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

ENERGY EXCHANGES IN THE WAKE OF A MULTISCALE SYSTEM

Neelakash Biswas
Department of Aeronautics
Imperial College London, UK
n.biswas20@imperial.ac.uk

Oliver R.H. Buxton
Department of Aeronautics
Imperial College London, UK
o.buxton@imperial.ac.uk

ABSTRACT

pler multiscale flow. Hence, we first discuss results of a simplified multiscale geometry consisting of a main cylinder anda leeward control rod. The control rod has a diameter an orderof magnitude smaller than the main cylinder, thereby coherence is introduced into the flow at significantly different lengthand time scales. Interestingly, for certain positions of the control rod, the interaction between the wakes of the two wake
generating bodies generates new coherent structures. In fact,
Cicolin et al. (2021) recently reported the observation of two
such new frequencies which exactly corresponded to the sum
and difference of the frequencies corresponding to the fundamental sheddings of the control rod and the main cylinder. In
this work, we develop a deeper understanding of the processof generation of these new coherent motions with the samedata set as Cicolin et al. (2021). In order to understand theinteraction between the different coherent structures, we employ the multiscale triple decomposition technique proposedby Baj et al. (2015) where the flow field is decomposed as the
combination of the mean component (ū), sum of contributions
from different coherent motions (∑l ũl ) and the incoherent or
stochastic part (u′ ) as follows

In this study, we investigate the energy exchanges
between coherent structures in the wake of a model multiscale
geometry consisting of a cylinder and a leeward control rod
using Particle Image Velocimetry (PIV) data. For certain
control rod positions, we observe new secondary coherent
motions with distinct characteristic frequencies in the flow
field aside from the frequencies associated with the sheddingsof the control rod, the main cylinder and its harmonics.
We employ the multiscale triple-decomposition techniqueproposed by Baj et al. (2015), to extract the spatio-temporal
structure of the coherent modes associated with each of thefrequencies and also study the energy exchanges to/fromthe coherent motions using the triple decomposed kinetic
energy budget equations derived by Baj & Buxton (2017). We
find almost the same sequence of energy transfers between
the primary and secondary modes as reported by Baj &Buxton (2017) for a different multiscale geometry, hinting at a
possible universality of these structures in any multiscale flow.
We perform a series of PIV experiments to look for similarenergy exchanges in the wake of a representative rotor whichis also a multiscale system due to the presence of multiple
length (and time) scales simultaneously introduced in the flowfrom the nacelle, tower and rotor blades. If the universality
in energy exchanges is upheld even for a multiscale systemwith rotation, it would prove to be a powerful tool for the near
wake-modeling of wind turbines.

′u(x, y,t) = ū(x, y) + ∑ ũl (x, y,t) + u (x, y,t)(1)lIn order to obtain the energy transfers to/from the coherent structures, we use the multiscale triple decomposed turbulent kinetic energy transport equations derived by Baj & Buxton (2017). Based on the coherent energy budget equations, we
identify the differences in the process of formation of different
coherent structures and also their roles in the energy cascade
process. Finally, some preliminary results from PIV experiments on a model wind turbine is shown and the multiscale
nature of the wind turbine wake is discussed.

INTRODUCTION
Multiscale flows are encountered everywhere around us,
for instance flow through offshore platforms, a forest or
cityscape or even flow through a wind turbine. In any multiscale flow different length (and time) scales are simultaneously
introduced into the flow. Understanding the multiscale interactions in these flows is not only of fundamental interest but
also is important in modelling industrially relevant flows suchas the wake of a wind turbine. In a wind turbine wake, forcings
are introduced at different length scales by the tower, nacelle,
tip vortices etc. The importance of the nacelle and tower in thedevelopment of the near wake has been highlighted in somestudies e.g. (Foti et al., 2018), but has not been quantified indetail. Understanding the energy exchanges in such a complex
multiscale geomerty requires prior understanding from a sim-

PIV experiments and data set
The PIV experiments of Cicolin et al. (2021) had a main
cylinder of diameter (D) 0.05m and the control rod’s diameter(d) was 0.1D. The control rod was mounted at a centre-tocentre distance (R) of 0.7D while the control rod angle (θ )was varied between 90◦ and 140◦ (see fig. 1). The free stream

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

Figure 1. A schematic of the multiscale geometry consistingof a main cylinder and a control rod (the image is reproduced
from Biswas et al. (2022))
Figure 2. (a) Side view wind turbine model along with motorand mounting system. The field of view of the two experiments
are shown as FOV 1 and FOV 2. Sub figure (b) shows the front
view of the wind turbine model.

velocity (U∞ ) was fixed at 0.4m/s giving a Reynolds number
ReD = 20 × 103 for the cylinder and Red = 2 × 103 for thecontrol rod. Readers are referred to Cicolin et al. (2021) for
further details of the experimental measurements.
New PIV experiments are conducted on a scaled wind turbine model in the recirculating water channel in the hydrodynamics flume at Imperial College London. The 3D printed
model wind turbine consisted of a nacelle and a tower to replicate the wake dynamics of an actual wind turbine to the extent possible. Fig. 2 shows a schematic of the model geometry along with important dimensions. The turbine is run bya stepper motor RS 829-3512 and the torque is transmitted tothe shaft connected to the wind turbine through a pulley andbelt mechanism, that passes through the wind turbine tower.
The freestream velocity was fixed to 0.2 m/s and maximum
chord based Re was 4 × 103 . Experiments were conducted at
tip speed ratios (λ ) varying from 3-6. 3 experiments were conducted over a large field of view (FOV) in the central cross sectional plane extending up to 3.5 rotor diameters in the streamwise direction. 3 Phantom v641 cameras were used to capture
the data at a frequency of 100Hz. The FOV of the first experiment (henceforth referred to as FOV 1) focused the region
below the nacelle (see fig. 2) (a)) to capture the evolution of
the tip vortices. In the second experiment, the FOV (FOV 2)
focused the region above the nacelle to understand the effect of
the tower. The third experiment focused on the nacelle wake.
In this paper, preliminary results are shown for λ = 4.5 only in
FOV 1 and FOV 2.

Figure 3. Frequency spectra of the transverse component of
the fluctuating velocity at x/D = 1 (a,c) and x/D = 2 (b,d) forθ = 90◦ and θ = 115◦

RESULTS AND DISCUSSIONS
precise position of the control rod. At θ = 115◦ , the control
rod is effectively shadowed by the separated shear layer fromthe main cylinder, hence fc is not prominent, nor are fc ± fm
(see fig. 3(c)), while the harmonics of fm are prominent at
x/D = 2 (fig.3(d)). Interestingly fig. 3(a) reveals a particularspatial configuration of the modes fc ± fm . fc + fm resides near
the separated shear layer between the main cylinder and thecontrol rod whereas, fc − fm resides more towards the opposite separated shear layer of the control rod, which is remarkably similar to the observation of Baj & Buxton (2017) fora fundamentally different multiscale flow, a one dimensional
multiscale static mixer.
The spatio-temporal velocity field (ũl (x, y)) associatedwith each of the frequencies identified in the flow field can beobtained by using the multi-scale triple decomposition shownin equation 1. We apply optimal mode decomposition (OMD),
first proposed by Wynn et al. (2013) to obtain (ũl (x, y)) for
each of the coherent motions. The OMD algorithm is a gen-

By definition, a multiscale flow contains coherent motionsof several distinct frequencies. To identify the important frequencies in the flow consisting of a main cylinder and a control rod considered herein, let us first observe the evolution of
the transverse velocity spectra for different values of θ withstreamwise distance. In fig. 3(a) see that the near field (x/D =1) for θ = 90◦ consists of a number of prominent frequencies.
The low frequency peak at Strouhal number (St = f D/U∞ ) ofaround 0.166 (associated frequency, fm ≈ 1.3Hz) corresponds
to the vortex shedding of the main cylinder, while the otherstrong peak at St = 2.45 (associated frequency, fc ≈ 19.6Hz)
corresponds to the fundamental shedding mode of the control
rod. Around fc , two additional prominent peaks are observed(indicated by green arrows) which correspond to the frequencies fc ± fm . At x/D = 2, fm is more energetic and harmonics
of fm are observed. However, fc and fc ± fm are not observedat this station, which shows that these modes are spatially lesspersistent. The strength of fc as well as fc ± fm depends on the2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
eralised version of the dynamic mode decomposition (DMD)
algorithm proposed by Schmid (2010) and can identify individual coherent motions in the flow field under investigation.
We apply the multiscale triple decomposition on the data set
and show the modes associated with fc , fm and fc ± fm forθ = 90◦ in fig. 4. The filled contours show the time averaged kinetic energies (defined as 21 (ũl ũl + ṽl ṽl )) of the modes,
while the contour lines represent the iso-phase lines of transverse velocity component obtained from the complex OMD
modes (see Biswas et al. (2022) for more details). Fig. 4(a)and (b) show the modes corresponding to the shedding of thecontrol rod and main cylinder respectively, which is evident by
the presence of a phase correlated region behind the respective wake generating bodies in which the kinetic energies associated with the modes are concentrated. The location where
the kinetic energy of the individual modes attain maximum isshown by a ‘+’ sign. The modes corresponding to the harmonics of the main cylinder’s shedding frequency were also
captured through OMD, but not shown here (see Biswas et al.
(2022) for details).

Figure 4. Energy and iso-phase lines associated with (a) control rod’s shedding mode ( fc ), and (b) main cylinder’s shedding mode ( fm ). The contour lines are shown from −π to π at
step size π/2 in (a) and π/15 in (b). The corresponding modes
for the low and high frequency secondary modes ( fc + fm ) are
shown in fig. (c) and (d) respectively. The contour lines areshown between −π to π at a step size of π/2 for both the secondary modes. The ‘+’ sign shows the location where kinetic
energy is maximum.

The secondary modes however are not captured by the
OMD spectrum even when the rank of the projection/dynamic
matrices was set to very high values while performing the analysis. However, since we have the phase reference of the secondary modes, i.e. fc ± fm , we can use phase averaging (seefor instance Reynolds & Hussain (1972); Cantwell & Coles(1983)) to extract the modes associated with the secondary
frequencies observed in fig. 3(a). Subsequently, we take thesecond Fourier mode of the phase-averaged flow field in order to produce a complex spatial mode that mimics the OMD
modes. We encourage readers to refer to Baj & Buxton (2017)
for an extensive description of identification of the secondarymodes using phase averaging. The secondary modes shown infig. 4(c-d) show a significantly large phase-correlated region
in the vicinity of the control rod. Note that the presence of aphase correlated region of velocity (vorticity) correlates wellwith the definition of a coherent structure proposed by Hussain (1986). For the lower secondary mode ( fc − fm ) in fig.
4(c), most of the energy of the mode is located towards thelower shear layer of the control rod, while for the higher secondary mode ( fc + fm ) in fig. 4(d), it is located more towards
the separated shear layer between the main cylinder and thecontrol rod. This spatial configuration of the secondary modesis the same as that observed in the frequency spectra in fig.
3(a). The kinetic energy of the secondary modes peaks further downstream compared to the control rod’s fundamental
shedding mode for which the peak was located very close to
the control rod. This difference in the location of peak kinetic
energy was also noted by Baj & Buxton (2017) and it hints
at the fact that a different mechanism is involved behind theformation of the secondary modes.

In equation 2, the term on the left hand side corresponds
to the unsteady term, whilst the sources/sinks are arranged on
the right hand side. The term −C̄l is the convection term and P̃ldenotes the coherent energy production term (from the meanfj

any coherent motion including the l th coherent motion. The
terms P̂l , εl and Dl represent the production of stochastic turbulent kinetic energy from the lth coherent motion, dissipation and diffusion terms respectively. The term denoted asT̃l+ − T̃l− is the nonlinear triadic energy transfer term and accounts for the net non-linear triadic energy transfer to/fromthe lth coherent mode from/to the other coherent modes. The
terms is defined asfT̃l+ − T̃l− = −

f

s
l
1
1
f f ∂uf f ∂uui l u jt i + ∑ ui s u jt i .
∑2 f s , ft∂xj
2 f s , ft
∂xj

(3)

Here, the frequencies fs and ft correspond to otherfrequencies which might influence the non-linear energy
gain/loss of fl . (Baj & Buxton, 2017) argued that this term
could be non-negligible only when the coherent structures involved in the energy exchange process formed a triad, i.e.
fl ± fs ± ft = 0.
Keeping this background in mind, let us now investigate
the energy budget terms for the coherent modes observed in
fig. 3(a) for θ = 90◦ and try to understand their dynamics.
Instead of reporting local profiles of the different kinetic energy
budget terms at different streamwise stations, we obtainRy ED/U∞ dy, where E is any kinetic energy budget term andshow its evolution with streamwise distance. See that for fm
and fc (fig. 5(a) and (b)), the main source term is the coherent energy production (P̃l ) compared to which the contribution

In order to understand the process of generation of thesemodes, it is important to track the energy transfers to/from theindividual coherent modes and the mean flow. Similar to thekinetic energy budget equations for the mean flow, Baj & Buxton (2017) derived separate kinetic energy budget equations
for each of the coherent modes present in a multiscale flow using the multiscale triple decomposition shown in equation 1.
The coherent energy budget equation is expressed in symbolic
terms as∂ k̃l
= −C̃l + P̃l − P̂l + T̃l+ − T̃l− − ε˜l + D̃l∂tfflow). Here, P̃l is defined as − ∑ fs ũi s ũ jl ∂∂ xūi , where s can be(2)3

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

Figure 5. Streamwise evolution of the kinetic energy budget terms for (a) the main cylinder’s and (b) the control rod’s
fundamental shedding modes at θ = 90◦ . Sub figure (c) and(d) shows the same for the low and high frequency secondray
modes respectively for θ = 90◦ . The horizontal (– · –) line
shows the 0 level for a reference.

Figure 6. Combined schematic energy transfer diagram forthe different modes identified within the flow. The modes
are arranged such that characteristic frequency of the modes
increases from bottom to top. The thick full headed arrows
show the net energy transfer between the various modes andthe mean flow. The single headed arrows show the nonlinear triadic energy transfer between two modes. A solid line
shows non-linear energy transfer from low-frequency to highfrequency, whilst the dashed lines show energy transfers fromhigh-frequency to low-frequency - an inverse cascade.

of the T̃l+ − T̃l− term is negligible. For fc ± fm (fig. 5(c,d)),the P̃l term is negligible and the major source of energy is
the T̃l+ − T̃l− term, which shows that fc ± fm are fundamentally different from either fc and fm . As these modes are notdirectly energised by the mean flow, it is imperative to term
them as ‘secondary modes’ as opposed to fc and fm which
are the ‘primary modes’. The energy budget of the harmonics
of fm are not shown here for brevity. However, it is worthwhile to note that the harmonics of fm behave in a mixed fashion, i.e. the energy transfer through P̃l and the T̃l+ − T̃l− term
are of similar magnitude for them (see Biswas et al. (2022)
for details). The entire energy exchange process is illustrated
through a flow diagram in fig. 6. The different coherent structures are represented as circles and placed in ascending orderof their frequency. The net energy transfer to the modes fromthe mean flow is shown by thick arrows while the thin arrows
represent non-linear energy transfers between different coherent modes. Note that the main source of non-linear energy
gain of the two secondary modes is the high frequency primary mode ( fc ). fc + fm gives away some of its energy to
fm , while almost a similar amount of energy is transferred tofc − fm from fm . These energy exchanges are similar to thefindings of Baj & Buxton (2017). Also, note that alongside theconventional energy cascade from large to small scales, there
is a significant amount reverse energy cascade present.
This analysis on a rather simple multiscale geometry lays
the foundation to understand more complicated industriallyrelevant flows such as the wake of a wind turbine. The wakeof a wind turbine is inherently multiscale due to the forcing
introduced at different length and time scales. For instance,
the nacelle, tower and vortices shed from the rotor blade canhave distinct characteristic frequencies and hence a distinct
spatio-temporal coherent mode associated with each frequency
as shown for the flow past a main cylinder and a control rod.
It is possible that these new frequencies can interact with eachother leading to new frequencies which can have significant
impact on the evolution of the turbine wake. We show an instantaneous non-dimensional vorticity field of a wind turbine
wake in fig. 7. FOV1 and FOV2 are obtained from different
experiments and are stitched together to aid visual representa-

tion. It can be seen that the near field of the turbine wake isrich with coherent structures and flow field is inherently complex. In FOV 1, the array of the tip vortices can be seen. The
vortices shed from the 3 blades start interacting at a streamwise distance around x/D = 2 and merge into a single structure. In FOV 2, on the other hand, the presence of the tower
causes the tip vortices to break down. Note that the vorticity
levels in FOV 2 are significantly enhanced compared to FOV
1 and the tower acts as an important source for this asymmetry in the flow. We next characterise the frequency content
in the flow. In fig. 8 fast Fourier transforms are evaluatedat selected points based on the fluctuating transverse velocity
component. The Strouhal number is calculated based on turbine diameter (0.2m) and freestream velocity (0.2m/s). For
the present case, the numerical value of the Strouhal number issame as that of the frequency. At x/D = 0.5, y/D = −0.5 (fig.
8(a)), The dominant frequency is the blade frequency, 4.22Hz.
The harmonics of the blade frequency are also observed at
this location. Further downstream at x/D = 3, y/D = −0.5
(fig. 8(c)), the dominant frequencies are the rotor frequency(1.41Hz) and its harmonics which correlates well with themerging of the tip vortices also seen in fig. 7. Interestingly,
in fig. 8(c), apart from the blade/rotor frequency and their harmonics, some new frequencies are also observed. Fig. 8(b)shows the frequency spectrum near the nacelle shear layer
at x/D = 0.5, y/D = −0.1. Here the dominant frequency isfound to be around 0.71Hz. If we non-dimensionalise thisfrequency based on the diameter of the nacelle, the Strouhal
number comes out to be approximately 0.11. This Strouhal
number correlates well with that reported for azimuthal shedding modes of an axisymmetric bluff body (Rigas et al., 2014).
Interestingly, at x/D = 3, y/D = −0.1, a low frequency dominant mode is observed at around St = 0.27. This frequency
correlates well with the wake meandering frequencies reported
in previous studies (Okulov et al., 2014; Chamorro & Porté-4

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

Figure 7. Instantaneous vorticity field of a wind turbine wake. FOV 1 and FOV 2 correspond to different experiments. They are
stitched together for a visual representation of the entire wake.

ating bodies. Hence, unlike the primary coherent motions (i.e.
the modes associated with the vortex shedding of the individual wake generating bodies), that are energised by the mean
flow, the secondary coherent modes are energised by the nonlinear traidic interaction term. In fact we find that most of theenergy gained by the secondary coherent motions comes fromthe high frequency primary mode forming a triad with the secondary modes.
This work raises the question of whether these energy
transfers are universal in any complex multiscale flow such asa wind turbine wake. Indeed, non-linear triadic interaction between different coherent structures such as the tip vortices andvortex shedding from the nacelle/tower can excite new structures in a wind turbine wake which might have a significant
influence on the spatio-temporal evolution of the wake. Indeed we show that the interaction between the vortices shedfrom the tower and the tip vortices makes the wake in FOV 2
look totally different from that observed in FOV 1. A number of new frequencies are observed apart from that associatedwith the shedding of the individual wake generating bodies.
Future work will focus on the origin of these new frequencies
using the triple decomposed coherent energy budget equations
discussed previously. Unlike the multiscale system consistingof a cylinder and a control rod, a wind turbine wake is significantly more complicated due to the flow being highly three
dimensional and the presence of rotation. High quality stereoscopic/tomographic PIV experiments are necessary to avail all
the terms in the energy budget equations. If the universality ofenergy exchanges is upheld even for such a complicated system, this will offer a promising strategy to improve the modelling of wind turbine wakes.

Figure 8. Transverse velocity spectra obtained at (a) x/D =0.5, y/D = −0.5, (b) x/D = 0.5, y/D = −0.1 (c) x/D = 3,
y/D = −0.5, and (d) x/D = 3, y/D = −0.1.

Agel, 2009). Apart from this frequency, a number of new nondominant frequencies are also observed at this station which
deserves further investigation.

CONCLUSIONS AND FUTURE WORKIn this paper we investigated the energy exchanges taking
place between different coherent motions present in a multiscale flow. The simplified multiscale system considered here
consists of a cylinder and a control rod and contained new secondary frequencies that showed remarkable similarity in termsof their spatial orientation and process of generation with aprevious study by Baj & Buxton (2017) involving a different
multiscale geometry. These secondary structures are produced
due to interaction between the wakes of different wake gener-

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