12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

EFFECT OF STREAMWISE DOMAIN SIZE ON THE POD MODE
CHARACTERISTICS IN AN ADVERSE PRESSURE GRADIENTTURBULENT BOUNDARY LAYER

Muhammad Shehzad, Bihai Sun, Daniel Jovic, Callum Atkinson, Julio Soria
Laboratory for Turbulence Research in Aerospace & Combustion (LTRAC),
Department of Mechanical and Aerospace Engineering
Monash University
Clayton, 3800, Victoria, Australia
Muhammad.shehzad@monash.edu, Bihai.Sun@monash.edu, Daniel.Jovic@monash.edu

Yasar Ostovan, Christophe Cuvier, Jean-Marc Foucaut
Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale Lille, UMR 9014
Laboratoire de Mécanique des Fluides de Lille (LMFL) - Kampé de Fériet
Lille, F-59000, France
yasar.ostovan@centralelille.fr, christophe.cuvier@centralelille.fr, jean-marc.foucaut@centralelille.fr

Christian Willert
Institute of Propulsion Technology
German Aerospace Center (DLR)
Cologne, Germany
chris.willert@dlr.de

ABSTRACT

1

Proper orthogonal decomposition (POD) is used to study coherent structures on measurements of wall-bounded turbulent
flows. In order to apply POD in the study of turbulent boundary layers, it is important to determine the appropriate size ofthe flow domain to be used in POD analysis. This study uses
the 2C-2D PIV measurements of an adverse pressure gradient (APG) turbulent boundary layer (TBL) with β ≈ 0 ∼ 3.74
and Reδ2 ≈ 1, 720 ∼ 23, 430 where Reδ2 is the momentum
thickness based Reynolds number and β is the Clauser’s pressure gradient parameter. The measurements were obtained in
the Laboratoire de Mécanique des Fluides de Lille (LMFL)
High-Reynolds-Number (HRN) Boundary Layer Wind Tunnel, Lille, France. Spanning over 20δ along the streamwise
direction (where δ is the boundary layer thickness in the middle of the field of view), these are appropriate to be used in
the study of the extent of the large-scale motions. POD analysis on the variable domain size along the streamwise direction(∆x) suggests that the POD mode shape changes with ∆x. In
this particular TBL, the shape of the first mode which represents the largest scales in the fluid flow, changes until ∆x is
equal to or greater than 8δ . It is also observed the smallerscales of higher-order POD modes take longer ∆x to converge
in the mode shape. The streamwise integral-length-scale of
6.68δ also confirms the size of the largest scales to be around8δ . Therefore, for this particular TBL, a minimum streamwise
domain size of 8δ is appropriate for the analysis of large-scale
motions using POD.

POD has been used as a tool to study the coherent structures
including large-scale motions (LSMs) since its first applicationin fluid dynamics by Lumley (1967). It is a generalization of
the conventionally used Fourier power spectral analysis and isused to investigate the TKE distribution as a function of scale
in a TBL flow that is inhomogeneous in the streamwise direction (Liu et al., 2001). POD has been used by Lengani
et al. (2017) to highlight the lift-up mechanism as well as
the instability of the low-speed streaks that initiate the transition in a boundary layer under a strong APG. Bakewell Jr& Lumley (1967) used POD to study the randomly distributed
counter-rotating eddies as LSMs in a turbulent pipe flow at a
ReD = Ub D/ν = 8, 700, where Ub is the bulk velocity and D
is the pipe diameter. Liu et al. (2001) used POD to evaluate the scales contributing toward the events that produce TKE
and Reynolds shear stress from 2D data of channel flows at
Reh = Ub h/ν = 5, 378 and 29, 935 (where h represents thechannel half-height) and concluded that the LSMs contain alarge fraction of the Reynolds streamwise stress and a small
fraction of Reynolds wall-normal stress. Wu (2014) studied
the intense LSMs in a ZPG-TBL at Reδ2 = 8200 and 12000
and performed POD to establish a connection between the firsttwo dominant POD modes and the instantaneous large-scale
structures. This study concluded that the Reynolds streamwise
stress, Reynolds shear stress and the spatial velocity correlation functions are reduced without the intense LSMs and deduced that the LSMs are significant contributors to the mostenergetic POD mode which is the first mode.

1

Introduction

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
The corresponding spatial mode φi can be computed as

In this paper, a brief description of proper orthogonal decomposition is presented, which is followed by its application
to the streamwise and wall-normal velocity fluctuations of aturbulent boundary layer obtained using 2C-2D PIV. The effectof the streamwise and wall-normal domain size on the properties of POD modes is investigated.
Throughout this paper, the x, y and z are taken as thestreamwise, wall-normal and spanwise directions, respectively. The instantaneous, mean and fluctuating velocities inthe x direction are referred to as u, U and u0 , respectively. Accordingly, the velocities in the y direction are represented byv.

2

1
φi = X ψi p .
λiThis can also be written as

Φ = X Ψ Λ −1/2 ,In this study, the snapshot-POD, which was introduced bySirovich (1987), is used to extract modes based on optimizing
the mean square of the fluctuating velocity. A brief descriptionof the POD method is as follows:
Consider a set of fluctuating velocity fields u 0 (x, y,t) givenbyu0 (x, y,t) = [uu0 (x, y,t1 ) u0 (x, y,t2 ) ... u0 (x, y,tN )] ∈ RM×N ,1
1 N
k = u 0 2 = ∑ λi .
2
2 i=1

M  N,(1)
where N is the number of snapshots, i.e. the velocity fields,
and M is the number of data points in each snapshot, which isequal to the number of velocity components multiplied by the
number of grid points, and3

u 0 (x, y,t) can be written as

N

(2)

i=1

where φi (x, y) is the ith spatial mode and ψi (t) is the set of thecorresponding temporal coefficients. Defining the covariancematrix R of the vector u 0 (x, y,t) as

R = X X T , R ∈ RM×M

(3)

with X = u 0 , the classical POD of Lumley (1967) yields

R φi = λi φi , φi ∈ RM ,(4)

where i = 1, 2, ... , M and λi represents the eigenvalue of the ith
POD mode. In the snapshot-POD method, the matrix X T X isused instead of X X T , which is much smaller in size but yields
the same nonzero eigenvalues (Sirovich, 1987). Hence, we canwrite

X T X ψi = λi ψi , ψi ∈ RN .

(8)

Experimental method

This study uses the 2C-2D PIV measurements of an APG-TBL
that were taken in the x − y plane in the LMFL HRN windtunnel. This facility has a 2-m wide, 1-m high and 20.6-m
long test section. A schematic diagram of the LMFL wind tunnel is shown in figure 1 with three sections that have differentpressure conditions: zero pressure gradient (ZPG), favourablepressure gradient (FPG) and APG. The APG-TBL measurements cover about 21δ in the streamwise direction and from0.02δ to 1.45δ in the wall-normal direction. These measurements were obtained using 16 sCMOS cameras in a 3.466 mlong continuous field of view (FOV) and were used in the characterization of a high-Reynolds-number APG-TBL developing
over the considerably long regions (Cuvier et al., 2017). The
FOVs of the LFOV measurements are also highlighted in figure 1. Since the measurements of the current APG-TBL wereoriginally presented by Cuvier et al. (2017), the reader is referred to that paper for the complete experimental details.
To compare the first- and second-order statistics of thecurrent APG-TBL with a ZPG-TBL, the 2C-2D PIV measurements of the latter taken in the x − y plane of the same experimental facility are also included. The ZPG-TBL measurements cover about 2.61δ in the streamwise direction, and from0.02δ to 3.12δ in the wall-normal direction. The experimental and PIV analysis parameters of the current APG-TBL arepresented in table 1.
The current measurements of both TBLs do not cover theregion from the wall to the end of the buffer layer. Therefore, the inner-layer measurements of two TBLs obtained in
the same location of the same facility as the current TBLs, havealso been included in the comparison of the first and secondorder statistics. These inner-layer measurements have a higher
spatial resolution (HSR) in the wall-normal direction and weretaken at a time different to the outer-layer measurements. The
HSR measurements of the ZPG-TBL and the APG-TBL, whencompared to outer-layer measurements, are approximately 5
times more spatially resolved in the streamwise direction andmore than 14 times in the wall-normal direction, respectively.

 0
u (x, y,t).
v0 (x, y,t)u 0 (x, y,t) = ∑ ψi (t)φi (x, y),(7)wherethe columns of Φ are the vectors of the spatial modesΦ = φ1 φ2 ... φN ∈ RM×N ), the columns of Ψ are the vec(Φtors oftemporal coefficients
corresponding to each POD mode
Ψ = ψ1 ψ2 ... ψN ∈ RN×N ) and Λ is the vector(Ψ of eigenvalΛ = λ1 λ2 ... λN ∈ues corresponding to each POD mode (ΛRN ).
The TKE of the boundary layer flow equals to half of thesum of the eigenvalues, i.e.

Proper orthogonal decomposition (POD)

u 0 (x, y,t) =

(6)

(5)

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

ZPG

U∞
y

width = 3.5m

width = 2.2m

Large FOV
x

APG

FPG
Large FOVx = 6.8m

Figure 1.

s=0

s = 5.6m

Schematic of the test section in the LML Wind Tunnel. Figure adapted from Cuvier et al. (2017).

Table 1. PIV analysis parameters. Source for the outer-layer
measurements of the APG-TBL: Cuvier et al. (2017).
Measurement

ZPG-TBL

Reynolds stress profiles are also scaled with outer variables
and shown in figure 2(b). As it is clear from these figures, theouter-layer measurements are consistent with the inner-layer
HSR measurements. The ZPG-TBL has one inner peak in
Reynolds streamwise stresses and no outer peak. The APGTBL has an outer peak located around y = 1.3δ1 that is asstrong as the inner peak. The Reynolds wall-normal and shear
stresses in the outer region have outer peaks in the APG-TBL
and plateaus in the ZPG-TBL.

APG-TBL

Inflow Velocity (m/s)

99Viscous length scale l + (µm)

4244

FOV (l + × l + )6, 342 × 7, 218

78, 847 × 5, 801(pixels × pixels)

2, 338 × 2, 786

32, 494 × 2, 391(mm × mm)

266 × 317

3, 466 × 254

Grid spacing (l + × l + )25 × 25

24 × 24(pixels × pixels)

7×7

10 × 10

IW size (l + × l + )57 × 57

58 × 58(pixels × pixels)

16 × 16

24 × 24

Frequency (Hz)

5

4

1.0

U/Ue

0.8

Number of samples

10,00030,000

Vector field size

334 × 398

3, 250 × 238

ZPG
ZPG (HSR)
APG
APG (HSR)

0.6
0.4

(a)

0.2
0.0
10−2

Edge velocity, Ue (m/s)

APG-TBL

9.64

11.90

Boundary layer thickness, δ (mm)

102

165

Displacement thickness, δ1 (mm)

16.4

26.9

Momentum thickness, δ2 (mm)

12.0

18.75

Shape factor, H

1.37

1.43

Momentum thickness based Reynolds number, Reδ2

7,750

14,830-2.11

Clauser’s pressure gradient parameter, β

100

101

u0iu0j /Ue2

0.008

Table 2. The boundary layer parameters at the middle of the
FOVs of the ZPG- and APG-TBLs.
ZPG-TBL

10−1
y/δ1

0.010

For the complete experimental details of the inner-layer (HSR)
measurements, the reader is referred to Shehzad et al. (2021).
The boundary layer parameters for both TBLs are adopted
from Cuvier et al. (2017) and presented in table 2.

u0 u0

0.006
0.004

(b)
v 0v 00.002
0.000−0.002

u0 v 0

10

−2

−1

10
y/δ1

10

0

101

Figure 2. The outer-scaled profiles of the (a) mean streamwise velocity and (b) the Reynolds stresses.

5

4

HSR

-5o

HSR

Effect of streamwise domain size on POD
modes

Among many of its applications, POD can be used to separate
the snapshots carrying intense LSMs from an overall ensembleof snapshots. It can also be used to separate the large-scale motions from the small-scale motions based on a threshold valueof the cumulative turbulent kinetic energy (CTKE). Before it isapplied to a turbulent boundary layer dataset, it is worthwhile

First- and second-order statistics

The mean velocity profiles of both TBLs scaled with the outervariables δ1 and Ue are shown in figure 2(a). Likewise, the312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

uφi
0.75

i=1

0.25
i=2

0.50
i=3

1.00
i=4

1.25
i=5

−5 0 5(x − xo )/δ

−5 0 5(x − xo )/δ

−5 0 5(x − xo )/δ

−5 0 5(x − xo )/δ

−5 0 5(x − xo )/δ

2δ

1.0
0.5

4δ

1.0
0.5

6δ

1.0
0.5

8δ

1.0
0.5

10 δ

1.0
0.5

12 δ

1.0
0.5

14 δ

1.0
0.5

16 δ

1.0
0.5

18 δ

1.0
0.5
1.0

20 δ

y/δ

y/δ

y/δ

y/δ

y/δ

y/δ

y/δ

y/δ

y/δ

y/δ

0.00

0.5

Figure 3. The contours of the first five POD modes as a function of ∆x that varies from 2δ to 20δ with increments of 2δ . Each mode
has been normalized with the peak value in the first mode for ∆x = 2δ . For ∆x ≥ 8δ , the iso-contour at the correlation level of C = 0.32
is highlighted with a dashed black line and its centroid with a red dot.

to investigate the effect of the domain size of the velocity fluctuations data in the streamwise direction, on the properties of
the POD modes and the result of any of its applications, forexample, the conditional statistics of the intense large-scale
motions.

have ∆x of about 21δ , these are suitable for this particularanalysis. Since the large-scale motions are dominant in theouter region (Hambleton et al., 2006), the wall-normal rangeof 0.02δ to 1.45δ of the APG-TBL measurements is also appropriate for this application.

To perform this analysis, one needs a dataset that has asufficiently long streamwise range to contain the longest of thecoherent structures reported in the previous studies. Hutchins& Marusic (2007) report that the large-scale structures areelongated in the streamwise direction with lengths of up to
20δ . Since the outer-layer measurements of the APG-TBL

Figure 3 shows the contours of the first five POD modes
as a function of the streamwise domain size ∆x which varies
from 2δ to 20δ with increments of 2δ . These modes have beennormalized with the maximum value of the first POD modefor ∆x = 2δ . The ∆x of 2δ to 6δ do not cover the full modeshape in the first mode for the iso-contour levels C of 0.2 to4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 20220∆θ o (%)∆Area (%)

3
−2
−4

2

1

−6

0

−8

0.00300.25
0.0025∆Cy /∆xmax∆Cx /∆xmax

0.20
0.15
0.10
0.05

0.00200.00150.00100.00050.00

0.0000
8

10

12

14

16

18

20

∆x

8

×δ

10

12

14
∆x

16

18

20

×δ

Figure 4. Properties of the iso-contour at C = 0.32 in the first POD mode as a function of ∆x. The solid lines represent the quadratic
curve fits.
all five POD modes are not converged. One can also observethat the higher the mode number, the longer ∆x it takes to converge the mode shape. This means that the smaller scales of
the higher mode numbers need a larger ∆x to converge. For∆x ≥ 8δ , the shapes of the second to the fifth POD modes
do not change much qualitatively but are expanded a little bitin the x direction as ∆x increases. Hence, it can be assumed
that the POD mode shapes have almost converged at ∆x = 8δ .
A larger ∆x would not change the scales size much but any
conditional averaging of the turbulent statistics on the velocity
fluctuations which are reconstructed using the POD modes orthe conditional averaging based on the temporal coefficientsof the first mode may be affected by the considerable rightwards shifting of the centroid of an iso-contour in the first POD
mode.

y/δ

1.0

0.5

0.0
−4

−3

−2

−1

0
(x − xo)/δ

1

2

3

4

Figure 5. Two-point correlation of the streamwise velocity
fluctuations ρuu at the streamwise middle of the large-fieldmeasurements of the APG-TBL. This streamwise location corresponds to β = 2.01. The iso-contour levels vary from 0.15
to 0.95 with increments of 0.1. In the wall-normal direction,
the correlation is centred at y = δ1 .

The first POD modes represent the LSMs, which populate
the outer region of the turbulent boundary layer (Vila et al.,2017). Since the ∆x ≥ 8δ contain the whole structure for the
first POD mode for the iso-contour levels of 0.2 to 1.0, ∆x = 8δis assumed to be an appropriate domain size for the subsequent
POD analysis on the APG-TBL dataset. It also indicates thatthe size of the large-scale structures in the APG-TBL is up to
8δ . The two-point correlation of the streamwise velocity fluctuations ρuu calculates the extent to which the scales are correlated in the streamwise direction, which is representative of
the streamwise size of the large-scale motions. ρuu is definedas1.0. ∆x = 8δ contains the whole structure for the iso-contour
levels of 0.2 to 1.0, and any larger ∆x does not change themode shape but shifts it rightwards. For ∆x ≥ 8δ , C = 0.32
is highlighted with a dashed black line and its centroid with ared dot. In this range, the mode shape does not change but thecentroid of C = 0.32 is shifted rightwards with an increase in∆x.
The geometrical properties of the first POD mode as afunction of ∆x are presented in figure 4. When ∆x increases
from 8δ to 20δ , the area of C = 0.32 decreases quadraticallyby almost 8% whereas its inclination angle increases by approximately 3.5%. The x-coordinate of the centroid is shiftedrightwards by approximately 25% of the maximum streamwise
domain size (∆xmax ) and the y-coordinate is shifted upwards byapproximately 3% of ∆xmax . The changes in the inclination angle and both coordinates of the centroid are weakly quadratic.
For the streamwise domain size of ∆x < 8δ , the shapes ofu0 (x, y)u0 (x + ∆x, y + ∆y)ρuu = q.
u0 2 (x, y)u0 2 (x + ∆x, y + ∆y)5(9)12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
take longer ∆x to converge in the mode shape. For ∆x ≥ 8δ ,the area and the angle of an iso-contour at C = 0.32 in the firstmode change only by approximately 8% and 3%, respectively,
for a 2.5 times increase in ∆x but its centroid is shifted rightwards by 25% of the maximum domain size. Hence, it is concluded that the size of the large-scale motions in this particular
TBL is up to 8δ , which is also confirmed by its streamwise
integral-length-scale of 6.68δ . It is also observed that an expanding flow domain shifts the energy contained in the lowerorder POD modes to the higher-order modes. Based on thisanalysis, it is concluded and recommended that whenever andwherever possible, it is worthwhile to perform the POD analysis on variable streamwise domain size to find the appropriate
size of the flow domain that covers the largest of the scales
therein.

Figure 5 shows iso-contours of the two-point correlationof the streamwise velocity fluctuations, centred at the middle ofR the x domain and y = δ1 . The integral-length-scale∞
luu = −∞
ρuu dx is calculated to be 6.68δ . This further validates the argument that the size of large-scale motions in thecurrent APG-TBL is up to 8δ . This is not consistent with thestreamwise extent of LSMs reported by Hutchins & Marusic(2007) in a ZPG-TBL which is up to 20δ . Although the current TBL is under an APG, most of its properties are considerably close to the ZPG-TBL because of the low β values. Notethat Hutchins & Marusic (2007) used Taylor’s frozen hypothesis and a convection velocity based on the local mean velocity,on the single-point measurements of a hot-wire rake to reconstruct the velocity signal along the x axis, which may haveaffected the true indication of the length of the large-scale motions.

k=1 (λk )

PN

∆x
10−3
10−4

10−1

Pi

k=1 (λk )/PN

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and adjacent wall region in turbulent pipe flow. The Physics
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R et al. 2017 Extensive characterisation of a high Reynolds
number decelerating boundary layer using advanced opticalmetrology. Journal of Turbulence 18 (10), 929–972.
Hambleton, WT, Hutchins, N & Marusic, Ivan 2006 Simultaneous orthogonal-plane particle image velocimetry measurements in a turbulent boundary layer .
Hutchins, N & Marusic, Ivan 2007 Evidence of very longmeandering features in the logarithmic region of turbulent
boundary layers. Journal of Fluid Mechanics 579, 1–28.
Lengani, Davide, Simoni, Daniele, Ubaldi, Marina, Zunino,
Pietro & Bertini, Francesco 2017 Experimental study offree-stream turbulence induced transition in an adverse
pressure gradient. Experimental Thermal and Fluid Science84, 18–27.
Liu, Z, Adrian, RJ & Hanratty, TJ 2001 Large-scale modes ofturbulent channel flow: transport and structure. Journal of
Fluid Mechanics 448, 53.
Lumley, John Leask 1967 The structure of inhomogeneous turbulent flows. Atmospheric turbulence and radio wave propagation .
Shehzad, Muhammad, Sun, Bihai, Jovic, Daniel, Ostovan,
Yasar, Cuvier, Christophe, Foucaut, Jean-Marc, Willert,
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Sirovich, Lawrence 1987 Turbulence and the dynamics of coherent structures. parts I-III. Quarterly of applied mathematics XLV, 561–590.
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∆x

10−2

λi /

REFERENCES

100

10−1

10−5

∆x
10

0

10

1

10
i

2

10

3

104

Figure 6. Effect of the streamwise domain size on POD
eigenvalues. The domain size ∆x is increasing from 8δ to 20δ
in the directions of the arrows. The empty circles represent∆x = 20δ and the solid lines represent the other cases.

Figure 6 presents the relative contributions (black profiles) and the cumulative relative contributions (red profiles) of
the POD modes to the total turbulent kinetic energy as a function of ∆x. When ∆x increases from 2δ to 20δ , the energy content of the first POD mode decreases from 18.22% to 4.28%
and of the second mode from 6.77% to 3.51%. Oppositely, forthe highest mode number, the energy content increases with anincrease in ∆x. The CTKE of the POD modes also decreases
with an increase in ∆x. This decrease is faster for the higher
values of ∆x and slower for the lower ∆x. This shows that withan increase in ∆x, the energy of the POD modes is shifted fromthe lower-order modes to the higher-order ones.

6

Conclusion

A change in the streamwise domain size of the measurementsof a TBL affects the properties of its POD modes. For the particular case of the present APG-TBL, the shapes of the first
five POD modes are not converged for a variable streamwise
domain size until it is equal to or greater than 8δ . It is also
observed that the smaller scales of higher-order POD modes

6