10TH INTERNATIONAL SYMPOSIUM ON PARTICLE IMAGE VELOCIMETRY - PIV13 Delft, The Netherlands, July 1-3, 2013
Loads and pressure evaluation of the flow around a flapping wing
Thibaut Tronchin1, Laurent David2and Alain Farcy11ISAE-ENSMA, Poitiers, France thibaut.tronchin@ensma.fr 2Universit ´e de Poitiers, France
ABSTRACT
An extension of nonintrusive approach based on an integral form of the momentum equation is proposed to evaluate instantaneous loads of a NACA0012 airfoil in flapping motion at Re = 1000 (The Reynolds number is based on the chord length and the velocity at 2/3 of the wingspan) in the case of the three-dimensional PIV measurements. Two major difficulties of this approach in 3D are the quality of the time-resolved volumic velocity fields and the estimation of the pressure field on the surface of the control volume. This last step needs a good spatial and temporal resolution of the velocity fields. Experimental data are issued from a scanning tomography PIV technique for a flapping wing in revolving motion. The volumes of particles are reconstructed using the 100 images of the acquisition, and the three-dimensional velocity fields around the airfoil are evaluated with an adaptive 3D cross-correlation. Phase-averaged velocity fields are estimated from the 25 velocity fields and are used by an iterative algorithm based on Navier-Stokes equations to deduce the pressure on the surfaces of the control volume. A special interest on the accuracy of the method is achieved. The results of the pressure fields and the load evaluation demonstrate the feasibility of the method and will provide new sets of data to better understand the effects of the 3D vortex structures on the wing lift in flapping motion.
1. INTRODUCTION
The interest for micro air vehicles (MAVs), for which the small dimensions imply a low Reynolds number, has led to the study of flapping flight. This type of motion involves an unsteady evolution of the three-dimensional vortex structures acting on the wing. The study of aerodynamic forces on immersed bodies generally can require two types of measurement methods. The first consists on a direct measurement of loads by means of sensors. The second is indirect and results from an integral form of the momentum equation on a carefully chosen volume. The difficulty to measure such small forces in experimental conditions has led to the evaluation of loads using momentum equation. This non intrusive method for evaluating aerodynamic loads [1] comes from momentum equation in integral form, applied on a volume located around the wing. The major difficulty for this approach is the calculation of the pressure field on the surface of the volume of integration. Different approaches have been implemented and tested in the litterature for two-dimensional flows [2].
In this paper, we apply the integration of the pressure gradient for the pressure evaluation and an extension of nonintrusive approach based on an integral form of the momentum equation is proposed to evaluate instantaneous loads of a NACA 0012 airfoil in flapping motion at Re = 1000 (The Reynolds number is based on the chord length and the velocity at 2/3 of the wingspan) in the case of the three-dimensional PIV measurements. Two major difficulties of this approach in 3D are the quality of the time-resolved volumic velocity fields and the estimation of the pressure field on the surface of the control volume. This last step needs a good spatial and temporal resolution of the velocity fields. Experimental data are issued from a scanning tomography PIV technique for a flapping wing in revolving motion. The volumes of particles are reconstructed using block acquisition of 100 images and the three-dimensional velocity fields around the airfoil are evaluated with an adaptive 3D cross-correlation. Phase-averaged velocity fields are estimated from the 25 velocity fields and are used by an iterative algorithm based on Navier-Stokes equations to deduce the pressure on the surfaces of the control volume. Direct Numerical Simulation are also provided for the validation of the methods. A special interest on the accuracy of the method is achieved.
2. EXPERIMENTAL SETUP
The experimental study was conducted in a water tank of octagonal section. The NACA 0012 airfoil of chord 0.06 m is put in flapping motion by the action of two motors controlled in position and speed. Those motors allow a rotation in revolution, of vertical axis located at the root of the wing, and a rotation of axis along the wingspan located at a quarter of chord from the leading edge allowing incidence variation. The kinematics of the wing is composed of two symmetrical movement phase. Each phase is composed three parts : an acceleration phase during which incidence evolves from 90◦to 45◦(t ∈ [0; 0.11T ]), a phase at constant speed and incidence (t ∈ [0.11T ; 0.39T ]), and a deceleration phase, during which the incidence varies from 45◦to 90◦(t ∈ [0.39T ; 0.5T ]).
2.1 Principle
Here, measurements of velocity fields are performed in a three-dimensional unsteady flow, at a low Reynolds number. A 3D3C PIV technical will be preferred over 2D PIV methods or stereo-PIV, which requires a large amount of measurements to reconstruct the flow over the volume, and many settings. Two methods currently allow the measurement of velocity in a volume by PIV. The first involves the use of several sensors, and by illuminating the entire measurement volume. Particle positions are then determined from images acquired by the various sensors by algebraic reconstruction algorithms [3]. However, this process requires a volume of a small thickness in order to separate the particles visible properly. The second method involves the use of a single sensor, and the scan of the measurement volume with light sheet [4]. This therefore enables the measurement of a larger volume, limited in the plane by the image sensor and in thickness by the the optical used (images must be in focus on the entire thickness of the volume). Note, however, that the acquisition of the volume is not done instantly. It lasts as long as the laser plane sweeps the depth of the volume. The chosen measurement technique is the second one, given the size of the region of the flow to be measured and the relatively low Reynolds number. The characteristic velocity of the flow imposes an acquisition frequency of the camera sufficiently high to freeze the flow during the measurement procedure of the volume. The acquisition frequency of volumetric images must be sufficiently low to allow the movement of the particles between the measurement of two consecutive volumes.
The illumination of the volume is done by means of a LASER Quantronix Darwin-Duo Nd:YLF whose power is 2 ∗ 18 mJ, coupled to an oscillating mirror. The laser beam passes through a converging lens to reduce the beam diameter to height of the mirror. The oscillating mirror enables a movement of the laser, which then passes through lens to ensure the parallelism of resulting laser planes. The oscillation of the mirror is imposed by a low frequency generator delivering an asymmetrical triangular periodic signal of frequency 20 Hz. The period of oscillation is fixed at 50 ms and consists of a forward motion (17 ms) during which the measurements are made, and a return movement, slower (33 ms), during which no action is performed. The mechanical oscillation of the mirror is at the origin of inertial effects disrupting the linearity of the movement. This phenomenon implies that the measurements must be made during the linear phases of motion of the mirror, so that equidistance between LASER plans is respected. Measuring a volume constituted of 100 planes thus takes 12.5 ms.
The octagonal section of the water tank impose to use a fixed measurement system. The acquisition is therefore performed in an absolute reference frame. In addition, the fact that measurements are done around a moving wing implies the presence of hidden areas close to the airfoil. These factors determine the choice of the measuring volume on a consecutive time interval centered on a position of the wing such that its wingspan is orthogonal to the laser planes, and small enough to reduce the influence of the areas masked by the wing. The system for moving the wing is then rotated to enable measurement of velocity fields in different position of the wing during its flapping motion. To ensure the representativeness of the observed phenomena, the measurement of different velocity fields is performed periodically during 25 cycles of motion in order to calculate a phase average for the flow. The measurement procedure is started after seven cycles of motion in order to obtain an established flow.
The Laser is synchronized with a fast camera Photron SA1, equipped with a Nikon lens of opening F#= 11 and focal 105 focal mm.
The measurement of 100 planes constituting a volume measurement of depth 8.3 cm are performed at a frequency 8 kHz. The resulting images, whose resolution are 1024 ∗ 752 pixels, cover an area of approximately 20 ∗ 15 cm2. The airfoil is located at the center of the picture in order to obtain a characterization of the flow around the wing.
2.2 Image processing
The pixel size of pictures that builds a measurement volume is function of the distance between the measurement plane and the camera. Planar images are resized to reconstruct volumetric images in an absolute frame. This method requires a calibration step in which measurements are made by placing a pattern in the measurement volume at different depths, in order to deduce the transformation matrix between the coordinate system related to the measurement and the reference frame. The volumic image is then resampled by a bicubic spline interpolator. Calculation of the velocity fields is carried out by two-passes subpixel correlation with an internally developed code based on the SLIP library [5]. The size of interrogation volumes, constant during the process, is 32 ∗ 32 ∗ 8 voxels with a recovery rate of 75%. Note that the size of interrogation volumes for the correlation process are chosen to obtain an isotropic mesh for the resulting velocity fields in the physical reference.. The calculation, carried out in two passes, incorporates a median filter (on average, less than 1% vectors are corrected).
The measurements of volumetric images are performed consecutively five times, in order to obtain a more suitable temporal resolution for the calculation of the acceleration field. It is calculated using second order central finite difference numerical schemes, using two velocity fields as shown in figure 1.
Figure 1: Representation of the method for calculating the velocity and acceleration fields.
2.3 Evaluation of uncertainties
Figure 2 is an example of an instantaneous velocity field obtained by PIV on the left, and a phase average of 25 periods on the right. The transverse velocities are plotted on a vertical plane approximately 3.2 chords of the wing’s root. An iso-surface VZ= V0= 0.016
m/s shows the flow on the upper surface toward the wing tip.
(a) Instantaneous field. (b) Phase average.
Figure 2: Velocity fields with respect to the motion of the wing, out of plane component of the velocity, and iso-surface Vz= 0.016
m/s.
Figure 3 shows fields of divergence of the velocity, for instantaneous data on the left and for the phase average on the right. The relatively random distribution of divergence for instant data shows the main sources of errors, located close to the wing and in the wake of the trailing edge.
(a) Instantaneous field. (b) Phase average.
Figure 3: Divergence of the velocity field obtained by 3D-3C PIV, and iso-surface of the absolute value of the divergence (|div(u)| = 2 s−1). The vertical plane is located at 5/6 chord of the wing tip.
To analyze the error obtained on the determination of the velocity field and due only to the correlation step, the following study is performed by masking the area of the measurement volume on which the profile is located during the recording process. Experimental uncertainties are characterized using a method described by [6] and based on the study of divergence. The average value of the divergence in the experimental tests is of the order of 0.001 pixels / pixel, and the averaged maximum is 0.025 pixels/pixel. This result indicates an uncertainty on the calculation of velocity fields halved compared to previous measurements made by [4], with a maximum uncertainty of less than 0.4 pixels. Sources of error obtained are mainly on planes at the end of the volume of measurements, due to degraded sharpness of volumetric images in these areas, and near the airfoil, given the relative motion of the wing relative to the measuring system and the light reflection close to the wing.
The 3D3C PIV acquisition system used here differs by a faster frequency of acquisition for planes constituting a volume (8 kHz here compared to 4 kHz) as well as by the use of an optical assembly to obtain focused images for each plane forming the measurement volume. These points explain the difference between the average and maximum differences of respectively 0.001 pixels / pixel and 0.025 pixels / pixel obtained here against respectively 0.002 and 0.06 pixels / pixel.
3. PRESSURE FIELD EVALUATION 3.1 Theoretical background
As part of the study of an immersed body in a three-dimensional flow, the application of the basic formulation for evaluating loads by the momentum equation approach requires knowledge of the pressure field on the surface of the volume of control used. The calculation by integrating the pressure field from the velocity fields then depends on the governing equations used to obtain the pressure gradient, and on the integration method used. The different methods for integrating the gradient generally depends on a direct integration, or in a resolution of a Poisson equation for pressure. Different approaches were tested and compared in the literature [7]. The reader can also refer to a recent review on the calculation of the pressure from the PIV measurements [8].
The pressure field is calculated on a three-dimensional mesh by integration of the pressure gradient on the computational domain. The first step is to determine the pressure gradient field, obtained from the absolute velocity fields by using Navier-Stokes equations as follows: ∂ p ∂xi = −ρ ∂ui ∂t + 3
∑
j=1 uj ∂ui ∂xj ! + µ 3∑
j=1 ∂2ui ∂x2j (1)The derivatives are approximated by the use of spatial or temporal second order central finite difference numerical schemes. The quality of the pressure gradient field depends on the quality of velocity fields used on the one hand, and on spatial and temporal resolutions on the other.
The pressure field is calculated by integrating iteratively the gradient over the domain, using as an initial condition a null pressure field over the whole domain. At each iteration, the values of the pressure field on the mesh is modified on each point according to the local pressure gradients and local values of the pressure field of the previous iteration. For a node of coordinates i jk and with n the iteration number, the equations governing the algorithm are noted :
pn=0i jk = 0 ∀i jk Pi, j,kn+1 = 1 i+1 ∑ f=i−1 j+1 ∑ g= j−1 k+1 ∑ h=k−1 δf gh i+1
∑
f=i−1 j+1∑
g= j−1 k+1∑
h=k−1 δf gh pf gh+ ∆xa ∂ p ∂xa f /2g/2h/2 ! (2) with : a= | f | + |g| + |h| ; δf gh = 1 if | f | + |g| + |h| = 1 & Mf gh∈ D(t) δf gh = 0 elseThe integration scheme does not take into account the pressure and pressure gradient on nodes located outside the domain D(t). The boundary conditions are implicit, and taken into account when calculating the gradient of pressure for nodes located on the surface ∂D(t) of the field which involve spatial derivatives of the velocty fields. These derivatives are determined on the surface of ∂D(t) by using a centered second order scheme. This implies the knowledge of the velocity of the wing over ∂DP(t), and the velocity of the fluid
outside the volume for ∂Df(t).
To accelerate the convergence of the algorithm, the pressure field at iteration n is not retained for the calculation of the values of the pressure field at iteration n + 1, but updated after each calculation of a node of the mesh. The integration of the pressure gradient then depends in part on integration path adopted. To avoid favoring one way to spread information and errors, the algorithm uses successively alternating directions for each iteration.
Note that calculating loads by using the integral form of momentum equation requires only the knowledge of the pressure field on the surface of the volume used. However, the calculation of the pressure field is performed on the whole domain where data are available, to incorporate into the process of calculating the most information possible.
The evaluation of the pressure field depends on local values of the velocity field through the calculation of the pressure gradient. This dependence implies a sensitivity to local errors, and their propagation during the calculation process. To limit this phenomenon, the computational domain D is divided into N sections Dndepending on the local data reliability. Here we consider a point Mi jk as
belonging to an area with fewer errors by comparing the Frobenius norm of the velocity gradient tensor relatively to the average value of this term on the field.
||grad−→V||F= v u u t 3
∑
i=1 3∑
j=1 ∂ui ∂xj 2 (3)DNrepresents the part of the domain in which the data are considered the most reliable, generally located where flow is less disturbed.
Part D0contains data that we assume the most noisy, usually close to the profile. The calculation of the pressure at each point belonging
to a subdomain of Dnof D is done by excluding from the calculation subdomains Dksuch that k < n. In this way, the transmission of
information on the integration domain in only made from Dnparts to Dkparts. The calculation of the subdomain DNis made separately
to the rest of the domain D (so it must exist a single subdomain DN).
This method requires as a first step to establish a criterion for defining the different subdomains. The aim is an application on PIV time-resolved 3D data, so the criterion should be defined in terms of the velocity fields. One can legitimately consider that the quality of measurements is altered in areas with high instantaneous velocity gradient. It is therefore proposed here to make it by comparing the local norm of the velocity gradient tensor (equation 3) to the average value over the domain. In addition, it is necessary that there is a common border area for each subdomain Dnwith a subdomain Dn+1, to use the surface of Dn+1as boundary condition for the
calculation of the pressure over Dn.
Integration is considered converged when the mean change of pressure value on the mesh over iterations is considered negligible. However, the discretization of the initial data introduce inaccuracies, generating errors which propagate during the process of integration of the pressure gradient. That is to say that even if the initial velocity fields are perfectly accurate, noise will be present during the integration. This phenomenon contributes to the fact that in many cases the mean change of pressure value on the mesh does not tend to 0. The aim is to determine a sufficiently low value for that information to be widely propagated during the integration process, but high enough to prevent excessive number of iterations. Indeed, from a certain number of iterations, only noise is transmitted during the calculation process, and the pressure field is then changed negligibly during on each iterations.
The evaluation of separating the domain for the calculation of pressure fields around a flapping wing has been made with DNS velocity fields. The comparison with numerical pressure fields showed a mean reduction of more than 50% of the error on resulting pressure fields with using sub domains of calculation.
3.2 Validation
Figure 4 represents iso-surfaces of dimensionless pressure at t = 0.16T obtained by direct numerical simulation on the left, and calculated by integration of the pressure gradient determined from DNS velocity fields on the right. If the results are qualitatively very similar, there is a slight difference in level.
Figure 4: Iso-surfaces of pressure coefficient Cp= −2 and pressure field on a slice orthogonal to the wingspan, located at 5/6 chords
of the wing tip (t = 0.16T ). The left side represent DNS results, and the right side pressure calculated by integrating its gradient
The figure represent surfaces between sub-domains used to limit the propagation of error of integration during the iterative process of calculation. The surface between D0and D1(a) is located close
to the wing over a large part of the wing, and extends in the wake where a tip vortex is form. The surface between sub-domains D1
and D2 (b) is also in a region near the wing and around the tip
vortex. The sub-domain D2represents 80% of the total size of the
domain of calculation. (a) ∂D0−1= D0∩ D1 (b) ∂D1−2= D1∩ D2
Figure 5: Representation of surfaces of intersection between the subdomains of calculation used for the iterative process of integrating the pressure gradient.
4. LOADS EVALUATION 4.1 Theoretical background
A balance equation in continuum mechanics quantifies the rate of change of an extensive quantity in a field, based on exchange between the field and outside, and sources in the field. The balance equation of momentum for a non-deformable and non-material domain D(t) of boundary ∂D(t) and animated by a proper motion−→W(x, y, z,t) is:
δ δt ZZZ D(t) ρ−→V dv | {z } rate of change + ZZ ∂D(t) ρ−→V−→Vr.−→n ds | {z } convective flux = ZZ ∂D(t) σ.−→n ds | {z } diffusive flux + ZZZ D(t) ρ−→f ds | {z } source (4)
with−→V the absolute velocity,−→Vr=
− →
V −−→Wthe velocity of the fluid in the frame of the motion of the domain, and−→n the outer normal to the surface ∂D(t). σ is the stress tensor and−→f represents the volume forces. the change over time of a quantity in the moving domain D(t) is noted δ δt . δ δt RRR D(t)
ρ−→V dvrepresents the change of momentum within the domain D(t) and especially if the domain is motionless then δ δt RRR D(t) ρ−→V dv= ∂ ∂t RRR D(t) ρ−→V dv.
Imposing a common border area between the domain and a rigid and impermeable to fluid solid, the surface ∂D(t) of the domain can be decomposed into ∂Df(t) + ∂Dp(t), with ∂Df(t) the surface area between made fluid zones, and ∂Dp(t) the surface between the fluid
and an immersed solid. The equation is then rewritten as follows:
δ δt ZZZ D(t) ρ−→V dv+ ZZ ∂Df(t) ρ−→V −→ Vr.−→n ds= ZZ ∂Df(t) σ.−→n ds+ ZZ ∂Dp(t) σ.−→n ds | {z } − → Fsolid/fluid + ZZZ D(t) ρ−→f ds (5)
with−→Fsolid/fluidthe resultant of the forces.
The scope of momentum balance D(t) is a parallelepiped located around the profile shown in Figure 6. During a motion of the wing without incidence variation, the profile is fixed relative to the control volume. The effects of gravity are neglected. The final equation used for calculating the forces−→F(t) transmitted to the wing is thus written:
− → F(t) = −ρδ δt ZZZ D(t) − → V dv | {z } unsteady term −ρ ZZ ∂Df(t) − → V −→ Vr.−→n ds | {z } convective term − ZZ ∂Df(t) p−→n ds | {z } pressure term + ZZ ∂Df(t) τ.−→n ds | {z } viscosity term (6) With : τ = µ −→ ∇ ⊗−→V +−→∇ ⊗−→Vt (7)
Figure 6: Representation of the control volume used for the balance equation of momentum.
In practice, unsteady, convective and viscous terms are directly deducted from the velocity fields. They respectively represent the change of momentum within the control volume, and the effect of convection and viscous forces on the surface of the control volume. Calculating the term of pressure requires the knowledge of the pressure field on the outer surface of the control volume, which is obtained by integration of its gradient (method described in the following paragraph).
The spatial and time derivative formulas used are centered and of order 2. As the same manner, the change of momentum in the volume is calculated by determining the quantity of momentum in the volume to times t + ∆tand t − ∆t.
4.2 Validation
We are now working to assess the error introduced by calculating the pressure field on the evaluation of loads. Figure 7 represents the evolution of loads calculated by using integral form of momentum equation, with the pressure field obtained by integrating the pressure gradient, and by using the pressure field resulting from DNS. This comparison is then representative of the influence of errors introduced by the integration process. Compared to figure 7, a fairly low difference is observed between loads calculated from pressure fields calculated by integration, and loads calculated using DNS pressure field.
0 0.5 1 1.5 0 0.11 0.39 0.5 Cx t/T pressure from integration
pressure from DNS 0 0.5 1 1.5 0 0.11 0.39 0.5 Cy t/T pressure from integration
pressure from DNS
Figure 7: Evolution of dimensionless forces (drag on the left and lift on the right), calculated by using momentum approach with the pressure field obtained by integration, and with the pressure field obtained by numerical simulation.
Figure 8 represents the evolution of dimensionless loads during a half period. Efforts are represented from direct numerical simulation in red (DNS) and using the integral form of momentum equation (BQM) in green. As an indication, loads are calculated by integrating the calculated pressure field over the surface of the wing. We note an overall qualitatively similar development for efforts obtained from DNS compared to results although a significant difference was observed in the case of the calculated pressure for the lift component, especially during the acceleration phase. Instead, loads obtained by integrating the pressure on the surface of the wing (still for the lift case) suggests a fairly low error rate of the pressure field recalculated following the method described above in the area nearby the wing. 0 0.5 1 1.5 0 0.11 0.39 0.5 Cx t/T DNS BQM pressure over the surface
0 0.5 1 1.5 0 0.11 0.39 0.5 Cy t/T DNS BQM pressure over the surface
Figure 8: Dimensionless loads obtained by DNS, by using integral form of momentum equation (BQM), and by integrating calculated pressure field over the surface of the wing.
5. RESULTS
Methods of calculation of pressure fields and loads are applied to the experimental case. 3D3C velocity fields were measured by PIV for five consecutive moments, allowing the determination of velocity and acceleration fields at the same time.
Experimental data are obtained for 17 positions of the wing along half a period of flapping, and for 25 cycles. Pressure fields and loads are calculated for each position and each period of flapping, and then averaged to obtain phase averages. This process allows the reconstruction of the evolution of the flow around the wing.
Figure 9 represents iso-surfaces of dimensionless pressure for the experimental case, at t = 0.18T . This figure shows similarly to the figure 4 the pressure obtained from PIV Measurements. The results shown are instantaneous for (b) (c) and (d) and a phase average calculated over 25 periods for figure 9 (a). Instantaneous results are characterized by a relative noise and unsteady effects, although consistency remains visible through larger values. Instead, the pressure field obtained by phase averaging seems less noisy, but the values are lower. This is probably due to the unsteadiness of the flow, including the area close to the wing tip, where leading edge vortices and tip vortices interact and are not located exactly in the same location for a period to another.
(a) phase average over 25 instantaneaous results. (b) Instantaneous results.
(c) Instantaneous results. (d) Instantaneous results.
Figure 9: Iso-surfaces of dimensionless pressure coefficient Cp= −2 and pressure field on a slice orthogonal to the wingspan, located
at 5/6 chords of the wing tip (t = 0.18T ).
If the calculated pressure fields around the wing for the experimental case can not be verified, the evaluation on DNS fields and the distribution of instantaneous pressure fields deduced from PIV measurements suggest a sufficient quality to study the topology of the flow.
Figure 10 represents the evolution of loads calculated by using the momentum equation for the experimental case compared to results obtained by DNS. If experimental results evolve qualitatively in a similar way during the acceleration phase, the phase of constant speed is characterized by a low peak of loads, a strong peak is observed at the beginning of the deceleration phase. The drag component has a significant shift compared with the numerical results. However, there is a similar evolution of the lift component, with levels during the phase of moving at a constant speed comparable.
Figure 11 shows loads calculated by integrating the pressure field on the surface of the wing from the PIV data, compared to DNS results. The evolution is similar to results obtained by the momentum approach, but the difference in level obtained for the drag component is not observed here. If the quality of the pressure field is not sufficient to determine accurate loads, the evolution of the phase average stay comparable to numerical results.
6. CONCLUSION
Three dimensional Particle image velocimetry is used to study the evolution of the flow generated by a flapping wing. The resulting loads are evaluated by using a momentum equation in an integral form on a volume located around the wing. The difficulty lies in the application on a three-dimensional unsteady flow, and in the calculation of the pressure field used on the surface of the volume to obtain loads. If the method of evaluation of loads currently does not obtain sufficient accuracy for a detailed analysis of aerodynamic loads today, the results are very promising.
−0.5 0 0.5 1 1.5 0 0.11 0.39 0.5 Cx t/T DNS Experimental −0.5 0 0.5 1 1.5 0 0.11 0.39 0.5 Cy t/T DNS Experimental
Figure 10: Phase averaged evolution of dimensionless forces (drag on the left and lift on the right), obtained from experimental results by using integral form of momentum equation.
−0.5 0 0.5 1 1.5 0 0.11 0.39 0.5 Cx t/T DNS expe −0.5 0 0.5 1 1.5 0 0.11 0.39 0.5 Cy t/T DNS expe
Figure 11: Phase averaged evolution of dimensionless forces (drag on the left and lift on the right), obtained from experimental results by integrating the calculated pressure field over the surface of the wing.
ACKNOWLEDGMENTS
The current work has been conducted as part of the AFDAR project, Advanced Flow Diagnostics for Aeronautical research, funded by the European Commission program FP7, grant n.265695
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