# Scalings of scale-by-scale turbulence energy in non-homogeneous turbulence

J. G. Chen1  J. C. Vassilicos1<EMAIL_ADDRESS>1Univ. Lille, CNRS, ONERA, Arts et Metiers Institute of Technology, Centrale
Lille,
UMR 9014 - LMFL - Laboratoire de Mécanique des Fluides de Lille - Kampé de
Fériet, F-59000 Lille, France

###### Abstract

A theory of non-homogeneous turbulence is developed and is applied to
boundary-free shear flows. The theory introduces assumptions of inner and
outer similarity for the non-homogeneity of two-point statistics and predicts
power law scalings of second order structure functions which have some
similarities with but also some differences from Kolmogorov scalings. These
scalings arise as a consequence of these assumptions, of the general inter-
scale and inter-space energy balance and of an inner-outer equivalence
hypothesis for turbulence dissipation. They reduce to usual Kolmogorov
scalings in stationary homogeneous turbulence. Comparisons with structure
function data from three qualitatively different turbulent wakes provide
support for the theory’s predictions but also raise new questions for future
research.

###### keywords:

##  1 Introduction

Eighty years ago, Kolmogorov (1941a, b) and Obukhov (1941) introduced the
theory of homogeneous equilibrium turbulence which became very influential
after Batchelor’s (1953) monograph on this theory was published twelve years
later. A further fifteen to twenty years down the line, this theory became a
theoretical pillar of one-point two-equation models (such as the very widely
used $k-\varepsilon$ model) and of two-point subgrid turbulence modelling for
Large Eddy Simulations (e.g. see Pope, 2000). These one-point and two-point
modelling approaches are used for a wide range of turbulent flows, well beyond
homogeneous equilibrium turbulence which is in fact of very limited practical
interest itself given that turbulent flows in nature and engineering are
typically non-homogeneous and out of Kolmogorov equilibrium.

The Kolmogorov-Obukhov theory predicts that, in the inertial range, the
turbulence energy spectrum is proportional to the $-5/3$ power of wavenumber
(Obukhov, 1941) and that the second order structure function is proportional
to the $2/3$ power of two-point separation distance (Kolmogorov, 1941b).
Quoting Kraichnan (1974), the $2/3$ and $-5/3$ power laws of Kolmogorov and
Obukhov have “achieved an embarassment of success” as they have “been found
not only where they reasonably could be expected” but also where the
Kolmogorov theory is not designed for, e.g. in significantly non-homogeneous
and near-field turbulence. Alves Portela et al. (2017) listed some examples of
papers dating from the 1960s onwards which reported observations of $-5/3$
spectra in conditions where the Kolmogorov theory’s premises and/or
assumptions do not hold. They also reported that the best-defined $-5/3$
spectra in the turbulent wake of a square prism are found in the most
inhomogeneous region of the flow’s near field, specifically at a distance
equal to about two square prism sizes from the wake-generating prism. Gomes-
Fernandes et al. (2014) observed with Particle Image Velocimetry (PIV) a very
well-defined $2/3$ second order structure function (see their figure 16) at
the very near field of a turbulence-generating grid where the turbulence is
not decaying yet with streamwise distance.

These observations suggest that the Kolmogorov theory of homogeneous
turbulence equilibrium cascade and of the resulting scale-by-scale energy must
be a special case of a more general turbulence theory. To break beyond the
Kolmogorov theory in a substantial way we need to break beyond its most
constraining premises: equilibrium and homogeneity. Some progress has been
made on non-equilibrium cascades and non-equilibrium dissipation and their
consequences on basic flow profiles over the past ten years, a most recent
development having been contributed by Ortiz-Tarin et al. (2021). The present
paper attempts to break beyond the Kolmogorov theory by taking substantial
account of non-homogeneity and of the various energy transport processes that
non-homogeneity involves.

Chen et al. (2021) found a turbulence dissipation non-homogeneity scaling
which is the same in three different turbulent wake flows. The mechanism of
turbulence dissipation in free shear flows being the non-linear turbulence
interscale energy transfer, their finding would suggest some potential
structure of some generality in cascades which occur in non-homogeneous
turbulence. Their work is therefore an important motivation for the present
paper.

Chen et al. (2021) used two Particle Image Velocimetry (PIV) set-ups, one with
small fields of view for turbulence dissipation measurements and one with a
large field of view for integral length scale measurements, to study cross-
stream profiles of turbulence dissipation $\varepsilon$, integral length scale
$L$ and turbulent kinetic energy $k$ in the wakes of three different side-by-
side pairs of identical square prisms. Depending on gap-ratio $G/H$, where $G$
is distance between the two prisms and $H$ is the size of the prisms, the flow
structures and dynamics of such flows can be very different. The three
different values of $G/H$ in the experiments of Chen et al. (2021) produced
three different flow regimes with significant qualitative differences in
dynamics, large-scale features and turbulent flow non-homogeneity. Yet, in
every location that they sampled in each one of these three flows, they found
that the local turbulence dissipation coefficient
$C_{\varepsilon}\equiv\varepsilon L/k^{3/2}$ and the local Reynolds number
$Re_{\lambda}\equiv\sqrt{k}\lambda/\nu$ (where $\lambda$ is the Taylor length-
scale and $\nu$ is the fluid’s kinematic viscosity) are not constant along the
cross-stream coordinate $y$ and vary in opposition to each other: where one
increases with $y$ the other decreases with $y$ and vice versa. After removal
of the large-scale coherent structures by a snapshot Proper Orthogonal
Decomposition, the turbulence dissipation coefficient
$C^{\prime}_{\varepsilon}$ of the remaining incoherent turbulence fluctuations
was found to be independent of viscosity and proportional to
${Re^{\prime}}_{\lambda}^{-3/2}$ (where ${Re^{\prime}}_{\lambda}$ is the
Taylor length-based Reynolds number of the incoherent turbulence) at all
streamwise positions tested, in all three flows for three different inlet
Reynolds numbers.

Such universal behaviour is reminiscent of the non-equilibrium dissipation
scaling $C_{\varepsilon}\sim(\sqrt{Re_{G}}/Re_{\lambda})^{n}$ which appears in
non-stationary conditions either directly in time as in decaying and more
generally time-evolving periodic turbulence (where $n=1$ (Goto & Vassilicos,
2015, 2016a, 2016b)) or in the streamwise direction of decaying grid-generated
turbulence (where $n=1$ (Vassilicos, 2015)), various bluff body wakes (where
$n=1$ (Vassilicos, 2015; Dairay et al., 2015; Obligado et al., 2016;
Chongsiripinyo & Sarkar, 2020)), turbulent jets (where $n=1$ (Cafiero &
Vassilicos, 2019)) and slender body wakes (where $n=4/3$ (Ortiz-Tarin et al.,
2021)). In all these cases, $C_{\varepsilon}$ and $Re_{\lambda}$ vary in time
or along the streamwise coordinate $x$ in opposition to each other: when one
increases the other decreases with $x$ and vice versa. This is similar to the
observation of Chen et al. (2021), except that their observation was made in
the transverse/cross-stream direction and is therefore characteristic of
turbulence non-homogeneity rather than non-stationarity. Also, the value of
the exponent $n$ found by Chen et al. (2021) after removal of the large-scale
coherent structures is $n=3/2$.

The non-constancy of $C_{\varepsilon}$ in non-stationary conditions and the
fact that $C_{\varepsilon}$ increases/decreases when $Re_{\lambda}$
decreases/increases was shown by Goto & Vassilicos (2016a) to be the
consequence of a non-equilibrium cascade. Could it be that the relation
between $C_{\varepsilon}$ and $Re_{\lambda}$ found by Chen et al. (2021) in
non-homogeneous conditions is a reflection of a “non-homogeneous turbulence
cascade”, i.e. a turbulence cascade which fundamentally concerns non-
homogeneous turbulence? This is the general question which we attempt to
address in the present paper. Previous scale-by-scale analyses in non-
homogeneous conditions include Kaneda’s (2020) linear response theory of the
inertial sublayer of wall-bounded turbulence and the study by Afonso &
Sbragaglia (2005) of passive scalars subjected to non-homogeneous scalar
forcings in a homogeneous gaussian, white-in-time and zero-mean velocity field
(see also Jurčišinová & Jurčišin, 2008).

The next section lays out a general theoretical framework for the study of
interscale energy transfers which is then applied to the energy cascade in the
special case of homogeneous turbulence. The Kolmogorov theory’s assumptions
for obtaining the second order structure function are spelled out in section 2
for ease of comparison with the assumptions of section 3 where we develop a
theory of energy transfers for non-homogeneous turbulence and derive
predictions for second order structure functions. In following two sections we
consider the non-homogeneous turbulent wake flows of Chen et al. (2021) and
use their data, which we describe in section 4, to test in section 5 the
structure function predictions of section 3. We conclude in section 6.

##  2 Theoretical framework and energy cascade in homogeneous turbulence

Turbulence energy cascades consist of interscale exchanges of turbulent energy
which can be formulated in terms of velocity differences $\delta{\bf
u}\equiv{1\over 2}[{\bf u}({\bf x}^{+},t)-{\bf u}({\bf x}^{-},t)]$ between the
fluid velocity ${\bf u}$ at position ${\bf x}^{+}$ and the fluid velocity
${\bf u}$ at position ${\bf x}^{-}$ at the same time $t$. Following Hill
(2001, 2002) and using the coordinate transformation ${\bf x}^{+}={\bf X}+{\bf
r}$, ${\bf x}^{-}={\bf X}-{\bf r}$, the following equation is derived from the
incompressible Navier-Stokes equation

${\partial\over\partial t}\delta{\bf u}+{\bf
u}_{X}\cdot{\bf\nabla}_{X}\delta{\bf u}+\delta{\bf
u}\cdot{\bf\nabla}_{r}\delta{\bf u}=-{\bf\nabla}_{X}\delta p+{\nu\over
2}({\bf\nabla}^{2}_{X}+{\bf\nabla}^{2}_{r})\delta{\bf u}$ (1)

where ${\bf u}_{X}({\bf X},{\bf r},t)\equiv{1\over 2}[{\bf u}({\bf
x}^{+},t)+{\bf u}({\bf x}^{-},t)]$, $\delta p({\bf X};{\bf r},t)\equiv{1\over
2}[p({\bf x}^{+},t)-p({\bf x}^{-},t)]$ in terms of the local pressure to
density ratio $p$; ${\bf\nabla}_{X}$ and ${\bf\nabla}^{2}_{X}$ are the
gradient and Laplacian in ${\bf X}$ space; ${\bf\nabla}_{r}$ and
${\bf\nabla}^{2}_{r}$ are the gradient and Laplacian in ${\bf r}$ space. Note
that ${\bf X}$ is the centroid between ${\bf x}^{+}$ and ${\bf x}^{-}$ and
that ${\bf r}$ is half the separation vector between ${\bf x}^{+}$ and ${\bf
x}^{-}$. Like ${\bf u}_{X}$ and $\delta p$, the velocity difference
$\delta{\bf u}$ can be expressed as a function of ${\bf X}$ and ${\bf r}$ (and
of course also time $t$).

An energy equation may be obtained from (1) by a scalar product with
$\delta{\bf u}$:

${\partial\over\partial t}|\delta{\bf u}|^{2}+{\bf\nabla}_{X}(\cdot{\bf
u}_{X}|\delta{\bf u}|^{2})+{\bf\nabla}_{r}\cdot(\delta{\bf u}|\delta{\bf
u}|^{2})=$ $-2{\bf\nabla}_{X}\cdot(\delta{\bf u}\delta p)+{\nu\over
2}({\bf\nabla}^{2}_{X}+{\bf\nabla}^{2}_{r})|\delta{\bf u}|^{2}-{1\over
2}(\varepsilon^{+}+\varepsilon^{-})$ (2)

where use is made of incompressibility and where
$\varepsilon^{+}\equiv\nu({\partial u_{i}({\bf x}^{+},t)\over\partial
x_{j}^{+}})^{2}$, $\varepsilon^{-}\equiv\nu({\partial u_{i}({\bf
x}^{-},t)\over\partial x_{j}^{-}})^{2}$ (with summations over $i,j=1,2,3$).
This is the equation which is effectively the basis of our arguments in the
following section but we use it for homogeneous turbulence in this section for
the sake of comparison between Kolmogorov’s theory of homogeneous turbulence
and the following section’s theory.

The energy cascade has in fact been mainly and mostly studied for homogeneous
turbulence (Batchelor, 1953; Tennekes & Lumley, 1972; Pope, 2000; Mathieu &
Scott, 2000; Sagaut & Cambon, 2018). Averaging the above equation over
realisations, which is equivalent to averaging over ${\bf X}$ in homogeneous
turbulence, one gets (see Frisch, 1995)

${\partial\over\partial t}\langle|\delta{\bf
u}|^{2}\rangle+{\bf\nabla}_{r}\cdot\langle\delta{\bf u}|\delta{\bf
u}|^{2}\rangle={\nu\over 2}{\bf\nabla}^{2}_{r}\langle|\delta{\bf
u}|^{2}\rangle-\langle\varepsilon\rangle$ (3)

where the brackets signify realisations average and
$\langle\varepsilon\rangle\equiv\nu\langle({\partial u_{i}({\bf
X},t)\over\partial X_{j}})^{2}\rangle$.

Equation (3) shows that the change in time of the turbulent kinetic energy
associated with the separation vector ${\bf r}$ occurs by turbulence
dissipation ($-\langle\varepsilon\rangle$ term in the equation), viscous
diffusion in ${\bf r}$ space (${\nu\over
2}{\bf\nabla}^{2}_{r}\langle|\delta{\bf u}|^{2}\rangle$ term) and energy
exchanges between separation vectors (term
${\bf\nabla}_{r}\cdot\langle\delta{\bf u}|\delta{\bf u}|^{2}\rangle$ which is
conservative in ${\bf r}$ space).

To make the link with the energy cascade more apparent, one can follow Zhou &
Vassilicos (2020) and define the average turbulent kinetic energy in scales
smaller than $r=|{\bf r}|$ as follows: $E(r,t)\equiv{3\over 4\pi
r^{3}}\int_{S(r)}d^{3}{\bf r}\langle|\delta{\bf u}|^{2}\rangle$. This is an
average over realisations (equivalently, over ${\bf X}$ in homogeneous
turbulence) and over a sphere of radius $r$ in ${\bf r}$ space. It is easy to
verify that $E=0$ at $r=0$, that $E$ tends to ${1\over 2}\langle|{\bf
u}|^{2}\rangle$ as $r\to\infty$ and that $E$ is a monotonically increasing
function of $r$. (An average energy similar to $E$ can be defined for non-
homogeneous flows by integrating over a volume that has a shape (not
necessarily spherical) which takes into account the shape of the flow and/or
the presence of walls. As this volume grows beyond correlation lengths, this
quantity tends to the one-point kinetic energy averaged over that volume. This
is an issue that needs to be addressed in detail and on its own in a
comparative way for various turbulent flows, and we do so in a dedicated
forthcoming paper.)

The following equation, valid for $r\gg\lambda$, is obtained by integrating
(3) and using the divergence theorem:

${\partial\over\partial t}E(r,t)+{3\over 4\pi}\int d\Omega_{{\bf
r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over r}|\delta{\bf
u}|^{2}\rangle\approx-\langle\varepsilon\rangle$ (4)

where $\hat{\bf r}\equiv{\bf r}/r$ and the integral $\int d\Omega_{{\bf r}}$
is over the solid angle in ${\bf r}$ space. The validity of this equation is
restricted to $r\gg\lambda$ because the integrated viscous diffusion term has
been omitted as it can be shown to be negligible compared to
$\langle\varepsilon\rangle$ for $r\gg\lambda$ (see Appendix B of Valente &
Vassilicos (2015)).

The interscale transfer rate ${3\over 4\pi}\int d\Omega_{{\bf
r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over r}|\delta{\bf u}|^{2}\rangle$
(scale-space flux if multiplied by $r^{3}$) vanishes at $r=0$ and tends to $0$
as $r\to\infty$, as expected from the concept of an interscale transfer. At a
given scale $r$, a scale-space flux and a cascade from large to small or from
small to large scales corresponds to a negative or positive ${3\over 4\pi}\int
d\Omega_{{\bf r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over r}|\delta{\bf
u}|^{2}\rangle$ and contributes a growth or decrease of $E(r,t)$ in time. A
forward cascade on average (from large to small scales) results from
predominance of compression, i.e. $\hat{\bf r}\cdot\delta{\bf u}<0$ in
${3\over 4\pi}\int d\Omega_{{\bf r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over
r}|\delta{\bf u}|^{2}\rangle$, whereas an inverse cascade on average (from
small to large scales) results from predominance of stretching, i.e. $\hat{\bf
r}\cdot\delta{\bf u}>0$ in ${3\over 4\pi}\int d\Omega_{{\bf
r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over r}|\delta{\bf u}|^{2}\rangle$
(Vassilicos, 2015). By virtue of incompressibility, $\int d\Omega_{{\bf
r}}\hat{\bf r}\cdot\delta{\bf u}=0$: this means that there are no
forward/inverse cascade events without inverse/forward cascade ones in
incompressible turbulence, whether the cascade is on average predominantly
forward or inverse.

The crucial step in Kolmogorov’s theory of homogeneous turbulence is the
hypothesis of equilibrium, namely $|{\partial\over\partial
t}E(r,t)|\ll\langle\varepsilon\rangle$ for $r$ much smaller than the integral
length-scale $L$ in the limit of very high Reynolds number. This hypothesis
leads to

${3\over 4\pi}\int d\Omega_{{\bf r}}\langle{\hat{\bf r}\cdot\delta{\bf u}\over
r}|\delta{\bf u}|^{2}\rangle\approx-\langle\varepsilon\rangle$ (5)

for $\lambda\ll r\ll L$ in the limit where $L/\lambda\to\infty$ (i.e. high
Reynolds number limit).

If the Kolmogorov equilibrium hypothesis can be stretched to scales $r$ as
close to $L$ as possible, then (5) yields the well-known Taylor-Kolmogorov
dissipation scaling

$\langle\varepsilon\rangle=C_{\varepsilon}k^{3/2}/L.$ (6)

It is to be noted that this extension of the Kolmogorov equilibrium hypothesis
to scales $r$ that are commensurate with $L$ leads to the conclusion that
$\langle\varepsilon\rangle$ is independent of viscosity at high enough
Reynolds numbers. Whilst it might be possible to argue the validity of
Kolmogorov’s hypothesis for very small scales $r$, it is much harder to do so
for $r\sim L$. We must therefore consider this extension as an additional
hypothesis which is effectively the hypothesis that turbulence dissipation is
independent of viscosity at high enough Reynolds numbers.

The Kolmogorov equilibrium cascade for homogeneous turbulence is a cascade
where the rate $k^{3/2}/L$ of turbulent energy lost by the largest turbulent
eddies into the cascade equals the interscale transfer rate at all scales $r$
of the inertial range which itself equals the turbulence dissipation
$\langle\varepsilon\rangle$ at the smallest scales, all this effectively
instantaneously. This demonstrates the central importance of
$\langle\varepsilon\rangle$ across all inertial range scales and is the basis
of Kolmogorov’s two similarity hypotheses which are therefore expressed in
terms of viscosity $\nu$ and turbulence dissipation
$\langle\varepsilon\rangle$ and no other quantity (except inner and outer
length-scales bounding the inertial range).

Kolmogorov’s two similarity hypotheses for equilibrium homogeneous turbulence
are the following –quoting from Pope (2000):

Kolmogorov’s first similarity hypothesis: “At sufficiently high Reynolds
number, the statistics of the small-scale motions ($r\ll L$) have a universal
form that is uniquely determined by $\nu$ and $\langle\varepsilon\rangle$.”

Kolmogorov’s second similarity hypothesis: “At sufficiently high Reynolds
number, the statistics of the motions of scale $r$ in the inertial range have
a universal form that is uniquely determined by $\langle\varepsilon\rangle$,
independent of $\nu$.”

It is then a matter of straightforward dimensional analysis to obtain

$\langle|\delta{\bf u}|^{2}\rangle\sim(\langle\varepsilon\rangle r)^{2/3}$ (7)

in the inertial range of scales $r$.

In the following section we introduce two similarity hypotheses for non-
homogeneous turbulence which replace the highly simplifying premise of
homogeneity and the two Kolmogorov similarity hypotheses. We also introduce an
inner-outer equivalence hypothesis for turbulence dissipation which replaces
Kolmogorov’s equilibrium hypothesis and we use the hypothesis that turbulence
dissipation is independent of viscosity at high enough Reynolds numbers and
the general inter-scale and inter-space energy balance. On this basis, and
with one extra rather weak assumption linking inner to outer scales, we obtain
scalings of second order structure functions in an inertial range of scales.
Our results have similarities with but are also fundamentally different from
(7) and cannot be obtained by dimensional analysis.

##  3 Energy transfers in non-homogeneous turbulence

In this section we consider non-homogeneous turbulence where interscale energy
transfers coexist with turbulent energy transport through physical space and
pressure-velocity effects.

Equation (1) is a vector equation. Our analysis in this section can be carried
out effectively unchanged for equation (2) but we want to compare our
conclusions with experimental data and so we concentrate on the first
component of equation (1) which, when multiplied by $\delta u_{1}$ (the first
component of $\delta{\bf u}$), yields an equation that is effectively the same
as (2) except that $|\delta{\bf u}|^{2}$ is replaced by $(\delta u_{1})^{2}$,
the pressure-velocity term is replaced by $-2\delta
u_{1}{\partial\over\partial X_{1}}\delta p$ and $-{1\over
2}(\varepsilon^{+}+\varepsilon^{-})$ is replaced by $-{1\over
2}(\varepsilon_{1}^{+}+\varepsilon_{1}^{-})$ where
$\varepsilon_{1}^{\pm}\equiv\nu({\partial u_{1}({\bf x}^{\pm},t)\over\partial
x_{j}})^{2}$ (with summation over $j=1,2,3$).

We now restrict our attention to turbulent flows, such as those of Chen et al.
(2021), which are statistically stationary in time at every location of the
flow, and we use the Reynolds decomposition ${\bf u}_{X}=\langle{\bf
u}_{X}\rangle+{\bf u}_{X}^{\prime}$ by averaging over realisations or
equivalently over time (but not over ${\bf X}$) in this statistically
stationary context. We therefore obtain the following equation for
$\langle(\delta u_{1})^{2}\rangle$:

$\langle{\bf u}_{X}\rangle\cdot{\bf\nabla}_{X}\langle(\delta
u_{1})^{2}\rangle+{\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta
u_{1})^{2}\rangle+{\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta
u_{1})^{2}\rangle$ $+2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta
p\rangle={\nu\over 2}({\bf\nabla}^{2}_{X}+{\bf\nabla}^{2}_{r})\langle(\delta
u_{1})^{2}\rangle-\varepsilon_{1}$ (8)

where $\varepsilon_{1}\equiv{1\over
2}\langle\varepsilon_{1}^{+}+\varepsilon_{1}^{-}\rangle$ and where use was
made of the incompressibility of $\langle{\bf u}_{X}\rangle$. This equation is
an energy balance involving mean flow advection of $\langle(\delta
u_{1})^{2}\rangle$ (the term $\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}\langle(\delta u_{1})^{2}\rangle$), turbulent
transport rate of $(\delta u_{1})^{2}$ in physical space (the term
${\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta u_{1})^{2}\rangle$),
interscale energy transfer rate ${\bf\nabla}_{r}\cdot\langle\delta{\bf
u}(\delta u_{1})^{2}\rangle$, the velocity-pressure gradient correlation term
$2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta p\rangle$, viscous
diffusion of $\langle(\delta u_{1})^{2}\rangle$ in ${\bf X}$ and ${\bf r}$
spaces, and turbulence dissipation rate $\varepsilon_{1}$. Note that no
Reynolds decomposition has been applied to $\delta{\bf u}$, similarly to
Klingenberg et al. (2020) who considered two-point moments of instantaneous
velocities, but that such a decomposition can of course be used provided that
this paper’s hypotheses and analysis are suitably reconsidered in the context
of knowledge, not currently available, of the range of applicability of these
hypotheses (see section 5). In the current setting, the term
${\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta u_{1})^{2}\rangle$
includes both the more traditional turbulent transport and turbulent
production by mean flow. Production may need to be treated separately if the
validity of some of this section’s hypotheses turn out to depend on how
significant turbulent production is. Kaneda (2020) used the Corrsin length
$l_{C}\equiv\langle\varepsilon\rangle^{1/2}/S^{3/2}$ to distinguish between
scales above $l_{C}$ where the mean shear $S$ is significant and scales below
$l_{C}$ where it is not. We use this length-scale $l_{C}$ in the data analysis
of section 5 to assess the impact of mean shear on our results.

At this point we limit our study to the longitudinal structure function and
therefore take ${\bf r}=(r,0,0)$, but our approach can also be applied to the
transverse structure function, i.e. $\langle(\delta u_{2})^{2}\rangle$ with
${\bf r}=(r,0,0)$. We start with our inner/outer similarity hypotheses for
non-homogeneous turbulence which is that there exists a substantial class of
turbulent flows with regions in them where each inter-scale/inter-space
transport process is similar to itself at different locations of the non-
homogeneous turbulent flow as long as it is rescaled with the appropriate
space-local velocity- and length-scales. These velocity- and length-scales are
different depending on whether the scales under consideration are small enough
for viscosity to be a significant influence (inner) or not (outer). We
therefore introduce an outer length-scale $l_{o}({\bf X})$ (some
integral/correlation length-scale independent of viscosity) and an inner
length-scale $l_{i}({\bf X})$ (dependent on viscosity) such that $l_{i}({\bf
X})\ll l_{o}({\bf X})$ for high enough Reynolds numbers. The mathematical
expression of our inner and outer similarity hypotheses, which replace
Kolmogorov’s statistical homogeneity and two similarity hypotheses in the
present non-homogeneous context, are therefore written for the two-point terms
in equation (8) as follows:

outer similarity for $r\gg l_{i}$:

$\langle(\delta u_{1})^{2}\rangle=V_{o2}^{2}({\bf X})f_{o2}(r/l_{o})$ (9)
${\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta
u_{1})^{2}\rangle={V_{o3}^{3}({\bf X})\over l_{o}}f_{o3}(r/l_{o})$ (10)
${\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta
u_{1})^{2}\rangle={V_{oX}^{3}({\bf X})\over l_{o}}f_{oX}(r/l_{o})$ (11)
$2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta
p\rangle={V_{op}^{3}({\bf X})\over l_{o}}f_{op}(r/l_{o})$ (12)

inner similarity for $r\ll l_{o}$:

$\langle(\delta u_{1})^{2}\rangle=V_{i2}^{2}({\bf X})f_{i2}(r/l_{i})$ (13)
${\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta
u_{1})^{2}\rangle={V_{i3}^{3}({\bf X})\over l_{i}}f_{i3}(r/l_{i})$ (14)
${\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta
u_{1})^{2}\rangle={V_{iX}^{3}({\bf X})\over l_{i}}f_{iX}(r/l_{i})$ (15)
$2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta
p\rangle={V_{ip}^{3}({\bf X})\over l_{i}}f_{ip}(r/l_{i})$ (16)

where we have introduced eight velocity scales $V_{o2}$, $V_{o3}$, $V_{oX}$,
$V_{op}$, $V_{i2}$, $V_{i3}$, $V_{iX}$, $V_{ip}$ which are explicitly
dependent on ${\bf X}$ and eight dimensionless functions of normalised $r$,
namely $f_{o2}$, $f_{o3}$, $f_{oX}$, $f_{op}$, $f_{i2}$, $f_{i3}$, $f_{iX}$,
$f_{ip}$ which are not explicitly dependent on ${\bf X}$. We warn that as one
progresses in the argument, a need appears to modify the inner similarity
assumptions (15)-(16). This is explained in the Appendix.

The turbulence dissipation rate $\varepsilon_{1}$ is also a two-point
statistic but a fundamentally different one because, unlike the ones listed
above, it does not tend to $0$ as $r\to 0$. In fact the simulations of Alves
Portela et al. (2017) suggest that it does not depend significantly on $r$,
which is increasingly evidently true as $r$ becomes smaller. One can of course
expect values of $r$ to exist which are large enough for $\varepsilon_{1}$ to
depend on $r$, but what we effectively assume here is that such high values of
$r$ are beyond our $r$-range of interest which we limit to values of $r$ much
smaller than $l_{o}$. We therefore define $C_{\varepsilon}$ by

$\varepsilon_{1}=C_{\varepsilon}({\bf X}){V_{o2}^{3}\over l_{o}}$ (17)

where the outer velocity and length scales $V_{o2}$ and $l_{o}$ are
independent of viscosity. In fact it is natural to take $V_{o2}\sim\sqrt{k}$
and $l_{o}\sim L$, and we are more precise about the definition of the
integral scale $L$ used in this paper’s data analysis later in the paper
(penultimate paragraph of section 4). As in Kolmogorov’s theory of homogeneous
turbulence, we make the hypothesis that $C_{\varepsilon}$ is independent of
viscosity at high enough Reynolds number.

###  3.1 Outer scale-by-scale energy balance

Injecting (9)-(10)-(11)-(12) and (17) into (8) we are led to

${2l_{o}\over V_{o2}^{2}}(\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}V_{o2})f_{o2}(r/l_{o})-(V_{o2}^{-1}\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}l_{o}){r\over l_{o}}f_{o2}^{\prime}(r/l_{o})$
$+{V_{oX}^{3}\over V_{o2}^{3}}f_{oX}(r/l_{o})+{V_{o3}^{3}\over
V_{o2}^{3}}f_{o3}(r/l_{o})+{V_{op}^{3}\over V_{o2}^{3}}f_{op}(r/l_{o})=$
$-C_{\varepsilon}+R^{-1}{l_{o}^{2}\over
V_{o2}^{2}}[{\bf\nabla}^{2}_{X}V_{o2}^{2}f_{o2}(r/l_{o})]+R^{-1}{\bf\nabla}^{2}_{r/l_{0}}f_{o2}(r/l_{o})$
(18)

where $R\equiv 2V_{o2}l_{o}/\nu$ is a naturally appearing local (in ${\bf X}$)
Reynolds number, ${\bf\nabla}^{2}_{r/l_{0}}$ is the Laplacian with respect to
${\bf r}/l_{o}$ rather than just ${\bf r}$, and $f_{o2}^{\prime}(r/l_{o})$ is
the derivative of $f_{o2}$ with respect to its argument $r/l_{o}$.

In the limit where $R\gg 1$, this outer balance simplifies to

${2l_{o}\over V_{o2}^{2}}(\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}V_{o2})f_{o2}(r/l_{o})-(V_{o2}^{-1}\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}l_{o}){r\over l_{o}}f_{o2}^{\prime}(r/l_{o})$
$+{V_{oX}^{3}\over V_{o2}^{3}}f_{oX}(r/l_{o})+{V_{o3}^{3}\over
V_{o2}^{3}}f_{o3}(r/l_{o})+{V_{op}^{3}\over V_{o2}^{3}}f_{op}(r/l_{o})\approx-
C_{\varepsilon}({\bf X})$ (19)

This is the normalised outer scale-by-scale energy balance and it involves
mean advection (first line in the equation), turbulent transport in space,
interscale turbulence transfer, the velocity-pressure gradient correlation
term and turbulence dissipation (second line in the equation). There is no
viscous diffusion in the outer scale-by-scale energy balance for $R\gg 1$.

The fact that the right hand side of (19) is independent of $r/l_{o}$ implies

${2l_{o}\over V_{o2}^{2}}(\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}V_{o2})\sim V_{o2}^{-1}\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}l_{o}$ $\sim{V_{oX}^{3}\over
V_{o2}^{3}}\sim{V_{o3}^{3}\over V_{o2}^{3}}\sim{V_{op}^{3}\over
V_{o2}^{3}}\sim C_{\varepsilon}({\bf X})$ (20)

because ${\bf X}$ and $r$ are independent variables. These proportionalities
imply that all four outer velocity scales are effectively the same, i.e. the
same functions of ${\bf X}$, if $C_{\varepsilon}$ is independent of ${\bf X}$.
However, if $C_{\varepsilon}$ does depend on spatial position ${\bf X}$ as has
been found to be the case in the turbulent wakes of Chen et al. (2021), then
there are effectively only two independent outer velocities, $V_{o2}$ and
$V_{oX}\sim V_{o3}\sim V_{op}\sim V_{o2}C_{\varepsilon}^{1/3}$, with two
different dependencies on ${\bf X}$. Furthermore, (20) implies that $V_{oX}$,
$V_{o3}$ and $V_{op}$ are independent of viscosity given our hypothesis on
$C_{\varepsilon}$ and $V_{o2}\sim\sqrt{k}$. The Reynolds number independence
of $V_{oX}/V_{o2}$, $V_{o3}/V_{o2}$ and $V_{op}/V_{o2}$ is used in the last
sentence of section 3.2, and the conclusion (20) of the outer scale-by-scale
balance is used in the Appendix and in section 3.4.

###  3.2 Inner scale-by-scale energy balance

Injecting (13)-(14)-(15)-(16) and (17) into (8) we are led to

$2V_{i2}(\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}V_{i2})f_{i2}(r/l_{i})-{V_{i2}^{2}\over
l_{i}}(\langle{\bf u}_{X}\rangle\cdot{\bf\nabla}_{X}l_{i}){r\over
l_{i}}f_{i2}^{\prime}(r/l_{i})$ $+{V_{iX}^{3}\over
l_{i}}f_{iX}(r/l_{i})+{V_{i3}^{3}\over l_{i}}f_{i3}(r/l_{i})+{V_{ip}^{3}\over
l_{i}}f_{ip}(r/l_{i})=$ $-C_{\varepsilon}{V_{o2}^{3}\over l_{o}}+{\nu\over
2}({\bf\nabla}^{2}_{X}+{\bf\nabla}^{2}_{r})(V_{i2}^{2}f_{i2}(r/l_{i}))$ (21)

where $f_{i2}^{\prime}(r/l_{i})$ is the derivative of $f_{i2}$ with respect to
its argument $r/l_{i}$.

The inner scales depend on viscosity, and we make the assumption that they are
related to the outer scales by the a priori general forms
$V_{i2}^{2}=V_{o2}^{2}g_{2}(R)$, $l_{i}^{2}=l_{o}^{2}g_{l}(R)$,
$V_{i3}^{3}=V_{o3}^{3}g_{3}(R)$, $V_{iX}^{3}=V_{oX}^{3}g_{X}(R)$ and
$V_{ip}^{3}=V_{op}^{3}g_{p}(R)$ where all functions $g$ decrease to $0$ as
$R\to\infty$. In terms of these relations between inner and outer scales,
equation (21) becomes

$\sqrt{g_{2}(R)}{2l_{o}\over V_{o2}^{2}}[\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}(V_{o2}\sqrt{g_{2}(R)})]f_{i2}(r/l_{i})$
$-g_{2}(R)[V_{o2}^{-1}g_{l}^{-1/2}(R)\langle{\bf
u}_{X}\rangle\cdot{\bf\nabla}_{X}(l_{o}\sqrt{g_{l}(R)})]{r\over
l_{i}}f_{i2}^{\prime}(r/l_{i})$ $+g_{l}^{-1/2}g_{X}{V_{oX}^{3}\over
V_{o2}^{3}}f_{iX}(r/l_{i})+g_{l}^{-1/2}g_{3}{V_{o3}^{3}\over
V_{o2}^{3}}f_{i3}(r/l_{i})$ $+g_{l}^{-1/2}g_{p}{V_{op}^{3}\over
V_{o2}^{3}}f_{ip}(r/l_{i})=-C_{\varepsilon}+R^{-1}[{l_{o}^{2}\over
V_{o2}^{2}}{\bf\nabla}^{2}_{X}(V_{o2}^{2}g_{2}f_{i2})]$
$+R^{-1}g_{2}g_{l}^{-1}{\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$ (22)

where ${\bf\nabla}^{2}_{r/l_{i}}$ is the Laplacian with respect to ${\bf
r}/l_{i}$.

Taking the limit $R\gg 1$, the mean flow advection terms in the first line of
equation (22) tend to $0$ because $g_{2}(R)\to 0$ and so does the viscous
diffusion term in ${\bf X}$ space on the right hand side. In this high
Reynolds number limit we are therefore left with

$g_{l}^{-1/2}g_{X}{V_{oX}^{3}\over
V_{o2}^{3}}f_{iX}(r/l_{i})+g_{l}^{-1/2}g_{3}{V_{o3}^{3}\over
V_{o2}^{3}}f_{i3}(r/l_{i})+g_{l}^{-1/2}g_{p}{V_{op}^{3}\over
V_{o2}^{3}}f_{ip}(r/l_{i})$
$=-C_{\varepsilon}+R^{-1}g_{2}g_{l}^{-1}{\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i}).$
(23)

Viscous diffusion and dissipation must both be present in the inner scale-by-
scale energy budget as $R\to\infty$ and therefore

$g_{2}(R)/g_{l}(R)\sim R.$ (24)

To ensure that non-linearity is also present in some form in this inner scale-
by-scale budget, and given that $V_{oX}/V_{o2}$, $V_{o3}/V_{o2}$ and
$V_{op}/V_{o2}$ are independent of viscosity, $g_{X}/\sqrt{g_{l}}$,
$g_{3}/\sqrt{g_{l}}$ and $g_{p}/\sqrt{g_{l}}$ must also tend to a finite
constant (independent of $R$) or to $0$ (perhaps some of the three but not all
three) as $R\to\infty$.

###  3.3 Intermediate scaling of the second order structure function

Both similarity forms (9) and (13) hold in the intermediate range of scales
$l_{i}\ll r\ll l_{o}$, i.e.

$\langle(\delta u_{1})^{2}\rangle=V_{o2}^{2}({\bf
X})f_{o2}(r/l_{o})=V_{i2}^{2}({\bf X})f_{i2}(r/l_{i}).$ (25)

Using $V_{i2}^{2}=V_{o2}^{2}g_{2}(R)$ and $l_{i}^{2}=l_{o}^{2}g_{l}(R)$, the
following outer-inner relation follows:

$f_{o2}(r/l_{o})=g_{2}(R)f_{i2}({r\over l_{o}}g_{l}^{-1/2}(R))$ (26)

which implies ${d\over dR}[g_{2}(R)f_{i2}({r\over l_{o}}g_{l}^{-1/2}(R))]=0$
and therefore ${{d\over dR}g_{2}\over g_{2}}=-{r\over
l_{o}}{f^{\prime}_{i2}\over f_{i2}}{d\over dR}(g_{l}^{-1/2})$ (where
$f^{\prime}_{i2}$ is the derivative of $f_{i2}$ with respect to its argument).
Hence ${r\over l_{o}}{f^{\prime}_{i2}\over f_{i2}}$ is independent of
$r/l_{o}$, which in turn implies

$f_{i2}(r/l_{i})\sim(r/l_{i})^{n}$ (27)

for $l_{i}\ll r\ll l_{o}$ with

$g_{2}(R)g_{l}^{-n/2}(R)=Const$ (28)

independent of $R$.

We have one exponent $n$ and two functions $g_{2}(R)$ and $g_{l}(R)$ to
determine, and two relations between them: (24) and (28). We therefore need
one more relation and for this we introduce an hypothesis which we term inner-
outer equivalence for turbulence dissipation. This hypothesis replaces in non-
homogeneous turbulence Kolmogorov’s equilibrium hypothesis for homogeneous
turbulence.

###  3.4 Hypothesis of inner-outer equivalence for turbulence dissipation

The inner-outer equivalence hypothesis states that the turbulence dissipation
should depend on inner variables in the same way that it depends on outer
variables. Applied to (17), it states that if the turbulence dissipation rate
(per unit mass) is equal to $C_{\varepsilon}V_{o2}^{3}/l_{o}$ in terms of
outer variables, then it should also be equal to
$C_{\varepsilon}V_{i2}^{3}/l_{i}$ in terms of inner variables. Taking account
of $V_{i2}^{2}=V_{o2}^{2}g_{2}(R)$ and $l_{i}^{2}=l_{o}^{2}g_{l}(R)$ we obtain
a third relation, namely

$g_{l}(R)=g_{2}^{3}(R).$ (29)

Relations (24), (28) and (29) yield

$g_{2}(R)\sim R^{-1/2},g_{l}(R)\sim R^{-3/2},n=2/3.$ (30)

We can also use $g_{l}^{-1/2}g_{3}=Const$ stated in the sentence under (24)
and also given in (41) to obtain $g_{3}\sim R^{-3/4}$. These are Kolmogorov-
looking power laws and exponents in a non-Kolmogorov situation where the non-
homogeneity of the turbulence is in fact of the essence.

The inner-outer equivalence for turbulence dissipation can be understood quite
broadly and does not strictly rely on (17). It can also be taken to mean that
if we have $C_{\varepsilon}\sim V_{o3}^{3}/V_{o2}^{3}$ as per (20), then we
must also have $C_{\varepsilon}\sim V_{i3}^{3}/V_{i2}^{3}$. The ratio between
the outer velocity scales $V_{o3}$ and $V_{o2}$ is therefore the same as the
ratio between the inner velocity scales $V_{i3}$ and $V_{i2}$, i.e.
${V_{o3}\over V_{o2}}={V_{i3}\over V_{i2}}$. Taking account of
$V_{i3}^{3}=V_{o3}^{3}g_{3}(R)$ and $V_{i2}^{2}=V_{o2}^{2}g_{2}(R)$ we obtain

$g_{2}(R)=g_{3}^{2/3}(R).$ (31)

Relations (24), (41), (28) and (31) yield the exact same power laws and
exponents (30) and $g_{3}\sim R^{-3/4}$.

###  3.5 Predictions

The power laws and exponents (30) and $g_{3}\sim R^{-3/4}$ obtained by our
attempt at a theory of non-homogeneous turbulence (of course restricted by the
yet unknown domain of validity of the assumptions we based it on) imply the
following relations between inner and outer variables:

$l_{i}=l_{o}R^{-3/4}$ (32) $V_{i2}=V_{o2}R^{-1/4}$ (33)
$V_{i3}=V_{o3}R^{-1/4}.$ (34)

Note that $l_{i}$ is the Kolmogorov length
$\eta\equiv(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$ and $V_{i2}$ is the
Kolmogorov velocity $u_{\eta}\equiv(\langle\varepsilon\rangle\eta)^{1/3}$ only
if $C_{\varepsilon}$ is independent of ${\bf X}$. However, $l_{i}$ and
$V_{i2}$ are different from $\eta$ and $u_{\eta}$, respectively, if
$C_{\varepsilon}$ varies in physical space, as is indeed the case in the non-
homogeneous turbulence experiments of Chen et al. (2021). To be more specific,
whilst $l_{i}$ and $V_{i2}$ have the same dependencies on viscosity as $\eta$
and $u_{\eta}$ respectively, they have different dependencies on ${\bf X}$
than $\eta$ and $u_{\eta}$ when the turbulence is non-homogeneous and
$C_{\varepsilon}$ varies with ${\bf X}$.

Note also that $n=2/3$ means $f_{i2}\sim(r/l_{i})^{2/3}$ in the intermediate
range $l_{i}\ll r\ll l_{o}$. Going back to (13), we can define
$f_{i2}^{*}(r/l_{i})\equiv(r/l_{i})^{-2/3}f_{i2}(r/l_{i})$ and therefore write

$\langle(\delta
u_{1})^{2}\rangle=V_{i2}^{2}(r/l_{i})^{2/3}f_{i2}^{*}(r/l_{i})$ (35)

for $r\ll l_{o}$. Making use of (32) and (33), equation (35) becomes

$\langle(\delta
u_{1})^{2}\rangle=V_{o2}^{2}(r/l_{o})^{2/3}f_{i2}^{*}(r/l_{i})\sim
k(r/L)^{2/3}f_{i2}^{*}(r/l_{i})$ (36)

and like (35), it is expected to be valid for $r\ll l_{o}$. In the
intermediate range $l_{i}\ll r\ll l_{o}$, $f_{i2}^{*}$ is expected to be
independent of $r/l_{i}$. We made use in (36) of $V_{o2}^{2}\sim k$ and
$l_{o}\sim L$.

It must be stressed that our theory’s prediction (36) is different from the
Kolmogorov form $\langle(\delta u_{1})^{2}\rangle=(\langle\varepsilon\rangle
r)^{2/3}f_{i2}^{*}(r/\eta)$ because the turbulence dissipation does not vary
in space as $k^{3/2}/L$, and $l_{i}$ does not vary in space as $\eta$ either.
Our theory’s prediction (36) would have been identical to Kolmogorov’s
prediction if the turbulence was homogeneous and therefore neither turbulence
dissipation nor $k^{3/2}/L$ varied in space.

Before closing the section it is worth mentioning once again that this
section’s arguments can also be applied to the transverse structure function,
i.e. $\langle(\delta u_{2})^{2}\rangle$ with ${\bf r}=(r,0,0)$, with identical
results.

Figure 1: (a) PIV set-up for the energy dissipation rate measurements. (b) PIV
set-up for the integral length scale measurements. (c) The coordinate system
normal to the prisms’ spanwise direction and definitions of $G$, $H$ and
$x_{c}$. All fields of view in (a), (b) and (c) and laser sheets in (a) and
(b) are in the horizontal $(x,y)$ plane. Figure courtesy of Chen et al (2021).

##  4 The experiment of Chen et al. (2021)

Given that the laboratory experiments of Chen et al. (2021) provided the
motivation for the developments in the previous section, it is natural to test
the previous section’s theory against data from that experiment. In this
section we give a brief reminder of the salient features of the experiment of
Chen et al. (2021) (a detailed description can of course be found in their
paper) and we describe the data from that experiment which we use in section
5.

Chen et al. (2021) experimented with three different turbulent wakes of two
side-by-side identical square prisms of side length/width $H=0.03$m. The three
different cases corresponded to three different gap ratios $G/H=1.25,2.4,3.5$
chosen because they give rise to three qualitatively different flow regimes in
terms of dynamics, large-scale features and inhomogeneity. $G$ is the centre-
to-centre distance between the prisms (see figure 1 reproduced here from Chen
et al. (2021)). The wind tunnel’s test section was 2m wide by 1m high and the
prisms were placed with their spanwise axis parallel to the tunnel’s height.
Chen et al. (2021) acquired data for three incoming velocities
$U_{\infty}=5,6,7.35$ m/s corresponding to global Reynolds numbers $Re$
$(\equiv U_{\infty}H/\nu)$ = 1.0, 1.2 and $1.5\times 10^{4}$, respectively.

Figure 2: Spatial distributions of the normalised turbulent kinetic energy
$k/U_{\infty}^{2}$ at Re = $1.0\times 10^{4}$ and the positions of the small
fields of view (dashed squares) for each gap ratio. (a) G/H = 1.25, (b) 2.4,
and (c) 3.5. Figure courtesy of Chen et al. (2021).

Two different 2D2C PIV set-ups were used, one designed for turbulent
dissipation measurements (see figure 1a) and the other for measurements of
integral length scale (see figure 1b), both measuring two horizontal fluid
velocity components in the horizontal $(x,y)$ plane normal to the vertical
span of the prisms. A dual-camera PIV system with small field of view (SFV)
was used for the turbulent kinetic energy dissipation rate (see figure 1a,c).
Two sCMOS cameras, one over the top and one under the bottom of the test
section, observed the same SFV and obtained two independent measurements of
the same velocity fields which were then used to reduce the noise in
estimating the energy dissipation rate (see Chen et al. (2021) for detailed
explanations). The SFV size was similar to the horizontal size of the prisms,
specifically about 1$H$ in streamwise direction by 0.9$H$ in cross-stream
direction (figure 1 a,c).

Table 1: Details of the small fields of view (SFV). $Re$ |  | $1.0\times 10^{4}$ |  | $1.2\times 10^{4}$ |  | $1.5\times 10^{4}$
---|---|---|---|---|---|---
$G/H$ |  | 1.25 | 2.4 | 3.5 |  | 1.25 | 2.4 | 3.5 |  | 3.5
Cases |  | SFV7 | SFV2.5 | SFV7 |  | SFV14 | SFV14 | SFV14 |  | SFV20
| SFV14 | SFV5 | SFV14 |  | SFV20 | SFV20 | SFV20 |  |
| SFV20 | SFV10 | SFV20 |  |  |  |  |  |
|  | SFV20 |  |  |  |  |  |  |

For each gap ratio $G/H$, Chen et al. (2021) took measurements with SFVs at
several downstream positions for two or three global Reynolds numbers $Re$.
The centre of all the SFVs was on the geometric centerline ($y=0$), as
sketched in figure 1c, and different SFVs differed by different streamwise
positions $x_{c}$ of the SFV centre. The measurement cases are summarized in
table 1. The different SFVs are referred to as SFV$\mathcal{N}$ where
$\mathcal{N}$ gives an idea in terms of multiples $\mathcal{N}$ of $H$ of the
approximate streamwise coordinate $x_{c}$ of the centre of the SFV. The
positions of the SFVs relative to the prisms can be seen in figure 2. In this
study we use data from all the cases listed in table 1 except $G/H=1.25$ and
$3.5$ at the smallest Reynolds number ($1.0\times 10^{4}$).

The other 2D2C PIV set-up used by Chen et al. (2021) was a system of four
sCMOS cameras arranged consecutively in streamwise direction to allow integral
length scale measurements in a large field of view (LFV) ranging from
$x=0.53H$ to $x=24.3H$ in streamwise direction and $y=-2.4H$ to $y=2.9H$ in
cross-stream direction for $Re=1.0\times 10^{4}$, and from $x=11.1H$ to
$x=24.9H$ and $y=-2.8H$ to $y=3H$ for $Re=1.2$ and $1.5\times 10^{4}$, see
figure 1b.

Figure 3: Comparison between $S_{2}^{u}/(\langle\varepsilon\rangle r)^{2/3}$
(blue, $r/\eta$ in abscissa where
$\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$) and
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ (red, $r/l_{i}$ in abscissa where
$l_{i}=LR^{-3/4}$, $R=\sqrt{k}L/\nu$) across the flow (different same-colour
curves are for different values of $y$) in each SFV for G/H = 2.4 at
$Re=1.0\times 10^{4}$. (a) SFV2.5, (b) SFV5, (c) SFV10, (d) SFV20.

The acquisition frequency was 5Hz for SFV and 4Hz for LFV. 20,000 velocity
fields were captured for each measurement, corresponding to about 67 mins for
SFV measurements and 83mins for LFV measurements. The final interrogation
window size of their PIV analysis was $24\times 24$ pixels with about $58\%$
overlap, which correspond to a $312\mu m$ interrogation window for SFV and
1.6mm for LFV. In the SFVs, the ratio of the interrogation window size to the
Kolmogorov length scale varied from 4.5 at the nearest position (SFV2.5) to
2.5 at the farthest position (SFV20), and was below 3.2 for $x/H>10$ (see
figure 2 in Chen et al. (2021)). The turbulent energy dissipation rate was
approximated based on the assumption of local axisymmetry in the streamwise
direction (George & Hussein, 1991). Lefeuvre et al. (2014) demonstrated that
the turbulent kinetic energy dissipation rate estimated based on this
assumption is a good representation of the full energy dissipation rate across
the stream in the wake of a square prism, in fact more accurate than the
turbulent energy dissipation rate estimated from the local isotropy
assumption.

The turbulent kinetic energy $k$ was estimated from the two horizontal
turbulent velocity components and showed, using direct numerical simulation
data from Zhou et al. (2019), that the ratio of this estimate to the full
turbulent kinetic energy remains about constant at 0.75 to 0.8 for streamwise
distances $x/H\geq 5$. They also calculated correlation length scales in the
streamwise direction of both streamwise and cross-stream fluctuating
velocities. Good convergence of the auto-correlation functions was achieved
for the cross-stream fluctuations whereas the streamwise fluctuations gave
rise to integral length scales close to $10H$ in those cases when it was
possible to extract values of this length scale from the auto-correlation
function (the auto-correlation function of streamwise velocity fluctuations
often did not converge to $0$). We therefore adopt their choice of integral
length-scale $L$ which is the integral length scale of the cross-stream
turbulence fluctuations in the streamwise direction. We use their data for $L$
in the following cases: $G/H=1.25$, $SFV14$ and $SFV20$, where $L/H$ ranges
from about $1.4$ to $1.6$; $G/H=2.4$ where $L/H$ ranges between $0.4$ and
$0.5$ in SFV2.5, $0.75$ to $0.8$ in SFV5, and hovers around 1 in SFV14 and
SFV20; and $G/H=3.5$, SFV14 and SFV20, where $L/H$ ranges between about 1/2
and about 2/3.

In section 5 we use the integral length scale data just mentioned and also
local streamwise and cross-stream velocity data obtained by Chen et al. (2021)
in SFVs as well as local turbulent dissipation rate $\varepsilon$ and local
turbulent kinetic energy data in SFVs.

Figure 4: Comparison between $S_{2}^{v}/(\langle\varepsilon\rangle r)^{2/3}$
(blue, $r/\eta$ in abscissa where
$\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$)) and
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ (red, $r/l_{i}$ in abscissa where
$l_{i}=LR^{-3/4}$, $R=\sqrt{k}L/\nu$) across the flow (different same-colour
curves are for different values of $y$) in each SFV for G/H = 2.4 at
$Re=1.0\times 10^{4}$. (a) SFV2.5, (b) SFV5, (c) SFV10, (d) SFV20.

##  5 Scalings of second order structure functions

In this section, we assess our theory’s prediction (36) for the longitudinal
structure function as well as the equivalent prediction for the transverse
structure function in the three turbulent wake flows described in the previous
section.

Figure 2 shows the spatial distribution of the turbulent kinetic energy in the
three flows and illustrates the qualitative differences between them. Each
flow represents one of the three different flow regimes obtained for different
values of $G/H$ (see Sumner et al., 1999; Alam et al., 2011): the ‘single-
bluff-body regime’ (G/H = 1.25), the ‘bistable regime’ (G/H = 2.4) and the
‘coupled vortex regime’ (G/H = 3.5). Consistently, the turbulent kinetic
energy field, as well as the mean flow and integral scale fields (not
reproduced here from Chen et al. (2021)), exhibit distinct spatial
distributions and different inhomogeneity structures which are described and
discussed in Chen et al. (2021). The inhomogeneity is also present within the
SFVs where the turbulence dissipation coefficient $C_{\varepsilon}$ is found
to vary systematically with spatial position in a way that is common to the
three different flows and the different SFVs even though the spatial
inhomogeneities of turbulent kinetic energy and the turbulence dissipation
rate vary from flow to flow and from SFV to SFV (see figure 18 in Chen et al.
(2021)). These qualitative differences in large-scale features and
inhomogeneity provide some variety for the testing of the predictions of
section 3.

We use the data of Chen et al. (2021) to calculate the longitudinal structure
function
$S_{2}^{u}\equiv<[u^{\prime}_{1}(x_{0}+r,y)-u^{\prime}_{1}(x_{0},y)]^{2}>$ and
the transverse structure function
$S_{2}^{v}\equiv<[u^{\prime}_{2}(x_{0}+r,y)-u^{\prime}_{2}(x_{0},y)]^{2}>$
where use is made of the Reynolds decomposition $u_{1}=U_{1}+u^{\prime}_{1}$
and $u_{2}=U_{2}+u^{\prime}_{2}$ into mean flow components $U_{1}$
(streamwise) and $U_{2}$ (cross-stream) and turbulent fluctuating velocity
components $u^{\prime}_{1}$ and $u^{\prime}_{2}$. The averaging operation is
over 20,000 velocity field snapshots and we checked that there is no
significant dependence on the choice of streamwise origin $x_{0}$ for all the
$G/H$, $SFV$ and $Re$ cases examined here, except perhaps $G/H=2.4$ at
$SFV2.5$, where changes of $x_{0}$ can create slight shifts of the curves in
figures 3a and 4a without significantly changing their shape.

Figure 5: $C_{v}(r/l_{i})$ versus $r/l_{i}$ (where $l_{i}=LR^{-3/4}$ with
$R=\sqrt{k}L/\nu$) for $G/H=2.5$ and $Re=1.0\times 10^{4}$ at (a) SFV2.5, (b)
SFV5, (c) SFV10 and (d) SFV20. The horizontal dashed line is $C_{v}=4/3$.
Different lines correspond to different $y$ and the red line is the average
curve.

The theory of section 3 was presented for the longitudinal and transverse
structure functions involving $u_{1}$ and $u_{2}$ rather than $u^{\prime}_{1}$
and $u^{\prime}_{2}$. In fact, these structure functions are insensitive to
this difference in all the plots presented in figures 3 to 11 except for two:
the plot in figure 3a which corresponds to the very near field SFV2.5 where
the streamwise mean flow varies appreciably in both the streamwise and the
cross-stream directions within the SFV; and, very slightly, the plot in figure
7a which is another case where the streamwise mean flow varies appreciably in
the streamwise direction within the SFV (though less in the cross-stream
direction in this case).

We start the presentation and discussion of our data analysis with figures 3
and 4 where we plot normalised $S_{2}^{u}$ and $S_{2}^{v}$, respectively, as
functions of normalised $r$ for different cross-stream coordinates $y$ in the
case $G/H=2.4$, $Re=10^{4}$. This is the case for which we have data from four
different SFV stations and therefore can get an impression of dependence on
streamwise distance from the pair of square prisms. Neither the Kolmogorov
scalings $S_{2}^{u}/(\varepsilon r)^{2/3}$ and $S_{2}^{v}/(\varepsilon
r)^{2/3}$ versus $r/\eta$ (which we show for comparison even though they
should not be expected to hold in locally non-homogeneous turbulence) nor our
scaling (36), i.e. $S_{2}^{u}/k(r/L)^{2/3}$ and $S_{2}^{v}/k(r/L)^{2/3}$
versus $r/l_{i}$, collapse the data well in the SFV stations closest to the
prisms, i.e. SFV2.5 and SFV5. However, our scaling returns a clearly better
collapse than Kolmogorov’s in SFV20 and the two different types of scaling may
be judged as comparable, perhaps with a slight preference for scaling (36), in
SFV10.

It is intriguing that the curves $S_{2}^{v}/k(r/L)^{2/3}$ versus $r/l_{i}$
appear to plateau at a value that is about $4/3$ the value where the curves
$S_{2}^{u}/k(r/L)^{2/3}$ versus $r/l_{i}$ appear to plateau in SFV10 and SFV20
(see figures 3c,d and 4c,d). The presence of such a $4/3$ multiplier is well
understood in cases where the turbulence is isotropic and locally homogeneous,
in which cases it is possible to prove the relation
$S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial S_{2}^{u}\over\partial r}$ (e.g. see
Pope, 2000). Indeed, if $S_{2}^{u}=A_{u}r^{2/3}$ in a certain range of scales,
then $S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial S_{2}^{u}\over\partial r}$
implies $S_{2}^{v}={4\over 3}A_{u}r^{2/3}$ in that same range of scales. There
is no local homogeneity in any SFV studied here given that the turbulence is
inhomogeneous within them and that their size is comparable to the local
integral length-scales (between under 1/2 to slightly over 3/2 the integral
scale $L$). The usual way to derive $S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial
S_{2}^{u}\over\partial r}$ can therefore not be applied here. Nevertheless,
our theory’s inner scaling predictions $S_{2}^{u}=k(r/L)^{2/3}f_{u}(r/l_{i})$
and $S_{2}^{v}=k(r/L)^{2/3}f_{v}(r/l_{i})$, if injected in
$S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial S_{2}^{u}\over\partial r}$, yield
$f_{v}={4\over 3}f_{u}+{(r/l_{i})\over 2}{\partial
f_{u}\over\partial(r/l_{i})}$, which means that $C_{v}$ defined as follows
should equal $4/3$, i.e.

$C_{v}\equiv{f_{v}-{(r/l_{i})\over 2}{\partial
f_{u}\over\partial(r/l_{i})}\over f_{u}}=4/3.$ (37)

Note that the value $4/3=1+(2/3)/2$ results from the exponent $2/3$. If
$C_{v}$ differs from $4/3$ then either our theory’s predictions are at fault
or $S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial S_{2}^{u}\over\partial r}$ does
not hold or both.

Figure 6: $\langle u,_{y}^{2}\rangle/\langle v,_{x}^{2}\rangle$ versus
normalised cross-stream coordinate $y/H$ in different SFVs for $Re=1.0\times
10^{4}$ and (a) $G/H=1.25,(b)2.4,(c)3.5$.

In figure 5 we plot $C_{v}$ versus $r/l_{i}$ obtained from the previous two
figures for $G/H=2.5$, $Re=10^{4}$. $C_{v}$ is clearly well below $4/3$ at all
scales in SFV2.5 where the departure from collapse of $S_{2}^{u}$ and
$S_{2}^{v}$ for different values of $y$ is the greatest. However, $C_{v}$
tends to $4/3$ gradually from below as the SFV moves further away from the
prisms, and at SFV20 one can say that $C_{v}$ is close to $4/3$ for all values
of $y$ over the entire range of $r/l_{i}$. This is a non-trivial observation
which will require future investigation because Chen et al. (2021) showed
quite clearly that there is no local homogeneity in SFV20 in terms of
turbulent kinetic energy and turbulent dissipation rate, which means that the
usual grounds for $S_{2}^{v}=S_{2}^{u}+{r\over 2}{\partial
S_{2}^{u}\over\partial r}$ are absent even though the mean flow may be at its
closest to local homogeneity in SFV20 compared to other SFV stations. At this
stage, we have no explanation for this result but we do note that it is
consistent with the theory’s 2/3 exponent prediction for the second order
structure functions’ power law behaviour in the intermediate range of scales.
Figures 3c,d and 4c,d may be giving some partial support to this 2/3 exponent
in SFV10 and SFV20 but over a range of scales that is much smaller than the
range of scales where $C_{v}$ effectively equals $4/3$ in SFV20 and where
$C_{v}$ is close to $4/3$ in SFV10.

Figure 7: Comparison between $S_{2}^{u}/(\langle\varepsilon\rangle r)^{2/3}$
(blue, $r/\eta$ in abscissa where
$\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$) and
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ (red, $r/l_{i}$ in abscissa where
$l_{i}=LR^{-3/4}$, $R=\sqrt{k}L/\nu$) across the SFV14 and SFV20 for all gap
ratios at $Re=1.2\times 10^{4}$. Different same-colour curves correspond to
different $y$ positions. (a) $G/H=1.25$, SFV14, (b) $G/H=1.25$, SFV20; (c)
$G/H=2.4$, SFV14, (d) $G/H=2.4$, SFV20; (e) $G/H=3.5$, SFV14, (f) $G/H=3.5$,
SFV20.

It may be that $C_{v}$ is actually more sensitive to anisotropy than
inhomogeneity. We therefore conducted a local isotropy test, reported in
figure 6, where we examined the ratio $\langle u,_{y}^{2}\rangle/\langle
v,_{x}^{2}\rangle$ ($u,_{y}\equiv\partial u^{\prime}_{1}/\partial y$ and
$v,_{x}\equiv\partial u^{\prime}_{2}/\partial x$) across the stream (different
$y$-positions) in each SFV for different values of $G/H$. This ratio was taken
at the same $x_{0}$ (at a distance of about $H/10$ from the upstream edge of
each SFV) where the structure functions reported in this paper have been
calculated, but we checked that figure 6 does not depend significantly on
$x_{0}$. Departures from $\langle u,_{y}^{2}\rangle/\langle
v,_{x}^{2}\rangle=1$ indicate departures from local isotropy and the biggest
such departures are found at SFV2.5 followed by SFV5 (see figure 6b). The case
SFV2.5, $G/H=2.4$ is indeed the case where our second order structure
functions are furthest from collapse, whether in Kolmogorov variables or along
the lines of our scaling (36) (see figures 3a and 4a). There is also
unsatisfactory collapse in SFV5 (see figures 3b and 4b), and figure
$\ref{fig:isotropy test}b$ shows that a significant departure from local
isotropy remains there. In fact, figure 6 suggests that $\langle
u,_{y}^{2}\rangle/\langle v,_{x}^{2}\rangle$ takes values closest to 1 (with a
tolerance of about 10%), and therefore does not indicate significant
departures from local isotropy, further downstream, beyond SFV7 for all three
$G/H$ values. Given that figures 5 and 6 suggest that isotropy may be a
prerequisite for our theory’s scalings to hold, a point which will also
require future investigation given that isotropy did not feature explicitly in
our theory’s assumptions, we limit the remainder of our data analysis to SFV14
and SFV20 in all three $G/H$ cases (Chen et al. (2021) did not take SFV10
measurements for $G/H=1.25$ and $G/H=3.5$). The data for these SFV stations
come with values of $Re$ higher than $10^{4}$ which is welcome given that our
theory has been developed for high Reynolds numbers.

Figure 8: Comparison between $S_{2}^{v}/(\langle\varepsilon\rangle r)^{2/3}$
(blue, $r/\eta$ in abscissa where
$\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$) and
$S_{2}^{v}/(k^{3/2}r/L)^{2/3}$ (red, $r/l_{i}$ in abscissa where
$l_{i}=LR^{-3/4}$, $R=\sqrt{k}L/\nu$) across the SFV14 and SFV20 for all gap
ratios at $Re=1.2\times 10^{4}$. Different same-colour curves correspond to
different $y$ positions. (a) $G/H=1.25$, SFV14, (b) $G/H=1.25$, SFV20; (c)
$G/H=2.4$, SFV14, (d) $G/H=2.4$, SFV20; (e) $G/H=3.5$, SFV14, (f) $G/H=3.5$,
SFV20.

Figure 9: $C_{v}(r/l_{i})$ versus $r/l_{i}$ (where $l_{i}=LR^{-3/4}$ with
$R=\sqrt{k}L/\nu$) for SFV14 and SFV20 at $Re=1.2\times 10^{4}$ for all $G/H$
values. Different curves correspond to different $y$-positions and the red
line is the average curve. The horizontal dashed line in all the plots
indicates the value $4/3$. (a) $G/H=1.25$, SFV14; (b) $G/H=1.25$, SFV20; (c)
$G/H=2.4$, SFV14$;(d)$G/H=2.4$,SFV20;(e)$G/H=3.5$,SFV14$; (f) $G/H=3.5$,
SFV20.

In figures 7 and 8 we plot, respectively, normalised $S_{2}^{u}$ and
$S_{2}^{v}$ as functions of normalised $r$ for different cross-stream
coordinates $y$ and all three gap ratios, in SFV14 and SFV20 at $Re=12,000$.
Our scaling (36), i.e. $S_{2}^{u}/k(r/L)^{2/3}$ and $S_{2}^{v}/k(r/L)^{2/3}$
versus $r/l_{i}$, collapses the data well in both SFV stations and for all
three gap ratios. The Kolmogorov scalings $S_{2}^{u}/(\varepsilon r)^{2/3}$
and $S_{2}^{v}/(\varepsilon r)^{2/3}$ versus $r/\eta$ (which we show for
comparison even though they should not be expected to hold here) returns a
much worse collapse in all cases. There is a general tendency towards a
plateau in $S_{2}^{u}/k(r/L)^{2/3}$ for $r/l_{i}\geq 40$ which supports the
theory’s intermediate range $r^{2/3}$ prediction for the $r$-dependence of
$S_{2}$. However, there is also a difference between $G/H=1.25$ and the other
two $G/H$ values which is most notable in figure 8: the normalised $S_{2}^{v}$
takes a turn towards higher values as $r/l_{i}$ increases beyond about 90 in
the $G/H=1.25$ case but not in the other two cases. Consistently with this
observation, $C_{v}$ is also qualitatively different for $G/H=1.25$ and for
the two other values of $G/H$ (see figure 9): it takes values significantly
above $4/3$ and in fact increasing with increasing $r$ for $r/l_{i}$ larger
than about $100$ in the $G/H=1.25$ case, whereas nothing of the sort happens
in the two other $G/H$ cases. In fact, $C_{v}$ is very close to $4/3$ for all
$y$ and all $r/l_{i}$ sampled at SFV20 for both $G/H=$ $2.4$ and $3.5$. The
same is the case at SFV14 for $G/H=2.4$ but slightly less so for $G/H=3.5$
where $C_{v}$ is close to $4/3$ for $r/l_{i}$ below about 40 and slightly
decreases gradually below $4/3$ with increasing $r/l_{i}$ beyond $r/l_{i}=40$.
We note once again that the close to $4/3$ values of $C_{v}$ may signify
indirect support of the $2/3$ exponent in the power-law dependence on $r$ of
the second order structure functions. A range with such a power law is more
visible, though, in $S_{2}^{u}$ (figure 7) than $S_{2}^{v}$ (figure 8).

Figure 10: Cross-stream profiles (along $y$ within an SFV) of $\langle
L/l_{c}\rangle_{x}$, where $\langle\cdot\rangle_{x}$ denotes streamwise
averaging within the SFV. (a) $G/H=2.4$, $Re=1.0\times 10^{4}$, (b) $G/H=1.25$
and $3.5$, $Re=1.0\times 10^{4}$. (c) $G/H=1.25$, $2.4$ and $3.5$,
$Re=1.2\times 10^{4}$.

Figure 11: (a) $S_{2}^{u}/r^{2/3}$ versus $r$ for 5 different cases: (i)
SFV14, $G/H=2.4$, $Re=1.2\times 10^{4}$; (ii) SFV14, $G/H=3.5$, $Re=1.2\times
10^{4}$; (iii) SFV20, $G/H=2.4$, $Re=1.2\times 10^{4}$; (iv) SFV20, $G/H=3.5$,
$Re=1.2\times 10^{4}$; (v) SFV20, $G/H=3.5$, $R=1.5\times 10^{4}$. Different
same-colour curves correspond to different $y$ positions. (b)
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ versus $r/l_{i}$ (where $l_{i}=LR^{-3/4}$ with
$R=\sqrt{k}L/\nu$) and $S_{2}^{u}/(\langle\varepsilon\rangle r)^{2/3}$ versus
$r/\eta$ (where $\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$) for cases
(i) and (ii), i.e. SFV14. (c) Same as (b) but for cases (iii), (iv) and (v),
i.e. SFV20. (d) Same as (b) and (c) but for all cases, SFV14 and SFV20.

The departures of $C_{v}$ from $4/3$ in the near field positions SFV2.5 and
SFV5 of the $G/H=2.4$ flow (figure 5) could be accounted for by the departures
from local isotropy evidenced in figure 6. However, the departures of $C_{v}$
from $4/3$ seen in figure 9 for the $G/H=1.25$ flow case cannot be explained
that way (see figure 6). We therefore investigate the possibility that mean
shear may be responsible for these departures. Following Kaneda (2020) we
estimate a Corrsin length $l_{C}\equiv\sqrt{\langle\epsilon\rangle/S^{3}}$ and
compare it to our estimate of the integral length-scale $L$ (see penultimate
paragraph of section 4). In figure 10 we plot $\langle L/l_{C}\rangle_{x}$ as
a function of cross-stream coordinate $y$ where $\langle\cdot\rangle_{x}$
symbolises averaging over streamwise coordinate $x$ within a small field of
view (SFV) and where $l_{C}$ is calculated by taking $S=|{\partial
U_{1}\over\partial x}|+|{\partial U_{1}\over\partial y}|+|{\partial
U_{2}\over\partial x}|+|{\partial U_{2}\over\partial y}|$ to avoid over-
estimating $l_{C}$. For SFV14 and SFV20 in the flow cases $G/H=2.4$ and $3.5$
where our theory agrees well with the experimental data, $\langle
L/l_{C}\rangle_{x}$ is well below $0.5$, suggesting that mean shear is absent
at the length-scales considered for the second order structure functions. This
agrees with our observation in this section’s fourth paragraph that these
structure functions are effectively the same at $SFV14$ and $SFV20$ for
$G/H=2.4$ and $3.5$ if calculated for the instantaneous or the fluctuating
velocities. Consistently perhaps, $\langle L/l_{C}\rangle_{x}$ is larger than
$1.5$ for SFV2.5, $G/H=2.5$ where neither Kolmogorov’s nor our scalings work
(figures 3 and 4). Turbulence production and anisotropy should be taken
somehow explicitely into account in this case, which the theory in section 3
does not. Such an extension of our theory may also be needed for SFV5,
$G/H=2.4$ where our theory’s scalings also do not work well (figures 3 and 4)
and $\langle L/l_{C}\rangle_{x}$ takes values up to about 0.75.

However, $\langle L/l_{C}\rangle_{x}$ cannot be, on its own, a reliable
criterion for the applicability of the theory in section 3\. It takes values
at SFV10, $G/H=2.4$ which are comparable to those that it takes at SFV5,
$G/H=2.4$, yet our theory’s collapse for the second order structure functions
is not as bad at SFV10 as it is at SFV5 (see figures 3 and 4). More
dramatically, $\langle L/l_{C}\rangle_{x}$ is larger than 1.5 at SFV14 in the
$G/H=1.25$ flow case, yet the collapse there is good (figures 7a and 8a) with
the only exception that the exponent $2/3$ may not be present in $S_{2}^{v}$.
At SFV20 in the same flow, $\langle L/l_{C}\rangle_{x}$ takes values between
$0.5$ and $1.1$, larger than the values it takes at SFV14 and SFV20 for
$G/H=2.4$ and $3.5$ where the theory works rather well (figures 7 and 8), but
well below $1.5$, the smallest value it takes at SFV14 for $G/H=1.25$. Yet, at
SFV20 of this $G/H=1.25$ flow, the exponent $2/3$ appears absent from both
$S_{2}^{u}$ and $S_{2}^{v}$ even though our scalings collapse them both very
well. Also, $C_{v}$ is similar at SFV14 and SFV20 of this flow case (figure
9a,b). All in all, we cannot quite conclude that the Corrsin length $l_{C}$
can be used on its own as the basis for an applicability criterion of our
theory (e.g. that the theory appplies where $\langle L/l_{C}\rangle_{x}$ is
smaller than 0.5 but does not apply where it is larger than 1). Clearly, the
determination of the range and criteria of applicability of the theory
requires more research which we must leave for future studies.

We now go one step further and explore the possibility that the scaling (36)
may be able to collapse second order structure functions across flows (i.e.
different values of $G/H$), at different streamwise stations in these flows
and for different Reynolds numbers. Even though we consider only two different
SFV stations (SFV14 and SFV20), only two Reynolds numbers ($Re=12,000$ and
$15,000$, see table 1) and only two of our three flows ($G/H=2.4$ and $3.5$),
it does remain worthwhile to test for such collapse. The $G/H=1.25$ flow case
is not included in this test because of the clear differences between
$G/H=1.25$ on the one hand and $G/H=2.4$, $3.5$ on the other hand which are
evidenced in figures 7 and 8.

In figures 11a and 12a we plot, respectively, $S_{2}^{u}/r^{2/3}$ and
$S_{2}^{v}/r^{2/3}$ versus $r$ (in meters) for a wide range of $y$ positions
(within the SFV) and five different cases: (i) SFV14, $G/H=2.4$, $Re=12,000$;
(ii) SFV14, $G/H=3.5$, $Re=12,000$; (iii) SFV20, $G/H=2.4$, $Re=12,000$; (iv)
SFV20, $G/H=3.5$, $Re=12,000$; (v) SFV20, $G/H=3.5$, $R=15,000$. There is no
collapse. As a second step, we plot in figures 11b and 12b,
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ and $S_{2}^{v}/(k^{3/2}r/L)^{2/3}$
respectively, both versus $r/l_{i}$, for cases (i) and (ii) which are for
SFV14. The collapse is defensible but better for $S_{2}^{u}$ than for
$S_{2}^{v}$. These two structure functions are also plotted with Kolmogorov
scalings in these two figures for comparison and it is clear that the
Kolmogorov scalings cannot achieve collapse, particularly for $S_{2}^{u}$. Our
third step is to test for collapse in SFV20. For this, we plot
$S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ and $S_{2}^{v}/(k^{3/2}r/L)^{2/3}$ versus
$r/l_{i}$ for cases (iii), (iv) and (v) which are for SFV20 (see figures 11c
and 12c). Conclusions are similar. In a final step, we plot all cases (i),
(ii), (iii), (iv) and (v) together in figures 11d and 12d. There is a
defensible collapse for $S_{2}^{u}$ across these two flows, two flow stations,
and two Reynolds numbers, much better than if a collapse was attempted in
Kolmogorov variables. One may say that there is also a more or less acceptable
collapse for $S_{2}^{v}$, but it is less sharp than the collapse of
$S_{2}^{u}$. However, a careful look at the plots, in particular figure 12b,c,
reveals that there remains some dependence on $G/H$ at SFV14 and SFV20, i.e.
some dependence of the similarity function $f_{v}(r/l_{i},G/H)$ on inlet
conditions at these stations.

##  6 Conclusions and new research directions

We developed a theory of non-homogeneous turbulence which we applied to free
turbulent shear flows. The theory predicts that second order structure
functions scale according to equation (36) in non-homogeneous turbulence, i.e.

$S_{2}^{u}=k(r/L)^{2/3}f_{u}(r/l_{i})$ (38)

and

$S_{2}^{v}=k(r/L)^{2/3}f_{v}(r/l_{i})$ (39)

where $l_{i}$ is an inner length-scale which differs from but has the same
dependence on viscosity as the Kolmogorov length-scale $\eta$. These scalings
take non-homogeneity explicitely into account and are different from
Kolmogorov scalings. However they become identical to Kolmogorov scalings in
homogeneous stationary turbulence.

Figure 12: (a) $S_{2}^{v}/r^{2/3}$ versus $r$ for 5 different cases: (i)
SFV14, $G/H=2.4$, $Re=1.2\times 10^{4}$; (ii) SFV14, $G/H=3.5$, $Re=1.2\times
10^{4}$; (iii) SFV20, $G/H=2.4$, $Re=1.2\times 10^{4}$; (iv) SFV20, $G/H=3.5$,
$Re=1.2\times 10^{4}$; (v) SFV20, $G/H=3.5$, $R=1.5\times 10^{4}$. Different
same-colour curves correspond to different $y$ positions. (b)
$S_{2}^{v}/(k^{3/2}r/L)^{2/3}$ versus $r/l_{i}$ (where $l_{i}=LR^{-3/4}$ with
$R=\sqrt{k}L/\nu$) and $S_{2}^{v}/(\langle\varepsilon\rangle r)^{2/3}$ versus
$r/\eta$ (where $\eta=(\nu^{3}/\langle\varepsilon\rangle)^{1/4}$) for cases
(i) and (ii), i.e. SFV14. (c) Same as (b) but for cases (iii), (iv) and (v),
i.e. SFV20. (d) Same as (b) and (c) but for all cases, SFV14 and SFV20.

The theory is based on concepts of inner and outer similarity which are used
to characterise the non-homogeneity of two-point statistics and on a new
inner-outer dissipation equivalence hypothesis. Future research is required to
establish the breadth of applicability of this hypothesis and of such
similarity assumptions across a variety of non-homogeneous turbulent flows.
Isotropy is not explicitly assumed in the theory, but our analysis of the data
of Chen et al. (2021) suggests that it may be required at some level.
Turbulence production is not explicitely taken into account either and our
data analysis also suggests that the theory requires modifications when
production cannot be neglected. These are issues which will need to be
addressed in future research along with the question of the particular
quantities which may obey similarity at the inner and/or outer level, and
those which may not. Another important related issue which also requires
future research is the issue of large-scale coherent structures and their
effects on or just relations to similarity and non-homogeneity and turbulence
interscale/interspace transfers. Are they, for example, responsible for the
inner-outer equivalence for turbulence dissipation as a property of the type
of non-homogeneity considered here? How general is this inner-outer
equivalence? We should start thinking in terms of different classes of non-
homogeneity depending, for example, on the degree of significance of
turbulence production and on differences in coherent structure dynamics in the
presence or absence of turbulence production. Alves Portela et al. (2020)
showed that non-homogeneity and coherent structures are in fact key for the
proper understanding of interscale turbulence transfers in near-field non-
homogeneous turbulent wakes. Future research should probably combine their
approach with the present one and investigate these new questions.

The data of Chen et al. (2021) which have been used here to test the theory’s
new scalings are from three different turbulent wakes of side-by-side
identical square prisms. These three non-homogeneous turbulent flows differ by
their gap ratio $G/H$ and represent three qualitatively different flow
regimes. Good agreement with the theory’s scalings (38) and (39) has been
found in all three flows, but far enough from the prisms where the turbulence
is clearly not locally homogeneous but nevertheless exhibits some indications
of local isotropy. Whereas the scalings (38) and (39) collapse the structure
functions rather well in all three flows at these far enough positions (which
are actually not so far as they are only between $10H$ and $20H$), the
intermediate $r^{2/3}$ power law predicted by the theory is more clearly
supported by the data for $S_{2}^{u}$ than for $S_{2}^{v}$. This may or may
not be a Reynolds number effect, which is another issue needing to be
addressed in future studies. However, there are also clear qualitative
differences between $S_{2}^{v}$ for $G/H=1.25$ and $S_{2}^{v}$ for $G/H=2.4$
and $3.5$ even if (39) collapses both cases. These differences point to a
power law for $S_{2}^{v}(r)$ which may be slightly different from $r^{2/3}$ in
the $G/H=1.25$ case. Differences of this sort can be exploited in future
investigations which could lead to a much deeper understanding of scale-by-
scale energy scalings and energy transfers in non-homogeneous turbulent flows.

Finally it is worth mentioning that (38) and (39) are also able to more or
less collapse structure functions for different cross-stream positions, two
far enough streamwise stations, two flow cases $G/H=2.4$, $3.5$ and two
Reynolds numbers (see figures 11 and 12). However, some dependence on inlet
conditions remains and one may need to take into account some dependence of
the similarity functions, particularly $f_{v}$, on $G/H$.

Acknowledgements This work was supported by JCV’s Chair of Excellence CoPreFlo
funded by I-SITE-ULNE/MEL/Région Hauts de France. We are grateful to the
authors of Chen et al. (2021) for their data, in particular Dr Christophe
Cuvier who has been essential in the setting up of their experiment and data
collection.

##  Appendix A Consistency between the inner scale-by-scale energy balance and
the inner similarity hypothesis

Using (20) (which we obtained from the outer scale-by-scale balance in section
3.1) and (24), the high Reynolds number inner scale-by-scale energy balance
(23) becomes

$g_{l}^{-1/2}g_{X}\alpha f_{iX}(r/l_{i})+g_{l}^{-1/2}g_{3}\gamma
f_{i3}(r/l_{i})+g_{l}^{-1/2}g_{p}\beta f_{ip}(r/l_{i})=$
$-1+C_{\varepsilon}^{-1}{\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$ (40)

in terms of the dimensionless proportionality constants
$\alpha=(V_{oX}/V_{o2})^{3}/C_{\varepsilon}$,
$\beta=(V_{op}/V_{o2})^{3}/C_{\varepsilon}$ and
$\gamma=(V_{o3}/V_{o2})^{3}/C_{\varepsilon}$ (some of these constants, though
not all, could in principle be zero). Irrespective of which transport/transfer
term we keep in this high Reynolds number inner energy budget, i.e. which of
$g_{l}^{-1/2}g_{X}$, $g_{l}^{-1/2}g_{3}$ and $g_{l}^{-1/2}g_{p}$ we let tend
to zero in the limit $R\to\infty$, we end up with a function of $r/l_{i}$
being equal to
$-1+C_{\varepsilon}^{-1}{\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$ which is
possible either if $C_{\varepsilon}$ is constant, i.e. independent of ${\bf
X}$, or if ${\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})=0$ which means that the
longitudinal second order structure function should be a harmonic function of
$r$, which is not realistic. As we want a theory for non-homogeneous
turbulence where $C_{\varepsilon}$ depends on ${\bf X}$, we need to modify
some of the inner similarity assumptions (14), (15) and/or (16).

Firstly, we keep the interscale turbulence energy transfer in the high
Reynolds number inner scale-by-scale energy balance. This requires

$g_{l}^{-1/2}(R)g_{3}(R)=Const$ (41)

independent of $R$. Secondly, we take $g_{l}^{-1/2}g_{X}=Const$ and
$g_{l}^{-1/2}g_{p}=Const$ to allow for the possibility of both or either of
the turbulent transport and the velocity-pressure gradient correlation terms
to be present in the high Reynolds number inner scale-by-scale energy balance.
(These terms can be made to be absent from this balance by artificially
setting $\alpha$ and/or $\beta$ equal to $0$ respectively.) We chose to modify
the inner similarity forms of these two terms as they concern statistics which
do not only involve two-point velocity differences whereas this is not the
case of the interscale transfer rate term which does. Looking at (40), the
modification that we are forced to make must ensure that
$C_{\varepsilon}(1+\alpha f_{iX}+\gamma f_{i3}+\beta f_{ip})$ is independent
of ${\bf X}$ (explicitely). We must therefore replace $f_{iX}$ and $f_{ip}$ in
(15) and (16) by the following functions of $r/l_{i}$ and ${\bf X}$:

$f_{iX}(r/l_{i},{\bf X})={F_{iX}(r/l_{i})\over C_{\varepsilon}({\bf
X})}-A-Bf_{i3}(r/l_{i})$ (42) $f_{ip}(r/l_{i},{\bf X})={F_{ip}(r/l_{i})\over
C_{\varepsilon}({\bf X})}-C-Df_{i3}(r/l_{i})$ (43)

where $A$, $B$, $C$ and $D$ are dimensionless constants and $F_{iX}$ and
$F_{ip}$ are functions of $r/l_{i}$ only. Note that if the turbulent transport
term is not present in the high Reynolds number inner scale-by-scale energy
balance then $F_{iX}=0$ and $A=B=0$, and if the velocity-pressure gradient
correlation term is not present in that balance then $F_{ip}=0$ and $C=D=0$.
All possibilities are therefore covered.

With (42) and (43), the requirement that $C_{\varepsilon}(1+\alpha
f_{iX}+\gamma f_{i3}+\beta f_{ip})$ must be independent of ${\bf X}$, and in
fact equal to ${\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$, yields
$\gamma=\alpha B+\beta D$, $1=\alpha A+\beta C$ and $\alpha F_{iX}+\beta
F_{ip}={\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$. Taking into account (14),
(15), (16) and (17), this latter equation represents the following high
Reynolds number balance for $r\ll l_{o}$:

${\bf\nabla}_{X}\cdot\langle{\bf u}_{X}^{\prime}(\delta
u_{1})^{2}\rangle+{\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta
u_{1})^{2}\rangle+2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta
p\rangle=$ $-\varepsilon_{1}+{V_{o2}^{3}\over
l_{o}}{\bf\nabla}^{2}_{r/l_{i}}f_{i2}(r/l_{i})$ (44)

Not only does the theory not lead to unrealistic conclusions with the
modifications (42) and (43) of the inner similarity forms, it also leads to
this, arguably interesting, high Reynolds number balance (44). The possibility
that this balance may in fact involve only ${\bf\nabla}_{X}\cdot\langle{\bf
u}_{X}^{\prime}(\delta u_{1})^{2}\rangle$ and
${\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta u_{1})^{2}\rangle$ or only
${\bf\nabla}_{r}\cdot\langle\delta{\bf u}(\delta u_{1})^{2}\rangle$ and
$2\langle\delta u_{1}{\partial\over\partial X_{1}}\delta p\rangle$ exists and
is covered by our approach (just take $F_{ip}=0$ and $C=D=0$ in the former
case and $F_{iX}=0$ and $A=B=0$ in the latter). One should not interpret (44)
to necessarily mean that interscale energy transfer and turbulence dissipation
are balanced by both turbulent transport in space and the velocity-pressure
gradient correlation term. They are balanced by at least one or the other or
both. This is a consequence of the choice we made not to modify the inner
similarity form of the interscale transfer rate and keep $f_{i3}$ as function
of $r/l_{i}$ only.

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