the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Local correlation-based transition models for high-Reynolds-number wind-turbine airfoils

### Yong Su Jung

### Ganesh Vijayakumar

### Shreyas Ananthan

### James Baeder

Modern wind-turbine airfoil design requires robust performance predictions for
varying thicknesses, shapes, and appropriate Reynolds numbers. The airfoils of
current large offshore wind turbines operate with chord-based Reynolds numbers
in the range of 3–15 million. Turbulence transition in the airfoil boundary
layer is known to play an important role in the aerodynamics of these airfoils
near the design operating point. While the lack of prediction of lift stall
through Reynolds-averaged Navier–Stokes (RANS) computational fluid dynamics
(CFD) is well known, airfoil design using CFD requires the accurate prediction
of the glide ratio ($L/D$) in the linear portion of the lift polar. The
prediction of the drag bucket and the glide ratio is greatly affected by the
choice of the transition model in RANS CFD of airfoils. We present the
performance of two existing local correlation-based transition
models – one-equation model (*γ*− SA) and two-equation model
($\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}-$ SA) coupled with the Spalart–Allmaras (SA)
RANS turbulence model – for offshore wind-turbine airfoils operating
at a high Reynolds number. We compare the predictions of the two transition
models with available experimental and CFD data in the literature in the
Reynolds number range of 3–15 million including the AVATAR project
measurements of the DU00-W-212 airfoil. Both transition models predict a
larger $L/D$ compared to fully turbulent results at all Reynolds numbers. The
two models exhibit similar behavior at Reynolds numbers around 3 million. However, at higher Reynolds numbers, the one-equation model fails to
predict the natural transition behavior due to early transition onset. The
two-equation transition model predicts the aerodynamic coefficients for
airfoils of various thickness at higher Reynolds numbers up to 15 million
more accurately compared to the one-equation model. As a result, the
two-equation model predictions are more comparable to the predictions from
*e*^{N} transition model.
However, a limitation of this model is observed at very high Reynolds numbers
of around 12–15 million where the predictions are very sensitive to the
inflow turbulent intensity. The combination of the two-equation transition
model coupled with the Spalart–Allmaras (SA) RANS turbulence model is a
good method for performance prediction of modern wind-turbine airfoils
using CFD.

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The aerodynamic design of increasingly large rotors (Veers et al., 2019) to satisfy the world's wind energy needs relies on robust and accurate performance predictions at all operating conditions. The airfoils of current large wind turbines operate at chord-based Reynolds numbers of 3–15 million, as shown in Fig. 1. Laminar–turbulent boundary layer transition is a complex phenomenon that affects the aerodynamics of airfoil boundary layers near the design operating point. Reynolds-averaged Navier–Stokes (RANS) modeling using computational fluid dynamics (CFD) is a common high-fidelity modeling tool used for airfoil design. Typical RANS-CFD solvers are augmented with transition models to improve accuracy of aerodynamic predictions of airfoils.

This work is part of a project to develop a machine-learning inverse-design capability for three-dimensional (3D) aerodynamic design of wind-turbine rotors. In the first phase of our project, we focus on inverse design of two-dimensional (2D) airfoils. Our goal is to develop a robust 2D airfoil capability with the appropriate transition model that can accurately predict the performance of airfoils of various thicknesses and shapes at different operating conditions to generate reliable training data for the machine-learning process. It is well known that 2D RANS CFD does not accurately capture the stall behavior of airfoils because it is an unsteady 3D phenomenon (Ceyhan et al., 2017b). Airfoils are typically designed to operate inside a range of angles of attack for maximum performance away from stall in the linear portion of the lift curve. Hence, the generation of training data for airfoil-design purposes requires the accurate prediction of the glide ratio inside the design range of angles of attack. The variation of the glide ratio near the design points is highly sensitive to the boundary layer transition onset location.

Transition modeling and simulation are divided into analytical models based on
stability theory and statistical models. The *e*^{N} model is a popular
analytical transition model based on the linear stability theory. However, the
application of the *e*^{N} method within a conventional RANS framework
that runs on massively parallel computers is difficult. This is because it
involves non-local search and line integration operations for boundary layer
quantities (e.g., displacement/momentum thickness and shape factor).
Also, additional efforts in communications between *e*^{N} and RANS methods are required (Sheng, 2017).
In wind-turbine applications, the *e*^{N} method has
been used in either 2D RANS flow solvers or a low-fidelity XFOIL
code (Sorensen et al., 2016; Ceyhan et al., 2017b). However, much more complex
infrastructure is required in coupling it with a full 3D
RANS-CFD method.

Statistical models like local correlation-based transition models (LCTMs) that
solve prognostic transport equations for transition variables are more suitable
for use with RANS-CFD models. Two major LCTMs are the two-equation
$\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ model developed by Langtry and Menter (2009), and the simplified
one-equation *γ* model by Menter et al. (2015). The one-equation model is
preferable for wind-turbine modeling, as it satisfies Galilean invariance, a
requirement for application to rotating physical systems. These transition
models were originally developed to be coupled with the shear stress transport
(*k*–*ω* SST) turbulence model that is widely used in the wind-turbine
community. However, different versions of LCTM coupled to the one-equation SA
turbulence model have also been
developed (Medida, 2014; Wang and Sheng, 2014; Nichols, 2019). The SA turbulence model
has advantages of robustness, reliability, and lower computational cost
than the (*k*–*ω* SST) model. The LCTM–SA models have been
successfully applied to a wide range of aerospace problems including rotorcraft.

Applications of RANS-CFD to wind-turbine modeling have mostly focused on using
the *k*–*ω* SST turbulence model coupled to the LCTM or the *e*^{N}-based
transition model (Sorensen et al., 2016; Ceyhan et al., 2017b). Sorensen et al. (2014) showed that the
two-equation $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ transition model fails to correctly predict
natural transition behaviors at high Reynolds numbers compared to the
*e*^{N}-based model. Two out of the four codes in the blind-test campaign
(Ceyhan et al., 2017b) to predict the performance of the DU00-W212 airfoil using
AVATAR data (Ceyhan et al., 2017a) also used the *e*^{N}-based
model. Hence, the above studies together show the superiority of
the *e*^{N}-based method over LCTM for predicting natural transition behavior
for high-Reynolds-number flows. However, there is a lack of studies using the
LCTM coupled with the SA turbulence model for wind-turbine applications.

In this paper, we quantify the performance of both one-equation and two-equation
LCTM coupled with the SA turbulence model in simulating wind-turbine airfoils at
a wide range of Reynolds numbers. We compare our simulation results to not only
experiments but also the other predictions using different transition models
(e.g., *e*^{N}-based). First, the formulations of correlation-based
transition models are presented briefly for completeness in
Sect. 2. Then, we show validation and code comparison
results for the SA turbulence model using fully turbulent approximation in
Sect. 3. Section 4 analyzes the differences
between the predictions from the one-equation and two-equation transition models
through comparison to experimental and other reference data in the
literature. This includes the comparison to the measurements from the AVATAR
project (Ceyhan et al., 2017a) on the DU00-W-212 airfoil at Reynolds numbers of
3–15 million. We then compare the predictions of the two LCTMs for airfoils from
three modern, open-source, megawatt-scale wind turbines: NREL 5 MW
(Jonkman et al., 2009), DTU 10 MW (Bak et al., 2013), and IEA 15 MW
(Gaertner et al., 2020). Our simulation results are compared with available
reference data from experiments and/or other simulations in the
literature. Finally, we conclude with a discussion of the transition models in
RANS-CFD solvers for airfoil design in modern wind turbines.

## 2.1 Reynolds-averaged Navier–Stokes solver

The Hamiltonian solver (HAM2D) is a Reynolds-averaged Navier–Stokes (RANS) flow
solver that was developed at the University of Maryland (Jung and Baeder, 2019). This
is a parallel solver for the solution of the two-dimensional compressible
Navier–Stokes equations on unstructured meshes using finite-volume
formulation. A fifth-order weighted essentially non-oscillatory (WENO) scheme is
used for spatial reconstruction, and Roe's approximate Riemann solver is used to
compute inviscid fluxes. Viscous fluxes are calculated using second-order
central differencing. For the steady-state solution, the preconditioned
generalized minimum residual (GMRES) is used as the implicit time-integration
method. The turbulent boundary layer is modeled using the one-equation SA
model. Both the one-equation *γ*−SA transition model (Lee and Baeder, 2021) and
the two-equation $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}-$ SA transition
model (Medida, 2014) have been coupled with the SA turbulence model to
predict boundary layer transition if necessary. In this study, the
incompressible flow condition was approximated in the compressible solver using
a free-stream Mach number of 0.1. The Reynolds number based on chord length and
angle of attack was adjusted for test flow conditions.

Our in-house automated airfoil mesh generation was used for various test
airfoils, which is designed to require relatively few control inputs from
airfoil coordinates (Costenoble et al., 2018). For efficient meshing, the surface
point distribution (clustering/stretching) is based on local sharp corners or
different surface curvatures along the airfoil. An O-type grid is used to allow
for a finite-thickness trailing edge. A strand-/advancing-front-based method is
used to generate a body-fitted mesh around the airfoil, and the triangle
elements are used to extend the domain to the outer boundary. All triangular
elements are transformed to obtain a pure quadrilateral mesh, which is
required by the flow solver. In previous studies, the current flow solver and
automated mesh generation have been validated through various canonical
problems (Costenoble et al., 2017, 2018; Jung and Baeder, 2019). For the simulations
in this paper, the number of nodes in the wraparound direction was fixed as 400 points, as determined through a grid convergence study
(Appendix B), and the initial wall-normal spacing was
varied according to the test Reynolds number, such that ${y}^{+}=\mathrm{1}$. The outer
boundary was placed 300 chord lengths away from the airfoil where the far-field
boundary condition was imposed. Figure 2 shows the mesh
generated using this method around the DU93-W-210LM airfoil at a Reynolds number of
9×10^{6} as an example. The convergence of the residuals and the
aerodynamic coefficients with solver iterations is shown in
Appendix C.

## 2.2 Two-equation laminar–turbulent boundary layer transition model

The two-equation LCTM model used in this study is the
$\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}-$ SA model, also known as the Medida–Baeder
transition model. A brief description of this transition model is presented in
this paper, and a detailed description can be found in the previous studies (Medida, 2014; Jung and Baeder, 2019). This transition model can predict natural
transition, separation-induced transition, and bypass transition and has been
validated through various canonical problems. The transition model uses the
concept of intermittency, *γ*, in order to trigger transition locally. The
intermittency is a scalar transport variable that varies between 0 (pure
laminar) and 1 (pure turbulent). The transport equation for the intermittency is
given by

where *P*_{γ} and *D*_{γ} denote the production and destruction term, respectively.

The transport equation for transition momentum thickness Reynolds number, $\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$, is used to account for history effects of pressure gradient on determining the onset of transition. This equation is given by

where ${P}_{\mathit{\theta}\mathrm{t}}=\mathrm{0.03}\frac{\mathit{\rho}}{t}({\mathit{Re}}_{\mathit{\theta}\mathrm{t}}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}})(\mathrm{1.0}-{F}_{\mathit{\theta}\mathrm{t}})$.

Once the distribution of $\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ in the computational domain is solved, the critical momentum thickness Reynolds number is obtained through
${\mathit{Re}}_{\mathit{\theta}\mathrm{c}}=\mathrm{0.62}\cdot \stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$. Then, the intermittency
production can be triggered based on the ratio of the local vorticity Reynolds
number, *Re*_{v}, to the *Re*_{θc}. For the production term, the
transition onset momentum thickness Reynolds number, *Re*_{θt}, is
computed through the empirical correlations in an iterative manner, which are
functions of the streamwise pressure gradient parameter, *λ*_{θ}, and the
inflow turbulent intensity. *λ*_{θ} is defined as

where $U=\sqrt{{u}^{\mathrm{2}}+{v}^{\mathrm{2}}+{w}^{\mathrm{2}}}$.

When this transition model is coupled with the SA turbulence model, the intermittency is used to control only the production term of the transported variable, $\stackrel{\mathrm{\u0303}}{\mathit{\nu}}$, as

Details of the current implementation of the transition model compared to $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ model by Langtry and Menter (2009) are shown in the previous study (Medida, 2014; Jung and Baeder, 2019)

## 2.3 One-equation laminar–turbulent boundary layer transition model

The one-equation transition model first proposed by Menter et al. (2015) uses
only the intermittency variable, *γ*: hence, only the transport equation
for the intermittency is required as shown in Eq (1). Both
production and destruction terms for the intermittency are different compared to $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ model. The transport equation for $\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ is
replaced with the empirical-based formulation as follows for obtaining
*Re*_{θc}:

Our implementation of the one-equation model uses modified coefficients of
*C*_{TU1}, *C*_{TU2}, and *C*_{TU3} compared to that by Menter et al. (2015),
which gives better correlation with the experiments than the original values, as
shown by Colonia et al. (2016). The modified values of the constants are

In the *Re*_{θc} formulation, *F*_{PG} is introduced to sensitize the transition onset to the streamwise pressure gradient. The pressure gradient parameter, *λ*_{θ}, in Eq. (4) is approximated as
*λ*_{θL} in the model; thus it becomes only a function of wall-normal direction velocity and coordinate in addition to the kinematic viscosity,
*ν*.

where *d*_{w} is the wall normal distance, and *V* and *y* are the wall-normal velocity and coordinate, respectively.

The one-equation transition model was coupled with the SA turbulence model first by Nichols (2019) using the equations as follows:

It should be noted that the intermittency is used to control both the production
and destruction terms of the SA model unlike the two-equation transition model.
Nichols (2019) also defined a re-scaled transition variable, *γ*_{s},
which goes from zero at the wall to one in turbulent regions of the flow as
below. This is because the SA model requires the production source term to go to
zero in laminar regions of the flow.

In addition, ${P}_{\stackrel{\mathrm{\u0303}}{\mathit{\nu}}}^{\mathrm{lim}}$ as an additional production term was
proposed to ensure the generation of turbulent kinetic energy at the transition
point for low free-stream turbulence intensity levels. Finally, for the local
turbulence intensity computation, Tu_{L}, turbulent kinetic energy, *k*, and
specific dissipation, *ω*, variables were replaced as shown in the
equations below because these variables are not available in the SA turbulence
model.

where *S* is the strain rate magnitude.

Based on the formulation for SA model by Nichols (2019), Lee and Baeder (2021) employed a constant free-stream turbulence intensity assumption in the entire flow field, which is valid for external aerodynamic flows. An input turbulence intensity is a measured value from an experiment. The modified formulation was validated through canonical problems in both two and three dimensions (Lee and Baeder, 2021). Therefore, in the current study, we used the same formulation as the work by Lee and Baeder (2021) for the one-equation transition model.

In this section, we show code-to-code comparisons and validation studies for the
SA turbulence model using our solver HAM2D. The performance of the SA turbulence
model using the current solver and mesh-generation approach has been validated
in previous studies (Jung et al., 2017; Jung and Baeder, 2019; Jung, 2019) through various test
cases from NASA Turbulence Modeling Resource (TMR, 2017). The test cases
include a 2D zero-pressure-gradient flat plate, 2D bump in channel, and NACA
0012 airfoil. The case of turbulent flow past a NACA0012 airfoil is shown in
this paper as an example. The flow condition is a free-stream Mach number of
0.15, a Reynolds number of 6 million, and three angles of attack (0, 10, and
15^{∘}). A structured airfoil C-type mesh (897×257) provided by the
TMR website is used for the current simulation. Table 1 shows
the comparison of the lift and drag coefficients of the NACA0012 airfoil using
the SA turbulence model. The force coefficients predicted by HAM2D are
comparable to the results predicted by well-established legacy codes.

For wind-turbine airfoils, we compare our predictions using the SA model with
those from EllipSys2D for FFA-301 and FFA-360GF airfoils using the
*k*–*ω* SST model at a Reynolds number of 10 million under a fully
turbulent flow assumption (Bak et al., 2013).
Both predictions used enough fine meshes for the fully
turbulent flow simulation; thus, there is minor mesh dependency on both
predictions.
Also, both simulations neglected compressibility because EllipSys2D is a
incompressible solver.

FFA-360GF is a very thick airfoil with
$t/{c}_{\mathrm{max}}=\mathrm{36}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$ and a gurney flap. Figure 3 compares
the lift coefficient, drag coefficient, and lift-to-drag ratio as a function of
angle of attack from each simulation. Both predictions show very good agreement
in the drag coefficient and lift-to-drag ratio over the test angles of attack
from −4 to 20^{∘}. Also, the linear portion of the lift polar is
well matched between the predictions. This shows that the one-equation SA model
provides similar performance compared to the two-equation *k*–*ω* SST model
under fully turbulent flow conditions.

We compare the aerodynamic load predictions from the one-equation and two-equation transition models on airfoils from modern wind turbines with available reference data from experiments and/or other simulation results in the literature. First we consider the S809 airfoil at a low Reynolds number of 2 million. Then, we compare the performance of the two transition models on the DU00-W-212 airfoil for which wind tunnel measurements are available through the AVATAR project (Ceyhan et al., 2017a) at Reynolds numbers of 3–15 million. We compare the air load with measurements in both fully turbulent and free-transition conditions. The effect of the choice of transition model on the prediction of the transition onset location is analyzed. Finally, the sensitivity of free-stream turbulent intensity on air load predictions using two-equation transition model is shown.

Next, we evaluate the effect of the transition model on other airfoils from three modern, open-source, megawatt-scale wind turbines: NREL 5 MW (Jonkman et al., 2009), DTU 10 MW (Bak et al., 2013), and IEA 15 MW (Gaertner et al., 2020). To improve the readability, we show the comparison of the prediction from the two transition models for three representative airfoils (DU91-W2-250 and NACA64-618 airfoils from NREL 5 MW and FFA-W3-301 airfoil from DTU 10 MW/IEA 15 MW) in this section and the rest in Appendix Sects. A1 and A2.

## 4.1 S809 airfoil

The two transition models considered in this study are evaluated through the S809 airfoil used in the NREL Phase VI wind turbine (Hand et al., 2001). We show validation of the aerodynamic performance prediction against experimental data (Somers, 1997) as well as previous simulation results using NASA's OVERFLOW code from Coder (2019) using the SA-neg turbulence model with the AFT2019 transition model. The AFT2019 transition model was developed based on linear stability theory, which is also widely used in aerospace problems. It solves two transport equations for amplification factor and intermittency. We also compare our results with the other predictions using the two-equation transition model in OVERFLOW by Hall (2018).

The test flow condition parameters are a free-stream Mach number of 0.1, Reynolds number of 2 million based on chord length, and a free-stream turbulence intensity of 0.05 %. We use the medium-resolution reference structured C-grid from the 2018 transition modeling workshop (Hall, 2018) with dimensions of 705×87 including 513 points on the surface and 97 points in the wake.

An angle-of-attack sweep was conducted from −8 to 15^{∘}.
Figure 4 compares the lift polar, drag polar, and the transition
location of the predictions from the fully turbulent and free-transition
simulations using both transition models against experimental data and
other simulation results (Coder, 2019; Hall, 2018).

Figure 4a shows that all simulations predict the lift coefficient well in the linear region of the lift polar. In detail, slightly higher lift coefficients from the untripped experiment than the tripped one are captured using either the one-equation model or two-equation model. Otherwise, the predictions significantly overpredict the maximum lift coefficient due to the known limitations of 2D CFD-RANS. Figure. 4b shows that the drag predictions from HAM2D using the fully turbulent approximation are in excellent agreement with the OVERFLOW simulation results (Coder, 2019) at the same flow condition over the full range of angle of attack while showing a slight underprediction in the drag bucket compared to the tripped boundary layer experimental data.

The HAM2D predictions using both transition models show a similar underprediction inside the drag bucket while having excellent agreement with results from the AFT2019 model (Coder, 2019). The two-equation transition model predicts a slightly lower drag than the one-equation model inside the drag bucket and also shows an earlier departure from the drag bucket near a lift coefficient of 0.6 similar to the results using two-equation model from Hall (2018).

Figure 4c compares the transition onset location, *X*_{T},
predicted by the current transition models with experimental data on both the
upper and lower sides of the airfoil. The transition onset location was
determined by picking up the point in the middle of a sharp increase in the
intermittency on the surface. As the angle of attack increases, the transition
point on the upper surface moves to the stagnation point due to an increasing
adverse pressure gradient. On the other hand, the transition onset on the lower
surface moves downstream due to an increasing favorable pressure gradient with
an increasing angle of attack. The transition occurs due to a short and intense
laminar separation bubble on both sides of the airfoil (separation-induced
transition). Overall, the transition onset locations predicted by both
transition models match well with the experimental data. The sharp movement on
the upper surface at the 6^{∘} angle of attack was well captured by the
two-equation model. However, this movement was predicted at 8^{∘} from the
one-equation model. This difference in the onset locations explains the earlier
departure from the drag bucket using the two-equation model compared to the
one-equation or the AFT2019 model.

## 4.2 DU00-W-212 airfoil – AVATAR

The AVATAR project from the European Union focused on aerodynamics of large rotors (Ceyhan et al., 2017a). The aerodynamic measurements on the DU00-W-212 airfoil from wind-tunnel experiments at conditions similar to those of 10 MW+ turbines were made publicly available through this project by Ceyhan et al. (2017a). We compare the effect of the one-equation and two-equation transition models against this data set as well as the results from the blind-test study by Ceyhan et al. (2017b) at Reynolds numbers of 3, 6, 9, 12, and 15 million. In the experiment, the lift and pitching moment coefficients were calculated using pressure taps along the airfoil, and the drag coefficient was calculated from the flow loss of momentum using the wake rake. It should be noted that the drag measurement can be inaccurate at post-stall region due to the nature of 3D flows which were measured using the wake rake at a fixed span location. Also, in the experiment, three different turbulent intensity levels were measured at the model location in different periods. Thus, we also study the sensitivity of the air load predictions to the inflow turbulence intensity level through the three different measured intensities shown in Table 2 and as performed by Ceyhan et al. (2017b).

The computational grid for the DU00-W-212 airfoil was generated using the automated mesh generation procedure described in Sect. 2. It has 500 points in the wraparound direction and the initial wall-normal spacing of $\mathrm{1.8}\times {\mathrm{10}}^{-\mathrm{6}}$ chord (${y}^{+}=\mathrm{1}$), which results in a grid with a resolution comparable to the meshes used by Ceyhan et al. (2017b).

We performed fully turbulent and free-transition flow simulations with both
transition models at the five different Reynolds numbers in
Table 2 and the angles of attack ranging from −4 to
20^{∘}. Figure 5 shows the comparison of the lift-to-drag
ratio and drag polar between the fully turbulent flow simulation results from
HAM2D with the experimental data with the tripped boundary
layer (Pires et al., 2016). Figure 5a shows that HAM2D
overpredicts the maximum lift-to-drag ratio, but the overall trend from the
experiment is captured well: the slope in the linear region increases as the
Reynolds number increases, and the maximum lift-to-drag ratio increases as the
Reynolds number increases. In Fig. 5b, the minimum drag
coefficients match well with experimental data in the linear region of the
lift polar at all Reynolds numbers and decreases as the Reynolds number
increases.

Figure 6 compares the lift-to-drag (glide) ratio predicted by HAM2D
using the one-equation and two-equation transition models with experimental data for
the untripped boundary layer (Ceyhan et al., 2017a) and the other simulations
from Pires et al. (2016). The other simulation results were obtained using the
*k*–*ω* SST turbulence model coupled with different transition models:
the semi-empirical *e*^{N} method by Drela–Giles for DTU EllipSys, the
*e*^{N} method combined with the linear stability solver for the Kiel University of Applied Sciences TAU code (Kiel TAU), and the
Granville–Schlichting model (Ceyhan et al., 2017b) for NTUA MapFlow. The lowest
turbulence intensity level (Ti3) was used at each Reynolds number for all the
computations, as shown in Table 2. The one-equation
transition model in HAM2D was able to capture a reasonable maximum lift-to-drag
ratio only at $\mathit{Re}=\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$; the prediction becomes progressively worse
compared to experimental data and all other simulation results at higher
Reynolds numbers. On the other hand, the two-equation transition model in HAM2D
shows fairly good agreement compared to both experiment and other simulation
results up to $\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$. The prediction of the linear slope and the
maximum $L/D$ value are comparable with those of the *e*^{N}-based
transition models from Pires et al. (2016). At $\mathit{Re}=\mathrm{12}\times {\mathrm{10}}^{\mathrm{6}}$ and
$\mathit{Re}=\mathrm{15}\times {\mathrm{10}}^{\mathrm{6}}$, the two-equation model in HAM2D predicts a lower linear
slope than all the reference results, and the angle of attack for the maximum
$L/D$ is delayed. However, the results from two-equation model are much more in
agreement with all reference data in Fig. 6 than the one-equation
model over the entire range of Reynolds numbers.

To find the reason for underprediction of the lift-to-drag ratio using both
transition models, we compare the drag polars from HAM2D predictions and the
reference results in Fig. 7. At $\mathit{Re}=\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$, the
HAM2D results using both transition models predict the laminar drag bucket
well. As the Reynolds number increases above 3×10^{6}, the one-equation
model consistently overpredicts the minimum drag while the range of angle of
attack of the drag bucket becomes much smaller than the reference data. This
explains the significant underprediction in lift-to-drag ratio at higher
Reynolds numbers by the one-equation model. On the other hand, the two-equation
model reasonably predicts the experimental drag values up to $\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$
compared to the other simulation results. At $\mathit{Re}=\mathrm{12}\times {\mathrm{10}}^{\mathrm{6}}$, the minimum
drag is overpredicted by six drag counts, and the sharp corner of the laminar
bucket is not properly captured. More deviation is observed at
$\mathit{Re}=\mathrm{15}\times {\mathrm{10}}^{\mathrm{6}}$.

For the DU00-W-212 airfoil, the drag coefficients at varying Reynolds number are
compared with the experiment at 4^{∘} angle of attack where the maximum $L/D$
ratio occurs as shown in Fig 8. It is shown that the drag
coefficient from the experiment decreases from 3 to 9 million Reynolds numbers,
and then it increases until 15 million Reynolds numbers. However, the variations between the Reynolds numbers are minor.
For the two-equation model, the variations between the Reynolds numbers are also minor as the experiment, though the drag increases as Reynolds number increases as shown in Fig. 8a.
Otherwise, the drag clearly increases as Reynolds number increases in the
one-equation model prediction, which is the opposite trend to the experiment as shown
in Fig. 8b.
Also, the predicted drags are broken down into viscous and pressure drag
components. As a result, the viscous drag component is dominant over the pressure
drag at all Reynolds numbers from both transition models. This also indicates
the importance of transition onset predictions because the skin
friction is much higher in a turbulent boundary layer than in a laminar boundary layer.

Figure 9 compares the transition onset location, *X*_{T},
predicted by HAM2D using both transition models on upper and lower sides of the
DU00-W-212 airfoil at two representative Reynolds numbers: 3×10^{6} and
9×10^{6}.
These predictions are also compared with those from EllipSys2D
using the *k*–*ω* SST turbulence model and different
transition models: $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ (LCTM), *e*^{N}
model, and the *e*^{N}-BP model with bypass transition (Sorensen et al., 2014).

At $\mathit{Re}=\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$, the predicted transition onset locations from HAM2D using both transition models compare well with results from EllipSys2D, as shown in Fig. 9a and b. As the angle of attack increases, the onset location moves due to the changes in the streamwise pressure gradient similar to the behavior in the S809 airfoil in Fig. 4.

However, at $\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$, larger deviations start to occur between the
predictions from the one-equation and two-equation model on both upper and lower
surfaces, as shown in Fig. 9c and d. Using the
one-equation model, the onset prediction rapidly moves to the stagnation point
at the 2^{∘} angle of attack on the upper surface while showing erratic
behavior on the lower surface. This explains the early escape of the laminar
drag bucket and significant overprediction in drag by the one-equation model
compared to the experimental data. Similarly, the larger deviations are also
observed among the EllipSys predictions (Sorensen et al., 2014) at
$\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$. The $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ (LCTM) predicts the onset earlier than
*e*^{N} and *e*^{N}-BP models on both upper and lower
surfaces. It is also observed that the bypass transition starts playing a role
over the natural transition at this higher Reynolds number by showing the
earlier onset prediction using the *e*^{N}-BP model instead of the
*e*^{N} model. The two-equation model predictions show a consistent
trend in the movement of the onset location, and the results are quite similar
to the results from the *e*^{N}-BP transition model on both surfaces.

Finally, the effect of different inflow turbulence intensity on the predictions
from the two-equation transition model is shown in
Fig. 10 at two different Reynolds numbers of 3×10^{6} and 12×10^{6}. The predicted air loads are compared from the three
different turbulent intensity levels from Ti1 to Ti3, as shown in
Table 2. The prediction of the lift-to-drag ratio is highly
sensitive to the inflow turbulent intensity level. Also, the sensitivity
becomes stronger at the higher Reynolds number, which results in the best
correlation with the experiment using the lowest intensity (Ti3) as observed in
a previous study using the *e*^{N} transition model (Ceyhan et al., 2017b).

## 4.3 DU series airfoils and NACA64-618

The predictions of HAM2D using both transition models for the airfoils in the NREL 5 MW turbine (Jonkman et al., 2009) are compared against data available in the DOWEC 6 MW pre-design report (Kooijman et al., 2003) for Reynolds numbers of 6 and 7 million. The reference data for the DU airfoils at $\mathit{Re}=\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$ are taken from experiments in the LTT wind tunnel of TU Delft. The results for the $\mathit{Re}=\mathrm{7}\times {\mathrm{10}}^{\mathrm{6}}$ are the result of a synthesis process, in which measured data for at $\mathit{Re}=\mathrm{3}\times {\mathrm{10}}^{\mathrm{6}}$ are translated to the higher Reynolds number using the airfoil-design code RFOIL (Van Rooij, 1996). According to the DOWEC 6 MW pre-design report, the reference data for the NACA64-618 airfoil is obtained from appendix IV of Abbott and von Doenhoff (1959). Also, the available reference data were corrected for a blade aspect ratio of 17 in the DOWEC 6 MW pre-design report (Kooijman et al., 2003). However, we believe the data are still valid as a reference in explaining any differences of model predictions.

The automated grid generation for these airfoils uses 400 points in the wraparound direction based on the grid-refinement study shown in Appendix Sect. B. The free-stream turbulence intensity is set to 0.1 %. Figures 11 show the comparison of fully turbulent and free-transition results for DU91-W2-250 airfoil at $\mathit{Re}=\mathrm{7}\times {\mathrm{10}}^{\mathrm{6}}$ and NACA64-618 airfoil at $\mathit{Re}=\mathrm{6}\times {\mathrm{10}}^{\mathrm{6}}$ against reference data from Kooijman et al. (2003).

All simulation results, both using the fully turbulent and transition models, show similar behavior and predict the lift coefficient well in the linear region including the lift-curve slope and the zero-lift angle of attack. Overall, the current simulations predicted delayed stall angles compared to the reference data. It should be noted that the same trend is also observed in the previous comparison with pure experimental data for the DU00-W-212 airfoil in Fig. 10, which is a typical challenge in 2D RANS-CFD modeling of airfoils. The unphysical linear increment in the drag coefficient is observed after the stall angle only in the reference data. This might be due to the synthesized process between RFOIL calculations and experimental data.

By using either the one- or two-equation transition model, lower drag
coefficients were predicted at around 0^{∘} as a result of laminar boundary
layer detection. This results in a better agreement in lift-to-drag ratio
against reference data compared to the fully turbulent simulations. The
prediction of the maximum lift-to-drag ratio is significantly improved using
the two-equation model compared to the one-equation model for both airfoils.
The one-equation model underpredicts the lift-to-drag ratio in the linear
portion of the lift curve due to early transition onset as the angle of attack
increases. Thus, the two-equation transition model is an appropriate choice for
wind-turbine airfoil simulations at high Reynolds numbers.

Similar comparison results for the other NREL 5 MW airfoils with different maximum relative thickness $(t/c{)}_{\mathrm{max}}$ are shown in Appendix Sect. A1. Overall, two-equation transition model improves the predictions for the other airfoils as well.

## 4.4 FFA series airfoils

We compare the predictions of HAM2D using both transition models for the FFA-W3
series of airfoils in the DTU 10 MW (Bak et al., 2013) and the IEA 15 MW
turbine (Gaertner et al., 2020) at a Reynolds number of 10 million. We also compare
our results against the publicly available simulation data from EllipSys2D for
these airfoils (Gaertner et al., 2020) using the *k*–*ω* SST turbulence model
with the semi-empirical *e*^{N} method (Drela and Giles, 1987). However, only
a combination of 70 % free-transition and 30 % fully turbulent polar is available
for these airfoils; i.e., the lift and drag values at each angle of attack are
linearly interpolated between the free-transition and fully turbulent results
using the 70:30 ratio. Therefore, we generate the same mixed polars in this
study by using one-equation and two-equation transition model for appropriate
data comparison. The automated grid generation for these airfoils uses 400
points in the wraparound direction based on the grid-refinement study shown in
Appendix Sect. B. The free-stream turbulence intensity is set
to 0.1 %.

For the FFA-W3-301 airfoil, our simulation results are compared with simulation results from EllipSys2D (Gaertner et al., 2020) as shown in Fig. 12. The predictions using HAM2D with the two-equation model show excellent agreement with the reference. The one-equation model predicted much lower lift-to-drag ratio than the predictions from other transition models in the linear portion of the lift curve due to earlier transition onset. This is similar to the behavior seen in the DU00-W-212 (Sect. 4.2) and the NREL 5 MW airfoils (Sects. 4.3 and A1).

Similarly, the comparison results for the other FFA-W3 airfoils with different maximum relative thickness $(t/c{)}_{\mathrm{max}}$ are shown in Appendix Sect. A2. Overall, the predictions only from two-equation model show excellent agreement with the reference for the other airfoils as well.

For a better understanding of the two-equation model in
HAM2D, we compare its behavior to the implementation of the $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ model in EllipSys2D using the *k*–*ω* SST turbulence model for the FFA-W3-301
airfoil (Bak et al., 2013) at $\mathit{Re}=\mathrm{10}\times {\mathrm{10}}^{\mathrm{6}}$. The skin friction
distribution predicted by HAM2D and EllipSys2D for this airfoil is compared at
four different angles of attack in Fig. 13. The sign of skin
friction is defined by the sign of the local streamwise velocity at each
point. The transition onset location is indicated by a sharp increase in the
skin friction value on both the upper and lower surfaces. The transition onset
prediction from the one-equation model in HAM2D rapidly moves to upstream at the
8^{∘} angle of attack. As a result, it predicts a delayed onset at the lower
angles of attack but earlier onset at the higher angles of attack compared
against the EllipSys2D result. On the other hand, the transition onset
predicted by the two-equation model is downstream of the predictions from the
one-equation model and from EllipSys2D at all angles of attack. A similar
behavior was also observed for the DU-00-W212 airfoil at $\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$ in
Fig. 9. The delayed onset locations from the two-equation model in
HAM2D than other LCTM predictions might explain the good air load agreement with *e*^{N} method as shown in Fig. 12.

We evaluated the performance of two local correlation-based transition models
within our in-house 2D compressible Reynolds-averaged Navier–Stokes (RANS)
solver HAM2D for applications to modern wind-turbine airfoils at high Reynolds
numbers. The one-equation transition model (*γ*− SA) and the two-equation
transition model ($\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}-$ SA) are coupled with the Spalart–Allmaras (SA) one-equation turbulence model. We compare the predictions of the two transition models with available experimental and computational fluid dynamics (CFD) data in the literature in the Reynolds number range of 3–15 million including the AVATAR project measurements of the DU00-W-212
airfoil (Ceyhan et al., 2017a) and for airfoils from three modern,
open-source, megawatt-scale wind turbines: NREL 5 MW (Jonkman et al., 2009), DTU
10 MW (Bak et al., 2013), and IEA 15 MW (Gaertner et al., 2020).

The two models exhibit similar behavior at Reynolds numbers around 3
million. The one-equation transition model fails to predict the natural
transition behavior at the high Reynolds numbers ranging from 6 million to 15 million due to early transition onset, as reported in a previous
study for the $\mathit{\gamma}-\stackrel{\mathrm{\u203e}}{{\mathit{Re}}_{\mathit{\theta}\mathrm{t}}}$ model (Sorensen et al., 2016).
The two-equation transition model presents much
better predictions in aerodynamic coefficients (e.g., stall angle, maximum lift
coefficient, and lift-to-drag ratio) than the one-equation transition model. As
a result, comparable performance with the *e*^{N}-based transition models
within RANS CFD is observed for the various thickness airfoils.
At high Reynolds numbers from 12 million, the two-equation model also
somewhat underpredicted the maximum lift-to-drag ratio compared to the results
from *e*^{N}-based transition models.

The one-equation transition model also fails to predict the correct trends of
the aerodynamic coefficients, especially the peak lift-to-drag ratio, with
the Reynolds number. On the other hand, the predictions in
aerodynamics coefficients at all Reynolds numbers from the two-equation transition
model are much closer to that of the experimental data and comparable to the
predictions from the *e*^{N}-based models in the
literature (Ceyhan et al., 2017b).
The predictions from the two-equation transition model exhibits a strong
sensitivity to the free-stream turbulence intensity at the high Reynolds
number, as previously observed from the *e*^{N}-based models.
Overall, the combination of the two-equation transition model coupled with the
Spalart–Allmaras RANS turbulence model is a good method for performance
prediction of modern wind-turbine airfoils using CFD.

The shortcomings of the one-equation transition model at high Reynolds numbers have been identified by comparing against the two-equation transition model. However, the formulation of one-equation transition model satisfies Galilean invariant, which is desirable in a simulation with rotating bodies (e.g., blade). Therefore, in the future, we plan to improve the performance of the one-equation transition model using the field-inversion and machine-learning approach which was validated for the SA turbulence model (Holland et al., 2019).

The current predictions of HAM2D using one- and two-equation transition models are further evaluated for NREL 5 MW (Jonkman et al., 2009), DTU 10 MW (Bak et al., 2013), and IEA 15 MW (Gaertner et al., 2020) airfoils which have different maximum relative thickness, $(t/c{)}_{\mathrm{max}}$.

## A1 DU series airfoils

Figures A1 show the comparison of fully turbulent and free-transition results for DU airfoils at $\mathit{Re}=\mathrm{7}\times {\mathrm{10}}^{\mathrm{6}}$ against reference data from Kooijman et al. (2003). Overall, much better agreements in lift-to-drag ratio against experimental data are observed by using the two-equation transition model for all airfoils. Also, the lift-curve slopes in the linear region are better matched with the experiment using the two-equation transition model for airfoils with larger maximum relative thickness $(t/c{)}_{\mathrm{max}}$.

The difference between the predictions of the maximum $L/D$ and lift-curve slope from the one-equation and two-equation models increases for larger thickness airfoils. This might be because the onset location typically moves towards the leading edge for the thicker airfoils due to the higher adverse pressure gradient at the same angle of attack. A more detailed discussion can be found in Sect. 4.3.

## A2 FFA series airfoils

For the FFA-W3 airfoils with different airfoil thickness, our simulation
results using both transition models are compared with simulation results from
EllipSys2D (Gaertner et al., 2020) using the semi-empirical *e*^{N}
method, as shown in Fig. A2. In this comparison, a
combination of 70 % free-transition and 30 % fully turbulent polars is used as
we already discussed in the Sect. 4.4. For all FFA
series airfoils, the predictions using HAM2D with the two-equation model show
excellent agreement with the reference except in the case of the very thick
FFA-W3-360 airfoil.
Like the comparison in the Sect. 4.4 for FFA-W3-301
airfoil, the one-equation model predicted much lower lift-to-drag ratio than
the predictions from the two-equation transition model for all airfoils. A more detailed
discussion can be found in Sect. 4.4.

A grid convergence study was conducted to measure the sensitivity of airfoil
performance to grid resolution to validate the current grid resolution. The
grid convergence study was performed for the airfoils using different number of
surface points: 300, 400, 500, and 600. The initial wall-normal spacing was
fixed with a small enough value such that ${y}^{+}=\mathrm{1}$. The test was focused at a
specific operating flow condition with *α*=4^{∘} and $\mathit{Re}=\mathrm{9}\times {\mathrm{10}}^{\mathrm{6}}$. The simulations are performed for both the fully turbulent and
free-transition boundary layer.
The results for the DU93-W-210LM airfoil are shown in Figs. B1,
B2, and B3. The *y*-axis limit is the
ranges of ±1% of each mean's changes in the number of grid points. It is
seen that the magnitude of the variation is less than 1 % of their actual
values, which results in minor variation compared to the variation from
different airfoils or flow conditions of the current interest.

Figures C1 and C2 show the solution
convergence history during the simulation for the representative flow condition
at two different Reynolds numbers of 3×10^{6} and 9×10^{6}. Both
lift and drag coefficients are converged within 1500 iterations, wherein the
solution residual drops by more than 3 orders of magnitude.

Data for any plots in this paper can be made available on request.

YSJ developed the simulation code and performed the simulations. YSJ wrote the manuscript with significant input from GV. GV provided the reference data set and developed the post-processing code for the simulation results. SA and JB provided guidance for the research and reviewed the manuscript.

The contact author has declared that neither they nor their co-authors have any competing interests.

The views expressed in the
article do not necessarily represent the views of the DOE or the U.S.
Government. The U.S. Government retains and the publisher, by accepting the
article for publication, acknowledges that the U.S. Government retains a
nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce
the published form of this work, or allow others to do so, for U.S. Government
purposes.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under contract no. DE-AC36-08GO28308. Funding provided by the Advanced Research Projects Agency – Energy (ARPA-E) Design Intelligence Fostering Formidable Energy Reduction and Enabling Novel Totally Impactful Advanced Technology Enhancements (DIFFERENTIATE) program. A portion of this research was performed using computational resources sponsored by the Department of Energy's Office of Energy Efficiency and Renewable Energy and located at the National Renewable Energy Laboratory. The transition modeling work was supported under the Vertical Lift Research Center of Excellence Grant at the University of Maryland with Mahendra Bhagwat as technical monitor.

This research has been supported by the Advanced Research Projects Agency – Energy (DIFFERENTIATE program).

This paper was edited by Gerard J. W. van Bussel and reviewed by Nando Timmer and one anonymous referee.

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- Abstract
- Introduction
- Methodology
- Validation of turbulence model
- Results: transition modeling
- Conclusions
- Appendix A: Additional results
- Appendix B: Grid convergence study
- Appendix C: Solution convergence study
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References

*e*

^{N}-based method showed its superiority over local correlation-based transition models (LCTMs) coupled with the SST turbulence model for predicting transition behavior at high-Reynolds-number flows (3–15 million). We evaluated the performance of two LCTMs coupled with the SA turbulence model. As a result, the SA-based two-equation transition model showed a comparable performance with the

*e*

^{N}-based method and better glide ratio (

*L/D*) predictions than the SST-based model.

*e*

^{N}-based method showed its superiority over local correlation-based transition...

- Abstract
- Introduction
- Methodology
- Validation of turbulence model
- Results: transition modeling
- Conclusions
- Appendix A: Additional results
- Appendix B: Grid convergence study
- Appendix C: Solution convergence study
- Data availability
- Author contributions
- Competing interests
- Disclaimer
- Acknowledgements
- Financial support
- Review statement
- References