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

INHOMOGENEOUS TURBULENCE SCALINGS IN A FREE SHEAR FLOW

F. Alves Portela
Laboratoire de Mécanique des Fluides de Lille
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille
Kampé de Feriet, 59000 Lille, France
felipe.alvesportela@univ-lille.fr

J. C. Vassilicos
Laboratoire de Mécanique des Fluides de Lille
Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille
Kampé de Feriet, 59000 Lille, Francejohn-christos.vassilicos@centralelille.fr

which reflects the non-equilibrium between inertial range energy exchanges and the turbulence dissipation rate.
The non-equilibrium relation given by eq. (1) has been reported in a broad variety of flows (see Vassilicos, 2015; OrtizTarin et al., 2021, and references therein) by probing the concurrent evolution of Cε and Reλ (the local Taylor length-based
Reynolds number) in the direction of the mean flow or in time.
Recently, however, Chen et al. (2021) reported Cε ∝ Re−1.5
λby measuring those two quantities along the breadth of planar wakes generated by pairs of square prisms. In a follow-up
work, Chen & Vassilicos (2022) presented a theory of inhomogeneous turbulence which retrieves the 2/3 scaling of thesecond order structure function by taking into account the spatial inhomogeneity of Cε and Reλ :ABSTRACT
We explore the dependence of the normalised turbulencedissipation rate Cε on the local Taylor length-based Reynolds
number Reλ and its relation to two-point statistics, using DNS
data of planar wakes generated by pairs of square prisms.
Power laws relating Cε and Reλ are found across the wholebreadth of the wake as well as in the direction of the mean
flow. A recent theory of inhomogeneous turbulence is confronted to this dataset. In the case where the two square prisms
are not too close to each other, the theory provides satisfactorycollapse of the second order structure functions even where
the turbulence is highly inhomogeneous. Our results suggest that the theory could benefit from explicitly incorporating
mean flow inhomogeneity in order to broaden its applicability,
since we find the largest Reynolds numbers precisely where
the mean flow is most inhomogeneous.

δ u2 = k

INTRODUCTION(2)This scaling is different from the Kolmogorov scaling(εr)2/3 f (r/η) because ε does not scale as k3/2 /L . The authors also argue that eq. (2) could hold both for the stream-wiseand cross-stream velocity components, respectively u1 and u2 ,which they confirm with their experiments, so long as mean
flow inhomogeneity is not significant.
The theory of Chen & Vassilicos (2022) makes use of ageneralised scale-by-scale energy budget which distinguishes
between exchanges in physical (X) and scale space (r)In free shear flows such as turbulent wakes, intense turbulence is produced by immersing solid bodies in an otherwise undisturbed flow. The resulting flows are often populated
by coherent structures which interact with the turbulence in a
non-trivial manner, introducing long range correlations, bothin space and time. The turbulent cascade that ensues is fundamentally different from the classical homogeneous, isotropic
and statistically stationary cascade (Thiesset & Danaila, 2020).
There is, however, some common ground between thesetwo classes of turbulence: most notably, the infamous −5/3
spectral power law of the velocity spectrum which has beenreported in a plethora of flows where the turbulence is demonstrably non-conforming with the assumptions involved in
the classical theory of Kolmogorov (Kraichnan, 1974; Alves
Portela et al., 2017).
Such departure from a classical cascade is encapsulated
in the dependence of the normalised dissipation Cε = εU 3 /L
on global and local Reynolds numbers
p
√
Re∞
,
Cε ∝
Reλ

 r 2/3  r 
f
.
L
ℓi

∂ ⟨σ u′i (δ u1 )2 ⟩∂ ⟨(δ u1 )2 ⟩ ∂ ⟨δ ui (δ u1 )2 ⟩⟨σ ui ⟩
+
=−
+
∂X
∂ ri
∂X
{z i } |
{z
} |
{z i
}
|
Au

∂δ p
1
+ 2⟨δ u1
⟩+ν
∂ X1
2
|
{z
} |
T p,u

Πu

Tu

∂
∂
+ 2
2
∂ Xi
∂ ri
{z
Du

!

⟨(δ u1 )2 ⟩ −σ ε1 ,(3)}where the σ and δ symbols denote the two-point half-sum
and half-difference, respectively, and ui = ⟨ui ⟩ + u′i . Inner and(1)

1

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

10d
U∞

g

y

We start by investigating how the normalised turbulencedissipation rate

20dx

Cε = ε

40d

U3
L

(4)

depends on the local Reynolds number
Figure 1: Problem dimensions in terms of the squares

size d.

Reλ =

Uλ
ν

(5)

q
2
with the Taylor micro-scale defined as λ = 15 νε U3 .
The choice of velocity and length scales for eqs. (4)and (5) follows that of Chenpet al. (2021). Namely, the velocity scale U is taken as ⟨ u′i u′i ⟩ whereas the length scale
L is obtained by integrating the stream-wise autocorrelation
function of u′2 up to its first zero crossing.
In Chen et al. (2021), the turbulence dissipation, which is
involved in eq. (4) directly and in eq. (5) through λ , was computed using the assumption of small-scale axisymmetry (seeGeorge & Hussein, 1991). Our DNS suggests (not shown here
for brevity) that this assumption yields very good estimates ofε, perhaps even slightly better than small-scale isotropy (forthe g/d = 2.4 case in particular) which also yields good dissipation estimates. We also found that computing L from thestream-wise autocorrelation function of u3 does not alter anyof our conclusions.
We find that both Cε and Reλ vary significantly along x
and y, for both gap ratios considered here. As seen in fig. 2,
the stream-wise evolution of Cε and Reλ is drastically different
between configurations: while Reλ increases with x for g/d =1.25 it decreases for g/d = 2.4, conversely Cε decreases withx for g/d = 1.25 and increases for g/d = 2.4. Looking at the
product Cε Reλ along x (fig. 3) reveals that the nonequilibrium
scaling Cε ∝ Re−1
holds only for g/d = 2.4 as for g/d = 1.25
λCε decreases faster with x than the increase in Reλ .
The y extent in Chen & Vassilicos (2022) where Cε andReλ were measured was restricted to |y| < 0.5d. Here we examine these two quantities throughout the whole breadth of
the wake. Specifically, at a given x location, we consider all
y locations where the mean enstrophy is larger than 1/4 themaximum mean enstrophy at that x location, as illustrated in
fig. 4.
In fig. 5 we plot Cε versus Reλ within the whole wakefor x > 10d; it can be clearly seen that these two quantities−p
are related through power laws of the type Cε ∝ Reλ . For
g/d = 1.25 we find p ≈ 1.3 from x = 10d onwards, whereas in
the g/d = 2.4 configuration p gradually increases from about
1.4 at x = 10d to about 1.5 at x = 20d. Figure 6 shows Cεversus Reλ at x = 20d for both gap ratios.
Notice that while in fig. 5 the data is coloured by x location, in fig. 6 it is coloured by the distance to the centreline(y). Figure 6 highlights the robustness of the power-law, inparticular for the g/d = 2.4 case where Reλ first increases andthen decreases with increasing distance to the centreline, all−p
the while in close agreement with the fit Cε ∝ Reλ .
Interestingly, Chen & Vassilicos (2022) also observed−pCε ∝ Reλ with some scatter in the values of p and foundp ≈ 1.5 after removing the dominant POD modes in their data.
Our value of p ≈ 1.3 for g/d = 1.25 is also remarkably close tothat found in the experiments of Chen & Vassilicos (2022) for
the same gap ratio at x = 7d (without the removal of dominant
POD modes).

outer length scales ℓi and L ) are then introduced to establish
inner and outer balances from eq. (3). By introducing someassumptions on the structure of these balances and requiring
dissipation to scale both in terms of inner and outer quantities (all of which effectively replace Kolmogorov’s similarity
hypotheses) one can then obtain eq. (2).
In the present contribution, DNS of planar wakes generated by pairs of square prisms at Re∞ = 5000, for two different
spacings between the prisms, were carried out. This databaseallows us to see some of the boundaries of applicability of thetheory introduced in Chen & Vassilicos (2022) by accessing
information not available to Chen et al. (2021) such as the full
velocity gradient tensor and the out-of-plane velocity component. This allows us to go beyond the experiments by directlycomputing all the terms involved in eq. (3).

NUMERICAL SETUP
The incompressible Navier-Stokes equations are solved
using Incompact3d, an open-source solver developed byBartholomew et al. (2020). The spatial discretisation uses 6thorder compact differences whereas the time integration is carried out with a 3rd order Adams-Bashfort scheme. Stabilityof the numerical solution is ensured by selectively introducing numerical dissipation as described in Lamballais et al.
(2011). The no-slip condition is enforced through the immersed boundary method as in Gautier et al. (2014).
The set-up used in this work is similar to that of Zhou
et al. (2019) with a geometry as illustrated in fig. 1. Uniform velocity U∞ is imposed at the inlet whereas the outlet is treated with a convective boundary condition (using the
maximum velocity normal to the outlet for the convective velocity). Periodicity is enforced at the cross-flow boundaries
which are 20d apart. The span-wise direction is πd wide andis also treated as periodic. The global Reynolds number is
Re∞ = U∞ d/ν = 5000, where ν is the kinematic viscosity. The
gap ratios g/d simulated are 1.25 and 2.4.
For both gap ratios considered, the above configuration
led to a resolution varying between 2η and 6η, typically ofO(3η), with η being the Kolmogorov length scale. The shedding frequency, taken as that where the power spectral densityof u′2 (fluctuating velocity in the y direction) is maximal, forthe g/d = 1.25 and g/d = 2.4 were found to be 0.08U∞ /d and0.16U∞ /d, respectively, leading to a sampling time of ca. 180
and 420 shedding cycles, respectively.

2

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

1

−3

0.5
0.25
0

2

−5
0

5

10

15

20

0

25

10

20

30

x/d
−3

800
600

−2

0

−1

1

2

5

400

y/d

Reλ

1

0

x/d

200
0

0

−1

5
y/d

Cε

0.75

−2

0
−5

0

5

10

15

20

0

25

10

20

30

x/d

x/d

Figure 4: Contour map of log (E ), where E = ⟨ωi ωi ⟩ isthe mean enstrophy (and ω = ∇ × u ). The dotted line
indicates the y location at which E / max(E ) = 0.25 ateach streamwise position x. Top: g/d = 1.25; bottom:
g/d = 2.4.

Figure 2: Cε (top) and Reλ (bottom) along x at y = 0 (fulllines) and y = g/2 (dashed lines) for g/d = 1.25 (blue)and g/d = 2.4 (red).

1.25

75

1

50

0.75

25
0

0

5

10

15

20

2
1.5
1
0.5
0

0.5

10
20
x/d

0.25

25

0

x/d
Figure 3: The product Cε Reλ along x (see caption offig. 2 for legend).

Second Order Structure Functions:and Budget

p

100

Cε

Cε Reλ

125

0

100

200

300 400
Reλ

500

600

700

Figure 5: Cε versus Reλ for 10 < x/d < 25 and for ywithin the dotted lines indicated in fig. 4; the points arecoloured by x from blue to red. The leftmost clusterof points corresponds to g/d = 2.4 and the rightmost to
g/d = 1.25. The inset show the exponents p along x for
g/d = 1.25 (blue) and g/d = 2.4 (red).

Scaling

Having established that Cε varies in space, following aclear power law dependence on Reλ , we now turn our attention
to the second order structure functions of the streamwise andcross-stream velocity components. These are shown in figs. 7
and 8 and it becomes immediately evident that the g/d = 1.25
case does not exhibit clear 2/3 power laws, whereas the g/d =2.4 case does. In fact, with increasing downstream distance,
the plateau in the compensated structure functions becomes
evermore evident for g/d = 2.4 (even if narrower) whereas for
g/d = 1.25 there is an increasing departure from any plateau.
Apart from a potential contribution from mean flow
(whose significance should reduce with increasing X), it islikely that coherent structures associated with vortex shedding
play an important role in figs. 7 and 8. Alves Portela et al.

(2017) showed how (but see also Thiesset et al., 2014) very
intense vortex shedding “contaminates” the scale-space distribution of second order structure functions, potentially obfuscating any underlying power laws (this is particularly true atmoderate Reynolds numbers). Indeed, we found (not shownfor brevity) that the peak power spectral density (normalised
by total energy) of u′2 is ca. three orders of magnitude larger
for g/d = 1.25 than g/d = 2.4. Furthermore, the magnitude ofthis peak decreases with increasing X/d significantly only for
g/d = 2.4.

3

1
Cε

0.75
0.5
0.25
0

0

100

200

300 400
Reλ

500

600

700

Figure 6: Cε versus Reλ at x = 20d, the full lines indicate

δ u2 /(ε r)2/3 , δ u2 /[k(r/L )2/3 ]8

δ v2 /(ε r)2/3 , δ v2 /[k(r/L )2/3 ]the fits 743Re−1.5(left) and 480Re−1.33. The leftmost
λλcluster of points corresponds to g/d = 2.4 and the rightmost to g/d = 1.25.

25

6
4

101
rx /η , rx /`i

102

5

4
3
2
1
0
100

101
rx /η , rx /`i

102

101
rx /η , rx /`i

102

4
3
2
1
0
100

101
rx /η , rx /`i

102

requires that the mean flow is homogeneous: otherwise thenon-linear terms in eq. (3) will feature production, which is unlikely to satisfy the same inner/outer scalings as “pure” interscale/space transfers.
Figures 9 and 10 shows the terms that feature in eq. (3),except for the viscous diffusion term which is negligible forrx ≳ λ (see Valente & Vassilicos, 2015). One can show thatinner similarity actually translates to ∝ σ ε1 , which is why
we have plotted (versus rx /ℓi ) all terms normalised by σ ε1 in
figs. 9 and 10.
Contrasting the budgets of ⟨(δ u)2 ⟩ in figs. 9 and 10 reveals that Πu and T p,u are qualitatively different between thetwo configurations. In particular, while for g/d = 2.4 T p,uremains small at all rx , X and Y investigated here, it is actually one of the dominant terms in the g/d = 1.25 case, acting
as a source of ⟨(δ u)2 ⟩. Conversely, in the g/d = 1.25 flowΠu actually changes sign for some rx (∼ 75ℓi at X = 15, 20d
and ∼ 200ℓi at X = 25d) whereas it remains negative at all rx
for g/d = 2.4. In both cases the spatial transport is comparable to the remaining terms in the budget, acting as a source of⟨(δ u)2 ⟩ - actually the only source term for the g/d = 2.4.
While the change in sign of Πu could suggest a reversal in direction of the non-linear cascade in the g/d = 1.25
flow, we have checked that δ u1 δ u21 is monotonic and negative for all rx , X and Y considered. This means that in thisflow, contributions to Πu associated with at least one of the∂ ⟨δ u3 (δ u1 )2 ⟩∂ ⟨δ u2 (δ u1 )2 ⟩andintroduce sufficient expanterms
∂ ry
∂ rz
sion in the cascade (see Vassilicos, 2015; Chen & Vassilicos,
2022, for an interpretation of the non-linear cascade in termsof compression/expansion events), along directions normal torx , to overcome the compression occurring along rx .
Let us now restrict our attention to the g/d = 2.4 case,where clear 2/3 power laws were observed in fig. 8. The
theory of Chen & Vassilicos (2022) predicts that these power

20
15
10
5
0
100

5

Figure 8: As fig. 7 but for g/d = 2.4.

2
0
100

δ u2 /(ε r)2/3 , δ u2 /[k(r/L )2/3 ]1.25

δ v2 /(ε r)2/3 , δ v2 /[k(r/L )2/3 ]12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 7: The second order structure functions ⟨(δ u)2 ⟩and ⟨(δ v)2 ⟩ normalised both in Kolmogorov variables(dashed lines) and as in eq. (2) (full lines), for g/d =1.25. The colours distinguish between the streamwise
locations X/d = 15 (red), X/d = 20 (green) and X/d =25 (blue); lines of the same colour correspond to different 0 < Y /d < 0.5.
Focusing on the g/d = 2.4 configuration, fig. 8 makes it
clear that the normalisation given by eq. (2) collapses more
successfully the structure functions at different locations (both
X and Y ) than Kolmogorov normalisation. We found that further increasing the range of Y over which the curves are plotted
deteriorates the collapse. Larger values of Y , however, were
found to be associated with stronger mean-flow inhomogeneity, whereas the theory of Chen & Vassilicos (2022) actually4

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

1.5

3
2

1

1
0

0.5

−1

0

−2
−3

0

100

200

−0.5

300

0

50

rx /`i
2

−1
0

100

200

laws are associated with self-similarity of the terms involvedin eq. (3) in terms of inner and outer scales, whereas thematching between inner and outer budgets further implies thatΠu /σ ε1 = const. in the intermediate range of scales, assuming of course that such an intermediate range exists and is wide
enough.
We do not observe a clear plateau in Πu /σ ε1 , except
perhaps for X = 15d; as X increases (recall from fig. 2 thatthe local Reynolds number decreases with increasing x), themagnitude of Πu /σ ε1 actually decreases, retaining a peak at
rx ∼ λz , a Taylor micro-scale that uses only span-wise information (which we found to better represent the scale above
which Du is negligible). Notice that the theory implies that
Au becomes negligible in the inner balance of ⟨(δ u)2 ⟩ withincreasing Reynolds number. It is certainly not the case that
Au is negligible here, which may be a Reynolds number effect. However, we also find that the spatial regions with largest
Reλ (e.g. x/d < 15 but also y/d > 0.5d) correspond to regions
where mean flow inhomogeneity becomes significant, which
is not accounted for in the theory. As already mentioned, thetheory deals with inhomogeneity of the turbulence but not of
the mean flow.
It is worth noting that Πv /σ ε2 (shown in fig. 11 and corresponding to v ≡ u3 ) does also not display a clear plateau,despite ⟨(δ v)2 ⟩ displaying clear 2/3 power laws. However, incontrast to Πu , we find that for rx /ℓi ≲ 100 the Πv /σ ε2 are all
independent of x which indicates that Πv is self-similar at the
inner scales.

300

rx /`iFigure 9: The various terms involved in eq. (3) nor-malised by σ ε1 for g/d = 1.25. Top: minus the interscale transfer −Πu (full line), pressure-velocity correlation T p,u (dotted line) and total spatial transport Au −Tu(dash-dotted line); bottom: mean flow advection Au(dashed line) and turbulent transport Tu (dotted line).
The colours distinguish between the streamwise locations X/d = 15 (red), X/d = 20 (green) and X/d = 25
(blue); lines of the same colour correspond to different
0 < y/d < 0.5.

2
1
0
−1
−2
−3
0

50

100
150
rx /`i

200

250

2

CONCLUSION
In this work we have performed DNS of two turbulent
wakes generated by pairs of prisms in configurations similar
to the experiments of Chen et al. (2021). Our results allowed
us to confirm some of the assumptions made in their data analysis (owing to experimental limitations), such as small scale
axisymmetry.
We have put the observations of Chen et al. (2021) to thetest by exploring the flow along much broader regions in space.
−p
Namely, we have found that power laws of the type Cε ∝ Reλexist over practically the whole breadth of the mean wake andthat the exponent p is actually quite robust, showing only little dependence on stream-wise location. The two configurations analysed yield different exponent p, but they also differ drastically in terms of the streamwise evolution of Cε and

1
0
−1
−2
−3

250

by σ ε2 . The colours distinguish between the streamwise locations X/d = 15 (red), X/d = 20 (green) andX/d = 25 (blue); lines of the same colour correspond todifferent 0 < y/d < 0.5.

0

−4

200

Figure 11: Minus the interscale transfer −Πv , normalised

1

−2

100
150
rx /`i

0

50

100
150
rx /`i200

250

Figure 10: As fig. 9 but for g/d = 2.4.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
nal of Fluid Mechanics 825, 315–352.
Bartholomew, P., Deskos, G., Frantz, R. A. S., Schuch, F. N.,
Lamballais, E. & Laizet, S. 2020 Xcompact3d: An opensource framework for solving turbulence problems on acartesian mesh. SoftwareX 12, 100550.
Chen, J. G., Cuvier, C., Foucaut, J-M., Ostovan, Y. & Vassilicos, J. C. 2021 A turbulence dissipation inhomogeneity
scaling in the wake of two side-by-side square prisms. Journal of Fluid Mechanics 924.
Chen, J. G. & Vassilicos, J. C. 2022 Scalings of scale-by-scale
turbulence energy in non-homogeneous turbulence (underreview). Journal of Fluid Mechanics .
Gautier, R., Laizet, S. & Lamballais, E. 2014 A dns studyof jet control with microjets using an immersed boundary
method. International Journal of Computational Fluid Dynamics 28 (6-10), 393–410.
George, W. K. & Hussein, H. J. 1991 Locally axisymmetricturbulence. Journal of Fluid Mechanics 233, 1–23.
Kraichnan, R. H. 1974 On kolmogorov’s inertial-range theories. Journal of Fluid Mechanics 62 (2), 305–330.
Lamballais, E., Fortuné, V. & Laizet, S. 2011 Straightforward
high-order numerical dissipation via the viscous term fordirect and large eddy simulation. Journal of Computational
Physics 230 (9), 3270–3275.
Ortiz-Tarin, J. L., Nidhan, S. & Sarkar, S. 2021 High-reynoldsnumber wake of a slender body. Journal of Fluid Mechanics
918.
Thiesset, F. & Danaila, L. 2020 The illusion of a Kolmogorov
cascade. Journal of Fluid Mechanics 902.
Thiesset, F., Danaila, L. & Antonia, R. A. 2014 Dynamicalinteractions between the coherent motion and small scales
in a cylinder wake. Journal of fluid mechanics 749, 201–
226.
Valente, P. C. & Vassilicos, J. C. 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Physics of Fluids 27 (4), 045103.
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annual
Review of Fluid Mechanics 47, 95–114.
Zhou, Y., Nagata, K., Sakai, Y. & Watanabe, T. 2019 Extreme
events and non-kolmogorov −5/3 spectra in turbulent flows
behind two side-by-side square cylinders. Journal of Fluid
Mechanics 874, 677–698.

Reλ : whereas a familiar non-equilibrium relation develops forx/d ≳ 10 in the g/d = 2.4 case (where Reλ decreases with increasing x), the smaller gap ratio (g/d = 1.25) actually leads
to Cε decreasing at a much faster rate with x than the increase
in Reλ .
We observed 2/3 power laws in ⟨(δ u1 )2 ⟩ and ⟨(δ u2 )2 ⟩,the second order structure functions associated with thestreamwise and cross-stream velocity components, respectively, for separations aligned with x only for the larger gap
ratio g/d = 2.4; we expect that the much stronger coherent
structures observed in the g/d = 1.25 case may be the sourceof this difference. The terms involved in the energy budget ofthese quantities are also qualitatively very different between
the two flows.
Concerning the different energy budgets of ⟨(δ u1 )2 ⟩ between the two flows, in the smaller gap ratio case there isan inversion in the sign of the non-linear inter-scale transfer attributable to expansion events in the directions normal
to rx . Further investigation of this phenomenon will require
analysing the cascade along the ry and rz directions.
In an assessment of the inhomogeneous turbulence theory of Chen & Vassilicos (2022) in the g/d = 2.4 case, wefind that mean flow advection is not negligible at small scales
owing to the moderate Reynolds numbers. In order to investigate regions of these flows with much larger local Reynolds
number it becomes necessary to either simulate higher global
Reynolds numbers, which is computationally very challenging, or extend the theory to account for inhomogeneity in themean flow given that the local Reynolds number is highest inthese regions.

ACKNOWLEDGEMENTS
This work is supported by JCV’s Chair of Excellence CoPreFlo funded by I-SITE-ULNE (grant number R-TALENT19-001-VASSILICOS); MEL (grant number CONVENTION219-ESR-06) and Region Hauts de France (grant number
20003862).

REFERENCES
Alves Portela, F., Papadakis, G. & Vassilicos, J. C. 2017 The
turbulence cascade in the near wake of a square prism. Jour-6