2. Theoretical framework and energy cascade in homogeneous turbulence

Turbulence energy cascades consist of interscale exchanges of turbulent energy that can be formulated in terms of velocity differences $\delta \boldsymbol {u} \equiv \frac {1}{2}[\boldsymbol {u} (\boldsymbol {x}^{+},t) - \boldsymbol {u} (\boldsymbol {x}^{-},t)]$ between the fluid velocity $\boldsymbol {u}$ at position $\boldsymbol {x}^{+}$, and the fluid velocity $\boldsymbol {u}$ at position $\boldsymbol {x}^{-}$ at the same time $t$. Following Hill (Reference Hill2001, Reference Hill2002) and using the coordinate transformation $\boldsymbol {x}^{+} = \boldsymbol {X} + \boldsymbol {r}$, $\boldsymbol {x}^{-} = \boldsymbol {X} - \boldsymbol {r}$, the following equation is derived from the incompressible Navier–Stokes equation:

(2.1)\begin{equation} \frac{\partial}{\partial t} \delta \boldsymbol{u} + \boldsymbol{u}_X\boldsymbol{\cdot} \boldsymbol{\nabla}_X \delta \boldsymbol{u} + \delta \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}_r \delta \boldsymbol{u} ={-}\boldsymbol{\nabla}_X\delta p + \frac{\nu}{2} (\boldsymbol{\nabla}^{2}_X + \boldsymbol{\nabla}^{2}_r) \delta \boldsymbol{u}, \end{equation}

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

An energy equation may be obtained from (2.1) by a scalar product with $\delta \boldsymbol {u}$:

(2.2)\begin{align} &\frac{\partial }{\partial t} \vert \delta \boldsymbol{u}\vert^{2} + \boldsymbol{\nabla}_X \boldsymbol{\cdot}(\boldsymbol{u}_X \vert \delta \boldsymbol{u}\vert^{2}) + \boldsymbol{\nabla}_r \boldsymbol{\cdot} (\delta \boldsymbol{u}\,\vert \delta \boldsymbol{u}\vert^{2}) \nonumber\\ &\quad ={-}2\boldsymbol{\nabla}_X \boldsymbol{\cdot} (\delta \boldsymbol{u}\,\delta p) + \frac{\nu} {2} (\boldsymbol{\nabla}^{2}_X + \boldsymbol{\nabla}^{2}_r) \vert \delta \boldsymbol{u}\vert^{2} -\frac{1}{2}(\varepsilon^{+} + \varepsilon^{-}), \end{align}

where use is made of incompressibility and where $\varepsilon ^{+} \equiv \nu ({\partial u_{i} (\boldsymbol {x}^{+}, t)}/{\partial x_{j}^{+}})^{2}$, $\varepsilon ^{-} \equiv \nu ({\partial u_{i} (\boldsymbol {x}^{-}, t)}/{\partial x_{j}^{-}})^{2}$ (with summations over $i,j=1,2,3$). This is the equation that 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 studied mainly and mostly for homogeneous turbulence (Batchelor Reference Batchelor1953; Tennekes & Lumley Reference Tennekes and Lumley1972; Mathieu & Scott Reference Mathieu and Scott2000; Pope Reference Pope2000; Sagaut & Cambon Reference Sagaut and Cambon2018). Averaging the above equation over realisations, which is equivalent to averaging over $\boldsymbol {X}$ in homogeneous turbulence, one gets (see Frisch Reference Frisch1995)

(2.3)\begin{equation} \frac{\partial}{\partial t} \langle\vert \delta \boldsymbol{u}\vert^{2}\rangle + \boldsymbol{\nabla}_r \boldsymbol{\cdot} \langle\delta \boldsymbol{u}\vert \delta \boldsymbol{u}\vert^{2}\rangle = \frac{\nu}{2} \boldsymbol{\nabla}^{2}_r \langle\vert \delta \boldsymbol{u}\vert^{2}\rangle - \langle\varepsilon\rangle, \end{equation}

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

Equation (2.3) shows that the change in time of the turbulent kinetic energy associated with the separation vector $\boldsymbol {r}$ occurs by turbulence dissipation ($-\langle \varepsilon \rangle$ term in the equation), viscous diffusion in $\boldsymbol {r}$ space ($({\nu }/{2})\boldsymbol {\nabla }^{2}_r \langle \vert \delta \boldsymbol {u}\vert ^{2}\rangle$ term) and energy exchanges between separation vectors (term $\boldsymbol {\nabla }_r \boldsymbol {\cdot } \langle \delta \boldsymbol {u}\,\vert \delta \boldsymbol {u}\vert ^{2}\rangle$, which is conservative in $\boldsymbol {r}$ space).

To make the link with the energy cascade more apparent, one can follow Zhou & Vassilicos (Reference Zhou and Vassilicos2020) and define the average turbulent kinetic energy in scales smaller than $r=\vert \boldsymbol {r}\vert$ as follows: $E(r,t) \equiv 3/(4 {\rm \pi}r^3)\int _{S(r)} {\rm d}^{3}\boldsymbol {r}\, \langle \vert \delta \boldsymbol {u}\vert ^{2}\rangle$. This is an average over realisations (equivalently, over $\boldsymbol {X}$ in homogeneous turbulence) and over a sphere of radius $r$ in $\boldsymbol {r}$ space. It is easy to verify that $E=0$ at $r=0$, that $E$ tends to $\frac {1}{2}\langle \vert \boldsymbol {u}\vert ^{2}\rangle$ 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) that 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 (2.3) and using the divergence theorem:

(2.4)\begin{equation} \frac{\partial}{\partial t} E(r,t) + \frac{3}{4{\rm \pi}}\int {\rm d}\varOmega_{\boldsymbol{r}} \left\langle \frac{\hat{\boldsymbol{r}} \boldsymbol{\cdot} \delta \boldsymbol{u}}{r}\,\vert \delta \boldsymbol{u}\vert^{2}\right\rangle \approx{-} \langle\varepsilon\rangle, \end{equation}

where $\hat {\boldsymbol {r}}\equiv \boldsymbol {r}/r$ and the integral $\int {\rm d}\varOmega _{\boldsymbol {r}}$ is over the solid angle in $\boldsymbol {r}$ 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 Reference Valente and Vassilicos2015).

The interscale transfer rate $3/(4 {\rm \pi})\int {\rm d}\varOmega _{\boldsymbol {r}}\,\langle {\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r} \vert \delta \boldsymbol {u}\vert ^{2}\rangle$ (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 $3/(4 {\rm \pi})\int {\rm d} \varOmega _{\boldsymbol {r}}\,\langle {\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r} \vert \delta \boldsymbol {u}\vert ^{2}\rangle$, 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. $\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u} < 0$ in $3/(4 {\rm \pi})\int {\rm d}\varOmega _{\boldsymbol {r}}\, \langle {\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r} \vert \delta \boldsymbol {u}\vert ^{2}\rangle$, whereas an inverse cascade on average (from small to large scales) results from predominance of stretching, i.e. $\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u} > 0$ in $3/(4 {\rm \pi})\int {\rm d}\varOmega _{\boldsymbol {r}}\, \langle {\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u}}/{r} \vert \delta \boldsymbol {u}\vert ^{2}\rangle$ (Vassilicos Reference Vassilicos2015). By virtue of incompressibility, $\int {\rm d}\varOmega _{\boldsymbol {r}}\,\hat {\boldsymbol {r}} \boldsymbol {\cdot } \delta \boldsymbol {u} = 0$; 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 $\vert ({\partial }/{\partial t}) E(r,t)\vert \ll \langle \varepsilon \rangle$ for $r$ much smaller than the integral length-scale $L$ in the limit of very high Reynolds number. This hypothesis leads to

(2.5)\begin{equation} \frac{3}{4{\rm \pi}}\int {\rm d}\varOmega_{\boldsymbol{r}} \left\langle \frac{\hat{\boldsymbol{r}} \boldsymbol{\cdot} \delta \boldsymbol{u}}{r}\,\vert \delta \boldsymbol{u}\vert^{2}\right\rangle \approx{-} \langle\varepsilon\rangle \end{equation}

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

If the Kolmogorov equilibrium hypothesis can be stretched to scales $r$ as close to $L$ as possible, then (2.5) yields the well-known Taylor–Kolmogorov dissipation scaling

(2.6)\begin{equation} \langle\varepsilon\rangle = C_{\varepsilon} k^{3/2}/L . \end{equation}

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 (Reference Pope2000):

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

(2.7)\begin{equation} \langle\vert \delta \boldsymbol{u}\vert^{2}\rangle \sim (\langle\varepsilon\rangle r)^{2/3} \end{equation}

in the inertial range of scales $r$.

In the following section we introduce two similarity hypotheses for non-homogeneous turbulence that replace the highly simplifying premise of homogeneity and the two Kolmogorov similarity hypotheses. We also introduce an inner–outer equivalence hypothesis for turbulence dissipation that 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 interscale and interspace 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 (2.7), 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 (2.1) is a vector equation. Our analysis in this section can be carried out effectively unchanged for (2.2), but we want to compare our conclusions with experimental data so we concentrate on the first component of (2.1), which, when multiplied by $\delta u_{1}$ (the first component of $\delta \boldsymbol {u}$), yields an equation that is effectively the same as (2.2) except that $\vert \delta \boldsymbol {u}\vert ^{2}$ is replaced by $(\delta u_{1})^{2}$, the pressure–velocity term is replaced by $-2 \delta u_{1} ({\partial }/{\partial X_{1}}) \delta p$, and $-\frac {1}{2} (\varepsilon ^{+} + \varepsilon ^{-})$ is replaced by $-\frac {1}{2} (\varepsilon _{1}^{+} + \varepsilon _{1}^{-})$, where $\varepsilon _{1}^{\pm } \equiv \nu ({\partial u_{1}(\boldsymbol {x}^{\pm },t)}/{\partial x_{j}})^{2}$ (with summation over $j=1,2,3$).

We now restrict our attention to turbulent flows, such as those of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021), that are statistically stationary in time at every location of the flow, and we use the Reynolds decomposition $\boldsymbol {u}_{X} = \langle \boldsymbol {u}_{X}\rangle + \boldsymbol {u}_{X}^{\prime }$ by averaging over realisations or equivalently over time (but not over $\boldsymbol {X}$) in this statistically stationary context. We therefore obtain the following equation for $\langle (\delta u_{1})^{2}\rangle$:

(3.1)\begin{align} &\langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X \langle (\delta u_{1})^{2}\rangle + \boldsymbol{\nabla}_X \boldsymbol{\cdot} \langle \boldsymbol{u}_X^{\prime} (\delta u_{1})^{2}\rangle + \boldsymbol{\nabla}_r \boldsymbol{\cdot} \langle \delta \boldsymbol{u} (\delta u_{1})^{2}\rangle+2\left\langle \delta u_{1} \frac{\partial}{\partial X_{1}} \delta p\right\rangle\nonumber\\ &\quad = \frac{\nu}{2} (\boldsymbol{\nabla}^{2}_X + \boldsymbol{\nabla}^{2}_r) \langle (\delta u_{1})^{2}\rangle -\varepsilon_{1}, \end{align}

where $\varepsilon _{1} \equiv \frac {1}{2}\langle \varepsilon _{1}^{+} + \varepsilon _{1}^{-}\rangle$ and use was made of the incompressibility of $\langle \boldsymbol {u}_X\rangle$. This equation is an energy balance involving mean flow advection of $\langle (\delta u_{1})^{2}\rangle$ (the term $\langle \boldsymbol {u}_X\rangle \boldsymbol {\cdot } \boldsymbol {\nabla }_X \langle (\delta u_{1})^{2}\rangle$), turbulent transport rate of $(\delta u_{1})^{2}$ in physical space (the term $\boldsymbol {\nabla }_X \boldsymbol {\cdot } \langle \boldsymbol {u}_X^{\prime } (\delta u_{1})^{2}\rangle$), interscale energy transfer rate $\boldsymbol {\nabla }_r \boldsymbol {\cdot } \langle \delta \boldsymbol {u} (\delta u_{1})^{2}\rangle$, the velocity–pressure gradient correlation term $2\langle \delta u_{1} ({\partial }/{\partial X_{1}}) \delta p\rangle$, viscous diffusion of $\langle (\delta u_{1})^{2}\rangle$ in $\boldsymbol {X}$ and $\boldsymbol {r}$ spaces, and turbulence dissipation rate $\varepsilon _{1}$. Note that no Reynolds decomposition has been applied to $\delta \boldsymbol {u}$, similarly to Klingenberg, Oberlack & Pluemacher (Reference Klingenberg, Oberlack and Pluemacher2020), 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 § 5). In the current setting, the term $\boldsymbol {\nabla }_X \boldsymbol {\cdot } \langle \boldsymbol {u}_X^{\prime } (\delta u_{1})^{2}\rangle$ 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 (Reference Kaneda2020) 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 § 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 $\boldsymbol {r} = (r,0,0)$, but our approach can also be applied to the transverse structure function, i.e. $\langle (\delta u_{2})^{2}\rangle$ with $\boldsymbol {r} = (r,0,0)$. We start with our inner/outer similarity hypotheses for non-homogeneous turbulence, that there exists a substantial class of turbulent flows with regions in them where each interscale/interspace 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 $l_{o} (\boldsymbol {X})$ (some integral/correlation length-scale independent of viscosity) and an inner length-scale $l_{i}(\boldsymbol {X})$ (dependent on viscosity) such that $l_{i} (\boldsymbol {X}) \ll l_{o} (\boldsymbol {X})$ for high enough Reynolds numbers. The mathematical expressions 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 (3.1) as follows.

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

(3.2)\begin{gather} \langle (\delta u_{1})^{2}\rangle = V_{o2}^{2}(\boldsymbol{X})\,f_{o2} (r/l_{o}), \end{gather}

(3.3)\begin{gather}\boldsymbol{\nabla}_r \boldsymbol{\cdot} \langle \delta \boldsymbol{u} (\delta u_{1})^{2}\rangle = \frac{V_{o3}^{3}(\boldsymbol{X})}{l_{o}}\,f_{o3} (r/l_{o}), \end{gather}

(3.4)\begin{gather}\boldsymbol{\nabla}_X \boldsymbol{\cdot} \langle \boldsymbol{u}_X^{\prime} (\delta u_{1})^{2}\rangle = \frac{V_{oX}^{3}(\boldsymbol{X})}{l_{o}}\,f_{oX} (r/l_{o}), \end{gather}

(3.5)\begin{gather}2\left\langle \delta u_{1} \frac{\partial}{\partial X_{1}} \delta p\right\rangle = \frac{V_{op}^{3}(\boldsymbol{X})}{l_{o}}\,f_{op} (r/l_{o}). \end{gather}

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

(3.6)\begin{gather} \langle (\delta u_{1})^{2}\rangle = V_{i2}^{2}(\boldsymbol{X})\,f_{i2} (r/l_{i}), \end{gather}

(3.7)\begin{gather}\boldsymbol{\nabla}_r \boldsymbol{\cdot} \langle \delta \boldsymbol{u} (\delta u_{1})^{2}\rangle = \frac{V_{i3}^{3}(\boldsymbol{X})}{l_{i}}\,f_{i3} (r/l_{i}), \end{gather}

(3.8)\begin{gather}\boldsymbol{\nabla}_X \boldsymbol{\cdot} \langle \boldsymbol{u}_X^{\prime} (\delta u_{1})^{2}\rangle = \frac{V_{iX}^{3}(\boldsymbol{X})}{l_{i}}\,f_{iX} (r/l_{i}), \end{gather}

(3.9)\begin{gather}2\left\langle \delta u_{1} \frac{\partial}{\partial X_{1}} \delta p\right\rangle = \frac{V_{ip}^{3}(\boldsymbol{X})}{l_{i}}\,f_{ip} (r/l_{i}). \end{gather}

Here, we have introduced eight velocity-scales $V_{o2}$, $V_{o3}$, $V_{oX}$, $V_{op}$, $V_{i2}$, $V_{i3}$, $V_{iX}$, $V_{ip}$, which are dependent explicitly on $\boldsymbol {X}$, 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 dependent explicitly on $\boldsymbol {X}$. We warn that as one progresses in the argument, a need appears to modify the inner similarity assumptions (3.8) and (3.9). This is explained in Appendix A.

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. (Reference Alves Portela, Papadakis and Vassilicos2017) 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 that are large enough for $\varepsilon _{1}$ to depend on $r$, but what we assume here effectively 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

(3.10)\begin{equation} \varepsilon_{1} = C_{\varepsilon}(\boldsymbol{X})\,\frac{V_{o2}^{3}}{l_{o}}, \end{equation}

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 § 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 (3.2)–(3.5) and (3.10) into (3.1), we are led to

(3.11)\begin{align} &\frac{2l_{o}}{V_{o2}^{2}}(\langle \boldsymbol{u}_X\rangle\boldsymbol{\cdot}\boldsymbol{\nabla}_X V_{o2})\,f_{o2}(r/l_{o}) - (V_{o2}^{{-}1} \langle \boldsymbol{u}_X\rangle\boldsymbol{\cdot} \boldsymbol{\nabla}_X l_{o})\,\frac{r}{l_{o}}\,f_{o2}^{\prime}(r/l_{o})\nonumber\\ &\qquad +\frac{V_{oX}^{3}}{V_{o2}^{3}}\,f_{oX}(r/l_{o}) +\frac{V_{o3}^{3}}{V_{o2}^{3}}\,f_{o3}(r/l_{o}) +\frac{V_{op}^{3}}{V_{o2}^{3}}\,f_{op}(r/l_{o}) \nonumber\\ &\quad ={-}C_{\varepsilon} + R^{{-}1} \frac{l_{o}^{2}}{V_{o2}^{2}}\, [\boldsymbol{\nabla}^{2}_X V_{o2}^{2}\,f_{o2}(r/l_{o})]+ R^{{-}1}\boldsymbol{\nabla}^{2}_{r/l_{0}} f_{o2}(r/l_{o}), \end{align}

where $R\equiv 2V_{o2}l_{o}/\nu$ is a naturally appearing local (in $\boldsymbol {X}$) Reynolds number, $\boldsymbol {\nabla }^{2}_{r/l_{0}}$ is the Laplacian with respect to $\boldsymbol {r}/l_{o}$ rather than just $\boldsymbol {r}$, 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

(3.12)\begin{align} & \frac{2l_{o}}{V_{o2}^{2}}(\langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X V_{o2})\,f_{o2}(r/l_{o}) - (V_{o2}^{{-}1} \langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X l_{o})\,\frac{r}{l_{o}}\,f_{o2}^{\prime}(r/l_{o})\nonumber\\ &\quad +\frac{V_{oX}^{3}}{V_{o2}^{3}}\,f_{oX}(r/l_{o}) +\frac{V_{o3}^{3}}{V_{o2}^{3}}\,f_{o3}(r/l_{o}) +\frac{V_{op}^{3}}{V_{o2}^{3}}\,f_{op}(r/l_{o}) \approx{-}C_{\varepsilon} (\boldsymbol{X}) . \end{align}

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 (3.12) is independent of $r/l_{o}$ implies

(3.13)\begin{equation} \frac{2l_{o}}{V_{o2}^{2}}(\langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X V_{o2}) \sim V_{o2}^{{-}1} \langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X l_{o} \sim \frac{V_{oX}^{3}}{V_{o2}^{3}}\sim \frac{V_{o3}^{3}}{V_{o2}^{3}}\sim \frac{V_{op}^{3}}{V_{o2}^{3}} \sim C_{\varepsilon} (\boldsymbol{X}) \end{equation}

because $\boldsymbol {X}$ and $r$ are independent variables. These proportionalities imply that all four outer velocity scales are effectively the same, i.e. the same functions of $\boldsymbol {X}$, if $C_{\varepsilon }$ is independent of $\boldsymbol {X}$. However, if $C_{\varepsilon }$ does depend on spatial position $\boldsymbol {X}$, as has been found to be the case in the turbulent wakes of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021), 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 $\boldsymbol {X}$. Furthermore, (3.13) 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 § 3.2, and the conclusion (3.13) of the outer scale-by-scale balance is used in Appendix A and in §§ 3.4 and 3.5.

3.2. Inner scale-by-scale energy balance

Injecting (3.6)–(3.9) and (3.10) into (3.1), we are led to

(3.14)\begin{align} & 2V_{i2}(\langle\boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X V_{i2})\, f_{i2}(r/l_{i}) - \frac{V_{i2}^{2}}{l_{i}} (\langle\boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X l_{i})\,\frac{r}{l_{i}}\,f_{i2}^{\prime}(r/l_{i})\nonumber\\ &\qquad +\frac{V_{iX}^{3}}{l_{i}}\,f_{iX}(r/l_{i}) +\frac{V_{i3}^{3}}{l_{i}}\,f_{i3}(r/l_{i}) +\frac{V_{ip}^{3}}{l_{i}}\,f_{ip}(r/l_{i}) \nonumber\\ &\quad ={-}C_{\varepsilon}\frac{V_{o2}^{3}}{l_{o}} + \frac{\nu}{2} (\boldsymbol{\nabla}^{2}_X + \boldsymbol{\nabla}^{2}_r) (V_{i2}^{2}\,f_{i2}(r/l_{i})), \end{align}

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, (3.14) becomes

(3.15)\begin{align} &\sqrt{g_{2}(R)}\,\frac{2l_{o}}{V_{o2}^{2}}\,[\langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X (V_{o2}\sqrt{g_{2}(R)})]\,f_{i2}(r/l_{i}) \nonumber\\ &\qquad - g_{2}(R)\,[V_{o2}^{{-}1}\,g_{l}^{{-}1/2}(R)\langle \boldsymbol{u}_X\rangle \boldsymbol{\cdot} \boldsymbol{\nabla}_X (l_{o}\sqrt{g_{l}(R)})]\,\frac{r}{l_{i}}\,f_{i2}^{\prime}(r/l_{i})\nonumber\\ &\qquad +g_{l}^{{-}1/2}g_{X}\,\frac{V_{oX}^{3}}{V_{o2}^{3}}\,f_{iX}(r/l_{i}) +g_{l}^{{-}1/2}g_{3}\,\frac{V_{o3}^{3}}{V_{o2}^{3}}\,f_{i3}(r/l_{i}) +g_{l}^{{-}1/2}g_{p}\,\frac{V_{op}^{3}}{V_{o2}^{3}}\,f_{ip}(r/l_{i})\nonumber\\ &\quad ={-}C_{\varepsilon} + R^{{-}1}\left[\frac{l_{o}^{2}}{V_{o2}^{2}}\boldsymbol{\nabla}^{2}_X (V_{o2}^{2}g_{2} f_{i2})\right] + R^{{-}1}g_{2}g_{l}^{{-}1}\boldsymbol{\nabla}^{2}_{r/l_{i}} f_{i2}(r/l_{i}), \end{align}

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

Taking the limit $R\gg 1$, the mean flow advection terms in the first line of (3.15) tend to $0$ because $g_{2}(R)$ tends to $0$ and so does the viscous diffusion term in $\boldsymbol {X}$ space on the right-hand side. In this high Reynolds number limit we are therefore left with

(3.16)\begin{align} &g_{l}^{{-}1/2}g_{X}\,\frac{V_{oX}^{3}}{V_{o2}^{3}}\,f_{iX}(r/l_{i}) +g_{l}^{{-}1/2}g_{3}\,\frac{V_{o3}^{3}}{V_{o2}^{3}}\,f_{i3}(r/l_{i}) +g_{l}^{{-}1/2}g_{p}\,\frac{V_{op}^{3}}{V_{o2}^{3}}\,f_{ip}(r/l_{i})\nonumber\\ &\quad ={-}C_{\varepsilon} + R^{{-}1} g_{2}g_{l}^{{-}1}\boldsymbol{\nabla}^{2}_{r/l_{i}} f_{i2}(r/l_{i}) . \end{align}

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

(3.17)\begin{equation} g_{2}(R)/g_{l}(R) \sim R . \end{equation}

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 (3.2) and (3.6) hold in the intermediate range of scales $l_{i}\ll r \ll l_{o}$, i.e.

(3.18)\begin{equation} \langle (\delta u_{1})^{2}\rangle = V_{o2}^{2}(\boldsymbol{X})\,f_{o2} (r/l_{o})=V_{i2}^{2}(\boldsymbol{X})\,f_{i2} (r/l_{i}). \end{equation}

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

(3.19)\begin{equation} f_{o2} (r/l_{o})=g_2(R)f_{i2}((r/l_o) g^{-1/2}(R)), \end{equation}

which implies

\begin{align*} \frac{{\rm d}}{{\rm d}R}\left[g_{2} (R)\,f_{i2} \left(\frac{r}{l_{o}}\,g_{l}^{-1/2}(R)\right)\right] =0\end{align*}

and therefore

\begin{align*} \frac{\dfrac{{\rm d}}{{\rm d}R}g_{2}}{g_{2}} = - \frac{r}{l_{o}}\frac{f'_{i2}}{f_{i2}} \frac{{\rm d}}{{\rm d}R}(g_{l}^{-1/2})\end{align*}

(where $f'_{i2}$ is the derivative of $f_{i2}$ with respect to its argument). Hence $({r}/{l_{o}} )({f'_{i2}}/{f_{i2}})$ is independent of $r/l_o$, which in turn implies

(3.20)\begin{equation} f_{i2} (r/l_{i}) \sim (r/l_{i})^{n} \end{equation}

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

(3.21)\begin{equation} g_{2}(R)\,g_{l}^{{-}n/2}(R) = {\rm const}. \end{equation}

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: (3.17) and (3.21). We therefore need one more relation, and for this we introduce a hypothesis that we term inner–outer equivalence for turbulence dissipation. This hypothesis replaces Kolmogorov's equilibrium hypothesis for homogeneous turbulence in non-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 (3.10), 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

(3.22)\begin{equation} g_{l}(R) = g_{2}^{3}(R). \end{equation}

Relations (3.17), (3.21) and (3.22) yield

(3.23a,b)\begin{equation} g_{2}(R)\sim R^{{-}1/2},\quad g_{l}(R)\sim R^{{-}3/2},\quad n=2/3. \end{equation}

We can also use $g_{l}^{-1/2}g_{3} = {\rm const.}$ stated in the sentence under (3.17) and also given in (A2) 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 rely strictly on (3.10). It can also be taken to mean that if we have $C_{\varepsilon } \sim V_{o3}^{3}/V_{o2}^{3}$ as per (3.13), 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. ${V_{o3}}/{V_{o2}} = {V_{i3}}/{V_{i2}}$. Taking account of $V_{i3}^{3} = V_{o3}^{3}\,g_{3}(R)$ and $V_{i2}^{2} = V_{o2}^{2}\,g_{2}(R)$, we obtain

(3.24)\begin{equation} g_{2}(R)=g_{3}^{2/3}(R). \end{equation}

Relations (3.17), (A2), (3.21) and (3.24) yield the exact same power laws and exponents (3.23a,b) and $g_{3}\sim R^{-3/4}$.

3.5. Predictions

The power laws and exponents (3.23a,b) 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 on which we based it) imply the following relations between inner and outer variables:

(3.25)\begin{gather} l_{i} = l_{o} R^{{-}3/4}, \end{gather}

(3.26)\begin{gather}V_{i2} = V_{o2} R^{{-}1/4}, \end{gather}

(3.27)\begin{gather}V_{i3} = V_{o3} R^{{-}1/4}. \end{gather}

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 $\boldsymbol {X}$. 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. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021). 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 $\boldsymbol {X}$ than $\eta$ and $u_{\eta }$ when the turbulence is non-homogeneous and $C_{\varepsilon }$ varies with $\boldsymbol {X}$.

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 (3.6), we can define $f_{i2}^{*} (r/l_{i}) \equiv (r/l_{i})^{-2/3}\,f_{i2}(r/l_{i})$ and therefore write

(3.28)\begin{equation} \langle (\delta u_{1})^{2}\rangle = V_{i2}^{2} (r/l_{i})^{2/3}\,f_{i2}^{*} (r/l_{i}) \end{equation}

for $r\ll l_{o}$. Making use of (3.25) and (3.26), (3.28) becomes

(3.29)\begin{equation} \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}), \end{equation}

and like (3.28), this 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 (3.29) of $V_{o2}^{2} \sim k$ and $l_{o} \sim L$.

It must be stressed that our theory's prediction (3.29) 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 (3.29) 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.

Our theory also makes a prediction for $\boldsymbol {\nabla }_r \boldsymbol {\cdot } \langle \delta \boldsymbol {u} (\delta u_{1})^{2}\rangle$. Using (3.10) in conjunction with (3.13), which was obtained from the outer scale-by-scale energy balance in § 3.1, we get $V_{o3}^{3}/l_{o}\sim \varepsilon _{1}$. From (3.25) and (3.27), $V_{i3}^{3}/l_{i}\sim V_{o3}^{3}/l_{o}$, hence $V_{i3}^{3}/l_{i}\sim \varepsilon _{1}$. The outer and inner similarity forms (3.3) and (3.7) become $\boldsymbol {\nabla }_r \boldsymbol {\cdot } \langle \delta \boldsymbol {u} (\delta u_{1})^{2}\rangle = \varepsilon _{1}\, f_{o3}(r/l_{o})$ and $\boldsymbol {\nabla }_r \boldsymbol {\cdot } \langle \delta \boldsymbol {u} (\delta u_{1})^{2}\rangle = \varepsilon _{1}\,f_{i3}(r/l_{i})$, respectively, together implying

(3.30)\begin{equation} \boldsymbol{\nabla}_r \boldsymbol{\cdot} \langle \delta \boldsymbol{u} (\delta u_{1})^{2}\rangle \sim \varepsilon_{1} \end{equation}

in the intermediate range $l_{i}\ll r \ll l_{o}$. This result may form the basis for explaining the observation by Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) of orientation-averaged nonlinear interscale transfer rates approximately equal to minus the turbulence dissipation rate over significant ranges of separation distances in a turbulent wake's near field, where all the inhomogeneity-related energy processes in the scale-by-scale energy balance are actually active. The proportionality (3.30) obtained for non-homogeneous turbulence where all inhomogeneity-related energy processes are active does not have the same physical foundation as (2.5) for homogeneous equilibrium turbulence where they are not.

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 $\boldsymbol {r} = (r,0,0)$, with identical results.

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

Given that the laboratory experiments of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) (a detailed description can of course be found in their paper), and we describe the data from that experiment that we use in § 5.

Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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. Here, $G$ is the centre-to-centre distance between the prisms (see figure 1, reproduced here from Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021)). The wind tunnel's test section was 2 m wide by 1 m high, and the prisms were placed with their spanwise axes parallel to the tunnel's height. Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) acquired data for three incoming velocities, $U_\infty = 5, 6, 7.35\ {\rm m}\ {\rm s}^{-1}$, corresponding to global Reynolds numbers $Re$ ($\equiv U_\infty H/\nu$) $1.0\times 10^{4}$, $1.2\times 10^{4}$ and $1.5 \times 10^{4}$, respectively.

Figure 1. (a) PIV set-up for the energy dissipation rate measurements. (b) PIV set-up for the integral length scale measurements. (c) Coordinate system normal to the prisms’ spanwise direction, and definitions of $G$, $H$ and $x_c$. All fields of view in (a–c) and laser sheets in (a,b) are in the horizontal $(x,y)$ plane. Figure courtesy of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021).

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. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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 1a,c).

For each gap ratio $G/H$, Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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 centreline ($y = 0$), as sketched in figure 1(c), and different SFVs differed by different streamwise positions $x_c$ of the SFV centre. The measurement cases are summarised 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 ($1.0 \times 10^{4}$).

Table 1. Details of the small fields of view (SFVs).

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 (SFV, dashed squares) for each gap ratio. (a) $G/H = 1.25$, (b) $G/H = 2.4$, and (c) $G/H = 3.5$. Figure courtesy of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021).

The other 2D2C PIV set-up used by Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) was a system of four sCMOS cameras arranged consecutively in the 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 the streamwise direction and from $y=-2.4H$ to $y=2.9H$ in the cross-stream direction for $Re = 1.0 \times 10^{4}$, and from $x = 11.1H$ to $x=24.9H$ and from $y=-2.8H$ to $y=3H$ for $Re = 1.2\times 10^{4}$ and $Re = 1.5 \times 10^{4}$; see figure 1(b).

The acquisition frequency was 5 Hz for SFV and 4 Hz for LFV. 20 000 velocity fields were captured for each measurement, corresponding to about 67 min for SFV measurements and 83 min for LFV measurements. The final interrogation window size of their PIV analysis was $24 \times 24$ pixels, with about $58\,\%$ overlap, which corresponds to a $312\ \mathrm {\mu } {\rm m}$ interrogation window for SFV and 1.6 mm 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. Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021). The turbulent energy dissipation rate was approximated based on the assumption of local axisymmetry in the streamwise direction (George & Hussein Reference George and Hussein1991). Lefeuvre et al. (Reference Lefeuvre, Thiesset, Djenidi and Antonia2014) 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. (Reference Zhou, Nagata, Sakai and Watanabe2019), that the ratio of this estimate to the full turbulent kinetic energy remains about constant at 0.75–0.8 for streamwise distances $x/H \geqslant 5$. They also calculated correlation length-scales in the streamwise direction of both streamwise and cross-stream fluctuating velocities. Good convergence of the autocorrelation functions was achieved for the cross-stream fluctuations, whereas the streamwise fluctuations gave rise to integral length scale close to $10H$ in those cases when it was possible to extract values of this length scale from the autocorrelation function (the autocorrelation 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, and between 0.75 and 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 § 5 we use the integral length scale data just mentioned and also local streamwise and cross-stream velocity data obtained by Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) in SFVs as well as local turbulent dissipation rate $\varepsilon$ and local turbulent kinetic energy data in SFVs.

5. Scalings of second-order structure functions

In this section, we assess our theory's prediction (3.29) 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. Reference Sumner, Wong, Price and Paidoussis1999; Alam, Zhou & Wang Reference Alam, Zhou and Wang2011): 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. Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021), exhibit distinct spatial distributions and different inhomogeneity structures that are described and discussed in Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021). The inhomogeneity is also present within the SFVs where the turbulence dissipation coefficient $C_{\varepsilon }$ is found to vary significantly and systematically with spatial position in different SFVs (see figure 3) in ways that are common in the three different flows, even though the spatial inhomogeneities of turbulent kinetic energy and the turbulence dissipation rate vary from flow to flow (see figure 18 in Chen et al. Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021). These qualitative differences in large-scale features and inhomogeneity provide some variety for the testing of the predictions of § 3.

Figure 3. Panels (a,c,e) show $\langle C_{\varepsilon } \rangle _{t,x}$, where the brackets $\langle \dots \rangle _{t,x}$ signify an average over time $t$ and streamwise coordinate $x$, versus $y/H$ within a small field of view. Panels (b,d,f) show $\langle l_{i} \rangle _{t,x}/\langle \eta \rangle _{t,x}$ versus $y/H$, where $l_{i}=LR^{-3/4}$ and $\eta = (\nu ^{3}/\langle \varepsilon \rangle )^{1/4}$. Red lines correspond to SFV20 and blue lines to SFV14 (a,b,e,f), or SFV10 (c,d). $Re = 1.0\times 10^{4}$. Plots (a,b) are for $G/H=1.25$; (c,d) for $G/H=2.4$; (e,f) for $G/H=3.5$.

We use the data of Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) to calculate the longitudinal structure function $S_{2}^{u}\equiv \langle[u'_{1} (x_{0} + r, y) - u'_{1}(x_{0}, y)]^{2}\rangle$ and the transverse structure function $S_{2}^{v}\equiv \langle[u'_{2} (x_{0} + r, y) - u'_{2}(x_{0}, y)]^{2}\rangle$ where use is made of the Reynolds decomposition $u_{1} = U_{1} + u'_{1}$ and $u_{2} = U_{2} + u'_{2}$ into mean flow components $U_1$ (streamwise) and $U_{2}$ (cross-stream), and turbulent fluctuating velocity components $u'_{1}$ and $u'_{2}$. 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 4(a) and 5(a) without significantly changing their shape.

Figure 4. 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 theory of § 3 was presented for the longitudinal and transverse structure functions involving $u_1$ and $u_2$ rather than $u'_1$ and $u'_{2}$. In fact, these structure functions are insensitive to this difference in all the plots presented in figures 4–13 except for two: the plot in figure 4(a), which corresponds to the very near field SFV2.5 where the streamwise mean flow varies appreciably in both the streamwise and cross-stream directions within the SFV; and, very slightly, the plot in figure 8(a), 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).

Figure 5. 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.

Figure 6. Plots of $C_v (r/l_i)$ versus $r/l_{i}$ (where $l_{i}=LR^{-3/4}$, with $R=\sqrt {k}L/\nu$) for $G/H = 2.4$ 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.

Figure 7. Plots of $\langle u,_y^{2}\rangle /\langle v,_x^{2}\rangle$ versus normalised cross-stream coordinate $y/H$ in different SFVs for $Re=1.0 \times 10^{4}$ and (a) $G/H = 1.25$, (b) $G/H = 2.4$, (c) $G/H = 3.5$.

Figure 8. 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 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. 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 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 10. Plots of $C_v (r/l_i)$ 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.

Figure 11. 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 12. (a) Plots of $S_{2}^{u}/r^{2/3}$ versus $r$ for five 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) Plots of $S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ 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,c) but for all cases, SFV14 and SFV20.

Figure 13. (a) Plots of $S_{2}^{v}/r^{2/3}$ versus $r$ for five 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) Plots of $S_{2}^{v}/(k^{3/2}r/L)^{2/3}$ 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,c) but for all cases, SFV14 and SFV20.

We start the presentation and discussion of our data analysis with figures 4 and 5, 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 (3.29), 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 (3.29), 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 $4/3$ of 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 4c,d and 5c,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 $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$ (e.g. see Pope Reference Pope2000). Indeed, if $S_{2}^{u} = A_{u} r^{2/3}$ in a certain range of scales, then $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$ implies $S_{2}^{v} = \frac {4}{3} A_{u} r^{2/3}$ 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 $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$ therefore cannot 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 into $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$, yield $f_{v} = \frac {4}{3} f_{u} + ({(r/l_{i})}/{2})({\partial f_{u}}/{\partial (r/l_{i})})$, which means that $C_v$ defined as follows should equal $4/3$, i.e.

(5.1)\begin{equation} C_{v} \equiv \frac{f_{v} - \dfrac{(r/l_{i})}{2}\,\dfrac{\partial f_{u}}{\partial (r/l_{i})}}{f_{u}} = 4/3 . \end{equation}

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 $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$ does not hold, or both.

In figure 6, we plot $C_{v}$ versus $r/l_{i}$ obtained from the previous two figures for $G/H=2.5$, $Re=10^{4}$. Here, $C_{v}$ 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 that will require future investigation because Chen et al. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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 $S_{2}^{v} = S_{2}^{u} + ({r}/{2})({\partial S_{2}^{u}}/{\partial r})$ 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 4(c,d) and 5(c,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.

It may be that $C_v$ is actually more sensitive to anisotropy than inhomogeneity. We therefore conducted a local isotropy test, reported in figure 7, where we examined the ratio $\langle u,_y^{2}\rangle /\langle v,_x^{2}\rangle$ (where $u,_y \equiv \partial u'_{1}/\partial y$ and $v,_x \equiv \partial u'_{2}/\partial x$) 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 7 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 7b). 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 (3.29) (see figures 4a and 5a). There is also unsatisfactory collapse in SFV5 (see figures 4b and 5b), and figure 7(b) shows that a significant departure from local isotropy remains there. In fact, figure 7 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 6 and 7 suggest that isotropy may be a prerequisite for our theory's scalings to hold – a point that 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. (Reference Chen, Cuvier, Foucaut, Ostovan and Vassilicos2021) 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.

In figures 8 and 9 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 (3.29), 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) return 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 $r/l_{i} \geqslant 40$, 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 9: 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 10): 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 $G/H=3.5$. 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 values of $C_v$ close to $4/3$ 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 8) than in $S_{2}^{v}$ (figure 9).

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 6) could be accounted for by the departures from local isotropy evidenced in figure 7. However, the departures of $C_v$ from $4/3$ seen in figure 10 for the $G/H=1.25$ flow case cannot be explained that way (see figure 7). We therefore investigate the possibility that mean shear may be responsible for these departures. We estimate a Corrsin length $l_{C} \equiv \sqrt {\langle \varepsilon \rangle /S^{3}}$ (see Sagaut & Cambon Reference Sagaut and Cambon2018), following Kaneda (Reference Kaneda2020), and compare it to our estimate of the integral length-scale $L$ (see penultimate paragraph of § 4). In figure 11 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 an SFV, and $l_C$ is calculated by taking $S = \vert {\partial U_{1}}/{\partial x}\vert + \vert {\partial U_{1}}/{\partial y}\vert + \vert {\partial U_{2}}/{\partial x}\vert +\vert {\partial U_{2}}/{\partial y}\vert$ 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, $G/H=2.4$, where neither Kolmogorov's nor our scalings work (figures 4 and 5). Turbulence production and anisotropy somehow should be taken explicitly into account in this case, which the theory in § 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 4 and 5) 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 § 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 4 and 5). 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 8a and 9a), 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 that it takes at SFV14 and SFV20 for $G/H=2.4$ and $3.5$, where the theory works rather well (figures 8 and 9), but well below $1.5$, the smallest value that 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 (figures 10a,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 applies 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 (3.29) 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 8 and  9.

In figures 12(a) and 13(a) we plot, respectively, $S_{2}^{u}/r^{2/3}$ and $S_{2}^{v}/r^{2/3}$ versus $r$ (in metres) 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 12(b) and 13(b), $S_{2}^{u}/(k^{3/2}r/L)^{2/3}$ 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 12c and 13c). Conclusions are similar. In a final step, we plot all cases (i), (ii), (iii), (iv) and (v) together in figures 12(d) and 13(d). 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 were 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 figures 13(b,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.