2 Triply decomposed velocity field

The Reynolds decomposition distinguishes between the mean field and the fluctuating field. When the flow exhibits a well-defined non-stochastic (e.g. periodic) flow feature, one can further decompose the fluctuating field into a coherent field and a stochastic field (Hussain & Reynolds Reference Hussain and Reynolds1970; Reynolds & Hussain Reference Reynolds and Hussain1972). The velocity field is therefore the sum of three fields: $\boldsymbol{u}_{\boldsymbol{f}\boldsymbol{u}\boldsymbol{l}\boldsymbol{l}}=\boldsymbol{U}+\tilde{\boldsymbol{u}}+\boldsymbol{u}^{\prime }$, where $\boldsymbol{U}$ is the mean velocity field obtained by time-averaging $\boldsymbol{u}_{\boldsymbol{f}\boldsymbol{u}\boldsymbol{l}\boldsymbol{l}}$, and where $\tilde{\boldsymbol{u}}$ and $\boldsymbol{u}^{\prime }$ are the coherent and stochastic parts, respectively, of the fluctuating velocity field. The coherent fluctuating velocity $\tilde{\boldsymbol{u}}$ is obtained by phase-averaging $\boldsymbol{u}_{\boldsymbol{f}\boldsymbol{u}\boldsymbol{l}\boldsymbol{l}}-\boldsymbol{U}$ and the stochastic fluctuating velocity is the remainder and is obtained from $\boldsymbol{u}^{\prime }=\boldsymbol{u}_{\boldsymbol{f}\boldsymbol{u}\boldsymbol{l}\boldsymbol{l}}-\boldsymbol{U}-\tilde{\boldsymbol{u}}$. If $\boldsymbol{u}_{\boldsymbol{f}\boldsymbol{u}\boldsymbol{l}\boldsymbol{l}}$ is incompressible, $\boldsymbol{U}$, $\tilde{\boldsymbol{u}}$ and $\boldsymbol{u}^{\prime }$ are incompressible too. With similar notation one also decomposes the pressure field, $p=P+\tilde{p}+p^{\prime }$. In the present work which is concerned with the planar wake of a square prism, both time- and phase-averaging operations also involve averaging in the spanwise direction, i.e. in the direction $x_{3}$ which is normal to the plane of the average wake flow. Fluid velocities and spatial coordinates in the streamwise direction are denoted by $U_{1}$, $\tilde{u} _{1}$, $u_{1}^{\prime }$ and $x_{1}$, respectively; in the cross-stream direction they are $U_{2}$, $\tilde{u} _{2}$, $u_{2}^{\prime }$ and $x_{2}$. The spanwise fluid velocity components are $U_{3}$, $\tilde{u} _{3}$, $u_{3}^{\prime }$.

The definitions of $\tilde{\boldsymbol{u}}$ and $\tilde{p}$ require a reference phase. One can obtain a reference phase from a pressure tap on the cylinder (see e.g. Braza, Perrin & Hoarau Reference Braza, Perrin and Hoarau2006) or from the fluctuating velocity signal, either from within the turbulent flow after appropriately filtering (see e.g. Thiesset et al. Reference Thiesset, Danaila and Antonia2014) or from the outside of the turbulent core (see e.g. Davies Reference Davies1976). Wlezien & Way (Reference Wlezien and Way1979) provide an extensive comparison of different methods with focus on experimental techniques.

In the present analysis, the phase angle $\unicode[STIX]{x1D719}$ used to compute phase averages is extracted from the Hilbert transform of the lift coefficient $C_{L}$ (see Feldman (Reference Feldman2011) for details on the Hilbert transform). This choice follows naturally from the fact that the lift on the square prism in our flow closely follows a sinusoid in time.

The data being discrete in time, $\unicode[STIX]{x1D719}$ was binned into $32$ groups. A smaller bin size would have improved phase resolution but would have reduced statistical convergence (as fewer samples would have fallen within each bin). Thus, each time instant is associated with a value $\unicode[STIX]{x1D719}=-\unicode[STIX]{x03C0}+n(2\unicode[STIX]{x03C0}/32)$ where $0<n<31$. The resulting phase-averaged lift and drag coefficients are plotted in figure 1 versus the phase angle $\unicode[STIX]{x1D719}$, where $\unicode[STIX]{x1D719}=0$ has been chosen such that $\tilde{C}_{L}(\unicode[STIX]{x1D719}=0)=0$.

Figure 1.

Evolution of phase-averaged lift and drag coefficients $\tilde{C}_{L}$ and $\tilde{C}_{D}$ along the normalised phase $\unicode[STIX]{x1D719}/\unicode[STIX]{x03C0}$.

The phase-averaged velocity field $\tilde{\boldsymbol{u}}$ is shown in figure 2 for four different values of $\unicode[STIX]{x1D719}$: $0$, $\frac{1}{4}\unicode[STIX]{x03C0}$, $\frac{1}{2}\unicode[STIX]{x03C0}$ and $\frac{3}{4}\unicode[STIX]{x03C0}$.

It clearly displays a structure similar to that of the von Kármán vortex street where the alternating vortices display opposite circulation, the positive ones travelling slightly above and the negative ones slightly below the centreline. Note that $\tilde{u} _{3}=0$ uniformly and that $\tilde{u} _{1}$ and $\tilde{u} _{2}$ depend on $x_{1}$ and $x_{2}$ but not on $x_{3}$.

Figure 2.

Contours of the magnitude of the phase-averaged velocity $\tilde{\boldsymbol{u}}$ (normalised by $U_{\infty }$) and white unit vectors locally parallel to $\tilde{\boldsymbol{u}}$. The large arrow on the left indicates the direction of the free-stream velocity $U_{\infty }$. Using the phase angles shown in figure 1: (a) $\unicode[STIX]{x1D719}=0$; (b) $\unicode[STIX]{x1D719}=\frac{1}{4}\unicode[STIX]{x03C0}$; (c) $\unicode[STIX]{x1D719}=\frac{1}{2}\unicode[STIX]{x03C0}$; (d) $\unicode[STIX]{x1D719}=\frac{3}{4}\unicode[STIX]{x03C0}$.

The coherent vorticity field $\unicode[STIX]{x1D735}\times \tilde{\boldsymbol{u}}$ is aligned with the spanwise direction and therefore has only one non-zero component $\tilde{\unicode[STIX]{x1D714}}_{3}$. As shown in Alves Portela, Papadakis & Vassilicos (Reference Alves Portela, Papadakis and Vassilicos2018) for this exact same flow (see their figure 3), lines of constant vorticity approximately coincide with streamlines of $\tilde{\boldsymbol{u}}$. As discussed in Hussain (Reference Hussain1983), the streamlines are not necessarily good indicators of the presence of coherent structures, but Lyn et al. (Reference Lyn, Einav, Rodi and Park1995) argue that, apart from the base region in the very near wake where the coherent structures are formed, there is indeed a correspondence between isovorticity and streamlines in identifying coherent structures.

The spectra of the full fluctuating velocity component $\tilde{u} _{1}+u_{1}^{\prime }$ and $\tilde{u} _{2}+u_{2}^{\prime }$ are compared to those of their stochastic counterparts $u_{1}^{\prime }$ and $u_{2}^{\prime }$ in figure 3. As is well known, the shedding frequency is double in the spectrum of $\tilde{u} _{1}+u_{1}^{\prime }$ compared to the spectrum of $\tilde{u} _{2}+u_{2}^{\prime }$, and we checked that it corresponds to the distance between coherent vortices in figure 2 (the distance between successive such vortices does not vary much). Note that the energetic peak present at the shedding frequency in the spectrum of $\tilde{u} _{1}+u_{1}^{\prime }$ is absent in the spectrum of $u_{1}^{\prime }$ and that the energetic peak present at the shedding frequency in the spectrum of $\tilde{u} _{2}+u_{2}^{\prime }$ is absent in the spectrum of $u_{2}^{\prime }$.

Figure 3.

Power spectrum densities normalised by $U_{\infty }d$ of (a) streamwise and (b) cross-stream fluctuating velocities, before (dashed lines) and after (full lines) removing the phase component, between $x_{1}/d=1$ (blue/top) and $x_{1}/d=8$ (dark green/bottom) offset by one decade every $d$. The dashed line indicates a slope of $-5/3$ and the dotted line indicates (a) $f=2f_{s}$ and (b) $f=f_{s}$.

Figure 4.

Profiles of kinetic energies $\tilde{k}$ and $k^{\prime }$ computed from the phase and stochastic components, respectively, along the centreline. The total kinetic energy $k=\tilde{k}+k^{\prime }$ is also shown for comparison.

In conclusion, the phase-averaged fluctuating velocity $\tilde{\boldsymbol{u}}$ is representative of the coherent structures in the present flow as it contains the shedding’s characteristic time signature, and its spatial distribution (figure 2) is one of approximately periodic large scale vortices.

In Hussain (Reference Hussain1983) and Hussain, Jeong & Kim (Reference Hussain, Jeong and Kim1987) it is argued that these coherent structures do not necessarily provide a large contribution to the turbulent kinetic energy. Of course, the regions of the flow considered by these authors are much further downstream than the region of the flow studied here. Figure 4 makes it clear that the coherent structures contribute most of the turbulent kinetic energy $k\equiv \frac{1}{2}\langle |\tilde{\boldsymbol{u}}+\boldsymbol{u}^{\prime }|^{2}\rangle$ in the near wake considered here and that their contribution ($\tilde{k}\equiv \frac{1}{2}\langle |\tilde{\boldsymbol{u}}|^{2}\rangle$) decreases, in the direction of the mean flow, at a faster rate than the kinetic energy associated with the stochastic motions ($k^{\prime }\equiv \frac{1}{2}\langle |\boldsymbol{u}^{\prime }|^{2}\rangle$) in line with Hussain (Reference Hussain1983) and Hussain et al. (Reference Hussain, Jeong and Kim1987). (The brackets $\langle \cdots \rangle$ symbolise combined time- and spanwise-average operations using approximately $10^{3}$ snapshots spanning just over $32$ shedding cycles. The additional spanwise average involves $150$ planes in the spanwise direction which is statistically homogeneous. This level of statistics proved sufficient to converge the averages of all the quantities presented in this paper.) Note that $k=\tilde{k}+k^{\prime }$. Note also that the Taylor length-based Reynolds number $Re_{\unicode[STIX]{x1D706}}$ varies on the centreline from approximately 120 at $x_{1}/d=2$ to approximately 170 at $x_{1}/d=10$ if it is defined on the basis of $\sqrt{\langle u_{1}^{\prime 2}\rangle }$ and from approximately 100 at $x_{1}/d=2$ to approximately 190 at $x_{1}/d=10$ if it is defined on the basis of $\sqrt{2k^{\prime }/3}$.

In the following section we introduce scale-by-scale energy budgets adapted to the triple decomposition of a velocity field into its mean and its coherent and stochastic fluctuations.

3 Scale-by-scale energy budgets

The most general forms of scale-by-scale energy budget have been derived by Hill (Reference Hill1997, Reference Hill2001, Reference Hill2002a) and Duchon & Robert (Reference Duchon and Robert2000) without making any assumption on the nature of the turbulence. Using the Reynolds decomposition and averaging over time in general but also in the spanwise direction for this paper’s particular flow, this equation (which we refer to as the Kármán–Howarth-Monin–Hill equation) follows from the Navier–Stokes equation and incompressibility and takes the form

(3.1)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{U_{i}^{+}+U_{i}^{-}}{2}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle }{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2}\rangle }{\unicode[STIX]{x2202}r_{i}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle }{\unicode[STIX]{x2202}r_{i}}=-2\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{j}}{\unicode[STIX]{x2202}r_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\langle (u_{i}^{+}+u_{i}^{-})\unicode[STIX]{x1D6FF}u_{j}\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{j}}{\unicode[STIX]{x2202}x_{i}}-\frac{\displaystyle \unicode[STIX]{x2202}\left\langle \frac{u_{i}^{+}+u_{i}^{-}}{2}\unicode[STIX]{x1D6FF}q^{2}\right\rangle }{\unicode[STIX]{x2202}x_{i}}-2\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}p\rangle }{\unicode[STIX]{x2202}x_{i}}+\unicode[STIX]{x1D708}\frac{1}{2}\frac{\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle }{\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{i}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,2\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle }{\unicode[STIX]{x2202}r_{i}\unicode[STIX]{x2202}r_{i}}-4\unicode[STIX]{x1D708}\left(\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}}{\unicode[STIX]{x2202}x_{i}}\right\rangle +\frac{1}{4}\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}}{\unicode[STIX]{x2202}r_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}}{\unicode[STIX]{x2202}r_{i}}\right\rangle \right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}q^{2}\equiv \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{i}$ in terms of the fluctuating velocity differences $\unicode[STIX]{x1D6FF}u_{i}\equiv (\tilde{u} _{i}^{+}+{u_{i}^{\prime }}^{+})-(\tilde{u} _{i}^{-}+{u_{i}^{\prime }}^{-})$ (for components $i=1,2,3$), $\unicode[STIX]{x1D6FF}U_{i}\equiv U_{i}^{+}-U_{i}^{-}$, $\unicode[STIX]{x1D6FF}p\equiv (\tilde{p}^{+}+p^{\prime +})-(\tilde{p}^{-}+p^{\prime -})$, and the superscripts $+$ and $-$ distinguish quantities evaluated at positions $\unicode[STIX]{x1D743}^{+}\equiv \boldsymbol{x}+\boldsymbol{r}/2$ and $\unicode[STIX]{x1D743}^{-}\equiv \boldsymbol{x}-\boldsymbol{r}/2$, respectively; e.g. $u_{i}^{+}\equiv \tilde{u} _{i}^{+}+{u_{i}^{\prime }}^{+}$ and $u_{i}^{-}\equiv \tilde{u} _{i}^{-}+{u_{i}^{\prime }}^{-}$ are the full fluctuating velocity components at $\unicode[STIX]{x1D743}^{+}$ and $\unicode[STIX]{x1D743}^{-}$, respectively. Equation (3.1) is written in a six-dimensional reference frame $x_{i},r_{i}$ where coordinates $x_{i}$ of $\boldsymbol{x}$ are associated with a location in physical space and the scale space is the space of all separations and orientations $\boldsymbol{r}=(r_{1},r_{2},r_{3})$ between two points (we refer to $r=|\boldsymbol{r}|$ as a scale). If the average operation is not over time but over realisations, then the extra term $\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}t$ can also be present on the left-hand side of (3.1). (Note that an even more general form of the KHMH equation can be obtained without any decomposition and without any averaging operation, see Duchon & Robert (Reference Duchon and Robert2000), Hill (Reference Hill2002a) and Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018).)

Following Valente & Vassilicos (Reference Valente and Vassilicos2015), Gomes-Fernandes, Ganapathisubramani & Vassilicos (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015) and Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017), each term in (3.1), re-written as

(3.2)$$\begin{eqnarray}{\mathcal{A}}+\unicode[STIX]{x1D6F1}+\unicode[STIX]{x1D6F1}_{U}={\mathcal{P}}+{\mathcal{T}}_{u}+{\mathcal{T}}_{p}+{\mathcal{D}}_{x}+{\mathcal{D}}_{r}-\unicode[STIX]{x1D700}_{r},\end{eqnarray}$$

is associated with a physical process in the budget of $\langle \unicode[STIX]{x1D6FF}q^{2}\rangle$ as follows:

1. (i) $4{\mathcal{A}}=((U_{i}^{+}+U_{i}^{-})/2)(\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}x_{i})$ is the mean advection term;

2. (ii) $4\unicode[STIX]{x1D6F1}=\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}r_{i}$ is the nonlinear interscale transfer rate which accounts for the effect of nonlinear interactions in redistributing $\unicode[STIX]{x1D6FF}q^{2}$ within the $r_{i}$ space and is given by the divergence in scale space of the flux $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2}\rangle$;

3. (iii) $4\unicode[STIX]{x1D6F1}_{U}=(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle )/\unicode[STIX]{x2202}r_{i}$ is the linear interscale transfer rate. (The term ‘linear’ used here does not mean that a linearisation of the Navier–Stokes equation has been performed.);

4. (iv) $4{\mathcal{P}}=-2\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}u_{j}\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{j}/\unicode[STIX]{x2202}r_{i})-\langle (u_{i}^{+}+u_{i}^{-})\unicode[STIX]{x1D6FF}u_{j}\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{j}/\unicode[STIX]{x2202}x_{i})$ can be associated with the production of $\langle \unicode[STIX]{x1D6FF}q^{2}\rangle$ by mean flow gradients (see Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) for more details);

5. (v) $4{\mathcal{T}}_{u}=-\unicode[STIX]{x2202}\langle ((u_{i}^{+}+u_{i}^{-})/2)\unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}x_{i}$ is the transport of $\unicode[STIX]{x1D6FF}q^{2}$ in physical space due to turbulent fluctuations;

6. (vi) $4{\mathcal{T}}_{p}=-2(\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}p\rangle /\unicode[STIX]{x2202}x_{i})$ is the pressure-velocity term, equal to $-2$ times the correlation between fluctuating velocity differences and differences of fluctuating pressure gradient;

7. (vii) $4{\mathcal{D}}_{x}=\unicode[STIX]{x1D708}(1/2)(\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}x_{i}\unicode[STIX]{x2202}x_{i})$ is the viscous diffusion in physical space;

8. (viii) $4{\mathcal{D}}_{r}=2\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}^{2}\langle \unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}r_{i}\unicode[STIX]{x2202}r_{i})$ is the viscous diffusion in scale space. This term is equal to the dissipation $\unicode[STIX]{x1D700}$ when the two points coincide ($r=0$) and can be shown (see appendix B in Valente & Vassilicos (Reference Valente and Vassilicos2015)) to be negligible for separations larger than the Taylor microscale; and

9. (ix) $4\unicode[STIX]{x1D700}_{r}=4\unicode[STIX]{x1D708}(\langle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}/\unicode[STIX]{x2202}x_{i})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}/\unicode[STIX]{x2202}x_{i})\rangle +\frac{1}{4}\langle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}/\unicode[STIX]{x2202}r_{i})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}/\unicode[STIX]{x2202}r_{i})\rangle )$ and $\unicode[STIX]{x1D700}_{r}$ is actually the two-point average dissipation rate $\unicode[STIX]{x1D700}_{r}=(\unicode[STIX]{x1D700}^{+}+\unicode[STIX]{x1D700}^{-})/2$ as it equals $\frac{1}{2}\unicode[STIX]{x1D708}(\langle (\unicode[STIX]{x2202}u_{j}^{+}/\unicode[STIX]{x2202}x_{i}^{+})(\unicode[STIX]{x2202}u_{j}^{+}/\unicode[STIX]{x2202}x_{i}^{+})\rangle +\langle (\unicode[STIX]{x2202}u_{j}^{-}/\unicode[STIX]{x2202}x_{i}^{-})(\unicode[STIX]{x2202}u_{j}^{-}/\unicode[STIX]{x2202}x_{i}^{-})\rangle )$.

With the triple decomposition introduced in § 2 one can decompose the second-order structure function $\langle \unicode[STIX]{x1D6FF}q^{2}\rangle$ into a stochastic and a coherent part, i.e. $\langle \unicode[STIX]{x1D6FF}q^{2}\rangle =\langle \unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle +\langle \unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$ where $\unicode[STIX]{x1D6FF}\tilde{q}^{2}\equiv \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\tilde{u} _{i}$ with $\unicode[STIX]{x1D6FF}\tilde{u} _{i}\equiv \tilde{u} _{i}^{+}-\tilde{u} _{i}^{-}$ and $\unicode[STIX]{x1D6FF}q^{\prime 2}\equiv \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{i}^{\prime }$ with $\unicode[STIX]{x1D6FF}u_{i}^{\prime }\equiv {u_{i}^{\prime }}^{+}-{u_{i}^{\prime }}^{-}$. The fluctuating pressure difference $\unicode[STIX]{x1D6FF}p$ is also decomposed in a similar way, i.e. $\unicode[STIX]{x1D6FF}p=\unicode[STIX]{x1D6FF}\tilde{p}+\unicode[STIX]{x1D6FF}p^{\prime }$ where $\unicode[STIX]{x1D6FF}\tilde{p}\equiv \tilde{p}^{+}-\tilde{p}^{-}$ and $\unicode[STIX]{x1D6FF}p^{\prime }\equiv p^{\prime +}-p^{\prime -}$.

This decomposition into stochastic and coherent fluctuations warrants new scale-by-scale energy budgets to be derived and this was done by Thiesset et al. (Reference Thiesset, Danaila and Antonia2014) by neglecting mean flow velocity differences $\unicode[STIX]{x1D6FF}U_{i}$. The resulting slightly more general equations for $\langle \unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$ and $\langle \unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle$ without neglecting $\unicode[STIX]{x1D6FF}U_{i}$ are, respectively,

(3.3)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{U_{i}^{+}+U_{i}^{-}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\langle \unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}U_{i}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =-\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}}{\unicode[STIX]{x2202}x_{j}}-2\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}}{\unicode[STIX]{x2202}r_{j}}-\left\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}}{\unicode[STIX]{x2202}x_{j}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -\,2\left\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}}{\unicode[STIX]{x2202}r_{j}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left\langle \frac{\tilde{u} _{i}^{+}+\tilde{u} _{i}^{-}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}q^{\prime 2}}{\unicode[STIX]{x2202}x_{i}}\right\rangle -\left\langle \frac{{u_{i}^{\prime }}^{+}+{u_{i}^{\prime }}^{-}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}q^{\prime 2}}{\unicode[STIX]{x2202}x_{i}}\right\rangle -2\left\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}x_{i}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D708}\left\langle \frac{1}{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}q^{\prime 2}}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}}\right\rangle +2\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}q^{\prime 2}}{\unicode[STIX]{x2202}r_{j}\unicode[STIX]{x2202}r_{j}}\right\rangle -4\unicode[STIX]{x1D708}\left(\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{i}}\right\rangle +\frac{1}{4}\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}^{\prime }}{\unicode[STIX]{x2202}r_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}u_{j}^{\prime }}{\unicode[STIX]{x2202}r_{i}}\right\rangle \right)\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and

(3.4)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{U_{i}^{+}+U_{i}^{-}}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\langle \unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle +\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle +2\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }\unicode[STIX]{x1D6FF}\tilde{u} _{j}\rangle +\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}\langle \unicode[STIX]{x1D6FF}U_{i}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =-\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}(\tilde{u} _{j}^{+}+\tilde{u} _{j}^{-})\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}}{\unicode[STIX]{x2202}x_{j}}-2\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\tilde{u} _{j}\rangle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}}{\unicode[STIX]{x2202}r_{j}}+\left\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}}{\unicode[STIX]{x2202}x_{j}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left\langle 2\unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}}{\unicode[STIX]{x2202}r_{j}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -\,\left\langle \frac{\tilde{u} _{i}^{+}+\tilde{u} _{i}^{-}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{q}^{2}}{\unicode[STIX]{x2202}x_{i}}\right\rangle -\left\langle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}[({u_{i}^{\prime }}^{+}+{u_{i}^{\prime }}^{-})\unicode[STIX]{x1D6FF}u_{j}^{\prime }\unicode[STIX]{x1D6FF}\tilde{u} _{j}]\right\rangle -2\left\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\frac{\unicode[STIX]{x2202}\tilde{p}}{\unicode[STIX]{x2202}x_{i}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D708}\left\langle \frac{1}{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}\tilde{q}^{2}}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}}\right\rangle +2\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}\tilde{q}^{2}}{\unicode[STIX]{x2202}r_{j}\unicode[STIX]{x2202}r_{j}}\right\rangle -4\unicode[STIX]{x1D708}\left(\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{j}}{\unicode[STIX]{x2202}x_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{j}}{\unicode[STIX]{x2202}x_{i}}\right\rangle +\frac{1}{4}\left\langle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{j}}{\unicode[STIX]{x2202}r_{i}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{j}}{\unicode[STIX]{x2202}r_{i}}\right\rangle \right).\qquad\end{eqnarray}$$

Evidently both equations (3.3) and (3.4) are rather similar to the KHMH equation (3.1) and we therefore make use of similar notation to identify the individual terms

(3.5)$$\begin{eqnarray}{\mathcal{A}}^{\prime }+\unicode[STIX]{x1D6F1}^{\prime }+\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }+\unicode[STIX]{x1D6F1}_{U}^{\prime }={\mathcal{P}}_{U}^{\prime }+{\mathcal{P}}_{\tilde{u} }^{\prime }+{\mathcal{T}}_{\tilde{u} }^{\prime }+{\mathcal{T}}_{u^{\prime }}^{\prime }+{\mathcal{T}}_{p^{\prime }}^{\prime }+{\mathcal{D}}_{x}^{\prime }+{\mathcal{D}}_{r}^{\prime }-\unicode[STIX]{x1D700}_{r}^{\prime }\end{eqnarray}$$

for (3.3) and

(3.6)$$\begin{eqnarray}\tilde{{\mathcal{A}}}+\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }+\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}+\tilde{\unicode[STIX]{x1D6F1}}_{U}=\tilde{{\mathcal{P}}}_{U}-{\mathcal{P}}_{\tilde{u} }^{\prime }+\tilde{{\mathcal{T}}}_{\tilde{u} }+\tilde{{\mathcal{T}}}_{{\mathcal{P}}_{\tilde{u} }}+\tilde{{\mathcal{T}}}_{\tilde{p}}+\tilde{{\mathcal{D}}}_{x}+\tilde{{\mathcal{D}}}_{r}-\tilde{\unicode[STIX]{x1D700}}_{r}\end{eqnarray}$$

for (3.4). Here, $4{\mathcal{A}}^{\prime }$, $4\unicode[STIX]{x1D6F1}^{\prime }$, $4\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }$ and $4\unicode[STIX]{x1D6F1}_{U}^{\prime }$ correspond to the first, second, third and fourth terms in the first line of (3.3) and $4\tilde{{\mathcal{A}}}$, $4\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }$, $4\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}$ and $4\tilde{\unicode[STIX]{x1D6F1}}_{U}$ correspond to the first, second, third and fourth terms in the first line of (3.4). Here, $4{\mathcal{P}}_{U}^{\prime }$ and $4\tilde{{\mathcal{P}}}_{U}$ correspond to the sum of the first and second terms in the second line of (3.3) and (3.4), respectively. For the same reasons given for $4{\mathcal{P}}$ by Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017), $4{\mathcal{P}}_{U}^{\prime }$ and $4\tilde{{\mathcal{P}}}_{U}$ are production terms of $\langle \unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$ and $\langle \unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle$, respectively, and $4{\mathcal{P}}=4{\mathcal{P}}_{U}^{\prime }+4\tilde{{\mathcal{P}}}_{U}$. The term $4{\mathcal{P}}_{\tilde{u} }^{\prime }\equiv -\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}/\unicode[STIX]{x2202}x_{j})\rangle -\langle 2\unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}/\unicode[STIX]{x2202}r_{j})\rangle$ appears with opposite signs in (3.3) and (3.4) and is therefore the production term which exchanges energy at given $\boldsymbol{x}$ and $\boldsymbol{r}$ between the stochastic and the coherent fluctuating motions. The spatial transport terms $4{\mathcal{T}}_{\tilde{u} }^{\prime }$ and $4{\mathcal{T}}_{u^{\prime }}^{\prime }$ are the first and second terms in the fourth line of (3.3) and the stochastic pressure-stochastic velocity term $4{\mathcal{T}}_{p^{\prime }}^{\prime }$ is the third term on this line. Similarly, the transport terms $4\tilde{{\mathcal{T}}}_{\tilde{u} }$ and $4\tilde{{\mathcal{T}}}_{{\mathcal{P}}_{\tilde{u} }}$ are the first and second terms in the fourth line of (3.4) and the coherent pressure-coherent velocity term $4\tilde{{\mathcal{T}}}_{\tilde{p}}$ is the third term on this line. The remaining terms are the diffusion terms $4\tilde{{\mathcal{D}}}_{x}$, $4\tilde{{\mathcal{D}}}_{r}$, $4{\mathcal{D}}_{x}^{\prime }$ and $4{\mathcal{D}}_{r}^{\prime }$ and the dissipation terms $4\tilde{\unicode[STIX]{x1D700}}_{r}$ and $4\unicode[STIX]{x1D700}_{r}^{\prime }$ which are defined exactly as the diffusion and dissipation terms in the KHMH equation (3.1) and (3.2) but for the coherent and stochastic velocity fields, respectively, rather than for the total fluctuating velocity field.

Adding (3.5) with (3.6) results in the KHMH equation by combining terms with tilde and primes together (e.g. ${\mathcal{A}}={\mathcal{A}}^{\prime }+\tilde{{\mathcal{A}}}$, $\unicode[STIX]{x1D6F1}_{U}=\unicode[STIX]{x1D6F1}_{U}^{\prime }+\tilde{\unicode[STIX]{x1D6F1}}_{U}$, etc.) but also by noticing that

(3.7)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6F1}^{\prime }+\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }+\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }+\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}\end{eqnarray}$$

and

(3.8)$$\begin{eqnarray}{\mathcal{T}}_{u}={\mathcal{T}}_{u^{\prime }}^{\prime }+{\mathcal{T}}_{\tilde{u} }^{\prime }+\tilde{{\mathcal{T}}}_{\tilde{u} }+\tilde{{\mathcal{T}}}_{{\mathcal{P}}_{\tilde{u} }},\end{eqnarray}$$

which are the nonlinear interscale and interspace transfer terms.

The terms $\unicode[STIX]{x1D6F1}^{\prime }$, $\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }$ and $\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }$ can be interpreted as interscale transfer terms of either $\unicode[STIX]{x1D6FF}q^{\prime 2}$ or $\unicode[STIX]{x1D6FF}\tilde{q}^{2}$. Here, $4\unicode[STIX]{x1D6F1}^{\prime }\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$ represents the interscale transfer of energy associated with the stochastic motions by the stochastic motions (i.e. interscale transfer of $\unicode[STIX]{x1D6FF}q^{\prime 2}$ by $\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }$). Similarly, $4\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$ represents the interscale transfer of the energy associated with the stochastic motions by the coherent motions (i.e. interscale transfer of $\unicode[STIX]{x1D6FF}q^{\prime 2}$ by $\unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}$), and $4\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle$ represents the interscale transfer of the energy associated with the coherent motions by the coherent motions (i.e. interscale transfer of $\unicode[STIX]{x1D6FF}\tilde{q}^{2}$ by $\unicode[STIX]{x1D6FF}\tilde{u} _{i}$). The term $4\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}$ can be written as the difference between two interscale transfer terms: the interscale transfer by the stochastic velocity field of the total fluctuating energy and $4\unicode[STIX]{x1D6F1}^{\prime }$, i.e. $4\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}=4\unicode[STIX]{x1D6F1}_{u^{\prime }}-4\unicode[STIX]{x1D6F1}^{\prime }$ where $4\unicode[STIX]{x1D6F1}_{u^{\prime }}\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }|\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }+\unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}|^{2}\rangle$. Hence, combining $\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}$ with $\unicode[STIX]{x1D6F1}^{\prime }$ results in the interscale transfer of the total fluctuating energy by the stochastic motions (i.e. interscale transfer of $\unicode[STIX]{x1D6FF}q^{2}$ by $\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }$) so that (3.7) can be written as

(3.9)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6F1}_{u^{\prime }}+\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }+\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }.\end{eqnarray}$$

This proves to be an important equation in § 5.

The terms ${\mathcal{T}}_{u^{\prime }}^{\prime }$, ${\mathcal{T}}_{\tilde{u} }^{\prime }$, $\tilde{{\mathcal{T}}}_{\tilde{u} }$ represent turbulent transport in physical space. Specifically, $4{\mathcal{T}}_{u^{\prime }}^{\prime }\equiv -\langle (({u_{i}^{\prime }}^{+}+{u_{i}^{\prime }}^{-})/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}q^{\prime 2}/\unicode[STIX]{x2202}x_{i})\rangle$ represents interspace transport of stochastic turbulent energy by stochastic fluctuations, $4{\mathcal{T}}_{\tilde{u} }^{\prime }\equiv -\langle (({\tilde{u} _{i}}^{+}+{\tilde{u} _{i}}^{-})/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}q^{\prime 2}/\unicode[STIX]{x2202}x_{i})\rangle$, represents interspace transport of stochastic turbulent energy by coherent fluctuations, and $\tilde{{\mathcal{T}}}_{\tilde{u} }\equiv -\langle (({\tilde{u} _{i}}^{+}+{\tilde{u} _{i}}^{-})/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{q}^{2}/\unicode[STIX]{x2202}x_{i})\rangle$ represents interspace transport of coherent fluctuating energy by coherent fluctuations. The term $-4\tilde{{\mathcal{T}}}_{{\mathcal{P}}_{\tilde{u} }}$ is the difference between $4{\mathcal{T}}_{u^{\prime }}^{\prime }$ and the spatial transport of the total fluctuating energy by the two-point-average stochastic velocity, i.e. $4\tilde{{\mathcal{T}}}_{{\mathcal{P}}_{\tilde{u} }}=4{\mathcal{T}}_{u^{\prime }}-4{\mathcal{T}}_{u^{\prime }}^{\prime }$ were $4{\mathcal{T}}_{u^{\prime }}\equiv \langle (({u_{i}^{\prime }}^{+}+{u_{i}^{\prime }}^{-})/2)(\unicode[STIX]{x2202}|\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }+\unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}|^{2}/\unicode[STIX]{x2202}x_{i})\rangle$. This allows rewriting (3.8) as follows:

(3.10)$$\begin{eqnarray}{\mathcal{T}}_{u}={\mathcal{T}}_{u^{\prime }}+{\mathcal{T}}_{\tilde{u} }^{\prime }+\tilde{{\mathcal{T}}}_{\tilde{u} }.\end{eqnarray}$$

In the following section we compare the signs and magnitudes of the orientation-averaged terms in (3.5) and (3.6) in the near field turbulent planar wake.

4 Orientation-averaged scale-by-scale energy budgets in the near wake of a square prism

Each term, $Q$, in (3.5) and (3.6) is an average in time and spanwise direction and is therefore a function of planar coordinates $(x_{1},x_{2})$ and two-point separation vector $\boldsymbol{r}$, i.e. $Q=Q(x_{1},x_{2},\boldsymbol{r})$. We set $r_{3}=0$ and define the orientation-averaged quantity $Q^{a}$ by integrating $Q$ over the angle $\unicode[STIX]{x1D703}$ defined by $r_{1}=r\cos \unicode[STIX]{x1D703}$, $r_{2}=r\sin \unicode[STIX]{x1D703}$ which also defines the radius (and length scale) $r$ as follows: $Q^{a}(x_{1},x_{2},r)\equiv (1/2\unicode[STIX]{x03C0})\int _{0}^{2\unicode[STIX]{x03C0}}Q\,\text{d}\unicode[STIX]{x1D703}$. Such scale-space orientation-averaging has already been used by Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) and Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015) to study the terms in the KHMH equation (3.2). We verified that the KHMH equation, (3.1), is sufficiently well balanced numerically, as the difference between its left-hand and right-hand sides is two orders of magnitude smaller than $\unicode[STIX]{x1D700}_{r}$ for all $r$ investigated here, and even smaller than that when the two sides are orientation-averaged in scale-space plane $r_{3}=0$. We also checked that every term in (3.1) is indeed equal to the sum of its two corresponding terms in (3.3) and (3.4), for example ${\mathcal{A}}={\mathcal{A}}^{\prime }+\tilde{{\mathcal{A}}}$, $\unicode[STIX]{x1D6F1}_{U}=\unicode[STIX]{x1D6F1}_{U}^{\prime }+\tilde{\unicode[STIX]{x1D6F1}}_{U}$, etc.

Figure 5.

Orientation-averaged terms of (3.5) (equivalently equation (3.3)) normalised by $\unicode[STIX]{x1D700}^{a}$ at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$ on the geometric centreline. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

In figure 5 we plot all the orientation-averaged terms in (3.5) versus $r/d$ in the range $0\leqslant r/d\leqslant 1.1$ at two centreline positions, $(x_{1},x_{2})=(2d,0)$ and $(8d,0)$. These terms are plotted normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ which, for $r$ not much larger than $d$, is approximately equal to $\unicode[STIX]{x1D700}_{r}^{\prime a}$ (see Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2018) and to the one-point dissipation rate $\unicode[STIX]{x1D700}$ in the region of the centreline that we study. The range $0\leqslant r\leqslant 1.1d$ has also been chosen because the average distance between consecutively shed coherent vortices is comparable to $3d$.

The first observation to make in figure 5 is that the near wake region is so inhomogeneous that most of the terms in the scale-by-scale energy budget equation (3.5) are active. The terms dominating the range $0.4\leqslant r/d\leqslant 1.1$ at $x_{1}/d=2$ are $-{\mathcal{A}}^{\prime a}$ (${\mathcal{A}}^{\prime }\equiv ((U_{i}^{+}+U_{i}^{-})/2)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})\langle \unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$) and ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ (${\mathcal{P}}_{\tilde{u} }^{\prime }\equiv -\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}/\unicode[STIX]{x2202}x_{j})\rangle -2\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}\tilde{u} _{i}/\unicode[STIX]{x2202}r_{j})\rangle$) which are both positive, and ${\mathcal{T}}_{\tilde{u} }^{\prime a}$ (${\mathcal{T}}_{\tilde{u} }^{\prime }\equiv -\langle ((\tilde{u} _{i}^{+}+\tilde{u} _{i}^{-})/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}q^{\prime 2}/\unicode[STIX]{x2202}x_{i})\rangle$) and ${\mathcal{T}}_{p^{\prime }}^{\prime a}$ (${\mathcal{T}}_{p^{\prime }}^{\prime }\equiv -2\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}(\unicode[STIX]{x2202}p^{\prime }/\unicode[STIX]{x2202}x_{i})\rangle$) which are both negative (positive/negative terms correspond to a gain/loss in the budget). These terms are closely followed by the production of stochastic turbulent fluctuations by mean flow gradients, ${\mathcal{P}}_{U}^{\prime a}$ (${\mathcal{P}}_{U}^{\prime }\equiv -\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }({u_{j}^{\prime }}^{+}+{u_{j}^{\prime }}^{-})\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}/\unicode[STIX]{x2202}x_{j})-2\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}u_{j}^{\prime }\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}/\unicode[STIX]{x2202}r_{j})$), which is positive, and by the interscale transfer of stochastic fluctuating energy by coherent motions, plotted with a minus sign as $-\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime a}$ ($\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$), which is negative. The term which is in fact the largest in this scale range at $x_{1}/d=2$ is ${\mathcal{P}}_{\tilde{u} }^{\prime a}$, the rate of energy transfer between the coherent and stochastic fluctuating motions. This term being positive for all values of $r$ in figure 5, the coherent motions feed energy to the stochastic ones at all these scales. At the same time, the coherent fluctuations are responsible for removing energy from the stochastic ones by spatial transport; ${\mathcal{T}}_{\tilde{u} }^{\prime a}$ is negative and dominant at all scales $r$ too. Recall that these scale-dependent energy exchanges happen at $x_{1}/d=2$ on the centreline where the energy spectra have a broad well-defined power law range with exponent close to $-5/3$ (see figure 3) as already shown by Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017).

Note that the orientation-averaged nonlinear interscale transfer rate, plotted with a minus sign as $-\unicode[STIX]{x1D6F1}^{\prime a}$ ($\unicode[STIX]{x1D6F1}^{\prime }\equiv (\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}u_{i}^{\prime }\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$), is not too significant in the range $0.4\leqslant r/d\leqslant 1.1$ at $(x_{1},x_{2})=(2d,0)$. However it is one of the four dominant terms in the range $\unicode[STIX]{x1D706}/d\leqslant r/d\leqslant 0.4$ at this location. These four dominant terms are $-{\mathcal{A}}^{\prime a}$, $-\unicode[STIX]{x1D6F1}^{\prime a}$, ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ and ${\mathcal{T}}_{u^{\prime }}^{\prime a}$, and $\unicode[STIX]{x1D706}$ is the Taylor microscale defined as $\unicode[STIX]{x1D706}^{2}\equiv 2\langle u_{3}^{2}\rangle /\langle ((\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{3})u_{3})^{2}\rangle$. At $(x_{1},x_{2})=(2d,0),\unicode[STIX]{x1D706}$ is $0.09d$ and at $(x_{1},x_{2})=(8d,0),\unicode[STIX]{x1D706}$ is $0.15d$. The diffusion terms ${\mathcal{D}}_{x}^{\prime a}$ and ${\mathcal{D}}_{r}^{\prime a}$ effectively vanish at length scales $r$ larger than $\unicode[STIX]{x1D706}$, and they equal $\unicode[STIX]{x1D700}^{\prime a}$ at $r=0$, as expected (see Valente & Vassilicos Reference Valente and Vassilicos2015).

It is worth stressing that, at $(x_{1},x_{2})=(2d,0)$, the orientation-averaged nonlinear interscale transfer rate $\unicode[STIX]{x1D6F1}^{\prime a}$ is mainly balanced by the advection term $-{\mathcal{A}}^{\prime a}$ and coherent motion production and transport processes, i.e. ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ and ${\mathcal{T}}_{\tilde{u} }^{\prime a}$, in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant 0.4d$. Even though energy spectra have well-defined power law ranges with exponents close to $-5/3$ at $(x_{1},x_{2})=(2d,0)$, $\unicode[STIX]{x1D6F1}^{\prime a}$ is not constant with length scale $r$.

Further downstream, at $(x_{1},x_{2})=(8d,0)$, the orientation-averaged nonlinear interscale transfer rate $\unicode[STIX]{x1D6F1}^{\prime a}$ is mainly balanced by the advection term $-{\mathcal{A}}^{\prime a}$ and coherent motion transport, i.e. ${\mathcal{T}}_{\tilde{u} }^{\prime a}$, in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant 0.3d$. All the other orientation-averaged terms in (3.5) are less significant in this scale range and at this position. The orientation-averaged impact of the coherent motions on the scale-by-scale budget equation (3.5) gradually diminishes with increasing distance from the square prism. In the range $0.3d\leqslant r\leqslant 1.1d$ at $(x_{1},x_{2})=(8d,0)$, the dominant terms are now $-{\mathcal{A}}^{\prime a}$, ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ and $-\unicode[STIX]{x1D6F1}^{\prime a}$ which are all still positive, and ${\mathcal{T}}_{p^{\prime }}^{\prime a}$ which is still negative. The term ${\mathcal{T}}_{\tilde{u} }^{\prime a}$ has greatly reduced in relative importance from $(x_{1},x_{2})=(2d,0)$ to $(x_{1},x_{2})=(8d,0)$, but the presence of the pressure-velocity term ${\mathcal{T}}_{p^{\prime }}^{\prime a}$ has remained significant and approximately the same, if not even grown a little. Perhaps most striking of all is the fact that $-\unicode[STIX]{x1D6F1}^{\prime a}$ has grown to become closer to an approximate constant fraction of $\unicode[STIX]{x1D700}_{r}^{a}$ in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant 1.1d$ at $(x_{1},x_{2})=(8d,0)$ which is downstream of the point where the near $-5/3$ power law spectra appeared.

Figure 6.

Orientation-averaged terms of (3.6) (equivalently equation (3.4)) normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ at $(a)~x_{1}/d=2$ and (b) $x_{1}/d=8$ on the geometric centreline. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

Production of stochastic fluctuation energy by mean flow gradients, namely ${\mathcal{P}}_{U}^{\prime a}$, is a minor contributor to the scale-by-scale stochastic fluctuation balance equation (3.5) at $(x_{1},x_{2})=(2d,0)$ and effectively non-existent at $(x_{1},x_{2})=(8d,0)$ (see figure 5). However, figure 6 shows that production of coherent scale-by-scale energy by mean flow gradients, specifically $\tilde{{\mathcal{P}}}_{U}^{a}$ ($\tilde{{\mathcal{P}}}_{U}\equiv -\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}(\tilde{u} _{j}^{+}+\tilde{u} _{j}^{-})\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}/\unicode[STIX]{x2202}x_{j})-2\langle \unicode[STIX]{x1D6FF}\tilde{u} _{i}\unicode[STIX]{x1D6FF}\tilde{u} _{j}\rangle (\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{i}/\unicode[STIX]{x2202}r_{j})$), is an important source of scale-by-scale energy in the coherent fluctuations balance equation (3.6) at both positions $(x_{1},x_{2})=(2d,0)$ and $(8d,0)$. A clear picture emerges whereby, in an orientation-averaged sense, the mean flow gradients do not significantly feed the stochastic fluctuations directly but do feed the coherent motions which, in turn, feed the stochastic fluctuations via ${\mathcal{P}}_{\tilde{u} }^{\prime a}$. Indeed, the term ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ appears as a dominant term in the orientation-averaged versions of both budget equations (3.5) and (3.6) (see figure 7, and also figures 5 and 6) but with opposite signs. This holds over a wide range of scales as small as $\unicode[STIX]{x1D706}$ for the transfer of energy from the coherent to the stochastic fluctuations at $(x_{1},x_{2})=(2d,0)$ and as small as approximately $2\unicode[STIX]{x1D706}$ or less for the production by mean flow gradients at $(x_{1},x_{2})=(2d,0)$ and for both ${\mathcal{P}}_{\tilde{u} }^{\prime a}$ and $\tilde{{\mathcal{P}}}_{U}^{a}$ at $(x_{1},x_{2})=(8d,0)$ (see figure 7).

Figure 7.

Orientation-averaged production terms normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$ on the geometric centreline. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

Figure 8.

Orientation-averaged spatial transport terms normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$ on the geometric centreline. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

The terms in (3.6) mostly decay with streamwise distance from the prism along the centreline (see figure 6), but they remain overall comparable to the terms in (3.5) at the two positions $(x_{1},x_{2})$ examined here, particularly at length scales $r\geqslant 0.2d$ or $0.3d$. Looking at equation (3.8) and figure 8 we can see that the orientation-averaged turbulent transport of $\unicode[STIX]{x1D6FF}q^{2}$ in physical space (${\mathcal{T}}_{u}^{a}\equiv -((\unicode[STIX]{x2202}\langle (u_{i}^{+}+u_{i}^{-}/2)\unicode[STIX]{x1D6FF}q^{2}\rangle )/\unicode[STIX]{x2202}x_{i})$) is dominated by the orientation-averaged transport of stochastic fluctuations by coherent flow, i.e. ${\mathcal{T}}_{\tilde{u} }^{\prime a}$, at $(x_{1},x_{2})=(2d,0)$ over all plotted length scales $r$ and at $(x_{1},x_{2})=(8d,0)$ up to $r/d\approx 0.5$. Indeed, the fluid between alternate coherent vortices (of opposite circulation) has large cross-stream velocities which dominate turbulent transport in space. Here, ${\mathcal{T}}_{\tilde{u} }^{\prime a}$ is negative because turbulent eddies smaller than the separation between these large-scale coherent vortices are transported away from the centreline. We expect this dominance of coherent flow transport to subside with downstream distance as the large coherent structures weaken.

Figure 9.

Distribution of $\unicode[STIX]{x1D6F1}^{\prime }$ and ${\mathcal{T}}_{p^{\prime }}^{\prime }$ normalised by $\unicode[STIX]{x1D700}_{r}$ in scale space on the geometrical centreline at $x_{1}/d=8$ in (a) and (b), respectively.

The results reported in this section concern orientation-averaged terms of (3.6) and (3.5). The picture is of course more complex if these orientation averages are lifted. For example, the orientation-averaged fully stochastic nonlinear interscale transfer rate $\unicode[STIX]{x1D6F1}^{\prime a}$ is negative at all length scales $r$ sampled here, yet $\unicode[STIX]{x1D6F1}^{\prime }$ can be either negative or positive in the $(r_{1},r_{2})$ plane, depending on orientation (see figure 9). Similarly, the orientation-averaged fully stochastic pressure-velocity term ${\mathcal{T}}_{p^{\prime }}^{\prime a}$ is also negative at all the length scales $r$ that we sampled, yet ${\mathcal{T}}_{p^{\prime }}^{\prime }$ can also be either negative or positive in the $(r_{1},r_{2})$ plane depending on orientation, as shown in figure 9. The study of the distribution in the $(r_{1},r_{2})$ plane of the various terms in (3.6) and (3.5) is beyond the scope of this paper, but it is worth noting the correlation that seems to exist between $\unicode[STIX]{x1D6F1}^{\prime }$ and ${\mathcal{T}}_{p^{\prime }}^{\prime }$: figure 9 shows a significant tendency for these two terms to be positive or negative together. A correlation between fluctuations of the nonlinear interscale transfer rate and the pressure-velocity term has also been observed in DNS of periodic turbulence by Yasuda & Vassilicos (Reference Yasuda and Vassilicos2018) where it is discussed in more detail.

5 Effects of the coherent motion and inhomogeneity on the interscale energy transfer

Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) showed how the average nonlinear interscale transfer rate of $\unicode[STIX]{x1D6FF}q^{2}$ is roughly constant when the orientations of $\boldsymbol{r}$ are averaged out in the $r_{3}=0$ plane, despite this transfer rate’s distribution being far from uniform in this plane. This was in fact observed in spite of the severe inhomogeneities and anisotropies evidenced in the previous section by the various non-zero terms in the KHMH equations (3.6) and (3.5), and even at $x_{1}/d=2$ (albeit for a small range of separations) where the coherent motions contribute a large portion of the total fluctuating kinetic energy (recall figure 4). In this section we start by determining how this constancy of $\unicode[STIX]{x1D6F1}^{a}$ observed in Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2017) and in Gomes-Fernandes et al. (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015) depends on contributions arising from the coherent and stochastic motions individually (§ 5.1), but also on statistical inhomogeneity (§ 5.2). We close the section by checking the signs of interscale fluxes in § 5.3.

5.1 Constant nonlinear interscale transfer as a combined effect

As mentioned in the previous section, the orientation-averaged interscale transfer rate of stochastic fluctuating energy by stochastic motions, $\unicode[STIX]{x1D6F1}^{\prime a}$, is not independent of length scale $r$ at $x_{1}/d=2$ on the centreline. However, figure 10 shows that $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$, the orientation-averaged interscale transfer rate of total fluctuating energy by the stochastic motions is close to being constant with $r$ in the range $\unicode[STIX]{x1D706}<r<0.3d$ at this point ($x_{1}/d=2$, $x_{2}/d=0$). Furthermore, this approximate constant is closer to $-\unicode[STIX]{x1D700}_{r}^{a}$ if $\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime a}$ is taken into account, i.e. at $(x_{1},x_{2})=(2d,0)$, $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}+\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime a}$ is also approximately constant in the range $\unicode[STIX]{x1D706}<r<0.3d$ and closer to $-\unicode[STIX]{x1D700}_{r}^{a}$ than $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$. In fact, at this location, $\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }^{a}\approx 0$ and equation (3.9) reduces to

(5.1)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}^{a}\approx \unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}+{\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}\end{eqnarray}$$

in this range where $\unicode[STIX]{x1D6F1}^{a}$ is approximately constant and close to $-\unicode[STIX]{x1D700}_{r}^{a}$ (which is, in fact, almost equal to $-\unicode[STIX]{x1D700}$ in this range). This equation (5.1) also holds further downstream on the centreline, but over a longer range of scales, e.g. $\unicode[STIX]{x1D706}<r<d$ at $(x_{1},x_{2})=(8d,0)$ (see figure 10).

The fact that a scale range exists where $\unicode[STIX]{x1D6F1}^{a}/\unicode[STIX]{x1D700}_{r}^{a}$ is approximately constant and relatively close to $-1$ would not have been possible without the presence of coherent structures at $(x_{1},x_{2})=(2d,0)$. Whilst these coherent structures are non-dynamic in this scale range, in the sense that $\tilde{\unicode[STIX]{x1D6F1}}_{\tilde{u} }^{a}\approx 0$, they contribute to this clearly non-Kolmogorov yet Kolmogorov-sounding approximately constant value of $\unicode[STIX]{x1D6F1}^{a}/\unicode[STIX]{x1D716}_{r}^{a}$ close to $-1$ in two ways: predominantly through $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ for the constancy of $\unicode[STIX]{x1D6F1}^{a}/\unicode[STIX]{x1D700}_{r}^{a}$, and through ${\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$, the interscale transfer rate of stochastic energy by coherent fluctuations which improves the proximity of $\unicode[STIX]{x1D6F1}^{a}/\unicode[STIX]{x1D700}_{r}^{a}$ to $-1$.

Figure 10.

Orientation-averaged nonlinear interscale transfer terms from (3.3) and (3.4) normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

Further downstream, at $(x_{1},x_{2})=(8d,0)$, $\unicode[STIX]{x1D6F1}^{a}\approx \unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ in the range $\unicode[STIX]{x1D706}<r<0.4d$. In this range and at this position, the orientation-averaged interscale transfer rate of stochastic energy by coherent fluctuations is zero, and the near constancy with scale $r$ of $\unicode[STIX]{x1D6F1}^{a}$ is in fact, to a significant extent, accountable to $\unicode[STIX]{x1D6F1}^{\prime a}$, the orientation-averaged interscale transfer rate of stochastic energy by stochastic fluctuations (see figure 5). But the coherent structures also contribute significantly because $\unicode[STIX]{x1D6F1}^{a}$ is slightly but not insignificantly different from $\unicode[STIX]{x1D6F1}^{\prime a}$, in such a way that $\unicode[STIX]{x1D6F1}^{a}\approx \unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ is markedly closer to a constant than $\unicode[STIX]{x1D6F1}^{\prime a}$ in this scale range; compare $\unicode[STIX]{x1D6F1}^{\prime a}$ to $\unicode[STIX]{x1D6F1}^{a}$ and $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ in figure 11.

Figure 11.

Orientation-averaged interscale transfer terms normalised by $\unicode[STIX]{x1D700}_{r}^{a}$ at $x_{1}/d=8$ on the geometric centreline. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

The orientation-averaged interscale transfer of total fluctuating energy by the stochastic fluctuations, $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$, ceases to be constant at scales $r$ larger than $0.4d$; indeed, at this point $(x_{1},x_{2})=(8d,0)$, $-\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ is an increasing positive function of $r$ in the range $0.4d<r<d$, mirroring the decrease of $-{\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$ as a function of $r$ towards increasingly negative values (see figure 10). These two contributions add up in (5.1) to give a total interscale transfer rate $\unicode[STIX]{x1D6F1}^{a}$ which is approximately constant over a range of scales extended well beyond $r=0.4d$, as evidenced in figure 10. The correcting action of $-{\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$ (orientation-averaged energy transfer rate of stochastic energy by coherent fluctuations), via its positive values at length scales $r>0.4d$, is the essential ingredient for the extension of the near constancy of $\unicode[STIX]{x1D6F1}^{a}$ and its near equality to $-\unicode[STIX]{x1D700}$ over a range of scales which reaches as far out as $r=d$. Note that the fully stochastic interscale transfer rate $\unicode[STIX]{x1D6F1}^{\prime a}$ also shows a tendency for being approximately constant over this range at $(x_{1},x_{2})=(8d,0)$ (see figure 5) but its values are less close to $-\unicode[STIX]{x1D700}$ and less constant than $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}+{\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$. The coherent structures play a definite role in bringing $\unicode[STIX]{x1D6F1}^{a}$ closer to a constant equal to $-\unicode[STIX]{x1D700}$ at larger separations in this near-field flow.

In summary, at both locations $(x_{1},x_{2})=(2d,0)$ and $(8d,0)$, the reference equality

(5.2)$$\begin{eqnarray}-\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}-{\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}\approx \unicode[STIX]{x1D700}_{r}^{a}\end{eqnarray}$$

is not too far from our observations in the range where $\unicode[STIX]{x1D6F1}^{a}\approx \text{const}$. This equality re-writes $\unicode[STIX]{x1D6F1}^{a}\approx \text{const}$ with more information and this range increases as one moves downstream along the centreline reaching at least $\unicode[STIX]{x1D706}<r<d$ at $(x_{1},x_{2})=(8d,0)$. We stress that (5.2) is not exactly true. It might be more accurate to introduce a coefficient multiplying the right-hand side $\unicode[STIX]{x1D700}_{r}^{a}$ that is slightly smaller than 1 and not perfectly constant with $r$; but equation (5.2) is an important reference formula for our discussion which is not concerned, at this stage, with exact details. The coherent structures play an important role in both terms of the left-hand side of (5.2) at both locations $(x_{1},x_{2})=(2d,0)$ and $(8d,0)$, but the stochastic fluctuations do too and more so at $(x_{1},x_{2})=(8d,0)$ than $(x_{1},x_{2})=(2d,0)$.

The approximate balance $\unicode[STIX]{x1D6F1}^{a}\approx -\unicode[STIX]{x1D700}$ may be reminiscent of a Kolmogorov equilibrium cascade but the Kolmogorov theory is applicable to statistically homogeneous equilibrium turbulence which is far from the kind of turbulence in the present near-field wake. This approximate balance follows here from the approximate balance equation (5.2) and is partly supported by the effects of the coherent motions on the interscale turbulent energy transfers. The interscale transfer rate $\unicode[STIX]{x1D6F1}^{a}$ must therefore depend on the inlet/boundary conditions because of the memory carried by the coherent motions, as it also of course depends on the kinetic energy and size of the local large scale turbulent eddies. It is therefore not possible to derive a scaling for $\unicode[STIX]{x1D6F1}^{a}$ dimensionally, which means that it is not so easy to use the approximate balance $\unicode[STIX]{x1D6F1}^{a}\approx -\unicode[STIX]{x1D700}$ to derive a scaling for $\unicode[STIX]{x1D700}$ either. One can derive a scaling for the turbulence dissipation rate in the context of Kolmogorov equilibrium turbulence precisely because the interscale transfer rate is taken to be independent of inlet/initial/boundary conditions in this context. Goto & Vassilicos (Reference Goto and Vassilicos2016) have proposed a dissipation balance from which to derive turbulence dissipation scalings in non-stationary turbulence with a non-equilibrium cascade, and Alves Portela et al. (Reference Alves Portela, Papadakis and Vassilicos2018) have successfully adapted and applied this balance to the present near-field turbulent wake.

5.2 Inhomogeneity contributions to the nonlinear interscale energy transfer

In interpreting our results, it is relevant to dissociate the potential contribution of inhomogeneity to the interscale transfer rates. Given that

(5.3)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}\boldsymbol{u}\unicode[STIX]{x1D6FF}q^{2}=\boldsymbol{u}^{+}|\boldsymbol{u}^{+}|^{2}-\boldsymbol{u}^{-}|\boldsymbol{u}^{-}|^{2}+\boldsymbol{u}^{+}|\boldsymbol{u}^{-}|^{2}-\boldsymbol{u}^{-}|\boldsymbol{u}^{+}|^{2}-2\unicode[STIX]{x1D6FF}\boldsymbol{u}(\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}),\end{eqnarray}$$

one can see that statistical inhomogeneity can make a contribution to the average of $\unicode[STIX]{x1D6FF}\boldsymbol{u}\unicode[STIX]{x1D6FF}q^{2}$, at the very least from a non-zero average of $\boldsymbol{u}^{+}|\boldsymbol{u}^{+}|^{2}-\boldsymbol{u}^{-}|\boldsymbol{u}^{-}|^{2}$. However, we are mainly concerned with the nonlinear interscale transfer rate $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2})$ which has the property of being $0$ at $\boldsymbol{r}=0$ because it is equal to $\unicode[STIX]{x1D6FF}u_{i}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\unicode[STIX]{x1D6FF}q^{2}$ by incompressibility. We seek a decomposition of $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2})$ into an inhomogeneity term and a term unaffected by inhomogeneity such that both vanish at $\boldsymbol{r}=0$. Given that $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(u_{i}^{+}|\boldsymbol{u}^{+}|^{2}-u_{i}^{-}|\boldsymbol{u}^{-}|^{2})=\frac{1}{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+})(u_{i}^{+}|\boldsymbol{u}^{+}|^{2})+\frac{1}{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-})(u_{i}^{-}|\boldsymbol{u}^{-}|^{2})$ where $\unicode[STIX]{x1D709}_{i}^{+}=x_{i}+r_{i}/2$ and $\unicode[STIX]{x1D709}_{i}^{-}=x_{i}-r_{i}/2$, it is clear that $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(u_{i}^{+}|\boldsymbol{u}^{+}|^{2}-u_{i}^{-}|\boldsymbol{u}^{-}|^{2})$ is not $0$ at $\boldsymbol{r}=0$. We must therefore complement the inhomogeneity term $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(u_{i}^{+}|\boldsymbol{u}^{+}|^{2}-u_{i}^{-}|\boldsymbol{u}^{-}|^{2})$ in such a way that the resulting inhomogeneity term cancels when $\boldsymbol{r}=0$. Starting from

(5.4)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}(\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2})=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}[\unicode[STIX]{x1D6FF}u_{i}(|\boldsymbol{u}^{+}|^{2}+|\boldsymbol{u}^{-}|^{2})]-2\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}(\unicode[STIX]{x1D6FF}u_{i}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}),\end{eqnarray}$$

it rigorously follows that

(5.5)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}(\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2})=\frac{1}{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}[u_{i}^{+}|\boldsymbol{u}^{+}|^{2}+u_{i}^{-}|\boldsymbol{u}^{-}|^{2}-u_{i}^{-}|\boldsymbol{u}^{+}|^{2}-u_{i}^{+}|\boldsymbol{u}^{-}|^{2}]-2\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}(\unicode[STIX]{x1D6FF}u_{i}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}),\end{eqnarray}$$

where both the inhomogeneity term $\frac{1}{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})[u_{i}^{+}|\boldsymbol{u}^{+}|^{2}+u_{i}^{-}|\boldsymbol{u}^{-}|^{2}-u_{i}^{-}|\boldsymbol{u}^{+}|^{2}-u_{i}^{+}|\boldsymbol{u}^{-}|^{2}]$ and the interscale transfer term $-2(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})(\unicode[STIX]{x1D6FF}u_{i}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+})$ vanish at $\boldsymbol{r}=0$ (by virtue of incompressibility in the case of the interscale transfer term). The average value of the inhomogeneity term, $4\unicode[STIX]{x1D6F1}_{I}\equiv \frac{1}{2}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}+u_{i}^{-}|\boldsymbol{u}^{-}|^{2}-u_{i}^{-}|\boldsymbol{u}^{+}|^{2}-u_{i}^{+}|\boldsymbol{u}^{-}|^{2}\rangle$ can be non-zero in inhomogeneous turbulence but equals zero in homogeneous turbulence. It is clear that $\unicode[STIX]{x1D6F1}_{I}=0$ when the turbulence is statistically homogeneous. Unlike $\unicode[STIX]{x1D6F1}_{I}$, the average value of the pure interscale term, $4\unicode[STIX]{x1D6F1}_{H}\equiv -2(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}u_{i}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle$ can take non-zero values when the turbulence is statistically homogeneous.

We therefore have the decomposition

(5.6)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6F1}_{I}+\unicode[STIX]{x1D6F1}_{H},\end{eqnarray}$$

where (i) all three terms ($\unicode[STIX]{x1D6F1}$, $\unicode[STIX]{x1D6F1}_{I}$ and $\unicode[STIX]{x1D6F1}_{H}$) vanish at $\boldsymbol{r}=0$; (ii) $\unicode[STIX]{x1D6F1}_{I}$ can only be non-zero in the presence of inhomogeneity; and (iii) $\unicode[STIX]{x1D6F1}_{H}$ has the exact same form as $\unicode[STIX]{x1D6F1}$ in the case of homogeneous turbulence because $\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}q^{2}\rangle /\unicode[STIX]{x2202}r_{i}=-2(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle \unicode[STIX]{x1D6FF}u_{i}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle$ in such turbulence. This decomposition distinguishes between a term, $\unicode[STIX]{x1D6F1}_{I}$, that is clearly directly accountable to spatial inhomogeneities, and an interscale transfer rate $\unicode[STIX]{x1D6F1}_{H}$ which we may conjecture to be unaffected by spatial inhomogeneities. In relation to such a conjecture, we must ask whether our decomposition is unique.

Other such decompositions should take the form

(5.7)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}=(\unicode[STIX]{x1D6F1}_{I}+\unicode[STIX]{x1D6F1}_{IH})+(\unicode[STIX]{x1D6F1}_{H}-\unicode[STIX]{x1D6F1}_{IH}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}_{IH}$ must meet two conditions: (i) it must equal zero at $\boldsymbol{r}=0$; and (ii) it must vanish when the turbulence is statistically homogeneous. On account of this second condition, we write $\unicode[STIX]{x1D6F1}_{IH}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})\unicode[STIX]{x1D6F7}_{i}^{x}$. Because we are dealing with third-order statistics we assume that $\unicode[STIX]{x1D6F7}_{i}^{x}$ can only be a sum of products of three velocity components and the most general way to write this is as follows:

(5.8)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{i}^{x} & = & \displaystyle \unicode[STIX]{x1D6FC}_{1}\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle +\unicode[STIX]{x1D6FC}_{2}\langle u_{i}^{+}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle +\unicode[STIX]{x1D6FC}_{3}\langle u_{i}^{+}|\boldsymbol{u}^{-}|^{2}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FD}_{1}\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle +\unicode[STIX]{x1D6FD}_{2}\langle u_{i}^{-}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle +\unicode[STIX]{x1D6FD}_{3}\langle u_{i}^{-}|\boldsymbol{u}^{+}|^{2}\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{1}$, $\unicode[STIX]{x1D6FC}_{2}$, $\unicode[STIX]{x1D6FC}_{3}$, $\unicode[STIX]{x1D6FD}_{1}$, $\unicode[STIX]{x1D6FD}_{2}$, $\unicode[STIX]{x1D6FD}_{3}$ are dimensionless constants. With some care it easily follows that the condition $\unicode[STIX]{x1D6F1}_{IH}=0$ for $\boldsymbol{r}=0$ implies

(5.9)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{2}+\unicode[STIX]{x1D6FC}_{3}+\unicode[STIX]{x1D6FD}_{1}+\unicode[STIX]{x1D6FD}_{2}+\unicode[STIX]{x1D6FD}_{3}=0.\end{eqnarray}$$

Given that $\unicode[STIX]{x1D6F1}_{IH}$ contributes to the part $(\unicode[STIX]{x1D6F1}_{H}-\unicode[STIX]{x1D6F1}_{IH})$ of the decomposition, it must be possible to express it in the form $\unicode[STIX]{x1D6F1}_{IH}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\unicode[STIX]{x1D6F7}_{i}^{r}$. To find the conditions for this to be possible, we use $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+})+(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-})$ and use $\unicode[STIX]{x1D6F1}_{IH}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})\unicode[STIX]{x1D6F7}_{i}^{x}$ to write

(5.10)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{IH} & = & \displaystyle \unicode[STIX]{x1D6FC}_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+}}\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle +\unicode[STIX]{x1D6FC}_{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+}}\langle u_{i}^{+}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle +\unicode[STIX]{x1D6FD}_{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+}}\langle u_{i}^{-}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FD}_{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+}}\langle u_{i}^{-}|\boldsymbol{u}^{+}|^{2}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FC}_{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-}}\langle u_{i}^{+}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle +\unicode[STIX]{x1D6FC}_{3}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-}}\langle u_{i}^{+}|\boldsymbol{u}^{-}|^{2}\rangle +\unicode[STIX]{x1D6FD}_{1}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-}}\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D6FD}_{2}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-}}\langle u_{i}^{-}\boldsymbol{u}^{-}\boldsymbol{\cdot }\boldsymbol{u}^{+}\rangle .\end{eqnarray}$$

Note that $\unicode[STIX]{x1D6FC}_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+})\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle =2\unicode[STIX]{x1D6FC}_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle$ because $(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-})\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle =0$ and $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i}=\frac{1}{2}((\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{+})-(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-}))$. For the same reason, $\unicode[STIX]{x1D6FD}_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{i}^{-})\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle =-2\unicode[STIX]{x1D6FD}_{1}(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle$. All the other terms and combinations of other terms cannot be rephrased in $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i}$ form. The necessary form $\unicode[STIX]{x1D6F1}_{IH}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{i})\unicode[STIX]{x1D6F7}_{i}^{r}$ then implies $\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}_{3}=\unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{3}=0$. From (5.9) follows $\unicode[STIX]{x1D6FC}_{1}=-\unicode[STIX]{x1D6FD}_{1}$ and therefore

(5.11)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{IH}=\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r_{i}}(\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle +\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle )=\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}(\langle u_{i}^{+}|\boldsymbol{u}^{+}|^{2}\rangle -\langle u_{i}^{-}|\boldsymbol{u}^{-}|^{2}\rangle ),\end{eqnarray}$$

where we also made use of $\unicode[STIX]{x1D6F1}_{IH}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{i})\unicode[STIX]{x1D6F7}_{i}^{x}$ with (5.8) and (5.9), and where we set $\unicode[STIX]{x1D6FC}\equiv 2\unicode[STIX]{x1D6FC}_{1}$.

In conclusion, the decomposition equation (5.6) is not unique as one can always use $\unicode[STIX]{x1D6F1}_{IH}$ given by (5.11) to obtain another equally valid decomposition equation (5.7). However, if one averages over scale-space orientations, the decomposition

(5.12)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}^{a}=\unicode[STIX]{x1D6F1}_{I}^{a}+\unicode[STIX]{x1D6F1}_{H}^{a}\end{eqnarray}$$

is unique because $\unicode[STIX]{x1D6F1}_{IH}^{a}=0$ given that $\unicode[STIX]{x1D6F1}_{IH}$ in (5.11) is such that $\unicode[STIX]{x1D6F1}_{IH}(\boldsymbol{r})=-\unicode[STIX]{x1D6F1}_{IH}(-\boldsymbol{r})$. The conjecture that the orientation-averaged interscale transfer rate $\unicode[STIX]{x1D6F1}_{H}^{a}$ may be unaffected by spatial inhomogeneities is more likely to hold than the conjecture that $\unicode[STIX]{x1D6F1}_{H}$ is unaffected by spatial inhomogeneities. This conjecture and the decomposition introduced in this subsection are an attempt at introducing a tool which can help make some analytic sense of the concept of an inhomogeneous turbulence cascade.

Figure 12.

Orientation-averaged interscale energy transfer terms $\unicode[STIX]{x1D6F1}_{I}^{a}$, $\unicode[STIX]{x1D6F1}_{H}^{a}$ and $\unicode[STIX]{x1D6F1}^{a}$ (see equations (5.5) and (5.6)) at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

In figure 12 we plot the orientation-averaged interscale transfer rates $\unicode[STIX]{x1D6F1}^{a}$, $\unicode[STIX]{x1D6F1}_{I}^{a}$ and $\unicode[STIX]{x1D6F1}_{H}^{a}$ at the two centreline positions $(x_{1},x_{2})=(2d,0)$ and $(8d,0)$. Inhomogeneity interscale transfer is present and positive at all scales, but may be considered negligible at dissipative scales $r$ smaller than $0.1d$, i.e. smaller than the Taylor microscale $\unicode[STIX]{x1D706}$. However, it does make a significant contribution to the total interscale energy transfer rate $\unicode[STIX]{x1D6F1}^{a}$ at scales $r$ larger than $\unicode[STIX]{x1D706}$, particularly at $(x_{1},x_{2})=(2d,0)$ where $\unicode[STIX]{x1D6F1}_{I}^{a}$ is commensurate throughout these scales with the negative interscale transfer $\unicode[STIX]{x1D6F1}_{H}^{a}$. In fact $\unicode[STIX]{x1D6F1}^{a}$ changes sign from negative to positive as $r$ increases beyond $r\approx 0.6d$ because of the influence of the positive inhomogeneity interscale energy transfer rate.

In figure 12 one can also see that the contribution of the inhomogeneity part of the interscale energy transfer weakens with downstream distance, while remaining positive throughout the scales. Here, $\unicode[STIX]{x1D6F1}^{a}$ and $\unicode[STIX]{x1D6F1}_{H}^{a}$ are both negative throughout the scales and significantly closer to each other than to $\unicode[STIX]{x1D6F1}_{I}^{a}$ at $(x_{1},x_{2})=(8d,0)$, which is not the case at $(x_{1},x_{2})=(2d,0)$.

It is particularly intriguing that $\unicode[STIX]{x1D6F1}^{a}$ would not have been approximately constant across the scales, from approximately $\unicode[STIX]{x1D706}$ to approximately $0.3d$ at $(x_{1},x_{2})=(2d,0)$ and from approximately $\unicode[STIX]{x1D706}$ to approximately $d$ at $(x_{1},x_{2})=(8d,0)$, without the inhomogeneity contribution coming from $\unicode[STIX]{x1D6F1}_{I}^{a}$. It is in fact this inhomogeneity contribution which returns a near constancy of $\unicode[STIX]{x1D6F1}^{a}$ all the way up to scales $r$ equal to $d$ at $(x_{1},x_{2})=(8d,0)$ and imparts on the orientation-averaged interscale energy transfer $\unicode[STIX]{x1D6F1}^{a}$ a Kolmogorov-sounding behaviour over a decade of scales $r$.

The results of these two subsections suggest that the approximate balance $\unicode[STIX]{x1D6F1}^{a}\approx -\unicode[STIX]{x1D700}$ observed in our turbulent wake’s very near field, even if reminiscent of a Kolmogorov equilibrium for homogeneous turbulence, is in fact possible in this near-field turbulence because of the presence of spatial inhomogeneity and coherent structures.

5.3 interscale fluxes

In order to interpret the interscale physics behind the negative sign of $\unicode[STIX]{x1D6F1}^{a}$ it is necessary to also consider the interscale flux $\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}\unicode[STIX]{x1D6FF}q^{2}\rangle$ given that $\unicode[STIX]{x1D6F1}$ is the divergence of this flux in scale space $\boldsymbol{r}$. In particular, it is necessary to consider the sign of the radial component of the orientation-averaged interscale flux. One cannot claim that the interscale energy transfer proceeds from large to small scales on average if this sign is not negative too.

The interscale flux vectors which correspond to each term in (3.7) are related by

(5.13)$$\begin{eqnarray}\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}\unicode[STIX]{x1D6FF}q^{2}\rangle =\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\langle \unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\langle \unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle +2\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }(\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}})\rangle .\end{eqnarray}$$

The flux vectors are placed in this equation in exactly the same way as their corresponding interscale transfer rates are placed in (3.7). The interscale flux identity which reflects $\tilde{\unicode[STIX]{x1D6F1}}_{{\mathcal{P}}_{\tilde{u} }}=\unicode[STIX]{x1D6F1}_{u^{\prime }}-\unicode[STIX]{x1D6F1}^{\prime }$ is $2\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }(\unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}})\rangle =\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\unicode[STIX]{x1D6FF}q^{2}\rangle -\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle$. Combined with (5.13) it yields

(5.14)$$\begin{eqnarray}\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}\unicode[STIX]{x1D6FF}q^{2}\rangle =\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}^{\prime }\unicode[STIX]{x1D6FF}q^{2}\rangle +\langle \unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle +\langle \unicode[STIX]{x1D6FF}\tilde{\boldsymbol{u}}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle\end{eqnarray}$$

which corresponds to (3.9).

Figure 13.

Orientation-averaged nonlinear interscale radial fluxes terms at (a) $x_{1}/d=2$ and (b) $x_{1}/d=8$. The vertical dotted line gives the position of $r=\unicode[STIX]{x1D706}$.

We are interested in the orientation-averaged radial components of these fluxes in the $r_{3}=0$ plane. In figure 13 we plot, as functions of $r$, the orientation-averaged radial components (in the $r_{3}=0$ plane) $\langle \unicode[STIX]{x1D6FF}{u^{\prime }}_{r}\unicode[STIX]{x1D6FF}q^{2}\rangle ^{a}$, $\langle \unicode[STIX]{x1D6FF}\tilde{u} _{r}\unicode[STIX]{x1D6FF}q^{\prime 2}\rangle ^{a}$ and $\langle \unicode[STIX]{x1D6FF}\tilde{u} _{r}\unicode[STIX]{x1D6FF}\tilde{q}^{2}\rangle ^{a}$. The latter is zero where (5.2) is relevant. Concentrating our attention on the scale range where (5.2) is relevant, the signs of these orientation-averaged radial fluxes and of the corresponding orientation-averaged interscale transfer rates therefore suggest the following: (i) concerning $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$, the stochastic fluctuations transfer, on average, total (stochastic and coherent) fluctuating energy from large to small scales in the range $r<0.3d$ at $(x_{1},x_{2})=(2d,0)$ and $r<d$ at $(x_{1},x_{2})=(8d,0)$; (ii) concerning ${\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$, the coherent fluctuations transfer, on average, stochastic energy from large to small scales at length scales $r<0.3d$ at both spatial locations, but from small to large scales at $(x_{1},x_{2})=(8d,0)$ in the range $0.4d<r<d$. The contribution of ${\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$ is the smallest of the two interscale transfer rate terms, $\unicode[STIX]{x1D6F1}_{u^{\prime }}^{a}$ and ${\unicode[STIX]{x1D6F1}_{\tilde{u} }^{\prime }}^{a}$, in (5.1) and (5.2). The interscale fluctuating energy transfer proceeds, therefore, from large to small scales on average, mostly because of the large to small scale transfer of total fluctuating energy by stochastic fluctuations.