4.1 Two-dimensional sub-manifold analysis

We need to specify five coordinates to identify the maximum of a function in phase space. Identifying maxima in a five-dimensional space is already non-trivial. In the present case, the issue is more difficult due to the statistical nature of our object. For example, a standard Newton iteration would require the evaluation of the average to be maximised and its five-dimensional gradient at an arbitrary phase space point. Two different strategies may be conceived: either evaluating statistical average and its derivatives at the corresponding coordinate on the fly or having function and derivatives pre-evaluated on a suitable five-dimensional lattice. Both choices are unaffordable given the dimension of the dataset to be manipulated.

As an alternative, we may exploit physical intuition and start by fixing the streamwise coordinate of the mid-point $\boldsymbol{X}$ to the cross-stream plane of maximum (single-point) kinetic energy production, namely $X^{0}=5.8$ and $X^{0}=5.1$ for the two cases, § 3. The remaining coordinates, $Y^{0},r_{x}^{0},r_{y}^{0},r_{z}^{0}$ , will be progressively found from the maxima on selected two-dimensional sub-manifolds. We deliberately choose to fix the streamwise coordinate since the shear layer is elongated in this direction and the dependence on $X$ is mild. The emerging picture from slightly different $X$ sections is therefore negligibly sensitive to the precise location. On the other hand, the dependence on $Y$ is strong and requires a careful determination.

Let us take $\boldsymbol{r}$ parallel to the streamwise direction, $\boldsymbol{r}=(r_{x},0,0)$ , keeping in mind that $\boldsymbol{X}$ belongs to the fixed cross-stream plane $X=X^{0}$ . Accounting for translational invariance in the spanwise direction, this manifold is described by the two independent variables $Y$ and $r_{x}$ (the reader may wish to have a further look at figure 3). We shall therefore first search for the maximum of $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle$ in this plane, figure 5(a). The blank areas in the plot correspond to the physical domain boundaries which limit the allowed range of separation vectors. For this specific case, only $r_{x}\geqslant 0$ is considered given the $r_{x}\rightarrow -r_{x}$ symmetry that holds since $r_{y},r_{z}$ are still zero. In this plane, the production peak is located at $Y^{0}=0.45$ and $r_{x}^{0}=0.45$ . Note that $Y^{0}$ corresponds to the distance of the shear layer from the bottom wall, $y_{sl}=0.45$ .

Figure 5. Contour plots of the net production $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle$ for the dataset at $Re=2500$ . (a) Plane $(Y,r_{x})|_{X=5.8,r_{y}=r_{z}=0}$ ; (b) plane $(Y,r_{y})|_{X=5.8,r_{x}=r_{z}=0}$ ; (c) plane $(Y,r_{z})|_{X=5.8,r_{x}=r_{y}=0}$ ; (d) plane $(Y,r_{y})|_{r_{x}=0.45,r_{z}=0.4}$ .

Keeping $X$ constant and taking the separation vector in the wall-normal direction, $\boldsymbol{r}=(0,r_{y},0)$ , the two independent variables are now $Y$ and $r_{y}$ , panel (b). Although two inclined bands of intense production (to be further discussed in the following) are apparent, a single definite maximum cannot unambiguously be identified in this plane.

For separations in the spanwise direction, $\boldsymbol{r}=(0,0,r_{z})$ , the maximum is determined from panel (c). The highest production occurs at $Y^{0}=0.45$ , again matching the height of the shear layer and, although the isolines are rather elongated, the maximum is found at $r_{z}^{0}=0.4$ . Instantaneous configurations of the velocity field coinciding with $x=X^{0}$ and $y=Y^{0}$ are shown in figure 6. From panels (a) and (c), a well-defined transversal scale is apparent. The minimum of the transversal correlation coefficient $R_{z}^{uu}=\langle u(x^{0},y^{0},z)u(x^{0},y^{0},z+r_{z})\rangle /\langle u(x^{0},y^{0},z)^{2}\rangle$ shown in panel (d) identifies this characteristic length scale at the shear layer (red curve) as $L_{T}\simeq 0.42$ . This value is consistent with the transversal scale where the maximum net production occurs $(r_{z}^{0}=0.4)$ .

Figure 6. Instantaneous streamwise velocity contours for $Re=2500$ . (a)  $(y,z)$ -plane at $x^{0}=5.8$ ; (b) $(x,y)$ -plane at arbitrary $z^{0}$ ; (c) $(x,z)$ -plane at $y^{0}=0.45$ . Panel (d) shows the spanwise correlation $R_{z}^{uu}=\langle u(x^{0},y^{0},z)u(x^{0},y^{0},z+r_{z})/\langle u(x^{0},y^{0},z)\rangle ^{2}\rangle$ for $x^{0}=5.8$ in the shear layer, $y^{0}=0.45$ (red) and at the centreline, $y^{0}=1.0$ (blue).

Recalling that $r_{y}$ is still undetermined, the plane of panel (b) is relocated, see panel (d) which shows the production in the plane $(Y,r_{y})|_{X=5.8,r_{x}=0.45,r_{z}=0.4}$ . The maximum is clearly found at the intersection of the inclined bands at $Y=0.45$ and $r_{y}^{0}=0$ .

All the planes considered so far can now be repositioned: the first one – $(Y,r_{x})$ – will now be the plane $X=X^{0}$ , $r_{y}=r_{y}^{0}$ , $r_{z}=r_{z}^{0}$ , while the second – $(Y,r_{y})$ – and the third – $(Y,r_{z})$ – will be $X=X^{0}$ , $r_{z}=r_{z}^{0}$ , $r_{x}=r_{x}^{0}$ and $X=X^{0}$ , $r_{x}=r_{x}^{0}$ , $r_{y}=r_{y}^{0}$ , respectively. Based on this approach, an iterative algorithm that looks for the maxima and updates the planes until convergence could in principle be set-up but, for the present cases, it is found to leave the result essentially unaltered.

For the dataset at $Re=10\,000$ (not shown), the behaviour is similar but everything occurs closer to the bump since the shear layer, together with the maxima of $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle$ , moves towards the bump at increasing Reynolds number as shown in Mollicone et al. (Reference Mollicone, Battista, Gualtieri and Casciola2017).

In summary, to a degree of accuracy sufficient for our next discussion, the maximum net production intensity $(\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle )_{\max }$ is found at $X^{0}=5.8$ , $Y^{0}=0.45$ , $r_{x}^{0}=0.45$ , $r_{y}^{0}=0.$ and $r_{z}^{0}=0.4$ for the case at $Re=2500$ and at $X^{0}=5.1$ , $Y^{0}=0.45$ , $r_{x}^{0}=0.17$ , $r_{y}^{0}=0.$ and $r_{z}^{0}=0.13$ for $Re=10\,000$ .

As discussed in § 2, a way to represent the five-dimensional fields involved in the GKE consists in selecting two-dimensional planes. Considering, e.g. the $(Y,r_{y})$ -plane, the in-plane component of the flux vector $\unicode[STIX]{x1D731}_{6}$ , $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{y}})$ , follows the equation

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\Vert }=\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }, & \displaystyle\end{eqnarray}$$

where the general six-dimensional gradient is expressed as the sum of in-plane and normal component, $\unicode[STIX]{x1D735}_{\Vert }=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}Y,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{y})$ and $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{z},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r_{x})$ , respectively. The actual number of dimensions of the phase space is reduced to five given the invariance to $z$ -translations, $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{Z}=0$ . Equation (4.1) is obtained from (2.5) after moving the orthogonal components of the flux, $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{z}},\unicode[STIX]{x1D6F7}_{r_{x}})$ , to the right-hand side.

Figure 7. $Re=2500$ , $(Y,r_{y})$ -plane at $X^{0}=5.8$ , $r_{x}^{0}=0.45$ , $r_{z}^{0}=0.4$ . Solid isolines represent $\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ . (a) In-plane component of the flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{y}})$ as vectors, equation (2.3). The coloured contour denotes the source terms $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }$ , equation (4.1), where $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}_{X},\unicode[STIX]{x2202}_{r_{z}},\unicode[STIX]{x2202}_{r_{x}})$ and $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{z}},\unicode[STIX]{x1D6F7}_{r_{x}})$ . (b) In-plane transport velocity $\boldsymbol{w}_{\Vert }=(w_{Y},w_{r_{y}})$ as vectors, equation (2.15), and effective net production in the plane ${\mathcal{M}}$ , equation (4.3), as coloured contour. The yellow symbols denote the images of the bubble separatrix: $f_{b}(\tilde{\boldsymbol{x}})=0$ (triangles, $\tilde{\boldsymbol{x}}=\boldsymbol{X}+\boldsymbol{r}$ on the separatrix) and $f_{b}(\boldsymbol{x})=0$ (circles, $\boldsymbol{x}=\boldsymbol{X}-\boldsymbol{r}$ on the separatrix). The inset shows where ${\mathcal{M}}$ is positive (red) or negative (blue). (c) Normal component of the in-plane transport contribution $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ to the flux (vectors) shown on the bubble separatrix. Coloured contour shows $\tilde{{\mathcal{M}}}$ , equation (4.5).

4.2 Sub-manifold $(Y,r_{y})$

Figure 7(a) shows $\unicode[STIX]{x1D731}_{\Vert }$ in the $(Y,r_{y})_{X^{0}=5.8,r_{x}^{0}=0.45,r_{z}^{0}=0.4}$ -plane which passes through the maximum of net production. A surface in the ordinary three-dimensional space, $f(\boldsymbol{x})=0$ , generates two images in phase space depending on which one of the two points, $\tilde{\boldsymbol{x}}$ or $\boldsymbol{x}$ , belongs to the surface, e.g. $f(\tilde{\boldsymbol{x}}/\boldsymbol{x})=f(\boldsymbol{X}\pm \boldsymbol{r}/2)=0$ . The diamond shaped region of the plot is delimited below by the two images of the (curved) lower wall, $f_{lw}(\tilde{\boldsymbol{x}}/\boldsymbol{x})=0$ , and above by those of the (straight) upper wall of the channel, $f_{uw}(\tilde{\boldsymbol{x}}/\boldsymbol{x})=0$ . Since the section is taken at $r_{x}^{0}\neq 0$ , the lower boundary of the phase space domain is asymmetric due to the presence of the bump. In the figure, the two curves highlighted with the yellow symbols represent the loci in the $(Y,r_{y})$ -plane corresponding to either $\tilde{\boldsymbol{x}}$ or $\boldsymbol{x}$ belonging to the separatrix defining the recirculation bubble, i.e. solid line in figure 8. In physical space, the separatrix is defined by an equation of the form $f_{b}(\boldsymbol{x})=0$ or equivalently, given the translational invariance in the $z$ direction, $y=g_{b}(x)$ . When $\tilde{\boldsymbol{x}}$ is on the separatrix, $f_{b}(\tilde{\boldsymbol{x}})=0$ , i.e. $f_{b}(\boldsymbol{X}+\boldsymbol{r}/2)=0$ since $\tilde{\boldsymbol{x}}=\boldsymbol{X}+\boldsymbol{r}/2$ . It is assumed that $f_{b}<0$ identifies the region ${\mathcal{B}}$ inside the recirculating bubble. Analogously, when the second of the two points across which the structure function is evaluated belongs to the separatrix, $f_{b}(\boldsymbol{X}-\boldsymbol{r}/2)=0$ . The two manifolds $f_{b}(\boldsymbol{X}\pm \boldsymbol{r}/2)=0$ split the phase space into four regions, namely ${\mathcal{D}}_{II}=\{\boldsymbol{X}(\tilde{\boldsymbol{x}},\boldsymbol{x}),\boldsymbol{r}(\tilde{\boldsymbol{x}},\boldsymbol{x}):\tilde{\boldsymbol{x}},\boldsymbol{x}\in {\mathcal{B}}\}$ , where both $\tilde{\boldsymbol{x}}$ and $\boldsymbol{x}$ are inside the bubble, ${\mathcal{D}}_{IO}=\{\boldsymbol{X}(\tilde{\boldsymbol{x}},\boldsymbol{x}),\boldsymbol{r}(\tilde{\boldsymbol{x}},\boldsymbol{x}):\tilde{\boldsymbol{x}}\in {\mathcal{B}},\boldsymbol{x}\notin {\mathcal{B}}\}$ , where only $\boldsymbol{x}$ is outside ${\mathcal{B}}$ , ${\mathcal{D}}_{OI}=\{\boldsymbol{X}(\tilde{\boldsymbol{x}},\boldsymbol{x}),\boldsymbol{r}(\tilde{\boldsymbol{x}},\boldsymbol{x}):\tilde{\boldsymbol{x}}\notin {\mathcal{B}},\boldsymbol{x}\in {\mathcal{B}}\}$ , where only $\tilde{\boldsymbol{x}}$ is outside ${\mathcal{B}}$ , and ${\mathcal{D}}_{OO}=\{\boldsymbol{X}(\tilde{\boldsymbol{x}},\boldsymbol{x}),\boldsymbol{r}(\tilde{\boldsymbol{x}},\boldsymbol{x}):\tilde{\boldsymbol{x}}\notin {\mathcal{B}},\boldsymbol{x}\notin {\mathcal{B}}\}$ , where both $\tilde{\boldsymbol{x}}$ and $\boldsymbol{x}$ are outside ${\mathcal{B}}$ .

In the $(Y,r_{y})$ -plane, shown in figure 7(a), the traces of the two manifolds reduce to the two (straight) curves $Y=g_{b}(X^{0}\pm r_{x}^{0}/2)\mp r_{y}/2$ shown by the yellow symbols in the plot, where triangles show the trace of the manifold $f_{b}(\tilde{\boldsymbol{x}})=0$ whilst circles correspond to $f_{b}(\boldsymbol{x})=0$ . The solid isolines show the magnitude of the second-order structure function $\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ of figure 4(c), the colour background gives the isolevels of the right-hand side of (4.1) (dubbed the source in the following description) whilst vectors represent $\unicode[STIX]{x1D731}_{\Vert }$ . Just above the two images of the separatrix, inside ${\mathcal{D}}_{OO}$ , the two elongated regions of intense source correspond to the shear layer. The in-plane flux $\unicode[STIX]{x1D731}_{\Vert }$ in these regions is directed toward decreasing mid-point wall distance $Y$ and decreasing wall-normal separations $r_{y}$ . The physical interpretation of this process will be provided later, after a more complete description of the flow in the phase space. Two (almost) symmetric spots of intense source are observed where the images of the separatrix touch the images of the walls (the one on the left is located around the intersection at $Y\simeq 0.25$ , $r_{y}\simeq -0.45$ , solution of the equation $g_{b}(X^{0}-r_{x}^{0}/2)+r_{y}/2=g_{w}(X^{0}+r_{x}^{0}/2)-r_{y}/2$ , where $y=g_{w}(x)$ is the explicit equation of the lower channel wall, implicitly described by $f_{lw}(\boldsymbol{x})$ ). In the high intensity red spot on the left, the point $\boldsymbol{x}$ is outside the bubble and within the shear layer (phase space region above the yellow circles). The other point, $\tilde{\boldsymbol{x}}$ , is inside the bubble (below the yellow triangles) and close to the wall. An almost symmetric arrangement is found for the right spot. The absolute maximum of the source is found when both $\tilde{\boldsymbol{x}}$ and $\boldsymbol{x}$ are in the shear layer, just above the intersection of the two images of the separatrix. On the other hand, in the region ${\mathcal{D}}_{II}$ (the rhomboidal region limited by lower wall and separatrix) a negative source is present. The fluxes enter this region from outside. The integration of (2.5) over the phase space domain ${\mathcal{D}}_{II}$ leads to

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \oint _{\unicode[STIX]{x2202}{\mathcal{D}}_{II}}\unicode[STIX]{x1D731}_{6}\boldsymbol{\cdot }\unicode[STIX]{x1D742}_{6}\,\text{d}S_{6}=\int _{{\mathcal{D}}_{II}}(\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle )\,\text{d}^{3}\boldsymbol{X}\,\text{d}^{3}\boldsymbol{r}. & \displaystyle\end{eqnarray}$$

In the physical sense, the bubble is expected to be of dissipative nature, hence one would guess $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle <0$ . Clearly, the sign of $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle$ in the entire five-dimensional domain corresponding to the bubble could not be directly checked. However, two-dimensional sections of such domain, like the one shown in figure 7(a), see also the successive figures 9, 10, 11 and 12, suggest that this is indeed the case. This confirms that, being the bubble a statistically steady dissipative feature, the second-order structure function in the bubble region is sustained in time by fluxes entering the domain.

Figure 8. $Re=2500$ , (a) Mean velocity field near the bump. Coloured contour denotes the (single-point) TKE production. (b) Enlargement of the recirculating region. The solid line is the stream line separating the bubble from the outer flow.

Figure 9. $Re=2500$ , $(Y,r_{x})$ -plane at $X^{0}=5.8$ , $r_{y}^{0}=0$ , $r_{z}^{0}=0.4$ . In this plane $\unicode[STIX]{x1D735}_{\Vert }=(\unicode[STIX]{x2202}_{Y},\unicode[STIX]{x2202}_{r_{x}})$ . Solid isolines represent $\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ . (a) In-plane component of the flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{x}})$ as vectors, equation (2.3). Coloured contour denotes the source terms $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }$ , equation (4.1), where $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}_{X},\unicode[STIX]{x2202}_{r_{y}},\unicode[STIX]{x2202}_{r_{z}})$ and $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{y}},\unicode[STIX]{x1D6F7}_{r_{z}})$ . (b) In-plane transport velocity $\boldsymbol{w}_{\Vert }=(w_{Y},w_{r_{x}})$ as vectors, equation (2.15) and effective net production in the plane ${\mathcal{M}}$ , equation (4.3), as coloured contour. The yellow symbols denote the images of the bubble separatrix, see text and figure 7. The inset shows the sign of ${\mathcal{M}}$ . (c) Same as (a) with the in-plane velocity magnified in the region ${\mathcal{D}}_{II}^{\Vert }$ , see text. (d) Normal component of the in-plane transport contribution $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ to the flux (vectors) shown on the bubble separatrix. Coloured contour shows $\tilde{{\mathcal{M}}}$ , equation (4.5).

Figure 10. $Re=2500$ , $(Y,r_{x})$ -plane at $X^{0}=5.8$ , $r_{y}^{0}=0.12$ , $r_{z}^{0}=0.4$ . Panels (a) and (b) same as corresponding panels of figure 9.

The Lagrangian theory in § 2.2 allows us to define a velocity field $\boldsymbol{w}_{6}$ which transports the structure function in phase space. This, in a way, provides the phase space equivalent of the advection of fluid particles by the velocity field in physical space. Again, the transport velocity field defined in (2.14), (2.15) is not easily visualised. Planar sections can provide an illuminating, though admittedly incomplete, view. In a planar section, the transport (2.16), (2.17) for the structure function can be rearranged as

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}_{\Vert }}{\text{D}t}\left(\frac{1}{2}\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \right)={\mathcal{M}}, & \displaystyle\end{eqnarray}$$

where $\text{D}_{\Vert }/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{w}_{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\Vert }$ is the transport derivative in the plane ( $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t=0$ for the present statistically steady conditions) and the source term on the right-hand side is

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{M}}=\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }-\unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\Vert }^{D}-{\textstyle \frac{1}{2}}\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }\boldsymbol{w}_{\Vert }, & \displaystyle\end{eqnarray}$$

with the superscript $D$ denoting diffusion, equation (2.6). The transport velocity $\boldsymbol{w}_{\Vert }=(w_{Y},w_{r_{y}})$ in the $(Y,r_{y})$ -plane is shown in figure 7(b). As in panel (a), the solid isolines represent the structure function and the yellow symbols the two images of the separatrix. The colour contours now provide ${\mathcal{M}}$ , the right-hand side of (4.3), explicitly defined in (4.4). The inset highlights the zero isolevel of ${\mathcal{M}}$ , with red and blue denoting positive and negative values respectively.

Figure 11. $Re=2500$ , $(X,r_{x})$ -plane at $Y^{0}=0.45$ , $r_{y}^{0}=0$ , $r_{z}^{0}=0.4$ . In this plane $\unicode[STIX]{x1D735}_{\Vert }=(\unicode[STIX]{x2202}_{X},\unicode[STIX]{x2202}_{r_{x}})$ . Solid isolines represent $\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ . (a) In-plane component of the flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{x}})$ as vectors, equation (2.3). Coloured contour denotes the source terms $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }$ , equation (4.1), where $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}_{Y},\unicode[STIX]{x2202}_{r_{y}},\unicode[STIX]{x2202}_{r_{z}})$ and $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{y}},\unicode[STIX]{x1D6F7}_{r_{z}})$ . (b) In-plane transport velocity $\boldsymbol{w}_{\Vert }=(w_{X},w_{r_{x}})$ as vectors, equation (2.15) and effective net production in the plane ${\mathcal{M}}$ , equation (4.3), as coloured contour. The inset shows the sign of ${\mathcal{M}}$ . The yellow symbols denote the images of the bubble separatrix, see text and figure 7. (c) Same as (a) with the in-plane velocity magnified in the region ${\mathcal{D}}_{II}^{\Vert }$ , see text. (d) Normal component of the in-plane transport contribution $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ to the flux (vectors) shown on the bubble separatrix. Coloured contour shows $\tilde{{\mathcal{M}}}$ , equation (4.5).

Figure 12. $Re=2500$ , $(X,Y)$ -plane at $r_{x}^{0}=0.45$ , $r_{y}^{0}=0$ , $r_{z}^{0}=0.4$ . In this plane $\unicode[STIX]{x1D735}_{\Vert }=(\unicode[STIX]{x2202}_{X},\unicode[STIX]{x2202}_{Y})$ . Solid isolines represent $\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ . (a) In-plane component of the flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{Y})$ as vectors, equation (2.3). Coloured contour denotes the source terms $\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle -\unicode[STIX]{x1D735}_{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D731}_{\bot }$ , equation (4.1), where $\unicode[STIX]{x1D735}_{\bot }=(\unicode[STIX]{x2202}_{r_{x}},\unicode[STIX]{x2202}_{r_{y}},\unicode[STIX]{x2202}_{r_{z}})$ and $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{r_{x}},\unicode[STIX]{x1D6F7}_{r_{y}},\unicode[STIX]{x1D6F7}_{r_{z}})$ . (b) In-plane transport velocity $\boldsymbol{w}_{\Vert }=(w_{X},w_{Y})$ as vectors, equation (2.15) and effective net production in the plane ${\mathcal{M}}$ , equation (4.3), as coloured contour. The yellow symbols denote the images of the bubble separatrix, see text and figure 7. (c) Same as (a) with the in-plane velocity magnified in the region ${\mathcal{D}}_{II}^{\Vert }$ , see text. (d) Normal component of the in-plane transport contribution $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ to the flux (vectors) shown on the bubble separatrix. Coloured contour shows $\tilde{{\mathcal{M}}}$ , equation (4.5).

Following the standard interpretation of substantial derivatives, equation (4.3) is interpreted by saying that the structure function is advected along the trajectories of the field $\boldsymbol{w}_{\Vert }$ and changes its magnitude due to the source term on the right-hand side. Overall, the structure function is advected toward decreasing mid-point distance from the lower wall $Y$ and toward decreasing wall-normal separation $r_{y}$ , indicating that most of the flux $\unicode[STIX]{x1D731}_{\Vert }$ taking place in the same region is associated with transport. Focusing on the shear layer, coincident with the two elongated regions where ${\mathcal{M}}$ is intense just above the two images of the separatrix, the advection velocity is directed almost parallel to the images of the separatrix, towards smaller wall-normal mid-point distances $Y$ and smaller wall-normal separations $r_{y}$ . A bias is added to this gross trend, such that the $\boldsymbol{w}_{\Vert }$ velocity is slightly more inclined downwards than the local tangent to the separatrix. This means that the structure function, advected by the velocity field, crosses the separatrix from region ${\mathcal{D}}_{OO}$ above to enter region ${\mathcal{D}}_{IO}$ (left half of the plot, crossing the triangles) or ${\mathcal{D}}_{OI}$ (right half of the plot, crossing the circles). In other words, one of the two points (e.g. $\tilde{\boldsymbol{x}}$ as concerning the left part) is entering the recirculation bubble, while the other one remains outside. Following the advection velocity, the former point eventually enters the bubble. When this happens the point in phase space crosses the second image of the separatrix (the circles) leaving region ${\mathcal{D}}_{IO}$ to enter ${\mathcal{D}}_{II}$ . Just before the second image of the separatrix is crossed (point $\boldsymbol{x}$ still in the shear layer) a positive value of ${\mathcal{M}}$ is encountered, implying that the intensity of the structure function increases while advection by $\boldsymbol{w}_{\Vert }$ occurs.

Equation (4.3) can be recast in conservative form as

(4.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }\left(\boldsymbol{w}_{\Vert }{\textstyle \frac{1}{2}}\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \right)=\tilde{{\mathcal{M}}}={\mathcal{M}}+{\textstyle \frac{1}{2}}\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \unicode[STIX]{x1D735}_{\Vert }\boldsymbol{\cdot }\boldsymbol{w}_{\Vert }. & & \displaystyle\end{eqnarray}$$

This equation can be integrated in the domain $D_{II}^{\Vert }$ , defined as the trace of the phase space domain ${\mathcal{D}}_{II}$ in the considered $(Y,r_{y})$ -plane, to yield

(4.6) $$\begin{eqnarray}\displaystyle \int _{\unicode[STIX]{x2202}{\mathcal{D}}_{II}^{\Vert }}\boldsymbol{w}_{\Vert }\frac{1}{2}\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \boldsymbol{\cdot }\boldsymbol{n}_{\Vert }\,\text{d}l_{\Vert }=\int _{{\mathcal{D}}_{II}^{\Vert }}\tilde{{\mathcal{M}}}\,\text{d}S_{\Vert }, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}_{\Vert }$ is the outward normal. Panel (c) of figure 7 provides the isolines of $\tilde{{\mathcal{M}}}$ and shows the normal component of the advective flux on the boundary of region ${\mathcal{D}}_{II}^{\Vert }$ shown as the vector field $\boldsymbol{n}_{\Vert }\boldsymbol{\cdot }\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \boldsymbol{n}_{\Vert }$ . The structure function is advected inside the bubble, with the corresponding flux balancing the negative value of the relevant source term $\tilde{{\mathcal{M}}}$ .

4.3 Sub-manifold $(Y,r_{x})$

Figure 9 addresses the $(Y,r_{x})_{X^{0}=5.8,r_{y}=0,r_{x}=0.4}$ -plane, which also crosses the maximum of net production. The interpretation of the different panels should now be familiar from the discussion of the previous plane. Panel (a) shows the in-plane component of the flux as vectors, $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{x}})$ , the source term of (4.1) as colour contours, where now $\unicode[STIX]{x1D731}_{\bot }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{y}},\unicode[STIX]{x1D6F7}_{r_{z}})$ , and the structure function as solid isolines. The two images of the separatrix in this plane, curves of the form $Y=g_{b}(X^{0}\pm r_{x}/2)\mp r_{y}^{0}/2$ , are represented by the yellow symbols with the same meaning discussed previously. The in-plane flux is significant in regions ${\mathcal{D}}_{IO}^{\Vert }$ and ${\mathcal{D}}_{OI}^{\Vert }$ (one point is in the shear layer and the second inside the bubble) and is directed towards larger streamwise separations, $|r_{x}|$ . This trend is similar in region ${\mathcal{D}}_{II}^{\Vert }$ (both points inside the bubble). The in-plane transport velocity $\boldsymbol{w}_{\Vert }=(w_{Y},w_{r_{x}})$ shown in panel (b) together with the colour isolines of ${\mathcal{M}}$ , equation (4.4), confirms the same picture and shows the advection of the structure function towards increasing streamwise separations in the regions ${\mathcal{D}}_{IO}^{\Vert }$ and ${\mathcal{D}}_{OI}^{\Vert }$ . Therefore, the separation vector $\boldsymbol{r}$ is stretched in the streamwise direction along the trajectory in phase space, at an almost constant mid-point distance from the lower wall. This effect corresponds to $\dot{\ell }>0$ in the language adopted in the last paragraphs of § 2.2. Additional features are observed inside ${\mathcal{D}}_{II}^{\Vert }$ , where the advection velocity reverses close to the wall images. The vectors are magnified in panel (c) for a better view. In this region, the structure function is transported in a recirculating way. It is advected to smaller streamwise scales close to the wall, then away from the wall towards the separatrix and finally towards larger streamwise separations to match the behaviour in the regions above. In Lagrangian terms, the process would correspond to stretching $\dot{\ell }>0$ in the higher part of the (physical) recirculating bubble and compressing $\dot{\ell }<0$ in the lower part. Panel (d) shows the normal component of the advection flux $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ across the boundaries of region ${\mathcal{D}}_{II}^{\Vert }$ . The flux enters the bubble to balance the relevant source term $\tilde{{\mathcal{M}}}$ .

It is instructive to consider the same kind of plane, now at a positive wall-normal separation $r_{y}^{0}=0.12$ , figure 10. This choice is motivated by the fact that flux vectors in figure 7 converge into a sink for this specific separation. The point $\tilde{\boldsymbol{x}}$ is now further from the lower channel wall than $\boldsymbol{x}$ ( ${\tilde{y}}-y=r_{y}>0$ ), a configuration that enhances the role of the shear. In panel (a), a substantial flux $\unicode[STIX]{x1D731}_{\Vert }$ is found in the region of intense production corresponding to the shear layer (region ${\mathcal{D}}_{OO}^{\Vert }$ , just above the yellow symbols). The flux is systematically directed from negative streamwise separations ( $r_{x}<0$ ) towards positive ones ( $r_{x}>0$ ), i.e. the flux vectors are oriented from left to right. The flux is mostly inertial, as seen in panel (b) which shows the corresponding transport velocity $\boldsymbol{w}_{\Vert }$ . One may imagine to reconstruct the kinematics of this transport process. Following the trajectories of $\boldsymbol{w}_{\Vert }$ , the motion is at more or less constant $Y$ , that is the mid-point distance from the wall remains almost constant. Starting from the left side where $r_{x}<0$ , $r_{x}$ progressively increases along the trajectory to eventually become increasingly positive. Keeping in mind that $r_{y}^{0}>0$ in this plane, the processes can be interpreted as a sequence where the separation vector follows kinematics described by $r_{x}=r_{x}[s(t)]$ , $r_{y}=r_{y}^{0}$ , $r_{z}=r_{z}^{0}$ , with $s(t)$ the arc length along the phase space trajectory increasing with time along the advection, ${\dot{s}}=|\boldsymbol{w}_{\Vert }|$ , with $\text{d}r_{x}/\text{d}s>0$ as apparent from the plot. This is consistent with a rotation of the separation vector induced by the shear, see § 2.2. It is however important to consider that the separation vector components outside the plane ( $r_{y}$ and $r_{z}$ ) are frozen in the present plane.

4.4 Sub-manifold $(X,r_{x})$

The downstream behaviour is addressed in figure 11 that shows the $(X,r_{x})_{Y^{0}=0.45,r_{y}^{0}=0,r_{z}^{0}=0.4}$ -plane. Outside the bubble ( ${\mathcal{D}}_{OO}^{\Vert }$ region), the in-plane flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{r_{x}})$ is directed in the streamwise direction $X$ , with a small component towards larger streamwise scales $|r_{x}|$ , see panel (a). The physical origin of this flux is basically convective, as shown by the transport velocity field $\boldsymbol{w}_{\Vert }$ of panel (b). The $r_{x}$ component of the transport velocity becomes dominant in regions ${\mathcal{D}}_{IO/OI}^{\Vert }$ and in ${\mathcal{D}}_{II}^{\Vert }$ , see panel (c) where the vectors in the latter region are magnified for better readability. The Lagrangian interpretation corresponds to the stretching of the separation vector, $\dot{\ell }>0$ , implying that the structure function is advected at larger scales in the streamwise direction. A peculiar aspect of this plane is that the convective flux $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ is directed outside the bubble region feeding the neighbouring regions, ${\mathcal{D}}_{IO/OI}^{\Vert }$ , a feature already apparent from the total in-plane flux $\unicode[STIX]{x1D731}_{\Vert }$ of panel (a).

4.5 Sub-manifold $(X,Y)$

Finally, figure 12 examines the dynamics of the combined shear layer/separation bubble system in the $(X,Y)$ -plane at fixed $r_{x}^{0}=0.45,\,r_{y}^{0}=0,\,r_{z}^{0}=0.4$ . The in-plane flux $\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{X},\unicode[STIX]{x1D6F7}_{Y})$ is particularly strong in the shear layer, above the yellow circles (region ${\mathcal{D}}_{OO}^{\Vert }$ ), where the intensity of net production is large (red background contours). The flux is mainly directed in the streamwise direction $X$ and roughly follows the profile of the separatrix (circles), bends downwards, and encounters a region of negative net production (blue). The in-plane velocity $\boldsymbol{w}_{\Vert }$ , panel (b), confirms a significant convective effect transporting the structure function downstream. The magnified vectors inside the bubble, region ${\mathcal{D}}_{II}^{\Vert }$ , panel (c), provide evidence of the recirculating advection of the structure function inside the bubble where the source term is mostly negative (blue region below the yellow triangles). In this plane, where the separation vector is constant, the mid-point where the structure function is evaluated is advected along a recirculating path. Along a closed phase space streamline, panel (c), the structure function is attenuated in intensity in the lower part of the path, where the advection is upstream (negative $X$ -direction) to regain intensity in the upper part of the circuit, just below the separatrix. Panel (d) shows the normal component of the advection flux $\boldsymbol{w}_{\Vert }1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$ across the boundaries of region ${\mathcal{D}}_{II}^{\Vert }$ . Clearly, the inward fluxes prevail, confirming the dissipative features of the (physical) recirculating bubble.

In the above statistical analysis of the shear layer/recirculation bubble dynamics based on the GKE, we find that the bubble exchanges fluxes of structure function intensity with the surroundings, the shear layer being identified as the corresponding physical source. The exchange is mostly associated with convection, as shown by the normal component of the advective flux. The convective flux exchanged between the neighbouring regions ${\mathcal{D}}_{OO}$ , ${\mathcal{D}}_{IO/OI}$ and ${\mathcal{D}}_{II}$ can easily be shown to be a genuine effect of fluctuations. Indeed, the transport velocity $\boldsymbol{w}_{6}$ , equations (2.14) and (2.15), follows from two distinct contributions and can be decomposed as

(4.7) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}\displaystyle \boldsymbol{w}_{\boldsymbol{X}}=\frac{\langle \boldsymbol{U}^{\ast }|\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }{\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }+\frac{\langle \boldsymbol{u}^{\ast }|\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }{\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }=\boldsymbol{w}_{\boldsymbol{X}}^{M}+\boldsymbol{w}_{\boldsymbol{ X}}^{f}\\[12.0pt] \displaystyle \boldsymbol{w}_{\boldsymbol{r}}=\frac{\langle \unicode[STIX]{x1D6FF}\boldsymbol{U}|\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }{\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }+\frac{\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}|\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }{\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle }=\boldsymbol{w}_{\boldsymbol{r}}^{M}+\boldsymbol{w}_{\boldsymbol{ r}}^{f},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where the superscripts $M$ and $f$ stand for mean, to recall the advection due to the mean flow $\boldsymbol{U}$ , and for fluctuation, to remind the advection of the turbulent fluctuations $\boldsymbol{u}$ , respectively. $\boldsymbol{U}^{\ast }=(\boldsymbol{U}(\tilde{\boldsymbol{x}})+\boldsymbol{U}(\boldsymbol{x}))/2$ and $\unicode[STIX]{x1D6FF}\boldsymbol{U}=\boldsymbol{U}(\tilde{\boldsymbol{x}})-\boldsymbol{U}(\boldsymbol{x})$ are non-fluctuating quantities, hence, from (4.7), $\boldsymbol{w}_{\boldsymbol{X}}^{M}=\boldsymbol{U}^{\ast }$ and $\boldsymbol{w}_{\boldsymbol{r}}^{M}=\unicode[STIX]{x1D6FF}\boldsymbol{U}$ . As a consequence, the phase space point $(\boldsymbol{X},\boldsymbol{r})$ advected by the field $\boldsymbol{w}_{6}$ cannot be made to cross the two manifolds $f_{b}(\boldsymbol{X}\pm \boldsymbol{r}/2)=0$ , images of the physical separatrix in the phase space, by solely pure mean convection. The reason being that neither one of the two points $\tilde{\boldsymbol{x}}$ and $\boldsymbol{x}$ can cross the separatrix by the sole advection of the mean flow given the definition of separatrix as streamline (of the mean flow velocity) separating the bubble region with closed streamlines from the outer field where the streamlines are open. In fact, the normal convective fluxes shown in the last panels of figures 7, 9, 11 and 12 should be interpreted as a consequence of the turbulent transport velocity $(\boldsymbol{w}_{\Vert }^{f}\boldsymbol{\cdot }\boldsymbol{n}_{\Vert })1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle \boldsymbol{n}_{\Vert }$ . In other words, using standard nomenclature, separatrix crossing should be associated with the so-called turbulent diffusion. This expression is traditional for single-point statistics, such as for example diffusion of turbulent kinetic energy. Here, the term is referred to a phenomenon of similar origin taking place in the advection of the two-point second-order structure function.