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

FLOW STRUCTURE CHARACTERISTICS OF TURBULENT BOUNDARYLAYER MODIFIED BY MINIATURE VORTEX GENERATORS

C. I. Chan
School of Mechanical Engineering
University of Adelaide
South Australia 5005, Australia
chiip.chan@adelaide.edu.au

R. C. Chin
School of Mechanical Engineering
University of Adelaide
South Australia 5005, Australia
rey.chin@adelaide.edu.au

duction (Fransson et al., 2006; Fransson & Talamelli, 2012;
Shahinfar et al., 2012, 2014; Sattarzadeh et al., 2014). However, the same may not be true for flow that is fully turbulent.
Recent experimental studies (Lögdberg, 2006;
Lögdberg et al., 2009) on MVGs were conducted to investigate flow separation controls by vortex generatorsin adverse-and zero-pressure gradient turbulent boundary
layers and it has been shown that the induced high andlow-momentum regions persist for up to 300ℎ (where ℎ is
the device height). It would be worthwhile to pursue a study
on the streamwise evolution and behaviour of the induced
high and low-momentum regions in more detail. In addition,
triple velocity decomposition should be considered when
flow exhibits spatial heterogeneity perpendicular to the flow
direction (e.g.
spanwise periodicity) (Coceal & Belcher,2004). The total velocity fluctuations can be decomposed into
a turbulent and coherent fluctuations. The coherent fluctuationarise due to spanwise periodicity introduced by MVG can betreated in a similar fashion, as demonstrated by Shahinfar et al.
(2013) and recently by Chan & Chin (2022). In the present
study, the velocity fluctuation fields of the low-speed andhigh-momentum regions are first decomposed based on the
triple decomposition. To further investigate their similarityand difference, the proper orthogonal decomposition is thenemployed.

ABSTRACTIn this study we investigate the passive control of a turbulent boundary layer using miniature vortex generator (MVG),
which consists of pairs of winglets or rectangular blades arranged in spanwise oriented arrays in the flow. A well-resolved
large-eddy simulation of rectangular MVGs in a spatially evolving moderate Reynolds number zero pressure gradient turbulent
boundary layer up to 𝑅𝑒 𝜏 = 1350 is performed. Preliminary
results are presented on the different flow characteristics related to the large streamwise extent high and low-momentum
regions (HSR and LSR) introduced by the MVG. We further
employ a triple velocity decomposition to investigate the turbulent boundary layer modifications. The instantaneous velocity
fluctuation is decomposed into a turbulent velocity component and a spatial component, where the latter represents aspatial variation of the time-averaged flow produced by the
MVG. The turbulent velocity fluctuations associated with the
HSR and LSR are further investigated using proper orthogonal
decomposition (POD). The POD analysis aims to decompose
the velocity fluctuation field in terms of a set of basis function, which contains the temporal and spatial information of
the dominant structures in the flow. We further examine basedon this, a reduced-order reconstruction of the HSR and LSR,
using one-dimensional and two-dimensional data.

INTRODUCTION
The miniature vortex generator (MVG) is a passive device capable of generating streamwise orientated vortices
and give rise to long and persistent streamwise streaks thatevolve downstream in boundary layer flows. The MVG
remains a possible solution for effective flow control because of its simplicity and cost effectiveness. The effects
of MVGs upon laminar boundary layers developed on flat
plates have been well explored and understood both experimentally and numerically (Fransson & Talamelli, 2012;
Shahinfar et al., 2012). They are well known to delay flowtransition from laminar to turbulent through stabilizing the
growth of Tollmien–Schlichting (TS) waves and oblique disturbance, thereby resulting in a substantial skin friction re-

NUMERICAL PROCEDURE
In the following, the streamwise, wall-normal and spanwise coordinates are denoted as 𝑥, 𝑦 and 𝑧, and their velocity
components are denoted by 𝑢, 𝑣 and 𝑤, respectively. Timeaveraged velocity is denoted by a (¯·) or a capital letter e.g.
𝑈 = 𝑢, and fluctuation is denoted by a prime (· ′ ). The symbol
h·i denotes a global mean value over the span.
A numerical simulation of miniature vortex generators(MVG) set up in a moderate Reynolds number zero-pressuregradient turbulent boundary layer (TBL) was performed.
This simulation was motivated by recent experimental studies of Lögdberg et al. (2009), Sattarzadeh et al. (2014) andSattarzadeh & Fransson (2015). The MVG configuration is1

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

replacements

𝑡𝑚

𝑧

𝑑
o

ℎ
𝐿
𝛼

𝑥∗ = 0𝑧𝑥∗ = 0Λ𝑧

Figure 1: Schematic of the MVG layouts considered in the present study. The MVG parameters are scaled by the inlet

displacement thickness 𝛿0∗ : ℎ = 4 is the blade height, 𝑡 𝑚 = 1 is the blade width, 𝐿 = 10 is the blade length, 𝛼 = 15◦ is theangle of attack of the MVG with respect to the flow direction, 𝑑 = 10 is the spanwise distance between the centroids of
blades in one pair and Λ𝑧 = 40 is the spanwise spacing between MVG pairs.
−0.2

0𝑢/𝑈∞ , 𝑈/𝑈∞
0.2

0.4

0.6

0.8

1.0

(𝑏)

(𝑎)

4
2
0
−2
−4

𝑧∗

4
2
0
−2
−4

0

𝑥∗

(𝑐)

25

LSR
HSR
LSR
0

(𝑑)

(𝑒)

25

𝑥∗

2
𝑦∗
1
0
−4

−2

0
𝑧∗

2

4 −4

−2

0
𝑧∗

2

4 −4

−2

0
𝑧∗

2

4

Figure 2: (𝑎) Instantaneous realisation of the streamwise velocity flow field, 𝑢/𝑈∞ , at 𝑦 ∗ = 0.25 past a pair of MVG. Topview (𝑏) and cross-section views of the time-averaged streamwise velocity flow field, 𝑈/𝑈∞ , at (𝑐) 𝑥 ∗ = −0.5, (𝑑) 𝑥 ∗ = 0
and (𝑒) 𝑥 ∗ = 0.5, where 𝑥 ∗ = (𝑥 − 𝑥 𝑀 )/ℎ and 𝑧 ∗ = 𝑧/ℎ. Rectangular box outlines the MVG cross section. In (𝑏), the dashedblack lines mark the spanwise locations of high-speed region (HSR) and low-speed region (LSR), with secondary flows ofcommon flow down and common flow up at 𝑥 ∗ = 25, respectively.
shown in figure 1. The MVG array is positioned at 𝑥M = 950𝛿0∗from the inlet (corresponds to 𝑅𝑒 𝜏 = 𝛿+ ≃ 430, where 𝛿0∗ is
the inlet displacement thickness and 𝛿 is the boundary layer
thickness). The MVG parameters that scaled by the inlet displacement thickness are respectively: ℎ = 4 is the blade height,𝑡 𝑚 = 1 is the blade thickness, 𝐿 = 10 is the blade length, 𝛼 = 15◦
is the angle of attack of the MVG with respect to the flow direction, 𝑑 = 10 is the spanwise distance between the centroidsof blades in one pair and Λ 𝑧 = 40 is the spanwise spacing
between MVG pairs. Large-eddy simulation of the turbulent
boundary layer was performed using a fully spectral numericalcode (Chevalier et al., 2007). A sub-grid-scale approximate
deconvolution model (ADM-RT) has been employed to compute approximations to the unfiltered solutions of the incompressible continuity and Navier-Stokes equations by a repeatedfilter operation, i.e.
𝜕 𝑢ˆ 𝑖
= 0,
𝜕𝑥𝑖

(1)

𝜕 𝑢ˆ 𝑖 𝜕 𝑝ˆ1 𝜕 2 𝑢ˆ 𝑖𝜕 𝑢ˆ 𝑖+ 𝑢ˆ 𝑗+−= −𝜒𝐻 𝑁 ⊛ 𝑢ˆ 𝑖 ,𝜕𝑡𝜕𝑥 𝑗 𝜕𝑥𝑖 𝑅𝑒 𝜕𝑥 𝑗 𝜕𝑥 𝑗

(2)

the convolution and the relaxation term −𝜒𝐻 𝑁 ⊛ 𝑢ˆ 𝑖 : 𝜒 isthe model coefficient; and 𝐻 𝑁 ⊛ 𝑢ˆ 𝑖 is the high-pass approximately deconvolved quantities. The ADM-RT model has been
widely used for performing incompressible transitional andturbulent flows simulations (Stolz et al., 2001; Schlatter et al.,
2004; Eitel-Amor et al., 2014; Schlatter et al., 2010). Spatial discretisation is based on a Fourier series with 3/2 zeropadding for de-aliasing in the streamwise (𝑥) and spanwise(𝑧) directions, and a Chebyshev polynomial is employed in
the wall-normal direction (𝑦). The computational domain
in the streamwise, wall-normal and spanwise directions arerespectively: 𝑥 𝐿 × 𝑦 𝐿 × 𝑧 𝐿 = 6000𝛿0∗ × 200𝛿0∗ × 360𝛿0∗ using
6144 × 513 × 768 spectral modes. This give uniform grid spacings of Δ𝑥 + ≈ 16.9 and Δ𝑧 + ≈ 8.1 in the streamwise and spanwise directions (the superscript+ refers to scaling with thepfriction velocity 𝑢 𝜏 = 𝜏𝑤 /𝜌 and kinematic viscosity 𝜈, where𝜏𝑤 is the wall shear stress and 𝜌 is the fluid density). In thewall-normal direction, there is a minimum of 15 Chebyshev
collocation points within the region 𝑦 + < 10. The first grid
point away from the wall is at 𝑦 + ≈ 0.03, and the maximum
spacing is Δ𝑦 +max = 10.6. The time advancement is carriedout by a second-order Crank-Nicolson scheme for the viscousterms and a third-order four-stage Runge-Kutta scheme for the
non-linear terms (Chevalier et al., 2007). Details on the numerical procedure for the MVG can be found in Chan & Chin(2022).

with superscripts ∧ refer to a resolved-scale, and ⊛ denotes212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Global(𝑏)5𝑦 + = ℎ+ /2

20

LSR

(𝑐)

HSR

LSR

𝐷 ≃ 0.15

𝐷 ≃ −0.14

8

4

0.1

-0.00517

50

10−1 100

𝑦+

101

102

103

0

-0.03

-0.0175-0.005

−4

−2

e
𝑢

−0.1

4

2

0.01
75
0.03
0.04
25

425

25
-0.04

-0.0

𝑢′

Global

0.005

3
-0.0

1

LSR

6

+

0

-0.0

2

10

-0

3

(𝑈 − h𝑈i)/𝑈∞

05

.0

𝑦∗

𝑈+

𝑢 ′′

HSR

+

LSR

+

HSR

𝑢 ′′2 , 𝑢 ′2 , e
𝑢2

(𝑎)

0

0.005

0
𝑧∗

2

4

10−1 100

𝑦+

101

102

103

(𝑎) Comparison of the time-averaged streamwise velocity at 𝑥 ∗ = 25. The thick grey lines denote the linear
and log-law regions 1/𝜅 log 𝑦 + + 𝐶 with 𝜅 = 0.41 and 𝐶 = 3.7, 5.2 and 6.2 in the arrow direction. (𝑏) The velocity excess(𝑈 − h𝑈i)/𝑈∞ > 0 associated with the high-speed region and the velocity deficit (𝑈 − h𝑈i)/𝑈∞ < 0 associated with the
low-speed region at 𝑥 ∗ = 25. Inset shows the local spanwise skin friction variation at 𝑥 ∗ = 25. (𝑐) Comparison of thetime-averaged streamwise velocity fluctuations: 𝑢 ′′2 , 𝑢 ′2 and e𝑢2 .

Figure 3:components, which reads as:

Figure 2 shows instantaneous and time-averaged streamwise velocity fields obtained in the streamwise–spanwise planeat 𝑦/ℎ = 𝑦 ∗ = 0.25 (figures 2𝑎 and 2𝑏), and spanwise-wallnormal plane at 𝑥 ∗ = −0.5, 0 and 0.5 of the time-averaged
streamwise velocity field (figures 2𝑐, 𝑑, 𝑒), where 𝑥 ∗ = (𝑥 −𝑥 𝑀 )/ℎ is defined at the centre of a MVG. Spanwise alternatinghigh and low-speed patterns are observed with a high-speed
region (denoted as HSR) formed along the centreline of theMVG, accompanied with low-speed region (LSR) to the sideby-side region.

𝑢 𝑖 (𝑥, 𝑦, 𝑧)𝑢 𝑖 (𝑥, 𝑦, 𝑧, 𝑡) = h𝑈𝑖 i(𝑥, 𝑦) + 𝑢 𝑖′ (𝑥, 𝑦, 𝑧, 𝑡) + e= 𝑈𝑖 (𝑥, 𝑦, 𝑧) + 𝑢 𝑖′ (𝑥, 𝑦, 𝑧, 𝑡),(3)where the 𝑢 𝑖′ and e𝑢 𝑖 on the right-hand side of equation (3)are the turbulent fluctuation and MVG-induced fluctuation, respectively. The MVG-induced fluctuation e𝑢 𝑖 = 𝑈𝑖 − h𝑈i𝑖 is thespatial variation of the time-averaged flow due to MVG. The to𝑢 𝑖 is equal to the turbulent fluctuation
tal fluctuation, 𝑢 𝑖′′ = 𝑢 𝑖′ + e(𝑢 𝑖′ ) for the smooth wall case since e
𝑢 𝑖 = 0. The streamwise total
stress (𝑢 ′′𝑢 ′′ ), Reynolds stress (𝑢 ′𝑢 ′ ) and the MVG-induced
stress (e𝑢e𝑢 ) are presented in figure 3(𝑐) for the HSR, LSR andthe global (overall). Comparison of the MVG-induced stresses(red and blue dotted-dash lines) over the HSR and LSR suggests that the flow modification introduced by the MVGs on the
HSR is quite different from that in the LSR, which also reflect
on the differences between the turbulent stresses (red and blue
dash lines). Further assessments are necessary to characterised
their different flow behaviours.

RESULTS AND DISCUSSIONSLSR and HSR
We present preliminary results of the mean and turbulent
flow statistics of the HSR and LSR, respectively. Figure 3(𝑎)shows differences of the time-averaged streamwise velocity between the HSR and LSR. The velocity defect in the mean flow
profile can be estimated by comparing the downward shift ofthe log-law constants, as shown by the grey lines in figure 3(𝑎).
Vertical solid line denotes the wall-normal location 𝑦 ∗ ≃ 0.5,
which yields Δh𝑈i + = −1.5 at the LSR and Δh𝑈i + = 1.0 at the
HSR, respectively. The velocity defect is known to be related
to the skin friction variation, where locally the skin friction
is modified over the high- and low-speed regions along the
spanwise direction. Low-speed fluid is lifted from the walland reduces the streamwise wall shear stress 𝜏𝑤 and resultsin substantial skin friction drag reduction. On the other hand,downwash motion transports the high-speed fluid towards thewall and increases the skin friction drag at the centre of an MVG
pair. Figure 3(𝑏) shows the velocity perturbation defined as(𝑈 − h𝑈i)/𝑈∞ , indicating the locations of the HSR and LSR
and showing the presence of streamwise roll-modes induced
downstream of the MVG. The inset in figure 3(𝑏) shows the
local spanwise skin friction variation at 𝑥 ∗ = 25, defined as
𝐷 (𝑥, 𝑧 ∗ ) = ( 𝜏¯𝑤 − 𝜏¯𝑤,0 )/𝜏¯𝑤,0 , where 𝜏¯𝑤 = 𝜈 (𝑑 𝑢/𝑑𝑦)|¯
𝑦=0 is thetime-averaged wall shear stress and 𝑧 ∗ = 𝑧/ℎ. The subscript
0 refers to the smooth case. The 𝐷 > 0 denotes the local increase and 𝐷 < 0 denotes the local drag reduction. We observethat HSR is associated with an increase of skin friction up to𝐷 ≃ 0.15 at 𝑧 ∗ = 0, accompanied with a similar skin friction
reduction rate centred at the LSR (𝑧 ∗ ≃ ±3). Finally, to further elucidate the physical modification of the MVG on the
turbulent statistics we employed a similar approach to analyseroughness surface flow by triple decomposition of the velocity

POD ANALYSIS OF LSR AND HSR
The proper orthogonal decomposition (POD) is used toinvestigate the differences in the flow characteristics associatedwith LSR and HSR. In the present study the data obtained
from our numerical simulation the number of spatial points islarger than the number of snapshots. Therefore, the snapshot
POD approach has been used and is briefly described here.
Practically, the turbulent fluctuation was approximated by afinite sum of 𝑁 expansion coefficients 𝑎 𝑖 and spatial modes 𝜙𝑖as:𝑢 ′ (𝑥𝑥 , 𝑡) ≃

𝑁
Õ
𝑖=1

𝜙 𝑖 (𝑥𝑥 ).
𝑎 𝑖 (𝑡)𝜙(4)

The spatial modes 𝜙𝑖 (𝑥𝑥 ) were obtained from solving an eigenvalue problem of the correlation matrix of 𝑢 ′ . The correlation
matrix was calculated between individual snapshots of streamwise velocity flucutuation (𝑢 ′ (𝑥𝑥 , 𝑡 𝑖 ) = 𝑢 ′ (𝑥𝑥 , 𝑡 1 ), 𝑢 ′ (𝑥𝑥 , 𝑡 2 ), ...)
based on their temporal correlations. The correlation (temporal) matrix is given by:

𝑀 = 𝑀𝑖 𝑗 =3

1[𝑢 ′ (𝑥𝑥 , 𝑡 𝑖 ), 𝑢 ′ (𝑥𝑥 , 𝑡 𝑗 )],𝑁𝑡 − 1(5)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
where 𝑁𝑡 denotes the total number of snapshots and [·] defines
a dot product in the present study. The eigenvalue problem tobe solved can be written as:

where 𝑁 𝑅 = 1, 2, ..., denotes the number of modes for reconstruction. Alternately, it is common to reconstruct a low-ordermodel using the Galerkin projection in the literature, where theNavier–Stokes equations are first expressed in terms of POD
basis, and by taking an inner product of resulting equations with
the POD modes, a system of ordinary differential equations forthe 𝑎 𝑖 (𝑡) is obtained (e.g. Rowley et al., 2004). Results showthat the first 10 modes reconstruction is a fairly good approximation of the intensity profile, where they contribute to almost
95% of the total energy (as shown in figure 4(𝑏)), i.e.,(6)𝑀 𝜓 𝑖 = 𝜆𝑖 𝜓 𝑖 .

The eigenvalues might be sorted in a descending order to relate
to the energy content of the corresponding modes. Because wesolve the eigenvalue problem based on temporal correlations,
we need to project all the eigenmodes 𝜓 𝑖 to the velocity field𝑢 ′ (𝑥𝑥 , 𝑡 𝑖 ) so that we can recover the first 𝑁𝑡 spatial POD modesas

𝜙 𝑗 (𝑥𝑥 ) =Í′ 𝑥, 𝑡 )𝑖𝑖 𝜓 𝑗 (𝑡 𝑖 )𝑢 (𝑥p √𝜆 𝑗 𝑁𝑡 − 1,[𝑢 ′𝑁 (𝑥𝑥 , 𝑡 𝑖 ), 𝑢 ′𝑁 (𝑥𝑥 , 𝑡 𝑖 )]𝑅

[𝑢 ′ (𝑥𝑥 , 𝑡 𝑖 ), 𝑢 ′ (𝑥𝑥 , 𝑡 𝑖 )](7)

𝑢 ′ (𝑦, 𝑡) ≃𝑖=1

𝑎 𝑖 (𝑡)𝜙𝑖 (𝑦),

𝑁
Õ
𝑖=1

𝑎 𝑖 (𝑡)𝜙𝑖 (𝑦, 𝑧),

(8)

(9)

1D PODThe first six one-dimensional POD modes of the HSR andLSR are shown in figure 4(𝑎). Results show that the difference in the decompositions between HSR and LSR is quite
small, which might be expected because both HSR and LSR
arise from a pair of symmetrical streamwise vortices generated
by the MVG. Figure 4(𝑏) shows percentage contribution of theeigenvalues of the first 100 one-dimensional POD modes of the
HSR and LSR. The first POD modes of the HSR and LSR contain roughly 35% of the total energy, mainly reside at the nearwall region 𝑦 + ≃ 15, as shown in figure 4(𝑎). Figure 4(𝑎, 𝑏)
suggest that it is difficult to distinguish between HSR and LSR
velocity fluctuations using only one-dimensional POD. Figure
4(𝑐) shows a simple low-order reconstructions of the turbulent
fluctuation in the HSR and LSR, using a reduced number of
POD modes (e.g. Weiss et al., 2022), i.e.,
𝑁𝑅
Õ
𝑖=1

𝜙 𝑖 (𝑥𝑥 ),𝑎 𝑖 (𝑡)𝜙𝑖𝑁Õ𝜆𝑖 / 𝜆𝑖 ,(11)𝑖

Next we consider POD in spanwise and wall-normal directions (9). It is not surprised to observe that the differences
between the flow characteristics in the HSR and LSR becomemuch clearer. The first six POD modes 𝜙𝑖 (𝑦, 𝑧) are presented
in figure 5 ranked according to the eigenvalues. The first POD
mode appear to be anti-correlated in between HSR and LSR,
associating with a symmetrical streamwise vortices pair originating from the MVG. A visual inspection of the spanwise
separation between the same sign of correlation regions implies that 𝜆 𝑧 ≃ Λ 𝑧 /2, reflecting the spanwise wavelength ofthe energy peak at the dominant spanwise mode observed in(Chan & Chin, 2022), even though the first POD mode itself
contributes to only 6% of the total energy. The percentage contribution of the first 100 POD modes is plotted in figure 4(𝑏) in
a comparison with the 1D case. It can be seen that the contribution of the first six modes to the total energy is approximately
25% and up to 80% for the first 100 modes. Figure 6 shows thevelocity reconstruction of the HSR (𝑧 ∗ = 0) and LSR (𝑧 ∗ ≃ −3)
obtained using 2D POD modes. The blue line in figure 6(𝑎)suggests that 𝑢 ′𝑁 =1 ≃ 0. This is attributed to 𝜙1 (𝑦, 𝑧 ∗ = 0) ≃ 0
𝑅
as shown in figure 5 (illustrated as vertical dashed line). On theother hand, the vertical solid line represents the reconstructionof the LSR based on 𝜙1 (𝑦, 𝑧 ∗ = −3), plotted as intensity profile
in figure 6(𝑏) (blue line).

where we further examined the correlation to include the spanwise direction 𝑢 ′ (𝑦, |𝑧 ∗ | ≤ 5, 𝑡).

𝑢 ′ ∼ 𝑢 ′𝑁𝑅 (𝑥𝑥 , 𝑡) =𝑁𝑅
Õ

2D POD

where we treat 𝑢 ′ (𝑦, 𝑧 ∗ = 0, 𝑡) and 𝑢 ′ (𝑦, 𝑧 ∗ = −3, 𝑡) as two independent time series data representing the HSR and LSR,
respectively. We tend to compare the similarities and differences of the POD modes in the wall-normal direction between
HSR and LSR. Next, we will proceed to the two-dimensionalcase, i.e.

𝑢 ′ (𝑦, 𝑧, 𝑡) ≃=and 𝑎 𝑖 (𝑡)𝑎 𝑗 (𝑡) = 𝜆 𝑖 𝛿𝑖 𝑗 . Figure 4(𝑐) also demonstrates how thereconstructed intensity profile vary with increasing number ofmodes, particularly at the near-wall peak 𝑦 + ≃ 15 and outer region 𝑦/𝛿 ≃ 0.2 (𝑦 + ≃ 120), where the latter is clearly associatedwith the first two POD modes. Interestingly, the intensity of theinner peak seems to shift towards 𝑦 + ≃ 15 when higher modes
are used in the reconstruction (marked as ⋄ in figure 4𝑐).

where the spatial POD modes are essentially orthonormal, i.e.
𝜙 𝑖 , 𝜙 𝑗 ] = 𝛿𝑖 𝑗 . Finally, the expansion coefficients 𝑎 𝑖 can be[𝜙obtained by projection of the velocity field on the spatial modes.
In what follows, we will consider two cases. First, POD analysis
results will be presented in figure 4 for a one-dimensional POD
basis case, i.e.
𝑁Õ𝑅

CONCLUSIONSIn this study we focused on the flow behaviour of thelong streamwise extended and spanwise periodic high andlow-momentum regions produced by the MVGs. A largeeddy simulation of rectangular MVGs positioned in a spatiallyevolving moderate Reynolds number zero pressure gradientturbulent boundary layer up to 𝑅𝑒 𝜏 = 1350 was performed(Chan & Chin, 2022). The fluctuating velocity field associatedwith the HSR and LSR are first extracted by a triple velocity decomposition, then we analysed their velocity fluctuations
using proper orthogonal decomposition in the (i) wall-normal
direction and (ii) spanwise and wall-normal directions. A comparison of the POD modes between the 1D and 2D cases showsthat in the 1D case the modes 𝜙𝑖 (𝑦) and eigenvalue ratio associated with the HSR and LSR are similar. This is because

(10)

4

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

HSR ⋄

(𝑏)
100

LSR

Re(𝜙𝑖 (𝑦))

0.2

(𝑐)

(%)

0

−0.1
101
𝑦+

60

𝑢 ′𝑁𝑅 2

40

+

0
100

102

LSR

1, 2, 6, 10, all

4
2

20

𝜙1 , 𝜙2 , 𝜙3 , 𝜙4 , 𝜙5 , 𝜙6
100

HSR ⋄

6

80

0.1

−0.2

Í
Í
Í
𝜆 𝑖 / 1𝑁 𝜆, 𝑖1 𝜆/ 1𝑁 𝜆

0

102

101
𝑖

102

101
𝑦+

100

Figure 4: (𝑎) The first six one-dimensional POD modes 𝜙𝑖 (𝑦) ranked by their eigenvalues. (𝑏) Eigenvalues (black)and cumulative sum of the eigenvalues (red) in 1D case for HSR () and LSR (⋄), respectively. The eigenvalues). (𝑐) Low-order(black) and cumulative sum of the eigenvalues (red) for two-dimensional POD modes are shown in (reconstruction of turbulent intensity using the first 𝑁 𝑅 one-dimensional POD modes: 𝑁 𝑅 = 1, 2, 6, 10 and all (in the arrow
direction).
𝜙2 , 𝜆2 = 4.9%

𝑦∗

6

6

6

4

4

4

2

2

2

0

−4

−2

0

2

4

0

−4

𝜙4 , 𝜆4 = 3.4%𝑦∗𝜙3 , 𝜆 3 = 3.9%

−2

0

4

2

0

𝜙5 , 𝜆 5 = 2.9%
6

6

4

4

4

2

2

2

−4

−2

0
𝑧∗

2

4

0

−4

−2

0

4

2

0

0

−2

4

2

0.04

0

𝜙6 , 𝜆 6 = 2.5%

6

0

−4

𝑧∗

Re(𝜙𝑖 (𝑦, 𝑧))

𝜙1 , 𝜆 1 = 6%

−0.04

−4

−2

0

4

2

𝑧∗

Figure 5: The first six two-dimensional POD modes 𝜙𝑖 (𝑦, 𝑧) with their eigenvalues ranked by percentage contributions.
(𝑎)

(𝑏)

8
𝑁 𝑅 = 1, 2, 6, 10 ... all

6𝑢 ′𝑁𝑅 2

+

8

4

4

2

2

0

100

0

102

101

𝑁 𝑅 = 1, 2, 6, 10 ... all

6

100

102

101
𝑦+

𝑦+

Figure 6: Low-order reconstruction of turbulent intensity using the first 𝑁 𝑅 two-dimensional POD modes: 𝑁 𝑅 =1, 2, 6, 10, 20, 50, 100 and all (in the arrow direction). (𝑎) HSR and (𝑏) LSR.
spanwise correlation was not considered in the 1D case. The
major difference between the HSR and LSR is that when taking
into account of the expansion coefficient, the low-order velocity

reconstruction are found to be quite different, as shown in figure
4(𝑐). Example of time traces of expansion coefficient 𝑎 1 (𝑡) is
plotted in figure 7. On the other hand, for the 2D case, the first5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
1.5
0.75
𝑎 1 (𝑡)

0

−0.75
−1.5
0

200

400
600
𝑡𝑈∞ /𝛿0∗

800

1000

Figure 7: Time traces of expansion coefficient 𝑎 1 (𝑡) for HSR (red) and LSR (blue).
few modes show similarly that alternating spatial patterns with
opposite sign spaced in the spanwise direction, where the HSR
and LSR are localised at the interface between these regions
and indicate that the velocity fluctuations are anti-correlated.

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ACKNOWLEDGEMENTS
This research was undertaken with the assistance of resources provided at the NCI NF through the Computational
Merit Allocation Scheme, supported by the Australian Government and the Pawsey Supercomputing Centre. Funds provided
by the Australian Research Council are gratefully acknowledged.

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