Transonic airfoil aerodynamic characterization by means of PIV

Transonic airfoil aerodynamic characterization by means of PIV

A. Ashok, D. Ragni, B.W. van Oudheusden, F. Scarano

Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands

Abstract The present investigation assesses the method to extract aerodynamic coefficients and surface pressure from a 2D airfoil in the transonic compressible regime (Ma = 0.6 – 0.8). Particle image velocimetry (PIV) experiments are performed and momentum integrals are evaluated to obtain lift and drag coefficients, which are in turn compared to those measured by a rake of Pitot-probes in the wake of the airfoil and by surface pressure orifices. The method applied in this study is an extension of that proposed by Oudheusden et al. (2006) to the compressible flow regime and takes into account the compressibility effects in absence of shock waves. The measurement is performed by a high-repetition rate PIV system to rapidly obtain a large recordings ensemble for the statistical evaluation of flow properties. Several magnification levels are used in order to assess the effect of the spatial resolution on the estimates of surface pressure and aerodynamic coefficients. The study includes a validation of the PIV based method by comparison with the conventional methods over a range of incidence and Mach number.

1. Introduction

Aerodynamic loads are conventionally determined by use of force balances or point wise pressure measurements for surface pressure distribution and Pitot-probe rakes for wake analysis. These measurement techniques are fast, accurate and reliable, however they are intrusive, provide information only at discrete points and may not be easily implemented in complex flow configurations such as for unsteady problems. In recent years non-intrusive measurement techniques have allowed for the determination of fluid dynamic quantities at spatial resolutions that were previously unavailable. For instance, pressure sensitive Paint (PSP) has been demonstrated to be feasible in the determination of the surface pressure in the high-subsonic flow regime (Klein et al. 2005). It can provide a surface measurement of a three-dimensional configuration a good spatial resolution. However, due to the limitations in the physical principle of the method, accurate measurements are nowadays possible only in high speed flows where the pressure variations along the model are fairly above 1000 Pa.

Velocity field information, as provided by PIV, has demonstrated good potential for the non-intrusive determination of the aerodynamic forces on airfoils (Sjors and Samuelsson 2005; De Gregorio 2006; Oudheusden et al. 2006). Such approaches involve the evaluation of integral momentum concepts to compute the aerodynamic forces. It should be retained in mind that this approach involves an intermediate step, which is the evaluation of the flow field pressure from the velocimetry data based on the flow governing equations (Unal et al. 1998). In case of incompressible flow, the pressure can be related directly to velocity through Bernoulli’s equation if the flow is irrotational. For rotational flow regions, however, the pressure gradient can be computed from the momentum equation in differential form and the pressure field is obtained by subsequent integration, for instance by means of a space-marching approach (Baur and Köngeter 1999). This procedure has been demonstrated valid for the determination of airfoil aerodynamic coefficients in the incompressible regime (Oudheusden et al. 2006). The specific implementation in the supersonic compressible flow regime at Mach 2 has been recently addressed by Souverein et al. (2007), which introduced as additional problem the treatment of shock waves in the flow field.

applications. Additionally to the measurement of integral forces, the evaluation of the surface pressure distribution is made to assess the adequacy of PIV measurements for this purpose.

2. Theoretical background

2.1 Integral force determination

The pressure and shear stress distributions on the body surface are responsible for the force exerted by the fluid flow. Applying the momentum conservation concept, however, integral forces may be computed without having to resolve the flow quantities close the model; a schematic of the approach is shown in Figure 1-left. The aerodynamic forces are computed in the contour integral approach (Anderson 1991) as:

∫∫

∫∫

⋅

+

−

+

⋅

−

=

S S

dS

n

pn

VdS

n

V

t

F

(

)

ρ

(

)

(

τ

)

₍₁₎

Here, S is the contour with n the outside pointing normal on it. All the flow quantities must be evaluated on the contour; which allows the viscous stresses to be neglected here, as they usually do not play any significant role on a contour at some distance from the surface.. Assuming 2D flow conditions, applying Reynolds decomposition and re-writing in scalar notation Equation (1) yields:

( )

( )

( )

( )

( )

uv dy

( )

vv dx

( )

u v dy

( )

vv dx pdx L d dy p dx v u dy u u dx v u dy u u D d − − + − = + − + − = ' ' ' ' ' ' ' ' ' '

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

(2)

PIV returns velocity data and therefore the first two sets of terms in Equation 2 may be directly evaluated. The pressure and density, however, need to be calculated (see section 2.2).

Figure 1: Schematic of the control volume approach to loads determination (left). Momentum integral approach to wake analysis (right)

The contour integral is to be evaluated around the whole body, which introduces uncertainties on the integral value. This is recognized as a likely problem for the drag coefficient in particular (Oudheusden et al. 2006). Such effect can be largely alleviated by following a wake analysis approach. In the present study the method proposed by Jones (1936) is followed, additionally taking into account compressibility effects. The method is explained in Figure 1-right. Index I denotes an x-station far behind the model where the static pressure has recovered to p∞. Direct application of

the control volume method could then be limited to consider only the momentum deficit at station I.

However, in experiments the pressure has not recovered to p∞ in the measurement plane at station

II. Invoking the assumptions of mass conservation and constant total pressure along streamlines and further neglecting turbulent stresses enables to relate the quantities at station I (subscript 1) to the quantities at station II (subscript 2). This results in the following equation for the compressible drag coefficient:                                               −                   − − ⋅                   −                   − ⋅         ⋅       = − ∞ − ∞ − ∞ − − ∞ ∞ ∞ ∞

∫

d y_c p p p p p p p p p p p p c t t t t t t d 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1 1 1 2 2 2 2 γ γ γ γ γ γ γ γ γ γ γ (3)

Note that this formulation only requires that the static and total pressure in the measured wake profile given by PIV measurement, as well as their values in the free stream. As a main consequence the drag coefficient determination only requires the flow measurement in the wake allowing to apply a high-resolution imaging configuration. Conversely, the determination of the lift coefficient still requires the evaluation of the full contour integral, which needs a field of view encompassing the entire model at necessarily lower resolution.

2.2 Pressure determination

Different strategies apply to the determination of the pressure from velocity depending on the particular flow condition. Crocco’s theorem (Anderson 2003) establishes that for steady, adiabatic, inviscid flow, the entropy is constant in irrotational regions. Therefore the isentropic relations are valid in regions where the vorticity is negligible and the pressure exhibits a one-to-one relation with the local velocity:

1 2 2 2 1 2 1 1 − ∞ ∞ ∞              − − + = γ γ

γ

V V M p p (4)

In wakes, across shocks and for more general flow conditions one needs to rely on the governing flow equations, which returns an implicit formulation for the pressure gradient evaluation, in turn requiring an iterative approach for its numerical solution. Keeping the assumption of adiabatic flow, and further neglecting the effects of the viscous terms an explicit solution is available (Equation 5) for the pressure gradient (Oudheusden and Souverein, 2007):

(

V V

)

(

V

)

V M V M p p p p ∇ ⋅ ⋅ − − + = −∇ = ∇ − ∞ ∞ ∞ ∞ ∞ 2 2 2 2 2 2 1 ) / ln( γ γ (5)

by advancing from the boundary condition regions throughout the flow field (Oudheusden and Souverein, 2007). In the present study the integration is typically restricted to the wake region.

3. Experimental setup

The experimental investigation was performed in the Transonic-Supersonic wind Tunnel (TST) of the Aerodynamics Laboratories at the Delft University of Technology. The facility is a blow-down type wind tunnel that can achieve Mach numbers in the range 0.5 to 4.2 in a test section of dimension 280 mm (width) × 250 mm (height). The wind-tunnel was operated under subsonic conditions at a stagnation pressure of 1.9 bars with the choke section set to obtain free-stream Mach numbers in the range 0.6 to 0.8. The model used was a NACA 0012–30 airfoil, with a chord of 100 mm. The angle of attack was varied in the range 0 to 8º. The Reynolds number based on the chord was 2.2 × 106 at Mach 0.6. The model was equipped with 36 pressure orifices to measure the surface pressure distribution in order to be able to compare these with the pressure distributions and lift coefficients computed using PIV. In addition, a wake rake was used to determine the drag coefficient such that comparison with PIV measurements was possible.

Figure 2 – Schematic of the PIV set-up

(a) Three fields of view depicted schematically (b) The flow illuminated by the seeding as seen from outside the wind tunnel during set-up (no flow)

Figure 3 – Fields of view as imaged by the camera

PIV was used to obtain a velocity field around the model. Figure 2 shows a schematic of the PIV-set-up, while Figure 3 shows the three fields of view used in the investigation (details in Table 1)

and a photo of the test section. The pink tubes are the pressure connections to the transducer that is placed outside the wind tunnel. Tracer particles with a mean diameter of 1 µm are distributed by a seeding rake placed in the settling chamber. The PivTec PIVpart 45 seeding generator is equipped with a Laskin nozzle creating small particles of Di(2-ethylhexyl) sebacate (DEHS). The system was operated at 2 bars of overpressure with all nozzles on. The flow is illuminated by a Quantronix Darwin Duo Nd-YLF laser rated at 20 mJ per pulse emitting light at a wavelength of 527 nm. The laser pulse duration was 200 ns and the repetition rate was set at 0.5 kHz in double exposure i.e. pairs of images are acquired every 1/500 of a second. The light sheet was introduced into the tunnel through a prism located just below the lower wall of the wind tunnel. Laser sheet nominal thickness is 2 mm in the test section. For the large field of view, the time separation between pulses was 6 µs producing a particle displacement of 1 mm (16 pixels) in the free stream and about 0.5 mm (8 pixels) in the near wake at a free stream Mach number of about 0.6. At a free stream Mach number of about 0.8 a particle displacement of about 1.3 mm (21 pixels) in the free stream and about 0.6 mm (10 pixels) in the wake is observed. The flow was imaged using a Photron FastCAM SA1 (12-bit) CMOS with 1024 × 1024 pixels. A Nikon lens with a focal length of 105 mm was used at f# =

2.8. At this f number and with a pixel size of 20 µm, peak locking may be expected. To mitigate peak locking defocusing of the particles was carried out. Table 1 below summarizes some of the relevant parameters of the PIV arrangement. The images were cross-correlated using the WIDIM (Scarano & Riethmuller) software, see Table 2 for processing settings.

Flow geometry Nearly two-dimensional flow aligned

with the light sheet

Recording medium

Photron FastCAM CMOS (1024 ×1024 pixel)

Average in-plane

velocity 250 m/s (M = 0.6) 330 m/s (M = 0.8) Recording rate 500 Hz Field of View (×3)

150 ×150 mm2 (Large field of view) 50 ×50 mm2 (Leading edge zoom)

30 ×30 mm2 (Wake zoom)

Camera Lens

Nikon lens f# = 2.8

f = 60 mm (Large field of view) f = 105 mm (Wake zoom)

Magnification (×3)

0.14 (Large field of view) 0.41 (Leading edge zoom)

0.68 (Wake zoom)

Illumination Quantronix Darwin Duo

80 W Pulsed Nd-YLF laser

Recording method Double frame/ Single exposure Pulse delay (×3)

∆t = 6µs (Large field of view) 4µs (Leading edge zoom)

3µs (Wake zoom)

Table 1: PIV parameters used in the present investigation

LFOV LE zoom Wake zoom Type of processing Standard Ensemble correlation Standard

Ensemble size 500 500 500

Initial interrogation window size [pixel] 127×127 64×64 127×127

Overlap [%] 75 50 75

Final interrogation window [pixel] 31×31 15 (x)×5(y) 31×31

Grid spacing [mm] 1.5 mm 0.2 mm 0.25 mm

The three fields of view used include a large field of view (LFOV) which captures the velocity field as a whole. The LFOV is used to place the integration contour around the model in order to compute the integral forces. The leading edge zoom (LE zoom) is used to compute the pressure coefficient over the surface of the airfoil. Finally the wake zoom is used to compute the drag coefficient using Equation 3, bypassing the need for an entire contour and with higher spatial resolution.

For an integral momentum approach it is necessary to have a single contour encompassing the model. Due to restrictions in optical access, shadow regions were present in the imaged flow field where the particles were not illuminated (compare the picture in Figure 3(b)). It was therefore necessary to exploit the symmetry of the flow around the NACA 0012 airfoil to obtain the entire velocity field without any gaps, by superimposing two corresponding pairs of angles of attack. The suction and pressure side obtained from the two angles of attack are shown in Figure 4. These are then superimposed to form the complete velocity field.

(a) Velocity field measured at 4º (b) Velocity field measured at -4º

Figure 4 – Separate velocity fields obtained from two separate runs

4. Results

This section presents the final results of the PIV experiments, employing the three different fields of view where appropriate. A representative sample has been chosen from the measurement matrix. The first section concentrates on the extraction of the surface pressure from velocity measurements whilst the second is devoted to the computation of integral forces from velocimetry data.

4.1: Surface pressure determination

shock wave. Under these circumstances it is expected that the isentropic assumption will fail for the flow region behind the shock.

(a) Velocity field: M = 0.6, α = 2° (b) Cp contour: M = 0.6, α = 2°

(c) Velocity field: M = 0.6, α = 6° (d) Cp contour: M = 0.6, α = 6°

Figure 5: - Velocity fields and Pressure coefficient contours LE zoom

The purpose of this investigation is to confirm if the trend in the pressure coefficient close to the surface is captured by PIV. This is also important to justify the choice for the isentropic flow model. To this effect, the pressure profiles of Figure 6 have been extracted on lines normal to the surface of the airfoil (blue symbols). For reference, surface measurements from the pressure transducer have been added (red). The results seem quite promising as the projected value of the pressure coefficient at the surface is in excellent agreement with the value measured by the corresponding pressure orifice, in the absence of shocks. In particular, if linear extrapolation of the pressure coefficient is acceptable, the PIV-based approach is very competitive with the pressure orifices.

of attack are given in Figure 7. Although theoretically the isentropic pressure should be correct up to very close to the surface provided no separation takes place and in the absence of shocks (at these high Reynolds numbers the boundary layer thickness is on the order of 0.1 mm at 10% of the chord), the PIV measurements close to the surface may be unreliable due to reflections and edge effects. Therefore the PIV-based surface pressure distributions given in Figure 7 have been taken at a relatively large distance from the surface (~ 1 mm). This distance introduces a significant deviation from the reference surface pressure, especially in the leading edge region because of the large pressure gradient normal to the surface, as can be seen from Figure 6. For selected points the pressure coefficient has been linearly extrapolated down to the surface making use of the trend visible in Figure 6. These values are much closer to the measurements from the pressure orifices, showing the potential of such a procedure to correct the PIV-based pressure distribution for the lack of resolution and edge effects in this region.

Finally, the pressure distribution for alpha = 6˚ illustrates the impact of non-isentropic flow conditions. Whereas the pressures upstream of the terminating shock (which occurs at approximately 20% chord) is well captured by the PIV approach provided that an extrapolation to the surface is used in the leading edge region, the pressure immediately downstream of it is affected by the total pressure losses which are not accounted for by the present isentropic theory. Further work will be devoted to investigate how the non-isentropic effects can be taken into account, notably by using a pressure integration scheme similar to what is applied in the airfoil wake (Oudheusden and Souverein 2007).

(i) x/c = 0.05 (ii) x/c = 0.10 (iii) x/c = 0.30 (iv) x/c = 0.40 (a) M = 0.6 α = 2º

Figure 6: Pressure coefficient extracted along line tangential to the local surface gradient, in blue PIV data, in red pressure orifices ones

4.2: Integral force determination

The integral forces can be computed when a field of view encompassing the airfoil is available. The large FOV as used in the present experiment allows for the placement of such a contour from which the force coefficients may be computed. Figure 8 presents velocity and pressure coefficient contours of the large field of view. The velocity field shows the classic characteristics expected in the flow over an airfoil. A suction peak occurs on the top surface as the air accelerates over the airfoil. This is visible in the pressure coefficient where the lowest values of the Cp are found on the top surface, close to the leading edge. At 6º angle of attack the expansion over the top surface is strong resulting in locally supersonic flow. As expected in a transonic flow, there is a curved terminating shock on the bottom right edge of the supersonic bubble. The flow becomes rotational and non-isentropic after the shock. The pressure coefficient partially recovers in the wake however the static pressure is still well above the free stream static pressure. This justifies the necessity to include the pressure terms in the computation of the force coefficients. It is clear that the spatial resolution of the obtained velocity field is not good enough to capture the defect in the wake. This implies that drag computation from the contour approach is likely to be compromised and which further explains why the wake approach has been chosen for drag determination, as further discussed below. The black dashed lines in Figure 8(a) represent artifacts from the superimposition of the partial images (compare Figure 4). The mismatch is due to slight differences in the flow conditions during the separate runs performed to obtain the complete velocity field. The red lines represent the borders of

the integration region where Equation 5 is used to compute the pressure by integrating the (a) M = 0.6, α = 2°

(b) M = 0.6, α = 4°

(c) M = 0.6, α = 6°

underlying pressure gradient field. Outside of these red lines the flow is assumed to be isentropic. In Figure 8(b) the black dashed lines represent an integration contour with the solid circle representing the start/end point of the clockwise contour integration routine. Equation 2 was evaluated along this contour in order to compute the force coefficients.

(a) Velocity field: M = 0.6, α = 2° (b) Cp contour: M = 0.6, α = 2°

(c) Velocity field: M = 0.6, α = 6° (d) Cp contour: M = 0.6, α = 6°

Figure 8 - Velocity fields and Pressure coefficient contours - LFOV

Figure 9: Lift and drag coefficient versus angle of attack

0012 airfoil by the ONERA in their S3 facility. The lift coefficient computed by PIV-based method shows good agreement with both the wake rake and the AGARD data for the lower angles of attack. At higher angles the agreement between the wake rake the PIV-based method is not as good. The discrepancy is attributed to the inaccuracy of the pressure determination especially in the presence of shocks and in large wakes. From a preliminary error analysis it was clear that errors in the velocity field were amplified in the computation of other quantities such as velocity and pressure derivatives. The discrepancies between the present experiments and the AGARD data at higher angles are attributed to the fact that the data have not been corrected for blockage. The blockage ratio, defined as the ratio between the projected thickness of the airfoil and the height of the wind tunnel test section, is 0.05 for the airfoil in the TST-27 wind tunnel whereas it was 0.003 at S3.

(a) Velocity field: M = 0.6, α = 2° (b) Cp contour: M = 0.6, α = 2°

(c) Velocity field: M = 0.6, α = 6° (d) Cp contour: M = 0.6, α = 6°

Figure 10: Velocity fields and Pressure coefficient contours - Wake zoom

where the extent of the wake is not clearly defined or not captured by the field of view, the uncertainties in the drag coefficient computation are much larger. For angles of attack up to and including 4°, isentropic flow conditions are assumed at a certain distance above and below the trailing edge of the airfoil. Then in the wake region Equation 5 is used to integrate the pressure gradient field in order to obtain the pressure. The two integration fronts, starting from the opposite sides of the wake meet in the wake center line which introduces a small pressure discontinuity there. At larger angles of attack the isentropic flow assumption can only be made below the trailing edge and only an upward integration can be carried out. The drag coefficient computed using the PIV wake-zoom, presented in Figure 9(right) shows excellent agreement with the wake rake at smaller angles and also with literature. Even at the larger angles of attack there is good agreement between the wake rake and PIV in the wake. As observed previously, the contour approach does not yield good results for the drag, as small errors in the flow quantities over the contour result in very large errors on the drag coefficient.

Investigating the different contributions to the lift and drag computation, it is clear that the main contributions come from the pressure and mean momentum terms on the top and bottom ‘legs’ of the rectangular contour. The turbulent stresses do not contribute significantly since they are a few orders of magnitude smaller than the other terms. It is interesting to note that the pressure term dominates the other terms. It is clear that the static pressure has not recovered to free stream conditions and a momentum deficit approach alone would have therefore resulted in a severe underestimation of the drag coefficient.

5. Conclusions

PIV experiments have been conducted in a transonic-supersonic wind tunnel on an airfoil model in the transonic flow regime. Three different fields of view have been used to compute the surface pressure coefficient, the lift and drag coefficient. The surface pressure shows good agreement with the pressure orifices for attached inviscid flow. In the presence of shocks there is disagreement however there is perspective in the case of weak, normal or oblique shocks to accurately compute the surface pressure coefficient. Lift and drag coefficients can be reliably obtained from PIV, though there is some discrepancy between the PIV-based approach and the conventional approach. They are currently both estimated to have 10% error with respect to the conventional loads determination apparatus. The pressure term is dominant for both force coefficients, which is to be expected since the measurement plane is within one chord length of the trailing edge. The alternative wake-based formulation for the drag coefficient is crucial for obtaining precise results.

6. References

Anderson JD (1991) Fundamentals of Aerodynamics, 2nd Edition, McGraw Hill

Anderson JD (2003) Modern compressible flow with historical perspective, 3rd Edition, McGraw Hill

Baur T, Köngeter J (1999) PIV with high temporal resolution for the determination of local pressure reductions from coherent turbulent phenomena. 3rd Int. Workshop on PIV, Santa Barbara, 671-676

De Gregorio F (2006) Aerodynamic performance degradation induced by ice accretion. PIV technique assessment in icing wind tunnel. 13th Int. Symp. Apl. Laser Techn. to Fluid Mech., Lisbon, Portugal

Jones BM (1936) Measurement of profile drag by the pitot-traverse method. ARC R&M 1688

Klein C, Engler RH, Henne U, Sachs WE (2005) Application of pressure-sensitive paint for determination of the pressure field and calculation of the forces and moments of models in a wind tunne.l Exp. in Fluids 39, pp. 475-483

Oudheusden BW van, Scarano F, Casimiri EWF (2006) Non-intrusive load characterization of an airfoil using PIV. Exp. in Fluids 40, pp. 988-992

Oudheusden BW van, Souverein LJ (2007) Evaluation of the pressure field from PIV in a shock wave boundary layer interaction. 7th Int. Symp. on Particle Image Velocimetry, Rome, Italy

Sjors K, Samuelsson I (2005) Determination of the total pressure in the wake of an airfoil from PIV data. PIVNET II International Workshop on the Application of PIV in Compressible Flows, Delft, The Netherlands

Souverein LJ, Oudheusden BW van, Scarano F (2007) Particle Image Velocimetry based loads determination in supersonic flows. AIAA-2007-0050, 45th AIAA Aerosp. Science Meeting & Exhibit, Reno, USA