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In this paper, the lift coefficients of SC-0414 airfoil are estimated by applying modified Yamana’s method to the flow visualization results, which are obtained by utilizing the smoke tunnel. The application of the modified Yamana’s method is evaluated with two calculation methods. Additionally, the lift estimation, wake measurements, and numerical simulations are performed to clarify the low-speed aerodynamic characteristics of the SC airfoil with flaps. The angle of attack was varied from
−5
° to 8
°. The flow velocity was 12 m/s and the Reynolds number was 1.6 × 10
^{5}. As a result, the estimated lift coefficients show a good agreement with the results from reference data and numerical simulations. In clean condition, the lift coefficients calculated from the two methods show quantitative agreement, and no significant difference could be confirmed. However, the slope of the lifts calculated from
*y*
_{s} is higher and closer to the reference data than those obtained from s
*c*, where
*y*
_{s} denotes the height where the distance from the streamline to the reference line is the largest, and s
*c* denotes the displacement of the center of pressure from the origin of the coordinate, respectively. In the case of flaps, the GFs have an observable effect on the aerodynamic performance of the SC-0414 airfoil. When the height of the flap was increased, the lift and drag coefficients increased. The installation of a GF with a height equal to 1% of the chord length of the airfoil significantly improved the low-speed aerodynamic performance of SC airfoils.

Several experimental investigations on estimating the lift coefficient of airfoils in two-dimensional (2D) smoke tunnels have been performed [_{s}) where the distance from the streamline to the reference line is the largest. Yamana et al. [

In the original method, a smoke line must be adjusted to intersect with the reference line at a specific distance of upstream. It requires a long test section of smoke tunnel to visualize and adjust the intersection. Yamaguchi et al. [_{s} are both included in the original Yamana’s method [_{s} can be measured within higher spatial resolution, because it basically tends to be larger measured value. It is considered that more accurate results can be obtained by the method using y_{s}.

Supercritical airfoils are widely used for civil transport aircraft. These airfoils first garnered attention when NASA made efforts to develop an airfoil that shows better performance in transonic flow while retaining acceptable characteristics in the low-speed flow [

In the present study, the flow visualizations around SC-0414 airfoil and estimation of lift coefficient are performed utilizing the low-speed smoke tunnel. The application of the modified Yamana’s method is evaluated with two calculation methods based on the measurement of sc and y_{s}. Furthermore, the lift estimation, wake measurements and numerical simulations are performed to clarify the low-speed aerodynamic characteristics of the SC airfoil with Gurney flaps.

_{0} is the height of the smoke line at x = −nc, y_{1} is the height of the smoke line, which closest to the airfoil, at x = −c, sc is the horizontal displacement of the center of pressure, and y_{s} is the height of the smoke line at x = sc. When a streamline passes through points A(−nc, y_{0}), B(−c, y_{1}), and C(sc, y_{s}), the following stream function is established:

ψ [ − ( n + s ) C , Y 0 ] = ψ [ − ( 1 + s ) C , Y 1 ] = ψ [ 0 , Y s ] (1)

where C = πc/h, Y_{0} = πy_{0}/h, Y_{1} = πy_{1}/h, and Y_{s} = πy_{s}/h. Then, with the inclusion of the flap effects, the lift coefficient can be calculated as:

c l ( Y 0 , Y 1 , s C ) = 4 π ( Y 1 / C − Y 0 / C ) G ( Y 0 , Y 1 , s C ) (2)

where:

G ( Y 0 , Y 1 , s C ) = l n ( s i n h 2 ( n + s ) C + s i n 2 Y 0 ) 1 2 ( c o s h ( n + s ) C + c o s Y 0 ) − l n ( s i n h 2 ( 1 + s ) C + s i n 2 Y 1 ) 1 2 ( c o s h ( 1 + s ) C + c o s Y 1 ) (3)

The lift coefficient can thus be calculated from the measured values of y_{0}, y_{1}, and sc.

Next, through the application of the modified Yamana’s method, the calculation of the lift coefficient from y_{s} is considered. With the same variable definitions as above, we have:

c l ( Y 0 , Y s , s C ) = 4 π ( Y s / C − Y 0 / C ) G ( Y 0 , Y s , s C ) (4)

c l ( Y 1 , Y s , s C ) = 4 π ( Y s / C − Y 1 / C ) G ( Y 1 , Y s , s C ) (5)

With Equations (4) and (5), with sc and c_{l} as the variables, the lift coefficient c_{l} can be calculated from the measured values of y_{0}, y_{1}, and y_{s}. The lift coefficient can thus be calculated from the measured values of y_{0}, y_{1}, and sc.

In this study, we focus on investigating the modified Yamana’s method by considering y_{s}, which intended to estimate the lift coefficient with higher accuracy.

In this investigation, the value of n is 1.7, and the airfoil chord length c is 200 mm. nc is a distance from the origin of the coordinates to point A, which is free to be determined in the front of point C, as described in _{0} of point A, we put n as a value of 1.7. In the Yamana’s experiments [_{0} = 0 and the value of n is about 4.5. Besides, s is determined as the ratio between the airfoil chord length and the distance from the origin to point C, where the height of the smoke line is largest. The value of s changes when the measurement condition changes.

The lift coefficients estimated by the Yamana’s method showed a good agreement with other experimental results when the smoke lines flow smoothly around the upper surface of the airfoil [

The drag can be obtained by comparing the momentum in the air upstream of the model with that downstream of the model [_{d} is then given by

c d = 1 c ∫ a b ( u 2 u 1 − u 2 2 u 1 2 ) d y (6)

where u_{1} is the inflow velocity, u_{2} is the outflow velocity, [a, b] represents the measurement range in the wake of the test model, c is the chord length, and y is the position in the vertical direction.

A hot-wire system was used to measure the outflow velocity in the wake of the airfoil. A hot-wire anemometer (System 7000, Kanomax) was used in this study, and a I-type hot film (0251R-T5) was selected to measure the velocity. The data measured by the hot-wire system were transmitted to a computer by a data logger (NR500, Keyence).

The experimental condition is shown in _{∞} is 12 m/s, and the Reynolds number Re based on the chord length was 1.6 × 10^{5}. In wake measurement, the probe was set up 1.0 c downstream from the trailing edge of the airfoil model. The probe was swept in the vertical direction over a measurement range [a, b] of 300 mm (−150 mm ≤ y ≤ 150 mm; where y = 0 is a point located in the reference line in

Numerical simulations are performed to compare with experimental results.

Experimental condition | ||
---|---|---|

Flow velocity | 12 m/s | |

Test model | SC-0414 with GFs | |

Angle of attack | Flow visualization | −5˚ - 8˚ (1˚ interval) |

Wake measurement | −4˚ - 8˚ (2˚ interval) | |

Wake measurement | Wake location | 1.0c from the trailing edge of the airfoil model |

Measurement range | −150 mm ≤ y ≤150 mm | |

Measurement interval | Δy = 5 mm | |

Measurement time at 1 point | 20 s |

SC-0414 airfoil model is 200 mm. ICEM software was used to construct a C-type structured grid with 340 and 400 points distributed on the surface of the airfoil alone and the airfoil with the GF, respectively.

Upper and lower boundaries of the simulation domain are 10 chord-length away the airfoil, as the velocity inlet condition and downstream outflow boundary. The y+ value of the grid is approximately 1, which means that the height of first grid element nearest the airfoil is approximately 2.4 × 10^{−5} m. Simulations were performed using the software Ansys Fluent 18.2. The Reynolds-averaged Navier–Stokes (RANS) equation and the k-ω shear stress transport (SST) model was applied in all conditions [^{−6} for the continuity and 10^{−5} for other terms.

When the smooth smoke lines can be observed, the estimated lift coefficients showed a good agreement with other experimental results [

_{l} at different angles of attack α in the case of no flap. In previous study of the modified Yamana’s method in this experimental system [^{5} (airfoil tools [_{0}, y_{1}, y_{s} were determined as their average values. The variation of these values in the three pictures was not significant. Then, the average values are applied to the equation (2), (3), (4), (5) in section 2. The slopes of the lift obtained from the reference data and numerical simulation with respect to the angle of attack are approximately 0.118 and 0.103, respectively. Additionally, the slopes for the experimental results obtained from sc and y_{s} are 0.113 and 0.117, respectively. This demonstrates that the experimental results show a good agreement with the numerical simulation and reference XFOIL data results. The simulation results quantitatively agreed with the reference data for angles of attack from 0˚ to 8˚. However, the XFOIL lift coefficients are less than those obtained from the numerical simulation at negative angles of attack. In contrast, the lift coefficients calculated from sc and y_{s} show quantitative agreement, and no significant difference between these results could be confirmed. However, the slope of the lift calculated from y_{s} is higher and closer to the reference data than those obtained

from sc. Since the distance of y_{s} is larger than sc, it is considered that the measurement of y_{s} can obtain a better resolution, and the measurement of y_{s} is easier than sc.

E r r o r = | s c c a l − s c m e a | c

where sc_{cal} is the displacement of the center of pressure obtained from the Equations (4) and (5). The value y_{s} can be directly measured by G3 data software from the pictures. Then, Equations (4) and (5) becomes a set of 2 equations with 2 variables (c_{l} and sc). The set of the two equations that can be solved by using the Matlab program. The value of “sc” calculated by the Matlab program is called sc_{cal}. The value of “sc” directly measured from point C to the origin of the coordinates is called sc_{mea}. This value is applied to equation (2) to calculate the lift coefficients, which are called the “results base sc”. Overall, the maximum error was within approximately 1% regardless of the angle of attack, demonstrating the good agreement between the calculation and measurement results. The results also show the accuracy of the modified Yamana’s method considering the displacement of center pressure in the case of the SC-0414 airfoil.

_{s}, as were the airfoil baseline’s results in _{s} of the smoke line averaged over three measurements.

The slope of the lift coefficient from the simulation results is approximately 0.118 and is nearly the same for all flap height conditions. Because the modified Yamana’s method is effective in steady flow, this investigation was conducted at angles of attack ranging from −5˚ to 8˚. The case of the airfoil with l/c = 0.01 shows that the addition of a flap significantly improves the lift. From 4˚ to 8˚, in all cases, the experimental results were slightly different from the corresponding simulation results; however, the results showed overall good qualitative agreement. The experimental results showed the same trend as the simulations. An increase in the height of the flap produced an increase in the lift coefficient. The results show that in the case of the SC-0414 airfoil, the GF has similar effects as other simple plain flaps, as has been discussed by Raymer [

In this study, the flow visualization around SC-0414 airfoil and estimation of lift coefficient are performed utilizing the low-speed smoke tunnel. The application of the modified Yamana’s method is validated with two calculation methods. Additionally, the lift estimation, wake measurements and numerical simulations are performed to clarify the low-speed aerodynamic characteristics of the SC airfoil with Gurney flaps. The results are as follows:

1) The visualization results from the smoke wind tunnel experiment and the numerical simulation results showed a qualitative agreement. When increasing the height of the flaps, the area of high pressure is expanded at the leading and trailing edge due to the effect of increasing the camber.

2) The lift coefficient calculated by modified Yamana’s method shows good agreement with the numerical simulation results and reference data. The methods of obtaining the lift coefficient from the height y_{s} where the distance from the streamline to the reference line is the largest can obtain closer results to reference data than the method based on the displacement sc of the center of pressure.

3) In modified Yamana’s method calculation, the maximum error of the value sc was approximately 1% regardless of the angle of attack, demonstrating the agreement between the calculation and measurement results.

4) When the height of the flap was increased, the lift and drag coefficients increased. The installation of a GF with a height equal to 1% of the chord length of the airfoil significantly improved the low-speed aerodynamic performance of SC airfoils.

The authors declare no conflicts of interest regarding the publication of this paper.

Nguyen, T.D., Kashitani, M. and Taguchi, M. (2020) Low-Speed Aerodynamic Characteristics of Supercritical Airfoil with Small High-Lift Devices from Flow Pattern Measurements. Journal of Flow Control, Measurement & Visualization, 8, 159-172. https://doi.org/10.4236/jfcmv.2020.84010

c = the airfoil chord length

c_{d} = drag coefficient

c_{l} = lift coefficient

h = the height of the test section

y_{0} = the height of the smoke line at x = −nc

y_{1} = the height of the smoke line at x = −c

Re = Reynolds number based on chord length

sc = the horizontal displacement of the center of pressure

u_{1} = the inflow velocity (=U_{∞})

u_{2} = the outflow velocity

x, y = Cartesian coordinated system

α = angle of attack

y_{s} = the height of the smoke line at x = sc

ψ = Stream function