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

Analysis of the Contribution of Large Scale Motions to
the Skin Friction of a Zero-Pressure-Gradient

Turbulent Boundary Layer Using the Renard-Deck
Decomposition

B. Sun1∗, M. Shehzad1, D. Jovic1, C. Cuvier2, C. Willert3, Y. Ostovan2,
JM. Foucaut2, C. Atkinson1, J. Soria1

1 Laboratory for Turbulence Research in Aerospace & Combustion (LTRAC), Department of Mechanical
and Aerospace Engineering, Monash University, Victoria 3800, Australia

2 Université Lille Nord de France, LMFL - Laboratoire de Mécanique des Fluides de Lille, FRE CNRS
2017, Centrale Lille, ENSAM, ONERA, Villeneuve d’Ascq, France

3 Institute of Propulsion Technology, German Aerospace Center (DLR), Cologne, Germany
∗ bihai.sun@monash.edu

Abstract
Coherent flow structures in turbulent boundary layers have been an active field of research for many decades,
as they might be the key to reveal the mechanics of turbulence production and transport in turbulent shear
flows. Renard and Deck (2016) proposed a theoretical decomposition for the mean skin-friction coefficient
based on the mean kinetic energy budget in the streamwise direction. This decomposition, referred to as
the Renard-Deck (RD) decomposition, decomposes the mean skin friction generation into three physical
mechanisms in an absolute reference frame, namely, direct viscous dissipation, turbulent kinetic energy
production, and spatial growth. In this study, the large scale motions (LSMs) are extracted using a proper
orthogonal decomposition (POD) of the velocity field based on high-spatial-resolution two-dimensional –
two-component particle image velocimetry (HSR 2C-2D PIV) of a zero-pressure-gradient turbulent bound-
ary layer (ZPG-TBL), and their effect on the skin friction via RD decomposition.

The high Reynolds number turbulent boundary layer experiment was performed at the Laboratoire de
Mécanique des Fluides de Lille (LMFL) in the LMFL High Reynolds Number Boundary Layer Wind Tun-
nel. The measurement took place in the streamwise – wall-normal plane along the centreline of the wind
tunnel, where the ZPG-TBL at the measurement location has a free-stream velocity of 9m/s, a Reynolds
number based on the momentum thickness of Reθ = 8,120, a boundary layer thickness of δ ≈ 103mm
and a viscous length of l+ ≈ 40µm. The PIV images were captured with a 47MP Imperx Tiger T8810
camera which has a 57mm-diagonal array of size 8,864× 5,288 pixels. This results in a field of view of
255mm×152mm, or 2.46δ×1.45δ in terms of the boundary layer thickness, with a spatial resolution of 21.7
wall units in the streamwise direction and 5.42 wall units in the wall-normal direction. A more detailed de-
scription of the experiment, as well as an analysis on the method to correct for the lens distortion introduced
by the large sensor size, can be found in Sun et al. (2021).

The LSMs present in the ZPG-TBL are extracted from the velocity field using POD by the snapshot
method (Sirovich, 1987). Due to the limited field of view of the acquired data, the LSMs in the present
study are defined as the flow structures that have a wall-normal extent of approximately the full boundary
layer, which are identified by the most energetic POD mode as shown in figure 1a. This mode accounts
for 16% of the turbulent kinetic energy in the flow. Velocity field snapshots containing stronger LSMs than
others are identified using the time coefficient of the most energetic POD mode, φ1, whose distribution
is presented in figure 1b. The distribution of φ1 is Gaussian-like, with its absolute value representing the
strength of the LSM present in each snapshot. Therefore, the following criterion is used to determine the
snapshots containing strong LSMs,

|φ1,n| ≥ Kσφ1 , (1)



where φ1,n represents the time coefficient of the most energetic POD mode for snapshot n, and σφ1 represents
the standard deviation of the time coefficient for all snapshots. A higher K value represents a tighter selection
criterion.

0 1 2
x/

0

1

y
/

(a)

200 0 200
Time coeffcients

0

500

1000

1500

N
u
m

b
e
r 

o
f 

sn
a
p

sh
o
ts

(b)

all 0.5 1.0 1.5 2.0 2.5
K

0.3

0.4

0.5

0.6

S
tr

e
a
m

w
is

e
 

a
v
e
ra

g
e
d

 C
f b
/C

f

(c)

Figure 1: (a) A vector plot of the fluctuating velocity of the most energetic POD mode. The colours of
the vectors represent the velocity fluctuation magnitude. The red dashed line represents the boundary layer
thickness. (b) Distribution of the time coefficient of the most energetic POD mode, φ1. The red dashed line
represents ±2.5σ, ±2σ, ±1.5σ, ±σ, ±0.5σ and the mean value of φ1. (c) Streamwise averaged C fb/C f for
different K values. The error bars in C fb/C f represents its standard deviation along the streamwise direction.

Renard and Deck (2016) decomposition (RD decomposition) of the skin friction coefficient based on the
mean kinetic energy budget of the fluid motion in an absolute frame of reference, which is given by

C f =
2

U3
e

∫
∞

0
µ
(

∂U
∂y

)2

dy︸ ︷︷ ︸
C fa

+
2

U3
e

∫
∞

0
−〈u′v′〉∂U

∂y
dy︸ ︷︷ ︸

C fb

+
2

U3
e

∫
∞

0
(U−Ue)

∂

∂y

(
τ

ρ

)
dy︸ ︷︷ ︸

C fc

,
(2)

where τ = µ(∂U/∂y)−〈u′v′〉, U denotes the mean streamwise velocity, Ue denotes the free stream velocity,
〈u′v′〉 denotes the Reynolds shear stress, µ denotes the dynamic viscosity and ρ denotes the density. The
RD decomposition decomposes the skin friction into physically interpretable terms that are local at each
streamwise position. The first term, C fa represents the viscous dissipation of the mean streamwise kinetic
energy, and it depends only on the mean values of the fluid field therefore has no direct contribution from
the LSMs. The second term, C fb , represents the production of the turbulent kinetic energy extracted from the
mean streamwise kinetic energy and is the term of relevance in the analysis of the contribution of LSMs. The
last term, C fc , accounts for the growth of the boundary layer effect, as the growth of the boundary layer is
slow for a ZPG-TBL, this term is negligible. A conditional statistics analysis is performed by calculating the
C fb term using the snapshots containing strong LSMs as identified by equation 1. The resulting streamwise
averaged value of C fb/C f for different threshold value K is presented in figure 1c. This results show that
LSM dominated snapshots results in an increase of C fb , indicating that the LSMs are a significant contributor
to the turbulent production term in the RD decomposition of the skin friction.

Acknowledgements
This research was supported by the Australian Government through the Australian Research Council’s Dis-
covery Projects funding scheme. The research was also benefited from computational resources provided
by the Pawsey Supercomputing Centre and through NCMAS, supported by the Australian Government.
The computational facilities supporting this project included the NCI Facility, the partner share of the NCI
facility provided by Monash University through an ARC LIEF grant and the Multi-modal Australian Sci-
enceS Imaging and Visualisation Environment (MASSIVE). Daniel Jovic and Bihai Sun are supported by a
Research Training Program (RTP) scholarship provided by the Australian government.

References
Renard N and Deck S (2016) A theoretical decomposition of mean skin friction generation into physical

phenomena across the boundary layer. Journal of Fluid Mechanics 790:339–367

Sirovich L (1987) Turbulence and the dynamics of coherent structures. I. Coherent structures. Quarterly of
Applied Mathematics 45:561–571

Sun B, Shehzad M, Jovic D, Cuvier C, Willert C, Ostovan Y, Foucaut JM, Atkinson C, and Soria J (2021)
Distortion correction of two-component – two-dimensional PIV using a large imaging sensor with appli-
cation to measurements of a turbulent boundary layer flow at Reτ = 2,386. Experiment in Fluids