Experimental and Numerical Study of the Unsteady Wake of a Supercritical Airfoil in a Compressible Flow

1.Amirkabir University of Technology – Aerospace Engineering Department – Tehran – Iran. 2.Center of Excellence on Computational Aerospace Engineering – Amirkabir University of Technology – Tehran – Iran. *Correspondence author: Mahmoud Mani | E-mail: mani@aut.ac.ir Received: Jul. 18, 2017 | Accepted: Nov. 26, 2017 Section Editor: T John Tharakan ABSTRACT: Experimental investigations were carried out to study the wake profi le of a supercritical airfoil at Mach numbers of 0.4 and 0.6 in a pitching motion. Both static and dynamic tests were conducted in a tri-sonic wind tunnel. Flow fi eld inside the wake was measured by hot wire anemometry at downstream distances of 0.25 and 0.5 times the chord length from trailing edge. All data were taken at mean incidence angle of 3°; the amplitude of oscillation was 3° and the oscillation frequencies were 3 and 6 Hz. Moreover, numerical study was applied for the same airfoil under similar experimental test conditions; fi nally, wake profi les obtained from both numerical and experimental methods were compared.


INTRODUCTION
It is important to consider the unsteady aerodynamic behavior of airfoils to understand the problems associated with fl utter and buff et.Th e problems are more complicated at high-speed fl ows due to shock waves and associated separation.
Hot wire anemometry has been used widely in subsonic fl ow and sometimes in supersonic fl ow.However, major problems such as wire breakage due to high dynamic pressures, dust particles and vibrations limit the application of hot wires in transonic fl ows.Signal interpretation is also very complicated in such fl ows as the output voltage depends on density, velocity, and temperature, each having independently variable sensitivity.
Although much research has been conducted into the study of unsteady fl ow wake, there is steel much to do to investigate the wake profi les by hot wire anemometry in compressible fl ows.Bodapatti and Lee (1984) studied wake profi les of an airfoil along with an oscillating fl ap in a transonic fl ow.Th ey used pressure sensors besides hot wires and compared the results.Park et al. (1990) experimentally studied the wake of a pitching airfoil using hot wire at two Reynolds numbers of 27000 and 47000.Th e airfoil oscillation amplitude was 7.4°, and the tests were carried out at two reduced frequencies of 0.1 and 0.2 in average angles of attack of 0 -4°.Reduced frequency is a parameter that defi nes the degree of unsteadiness and is written as , ω is angular velocity, c is airfoil chord and U is freestream velocity.Th e tests results showed that when the mean incidence was 4°, the airfoil experienced a deep stall.Th e near wake measurements made it possible to detect and characterize the emergence of unsteady separation; as the reduced frequency increased, the phase angle at which unsteady boundary layer separation occurred was found to be raised.

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It is shown that the wake study and its behavior in compressible flow could be achieved by statistical and frequency analysis of sensors voltages when the accurate velocity value is not the issue.

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Calibration of sensor voltage is carried out by in-direct method and velocity profiles are obtained inside the wake.

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Mach contours and velocity profiles are obtained from numerical analysis and compared with experimental results.

EXPERIMENTAL SETUP
The test model was a Sc(2)0410 supercritical airfoil made of 2 cm thick VCN steel with chord and span lengths of 0.2 and 0.6 m, respectively.

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The tests were carried out in a high-velocity wind tunnel under the following conditions: a test section of 0.6 × 0.6 m, oscillation amplitude of 3°, mean angle of attack of 3°, Mach numbers of 0.4 and 0.6, and reduced frequencies of 0.0143 and 0.0286.
Walls of test chamber were porous and capable of up to 6% adjustment.The maximum free stream turbulence level (measured by hot wire sensors, 3 layers of wire screen, and 1 honeycomb) was 0.5%.Schematic of wind tunnel is shown in Fig. 1.   Figure 3 shows the rake including fourteen hot wire sensors.The sensors location detail is listed in Table 1.The sensors, 100 micron in diameter and 0.005 m in length, were made of nickel.In order to record hot wire sensors results, Dantec CTA constant temperature system was applied with overheat of 0.8 and a sampling frequency of 1200 kHz.For each sensor, this frequency was taken to be 5 kHz.CTA is Constant Temperature Anemometry.
Figure 5 shows the block diagram of data acquisition system.
To eliminate the white noise from results, a low-transition filter was used with a cut-off frequency of 300 Hz considering the PSD of curves and the Weiner method.This method is described in detail by Weiner (1949).
Figure 6 shows the energy level curve for a single hot wire sensor to determine the cut-off frequency.
The oscillation mechanism was capable of pitching the model at various amplitudes, mean incidence angles of attack and oscillation frequencies.Since x/c = 0.4 was the airfoil aerodynamic center, the pitch rotation axis was fixed at the x/c = 0.4 airfoil chord.A hydraulic vibrating system was used to create pitch oscillations that were transmitted to the airfoil through a shaft and extruded from the test chamber window (Fig. 7).The average and instantaneous angle of attack were measured by a linear 0.1° precision potentiometer.

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During the tests, Mach number measurements with maximum calculation error of 1% were obtained by means of a pitotstatic tube fixed at model's upstream.Pressures were measured using differential pressure sensors with a measurement range of up to 15 psi and maximum error of 0.15% span.05/21

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The tests error sources included the flow field (both non-uniformity and turbulence level in the wind tunnel), model installation and α-setting, pressure sensors, airfoil model structure, linear potentiometer, A/D range, hot wires, effects of walls and model blockage.According to Amiri et al. (2013), a maximum uncertainty of 4% was calculated.
Wind tunnel was the same for this research and for Amiri's but the NACA0012 airfoil was chosen in Amiri's research because its standard experimental data was available for verification and also to evaluate equipment and wind tunnel precision.

NUMERICAL METHOD
The flow structure around a supercritical airfoil was predicted employing Ansys CFX software and the results were compared with experimental tests.
Turbulence model was Shear Stress Transport (SST k-ω), which was proposed by Menter et al. (2003) to combine the precise, powerful k-ω model (for near wall regions where the Reynolds number is low) with free-stream independent k-ε model (for regions far from the walls where the Reynolds number is high).
Two dimensional simulations were applied as the case study was the main goal and to reduce computational costs.In addition, numerical analysis validation in 2D case is confirmed by some references such as Thiery and Coustols (2006) and Ol et al. (2009).
Calculation domain was selected by reviewing some references as Moreau et al. (2008) and Ang et al. (2004).Dimensions of domain are considered as 4 times of chord length in front, above, and below the leading edge, and 9 times of chord length behind the trailing edge.Moreover, it is sufficiently distanced from airfoil surface to minimize its influence on the solution.
Geometry and computational domain are shown in Fig. 8.After grid independency, fine grid was used to predict the flow field around Sc(2)0410 airfoil.Total element of grid is 187220 and total node is 376068.
Figure 9 shows the structured grid used to predict the flow field around the airfoil.

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The time step used in unsteady solutions was 0.0005 s.To achieve stable conditions, simulations were carried out for almost 4200 time steps to pass the inlet flow through the entire computational domain for approximately 7 times.The present results are obtained by statistical time averaging of the last 1800 time steps.
The method validity to reach the stable conditions could be carried out by referring to Moghadam et al. (2016) and Kiani and Javadi (2016).
Boundary conditions include time-independent uniform-velocity profile for inlets and static pressure for outlets.The airfoil surface is assumed to be non-slip and the reference pressure is considered as zero.

GOVERNING EQUATIONS
Governing equations for this problem are conservation of mass, momentum, and energy for the case of unsteady compressible flow as follows: Continuity (Eq.1): x Momentum (Eq.2): y Momentum (Eq.3): Energy (Eq. 4): where:   is the sum of kinetic and internal energy.
The shear stress (  ) and normal stress components (  and   ) can be expressed in terms of velocity gradients as follows: Shear stress (Eq.5): X-normal stress (Eq.6): Y-normal stress (Eq.7): where:  is the dynamic viscosity of the fluid.These equations are described in Detail by Drazin and Riley (2006).Several researches, as those carried out by Mojiri and Datta (2009) and Eleni et al. (2012), showed that by proper grid generation in boundary layer and Y + close to 1, SST k-ω model will give more precise results.SST model was applied here since capturing the wake was the main goal of the research in which vortex observation, flow behavior close to walls, and boundary layer analysis were to be focused on.
In SST turbulence model, K-ω and K-ε are both multiplied by a blending function and then summed.Blending function is designed so that it equals to 1 in near-wall regions (which activates K-ω model in that region) and equals to zero in regions far from the wall (which activates K-ε model in that region).This turbulence model is common for airfoil flow analysis and behaves properly in flow separation and in reverse pressure gradient.averaging of the last 1800 time steps.
The method validity to reach the stable conditions could be carried out by referring to Moghadam et al. (2016) and Kiani and Javadi (2016).
Boundary conditions include time-independent uniform-velocity profile for inlets and static pressure for outlets.The airfoil surface is assumed to be non-slip and the reference pressure is considered as zero.
computational domain for approximately 7 times.The present results are obtained by statistical time averaging of the last 1800 time steps.
The method validity to reach the stable conditions could be carried out by referring to Moghadam et al. (2016) and Kiani and Javadi (2016).
Boundary conditions include time-independent uniform-velocity profile for inlets and static pressure for outlets.The airfoil surface is assumed to be non-slip and the reference pressure is considered as zero.

Governing Equations
Governing equations for this problem are conservation of mass, momentum, and energy for the case of unsteady compressible flow as follows: where:  is the sum of kinetic and internal energy.
computational domain for approximately 7 times.The present results are obtained by statistical time averaging of the last 1800 time steps.
The method validity to reach the stable conditions could be carried out by referring to Moghadam et al. (2016) and Kiani and Javadi (2016).
Boundary conditions include time-independent uniform-velocity profile for inlets and static pressure for outlets.The airfoil surface is assumed to be non-slip and the reference pressure is considered as zero.
computational domain for approximately 7 times.The present results are obtained by statistical time averaging of the last 1800 time steps.
The method validity to reach the stable conditions could be carried out by referring to Moghadam et al. (2016) and Kiani and Javadi (2016).
Boundary conditions include time-independent uniform-velocity profile for inlets and static pressure for outlets.The airfoil surface is assumed to be non-slip and the reference pressure is considered as zero.
Several researches, as those carried out by Mojiri and Datta (2009) and Eleni et al. (2012), showed that by proper grid generation in boundary layer and Y + close to 1, SST k-ω model will give more precise results.SST model was applied here since capturing the wake was the main goal of the research in which vortex observation, flow behavior close to walls, and boundary layer analysis The shear stress () and normal stress components ( and ) can be expressed in terms of velocity gradients as follows: Shear stress (Eq.5): == (u/y +/) (5) X-normal stress (Eq.6): Y-normal stress (Eq.7):  (.)+2/ (7) where:  is the dynamic viscosity of the fluid.These equations are described in Detail by Drazin and Riley (2006).
Several researches, as those carried out by Mojiri and Datta (2009) and Eleni et al. (2012), showed that by proper grid generation in boundary layer and Y + close to 1, SST k-ω model will give more precise results.SST model was applied here since capturing the wake was the main goal of the research in which vortex observation, flow behavior close to walls, and boundary layer analysis The shear stress () and normal stress components ( and ) can be expressed in terms of velocity gradients as follows: Shear stress (Eq.5): == (u/y +/) (5) X-normal stress (Eq.6): Y-normal stress (Eq.7):  (.)+2/ (7) where:  is the dynamic viscosity of the fluid.These equations are described in Detail by Drazin and Riley (2006).
Several researches, as those carried out by Mojiri and Datta (2009)  activates K-ɛ model in that region).This turbulence model is common for airfoil flow analysis and behaves properly in flow separation and in reverse pressure gradient.
Equations applied for such turbulence model are as follows (Eq.8): where the F1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): and behaves properly in flow separation and in reverse pressure gradient.
Equations applied for such turbulence model are as follows (Eq.8): where the F1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): and behaves properly in flow separation and in reverse pressure gradient.
Equations applied for such turbulence model are as follows (Eq.8): where the F1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): and behaves properly in flow separation and in reverse pressure gradient.
Equations applied for such turbulence model are as follows (Eq.8): where the F1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): where the F1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): where: S is the shear stress value and F2 is the second blending function that is defined as follows (Eq.11): This method is described in detail by Menter et al. (1993Menter et al. ( ,1994)).
Figure 10 shows the calculated Y + in airfoil wall simulation.where the F 1 blending function is (Eq.9): where y is the distance from closest wall.
Turbulence viscosity is defined as follows (Eq.10): where: S is the shear stress value and F 2 is the second blending function that is defined as follows (Eq.11): This method is described in detail by Menter et al. (1993Menter et al. ( ,1994)).
Figure 10 shows the calculated Y + in airfoil wall simulation.High-resolution second order upwind method was used for numerical solution of partial differential equations where high accuracy was required in the presence of shocks or discontinuities.

RESULTS
All tests were carried out under static and dynamic conditions both experimentally and numerically.Velocity profiles, Mach contours, statistical and frequency analysis of hot wire outputs were used to identify the wake behavior.
The research results are categorized in two sections: statistical and frequency analysis of hot wire sensors voltages and velocity profiles (derived from calibration).
Before exploring the mentioned parts, it is preferred to show the lift coefficient diagrams in Figs.11 and 12 for flows with Mach numbers of 0.4 and 0.6, which are sketched to prove the results consistency between numerical analysis and experimental tests, and also to determine the angle of static stall.As it is shown in Figs. 5 and 6, the angle of static stall for Mach number of 0.4 is 9° and for Mach number of 0.6 is 7°.Therefore, experimental tests are carried out below stall angle for both Mach numbers of 0.4 and 0.6; a near stall state occurs only in the case of 0.6, oscillation amplitude of 3° and mean angle of attack of 3°.

STATISTICAL AND FREQUENCY ANALYSIS OF HOT WIRE SENSORS VOLTAGES
Results in this section show that the study of wake and its thickness in various cases are possible by statistical and frequency analysis of hot wires output voltages.

Static Part
Figure 13 indicates linear correlation coefficient diagram for hot wire sensors.Correlation coefficient is a value between +1 and −1 calculated so as to represent the linear interdependence of two variables or sets of data.Therefore, in this research, correlation coefficient between two adjacent sensors shows the flow behavior at sensor locations.As it is shown in Fig. 13, the coefficient is maximized for a couple of sensors located outside the wake and decreases when they are not at the same region (i.e., one sensor locates inside the wake region and the other locates outside).or angle of 4° for example, sensor S6 achieves minimum correlation with S7 and S5 since it is located inside the wake at the mentioned angle.This also occurs for sensor S5 at angle of 8°.
As mentioned before, the wake region could be estimated by depicting the Correlation coefficient parameter for adjacent sensors and studying their behavior.
Figure 14 shows the time history of hot wires signals at AOA = 4°; qualitative investigation of signals reveals that the wake sensor fluctuations are by far higher than those of other sensors; therefore, the wake location determination becomes possible through a voltage qualitative investigation.As suggested before, entry of one sensor into the wake region could be observed by means of signal analysis of the sensor voltage.Signal energy or PSD is also a significant parameter.Energy level for angles of attack of 3° and 6° are shown in Figs. 15 and 16 for three sensors located at y/c = 0, y/c = -0.035,and y/c = -0.07.As observed in Figs. 15 and 16, S6 sensor gains maximum energy level by entering the wake region at angle of 3°, whereas, at angle of 6°, by entry of S5 sensor into the region, it gains the maximum energy level and S6 sensor scores the second place.

Dynamic Part
Focus in this section is on the airfoil wake in a pitch sinusoidal motion.Figure 17 shows the instantaneous voltages of three sensors in α(t) = 3 + 3 sin (2πt/T -π/2) pitch oscillation motion at frequency of 3 Hz.Comparing consecutive cycles reveals their repeatability with an appropriate precision; if a complete cycle is considered as a pitch up-pitch down sequence, then every sensor will cover the same pattern through all cycles.Deviations may appear only in very small fluctuations, which are likely related to noise or any other unsteadiness that may instantaneously and irregularly affect the sensors voltages, but not to the flow physics.
Figure 18 shows the RMS variations in a complete cycle.Lines L1, L2, and L3 in a pitch up, are related to states when the wake enters y/c = 0.0, minimum voltage level, and when the wake leaves that point, respectively.Lines L4, L5, and L6 show the same states in a pitch down.Related times are printed against each line.RMS value for each signal increases with an increase in randomness and in amplitude of oscillations around the average value; hence, RMS could be considered equivalent to turbulence intensity in the flow.Variations in Fig. 18 confirm this equivalency; when the sensor enters the wake at L1, RMS value starts increasing and reaches its maximum at point I, then it starts to decrease, getting an extremum at L2, where the voltage is minimum.Afterwards, it starts to increase again, goes up to point II, and finally falls down by getting close to the boundary layer border.Similar behavior is observed in a pitch down.
Figure 19 shows the coefficient of linear correlation between two sensors at y/c = 0.0 (inside the wake) and at y/c = 0.245 (outside the wake).The coefficient of correlation between S7 and the flow outside the wake does not decrease immediately after the sensor locates inside the wake; rather, it decreases remarkably when it gets close to the minimum voltage zone and finally reaches a value of about -0.9 at L2.
Analysis of results from Figs. 13 to 19 confirms that the study of wake is possible by means of sensor voltage analysis even for oscillating airfoil.

VELOCITY PROFILES
Voltage calibration for hot wire sensors was carried out in this research and wake velocity profiles were obtained from hot wire anemometry.

Calibration
According to Motallebi (1994), in high speed flow, due to the compressibility effects, the anemometer is affected not only by changes in flow velocity, but also by density and temperature fluctuations.In other words, for a single normal wire, the output voltage of anemometer is a function of flow velocity, flow density and total temperature (Eq.12).( 12) This equation can be written as (Eq.13):

Velocity Profiles
Voltage calibration for hot wire sensors was carried out in this research and wake velocity profiles were obtained from hot wire anemometry.

Calibration
According to Motallebi (1994), in high speed flow, due to the compressibility effects, the anemometer is affected not only by changes in flow velocity, but also by density and temperature fluctuations.In other words, for a single normal wire, the output voltage of anemometer is a function of flow velocity, flow density and total temperature (Eq.12).

𝐸𝐸 = 𝐸𝐸(𝑈𝑈. 𝜌𝜌. 𝑇𝑇) (12)
This equation can be written as (Eq.13): ó % −  y % ò =  ô Log(U) + S ü Log(ρ) +  °Log(T) +  (13) If ρ and T are constants (Eq.14): If U and T are constants (Eq.15): If ρ and U are constants ( 16): If U and T are constants (Eq.15): If ρ and U are constants ( 16): Therefore, several tests shall be necessary to calibrate sensor's output and its conversion into velocity.Those tests require to obtain Su by velocity variation while temperature and density are constants.Also to obtain Sρ by density variation while temperature and velocity are constants and to obtain ST by temperature variation while velocity and density are constants.

Generally, experimental calculation of calibration coefficients could be achieved by two methods:
-Direct experimental method ó % −  y % ò =  ô Log(U) + S ü Log(ρ) +  °Log(T) +  (13) If ρ and T are constants (Eq.14): If U and T are constants (Eq.15): If ρ and U are constants ( 16): Therefore, several tests shall be necessary to calibrate sensor's output and its conversion into velocity.Those tests require to obtain Su by velocity variation while temperature and density are constants.Also to obtain Sρ by density variation while temperature and velocity are constants and to obtain ST by temperature variation while velocity and density are constants.

Generally, experimental calculation of calibration coefficients could be achieved by two methods:
-Direct experimental method ó % −  y % ò =  ô Log(U) + S ü Log(ρ) +  °Log(T) +  (13) If ρ and T are constants (Eq.14): If U and T are constants (Eq.15): If ρ and U are constants ( 16): Therefore, several tests shall be necessary to calibrate sensor's output and its conversion into velocity.Those tests require to obtain Su by velocity variation while temperature and density are constants.Also to obtain Sρ by density variation while temperature and velocity are constants and to obtain ST by temperature variation while velocity and density are constants.

Generally, experimental calculation of calibration coefficients could be achieved by two methods:
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Direct experimental method
If ρ and T are constants (Eq.14): If U and T are constants (Eq.15): If ρ and U are constants ( 16): Therefore, several tests shall be necessary to calibrate sensor's output and its conversion into velocity.Those tests require to obtain S u by velocity variation while temperature and density are constants.Also to obtain S ρ by density variation while temperature and velocity are constants and to obtain S T by temperature variation while velocity and density are constants.
Generally, experimental calculation of calibration coefficients could be achieved by two methods: • Direct experimental method

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Indirect or semi-experimental method In direct method, hot wire is placed in free stream of tunnel and the stagnation pressure is varied while the Mach number and stagnation temperature are kept unchanged.By time averaging, it is then possible to obtain the mean output voltage of the anemometer as a function of mean mass flux.
The slope of a plot of ln (E) versus ln (ru) will determine De ru .The De To sensitivity coefficient can be determined by a controlled change of gas total temperature while gas density and velocity are kept unchanged.This method is not practiced in conventional wind tunnels, since it requires several tests in special tunnels.
When the possibility of calibrating the hot wire in a controlled flow environment is not feasible in a given test facility, then it is advisable to calibrate the hot wire in wind tunnel by simultaneous variation of density, velocity and temperature.
An indirect calibration method is employed in this research, which was first introduced by Stainback and Nagabushana (1993) and Stainback et al. (1983).Applying the mentioned method they calculated, at first, the calibration coefficients by several tests in wind tunnel, and then estimated a statistically optimized equation (Eq.17): log(E) =  `+  % log (u) +  ± log (r) +  t log () +  â log (u) log (r) +  ¥ log (u) log () +  µ log (r) log () +  ∂ log (u) log (r) log () Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.
Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A1 to A8. Equations 18 to 20, for log(E) =  `+  % log (u) +  ± log (r) +  t log () +  â log (u) log (r) +  ¥ log (u) log () Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for data sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.
Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A1 to A8. Equations 18 to 20, for Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for data sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.
Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A1 to A8. Equations 18 to 20, for (17) Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A1 to A8. Equations 18 to 20, for log(E) =  `+  % log (u) +  ± log (r) +  t log () +  â log (u) log (r) +  ¥ log (u) log () +  µ log (r) log () +  ∂ log (u) log (r) log () Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): Equations for sensitivity coefficients are as follows (Eqs. 18,19,and 20): In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for data sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.
Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A 1 to A 8 .Equations 18 to 20, for sensitivity coefficients, were solved to give S u , S ρ , and S To , consequently.Applying these coefficients in tests helped to benefit from converting the voltage into velocity.

Static Part
Figures 20 to 22 show the velocity diagrams in wake region for Mach number of 0.4 at downstream distance of 0.25 chord.A significant velocity decrease in wake region is obvious in diagrams.Location of maximum velocity reduction shifts downward as AOA increases.S7 sensor indicates maximum voltage reduction at angle of 0°, which suggests minimum velocity in turn.Mentioned sensor leaves the wake region as the angle of attack varies from 0° to 3° and S6 sensor indicates the minimum velocity instead.That means the wake tends downward.
S5 sensor also enters the wake region by increasing the angle of attack to 6°, which indicates a large velocity reduction.The wake thickness increases at this angle of attack and both S5 and S6 sensors are placed inside the wake region.
Figures 23 to 25 show the velocity diagrams in wake region for Mach number of 0.6 at downstream distance of 0.25 chord.Velocity reduction is observed in the wake.Like in the previous case, the wake rotates clockwise by increasing the angle of attack.Wake thickness and velocity reduction at Mach number of 0.6 is more than those of flow with Mach number of 0.4.This is obvious by comparing Figs.22 and 25.
Velocity behavior at x/c = 0.5 is almost similar to that of x/c = 0.25, as shown in Fig. 26; deviations are the rate and slope of voltage reduction which is higher at x/c = 0.25, and the wake thickness which is greater at x/c = 0.5.This is a result of wake effect weakening as departing longitudinally from airfoil.As observed from velocity profiles, results obtained from both experimental and numerical analysis are in a good consistence.

Dynamic Part
Figures 27 to 30 show the wake velocity profiles in α(t) = 3 + 3 sin(wt) pitch oscillation.In order to achieve velocity profiles for oscillating airfoil, output signal for each sensor was recorded at each instantaneous angle of attack and averaged for 15 complete oscillating cycles.
At angle of attack of 3°, S6 and S7 sensors are both placed in wake region, while at angle of attack of 6°, S5 sensor indicates velocity reduction from free stream velocity, which implies a notable increase of wake thickness.
As observed from velocity profiles, the wake rotates clockwise and tends downward by increasing the angle of attack.At angle of attack of 3°, S6 sensor shows a velocity decrease, while at 6° the wake tends down and S5 sensor shows a velocity decrease instead.Wake thickness also increases due to flow separation at trailing edge and rise of a shock on the airfoil.Figure 31 shows the pressure coefficient diagram on airfoil surface for angles of 3.8° and 4° in a pitch up motion.The diagram indicates a shock for angles beyond 4°.
A shock occurs on the airfoil for angles above 4°.This is also noticeable by studying the Mach number contours for oscillating airfoil.
As indicated in the diagrams, the vortex created at leading edge moves towards trailing edge by increasing the angle of attack and finally sheds from trailing edge into the wake.
Maximum oscillation of 6° stays far from stall since static stall angle for a flow with Mach number of 0.4 is 9°, as described in lift coefficient diagrams.Although at Mach number of 0.4, separation occurs at trailing edge by increase in oscillation angle, but the airfoil would not experience a shallow stall.
In a flow with Mach number of 0.6, static stall angle is 7°.Oscillating shock occurs on the airfoil in this case, which shows a shallow stall.As indicated in the diagrams, the vortex created at leading edge moves towards trailing edge by increasing the angle of attack and finally sheds from trailing edge into the wake.
Maximum oscillation of 6° stays far from stall since static stall angle for a flow with Mach number of 0.4 is 9°, as described in lift coefficient diagrams.Although at Mach number of 0.4, separation occurs at trailing edge by increase in oscillation angle, but the airfoil would not experience a shallow stall.
In a flow with Mach number of 0.6, static stall angle is 7°.Oscillating shock occurs on the airfoil in this case, which shows a shallow stall.Finally, by moving downward and decreasing the angle of attack, the separated flow reattaches to the airfoil surface for both Mach numbers of 0.4 and 0.6.

CONCLUSION
Wake of a supercritical airfoil in a pitching motion was investigated experimentally and numerically in case of compressible flow.Static stall angle was obtained for Mach numbers of 0.4 and 0.6 by lift coefficient diagram in both experimental and numerical cases.Moreover, results consistency for both cases was indicated by lift coefficient diagram.
By depicting sensors output diagrams like RMS, time history, correlation coefficient and energy level, it was shown that the wake investigation and its behavior in compressible flow could be achieved by frequency and statistical analysis of sensors voltages if the precise velocity value does not matter.
Calibration was carried out despite the troubles in compressible flow and velocity curves were obtained for both fixed and oscillating airfoils.

Figure 2
Figure2shows the model mounted in test chamber.

Figure 2 .
Figure 2. Model in test chamber.

Figure 4
Figure4shows the airfoil and the rake inside the test chamber.

Figure 4 .
Figure 4. Airfoil, rake and sensors fixed in the test chamber.

Figure 5 .
Figure 5. Block diagram of data acquisition system.
. Technol.Manag., São José dos Campos, v11, e0319, 2019 Experimental and Numerical Study of the Unsteady Wake of a Supercritical Airfoil in a Compressible Flow

Figure 10 .
Figure 10.Y + calculated on the airfoil wall.

Figure 10 .
Figure 10.Y + calculated on the airfoil wall.

Figure 13 .
Figure 13.Linear correlation coefficient diagram for hot wire sensors.

Figure 14 .
Figure 14.Time history of hot wires output signals at AOA = 4°.

Figure 15 .
Figure 15.Sensors energy level at angle of attack of 3°.

Figure 16 .
Figure 16.Sensors energy level at angle of attack of 6°.

Figure 18 .
Figure 18.RMS variations in a complete cycle.

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tests shall be necessary to calibrate sensor's output and its conversion into velocity.Those tests require to obtain Su by velocity variation while temperature and density are constants.Also to obtain Sρ by density variation while temperature and velocity are constants and to obtain ST by temperature variation while velocity and density are constants.Generally, experimental calculation of calibration coefficients could be achieved by two methods: Direct experimental method ó % −  y % ò =  ô Log(U) + S ü Log(ρ) +  °Log(T) +  (13) If ρ and T are constants (Eq.14): logr) +  ¥ log () +  ∂ log (r) ln () (18) r =  ± +  â log (u) +  µ log () +  ∂ log (u) log ()) (19)  °∑ =  t +  ¥ log (u) +  µ log (r) +  ∂ log (u) log (r) (20)In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for data sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.

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In this research, output voltages for each sensor were obtained for eight values of Mach number during several tests in wind tunnel.Fourteen sensors mounted on a rake were placed in wind tunnel for data sampling in various Mach numbers.Mach number varied from 0.35 to 0.75 and sensors outputs were recorded by sampling frequency of 5 kHz.Obtaining output voltages, a system of equations comprising eight equations and eight unknowns was configured and solved to give the coefficients A1 to A8. Equations 18 to 20, for log(E) =  `+  % log (u) +  ± log (r) +  +  µ log (r) log () +  ∂(17)    Equations for sensitivity coefficien ô =  % +  â (log (18)  r =  ± +  â log (u) (19)  °∑ =  t +  ¥ log(20) In this research, output voltages fo number during several tests in wind tunnel wind tunnel for data sampling in various M and sensors outputs were recorded by sampl Obtaining output voltages, a syste unknowns was configured and solved to g

Table 1 .
Hot wire sensors location.