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

Near Wake Coherent Structures of a Turbulent Axisymmetric Wake

Taihang Zhu
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
taihang.zhu17@imperial.ac.uk

Georgios Rigas
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
g.rigas@imperial.ac.uk

Jonathan Morrison
Department of Aeronautics
Imperial College London
London SW7 2AZ, UK
j.morrison@imperial.ac.uk

average. Here, the mean pressure variation near the base edge
is kept within 0.2 Pa. To measure the base pressure, 64
pressure tappings are distributed uniformly on the base (shownon the lower side of figure 1). They are connected to the Chell
ESP-DTC pressure scanner to measure the pressure signal
with 0.041 Pa resolution. A Pitot tube is mounted at (2D, 1.3D, 0) to measure the freestream velocity and the reference
static pressure.

ABSTRACTThe coherent structures of a turbulent axisymmetric bluff
body wake are analysed using proper orthogonal
decomposition (POD) and conditional POD based onsynchronized near-wake velocity and base pressure
measurements. The analysis confirms the persistence of thelaminar unstable eigenmodes at high-Reynolds numbers (here𝑅𝑒# = 1.88 × 10* ). These correspond to a steady spatialsymmetry breaking mode and asymmetric unsteady vortex
shedding modes in the near wake. Additionally, an unsteadyaxisymmetric bubble pumping mode representing thestreamwise pulsation of the wake is found in the turbulent
wake. The asymmetric modes are extracted in the wake of theaxisymmetric body by performing conditional POD. The
asymmetry of the wake, quantified using the centre of pressure(CoP), is correlated to the base pressure using conditional
averaging. At the limits of the highly asymmetric wake or aperfectly symmetric wake, a high average pressure coefficient(low drag) is obtained. For the most probable state of thewake, a low-pressure coefficient (high drag) is obtained.
The above results provide insight on the coherent
structures for canonical 3D bluff body geometries andguidance for drag reduction strategies.

(a)

U∞, p∞ y

PIV field of view

D = 196.5

y

z

250.7

400.1

64 Pressure tapsx
Unit: mm(b)

EXPERIMENTAL SETUP
Figure 1. (a) A schematic of the experimental setup. The basepressure taps and the instantaneous base pressure distribution
are shown on the lower side. (b) A 3D view of the model with
the instantaneous PIV flow field and the synchronized basepressure measurement.

The experiment was conducted in the T2 wind tunnel inthe Department of Aeronautics, Imperial College London. The
test section is 1.15 m high, 1.61 m wide, and 4 m long, andequipped with a high-precision three-axis traverse. The inlet
velocity is 𝑈, = 15 m/s and the Reynolds number based on
the diameter of the bluff body model (D = 0.1965 m) is 𝑅𝑒# =1.88 × 10* . The turbulence intensity is less than 1% (10 kHz
lowpass filtered) and the mean velocity variation on the crosssection of the wind tunnel is less than 1%. Figure 1 presents aschematic of the experimental setup and a 3D view of themodel. A bullet-shaped axisymmetric body with a blunt
trailing edge is located on the centre-line of the wind tunnel.
The length-to-diameter ratio is L/D = 6.48, identical to the one
studied in Oxlade et al. (2015) and Rigas et al. (2014). The
model is carefully aligned with the inlet flow to achieve
axisymmetric base pressure distribution after a long-timeTime-resolved two-dimensional two-component PIV is
used to measure the near wake velocity fields of theaxisymmetric body. The field of view (FOV) and an exampleof the instantaneous velocity magnitude field are shown infigure 1(a). The FOV covers the near-wake area with a 400.08
mm x 250.68 mm rectangular plane pass through the bodyaxis. A dual-port Litron LDY300 laser is used to provide alaser sheet for visualizing the seeding particles. A Phantom
VEO 640L camera (with a resolution of 2560 x 1600 pixels),
combined with a Nikkor 135 lens, is used to capture the FOV1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
images with a sampling frequency of 1440 Hz. LaVision
DaVis 10 software is used to perform cross-correlation of theimage pairs, producing the vector fields with a sampling
frequency of 720 Hz. The velocity field is recorded on a 164
x 187 uniform grid, which gives a vector resolution of 2.53
mm. The velocity and pressure data are stored in 13 databases.

Each database consists of 5990 velocity snapshots separated
by ∆𝑈,/𝐷 = 0.106 (720 Hz sampling frequency). Eachdatabase is 8.32 s long giving a total sampling time of 108 s.
RESULTS-0.12 -0.1 -0.08 -0.06

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Figure 2. (a) The instantaneous and (b) the time-averaged PIV velocity field and the synchronized base pressure distribution. The left
column shows the 2D base pressure distribution, with the black dashed line showing the base edge, and the red dashed line showing
the PIV plane position. The middle column shows the base pressure at the PIV plane position. The right column shows the PIV velocity
magnitude.

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5 5
6666666
Figure 3. The second order properties of the wake: (a) Reynolds stress 666666
𝑢45 𝑢45; (b) Reynolds stress 6666666
𝑢75 𝑢57; (c) Reynolds stress 𝑢
4 𝑢7 ; (d)the RMS value of the base pressure fluctuating component. (a), (b), and (c) shows the PIV plane, and (d) shows the base.

The observability of coherent structures is limited by the
turbulent nature of high-Reynolds number wake. As is shownin figure 2(a), the turbulent wake is dominated by separationin the vicinity of the trailing edge. A turbulent shear layer isformed due to the high velocity gradient between the inlet flow
and the wake. The base pressure also fluctuates with the wakemotion, in which a wide range of coherent structures is visible.
A time-averaged velocity field and base pressure (figure 2b)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022show that after time-averaging, both the velocity field and thebase pressure distribution become symmetric. The second
order properties of the wake (the Reynolds stress and the rms
of the base pressure fluctuating component) are shown infigure 3. These are also symmetric. Therefore, the database islong enough to guarantee convergence, and the alignment of
the body with the inlet flow is satisfactory.

velocity fields, after which the power spectral density (PSD)
is used to analyse the time series of the mode amplitude (figure4). The energy ratio and normalized cumulative energy of each
POD mode are shown on figure 4 (a) and 4 (b), respectively.
They show that the first POD mode takes up to 21% of thetotal energy, which is three times as high as the second POD
mode (7%). The first four POD modes contribute about 40%of the total energy and represent the larger wake structures.
The premultiplied spectral density is used to analyse thetime series of the mode amplitude (figure 4 c). There are
multiple peaks near three distinct frequencies in the spectrum
and show low-rank dynamics. The first peak is close to zeroand the POD mode is of large scale, the shape resembling theunstable steady eigenmodes found in the laminar linear
stability analysis (LSA) (Rigas et al. 2016). This mode is a
steady spatial-symmetry-break mode with 𝑆𝑡# ~ 0 andrepresents the stochastic switching of the wake between what
appears to be a pair of quasi-steady states. A similar POD
mode was also identified in a rectangular-sectioned bluff bodyby Fan et al. (2020) and Pavia et al. (2019) using timeresolved stereoscopic PIV. The peak close to 𝑆𝑡# = 0.06 is the
bubble-pumping mode. Mode 2 on the upper side has the same
sign as that on the lower side, thus this mode is axisymmetricand represents the wake pulsation in the streamwise direction.
Near the vortex shedding frequency (𝑆𝑡# = 0.2) there are threepeaks, all of which are antisymmetric modes. The mode 2 andmode 4 are a pair of downstream travelling modes,
representing the main coherent structures of the vortex
shedding. These modes also resemble the unstable, unsteadymodes found in the LSA. Since mode 1 represents thestochastic switching of the vortex shedding plane, there issome projection of mode 1 on the vortex shedding frequency.

(a)

(b)

Figure 5. A time series of the first POD mode amplitude 𝑎< . A
PDF of 𝑎< is shown on the right-hand side, with A and B
showing the quasi-stable positions.
The LSA result shows that the unstable mode triggering
the transition from the steady wake to the unsteady one withvortex shedding is a spatial-symmetry-breaking mode (Rigas
et al. 2016). In figure 3, however, the spatial symmetry ispreserved when considering the absolute magnitude of eachmode. Figure 5 shows a time series of first POD mode
amplitude 𝑎< . A PDF of 𝑎< is shown on the right-hand side,with A and B showing the quasi-stable positions. It shows thatthe mode amplitude is not stabilized near the centre. Instead,the mode amplitude is shifting between two states (A and B)
resembling the bi-stability in the Ahmed body wake. It showsthat the wake is asymmetric in the stable state and the spatialsymmetry shown in the POD modes (figure 3c) is acombination of two asymmetric states.

(c)Figure 4. (a) The energy ratio and (b) normalized cumulative
energy of each POD mode. (c) The pre-multiplied PSD
spectrum of each POD mode amplitude and the main POD
modes.
To extract coherent structures from the turbulent wake,
POD is performed based on the horizontal and the vertical3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
eigenmodes at 𝑅𝑒# ~100 to show that the laminar LSA modes
persist in turbulent POD modes. Similar to the LSA, we willshow that the conditional POD modes based on the flow fieldnear the stable equilibrium is a more physical representationof the vortex shedding than the POD.
Figure 7(a) shows that the first unstable steady LSA
eigenmode closely resembles the first POD mode. In thelaminar wake, the first LSA mode breaks the 2D axisymmetricwake topology and transforms it into a 3D asymmetric steadywake (Rigas et al. 2016). Similarly, the first POD modes
represent the stochastic switching of the wake in symmetrybreaking quasi-stable positions. Thus, the unstable steadylaminar LSA mode persists in the high-Reynolds number
wake and they represent the same symmetry-breaking
phenomenon.
The main structures of the second LSA mode, whichtransforms the steady base flow into the unsteady wake, is also
found in the second POD mode in both 2D and 3D wake(figure 7b and c). The LSA and POD modes shown in figure6(b) are based on the 2D symmetric base flow. Rigas et al.
(2016) shows that this LSA eigenmode is not an accurate
representation of the second bifurcation and leads to anincorrect prediction of the bifurcation threshold, since thestability of the 2D base flow is broken by the first bifurcation.
Similarly, the POD mode based on the 2D flow is not an
accurate representation of the vortex shedding phenomenon,
since the 2D flow is not stable. The true LSA eigenmode forthe second bifurcation is calculated with the 3D base flow andaccurately predicts the Hopf bifurcation (Rigas et al. 2016). A
similar POD mode is found in the turbulent wake by theconditional POD based on the flow near the stable equilibrium(figure 6c). Compared to the symmetric POD mode (figure 6b),the symmetry-breaking POD mode is based on flow nearstable equilibrium and closely resembles the true LSA mode,thus representing a more physical vortex shedding
phenomenon.

Figure 6. The conditional POD modes based on A position andthe pre-multiplied PSD spectrum of each mode amplitude.
To extract the stable asymmetric POD modes, aconditional POD based on the velocity fields near one of thestable locations (A) is performed. Figure 6 shows theconditional POD modes and the corresponding PSD spectrum.
It is clear that when the wake is in a stable state, all the POD
modes lose symmetry. The most energetic mode near thevortex shedding frequency (𝑆𝑡# = 0.2) is biased toward thelower side and the upper side has largely disappeared,
indicating the vortex shedding is asymmetric in nature. The
bubble pumping mode (𝑆𝑡# = 0.07) is also asymmetric since
the flow field is biased, whereas the upper side and the lower
side of the mode retain the same sign, indicating the bubble
pumping is the pulsation of the wake. The 𝑆𝑡# ~0 mode is aquasi-steady, spatial symmetry-breaking mode and shows thatthe wake is asymmetric in the stable state. These results showthat the turbulent wake in the stable state is spatial-symmetry
breaking and consists of asymmetric POD modes.
(a)Unstable steady LSA eigenmode

First POD mode(b)

Unstable unsteady LSA eigenmode (2D)

Second POD mode(c)

Unstable unsteady LSA eigenmode (3D)

Second conditional POD mode

Figure 8. The relationship between the base pressure
coefficient and the CoP.

Figure 7. A direct comparison of the LSA eigenmodes (leftcolumn) with the POD and conditional POD modes (rightcolumn). (a) The first LSA mode (unstable steady mode) andthe first POD mode; (b) the second LAS mode based on 2D
axisymmetric base flow and the second POD mode; (c) the
second LAS mode based on 3D asymmetric base flow and thesecond POD mode.

The symmetry of the wake is quantified based on thecentre of pressure (CoP) and correlated to the base pressure(and thus drag) using conditional averaging. As is shown infigure 8, near the large probability (around 𝑟/𝐷 = 0.03), thebase pressure is low, indicating that for the most probable stateof the wake, low base pressure and high drag are obtained.
Since the concentration of the CoP represents the multistability nature of the wake, breaking the multi-stability is a

Here, we provide a direct comparison of the first two POD
modes at 𝑅𝑒# = 1.88 × 10* with the first two LSA

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
promising way of increasing base pressure and reducing drag.
Away from 𝑟/𝐷 = 0.03, the base pressure increases toward
the base centre and the base edge, showing that both the(a) 0.5

symmetric wake and the highly-asymmetric wake possess
higher base pressure and lower drag.

1(b) Lab reference frame

Rotating reference frame
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-0.115Figure 9. Conditional averaged flow fields, base pressure, and base pressure in a rotating frame of reference for (a, b) CoP > 0.05, (c,d) 0.023 < CoP < 0.027, and (e, f) CoP < 0.015.
Further insight into the relationship between the symmetryof the wake and the base pressure is provided by conditionallyselecting the PIV snapshots and the synchronized basepressure according to the CoP. Figure 9 shows conditional
averaged flow fields, base pressure, and base pressure in arotating frame of reference for CoP > 0.05, 0.023 < CoP <0.027, and CoP < 0.015. The synchronized PIV and basepressure shown on figure 9 (a) (c) and (e) are further filteredwith 𝐶𝑜𝑃<# > 0, where 𝐶𝑜𝑃<# is the 1D base pressure
measured simultaneously with the PIV flow fields. Following
Rigas et al. (2014) and Grandemange et al. (2013), the basepressure in a rotating reference frame is achieved by rotating
the base pressure according to the azimuthal component of the
CoP (𝐶𝑜𝑃D). The instantaneous base pressure fields are rotatedto move 𝐶𝑜𝑃D to zero, after which they are averaged to get thebase pressure in a rotating reference frame.
Figure 9(a, c, e) shows two asymmetric circulation
bubbles in the near wake with a low-pressure region near thelarger one. In the vicinity of the stagnation point on the base,
there is a high-pressure region that shifts with the level ofwake asymmetry. In the highly-asymmetric wake (figure 9a),the stagnation point on the base and the high-pressure region
are close to the edge. The large circulation bubble on the otherside leads to a low base pressure region, especially near the
bubble centre. As the CoP decreases, the high-pressure region
near the stagnation point on the base drops, while the basepressure rise near the large circulation bubble does notcompensate for the base pressure drop (figure 9c). Thus, theoverall base pressure drops and leads to a low-pressure, high-drag case near 0.023 < CoP < 0.027. As the wake becomes
more symmetric (figure 9f), both the base pressure near thestagnation point and the base pressure near the large
circulation bubble rises, which leads to a high-pressure, lowdrag case. Thus, the transition of the base pressure with the
CoP is the combined effect of the low-pressure zone near thecirculation bubbles and the high-pressure zone near thestagnation point on the base.
A two-dimensional base pressure distribution in thelaboratory reference frame and in the rotating reference frame
is shown on figure 9(b, d, f). Like figure 9(a), in the rotatingreference frame, there exists a high-pressure zone near close
to the stagnation point location on the base. As the CoP
decreases, the base pressure in a rotating reference frame
becomes more symmetric, with the high-pressure region
gradually moving from the base edge towards the centre.
CONCLUSIONSThe turbulent axisymmetric bluff body wake ( 𝑅𝑒# =1.88 × 10* ) is analysed using POD, conditional POD, andconditional average, based on the synchronized PIV and basepressure measurement of the near wake velocity field on aplane parallel to the streamwise direction.
The POD modes uncover three kinds of motion in thewake. The first POD mode is a steady symmetry-breaking
mode (𝑆𝑡# ~0) representing the stochastic switching of thevortex shedding plane in the azimuthal direction. The secondand fourth POD modes, coupled with the first POD mode,
represent the unsteady vortex shedding ( 𝑆𝑡# = 0.2 ). An

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
axisymmetric POD mode is found near the bubble pumping
frequency ( 𝑆𝑡# = 0.07 ), which represents the global
streamwise pulsation of the wake. By performing conditional
POD with the flow field near the stable equilibrium, theasymmetric modes are extracted from the near wake ofaxisymmetric body.
A direct comparison of the laminar LSA modes with the
turbulent POD modes shows that the unstable laminar modes
persist in the turbulent wake. The first LSA eigenmode
corresponds to the first POD mode, both of which represent
the steady spatial-symmetry breaking phenomenon in thewake. The second LSA eigenmode based on the unstable 2Dbase flow and the 3D stable base flow can be found in thesecond POD mode based on the unstable symmetric flow andthe stable asymmetric flow. The conditional POD mode basedon the flow near the stable equilibrium is a more physicalrepresentation of vortex shedding.
The symmetry of the wake is quantified based on the CoP
and correlated with the base pressure (and thus drag) using
conditional averaging. For the most probable state of thewake, a low-pressure coefficient is obtained (high drag). At
the limits of the highly asymmetric wake or a perfectly
symmetric wake, a high average pressure coefficient (lowdrag) is obtained. The transition of the base pressure with the
CoP is the combined effect of the low-pressure zone near thecirculation bubble and the high-pressure zone near thestagnation point on the base.
The above results provide insight to the coherent
structures for canonical 3D bluff body geometries and help to
guide the development of drag reduction strategies.

REFERENCE
Fan, Yajun, Xia, Chao, Chu, Shijun, Yang, Zhigang &Cadot, Olivier 2020 Experimental and numerical analysis of
the bi-stable turbulent wake of a rectangular flat-backed bluff
body. Physics of Fluids 32, 105111.
Grandemange, Mathieu, Gohlke, Marc & Cadot, Olivier
2013b Turbulent wake past a three-dimensional blunt body.
part 1. global modes and bi-stability. Journal of Fluid
Mechanics 722, 51–84.
Oxlade, A. R., Morrison, J. F., Qubain, A. & Rigas, G.
2015 High-frequency forcing of a turbulent axisymmetricwake. Journal of Fluid Mechanics 770, 305–318.
Pavia, Giancarlo, Varney, Max, Passmore, Martin &Almond, Mathew 2019 Three dimensional structure of theunsteady wake of an axisymmetric body. Physics of Fluids 31(2), 025113.
Rigas, G., Oxlade, A. R., Morgans, A. S. & Morrison, J.
F. 2014 Low-dimensional dynamics of a turbulent
axisymmetric wake. Journal of Fluid Mechanics, 755 R5.
Rigas, G., Esclapez, L. & Magri, L. 2016 Symmetry
breaking in 3D wakes. In Proceedings of the Summer
Program. Center for Turbulence Research, Stanford
University, arXiv:1703.07405.

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