14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Gridless Determination of Aerodynamic Loads Using
Lagrangian Particle Tracks

Christoph Mertens∗, José L. Costa Fernández, Andrea Sciacchitano,
Bas W. van Oudheusden, Jurij Sodja

Delft University of Technology, Faculty of Aerospace Engineering, 2629HS Delft, The Netherlands
∗ C.Mertens@tudelft.nl

Abstract
The aerodynamic loads on a flexible wing in terms of the surface pressure distribution and the lift force
along the span are determined experimentally based on non-intrusive Lagrangian particle tracking (LPT)
measurements. As the flexible wing deforms under the aerodynamic loads, its deformed shape is first re-
constructed based on structural LPT measurements conducted together with the flow measurements in an
integrated approach. Based on the reconstructed wing shape, flow tracers data are collected along surface
normals to evaluate the surface pressure, as well as along elliptic paths around the wing to determine the
circulation. The lift force is calculated from the surface pressure by integrating the pressure difference along
the chord, as well as from the circulation using the Kutta-Joukowski theorem. The circulation-based lift re-
sults are in very good agreement with reference measurements from a force balance, with differences in the
total lift force on the wing of less than 5%. The lift estimation based on the extrapolated surface pressure is
consistently lower than the circulation-based lift, by about 10%, due to the limited accuracy of the pressure
extrapolation near the leading edge region, where a considerable fraction of the lift is generated.

1 Introduction
The experimental determination of the aerodynamic loads that act on flexible wings is relevant for aeroe-
lastic research and development activities, in particular for providing reference data to validate numerical
aeroelastic models. The situation of aerodynamic loads on highly flexible wings is particularly relevant be-
cause these wings can exhibit nonlinear structural behavior, which increases the complexity of the numerical
aeroelastic model and as such the need for experimental validation data (Dimitriadis (2017)).

The installation of pressure sensors to measure the aerodynamic load on such highly flexible wings
is difficult, or, depending on the respective design of the flexible wing skin, may even be impossible. It is
therefore preferred to determine the aerodynamic loads non-intrusively, which can be achieved by measuring
the flow field around the wing and then inferring the aerodynamic loads from these measurements by using
the governing equations of the flow (van Oudheusden (2013); Rival and van Oudheusden (2017)).

Lagrangian particle tracking (LPT) via the Shake-The-Box algorithm (Schanz et al. (2016)) in combina-
tion with the use of helium-filled soap bubbles (HFSB) as flow tracers (Scarano et al. (2015)) facilitates the
performance of volumetric flow field measurements at a relatively large scale, which is required to obtain the
aerodynamic loads in the relevant aeroelastic regimes. A further advantage of this technique is that it facil-
itates the simultaneous characterization of the structural response and the three-dimensional flow topology
with an integrated approach, as described in Mertens et al. (2021).

However, a drawback of the LPT flow field measurements is its relatively poor spatial resolution, caused
by the relatively large inter-particle distance, especially when large-scale measurements with HFSB are
conducted. A method to mitigate this limitation is to perform a statistical analysis of the LPT data throughout
the flow field by ensemble-averaging the track data in time on a Cartesian grid (Agüera et al. (2016)). A
considerable disadvantage of this procedure for the application in a loads determination procedure on flexible
wings is that the grid of the flow field does not automatically comply with the deformed shape of the wing,
which impedes the loads determination.

In this study, it is demonstrated how the particle track data from the LPT measurements of the flow field
and the structure can be used to determine the aerodynamic loads on a flexible wing without requiring the



additional step of ensemble averaging the LPT flow measurements on a regular grid. An important first
step in this approach is the determination of the deformed shape of the wing under the aerodynamic load,
which is performed based on the LPT measurements of structural markers. Subsequently, two different
approaches are considered to determine the aerodynamic load from the flow measurements. In the pressure-
based approach, the measured shape of the wing is used to collect LPT data in the flow field along surface
normals to find the pressure on the wing surface by extrapolation. In the circulation-based approach, the
flow field LPT data is collected on elliptic contours around the wing to determine the lift force across the
span. The loads are calculated using relations from potential flow theory, which only require measurements
of the velocity, that are directly available from the individual LPT data points. Results are compared to
reference measurements of the overall lift force across the span from a force balance.

2 Experimental Procedures

2.1 Wind Tunnel Setup
The experiments were conducted in the Open Jet Facility at Delft University of Technology, which is an
open test section, closed return wind tunnel with an octagonal outlet of 2.85m×2.85m. The test object was
a highly flexible wing that is similar to the aeroelastic benchmark model design that is described in Avin
et al. (2021), which can sustain very large deformations, with wingtip displacements of up to 50% of its
span width. The wind tunnel was operated at a freestream velocity of U∞ = 18.3m/s, corresponding to a
Reynolds number of Re = 122000 based on the wing chord. Optical measurements of the flow and the
position of the wing were conducted simultaneously with an integrated measurement approach. A sketch of
the flexible wing and a photo of the measurement setup in the wind tunnel are shown in Fig. 1.

30

38
.2
5

Ø1.5

55
0

100

300

Ø10

(a)

𝑈∞

x

yz

1

2

3

4

(b)

Figure 1: Experimental model and wind tunnel setup, (a) sketch with dimensions (in millimeter) of the
flexible wing and structural markers grid, (b) photo of the experimental setup, 1: flexible wing, 2: high-
speed cameras, 3: LED illumination units, 4: stream of helium-filled soap bubbles

The flexible wing model that is illustrated in Fig. 1(a) has a chord length of c = 100mm and a span width
of s = 550mm with a NACA 0018 airfoil section. The wing structure is a 3D printed Nylon 12 chassis, with
a mounting base and a 300mm long tip rod, and an Aluminum 7075 spar with a thickness of 1.5mm that is
bonded to the chassis using Loctite 495 glue. The wing skin is made out of black Oralight polyester foil.



Structural measurements of the wing were performed by tracking reflective markers on the surface. For
this purpose, white circular markers with a diameter of 1.5mm were painted on the wing surface using a
laser-cut mask, markers were painted on the tip rod and at the spanwise locations of the ribs, which have a
spacing of 38.25mm, with a chordwise spacing of the markers of 30mm.

As visible in Fig. 1(b), the wing was mounted vertically in the wind tunnel test section. It was mounted
on a six-component force balance that was connected to a rotating table, which was used to set the geometric
angle of attack α of the wing. In this study, two different angles are investigated, α1 = 5° and α2 = 10°, and
force measurements acquired with the balance are used as a reference for the aerodynamic loads that are
determined based on the optical LPT measurements.

The optical measurement system that is shown in Fig. 1(b) consists of three Photron Fastcam SA1.1
high-speed cameras with a sensor size of 1024× 1024 pixels and an image sampling rate of 5.4kHz. The
flow measurements were performed using helium-filled soap bubbles (HFSB) as tracer particles (Scarano
et al. (2015)). The HFSB were produced with a seeding generator that is placed in the settling chamber of
the wind tunnel, upstream of the contraction of the wind tunnel nozzle. The seeding generator consists of
200 bubble-producing nozzles distributed over an area of 500mm×1000mm. The working principle of the
bubble-producing nozzles is described in Faleiros et al. (2019). The resulting HFSB seeding concentration
was around 1cm−3. The HFSB flow tracers were illuminated with three LaVision LED-Flashlight 300
modules, and the image acquisition and processing were performed with the LaVision Davis 10 software.

2.2 Data Acquisition and Processing
The data acquisition and processing to obtain the LPT measurements of the flow and the structure closely
follows the procedure of the integrated measurement approach described in Mertens et al. (2021). This
procedure begins with the optical calibration of the system by performing a volume self-calibration (Wieneke
(2008)) and generating an optical transfer function (Schanz et al. (2013)). Afterward, the integrated flow and
structural measurements are separated, as illustrated in Fig. 2. Optical measurements of the flow tracers and
the structural markers are acquired by the cameras, as shown in Fig. 2(a). To obtain the isolated information
of only the flow or the structure, temporal filters are applied to the time series of the images; the application
of a temporal high pass filter produces the isolated flow tracers information (Sciacchitano and Scarano
(2014), see Fig. 2(b)), and a temporal low-pass filter produces the isolated structural information (Mitrotta
et al., see Fig. 2(c)). Afterward, the LPT measurements of the structure and the flow are generated by
applying the Shake-The-Box algorithm (Schanz et al. (2016)) to the separate image data sets.

(a) (b) (c)

Figure 2: Image data processing procedure: (a) integrated optical measurement of flow and structure, (b) im-
age data of the flow tracers, (c) image data of the structure

The size of the measurement volume that was achieved with the optical measurement setup was around
27 liters (300mm× 300mm× 300mm), although part of this volume was obstructed by the presence of
the wing, such that the LPT measurements were obtained within a volume of around 22 liters. For the
aerodynamic loads determination, measurements from both sides of the wing and along the entire span
are necessary. To obtain this data with the described measurement setup, the LPT measurements were
performed in four separate acquisitions. First, two acquisitions were performed for the bottom half of the
wingspan, where the angle of attack of the wing was set to +α and −α, respectively. The data acquired at



−α (pressure side) was later mirrored in post-processing to the opposite side of the wing and merged with
the data acquired at +α (suction side). Thereby, the symmetry of the wing was exploited to simplify the
LPT data acquisition, because, with this approach, it is not necessary to move and recalibrate the optical
measurement setup. The same procedure is repeated for the top half of the wingspan.

The resulting four separate acquisitions are merged by transforming the LPT data from the measurement
coordinate system to the wind tunnel coordinate system with a rigid-body transformation. The translation
and rotation matrices for this rigid-body transformation are determined based on reference LPT measure-
ments of the structural markers on the wing, performed without wind tunnel operation before each manip-
ulation of the measurement volume. The accuracy of the employed merging procedure can be assessed by
comparing the flow velocity at corresponding positions with respect to the wing from different acquisitions.
Typical values of these differences were between 1% and 3% of the local velocity magnitude. Since these
values are of the same order of magnitude as when the merging of different flow measurement acquisitions
is performed automatically based on position measurements with a robotic arm as in Jux et al. (2018), these
differences are considered acceptable and not further investigated.

3 Wing Shape Reconstruction
After the LPT measurements of the structural markers are transformed from the measurement coordinate
system to the wind tunnel coordinate system, the marker position measurements are used to reconstruct the
deformed shape of the wing. This is achieved by calculating the wing deflection as an average value of
the marker positions for each spanwise section where the markers were painted (i.e., the ribs of the wing
and the tip rod) and then fitting a polynomial through these measurements along the spanwise direction.
Following this procedure, the result of the polynomial curve fit is used as a reference spanwise axis to
calculate the deformed wing shape. A fourth-order polynomial is used in this study to perform the curve fit,
which satisfies the geometric boundary conditions of a wing that is clamped at the root, i.e. d(z = 0) = 0
and d′(z = 0) = 0, and is defined as

d(z) = Az4 +Bz3 +Cz2, (1)
where the coefficients A, B, and C are determined with an optimization to provide the best fit to the ex-
perimental measurements in a least-squared sense. The individual marker measurements from 100 acquired
images are averaged to reduce the effect of measurement noise and small-scale vibrations of the wing during
the experiment. The results of the polynomial curve fit to the marker measurements for the two different
values of α are shown in Fig. 3. The standard deviation of the residual between the 15 measurement loca-
tions along the span and the curve fit is σ1 = 0.13mm for α1 and σ2 = 0.25mm for α2, corresponding to
0.28% and 0.29% of the respective wingtip displacement. Note that the value of the spanwise position of
the wingtip can exceed ztip/s = 1 because there are markers on the wingtip rod, which is not accounted for
in the definition of the span length s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.05

0.1

0.15

0.2

Figure 3: Displacement measurements along the span with polynomial curve fit

To further characterize the deformed wing shape, knowledge about the torsional deformation is required
as well. The torsional deformation can be expressed in terms of a twist angle ε of the wing around the
reference axis. The twist angle ε of the wingtip can be estimated from the structural marker measurements
on the tip rod. The experimentally determined twist angles were found to be ε1 = 0.26° for α1 and ε2 = 0.45°
for α2. Based on these small values of ε and to simplify the analysis, the twist has been neglected in
the further analysis, assuming that the deformed wing shape can be reconstructed with sufficient accuracy
directly from the polynomial curve fit of the deflection along the span.



4 Loads Determination Methods

4.1 Surface Pressure Determination
In incompressible, inviscid, and irrotational steady flow, the local flow velocity and pressure in the entire
field are related through Bernoulli’s equation, yielding

Cp =
p− p∞

1
2 ρU2

∞

= 1− V 2

U2
∞

, (2)

where p is the pressure, ρ is the fluid density, and V is the velocity magnitude. For attached flow around
wings, the aforementioned conditions for the application of Bernoulli’s equation are generally justifiable
assumptions, except for the viscous flow inside the boundary layer. The presence of the boundary layer and,
in view of random measurement errors, the prevalence of relatively strong velocity gradients near the surface
obstruct a direct application of Eq. 2 to the LPT measurements in the vicinity of the surface to determine
the local surface pressure. It is therefore necessary to perform an extrapolation of the pressure obtained with
Eq. 2 from the flow field to the surface, as demonstrated in Mertens et al. (2021).

The LPT flow measurements provide discrete values for the pressure in the flow field by applying Eq. 2 to
each particle track. For the surface pressure extrapolation in this study, the LPT measurements within a given
radius around a finite number of surface normals from the wing are used, instead of first interpolating them
onto a regular grid. The surface normals are computed from the reconstructed wing shape by discretizing the
wing shape with planar rectangular elements and then calculating the normal direction n and the tangential
direction t from a sampling point at the center of each element. The coordinates of the LPT measurements
are then transformed into the local coordinate system of the wing so that the data within a given normal
and tangential distance around each surface normal can be used for the extrapolation of the surface pressure
at the respective sampling point on the wing. The data collection procedure is illustrated schematically in
Fig. 4.

Figure 4: Sketch of the LPT data collection procedure for a sampling point at x/c = 0.86. Blue dots: LPT
track data considered for the pressure extrapolation, red dots: LPT track data that is not considered

In Fig. 5, the LPT measurements along 100 surface normals with a length of nmax = 15mm (0.15c)
are shown for a cross-section of the wing at α2 = 10° for the spanwise location z/s = 0.6. There are 50
normals on the suction (top) and pressure (bottom) side, respectively. The sampling points on the surface
are distributed using a half-cosine spacing along the chord to concentrate the sampling points in the leading
edge region of the airfoil where high pressure gradients are expected. The LPT data within a spanwise
region that extends over 11mm (±0.01s) around the spanwise location z/s = 0.6 is considered. Within
this spanwise region, the LPT data with a maximum tangential distance of tmax = 2.5mm (0.025c) to the
respective surface normal is collected and plotted along the normal in Fig. 5 as a function of its respective
wall distance n.



Figure 5: LPT-based pressure from Bernoulli’s equation along surface normals at z/s = 0.6 for α2 = 10°.
The labels (a-f) in the figure indicate the normals that are plotted in Fig. 6

As visible in Fig. 5, a considerable amount of random measurement noise and artifacts obstruct the
application of Eq. 2 in the direct vicinity of the surface. In particular, an unphysical increase in the pressure
obtained with Eq. 2 is observed close to the wing surface, most prominently near the trailing edge on
the suction side. In this region, the presence of the boundary layer decelerates the flow near the wing
surface, which according to Eq. 2 implies an increase in pressure, whereas it is known from boundary layer
theory that the pressure does not vary across the boundary layer (see Schlichting and Gersten (2017)). To
determine the surface pressure from the pressure in the flow field as obtained from the LPT measurements
with Eq. 2 despite the presence of the boundary layer and the random measurement errors, an extrapolation
of the surface pressure is performed by fitting a polynomial through the LPT data points in the wall-normal
direction using a least-squares method. A second-order polynomial is used and data points with a wall-
normal distance n up to nmax = 0.15c are considered for the fit. As for the illustration in Fig. 4 and the
measured data plotted in Fig. 5, LPT data from a spanwise region of ±0.01s around the reference position,
with a maximum tangential distance of ±tmax = 0.025c from the LPT data point to the respective surface
normal in the cross-sectional plane is considered for the pressure extrapolation. The pressure extrapolation
is illustrated in Fig. 6 for the six example locations at z/s = 0.6 of which the locations are indicated in Fig. 5.

In general, the value of the polynomial at the wall is considered as the extrapolated surface pressure.
The LPT data, the fitted polynomial, and the extrapolated surface pressure are indicated for x/c = 0.3 on
the pressure side in Fig. 6(a). The pressure extrapolation works as expected, despite the presence of a
considerable amount of random error. The presence of random measurement errors is more critical when
the error is not normally distributed, as near the stagnation point, which is illustrated in Fig. 6(b), where
all the outliers are associated with lower pressures. The effect of these outliers is diminished by using a
robust least-squares polynomial fit with bisquare weights1, which is used at all chordwise locations. An
extraordinarily challenging measurement region is the leading edge region on the suction side, which is the
flow region in between the stagnation point and suction peak, where the spatial gradients of the flow velocity
and the pressure are maximal. In this region, the measured pressure data near the wall exhibits a bifurcation
behavior, as shown in Figs. 6(c) and 6(d), which can be expected as a consequence of the non-zero value of
tmax and the large pressure gradients in the wall-tangential direction in this region. As a result of the robust
fitting approach, the polynomial fit follows the branch with the larger amount of measured data points in
Figs. 6(c) and 6(d). A different issue of the LPT-based pressure measurement estimation with Bernoulli’s
equation is the presence of the boundary layer where Eq. 2 incorrectly implies an increase in pressure, in
particular on the suction side. The effect of this issue on the curve fit is still negligible at x/c = 0.3 as shown
in Fig. 6(e), but clearly relevant further downstream, as shown in Fig. 6(f). Because the actual pressure does
not vary across the boundary layer, the pressure outside of the boundary layer, where Bernoulli’s equation is
applicable, should be used for the extrapolation of the surface pressure, instead of the value of the polynomial

1https://mathworks.com/help/curvefit/least-squares-fitting.html

https://mathworks.com/help/curvefit/least-squares-fitting.html


fit at the wall. In Figs. 6(e) and 6(f), it is shown that this value can be determined on the suction side by
considering the minimum value of the polynomial fit in the wall-normal direction. This approach is used
for all sampling points on the suction side of the wing downstream of x/c = 0.2, as these locations are most
critically affected by the presence of the boundary layer.

0 0.05 0.1 0.15

-2

-1.5

-1

-0.5

0

0.5

1

(a) x/c = 0.30, pressure side

0 0.05 0.1 0.15

-1.5

-1

-0.5

0

0.5

1

(b) x/c = 0.028, pressure side

0 0.05 0.1 0.15

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

(c) x/c = 0.036, suction side

0 0.05 0.1 0.15

-2.5

-2

-1.5

-1

-0.5

0

0.5

(d) x/c = 0.065, suction side

0 0.05 0.1 0.15

-4

-3

-2

-1

0

1

(e) x/c = 0.30, suction side

0 0.05 0.1 0.15

-2

-1.5

-1

-0.5

0

0.5

1

1.5

(f) x/c = 0.80, suction side

Figure 6: Surface pressure extrapolation at six different example locations at z/s = 0.6 for α2 = 10°



4.2 Lift Force Determination
For attached flow around wings at small angles of attack α, the lift force perpendicular to the inflow can
be assumed as equivalent to the normal force on the wing, which acts perpendicular to the chord. For thin
objects like airfoils, the sectional lift force can therefore be obtained by integrating the pressure difference
between the suction and pressure sides along the chord, with the section lift in coefficient form given as

C` =
L′

1
2 ρU2

∞c
=

∫ c

0
(Cp,pressure−Cp,suction)dx, (3)

where L′ is the sectional lift force per unit span, and Cp can be determined from the flow field as described
before. If the determination of the pressure distribution itself is not of primary interest, the section lift L′ can
alternatively be determined by using the Kutta-Joukowski theorem:

L′ = ρU∞Γ, (4)

where Γ is the circulation around the wing, which can be determined by performing a line integral of the
flow velocity along a closed contour C around the wing:

Γ =−
∮

C
~u ·d~s. (5)

Although the Kutta-Joukowski theorem is derived based on the assumption of potential flow, previous
studies have shown that it can be used to determine the lift accurately from experimental data for flows that
exhibit moderate effects of viscosity, and when the circulation is calculated from velocity measurements
outside of the region affected by these viscous effects (Mertens et al. (2021); Sharma and Deshpande (2012);
Olasek and Karczewski (2021)).

In this study, the circulation is determined from the LPT measurements by defining an elliptic path
around the wing section of interest and collecting the LPT data within a prescribed distance around this
path, which is then integrated along the tangential direction to determine the circulation. Because the lift
force acts perpendicular to the freestream and the spanwise axis of the wing, the elliptic integration paths
are defined in a local coordinate system of the respective wing section of interest, which is defined using the
experimentally reconstructed wing shape. The ellipse has its origin at the mid-chord position of the wing
section, the major axis is aligned with the chord and the minor axis is perpendicular to the chord in the
cross-sectional plane. An example of the circulation determination procedure is visualized for α2 = 10° at
z/s = 0.6 in Fig. 7.

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

-0.4

-0.2

0

0.2

0.4

-1.5

-1

-0.5

0

0.5

1

1.5

(a) LPT data on an elliptic contour, colored by tangential velocity

0 0.2 0.4 0.6 0.8 1

-30

-20

-10

0

10

20

30

(b) Tangential velocity along the elliptic contour

Figure 7: Illustration of the circulation determination procedure at z/s = 0.6 for α2 = 10°

In Fig. 7(a), the LPT data along an elliptic contour around the wing at z/s = 0.6 is visualized. The
elliptic contour has a major semi-axis length of a/c = 0.75 and a minor semi-axis length of b/c = 0.5. As



for the pressure extrapolation, LPT data within a spanwise region that extends over 11mm (±0.01s) around
the respective spanwise location is considered. The maximum distance in the cross-sectional plane of the
LPT data point to the ellipse to be considered for the circulation determination is taken as 1mm (0.01c).
The tangential velocity component ut of the LPT data at z/s = 0.6 along the elliptic contour is plotted in
Fig. 7(b). The origin of the contour is located upstream of the wing, aligned with the chordwise direction.
The positive direction along the contour is counter-clockwise around the wing. To determine the circulation
from this data, the tangential velocity of the individual LPT data points is filtered with a smoothing spline2.
The smoothed data is then integrated along the length of the contour to obtain the circulation.

While in the theoretical case of a potential flow the circulation would be independent of the contour as
long as the object is fully enclosed by it, this assumption cannot be directly made for the present case that
includes some effects of viscosity as well as due to the occurrence of measurement errors. Therefore, the
effect of the ellipse geometry parameters a and b on the circulation Γ is analyzed as shown in Fig. 8 for α1
and α2 at z/s = 0.6.

(a) α1 = 5° (b) α2 = 10°

Figure 8: Effect of the elliptic contour parameters on the circulation at z/s = 0.6

A relatively strong sensitivity of Γ to b is observed when the elliptic contour is in the immediate vicinity
of the wing, i.e. when the contour is located within the boundary layer. This effect is amplified for α2 =
10° as seen in Fig. 8(b), where the boundary layer thickness on the suction side is increased due to a
stronger adverse pressure gradient, when compared to α1 = 5° in Fig. 8(a). Further away from the wing,
the circulation does not vary systematically with a or b, but it is subject to some level of variation, likely
due to random measurement errors. To minimize the influence of these random errors, the circulation Γ

for a given spanwise location of the wing is determined in this study by averaging the value of Γ from 100
different elliptic contours at that spanwise location, where the parameters of the ellipse are varied between
0.65≤ a/c≤ 1.1 and 0.25≤ b/c≤ 0.7.

5 Results
The results for the LPT-based pressure determination at four different spanwise locations for α1 and α2 are
shown in Fig. 9. As expected, the larger angle of attack α2 produces higher pressures on the bottom and
lower pressures on the top side. Within the range of z shown in Fig. 9, no large variation of the pressure
distributions with the spanwise position is observed. As an effect of the bifurcation in the data used for
the pressure extrapolation near the leading edge that was discussed in the context of Fig. 6, the pressure
coefficient Cp is positive on the suction side in the region from the leading edge until around x/c = 0.05 in
all pressure distributions in Fig. 9.

2https://mathworks.com/help/curvefit/smoothing-splines.html

https://mathworks.com/help/curvefit/smoothing-splines.html


0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

(a) z/s = 0.2

0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

(b) z/s = 0.4

0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

(c) z/s = 0.6

0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

(d) z/s = 0.8

Figure 9: LPT-based surface pressure distribution at four different spanwise locations

The lift distributions obtained by integrating the pressure difference along the chord, as well as from
the circulation-based method are compared in Fig. 10. The lift distributions from both LPT-based methods
show a plateau of the lift along most of the span and a drop towards the wing root and tip. This behavior is
expected from aerodynamics theory due to the presence of the wing fixture at the root and the tip vortex. The
pressure-based lift is consistently lower than the circulation-based lift, which is likely due to the reduction
of the suction due to experimental errors near the leading edge, which were observed in Fig. 9. Furthermore,
the pressure-based lift shows a larger level of random variation along the span compared to the circulation-
based method, which is expected because the pressure-based method uses only LPT data measured in the
vicinity of the wing, whereas, for the circulation-based method, several different integration paths were used
and averaged.

A quantitative assessment of the accuracy of the LPT-based loads determination is performed by inte-
grating the LPT-based lift result from the pressure- and circulation-based approaches along the span. These
results are compared with the balance measurements in Tab. 1. The differences of the circulation-based
method to the force balance measurements are considerably smaller than for the pressure-based method, by
around 10%. The difference to the force balance for both methods is increased for α2 compared to α1, but
overall it remains below 5% for the circulation-based and below 15% for the pressure-based method.



0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

(a) α1 = 5°

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

(b) α2 = 10°

Figure 10: Comparison of the lift distribution across the span with different methods

Table 1: Comparison of the lift force from the LPT-based methods with the force balance measurements

α1 = 5° α2 = 10°
Method Lift force Difference Lift force Difference

Force balance 4.12 N 7.83 N
Pressure-based 3.74 N -9.3% 6.75 N -13.7%
Circulation-based 4.00 N -2.9% 7.48 N -4.5%

6 Conclusion
This study has demonstrated the non-intrusive flowfield-based determination of aerodynamic loads on a flex-
ible wing with a gridless approach based on discrete LPT measurement data. Contrary to most approaches
in the published literature, the loads were not determined from ensemble-averaged data on a Cartesian grid,
but instead by first determining the deformed shape of the wing from LPT measurements of structural mark-
ers and then analyzing the discrete LPT flowfield data in a local coordinate system, which is given by the
measured wing shape. The aerodynamic loads were evaluated for two different angles of attack, α1 = 5°
and α2 = 10° and with two different methods, which are the pressure-based and the circulation-based ap-
proaches. For the pressure-based approach, flow measurements in the immediate vicinity of the wing are
necessary. The LPT data in this region is associated with a considerable amount of random error, and it is
furthermore more challenging to analyze due to the presence of the viscous boundary layer. These chal-
lenges were addressed in this study by performing an extrapolation of the surface pressure along surface
normals. However, the lift estimation based on a chordwise integration of the pressure difference was up to
13.7% lower than the reference lift measured with the force balance, mostly due to the erroneous surface
pressure evaluation in the leading edge region of the wing. In contrast, the circulation-based approach does
not require measurements in the near vicinity of the wing, as such it suffers to a lesser extent from the
measurement-related issues. When comparing the integral lift force along the span from this approach with
force balance measurements, a very good agreement is observed, with differences in the lift force of less than
5%. Therefore, the circulation-based approach is preferred for a more accurate and robust lift determination,
if the pressure distribution itself is of no direct interest. A topic for future research is a performance assess-
ment of the gridless approach by comparison to results obtained with an ensemble-averaging approach.
Additional topics include the improvement of the pressure extrapolation in the leading edge region and the
application of the gridless approach to unsteady flows and flow situations in which the aerodynamic loads
are more critically affected by viscous effects.



Acknowledgements
The authors gratefully acknowledge the help of Adrián Grille Guerra with the Lagrangian particle tracking
measurements during the wind tunnel experiments. This work has been carried out in the context of the
HOMER (Holistic Optical Metrology for Aero-Elastic Research) project that has received funding from the
European Union’s Horizon 2020 research and innovation programme under grant agreement No. 769237.

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	Introduction
	Experimental Procedures
	Wind Tunnel Setup
	Data Acquisition and Processing

	Wing Shape Reconstruction
	Loads Determination Methods
	Surface Pressure Determination
	Lift Force Determination

	Results
	Conclusion