly representative of the low-speed, highly disturbed environments in which urban micro-scale turbines are expected to operate.

Despite growing interest in urban VAWTs, their aerodynamic design remains challenging due to the combined effects of low Reynolds numbers, cyclic variations in angle of attack, and strong sensitivity to airfoil characteristics. Existing studies employ a wide range of modeling approaches, including momentum-based methods such as the Double Multiple Streamtube (DMST) model, vortex-based methods, and high-fidelity CFD simulations [15,16]. While CFD provides detailed insight into unsteady flow phenomena, its computational cost often limits its applicability in early-stage design and parametric studies [17,18]. Consequently, low- and medium-fidelity models remain widely used but rely heavily on airfoil polar data and modeling assumptions, particularly under low-Reynolds-number and high-incidence conditions.

As a result, airfoil selection studies for VAWTs often focus on either symmetric profiles or general-purpose low-Reynolds-number airfoils, whereas fewer works systematically assess VAWT-specific airfoils designed to sustain large angles of attack and mitigate dynamic stall [19]. Moreover, many investigations emphasize optimal operating conditions, while self-starting behavior at low tip-speed ratios critical for urban micro-scale applications is less consistently addressed [20,21]. This motivates the need for comparative studies that assess airfoil performance trends across different aerodynamic models and operating regimes relevant to urban wind environments.

The present study addresses this gap through a systematic comparative analysis of six airfoil profiles commonly considered for small-scale, low-Reynolds-number VAWTs operating under typical urban wind speeds (2–8 m/s). Two aerodynamic modeling approaches of differing fidelity are employed: the Double Multiple Streamtube (DMST) model and the Lifting Line Free Vortex Wake (LLFVW) method, both implemented within the open-source QBlade framework. While several low- and medium-fidelity solvers such as CACTUS [22], OpenFAST, and other BEM-based tools reported in the literature [23,24] are available for VAWT performance prediction, QBlade is selected because it enables multiple aerodynamic models to be applied within a unified environment using identical airfoil polar data and rotor definitions. Rather than introducing new aerodynamic formulations, this work focuses on evaluating the consistency and robustness of airfoil performance trends across modeling approaches, with particular emphasis on the trade-off between self-starting capability at low tip-speed ratios and power efficiency under nominal operating conditions.

The analysis is intentionally restricted to lift-driven, Darrieus-type VAWTs, for which aerodynamic performance is governed primarily by airfoil characteristics and can be consistently assessed using blade-element-based models. Drag-based configurations, such as Savonius or hybrid rotors, rely on fundamentally different operating principles and performance metrics, and are therefore excluded to preserve methodological consistency and ensure that the conclusions remain directly applicable to airfoil selection and early-stage aerodynamic design.

Although urban wind environments are characterized by high turbulence intensity and strong temporal variability [25], these effects are not explicitly modeled in the present simulations. This simplification is adopted to isolate baseline aerodynamic trends associated with airfoil geometry and to enable consistent comparisons across airfoils and modeling approaches. Neglecting atmospheric turbulence is expected to primarily influence unsteady loading and dynamic stall intensity, while mean power coefficient trends and relative airfoil ranking are expected to remain qualitatively robust. The implications of this assumption are discussed in Section 5.5.

The main contributions of this work are threefold: (i) a comparative evaluation of startup torque and power coefficient for six airfoils representative of low-Reynolds-number, VAWT-specific, and symmetric profiles under urban wind conditions; (ii) a cross-model assessment of airfoil performance using both DMST and LLFVW, highlighting model-dependent and model-independent conclusions; and (iii) validation of LLFVW predictions against experimental benchmark data to establish confidence bounds for design-oriented simulations.

This article is structured as follows. Section 2 introduces the aerodynamic principles governing VAWT operation and key performance coefficients. Section 3 presents the airfoil profiles, turbine geometry, and simulation setup. Section 4 compares numerical predictions against experimental data. Section 5 reports the main findings, and Section 6 summarizes the conclusions of the study.

2. Background: Fundamental VAWT Aerodynamics

This section summarizes the governing aerodynamic principles of Darrieus-type vertical-axis wind turbines (VAWTs) to provide the physical foundation for the modeling and performance analysis presented in later sections.

A VAWT extracts kinetic energy from the wind through the aerodynamic forces acting on its rotating blades. These forces depend on the free-stream wind speed $U_{\infty }$, the turbine angular velocity $\omega$, and the instantaneous azimuthal position of the blade $\theta$. The blade element experiences a relative velocity that can be decomposed into tangential $U_c$ (chordwise) and normal components $U_n$ with respect to the rotor trajectory. These components are given by

$$\begin{array}{cc}\hfill U_c& =\omega R+U_{a}cos\theta ,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill U_n& =U_{a}sin\theta ,\hfill \end{array}$$

(2)

where $U_a$ is the local axial (induced) velocity, R is the rotor radius, and $\theta$ is the blade azimuthal angle. The magnitude of the relative velocity is

$$U_r=\sqrt{U_c^2+U_n^2}.$$

(3)

These velocity relationships are illustrated in Figure 2. The relative velocity defines the instantaneous angle of attack,   

$$\alpha ={tan}^{-1}\left[{\displaystyle \frac{U_{a}sin\theta }{U_{a}cos\theta +\omega R}}\right]-\gamma .$$

(4)

where $\gamma$ is the pitch angle (set to zero in the present study). Introducing the tip-speed ratio (TSR),

$$\lambda ={\displaystyle \frac{\omega R}{U_{\infty }}},$$

(5)

and the axial induction factor $a=U_a/U_{\infty }-1$, Equation (4) can be expressed in nondimensional form as

$$\alpha ={tan}^{-1}\left[{\displaystyle \frac{(1-a)sin\theta }{(1-a)cos\theta +\lambda }}\right]-\gamma .$$

(6)

2.1. Aerodynamic Forces on the Blade

The lift L and drag D generated by the blade element are

$$\begin{array}{cc}\hfill L& =\frac{1}{2}\rho U_r^{2}S{C}_l\left(\alpha \right),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill D& =\frac{1}{2}\rho U_r^{2}S{C}_d\left(\alpha \right),\hfill \end{array}$$

(8)

where $S=cH$ is the blade planform area, and $C_l$ and $C_d$ are the sectional lift and drag coefficients. These forces are resolved into tangential and normal components with respect to the rotor trajectory:

$$\begin{array}{cc}\hfill F_t& =Lsin\alpha -Dcos\alpha ,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill F_n& =Lcos\alpha +Dsin\alpha .\hfill \end{array}$$

(10)

The tangential force $F_t$ is the component responsible for torque generation and directly governs both the startup behavior and the nominal efficiency of the turbine. Because $\alpha \left(\theta \right)$ varies strongly over the revolution, the VAWT operates under inherently unsteady aerodynamic conditions involving dynamic stall, flow curvature effects, and Reynolds number sensitivity.

2.2. Time-Averaged Aerodynamic Loads and Power Output

The time-averaged thrust force T and torque Q are obtained by integrating the instantaneous aerodynamic loads over one full rotor revolution:

$$\begin{array}{cc}\hfill T& ={\displaystyle \frac{N}{2\pi }}{\int }_0^{2\pi }\left(F_{t}cos\theta -F_{n}sin\theta \right)d\theta ,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill Q& ={\displaystyle \frac{NR}{2\pi }}{\int }_0^{2\pi }{F}_t\left(\theta \right)d\theta ,\hfill \end{array}$$

(12)

The mean mechanical power is

$$P=Q\omega .$$

(13)

2.3. Aerodynamic Coefficients

The aerodynamic performance of a VAWT is expressed using the nondimensional coefficients

$$\begin{array}{cccccc}\hfill C_p& ={\displaystyle \frac{P}{\frac{1}{2}\rho U_{\infty }^3{A}_s}},\hfill & \hfill C_{Ft}& ={\displaystyle \frac{F_t}{\frac{1}{2}\rho U_{\infty }^2{A}_s}},\hfill & \hfill C_Q& ={\displaystyle \frac{Q}{\frac{1}{2}\rho U_{\infty }^2{A}_{s}R}},\hfill \end{array}$$

(14)

where $A_s=2RH$ is the rotor swept area. The power coefficient $C_p$ represents the ratio of the extracted mechanical power to the available wind power. The tangential force coefficient $C_{Ft}$ describes the instantaneous nondimensional torque-producing load on a blade. The torque coefficient $C_Q$ quantifies the nondimensional torque delivered to the shaft.

The applicability and assumptions of the quasi-steady framework are discussed in Section 3.2. The relationships summarized in this section provide the aerodynamic framework used by the numerical models described in Section 3.3.

3. Methods and Procedures

3.1. Airfoil Selection and Turbine Geometry

Previous studies on vertical-axis wind turbines (VAWTs) have established that airfoil geometry strongly influences self-starting capability and power production, particularly under the low Reynolds number and highly unsteady inflow conditions characteristic of urban environments [9,25]. However, many existing investigations are restricted to a narrow set of airfoils, focus on a single operating regime, or rely on a single aerodynamic modeling approach. These limitations hinder the extraction of generalizable trends suitable for early-stage design [19,26,27].

To address these limitations, the present study examines six airfoil profiles selected to represent distinct aerodynamic design strategies relevant to small-scale Darrieus VAWTs (

Table 1. Airfoils evaluated in this study.

Airfoil	Type	Thickness [%]	Camber [%]	Remarks
NACA 0012	Symmetric	12.0	0.0	Symmetrical
NACA 0015	Symmetric	15.0	0.0	Symmetrical
NACA 0018	Symmetric	18.0	0.0	Symmetrical
S1210	Cambered	12.01	7.27	Optimized for low Reynolds numbers
DU 06-W-200	Cambered	19.78	0.63	Designed for small-scale VAWTs [28]
Eppler E387	Cambered	9.07	3.80	Optimized for low Reynolds numbers

). The set includes three classical symmetric NACA airfoils (NACA 0012, 0015, and 0018; Figure 3), which are widely used as reference profiles in VAWT research, as well as three cambered airfoils developed for low-Reynolds-number or VAWT-specific applications (S1210, Eppler E387, and DU 06-W-200; Figure 4). This combination enables a systematic evaluation of the influence of airfoil thickness, camber, and low-Reynolds-number optimization on startup torque, dynamic stall behavior, and nominal power efficiency.

Rather than conducting an exhaustive survey of airfoil designs, the objective is to identify whether consistent performance trends emerge across airfoils with fundamentally different aerodynamic characteristics when analyzed using two commonly employed low- and medium-fidelity models (DMST and LLFVW). This comparative framework supports the identification of robust trade-offs and design-relevant tendencies while remaining compatible with early-stage VAWT development.

The symmetric NACA airfoils serve as a baseline set due to their extensive use in prior VAWT studies [29,30,31,32,33]. Their geometric symmetry ensures comparable upwind and downwind behavior, allowing a controlled assessment of thickness effects. To isolate the influence of camber and low-Reynolds-number tailoring, three additional airfoils were included (Figure 4). The Eppler E387 and Selig S1210 represent profiles optimized for high lift at low Reynolds numbers, while the DU 06-W-200 airfoil was specifically developed for small-scale VAWTs and exhibits enhanced torque generation and delayed stall under unsteady inflow conditions typical of micro-VAWT operation [28].

Together, these six profiles span a broad range of geometric and aerodynamic characteristics, enabling a focused investigation of which design features most strongly influence startup performance and efficiency of small VAWTs operating in typical urban wind speeds (2–8 m/s).

To isolate the impact of airfoil geometry, a fixed turbine configuration was employed for all cases. The rotor geometry follows a standard micro-VAWT design reported in the literature [34,35] and consists of a three-bladed, straight-bladed Darrieus H-rotor. The emphasis on straight blades serves to further isolate the aerodynamic effects of the airfoil by circumventing the introduction of additional geometric variables. This is achieved by avoiding the consideration of phenomena such as helical blade-induced wake interactions, which have the potential to obscure trends in airfoil performance. The corresponding geometric parameters are summarized in

Table 2. Rotor geometric and operational parameters.

Parameter	Symbol	Value
Rotor radius	R	0.515 m
Blade chord	c	0.085 m
Blade height	H	1.5 m
Number of blades	N	3
Solidity	$\sigma ={\displaystyle \frac{Nc}{2\pi R}}$	0.079
Solidity (Sandia) 1	$\sigma$	0.247
Air density	$\rho$	1.225 kg/m3
Wind velocity range	$U_{\infty }$	2–8 m/s

¹ Sandia Lab definition for solidity $\sigma ={\displaystyle \frac{NcH}{A_s}}$ [36].

and illustrated in Figure 5.

3.2. Applicability and Assumptions of the Quasi-Steady Framework

The aerodynamic formulations employed in this study are based on a quasi-steady blade-element framework, in which the instantaneous aerodynamic forces acting on each blade section are evaluated from static airfoil polars as a function of the local angle of attack and relative velocity. This approach is widely adopted in low- and medium-fidelity models for vertical-axis wind turbines and is commonly used for parametric and comparative analyses in early-stage design.

In the present work, the quasi-steady formulation is applied over wind speeds of $U_{\infty }$ = 2–8 m/s and tip-speed ratios $0.5\le \lambda \le 5$, corresponding to Reynolds numbers on the order of $\mathcal{O}\left({10}^5\right)$, which are representative of micro-scale VAWT operation in urban environments. Within this range, quasi-steady models have been shown to provide reasonable predictions of mean aerodynamic loads and performance trends, particularly when the objective is to compare relative airfoil behavior rather than to resolve detailed transient flow physics.

It is acknowledged that VAWTs operate under inherently unsteady conditions due to cyclic variations in angle of attack, dynamic stall, and blade-wake interactions. These effects are not explicitly resolved within the quasi-steady formulation itself. Instead, their influence is accounted for at the modeling level through the incorporation of dynamic stall modeling and wake-resolving approaches in the LLFVW simulations, as described in Section 3.3.

No additional engineering corrections related to structural components, such as strut or support-structure interference models or tower-shadow effects, were included in the present study. The rotor is therefore modeled as an isolated set of blades in both aerodynamic approaches in order to isolate the influence of airfoil characteristics and ensure a consistent basis for comparison.

Within the quasi-steady DMST framework implemented in QBlade, standard finite blade length and three-dimensional correction models are applied to account for tip-loss and spanwise flow effects, as is customary in streamtube-based VAWT modeling. These corrections are intrinsic to the DMST formulation and improve the prediction of mean aerodynamic loads without introducing airfoil-dependent tuning parameters. In contrast, no equivalent empirical tip-loss or three-dimensional correction models are applied in the LLFVW simulations, where three-dimensional effects are inherently represented through the explicit resolution of the vortex wake.

The omission of struts, support structures, and tower effects is consistent with common practice in both CFD-based and QBlade-based VAWT studies, where blades are frequently modeled in isolation to facilitate systematic aerodynamic comparisons and parametric analyses [27,35]. As a result, the present analysis emphasizes relative airfoil performance trends rather than absolute load or power prediction for a specific turbine installation.

3.3. Aerodynamic Modeling Approaches

Based on the aerodynamic framework outlined in Section 2, two numerical models of different fidelity are employed to predict the performance of the VAWT configuration considered in this study: (i) a low-fidelity Double-Multiple Streamtube (DMST) model and (ii) a medium-fidelity Lifting Line Free Vortex Wake (LLFVW) model. These approaches were deliberately selected to balance computational efficiency and physical fidelity, and to enable a comparative assessment of airfoil performance trends across modeling assumptions commonly used in early-stage VAWT design.

3.3.1. Double-Multiple Streamtube Model (DMST)

The Double-Multiple Streamtube (DMST) model implemented in QBlade [37,38] simulates VAWT performance using a momentum-theory-based approach. As illustrated in Figure 6, the rotor is divided into several independent streamtubes, each represented by an actuator disk. This allows separate treatment of upwind and downwind induction regions, improving accuracy over simpler single-streamtube models [39,40]. Despite its efficiency, DMST may overpredict power in high-solidity turbines [41].

The induced velocities at the upwind ($U_u$) and downwind ($U_d$) sides are:

$$\begin{array}{cc}\hfill U_u& =(1-a_u)U_{\infty }=\frac{U_{\infty }+U_e}{2},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill U_d& =(1-2a_u)(1-a_d)U_{\infty }=\frac{U_e+U_w}{2},\hfill \end{array}$$

(16)

where $a_u$ and $a_d$ are the induction factors, and $U_e$ and $U_w$ are the velocities exiting the upwind and downwind actuators.

From one-dimensional momentum theory, the thrust coefficient used in the DMST formulation is

$$C_t=4a(1-a),$$

(17)

while from blade element theory,

$$C_t={\displaystyle \frac{\overline{T}}{\frac{1}{2}\rho U_{\infty }^2{A}_{st}}},$$

(18)

with $\overline{T}=\frac{N\Delta \theta }{2\pi }T$ and $A_{st}=R\Delta H\Delta \theta sin\theta$. The elemental thrust is:

$$T=\frac{1}{2\pi }\rho U_r^{2}c\left(-C_{t}cos\theta +C_{n}sin\theta \right).$$

(19)

For both upwind and downwind regions, the thrust coefficients from momentum and blade element theories are iteratively matched:

$$\begin{array}{cc}\hfill C_t^u& =4a_u(1-a_u)={\displaystyle \frac{Nc}{2\pi }}{\left({\displaystyle \frac{U_{r,u}}{U_{\infty }}}\right)}^2\left(-C_t{\displaystyle \frac{cos\theta }{sin\theta }}+C_n\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill C_t^d& =4a_d(1-a_d)={\displaystyle \frac{Nc}{2\pi }}{\left({\displaystyle \frac{U_{r,d}}{U_{\infty }}}\right)}^2\left(-C_t{\displaystyle \frac{cos\theta }{sin\theta }}+C_n\right).\hfill \end{array}$$

(21)

This iterative process continues until convergence between momentum and blade-element formulations, ensuring consistency between aerodynamic loading and the induced flow in each streamtube.

3.3.2. Lifting Line Free Vortex Wake (LLFVW)

The Lifting Line Free Vortex Wake (LLFVW) method combines classical lifting line theory with an explicit representation of the rotor wake using discrete vortex elements. Each blade is modeled as a lifting line, represented by a bound vortex located at the quarter chord position along the blade span. The bound vortex strength varies with local aerodynamic loading and is determined from sectional airfoil polars.

Temporal and spatial variations in bound circulation lead to the formation of wake vortices. Time varying circulation at the blade trailing edge produces shed vortices, while spanwise gradients in circulation generate trailing vortices. These vortical structures are convected downstream and together form a fully three dimensional free vortex wake that evolves in time.

The induced velocity field acting on the blades is obtained by superposing the contributions from bound, shed, and trailing vortices using a Biot Savart formulation. Aerodynamic forces are then computed from the local circulation using the Kutta Joukowski relation. Tabulated airfoil polars are used to relate circulation to the local angle of attack [42,43].

In comparison with blade element momentum or double multiple streamtube models, LLFVW explicitly resolves wake kinematics and captures transient and three dimensional flow phenomena such as dynamic stall, vortex shedding, and tip vortex formation. This leads to improved accuracy in unsteady conditions or at high tip speed ratios, where momentum based models lose validity. At the same time, LLFVW remains computationally more efficient than higher fidelity approaches such as RANS, LES, or Lattice Boltzmann methods [35].

Within the lifting line formulation, the sectional circulation $\partial \Gamma$ is related to the lift force through

$$\partial F_L=\rho U_r\partial \Gamma ,$$

(22)

where $U_r$ is the relative velocity from free-stream, blade motion, and induced components. The induced velocity is computed from bound and wake elements:

$$U_i=-{\displaystyle \frac{1}{4\pi }}\int \Gamma {\displaystyle \frac{\overrightarrow{r}\times d\overrightarrow{l}}{r^3}}.$$

(23)

Wake convection is performed explicitly in time. Newly shed and trailing vortices are released at each time step and convected according to the local induced velocity field. A desingularized Biot-Savart formulation is used to account for viscous diffusion through a finite vortex core radius model [44,45]:

$$U_i=-{\displaystyle \frac{1}{4\pi }}\int \Gamma {\displaystyle \frac{\overrightarrow{r}\times d\overrightarrow{l}}{{\left(r^2+r_c^2\right)}^{3/2}}},$$

(24)

with the core radius $r_c$ evolving through empirical diffusion laws.

Overall, the LLFVW method provides a favorable balance between computational efficiency and physical fidelity. This makes it well suited for modeling the unsteady aerodynamics and complex wake dynamics characteristic of vertical axis wind turbines.

Dynamic stall effects within the LLFVW framework are represented using semi-empirical dynamic stall models derived from the Beddoes–Leishman (BL) family. Classical BL-type models describe unsteady lift, drag, and pitching moment through a combination of attached-flow, separated-flow, and vortex-induced contributions, typically represented using multiple coupled state variables and airfoil-dependent empirical parameters [46].

In QBlade, a more detailed BL-type implementation is available through the ATEFlap model, which explicitly represents leading-edge vortex formation, convection, and shedding [47,48]. This approach is well suited for applications focused on detailed transient load analysis, but it can become sensitive to parameter tuning and airfoil-specific calibration, particularly under deep-stall conditions.

In the present work, dynamic stall effects are modeled using the IAG dynamic stall model implemented within the LLFVW solver. The IAG model represents an improved formulation of the Beddoes–Leishman approach, with enhanced treatment of separated-flow and vortex-induced effects [49]. It employs an indicial (state-space) formulation in which unsteady lift, drag, and moment responses are described using a limited number of first-order lag states [50]. This formulation captures key dynamic stall features such as hysteresis, delayed stall onset, and post-stall lift recovery, while maintaining numerical robustness.

The IAG model was selected primarily because it has been shown to provide more accurate predictions in the post-stall and deep-stall regimes compared to standard Beddoes–Leishman implementations [49], particularly at low and moderate Reynolds numbers relevant to small-scale VAWTs.

As a result, LLFVW simulations are used not only to predict turbine performance, but also to assess whether airfoil performance rankings obtained using the DMST approach persist when unsteady wake effects and dynamic stall phenomena are explicitly resolved.

3.4. Simulation Setup

All aerodynamic simulations were performed using QBlade v2.0.9.1, an open-source software for airfoil analysis and wind turbine performance prediction. QBlade integrates the Blade Element Momentum (BEM) method for horizontal-axis wind turbines (HAWT) and provides both the Double-Multiple Streamtube (DMST) and Lifting Line Free Vortex Wake (LLFVW) models for VAWT applications. The workflow adopted in this study is summarized in Figure 7 and consists of five stages:

1.

Airfoil definition: Airfoils were either generated using QBlade’s built-in NACA generator (4- and 5-digit series) or imported from external .dat geometry files.

2.

Polar generation (XFOIL): The integrated XFOIL module in QBlade was employed to compute lift and drag polars at Reynolds numbers of $3\times {10}^4$, $5\times {10}^4$, $1\times {10}^5$, and $2\times {10}^5$. XFOIL is based on a coupled inviscid–viscous formulation, in which the boundary layer is solved using an integral method and coupled to a panel-based potential-flow solver. Laminar–turbulent transition is predicted using an $e^N$ envelope method, whereby transition is assumed once the integrated amplification of Tollmien–Schlichting disturbances reaches a critical value $N_{\mathrm{crit}}$. This parameter is empirically related to the free-stream turbulence intensity $Tu$ through correlations such as that proposed by van Ingen [51]. Following common practice in low-Reynolds-number airfoil studies and prior VAWT-related investigations, a value of $N_{\mathrm{crit}}=9$ was adopted, representative of smooth surfaces under low-to-moderate turbulence conditions [52]. Although urban environments are characterized by elevated turbulence levels, XFOIL does not explicitly resolve atmospheric turbulence effects; the selected value therefore serves as a conventional baseline for comparative analysis.

3.

360° polar extrapolation: Because VAWT blades routinely experience dynamic stall and reverse-flow conditions, all airfoil polars were extended to a full ${360}^{\circ }$ angle-of-attack range using the Montgomerie model [53]. This ensures a physically consistent representation of deep-stall and post-stall behavior.

4.

Rotor design: The Aerodynamic Blade Design module was used to define turbine geometry, including airfoil selection, rotor diameter, aspect ratio, chord distribution, and blade height.

5.

Performance simulation: Steady-state analyses were performed using the DMST model, while transient simulations employed the LLFVW solver. Both models were evaluated over a range of operating conditions (wind speed 2–8 m/s, TSR 0.5–5).

Although XFOIL-based polars are known to exhibit uncertainty in deep stall, particularly at low Reynolds numbers, their use is justified by the focus on comparative airfoil performance under identical aerodynamic inputs, such that any systematic errors introduced during polar generation are expected to affect all airfoils similarly. The influence of polar uncertainty on global turbine performance is further assessed indirectly through validation of DMST and LLFVW predictions against experimental power coefficient data. In this context, each LLFVW data point shown in the $C_P$–TSR plots of Section 5 corresponds to an independent time-resolved simulation at a fixed operating condition, rather than a post-processed sweep of tip-speed ratio.

The Montgomerie polar extrapolation employed in this work has been widely adopted in VAWT studies involving both symmetric and cambered airfoils, particularly when the objective is to compare relative aerodynamic trends rather than to predict absolute loads [54,55]. This approach is therefore consistent with the scope of the present study, which focuses on robustness of performance rankings across modeling fidelities. It is nevertheless acknowledged that cambered, low-Reynolds-number airfoils such as the S1210, E387, and DU 06-W-200 may exhibit pronounced aerodynamic asymmetry and earlier stall onset, increasing uncertainty in the post-stall regime. Accordingly, results involving these airfoils are interpreted in a qualitative and comparative sense. This limitation does not compromise the main conclusions of the study, particularly the observed trade-off between self-starting capability at low tip-speed ratios and nominal power efficiency.

The main environmental conditions and numerical parameters adopted in the DMST and LLFVW simulations are summarized in

Table 3. Simulation parameters used in QBlade for DMST and LLFVW analyses.

Parameter	Value/Setting
Environmental and operating conditions
Air density, $\rho$	1.225 kg/m3
Kinematic viscosity, $\nu$	1.647 × 10−5 m2/s
Wind velocity range	2–8 m/s
Tip-speed ratio range	0.5–5
DMST (Multi-DMS) settings
Blade elements	20
Maximum iterations	500
Convergence tolerance	${10}^{-5}$
Tip-loss correction	Enabled
Variable induction factors	Enabled
LLFVW settings
Blade panels	20
Dynamic stall model	IAG model
Wake formulation	Free vortex wake
Velocity integration scheme	2nd-order predictor-corrector (backward)
Azimuthal discretization	$4^{°}$
Simulation length	10 revolutions

.

4. Validation

4.1. Aerodynamic Coefficient Extrapolation

Because XFOIL does not converge reliably beyond static stall, the computed aerodynamic coefficients were extrapolated to obtain full ${360}^{°}$ polars for both the DMST and LLFVW simulations. QBlade provides two commonly used post-stall extrapolation approaches for this purpose: the Montgomerie method [53] and the Viterna–Corrigan model [56]. Although both methods extend pre-stall airfoil data into the fully separated flow regime, they rely on different physical assumptions and are suited to different operating conditions.

The Viterna–Corrigan model was originally developed for horizontal-axis wind turbines operating at high angles of attack and relatively high Reynolds numbers. Beyond stall, lift and drag are extrapolated using empirical relationships derived from flat-plate theory and momentum considerations, with the drag coefficient asymptotically approaching a maximum value governed by airfoil aspect ratio. This approach has been shown to perform well for mildly cambered airfoils with gradual stall characteristics and is therefore widely adopted in blade element momentum solvers for HAWTs. However, under the operating conditions typical of vertical-axis wind turbines characterized by large and rapid angle-of-attack variations, frequent flow reversal, and pronounced dynamic stall the quasi-steady flat-plate assumptions underlying the Viterna–Corrigan model can lead to unrealistic force predictions, particularly at low Reynolds numbers. These include overestimated lift recovery and underestimated drag during deep stall phases.

An alternative approach is provided by the Montgomerie post-stall extrapolation method, which is widely implemented in VAWT performance tools, including QBlade [57]. The Montgomerie method assumes that, beyond static stall, airfoil behavior gradually approaches that of an idealized thin plate, with lift decreasing smoothly and drag increasing toward limiting post-stall values. Using geometric and aerodynamic constraints, the method constructs a continuous and physically plausible ${360}^{°}$ polar by smoothly connecting pre-stall XFOIL data to asymptotic post-stall estimates.

The Montgomerie method was selected for the present study because it provides consistent and smooth aerodynamic coefficients over the full angle-of-attack range without requiring extensive airfoil-specific calibration. In contrast to the Viterna–Corrigan model, which is best suited to high-Reynolds-number HAWT applications, the Montgomerie approach is more robust under the large-amplitude, rapidly varying incidence conditions characteristic of vertical-axis rotors. Its physically consistent polar extension is well aligned with the quasi-steady aerodynamic inputs required by both DMST and LLFVW formulations, making it an appropriate choice for comparative airfoil performance analysis in VAWT applications.

To assess the reliability of the extrapolated polars, the QBlade-generated $C_L$ and $C_D$ curves for the NACA 0012 airfoil at $\mathrm{Re}=5\times {10}^5$ were compared with the experimental measurements of Sheldahl and Klimas [58]. As shown in Figure 8, the extrapolated lift and drag curves reproduce the pre-stall, stall, and post-stall trends of the experimental data, including the characteristic inverted parabolic shape of the drag coefficient associated with fully separated flow.

The relative deviation was quantified using

$$e=\left|{\displaystyle \frac{Y_{\mathrm{QBlade}}-Y_{\mathrm{Exp}}}{Y_{\mathrm{Exp}}}}\right|\times 100,$$

(25)

with the results summarized in

Table 4. Aerodynamic coefficient extrapolation (NACA 0012, $\mathrm{Re}=5\times {10}^5$): comparison of QBlade and experimental data [58].

	Experimental		QBlade		Error
Point	$\alpha$ [deg]	$C_L$ [-]	$\alpha$ [deg]	$C_L$ [-]	$e\left(\alpha \right)$ [%]	$e\left(C_L\right)$ [%]
I	10.22	0.929	13.82	1.231	35.23	32.44
II	44.47	1.075	47.70	0.862	7.28	19.80
III	139.49	−0.926	138.00	−0.855	1.07	7.66
Point	$\alpha$ [deg]	$C_D$ [-]	$\alpha$ [deg]	$C_D$ [-]	$e\left(\alpha \right)$ [%]	$e\left(C_D\right)$ [%]
IV	89.98	1.795	89.99	1.716	0.02	4.41

. The Montgomerie method reproduces extreme lift and drag values within 5–35%, which is consistent with previously reported uncertainty ranges for semi-empirical extrapolation methods in VAWT modeling [59]. This level of accuracy is sufficient for the comparative DMST and LLFVW analyses conducted in this study.

Although only the NACA 0012 airfoil is directly compared against experimental polar data, this validation is intended to assess the reliability of the extrapolation methodology rather than to individually validate each airfoil. Accurate post-stall aerodynamic data extending to $\pm {180}^{°}$ are essential for VAWT analysis, particularly at low tip-speed ratios encountered during startup; however, such data remain scarce. Consequently, semi-empirical extrapolation methods are widely employed in VAWT modeling, often under the assumption that fully separated post-stall behavior is only weakly dependent on airfoil shape [60]. While cambered low-Reynolds-number airfoils may exhibit increased uncertainty in deep stall, the present results are therefore interpreted in a comparative sense, consistent with the scope of the study and the objective of identifying robust performance trends rather than absolute load predictions.

4.2. DMST and LLFVW Model Validation

The performance of the DMST and LLFVW approaches was evaluated by comparing the predicted power coefficient $C_p$ against the experimental measurements of Battisti et al. [34]. Figure 9 presents the validation curves, and

Table 5. Power coefficient validation QBlade’s DMST and LLFVW predictions with the experimental data of Battisti et al. [34].

	Experimental	DMST	LLFVW	Error DMST	Error LLFVW
$\lambda$	$C_p$ [-]	$C_p$ [-]	$C_p$ [-]	$e\left(C_p\right)$ [%]	$e\left(C_p\right)$ [%]
1.50	0.0580	0.0553	0.0589	4.62	1.57
2.70	0.2842	0.3978	0.2800	39.98	2.50
3.52	0.0713	0.3686	0.1068	417.00	49.80

summarizes representative numerical values and corresponding percentage deviations.

Both models reproduce the onset of power generation and the low-TSR behavior. At $\lambda =1.50$, the DMST and LLFVW predictions differ from the experimental value by only 4.62% and 1.57%, respectively, demonstrating good agreement in the regime dominated by attached-flow aerodynamics.

As the TSR increases, the differences between the two approaches become more evident. Near the performance peak ($\lambda =2.70$), the LLFVW model maintains a close match to the experiment, with an error of 2.50%, while DMST overpredicts the power coefficient by nearly 40%. This behavior is consistent with the trend in Figure 9, where the DMST curve exhibits a sharper rise relative to the experimental data.

At higher TSR values, where wake expansion, induction losses, and unsteady effects become increasingly dominant, the limitations of DMST are particularly pronounced. At $\lambda =3.52$, the DMST prediction exceeds the experimental value by over 400%, while the LLFVW method although still overpredicts it shows a markedly lower deviation of 49.80%. The LLFVW curve also captures the experimentally observed drop in $C_p$ more faithfully, reflecting its more accurate representation of vortex dynamics and wake development.

Overall, the LLFVW method provides consistently superior agreement across the full operating range, reproducing both the magnitude and the location of the performance peak more accurately than DMST. For this reason, the LLFVW model is deemed the more reliable predictor of turbine performance in this study.

The markedly improved agreement of LLFVW with experimental data, relative to DMST, indicates that wake dynamics and unsteady induction effects play a dominant role in determining turbine performance, outweighing uncertainties associated with the XFOIL airfoil polar generation.

5. Results and Discussion

This section provides a thorough evaluation of the aerodynamic performance of six selected airfoil profiles using QBlade DMST and LLFVW simulations across wind speeds representative of urban environments. In line with the study’s objective of comparatively evaluating startup torque generation and nominal power performance for classical symmetric NACA, low-Reynolds number, and VAWT-specific profiles, the results are organized around three complementary operating regimes: (i) overall $C_p$-TSR trends at fixed inflow velocity, (ii) startup ($\lambda <2$), and (iii) nominal operation ($2\le \lambda \le 4$). Particular emphasis is placed on identifying the extent to which low-Re profiles facilitate self-starting under the weak winds typical of urban micro-turbines, as well as which airfoils offer increased efficiency once operational TSRs are reached. These analyses are interpreted alongside the tangential force as a function of the azimuth in Section 5.4, offering a comprehensive understanding of the mechanisms behind the observed performance differences.

5.1. Overall Performance Trends ($C_p$–TSR)

The aerodynamic performance of the six selected airfoil is evaluated by examining the power coefficient ($C_p$) as a function of tip-speed ratio ($\lambda$) across several free-stream velocities ($U_{\infty }$). Each figure is divided into two panels: panel (a) reports Double-Multiple Streamtube (DMST) predictions, representing a quasi-steady design-oriented model, whereas panel (b) presents Lifting-Line Free Vortex Wake (LLFVW) results, which incorporate unsteady aerodynamic effects and wake-induced velocity deficits. The latter therefore provides a more realistic estimate of operational performance, particularly in regimes characterized by dynamic stall and strong induction. These results complement the model-verification study reported in Section 4.2.

Figure 10 summarizes the behavior at $U_{\infty }=2.0$ m/s, corresponding to a low Reynolds-number regime $\mathcal{O}\left({10}^4\right)$. Under these conditions both models predict severe performance degradation: $C_p<0$ for $\lambda \gtrsim 2$ possibly to early laminar separation. The LLFVW curves lie below their DMST counterparts, consistent with the expected increase in unsteady losses. Notably, LLFVW predicts positive $C_p$ for the DU 06-W-200 and S1210 at $\lambda <2$, suggesting that these airfoils are capable of generating sufficient torque to overcome inertial and bearing losses during startup. In contrast, the symmetric NACA profiles show limited power extraction at this velocity, with NACA 0018 producing only a narrow positive peak near $\lambda \approx 3$.

At the intermediate velocity case ($U_{\infty }=4.0$ m/s), shown in Figure 11, both models converge to similar qualitative rankings: the NACA 0012, 0015, and 0018 yield the highest performance, whereas the E387 consistently underperforms. As expected, DMST predicts larger peak values (up to $C_p\approx 0.35$) and a broader optimal operating range. LLFVW predictions are more conservative, producing peaks of approximately $0.20$–$0.22$ and exhibiting a steeper decline beyond $\lambda \gtrsim 3.5$, reflecting stronger induction and pronounced dynamic-stall effects at increasing rotational speeds. The DU 06-W-200 and S1210 achieve positive $C_p$ only for low TSRs and experience a rapid decay thereafter.

Figure 12 highlights the divergence between the models at $U_{\infty }=6.0$ m/s. DMST continues to overpredict performance with broadened peaks reaching up to $C_p\approx 0.45$. In contrast, LLFVW predicts narrower maxima of approximately $0.25$–$0.27$ and a sharp drop beyond $\lambda \approx 3$, consistent with the onset of strong dynamic stall. For LLFVW, the three symmetric NACA airfoils again exhibit superior performance. The DU 06-W-200 and E387 provide moderate performance with peaks near $0.19$–$0.21$, while the S1210 shows substantial degradation, remaining below $C_p<0.05$ for most of the TSR range.

The highest velocity case, $U_{\infty }=8.0$ m/s, is shown in Figure 13. Here, the discrepancies between DMST and LLFVW become most pronounced. The DMST model predicts high peak efficiencies between $C_p\approx 0.3$–$0.4$ at $\lambda \approx 3$, and maintains unrealistically elevated values at large TSRs due to its quasi-steady formulation. LLFVW predictions, however, indicate that the true operational peaks occur earlier (around $\lambda \approx 2.5$) with maximum values between $0.23$ and $0.29$. Across both models, NACA 0018 remains the best performer, although the LLFVW curves show relatively small differences among the NACA family and DU 06-W-200. In contrast, DMST suggests diminished performance for the DU 06-W-200, while LLFVW indicates a more competitive behavior. The E387 provides intermediate performance in both models. The S1210 persists as the weakest airfoil, with LLFVW predicting early stall and negative $C_p$ for $\lambda >3$.

Overall, results across all velocities demonstrate (i) the consistent overprediction of performance by the DMST method, especially at higher TSRs; (ii) the superior behavior of the NACA family across these wind velocities; and (iii) the clear underperformance of the S1210 profile under all conditions examined.

5.2. Startup Performance ($\lambda <2$)

To complement the $C_p$-TSR curves presented in Section 5.1, the startup behavior of each airfoil is assessed using the mean power coefficient, $\overline{C_p}$, computed from LLFVW simulations over the interval $\lambda <2$. Figure 14 summarizes the resulting trends for inflow velocities between 2 and 8 m/s.

It is noted that the present startup analysis considers aerodynamic torque only and neglects the rotational inertia of the rotor–generator system. This modeling choice is intentional and is adopted to isolate the influence of airfoil aerodynamic characteristics on self-starting behavior under identical geometric and mass assumptions. Rotational inertia primarily governs the transient acceleration rate and the time required to overcome static equilibrium, rather than the sign and relative magnitude of the aerodynamic torque available during startup.

Similar aerodynamic-only startup analyses have been adopted in recent VAWT studies focused on low-TSR performance and self-starting trends, where the objective is to compare relative aerodynamic behavior rather than to predict startup time histories or cut-in dynamics [61,62]. Under such conditions, inertial effects act as a common scaling factor and do not alter the qualitative ranking of airfoils when rotor mass and geometry are held constant. Accordingly, the present results are interpreted in a comparative sense, emphasizing relative self-starting potential rather than absolute startup duration.

At $U_{\infty }=2.0$ m/s, most airfoils exhibit negative or near-zero $\overline{C_p}$, indicating insufficient torque to overcome aerodynamic and mechanical resistance. The DU 06-W-200 and S1210 represent clear exceptions, both generating consistently positive mean torque and thus demonstrating superior low-wind self-starting capability. By contrast, the NACA 0012, 0015, and 0018 profiles remain unable to initiate rotation at this speed.

When the wind velocity increases to 4.0 m/s, all airfoils achieve positive $\overline{C_p}$ within the startup regime, though their relative performance diverges. The DU 06-W-200 retains the highest mean torque, while the S1210 shows a moderate but steady improvement. The NACA 0012 and the low-Re E387 display the greatest increase from their low wind values, whereas the NACA 0015 and 0018 remain comparatively modest.

At higher inflow velocities (6–8 m/s), all profiles produce positive startup torque. The NACA 0015, NACA 0018, and S1210 exhibit the largest increases in $\overline{C_p}$, with the S1210 achieving the highest value at 8.0 m/s. The DU 06-W-200 maintains favorable performance but is less sensitive to velocity increases, consistent with its broad low-Re operating envelope. The E387 improves incrementally but remains moderate across all speeds, while the NACA 0012 consistently delivers the lowest $\overline{C_p}$ values.

The consistently poor startup performance of the NACA 0012 can be attributed to its symmetric and relatively thin geometry, which is unfavorable under the low tip-speed ratio and low Reynolds number conditions characteristic of the startup phase. At $\lambda <2$, the blades operate at large instantaneous angles of attack and experience highly unsteady inflow, where symmetric airfoils generate limited lift and undergo early flow separation. This results in intermittent torque production with frequent sign changes, as reflected by the negative $C_Q^{min}$ values and the highest torque ripple index (TRI) reported in

Table 6. Summary of the startup torque behavior ($\lambda <2$) for all airfoils (LLFVW data).

Airfoil	$\overline{{\mathit{C}}_{\mathit{Q}}}$1	${\mathit{C}}_{\mathit{Q}}^{max}$	${\mathit{C}}_{\mathit{Q}}^{min}$	${\mathit{C}}_{\mathit{Q},\mathbf{std}}$2	TRI 3
DU 06-W-200	0.0198	0.0394	0.0120	0.0077	1.3841
E387	0.0120	0.0281	−0.0041	0.0079	2.6866
NACA 0012	0.0056	0.0212	−0.0115	0.0085	5.8400
NACA 0015	0.0149	0.0475	−0.0181	0.0180	4.4023
NACA 0018	0.0133	0.0484	−0.0072	0.0147	4.1946
S1210	0.0175	0.0529	−0.0006	0.0150	3.0492

¹ Mean torque coefficient. ² Standard deviation of the torque coefficient. ³ Torque ripple index.

. Consequently, the NACA 0012 fails to sustain a stable positive aerodynamic moment, leading to systematically low $\overline{C_p}$ values across all wind speeds.

Table 6 summarizes the corresponding torque behavior. The DU 06-W-200 exhibits the highest and most stable torque production, characterized by the largest mean $\overline{C_Q}$ and the lowest torque ripple index (TRI), indicating smooth and reliable self-starting. In contrast, the NACA profiles and the S1210 show pronounced cycle-to-cycle variability, including intermittent negative torque, resulting in relatively high TRI values. The E387 demonstrates the lowest mean torque and frequent negative excursions, confirming its limited suitability for autonomous startup. Overall, the DU 06-W-200 emerges as the most favorable airfoil for consistent torque generation in the startup regime.

5.3. Nominal Operating Performance ($2\le \lambda \le 4$)

To provide a consolidated view of the airfoil performance within the typical operating TSR range of a VAWT, Figure 15 presents the mean power coefficient $\overline{C_p}$ obtained from the LLFVW simulations for $2\le \lambda \le 4$ across the inflow velocities considered (2–8 m/s). At $U_{\infty }=2.0$ m/s, all airfoils exhibit negative $\overline{C_p}$ values, confirming that none of the configurations can extract net positive power at such low wind speed. As the velocity increases, the NACA 0012, 0015, and 0018 profiles show a consistent improvement, with the 0015 and 0018 achieving the highest $\overline{C_p}$ values at 6–8 m/s.

In contrast, both the S1210 and the dedicated VAWT airfoil DU 06-W-200 produce substantially negative $\overline{C_p}$ at 4.0 m/s, indicating that they function as net drag devices and dissipate rather than generate energy within this TSR range. The DU 06-W-200 transitions to positive performance only at velocities above 4.0 m/s, reaching $\overline{C_p}$ levels comparable to those of the NACA airfoils at 8.0 m/s. The E387 remains inefficient throughout the nominal TSR range, exhibiting negative coefficients at low velocities and failing to match the performance of the NACA family at higher wind speeds.

A more detailed view of the nominal regime is provided in

Table 7. Summary of the nominal torque behavior ($2<\lambda <4$) for all airfoils (LLFVW data).

Airfoil	$\overline{{\mathit{C}}_{\mathit{Q}}}$1	${\mathit{C}}_{\mathit{Q}}^{max}$	${\mathit{C}}_{\mathit{Q}}^{min}$	${\mathit{C}}_{\mathit{Q},\mathbf{std}}$2	TRI 3
DU 06-W-200	0.0152	0.1116	−0.1534	0.0792	17.4006
E387	0.0364	0.0907	−0.0194	0.0350	3.0262
NACA 0012	0.0491	0.1071	−0.0216	0.0453	2.6219
NACA 0015	0.0602	0.1177	−0.0262	0.0510	2.3877
NACA 0018	0.0599	0.1164	−0.0116	0.0446	2.1375
S1210	−0.0337	0.0484	−0.1806	0.0639	6.7977

¹ Mean torque coefficient. ² Standard deviation of the torque coefficient. ³ Torque ripple index.

, which summarizes the torque behavior over the same TSR interval. The torque ripple index (TRI) is defined as

$$\mathrm{TRI}={\displaystyle \frac{C_{Q,\mathrm{std}}}{|\overline{C_Q}|}},$$

where $C_{Q,\mathrm{std}}$ is the standard deviation of the torque coefficient over one revolution and $\overline{C_Q}$ is its mean value. The NACA 0012, 0015, and 0018 profiles deliver the highest mean torque coefficients and exhibit relatively low torque-ripple indices, indicating efficient and stable power production. The E387 performs moderately, remaining below the NACA series but maintaining predominantly positive torque throughout this regime.

The DU 06-W-200, which excelled in the startup region (Section 5.2), displays highly inconsistent nominal behavior characterized by frequent negative torque excursions, large standard deviations, and the highest TRI value among all airfoils. This confirms that although DU 06-W-200 supports self-starting, it provides poor efficiency and stability once the turbine reaches operational TSRs. The S1210 performs worst overall: it exhibits negative mean torque, pronounced deep-stall characteristics, and highly asymmetric torque fluctuations, further reinforcing its unsuitability for sustained power production.

These trends highlight a clear decoupling between startup and nominal performance. Airfoils that are adequate for self-starting (such as DU 06-W-200 and S1210) tend to underperform once operating TSRs are reached, whereas the NACA 0015 and 0018 provide superior efficiency and torque stability during nominal operation at the expense of limited startup capability. This reinforces the need to balance both regimes when selecting airfoils for small scale VAWT applications.

Negative TRI values are non-physical and resulted from a sign inconsistency in the original post-processing; all TRI values are reported here as non-negative fluctuation metrics.

5.4. Aerodynamic Loads on the Rotor Blades

To further interpret the power-generation trends discussed in the previous subsections, this section examines the azimuthal variation of the tangential force coefficient, $C_{Ft}$, which directly governs torque production. Figure 16 and Figure 17 show $C_{Ft}\left(\theta \right)$ for all airfoils at two representative operating conditions: $\lambda =1.0$, characteristic of the startup regime, and $\lambda =3.0$, representative of nominal operation. Results correspond to the LLFVW simulations. Azimuth angles between $0^{°}$ and ${180}^{°}$ denote the upwind half-cycle, whereas ${180}^{°}$–${360}^{°}$ correspond to the downwind (leeward) half-cycle.

Positive values of $C_{Ft}$ indicate net driving torque; negative values denote torque loss, typically associated with drag-dominated or stalled flow. Across all profiles, energy extraction occurs primarily within a narrow sector around $\theta \approx {70}^{°}$–${90}^{°}$, consistent with the classical Darrieus torque-production mechanism [63,64].

• Startup regime ($\lambda =1$).

At $U_{\infty }=2$–4 m/s, the S1210 and DU 06-W-200 profiles exhibit the largest positive $C_{Ft}$ values on the upwind side, reflecting their design optimization for low Reynolds numbers. The E387 follows with moderate, yet consistently positive, tangential force. In contrast, the NACA 0012, 0015, and 0018 airfoils produce $C_{Ft}$ values near zero or weakly negative over much of the cycle, explaining their limited self-starting capability noted in Section 5.2.

Over the complete azimuthal cycle, all profiles except DU 06-W-200 experience extended intervals of negative $C_{Ft}$, particularly on the downwind side, indicating strong drag penalties at low TSR. As the inflow velocity increases to 6–8 m/s, all airfoils generate higher positive $C_{Ft}$ in the upwind half-cycle. In this regime, the S1210 displays the strongest peak values, while the DU 06-W-200 maintains the broadest positive interval. These characteristics are consistent with the high $\overline{C_p}$ values observed in the startup regime for both profiles, and with the DU 06-W-200’s notably low torque-ripple index.

• Nominal operating regime ($\lambda =3$).

The trend of $C_{Ft}\left(\theta \right)$ changes markedly. At $U_{\infty }=2$ m/s, all airfoils show irregular and low-magnitude tangential forces, consistent with the negative $\overline{C_p}$ values reported earlier. For higher velocities (4–8 m/s), the NACA family exhibits well-defined double-peaked positive $C_{Ft}$ distributions, indicating increased aerodynamic efficiency and reduced flow separation. These trends correlate with their superior nominal performance documented in Section 5.3.

The DU 06-W-200 also develops a coherent double-peaked structure at these velocities, though with slightly weaker magnitudes than the NACA profiles. In contrast, the S1210 and E387 show only a single positive peak in the upwind half-cycle and notably negative values over most of the downwind region. This behavior explains their low or negative $\overline{C_p}$ values in the nominal TSR interval despite their good startup performance.

Among the NACA series, performance improves with increasing thickness: the NACA 0018 produces the highest upwind $C_{Ft}$ peak, followed by the NACA 0015 and NACA 0012. This hierarchy mirrors the ordering observed in both the startup ($\lambda <2$) and nominal ($2\le \lambda \le 4$) mean-power trends. Although the S1210 exhibits the single largest upwind peak among all profiles, its strongly negative downwind $C_{Ft}$ fundamentally limits its ability to generate net positive torque at high TSR, consistent with the deterioration of its $C_p$–TSR curve.

Overall, the azimuthal force distributions confirm the key conclusions of the preceding sections: the DU 06-W-200 and S1210 provide strong low-speed torque but suffer from downwind losses at higher TSR, while the NACA 0015 and 0018 develop balanced and sustained positive tangential loading across a wide operating range, resulting in superior nominal efficiency.

5.5. Discussion

Aerodynamic Mechanisms Governing Startup and Nominal Performance

The observed differences in startup capability and nominal efficiency among the airfoil families can be interpreted in terms of their pressure distribution characteristics and flow-separation behavior under the unsteady angles of attack experienced by VAWT blades. During the startup regime ($\lambda <2$), blades encounter large instantaneous angles of attack and frequent flow reversal, leading to early leading-edge separation and predominantly separated flow over much of the azimuthal cycle. Under these conditions, airfoils designed for low Reynolds numbers, such as the DU 06-W-200 and S1210, maintain relatively stable pressure-side suction and delayed separation due to increased camber and favorable pressure gradients near the leading edge. This results in broader azimuthal regions of positive tangential force and reduced sensitivity to intermittent stall, explaining their superior self-starting performance.

In contrast, symmetric NACA airfoils exhibit limited pressure asymmetry at low Reynolds numbers and large angles of attack, leading to rapid leading-edge separation and weak lift generation during startup. The resulting flow is largely drag-dominated, with frequent sign changes in the tangential force, which manifests as low mean torque and high torque-ripple indices. These characteristics fundamentally limit their ability to sustain positive aerodynamic torque during the initial acceleration phase.

As the turbine transitions to nominal operation ($2\le \lambda \le 4$), the effective angles of attack decrease and the flow becomes more periodically attached during the upwind passage. Under these conditions, thicker symmetric NACA profiles develop more stable suction peaks and smoother pressure recovery, delaying large-scale separation and reducing dynamic stall severity. This leads to coherent double-peaked tangential force distributions and relatively small downwind penalties, consistent with their higher mean power coefficients and lower torque ripple.

Conversely, the DU 06-W-200 and S1210 airfoils experience extensive flow separation during the downwind half-cycle at nominal TSRs. Although these airfoils can generate high instantaneous lift peaks on the upwind side, their strongly cambered geometry promotes asymmetric pressure recovery and deep stall in the leeward region, resulting in large negative tangential forces that offset the upwind gains. This explains why airfoils optimized for low-Re startup performance exhibit diminished efficiency and stability once operational tip-speed ratios are reached.

Together, the results highlight a clear distinction between low-speed self-starting capability and nominal aerodynamic efficiency, which is central for VAWTs intended for deployment in low-wind urban environments. The dedicated VAWT airfoil DU 06-W-200 and the low-Re S1210 consistently generate positive torque at $\lambda <2$ and $U_{\infty }\approx 2$ m/s, demonstrating the strongest self-starting potential. Conversely, the symmetric NACA airfoils exhibit insufficient torque at low velocities and LLFVW results indicate negligible or negative $\overline{C_p}$ in the startup regime. These trends are consistent with the azimuthal distributions presented in Section 5.4, where the DU 06-W-200 and S1210 demonstrate extended upwind intervals of positive $C_{Ft}$ and limited downwind losses at low TSRs.

A fundamentally different behavior is observed once the rotor reaches its nominal operating regime ($2\le \lambda \le 4$). In this case, the NACA 0015 and NACA 0018 airfoils demonstrate the highest mean power coefficients and the lowest torque-ripple indices. This indicates that they are capable of efficient energy extraction and reduced unsteady penalties. Conversely, the DU 06-W-200 and S1210 suffer pronounced negative torque excursions and deep-stall behavior, indicating that their low-Re benefits did not translate into consistent nominal performance. The E387 exhibits moderate performance in both regimes but is clearly inferior to the NACA family at operational TSR.

The comparison between DMST and LLFVW across all velocities further reveals the limitations of quasi-steady models in predicting high-TSR efficiency, with DMST systematically overestimating $C_p$ and extending the apparent operational envelope. LLFVW predictions indicate earlier peak locations and sharper performance deterioration at large $\lambda$, consistent with increased induction and dynamic stall. Under the more realistic LLFVW model, the NACA airfoils maintain superior ranking, while the S1210 remains the weakest performer.

These analyses confirm the necessity of balancing startup and nominal performance when selecting airfoils for small-scale urban VAWTs. Airfoils optimized exclusively for low Reynolds numbers facilitate self-starting at low wind speeds, an essential requirement given the characteristic urban wind velocities summarized in Section 1. However, these same airfoils often exhibit suboptimal efficiency once design TSRs are reached. In contrast, thicker, symmetric NACA airfoils possess robust nominal efficiency but limited self-starting capability. The integration of $C_p$-TSR curves, torque metrics, and azimuthal load distributions provides a coherent explanation of the observed performance hierarchy across the operating regimes relevant to urban deployment.

In the present analysis, atmospheric turbulence characteristic of urban environments was intentionally excluded in order to isolate baseline aerodynamic trends and enable a consistent comparison across airfoils and modeling approaches. Similarly, the present startup assessment neglects system inertia; this simplification is consistent with the aerodynamic focus of the study and does not affect the comparative trends reported herein. This assumption primarily affects the magnitude of unsteady aerodynamic phenomena rather than the qualitative ranking of airfoil performance. Recent URANS-based CFD studies have demonstrated that increased turbulence intensity can significantly enhance the power coefficient of small-scale VAWTs by more than 20% in some cases while exhibiting negligible influence on large-scale turbines, highlighting the scale-dependent nature of turbulence effects [65]. Consequently, the absolute values of startup torque and power coefficient reported here may be optimistic, especially for airfoils already prone to deep-stall behavior, such as the DU 06-W-200 and S1210. However, because all airfoils were evaluated under identical inflow conditions, the observed relative trends namely, the superior self-starting capability of low-Re airfoils and the higher nominal efficiency of symmetric NACA airfoils are expected to remain qualitatively robust. The incorporation of stochastic inflow or prescribed turbulence intensity is therefore anticipated to shift performance levels but not alter the fundamental trade-off identified between startup and nominal efficiency. Quantifying these effects represents a significant extension of the present work.

High-fidelity numerical studies have demonstrated that unsteady and three-dimensional effects play a significant role in the aerodynamic behavior of VAWTs operating at low tip-speed ratios. For example, Gharaati et al. [66] employed large-eddy simulation coupled with an actuator line method to investigate the turbulent wake dynamics of straight- and helical-bladed VAWTs at TSRs of 0.4 and 0.6, showing that blade geometry strongly influences wake topology, turbulence transition, and the temporal variability of torque and power coefficients. Such studies provide detailed insight into wake evolution and load fluctuations but come at a substantial computational cost. In contrast, the present work does not aim to resolve wake turbulence or blade–wake interactions; instead, it focuses on assessing the consistency of airfoil performance trends across low- and intermediate-fidelity aerodynamic models, which remain valuable for early-stage design and parametric studies of small-scale VAWTs.

Finally, it is important to note that aerodynamic airfoil selection constitutes only one aspect of realizing effective small-scale VAWT deployment in urban environments. The heterogeneous and often sub-optimal wind conditions typical of cities, characterized by low mean speeds, high turbulence, and flow blockage, suggest that turbine siting plays a crucial role in performance. Recent studies support rooftop or elevated installations as a promising strategy to enhance exposure to usable wind, mitigate ground-level turbulence, and improve overall energy yield [3,67,68]. Integrating the airfoil performance results of the present study with thoughtful placement strategies may thus enable more realistic and efficient use of VAWTs in urban buildings or dense metropolitan contexts.

6. Conclusions

This study evaluated the aerodynamic performance of six airfoil profiles for a small-scale Darrieus VAWT operating under the low wind speeds characteristic of urban environments, using QBlade simulations based on DMST and LLFVW models. Across the operating envelope considered (2–8 m/s), the results consistently reveal a trade-off between self-starting capability at low tip-speed ratios and aerodynamic efficiency at nominal operating conditions.

Low-Reynolds-number airfoils such as the S1210 and the VAWT-oriented DU 06-W-200 exhibit superior starting torque and are the only airfoils capable of producing positive mean power at $\lambda <2$. This behavior is primarily attributed to their camber and low-Re optimization, which promote higher lift at low relative velocities. However, at higher TSRs these same features lead to earlier stall onset, pronounced hysteresis, and increased negative torque excursions, ultimately reducing their mean power coefficient under nominal operating conditions.

In contrast, symmetric NACA airfoils display limited self-starting capability but outperform the low-Re airfoils once operational TSRs are reached. Within the nominal operating range ($2\le \lambda \le 4$), the NACA 0015 and NACA 0018 achieve the highest mean $C_p$ and reduced torque ripple, confirming their suitability for sustained energy extraction. While the DU 06-W-200 reaches power coefficients comparable to those of the NACA airfoils at the highest wind speed considered (8.0 m/s), this occurs at the expense of increased torque unsteadiness. The E387 remains less efficient across all regimes and does not reach the performance levels of the NACA airfoils.

A comparison between DMST and LLFVW indicates that DMST systematically overpredicts performance at intermediate and high TSRs, highlighting the importance of resolving unsteady aerodynamic effects when dynamic stall and induction play a significant role. Validation against experimental reference data suggests that the LLFVW approach captures the overall power coefficient trends with reasonable accuracy, although discrepancies remain near peak efficiency and in stall-dominated conditions.

Within the set of airfoils examined in this study, no single profile simultaneously maximizes self-starting capability and nominal power output. For urban applications, where wind speeds typically range from 2 to 8 m/s, airfoil selection must therefore prioritize either autonomous startup under weak winds or higher efficiency once operational TSRs are achieved. Future work will focus on experimental validation and on hybrid blade concepts, defined here as spanwise airfoil variation along a single blade, combining sections optimized for startup near the blade root with profiles optimized for nominal efficiency toward the tip. Such configurations offer a physically motivated pathway to mitigating the identified startup–efficiency trade-off in micro-scale urban VAWTs.

Limitations and future work:

• Urban turbulence characteristics, including boundary layer effects, inflow variability, and rooftop–induced turbulence, were not considered and should be incorporated in future studies.
• The aerodynamic input polars were generated using XFOIL and extended using empirical post-stall extrapolation methods. While this approach is suitable for comparative studies, XFOIL exhibits limited accuracy in strongly separated and transitional flows, and post-stall extrapolation can introduce significant uncertainty in absolute load predictions. As a result, the present analysis emphasizes relative performance trends rather than precise force estimation.
• The coupling between aerodynamic performance, structural design, and noise constraints requires further investigation for practical micro-scale deployment.
• High fidelity CFD and experimental measurements are needed to validate LLFVW predictions and to quantify dynamic stall and induction mechanisms under realistic turbulence conditions.