Determining aerodynamic drag of transiting objects by time...

18th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 4 – 7, 2016

Aerodynamic drag of transiting objects by large-scale tomographic-PIV

W. Terra1,*, A. Sciacchitano1, F. Scarano1

1: Aerospace Engineering Department, TU Delft, The Netherlands * Correspondent author: [email protected]

Keywords: Tomographic PIV, aerodynamic drag, HFSB, large-scale measurements

ABSTRACT

Experiments are conducted that obtain the aerodynamic drag of a sphere towed within a rectangular duct from PIV.

The drag force is obtained invoking the time-average momentum equation within a control volume in a frame of

reference moving with the object. The sphere with 0.1 m diameter is towed at velocity of 1.5 m/s, corresponding to

Re = 10,000. The tomographic PIV measurements are conducted at 500 Hz in a volume of approximately 3 x 40 x 40

cubic centimeters. The large-scale measurement is attained making use of neutrally buoyant Helium-filled soap

bubbles of approximately 0.3 mm diameter as air-flow tracers, preluding to a potential upscale of the technique. The

measured drag depends upon three terms, namely the flow momentum, the pressure and the velocity fluctuations.

These individual terms vary largely at different distances behind the sphere, while the sum attains a relatively

constant value. More than two diameters behind the object the drag only varies by about 1%, yielding a practical

criterion for the drag evaluation of bluff objects with this technique.

1. Introduction

Aerodynamic forces result from the interaction between an object and a fluid in relative motion.

The aerodynamic drag is the component of the resulting aerodynamic force along the direction

of motion. The determination of the aerodynamic drag is relevant for many engineering

applications, e.g. to enhance the performance and reduce the fuel consumption of aircraft and

road vehicles, as well as for more general applications as speed sports and biomechanics for the

study of human and animal locomotion and flight.

Among the different techniques existing to quantify the aerodynamic drag, some directly

measure the force through a force balance connected to the object (Zdravkovich, 1990). Other

methods infer it indirectly, either by measuring the deficit of momentum in the wake of the


object (Selig et al., 2011) or by integrating the distribution of the fluid flow pressure over the

object surface (Neeteson, 2015). The latter approach only considers the pressure drag of an

object, and does not include the friction drag, yielding an underestimate for the total

aerodynamic drag.

Another relevant distinction is based on applications featuring either a stationary object (e.g.

immersed in a wind tunnel) or a moving one (catapult, towing tank). In the first case the model

position and flow conditions can be controlled accurately resulting in well repeatable

measurements. However, in some cases wind tunnel experiments are unsuited, for example due

to the interference of model supports, large models causing blockage effects, vortex development

hindered by wall interactions, impossibility of measuring the flow around an accelerating object.

Moreover a moving floor is needed for accurate ground vehicle aerodynamics. Bouard and

Coutenceau (1980) studied the early stage development of the wake behind an impulsively

started cylinder by towing it through a water channel. A towing approach in a water channel

was also selected to study the wing-tip vortices development of an Airbus A340-300 at large

distances behind the airplane (Scarano et al., 2002). This approach is potentially attractive for

measurements on objects that move autonomously, like transport vehicles, athletes of speed

sports and animals. In the specific case of professional cycling, the drag is routinely estimated

with the mechanical power generated by the rider, which however includes also the friction

force between wheels and ground and other mechanical resistance. Assumptions are required to

extract the aerodynamic resistance from the latter (Grappe et al. 2007). The same holds for the

constant speed torque measurement (Fontaras and Dilara, 2014) and coast down test (Howell et

al., 2002) on cars and trucks.

A general approach for the determination of the drag of transiting objects in air is relevant to the

above mentioned applications. The drag force can be obtained from non-intrusive measurements

of the velocity field in the wake of the object. Phase-locked particle image velocimetry (PIV)

measurements over propeller blades have demonstrated the principle for rotor aerodynamics

(Ragni et al. 2011). More recently, Neeteson et al. (2015) have extended the approach to transiting

objects for the estimation of the drag of a sphere freely falling in water. The latter approach,


however, relies upon the measurement of the surface pressure of a transparent object with index

of refraction matched with that of the medium, making it unsuitable for experiments in air.

An approach to measure the aerodynamic drag of transiting objects is currently missing. The

present work discusses the use of large-scale tomographic PIV for the determination of the

aerodynamic drag of transiting objects. The aerodynamic drag is evaluated by the application of

a control volume approach from the flow velocity measured in the wake of the object. The

approach is demonstrated by estimating the drag of a transiting object of moderate size (10 cm

diameter). Helium filled soap bubbles (HFSB) are used as flow tracers, due to their high light

scattering efficiency and tracing fidelity (Bosbach et al., 2009, and Scarano et al., 2015, among

others). The drag force acting on a moving body is derived from the equations of conservation of

momentum. Similar to Ragni et al. (2011), the time-average aerodynamic drag is expressed in

terms of momentum deficit, fluctuating velocity and time-average pressure. The results are

discussed in terms of these individual components computed at different distances behind the

object. The discussion explains the contribution of the different terms to the aerodynamic drag

and the change of contributions at increasing distances behind the model. Finally, the current

work is intended as a milestone towards larger scale applications for the study of the

aerodynamics of athletes in speed sports.

2. Working principle

Drag from a control volume approach

The integral drag force acting on a body can be derived through the application of the

conservation of momentum in a control volume containing this body (Anderson, 1991), which is

visualized in Fig. 1. For incompressible flow, the time dependent drag acting on the body can be

written as

𝐷(𝑡) = −𝜌 ∰𝜕𝑢

𝜕𝑡𝑑𝑉 − 𝜌 ∯(𝑽 ∙ 𝒏)𝑢

SV

𝑑𝑆 − ∯((𝑝𝒏 − 𝜏 ∙ 𝒏)𝑑𝑆)𝑥S

(1)

where V is the velocity vector, ρ is the density, p pressure and τ the viscous stress. V is the control

volume, with S its outer contour and n is the outward normal vector of S.


Fig. 1: Schematic description of the control volume approach.

The viscous stress is negligible with respect to the other contributions when the control surface is

sufficiently far away from the body surface (Kurtulus et al., 2007). Furthermore, the contour S is

defined as the contour abcd, with segments ab and cd approximating streamlines. When the

segments ab, cd and ad are taken sufficiently far away from the model, the expression of the drag

becomes:

𝐷(𝑡) = −𝜌 ∰𝜕𝑢

𝜕𝑡𝑑V +

V

𝜌 ∬ (𝑈∞ − 𝑢)𝑢Sbc

𝑑S + ∬ (𝑝∞ − 𝑝)Sbc

𝑑S (2)

The evaluation of the volume integral typically poses problems due to limited optical access all

around the object. Evaluating this integral can be avoided by considering the time-average drag

instead of the instantaneous one. When decomposing equation (2) into the Reynolds average

components and averaging both sides of the equation, the time-average drag force is obtained:

�̅� = 𝜌 ∬ (𝑈∞ − �̅�)�̅�Sbc

𝑑S − 𝜌 ∬ 𝑢′2̅̅ ̅̅Sbc

𝑑S + ∬ (𝑝∞ − �̅�)Sbc

𝑑S (3)

where ū is the time-average streamwise velocity and u’ the fluctuating streamwise velocity. This

expression allows us to derive the time-average drag of a model from the velocity and pressure

statistics in a stationary wake volume.

According to the principle of Galilean invariance, Equation (3) holds in any reference frame

moving at constant velocity. Following Ragni et al. (2011), the following expression can be

derived for the time-average drag acting on a model at constant speed UM:


�̅� = 𝜌 ∬ (𝑈∞ − (�̅� − 𝑈𝑀)(�̅� − 𝑈𝑀))Sbc(𝑡)

𝑑S − 𝜌 ∬ 𝑢′2̅̅ ̅̅Sbc(𝑡)

𝑑S + ∬ (𝑝∞ − �̅�)Sbc(𝑡)

𝑑S (4)

where ū is the streamwise velocity measured in the stationary frame of reference and UM is the

constant velocity at which the model moves.

The time-average pressure on the right hand side of the equation is evaluated from the tomo-PIV

data solving the Poisson equation for pressure, according to van Oudheusden (2013).

Appropriate boundary conditions must be prescribed to solve the Poisson equation for pressure.

Finally, this approach allows to derive the time-average drag on a moving body at constant

speed from the velocity statistics in a wake plane. However, instead of a plane, in this work a

thin wake volume is considered for a more accurate reconstruction of the pressure.

3. Experimental apparatus and procedure

Measurement system and conditions

A schematic of the system devised for the experiments is shown in Fig. 2 and a picture taken of

the setup in Fig. 3. The apparatus consists of a duct where a model is towed, 170 cm long with a

squared cross section of 50x50 cm2. The duct is closed in order to confine the seeding particles

within a limited region and has transparent walls to allow access for illumination and imaging

(Fig. 3).

Fig. 2: Schematic views of the experimental setup.


The model is a sphere of 10 cm diameter, which is towed through the tunnel at a constant speed

of 1.5 m/s. The model is supported by an aerodynamic strut, which has a thickness to chord

ratio of 15%, is 3 mm thick, 20 cm long and is mounted on a carriage moving on a rail beneath

the bottom wall of the duct. The carriage is mounted to the shaft of a motor, which is digitally

operated allowing to accurately control the speed of the model. The sphere is marked to track its

position during the transit. During each experiment, the sphere begins its motion at a distance of

about 25 diameters from the entrance to the duct to ensure a fully developed wake flow regime

past the sphere within the measurement domain.

Fig. 3: Overview of the developed system.

Tomographic system

The time-resolved tomo-PIV measurements are conducted using HFSB as tracer particles

(diameter ~ 300 μm). The HFSB are introduced into the tunnel by a rake of ten nozzles that

generate about 30,000 particles per second each. The seeding system (nozzles and a control panel

that regulates the air, helium and soap fluid flow rates) is provided by LaVision GmbH.


The illumination is provided by a Quantronix Darwin Duo Nd:YAG laser (nominal pulse energy

of 25 mJ at 1 KHz). The laser beam is diverged by spherical and cylindrical lenses. The oval cross

section is cut into a rectangular one via light stops (Fig. 3). The size of the measurement volume

is about 3 cm x 35 cm x 35 cm in x, y and z direction, respectively. The imaging system consists

of three Photron Fast CAM SA1 cameras (CMOS, resolution of 1024 x 1024 pixels, pixel pitch of

20 μm, 12 bit). Each camera is equipped with a 60 mm Nikkor lens set to f/8. The magnification

is approximately 0.07. In the present conditions the seeding density is approximately 3

particles/cm3 and 0.04 particles/pixel. PIV acquisition is performed within LaVision Davis 8.3 at

a frequency of 500 Hz.

Measurement procedure

The tunnel entrance and exit are closed to contain the HFSB seeding before the spheroid transits

through the tunnel. The door of the tunnel exit is of porous material to prevent creating over-

pressure in the container. The HFSB nozzles are operated until the concentration within the

measurement volume reaches steady-state conditions (typically two minutes). Then they are

switched off for the medium to become quiescent (typically 30 seconds). The tunnel entrance and

exit walls are opened, the model is put in motion through the tunnel and data is acquired. Fig. 4-

left depicts a sequence of the three raw images, separated by 30 ms, taken during a transit of the

model through the measurement domain. The black silhouette of the sphere is clearly visible

against a background of illuminated particles.


Fig. 4: (left) A sequence of three raw images of the sphere passing the measurement domain separated by a time increment of 30 ms; (right-

top) Illumination distribution along the measurement depth in an area of 25 x 25 cm2; (right-bottom) Illustration of the reconstruction of the

time-average velocity from sequential phase-average velocity fields.

Data reduction

The tomo-PIV data analysis is performed in LaVision Davis 8.3 and consists of standard image

pre-processing and a velocity reconstruction by sequential MTE-MART (Lynch and Scarano,

2014). Volume reconstruction is performed on a discretized domain of 1074 x 1050 x 72 voxels

and correlation volumes of 32 x 32 x 32 voxels with an overlap of 75% are used. This results in a

sequence of instantaneous velocity vector fields with a density of 3 vectors/cm. Fig. 4 (right-top)

presents the averaged distribution of particles over 100 reconstructed objects, showing the

excellent signal-to-noise ratio of the reconstruction.

A Galilean transformation of the instantaneous velocity data is performed in order to reduce the

data in a frame of reference consistent with the moving object. In this frame of reference, phase-


average velocity fields are obtained by averaging the data acquired at different transits of the

model. Afterwards, the phase-average velocity fields at the different time increments are

combined, constructing a time-average velocity field that is elongated in streamwise direction.

This approach is illustrated in Fig. 4 (bottom-right).

To reconstruct time-average pressure from the measured velocity, Dirichlet boundary conditions

are prescribed, which are derived from the isentropic flow relation.

4. Results

Instantaneous flow field

Fig. 5 shows the instantaneous velocity field in the center YZ-plane at four consecutive time

instants. Non-dimensional time is defined as t∗ = t U∞ D⁄ , where D is the diameter of the sphere.

At t* = 0 the back of the sphere is located at x = 0. Each increment in time corresponds to a

translation in space of one sphere diameter in negative x-direction. A peak of negative

streamwise velocity is clearly present in the near wake of the sphere at t* = 0. Furthermore, an

area of accelerated flow is visible at the periphery of the wake. When time evolves, the sphere

moves further along the negative x-direction and the velocity deficit in the measurement region

becomes less pronounced. This is consistent with the characteristics of the mean flow past a

sphere (Jang and Lee, 2008, Constantinescu and Squires, 2003). The wake of the supporting strut

is also visible in the different velocity fields as a negative streamwise velocity below the sphere

(z~0, y<0). Its effect on the drag estimate is discussed in the later sections.


Fig. 5: Instantaneous streamwise velocity u in the YZ-plane at four time instants, t* = 0, t* = 1, t* = 2 and t* = 3 for a

sphere velocity of 1.5 m/s

Time-average flow field

In the next sections the time-average velocity, fluctuating velocity and time-average pressure

fields are presented and compared to literature to understand how the individual terms from

Equation (4) contribute to the aerodynamic drag. Twenty instantaneous velocity fields, obtained

from twenty different passages of the sphere through the measurement volume, are used to

estimate the statistical average. The time-average data is presented in the reference frame of the


sphere in non-dimensional variables. Fig. 6 presents The streamwise velocity distribution in the

central XY-plane (left), the central XZ-plane (middle) and Fig. 7 shows this velocity distribution

in the YZ-plane (right) at x/D = 0.75. The velocity field is rather axis-symmetric, which is to be

expected for the flow past a sphere. The circular shape of the wake is altered only at the bottom

of the measurement domain due to the presence of the supporting strut.

From the streamlines in the wake of the sphere it is observed that the reattachment point is

located at about x/D = 1.3, which is consistent with values from literature. Table 1 lists the

results of four other works: Jang and Lee report a recirculation length, L/D, of 1.05 (Re = 11,000),

Ozgoren et al. (2011) a value of about 1.4 (Re = 10,000) and Yun et al. (2006) and Constantinescu

and Squires (2003) report 1.86 (Re = 10,000) and 2.2 (Re = 10,000), respectively. The wide range of

values can be ascribed to differences in the experimental setup, and, in the case of numerical

simulations, differences in turbulence models.

The maximum reverse flow velocity is about -0.5 occurring at x/D = 0.6 and y/D ~ 0 (Fig. 6-top),

which compares fairly well to the value of -0.4 reported by Constantinescu and Squires. The

location of maximum reverse flow differs from the results of Constantinescu and Squires: x/D =

1.41 and y/D = 0. This, however, is consistent with the larger recirculation length that they

report. In general the location of the maximum reverse flow compares well to that of the other

authors listed in Table 1.

Finally, the recirculation vortices show rather clearly in the central vertical plane (Fig. 6 bottom-

left) and are located at about x/D = 0.75 and y/D = ±0.45. This again is consistent with literature.

The presence of these vortices is less obvious from the horizontal center plane (Fig. 6 bottom-

right), which is most likely caused by a lack of statistical convergence.


Fig. 6: Non-dimensional time-average streamwise velocity in the wake of the sphere in the center XY-plane (left)

and the center XZ-plane (right); contours (top) and streamlines (bottom)


Fig. 7: Non-dimensional time-average streamwise velocity in the wake of the sphere in the YZ-plane at x/D = 0.75;

contours (left) and vectors (right).

Present data and

literature

center of

recirculation

max reverse velocity

(�̅� 𝑈∞⁄ )

max(√𝑢′2̅̅ ̅̅ 𝑈∞⁄ )

Re L/D Position

(x/D, y/D) value

Position

(x/D, y/D) value

Position

(x/D, y/D)

Present work 10,000 1.3 0.75 ±0.45 -0.5 0.85 0 0.35-0.4 1 ±0.4

Jang and Lee (2008)

(PIV) 11,000 1.05 0.75 ±0.25 0.6-0.9 0 0.65* 1.0* ±0.3*

Ozgoren et al. (2011)

(PIV) 10,000 1.4* 0.7* ±0.4*

Constantinescu et al.

(2003) (LES) 10,000 2.2 1.22 0.41 -0.40 1.41 0 0.5 1.78 0.46

Yun et al. (2006)

(LES) 10,000 1.86 0.25* 1.5* 0.45*

Table 1: Comparison between present experimental results and those of Jang and Lee and Ozgoren and numerical results of Constantinescu and Yun. *Value is an estimation from presented figures in literature


Reynolds stress field

Fig. 8 shows the contour plots of streamwise normal Reynolds stress, Rxx, in the center XY-plane

(left) and the center XZ-plane (right). It is observed that the field of the normal Reynolds stress is

rather symmetric in both planes. The distribution in the XZ-plane compares rather well to

literature (Jang and Lee, 2008; Constantinescu et al., 2003; Yun et al., 2006), with maxima around

x/D = 1 and z/D = ±0.4 and the presences of two branches of high local normal Reynolds stress,

diverging from the streamwise axis and decreasing in strength for x/D > 1. The distribution in

the XY-plane shows less similarity, likely due to the disturbance of the supporting strut. The

local maxima of Rxx are between 0.35 and 0.4, which is within the range that is listed in literature

(Table 1). Further statistical convergence of the results most likely allows pinpointing the

location of the peaks of the Reynolds stresses more accurately.

Fig. 8: Contours of the normal Reynold stress in the center XY-plane (left) and the center XZ-(right)

Pressure reconstruction

Fig. 9 depicts the distribution of the mean pressure coefficient in the center plane XY-plane (left)

and the center XZ-plane (right). The general distribution of the pressure coefficient is as

expected: The location of the pressure coefficient minima corresponds to the location of the

recirculation vortices (Fig. 6 bottom-left) and a peak of high CP is observed right after the


reattachment point. The distribution of pressure shows a slight asymmetry in the XY-plane,

which originates from similar asymmetries in the velocity statistics presented in Fig. 6 (left) and

Fig. 8 (left). To the best knowledge of the authors, no reference exists that shows the pressure

field in the wake of a sphere. For a comparison with literature only the base pressure coefficient

can be evaluated, which is about -0.5 in the present work. Yun et al. (2006) and Constantinescu

and Squires (2003) report a value of -0.27 and Bakic et al. (2006) a value of -0.3 at Re = 50,000.

With respect to these values we overestimate the base pressure. This work, however, does not

aim to accurately measure the base pressure. The next section will discuss how the integral

values of time-average velocity and pressure and the velocity fluctuations evolve in the wake of

the sphere. This will provide more insight into the reliability of the reconstructed pressure.

Fig. 9: Distribution of time-average pressure coefficient in the center XZ-plane (left) and the YZ-plane at x/D = 0.75

(right)

Aerodynamic drag

The time-average aerodynamic drag, derived from the velocity statistics measured in the wake

of the model, is expected to be independent of the distance of the measurement plane behind the

sphere. Therefore, the sum of the three right hand side terms from the drag Equation (4) is

expected to yield a constant value, while these terms individually are expected to vary with x/D

based on the results of time-average velocity, velocity fluctuations and time-average pressure


depicted in Fig. 7, Fig. 8 and Fig. 9, respectively. Fig. 10-left presents the drag coefficient

computed in the wake of the model from YZ-planes between x/D = 0.5 and x/D = 3.5 including

the three right hand side terms in the drag equation. As expected, the physical location at which

the drag is computed does hardly affect the resulting drag coefficient value, which confirms the

good quality of the data. The momentum term is strongly negative close to the sphere, with a

peak at x/D = 0.6, and increases quickly afterwards to reach a relatively constant value after

x/D = 2. The negative contribution of the momentum deficit at small x/D is mostly compensated

by the pressure term, which is large close to the sphere. The pressure term is expected to reach

zero value at large distance behind the model. It is observed that, instead, it reaches small

negative values. Measurement errors of the velocity may be the reason for this behavior. These

errors in the velocity are amplified in the velocity gradients, through which they propagate into

the reconstructed pressure, making the pressure relatively sensitive to these measurement errors.

The Reynolds stress term contributes negatively to the drag by definition. A negative peak is

located at x/D = 1, which corresponds to the location of the peaks of Rxx in Fig. 8. As expected,

this term slowly increases towards small values close to zero.

Fig. 10-right presents the drag coefficient in the wake of the sphere together with a range of

values reported in literature. The lower bound of this range is 0.39 reported by Yun et al. (2006)

and Constantinescu and Squires (2003) and the upper bound is 0.44 from Achenbach (1972) (Re =

20,000). The drag coefficient of the present work varies by 0.17, from 0.33 at x/D = 0.95 and 0.5 at

x/D = 0.5. However, both extrema are located close to the sphere. Considering only the region of

x/D > 2, the computed drag is relatively constant and fluctuates between 0.47 at x/D = 2 and

0.48 at x/D = 2.6, which is about 1% of the average of 0.475.

A CD of 0.475 overestimates the values in literature by 8 – 20%. This is, at least partly, caused by

the supporting strut of the sphere. The contours of streamwise velocity in Fig. 6, show the effect

of the strut on the flow in the wake, contributing to an increase in momentum deficit and,

therefore, drag.

Finally, a practical criterion is proposed to measure the aerodynamic drag of a transiting bluff

body by the presented approach: the drag should be measured at least two diameters behind the

model.


Fig. 10: Mean drag coefficient computed at different distances behind the sphere; CD and the individual momentum,

pressure and Re stress term (left), and, the present result relative to the range of values reported in literature (right).

5. Conclusions

A control volume approach is applied to determine the aerodynamic drag of transiting objects

by means of tomo-PIV measurements in the wake of the model. The concept is demonstrated

using a newly developed system to measure the flow over a sphere with a diameter of 10 cm

moving at 1.5 m/s. Velocity statistics in the wake of the sphere have been obtained from a set of

twenty model transits and the obtained time-average velocity, fluctuating velocities and time-

average pressure compare well to literature. The aerodynamic drag is computed as the sum of

these three components at different distances behind the sphere. As expected, the resulting drag

is relatively constant. Relatively large variations of 0.17 are observed at x/D < 2. However,

measuring the drag between x/D = 2 and x/D = 3.5, the results vary only about 1% from an

average of 0.475. This provides a practical criterion to evaluate the drag of bluff bodies with the

presented approach.


Acknowledgements

This work is partly funded by the TU Delft Sports Engineering Institute and the European

Research Council Proof of Concept Grant “Flow Visualization Based Pressure” (no. 665477).

Andrea Rubino is kindly acknowledged for the support on the experiments.

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