#### 4.1. Lift and Drag Airfoil Characteristics

Figure 5 shows the aerodynamic characteristics of the DU-91 W2-250 airfoil for two Reynolds numbers

$3\times {10}^{6}$ and

$6\times {10}^{6}$. These characteristics were calculated using the numerical procedures described in

Section 3 of this paper. Additionally, the characteristics obtained with full-turbulence models (the k-ω SST and the k-ε RNG) published in [

83,

84] were used for comparison. These numerical predictions were compared with the experimental measurements [

15,

58].

For the qualitative assessment of numerical approaches, the relative error,

$\delta $, was calculated from the formula:

For the zero angle of attack, the values of the drag coefficients measured in the wind tunnel are 0.0072 for a Reynolds number of $3\times {10}^{6}$ and 0.0069 for a Reynolds number of $6\times {10}^{6}$. For both analyzed Reynolds numbers, the relative error of all numerical approaches used, except for the k-ω SST and k-ε RNG models, was less than 12%. The lowest relative error was obtained for the STAR-CCM + code; it was 3% for Reynolds number of $3\times {10}^{6}$ and 4% for Reynolds number of $6\times {10}^{6}$. The obtained results showed that the relative error increases with the increase of the Reynolds number for all the used transition models, including the XFOIL code. The reverse tendency was observed for full-turbulence models. However, for these models the relative error of 60–70% for a Reynolds number of $6\times {10}^{6}$ is quite large.

For experimental data, the aerodynamic derivative

$a=d{C}_{L}/d\alpha $ was 6.88 for a Reynolds number of

$3\times {10}^{6}$ and 7.08 for a Reynold numbers of

$6\times {10}^{6}$. These derivatives are estimated for angles of attack in the range from 0° to 8.1°. For the same angles of attack range, the RANS approach together with the Transition SST turbulence model provided the aerodynamic derivatives of 7.03 and 6.95 for Reynolds numbers of

$3\times {10}^{6}$ and

$6\times {10}^{6}$, respectively. Values of the relative error of the aerodynamic derivative are 2.25% for

$Re=3\times {10}^{6}$ and

$1.9\%$ for

$Re=6\times {10}^{6}$. In the case of the URANS approach with the Transition SST model, the numerical results of the lift force coefficients are obtained for angles of attack in the range from −0.04° to 10.4°. Since, for the angle of attack equal to 10.4°, the calculated value of the

${\overline{C}}_{L}$ coefficient differs significantly from the experimental one (as can be seen with the naked eye in

Figure 5), the qualitative assessment of the lift coefficient characteristics by using the aerodynamic derivative is limited to the range of angles of attack from 0° to 8.1°. For the URANS approach, the relative error of the aerodynamic derivative is 6.1% for the Reynolds number of

$3\times {10}^{6}$ and 1.5% for the Reynolds number of

$6\times {10}^{6}$. In comparison with the experiment, only for the angle of attack of 10.4°, the relative error is 21.5% for the Reynolds number of

$3\times {10}^{6}$ and 12.4% for a Reynolds number of

$6\times {10}^{6}$. Although the largest differences between the experimental and numerical values are observed for the angle of attack equal to 10.4°, with the increase of the Reynolds number, the value of the relative error decreases. The increasing relative error of both components of the aerodynamic force starting from the angle of attack equal to 8.1° for both analyzed Reynolds numbers results from the simplifying assumptions used. Using the DES technique, Rogowski et al. proved that even at a small angle of attack, significant velocity fluctuations in spanwise direction are observed [

58]. These fluctuations increase with the angle of attack; however, their quantitative analysis using the DES technique is extremely computationally expensive.

The lift-to-drag ratio is the amount of lift generated by a wing divided by the aerodynamic drag:

Figure 5 shows the lift-to-drag ratio,

K, as a function of the angle of attack,

α, for the Reynolds numbers of

$3\times {10}^{6}$ and

$6\times {10}^{6}$. The results shown in this figure clearly favor the Transition SST approach. The characteristics of the lift-to-drag ratio calculated using the classical two-equation turbulence models (the RNG k-ε and the SST k-ω models) that are widely used to simulate the flow field in engineering applications, differ significantly from predictions given by the experiment and the Transition SST models. It can be observed that for both analyzed Reynolds numbers the two-equation turbulence models underestimate the values of the lift-to-drag ratio whereas the XFOIL method overestimates the characteristics.

Table 2 summarizes the derivative of the lift-to-drag ratio with respect to the angle of attack,

$dK/d\alpha $ for the various aerodynamic methods and for the linear range of the lift-to-drag ratio characteristics up to

$\alpha =4\xb0$. The results shown in this table confirm the similar compliance of all CFD codes with the implemented Transition SST turbulence model. Moreover, the aerodynamic characteristics obtained by means of classical turbulence models prove the critical role of transition phenomena in the boundary layer.

Figure 6 shows characteristics of the aerodynamic force coefficients,

${\overline{C}}_{D}$ and

${\overline{C}}_{L}$ and the torque coefficient

${\overline{C}}_{M}$, as a function of the Reynolds number for the six angles of attack. Numerical calculations are made for four Reynolds numbers:

$3\times {10}^{6}$,

$4\times {10}^{6}$,

$5\times {10}^{6},$ and

$6\times {10}^{6}$. These investigations were only conducted using the ANSYS Fluent CFD code and URANS approach together with the Transition SST turbulence model. As it can be seen from this figure, the lift coefficients are practically constant over the entire investigated range of both Reynolds numbers and angles of attack (

Figure 6c,d). The largest differences in the results are seen for the drag coefficient (

Figure 6a,b). The drag coefficient is a function of both the Reynolds number and the angle of attack. Charts illustrated in

Figure 6a,b show that up to an angle of attack equal to 6 degrees the drag decreases with increasing the Reynolds number. For angles of attack of 8 and 10 degrees the

${\overline{C}}_{D}$ characteristics follow a slightly different trend: at first, the drag decreases, and then it begins to increase. Munson et al. provided a typical character of the drag coefficient of an airfoil [

85]. The airfoil drag coefficient decreases as the Reynolds number increases. At first, the drag decreases faster and then its changes are not very important. According to Munson et al., a faster decrease in drag is seen in the range of Reynolds numbers from

$~{10}^{4}$ to

$~{10}^{6}$ [

85]. Then, up to the value of

$Re\approx {10}^{7}$, the drag coefficient changes little. It has been proven in these studies that starting from a Reynolds number of

$4\times {10}^{6}$, the drag becomes almost constant at zero angle of attack. As the angles of attack increase, the airfoil begins to behave like a blunt body and pressure drag plays a more significant role. This explains the increase in drag for angles of attack of 8 and 10 degrees.

Figure 6e,f show the torque coefficients for different angles of attack and as a function of the Reynolds number. The torque coefficient increases with the angle of attack, however, it is independent of the Reynolds number up to the angle of attack of 10 degrees.

#### 4.2. Static Pressure Coefficients

When an airfoil moves through the air, an interaction between the airfoil and the air occurs. In the previous subsection of this paper, this effect is described in terms of the aerodynamic forces. This subsection discusses this effect using the static pressure coefficient distributions along the airfoil. Static pressure coefficient is defined as:

where

$P$ is the static pressure,

${P}_{ref}$ is the reference pressure, and

${q}_{ref}$ is the dynamic reference pressure defined as:

where

${\rho}_{\infty}$ is the free stream density and

${V}_{\infty}$ is the free stream velocity.

The distribution of static pressure along an airfoil depends on many factors. These are both the shape of the airfoil itself and the flow parameters. This paper compares the pressure distributions obtained by different analytical approaches. Moreover, the influence of the Reynolds number and the angle of attack on the pressure distributions is shown here. In

Section 2 of this paper, it was proved that the oscillations of the aerodynamic forces obtained using the transient momentum equations and the Transition SST turbulence model are not large (

Figure 3). This Section also shows the effect of transient flow conditions on pressure distributions.

Figure 7 shows the validation of various numerical approaches based on experimental data. For clarity, the first four plots in

Figure 7 show the results obtained with the Transition SST approach implemented in various CFD codes, while the following four graphs in the same figure illustrate the results obtained with the two fully turbulent models.

Figure 7 also compares the results for the XFOIL code. The validation of all analytical approaches has been shown for one angle of attack of 4 degrees and for two Reynolds numbers of

$3\times {10}^{6}$ and

$6\times {10}^{6}$. A similar tendency in the difference between the experimental and numerical results was observed for the remaining angles of attack taken into account in this paper; therefore, they are not presented in this paper. The pressure distributions calculated using the Transition SST model and ANSYS Fluent CFD code shown in

Figure 7 are averaged over the same time interval as the aerodynamic forces discussed in the previous

Section 3. Due to the fact that practically no oscillations of the components of the aerodynamic forces appear, the instantaneous pressure distributions differ slightly from the instantaneous values. The obtained results show clearly greater accuracy of the Transition SST model with the k-ω SST and k-ε RNG models. The obtained results also show a much greater discrepancy with the experimental results and with each other for the largest Reynolds number studied in this paper. In particular, these differences are visible on the suction side of the airfoil for

$\mathrm{x}/\mathrm{c}$ ranging from 0.12 to 0.5. Rogowski and Hansen showed that for this case, which corresponded to a Mach number of 0.3, the local Mach number in the vicinity of the blade edge reaches the value of 0.5 [

58]. With the increase of the angle of attack on the suction edge of the airfoil, the area of the large Mach number field also increases. However, numerical calculations taking into account the compressibility of a continuous medium are definitely more expensive [

58].

Validation of the Transition SST approach more broadly is shown in

Figure 8. In this case, for better clarity of the graphs, the results are given only obtained with the ANSYS Fluent CFD code. These plots clearly show that the effects of compressibility can affect the accuracy of static pressure distributions.

As already mentioned above, the static pressure distributions obtained using the in ANSYS Fluent CFD code together with the URANS approach and the Transition SST turbulence model show a high agreement with the experimental results. In particular, very good agreement with the experimental results is obtained for the lower Reynolds number of

$3\times {10}^{6}$. In order to qualitatively compare both data sets, experimental and numerical, a quantitative analysis was performed using the definition of relative error:

where

${C}_{Pexp}$ is the measured static pressure coefficient, whereas

${C}_{PCFD}$ is the static pressure coefficient calculated using CFD. Since, as already mentioned, the experimental and numerical results behave in a similar manner for both analyzed Reynolds numbers (for a lower Reynolds number the better agreement of the numerical results with the experimental ones, and for a larger Reynolds number the agreement is smaller), an angle of attack equal to 4 degrees was selected for the quantitative analysis. To perform such a quantitative comparative analysis, the

${C}_{P}$ results obtained in the CFD analysis are interpolated into the coordinates for which the pressures were measured during the experiment. The relative error obtained from this analysis, averaged over the entire airfoil, is 9.8% for the Reynolds number of

$3\times {10}^{6}$ and 17.8% for the Reynolds number of

$6\times {10}^{6}$. The relative error was calculated separately for the suction side of the airfoil and for the pressure side to make this analysis even more useful. This analysis showed that for the lower analyzed Reynolds number, the relative error is 8.6% for the suction side of the airfoil and 11.3% for the pressure side. For the Reynolds number equal to

$6\times {10}^{6}$, the following results are obtained: 17% for the suction side of the profile and 18.8% for the pressure side. Based on this quantitative analysis, we can clearly see that as the Reynolds number increases, the relative error increases, especially for the pressure side. There are two possible reasons for this increase. One of the reasons may be the high Mach number locally present on the profile surface. In the numerical calculations performed in this work, the Mach number of undisturbed flow was 0.15 for a Reynolds number equal to

$3\times {10}^{6}$ and 0.3 for

$Re=6\times {10}^{6}$ [

58]. The second reason that may cause a larger error on the pressure surface of the airfoil is the neglect of the effects that occur in the spanwise direction [

15].

Figure 7 and

Figure 8 illustrate the validation of the numerical results of static pressure distributions with the experiment. However, due to the very high similarity, these figures do not show the effect of the Reynolds number on the static pressure coefficient characteristics.

Figure 9 presents a comparison of

${C}_{P}$ characteristics as a function of

$\mathrm{x}/\mathrm{c}$ for four Reynolds numbers:

$3\times {10}^{6}$,

$4\times {10}^{6}$,

$5\times {10}^{6}$, and

$6\times {10}^{6}$. As the differences in pressure distributions for the analyzed Reynolds number values are practically invisible at the suction side of the airfoil, only pressure distributions on the pressure side were considered in this analysis. Additionally, since the nature of these functions is similar for all analyzed angles of attack, only the results for one angle of attack equal to 4° are compared.

Figure 9 shows the Reynolds number effect at only two locations on the profile. These locations are marked with dashed lines for which other colors have been selected: detail one (magenta) and two (blue).

Figure 9c,d show zooms of these details. As shown in

Figure 9d, as the Reynolds number increased, a slight pressure increase is observed over the aft part of the airfoil, for

$\mathrm{x}/\mathrm{c}=0.88$.

From

Figure 7 and

Figure 8 it can be seen that the static pressure distribution depends primarily on the angle of attack. This was also proved above by analyzing the effect of the Reynolds number on static pressure distributions. Therefore, in this work, the effect of the angle of attack on the static pressure distribution was investigated for only one Reynolds number.

Figure 10 presents the static pressure distributions on the airfoil depending on the angle of attack for

$Re=6\times {10}^{6}$. The obtained results are compared for six different angles of attack ranging from 0 to 10 degrees. The analysis of these curves showed that in the case of the pressure side of the airfoil, the local peak of the

${C}_{P}\left(\mathrm{x}/\mathrm{c}\right)$ curve near the leading edge of the profile is observed. This peak has a very similar value equal to approximately 1 for all investigated angles of attack, however, as the angle of attack increases, the peak moves towards the back of the airfoil. The analysis of the location of this peak as a function of the angle of attack showed that this change is not linear but is a second-degree quadratic function with the equation:

${\left(\mathrm{x}/\mathrm{c}\right)}_{max}=0.00020\xb7{\alpha}^{2}+0.00118\xb7\alpha +0.0004$.

In the case of the suction side of the airfoil, the shape of the ${C}_{P}\left(\mathrm{x}/\mathrm{c}\right)$ curve has a slightly different course in two ranges of the angle of attack: up to an angle equal to 6 degrees and above it. Starting at an angle of attack of 6 degrees, a characteristic peak near the leading edge of the airfoil becomes visible on the ${C}_{P}$ curve. This peak increases as the angle of attack increases from the value of 1.76 for an angle of attack of 6 degrees to the value of 4 for an angle of attack of 10 degrees. The location of this peak slightly shifts towards the leading edge from the value of $\mathrm{x}/\mathrm{c}=0.0056$ for the angle of attack equal to 6 degrees to the value of 0.0025 for the angle of 10 degrees.

As mentioned at the beginning of this subsection, the nature of the transient flow parameters was also studied in this paper. Since, as already discussed in

Section 3 of this paper, the nature of the

${\overline{C}}_{L}$ and

${\overline{C}}_{D}$ curves is almost constant, therefore the instantaneous static pressure distributions differ very slightly from the averaged distributions illustrated above. In order to study quantitatively the dispersion of the instantaneous distributions from the mean value, the standard deviation (STD) distributions of the pressure coefficients have been calculated. Standard deviation is a parameter that is used much more often to evaluate the scatter of measurement results. However, since the obtained results of

${C}_{P}$ differed very little, it was decided to use this classic measure of variation. The following definition was used to calculate the standard deviation:

where

${N}_{p}$ is the number of population samples (in these simulations, the values of

${N}_{p}=33,334$),

${C}_{Pi}$ is the instantaneous pressure coefficient, and

${C}_{P}$ is the averaged pressure coefficient.

As is mentioned in

Section 3 of this paper, the mesh that provides an independent numerical solution has 620 nodes on the airfoil surface. In this work, the standard deviation was calculated separately for each node on the airfoil surface and separately for the suction and pressure sides of the airfoil. Standard deviation analysis was performed for two cases: (a) constant Reynolds number and variable angle of attack and (b) constant angle of attack and variable Reynolds number. The investigation showed that the standard deviation is a function of both these quantities. The performed simulations also proved that the obtained values of the standard deviation of the static pressure coefficient are much higher for the suction side of the airfoil compared to the pressure side. Therefore, only the STD distributions for the suction side of the airfoil are discussed in the further part of this paper.

Figure 11a shows a contour map of the standard deviation of the static pressure depending on the Reynolds number and the angle of attack of the airfoil. This figure shows that with the increase in the angle of attack, an increase in the value of STD is observed up to the angle of attack from 2° to 4°, then, the decrease in the value of STD with the angle of attack is observed. The angle at which the STD reaches its maximum depends on the Reynolds number; as the Reynolds number increases, the peak values of the STD function move towards higher angles of attack.

As already mentioned above, the standard deviation of the static pressure coefficients also has a distribution along the airfoil surfaces. Since the character of the STD is the same for all Reynolds numbers taken into account in this paper, case b (constant angle of attack and variable Re value) is considered for one angle of attack. An angle of attack equal to 4 degrees is selected for the analysis, for which the standard deviations reach maximum values.

Figure 11b shows a contour map of the standard deviation of the static pressure coefficients in terms of Reynolds number and x/c. The results shown in this figure apply only to the suction side of the airfoil since, as highlighted above, the largest changes in the STD values are observed on this surface. The contour map clearly shows that the maximum values of STD are concentrated in a narrow strip around the position x/c = 0.4. Additionally, it was observed that the width of this strip decreased with increasing Reynolds number. The location of the maximum values of STD is not accidental. It is related to the location of the laminar-turbulent transition.

In general, the values of the standard deviation obtained in all analyzes are minimal. The obtained minimum value of this parameter was $1.9\times {10}^{-5}$ and the maximum value was 0.003. Low standard deviation values indicate that the values tend to be close to the mean static pressure coefficients.

#### 4.3. Skin Friction Coefficient

The time-averaged skin friction depends on the same factors as static pressure. The formulation of the transition SST approach makes it possible to obtain such a distribution of the skin friction that it is possible to study the laminar-turbulent transition.

In this work, various numerical approaches were used to determine the aerodynamic performance of the DU 91-W2-250 profile. Basically, for three different numeric codes, the same turbulence model was used—the Transition SST approach developed by Langtry and Menter [

78]. The other tools were two fully turbulent models and XFOIL code. As the results of both forces, pressures and skin friction distributions obtained by various implementations of the Transition SST approach were very similar, this subsection of the paper was limited only to the analysis of the results obtained using the ANSYS Fluent code together with the URANS method.

In the first part of this subsection, the distributions of the skin friction coefficient for the angle of attack equal to 4 degrees and for two Reynolds numbers

$3\times {10}^{6}$ and

$6\times {10}^{6}$ obtained with this method were compared only with the results of the XFOIL code and the k-ε RNG turbulence model (

Figure 12). As expected, the results obtained with the k-ε RNG turbulence model are of a completely different nature compared to the transition models. This is due to a different formulation of this turbulence model, whereby the entire boundary layer is taken into account as turbulent. Such an assumption, however, causes that the distribution of the skin friction coefficient differs significantly from the real one [

35]. The

${C}_{f}$ characteristics obtained by the two approaches that take into account the transition effect are similar; in particular, it concerns the Reynolds number equal to

$6\times {10}^{6}$. For the Reynolds number of

$3\times {10}^{6}$, an overestimated value of the

${C}_{f}$ coefficient, computed using the XFOIL code, was observed in comparison with the numerical results obtained using the Transition SST approach. The results of the e

^{N} transition method based on linear stability and implemented in the XFOIL code are dependent on the N-factor, which should be determined by wind tunnel or flight test calibration [

35].

Figure 12 shows the results for both the suction side and the pressure side of the airfoil. As it can be seen from this figure, the values of the

${C}_{f}$ coefficient also depend on the airfoil side; this effect will be discussed in detail below.

In this part of the paper, the effects of the Reynolds number and angle of attack on the distributions of the ${C}_{f}$ coefficient will be discussed. As mentioned above, the ${C}_{f}$ distributions are different on both the suction and the pressure side of the airfoil. In this paper, the obtained results are illustrated separately for both sides of the profile to make these effects more visible.

Characteristics of the

${C}_{f}$ coefficient separately for the pressure and the suction side of the airfoil are shown in

Figure 13. Since the nature of the graphs obtained in this work is similar for all angles of attack analyzed here, the figure only illustrates the results for the angle of attack of 4 degrees. The characteristics shown in this figure are given for four Reynolds numbers:

$3\times {10}^{6}$,

$4\times {10}^{6}$,

$5\times {10}^{6},$ and

$6\times {10}^{6}$. Contrary to the static pressure coefficients (

Figure 9), the skin friction coefficient distributions significantly depend on the Reynolds number. However, it should be emphasized that the influence of the Reynolds number is mainly visible on the suction side of the airfoil (

Figure 13b). As the Reynolds number increases, the maximum value of the

${C}_{f}$ coefficient increases in the middle part of the airfoil, where the laminar-turbulent transition occurs. Moreover, as the Reynolds number increases, the jump in

${C}_{f}$ shifts to the leading edge. For the suction side of the airfoil and for the Reynolds number of

$6\times {10}^{6}$, the maximum value of the skin friction coefficient

${C}_{fmax}$ increased by 40.8% compared to the

${C}_{fmax}$ for

$Re=3\times {10}^{6}$. At the same time, the maximum value of the

${C}_{f}$ coefficient for

$Re=3\times {10}^{6}$ was located at

$\mathrm{x}/\mathrm{c}=0.60$ whereas for the

$Re=6\times {10}^{6}$, the

${C}_{fmax}$ has shifted towards the leading edge to the value

$\mathrm{x}/\mathrm{c}=0.44$. For a Reynolds number of

$3\times {10}^{6}$, the maximum value of the

${C}_{f}$ coefficient for the suction side of the airfoil is 28.1% larger compared to the

${C}_{fmax}$ for the pressure side. However, for the Reynolds number of

$6\times {10}^{6}$, this value increases to 71.1%. The increase of the

${C}_{fmax}$ with the increase of the Reynolds number is linear on the suction surface of the airfoil.

The effect of the Reynolds number on the distribution of the skin friction coefficient is discussed above. The second important factor influencing this physical quantity is the angle of attack.

Figure 14 discusses the

${C}_{f}$ distributions for two Reynolds numbers and for six angles of attack ranging from 0 to 10 degrees. Additionally, as already stated above, the

${C}_{f}$ distributions are different on the suction and pressure side of the airfoil, therefore

Figure 14 compares the

${C}_{f}$ distributions for both sides of the airfoil separately.

As mentioned above and shown in

Figure 13, the Reynolds number effect is more visible on the suction side of the airfoil. As shown in

Figure 14, this conclusion applies to all Reynolds numbers analyzed in this work. Moreover, as documented in

Figure 14, the effect of the angle of attack is also larger on the suction side of the airfoil as compared to the pressure side. The

${C}_{f}$ coefficient for both sides of the airfoil differs both in its maximum value and in its

$\mathrm{x}/\mathrm{c}$ location. As the angle of attack increases on the suction side of the airfoil, the maximum value of the

${C}_{f}$ coefficient moves towards the leading edge and towards the trailing edge on the pressure side of the airfoil. Moreover, as the angle of attack increases, the

${C}_{fmax}$ increases on the suction side of the airfoil and decreases on the pressure side. For a Reynolds number equal to

$3\times {10}^{6}$, with an angle of attack increment equal to 10 degrees (

$\Delta \alpha =10\xb0$), the absolute value of the

$\Delta \mathrm{x}/\mathrm{c}$ increment corresponding to the

${C}_{fmax}$ is 0.3126 on the suction surface and 0.1148 on the pressure surface. For a Reynolds number of

$6\times {10}^{6}$, the same values are respectively 0.3166 for the suction side and 0.0647 for the pressure side. It can be concluded from this analysis that with the increase of the Reynolds number, the value of the

$\Delta \mathrm{x}/\mathrm{c}$ on the suction side is almost constant and it has been shortened by half on the pressure side. For all the angles of attack analyzed in this paper, on the pressure side of the airfoil, the decrease in the maximum value of the

${C}_{f}$ coefficient with the increase in the angle of attack is approximately linear whereas on the suction side of the airfoil, the increase in the maximum value of the

${C}_{f}$ coefficient with the increase in the angle of attack is an exponential function.

As in the case of static pressure distributions (

Section 4.2 and

Figure 11), this work also examined the deviations of the instantaneous values of the skin friction coefficient from the average value (

Figure 15). This paper also uses the standard deviation formula (Equation (8)) to evaluate this deviation. As with the static pressure distributions, the values of the standard deviation for the suction side of the airfoil are much larger than for the pressure side (please see

Section 4.2). In the case of the

${C}_{f}$ coefficient, the STD results for the suction side of the airfoil are on average 11 times higher compared to the pressure side. The values of the standard deviation of the skin friction coefficient turned out to be small of the higher order and almost 99 times lower compared to the values of the standard deviation calculated for the static pressure coefficients.

#### 4.4. Variation of Transition Location with the Angle of Attack

The location of the transition is a very important engineering issue in modern fluid mechanics; however, it is not easy to determine. For the XFOIL code, this position is one of the simulation results. In the case of CFD analyzes, this location can be found on the basis of intermittency,

$\gamma $. In our CFD studies, however, this position was estimated from the distribution of the skin friction coefficient

${C}_{f}$, discussed in the previous subsection. In order to estimate the transition location

${X}_{tr}$ the point where the skin friction increases sharply, e.g., as shown in

Figure 14, was chosen. This is more consistent with the way the transition is determined experimentally where the intermittency is not known, e.g., using hot film sensors to directly measure the skin friction or a take thermal image of the airfoil surface and see when the heat flux often changes very sharply. Therefore, in this work, to determine this value, a criterion was used, which was established on the basis of the

${C}_{f}$ results obtained from the XFOIL approach. It was found that the position of

${X}_{tr}$ corresponds to an increase of the

${C}_{f}$ coefficient by 80% compared to the minimum value.

Figure 16 compares the location

${X}_{tr}$ of the laminar-turbulent transition predicted by the two analytical approaches and with the experiment. In this work, the analytical approaches used for the comparison are the Transition SST turbulence model implemented in the ANSYS Fluent CFD code and the XFOIL code. All simulations were carried out for the range of angles of attack in the range from 0 to 10 degrees. The authors of this work had only experimental data for the suction side of the airfoil.

In

Figure 16, it can be seen that two distinct areas are visible on the suction side of the airfoil. The

${X}_{tr}\left(\alpha \right)$ curves shown in this figure have two derivatives: in the range of angles of attack from 0 to 6 degrees for the first area, and in the range of angles of about 6 to 10 degrees for the second area. On the other hand, the relationship of the

${X}_{tr}$-coordinate for the pressure side of the profile is almost linear over the entire range of angles of attack taken into account in this paper.