## 1 Introduction

The language of turbulence is traditionally dichotomous. On the one hand there are scales and wavelengths and the notion of forward or reverse cascades of energy which are best suited to dealing with homogenous conditions, see Monin & Yaglom (Reference Monin and Yaglom1971*a*
). On the other hand, transport processes are described in physical space where inhomogeneity, Reynolds stresses and local turbulent kinetic energy production are the keywords, see Monin & Yaglom (Reference Monin and Yaglom1971*b*
). Contrary to quantum mechanics, where the complementarity is intrinsic to the microscopic world, in turbulence this duality is substantially an artefact induced by the formal tools used to study the problem. This is clearly seen by considering the well-known example of canonical wall bounded flows where the two aspects coexists. The equilibrium (logarithmic) layer is ideally traversed by a constant wall-normal flux of turbulent kinetic energy that sustains the fluctuations in the bulk. At the same time, the local production
$u_{\unicode[STIX]{x1D70F}}^{3}/(\unicode[STIX]{x1D705}y)$
is confined to a range of scales limited below by the characteristic length
$L_{shear}=\unicode[STIX]{x1D705}y$
, see Corrsin (Reference Corrsin1958), with energy cascading to small scale towards local dissipation, where
$\unicode[STIX]{x1D705}$
is the Kármán constant,
$y$
is the distance from the wall and
$u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$
is the friction velocity, with
$\unicode[STIX]{x1D70F}_{w}$
the average wall shear stress and
$\unicode[STIX]{x1D70C}$
the fluid density, see Townsend (Reference Townsend1980).

Overcoming the scale/position duality is particularly important for strongly non-equilibrium conditions, as in the buffer layer of the example above where the wall-normal flux originates, or in high Reynolds number flows around bluff bodies or behind abrupt section variations in channels or pipes. In the latter cases, recirculating regions form behind the obstacle, separated by intense shear layers from the external flow with open streamlines, see e.g. Landau & Lifshitz (Reference Landau and Lifshitz1987). Such flows are strongly inhomogeneous, with pronounced peaks of turbulent kinetic energy production. The local dissipation is insufficient to balance production and spatial fluxes move the energy downstream of the obstacle and into the recirculating bubble, see Mollicone *et al.* (Reference Mollicone, Battista, Gualtieri and Casciola2017) for a recent direct numerical simulation (DNS). Several questions naturally come to mind: What are the scales involved in these processes? How is the turbulent activity in the different ranges of scale distributed in space? Which mechanisms sustain eddies of different scale? For obvious reasons, single-point statistics cannot address these fundamental questions, but at the same time, the complementary balance in spectral space, Kolmogorov–Onsager–Heisenberg–von Weiszäcker–Lin equation, see e.g. Eyink & Sreenivasan (Reference Eyink and Sreenivasan2006), does not uncover the energy transfer in space, due to projection on non-local Fourier modes.

de Karman & Howarth (Reference de Karman and Howarth1938) and Kolmogorov (Reference Kolmogorov1941) originally devised their description of turbulence in terms of two-point observables (correlations and structure functions, respectively), see e.g. Frisch (Reference Frisch1995) for a comprehensive introduction, as a scale-by-scale approach to develop their theory of homogeneous and isotropic turbulence. Only recently, their theory has been progressively extended to more complex flows by using a generalised form of the Kolmogorov equation. The generalised Kolmogorov equation (GKE), see Hill (Reference Hill2002) for the derivation of the complete equation, accounts for anisotropic and inhomogeneous conditions and has been used in various studies of free shear by Casciola *et al.* (Reference Casciola, Gualtieri, Benzi and Piva2003, Reference Casciola, Gualtieri, Jacob and Piva2005), wall-bounded flows by Danaila *et al.* (Reference Danaila, Anselmet, Zhou and Antonia2001), Danaila, Anselmet & Zhou (Reference Danaila, Anselmet and Zhou2004), Marati, Casciola & Piva (Reference Marati, Casciola and Piva2004), Cimarelli, Angelis & Casciola (Reference Cimarelli, Angelis and Casciola2013) and wake turbulence by Gomes-Fernandes, Ganapathisubramani & Vassilicos (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015). Alternatively to GKE, other approaches are available to study the energy behaviour across the scales, see e.g. Cardesa, Vela-Martín & Jiménez (Reference Cardesa, Vela-Martín and Jiménez2017). Casciola *et al.* (Reference Casciola, Gualtieri, Benzi and Piva2003) address the link between the intermittency and anisotropy in homogeneous shear flows and employ the GKE to distinguish the shear-dominated scales by the small isotropic scales dominated by dissipation. Numerical data show that the dissipation scales are independent of the mean shear, thus intermittency corrections are universal. Danaila *et al.* (Reference Danaila, Anselmet, Zhou and Antonia2001) provide a generalised form of the Kolmogorov equation adding additional terms to account for the ‘large-scale turbulent diffusion acting from the walls through to the centreline of the channel’. The scale-by-scale budget in a planar turbulent channel flow is numerically studied by means of the GKE by Marati *et al.* (Reference Marati, Casciola and Piva2004) and Cimarelli *et al.* (Reference Cimarelli, Angelis and Casciola2013). Such understanding is fundamental to tune innovative techniques to control boundary layer transition to turbulence and to design innovative turbulence models, see Thiesset *et al.* (Reference Thiesset, Antonia, Danaila and Djenidi2013), since it is able to capture the correct dynamics of the fluctuations for inhomogeneous flows. Danaila *et al.* (Reference Danaila, Anselmet and Zhou2004) show the effects of turbulent diffusion and shear on the scale-by-scale budget far from the wall in a turbulent channel. The generalised equation is in agreement with hot-wire measurements in such regions, considered to be nearly sheared homogeneous turbulence. A more recent study by Cimarelli *et al.* (Reference Cimarelli, De Angelis, Schlatter, Brethouwer, Talamelli and Casciola2015), still concerning turbulent channel flow, focuses on peaks of scale energy found in both the near-wall region and overlap layer and the interaction between them. Gomes-Fernandes *et al.* (Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015) address the energy transfer across the scales in a highly non-homogeneous, anisotropic turbulent flow generated by a fractal grid. They show that the inter-scale energy transfers are characterised by the combination of both inverse cascade in the streamwise direction and forward cascade in the spanwise direction, whilst the overall cascade is forward.

The aim of the present work is to understand the production and transfer of turbulent kinetic energy (TKE) in the combined space of positions and scales for a definitely more complex geometry. We introduce a bulge (also referred to as bump) at one of the walls in a periodic turbulent channel flow in order to induce massive flow separation and strongly localised TKE production. The database is taken from a DNS recently performed by the authors (Mollicone *et al.*
Reference Mollicone, Battista, Gualtieri and Casciola2017) using the NEK5000 code (Fischer, Lottes & Kerkemeier Reference Fischer, Lottes and Kerkemeier2008) which is based on the spectral element method, see Patera (Reference Patera1984). Our focus is on the intense shear layer and the recirculation bubble that forms just after the bump. The GKE is applied at these specific features of the flow to study the scale-by-scale energy production, transport and dissipation. The second-order structure function, which is the basis of such an equation, is used as a proxy to define a scale-energy content, that is an interpretation of the energy associated with a given scale. The results show how the GKE, a five-dimensional equation in our anisotropic and strongly inhomogeneous flow, can describe in detail the turbulent flow behaviour and related energy mechanisms. As will be shown, the result of this analysis is readily interpreted in terms of the coherent structures of the shear layer.

## 2 The generalised Kolmogorov equation

### 2.1 General theory

In a statistically stationary turbulent flow, instantaneous fields
$\boldsymbol{u}_{T}(\boldsymbol{x},t_{i})$
– the subscript
$T$
meaning total, as opposed to average and fluctuation – sampled at
$N$
time instants
$t_{i}$
separated by more than the relevant correlation time, can be considered as elements of a statistical ensemble of (turbulent) velocity fields. In this context, the average over the ensemble,
$\boldsymbol{U}(\boldsymbol{x})=(1/N)\sum _{i}\boldsymbol{u}_{T}(\boldsymbol{x},t_{i})$
, is the mean flow while
$\boldsymbol{u}(\boldsymbol{x},t_{i})=\boldsymbol{u}_{T}(\boldsymbol{x},t_{i})-\boldsymbol{U}(\boldsymbol{x})$
is the fluctuation. Such (Reynolds) decomposition can be used for the other quantities of interest, e.g. the pressure field, where
$p(\boldsymbol{x},t)$
is the fluctuation and
$P(\boldsymbol{x})$
the average. In the following, we shall be dealing with the ensemble obtained by sampling the DNS fields of a turbulent channel flow with a bulge in one of the otherwise planar and parallel walls, see Mollicone *et al.* (Reference Mollicone, Battista, Gualtieri and Casciola2017). The flow is statistically stationary, with a single direction of spatial homogeneity corresponding to the spanwise direction. All quantities are made dimensionless with respect to the bulk velocity
$U_{b}=Q/(2h_{0})$
, where
$Q$
is the flow rate per unit width and
$h_{0}$
is the half-channel height from the bottom wall. Beside the bump geometry, here kept fixed, the Reynolds number
$Re=Q/(2\unicode[STIX]{x1D708})$
is the only control parameter of the system.

The focus will be on the second-order structure function,
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle =\langle \unicode[STIX]{x1D6FF}\boldsymbol{u}\cdot \unicode[STIX]{x1D6FF}\boldsymbol{u}\rangle$
, where angular brackets denote ensemble average and
$\unicode[STIX]{x1D6FF}\boldsymbol{u}=\tilde{\boldsymbol{u}}-\boldsymbol{u}=\boldsymbol{u}(\tilde{\boldsymbol{x}})-\boldsymbol{u}(\boldsymbol{x})$
is the fluctuation velocity increment between two points,
$\boldsymbol{x}=(x,y,z)=(x_{1},x_{2},x_{3})$
and
$\tilde{\boldsymbol{x}}$
. The velocity is expressed in one of the forms
$\boldsymbol{u}=(u,v,w)=(u_{1},u_{2},u_{3})$
. Due to the inhomogeneity of the flow, the second-order structure function depends on both
$\boldsymbol{x}$
and
$\tilde{\boldsymbol{x}}$
or, alternatively, can be considered as a function of the mid-point
$\boldsymbol{X}=(\tilde{\boldsymbol{x}}+\boldsymbol{x})/2$
and the separation vector
$\boldsymbol{r}=\tilde{\boldsymbol{x}}-\boldsymbol{x}$
, namely
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle =f(\boldsymbol{X},\boldsymbol{r})$
. Starting from the Navier–Stokes equations, the equation obeyed by the second-order structure function, hereafter called the generalised Kolmogorov equation (GKE), can be straightforwardly derived, see Hill (Reference Hill2002). A possible procedure is to consider the Navier–Stokes equation for the fluctuating field at point
$\boldsymbol{x}$
, scalar multiplying it by the velocity
$\tilde{\boldsymbol{u}}$
at
$\tilde{\boldsymbol{x}}$
, exchanging the roles of
$\boldsymbol{x}$
and
$\tilde{\boldsymbol{x}}$
, summing the result and averaging. This procedure leads to the Kármán–Howarth equation, see de Karman & Howarth (Reference de Karman and Howarth1938), for the correlation
$\langle \boldsymbol{u}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}\rangle$
. It can be restated in terms of velocity increments, leading to the equation for
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
. In manipulating the equation, one may take advantage of expressing the derivatives with respect to the position of the two points,
$\boldsymbol{x}$
and
$\tilde{\boldsymbol{x}}$
, in terms of mid-point
$\boldsymbol{X}$
and increment
$\boldsymbol{r}$
, e.g.
$\unicode[STIX]{x1D735}_{\boldsymbol{x}/\tilde{\boldsymbol{x}}}=1/2\unicode[STIX]{x1D735}_{\boldsymbol{X}}\mp \unicode[STIX]{x1D735}_{\boldsymbol{r}}$
. In doing so, the mid-point average velocity,
$\boldsymbol{u}^{\ast }=(\tilde{\boldsymbol{u}}+\boldsymbol{u})/2$
, naturally appears. Contrary to homogenous turbulence, the terms associated with the mean field inhomogeneity do not cancel out and play a crucial role in the dynamics of
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
, inducing a strong deviation from the classical Kolmogorov view of turbulence. The GKE finally reads

In the above equation,
$\unicode[STIX]{x1D6FF}q$
is the increment of a generic quantity
$q$
and
$q^{\ast }$
is the mid-point average. The symbol
$\otimes$
denotes the diadic product, the colon
$:$
denotes the double tensor contraction and
$\unicode[STIX]{x1D735}_{\boldsymbol{X}/\boldsymbol{r}}\cdot$
is the divergence with respect to
$\boldsymbol{X}$
or
$\boldsymbol{r}$
variables. In order to slightly simplify the equation, the so-called pseudo-dissipation
$\unicode[STIX]{x1D700}=1/Re\unicode[STIX]{x1D735}\boldsymbol{u}:\unicode[STIX]{x1D735}\boldsymbol{u}$
is used, see Hill (Reference Hill2002) for the corresponding expression in terms of dissipation
$\bar{\unicode[STIX]{x1D700}}=1/(2Re)(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}:(\unicode[STIX]{x1D735}\boldsymbol{u}+\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}$
. The GKE can be re-expressed to highlight its conservative structure as

where

are fluxes taking place at (mid-point) position and separation space respectively, and

are the corresponding production terms. Although we shall not dwell longer on the issue, it should be noted that our choice of associating the pressure–velocity correlation with the flux
$\unicode[STIX]{x1D731}_{\boldsymbol{X}}$
is to a large extent arbitrary. The corresponding term could be easily transformed from a divergence in
$\boldsymbol{X}$
-space into one in
$\boldsymbol{r}$
-space, making the alternative association of
$\langle \unicode[STIX]{x1D6FF}p\unicode[STIX]{x1D6FF}\boldsymbol{u}\rangle$
with
$\unicode[STIX]{x1D731}_{\boldsymbol{r}}$
perfectly legitimate.

Combining position and separation space in a six-dimensional space
$(\boldsymbol{X},\boldsymbol{r})$
the equation becomes

where the subscript recalls that, in principle, six independent coordinates are involved. For future convenience, two contributions to the flux are identified, one associated with convection,
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\boldsymbol{u}_{T}^{\ast }\rangle$
,
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\unicode[STIX]{x1D6FF}\boldsymbol{u}_{T}\rangle$
, and one associated with pressure and molecular diffusion,

The interpretation of the GKE is clear from its structure: the second-order structure function changes in time due to (i) net production, that is the difference between production,
$\unicode[STIX]{x1D6F1}_{\boldsymbol{X}}$
and
$\unicode[STIX]{x1D6F1}_{\boldsymbol{r}}$
, and dissipation,
$\langle \unicode[STIX]{x1D700}^{\ast }\rangle$
, and (ii) redistribution due to the fluxes,
$\unicode[STIX]{x1D731}_{\boldsymbol{X}}$
and
$\unicode[STIX]{x1D731}_{\boldsymbol{r}}$
, both in position and separation space, respectively. The high dimensionality (six spatial coordinates and one time) can be considerably reduced in presence of symmetries. The extreme case is stationary, homogeneous, isotropic turbulence, where the GKE reduces to the classical Kolmogorov equation with only one independent variable,
$r=|\boldsymbol{r}|$
. Removing the constraint of isotropy, the number of variables rises to the three components of the separation vector
$\boldsymbol{r}$
. For a steady, planar channel flow, whose fluctuations are statistically invariant in the streamwise and spanwise directions, the independent variables add to four: one wall-normal position coordinate,
$Y$
and the separation vector
$\boldsymbol{r}$
, see Marati *et al.* (Reference Marati, Casciola and Piva2004), Cimarelli *et al.* (Reference Cimarelli, Angelis and Casciola2013). Since streamwise translational invariance is broken by the bump for our present configuration, the independent coordinates are now five: the wall-normal and streamwise positions
$X$
and
$Y$
, and the separation vector
$\boldsymbol{r}$
.

Based on the expression
$\langle \boldsymbol{u}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}\rangle =2k^{\ast }-1/2\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
, where
$k=1/2\langle |\boldsymbol{u}|^{2}\rangle$
is the turbulent kinetic energy, one obtains the equation for
$k^{\ast }$
,

where
$\unicode[STIX]{x1D74D}=|\boldsymbol{u}|^{2}\boldsymbol{u}+p\boldsymbol{u}+|\boldsymbol{u}|^{2}\boldsymbol{U}-\unicode[STIX]{x1D708}\unicode[STIX]{x1D735}\boldsymbol{u}$
is the (ordinary) flux of turbulent kinetic energy and
$-\langle \boldsymbol{u}\otimes \boldsymbol{u}\rangle$
is the Reynolds stress. The combination of (2.1) and (2.7) is equivalent to the Kármán–Howarth equation for the correlation. Hence, the information provided is equivalent to the classical spectral description of turbulence (Eyink & Sreenivasan Reference Eyink and Sreenivasan2006), whenever the latter applies.

Two additional comments are useful. First, dependences on
$\boldsymbol{X}$
and the associated flux
$\unicode[STIX]{x1D731}_{\boldsymbol{X}}$
account for the broken translational symmetry (statistical inhomogeneity), dependence on the direction of the separation vector
$\boldsymbol{r}$
concerns statistical anisotropy and that on the length
$|\boldsymbol{r}|$
describes scale dependence. The second comment is about the nature of the second-order structure function. Although it can be understood as rough indication of the energy content associated with a given scale, this interpretation is technically incorrect. Energy is actually an extensive concept (i.e. it is additive) while adding second-order structure function is meaningless. This lack of additivity makes
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
somewhat less intuitive than the energy spectrum. The big intuition beside the spectrum (Wiener–Khintchine–Einstein theorem: Wiener (Reference Wiener1930), Khintchine (Reference Khintchine1934), Jerison, Singer & Stroock (Reference Jerison, Singer and Stroock1997)) is that the Fourier transform of the correlation provides the (average) energy density in wavenumber space, i.e. an additive quantity. The price to pay for this big advantage is the lack of positional information. Whilst this can be sacrificed in some cases, positional information is crucial in strongly out-of-equilibrium flows such as the one to be discussed below. In the physical space, the second-order structure function is the energy up to the considered scale, i.e. it accounts for the contribution of all the scales ranging from vanishing separation to the scale
$|\boldsymbol{r}|$
, see e.g. Thiesset *et al.* (Reference Thiesset, Danaila, Antonia and Zhou2011), Davidson (Reference Davidson2015).

### 2.2 Lagrangian interpretation

The evolution of two Lagrangian points, identified by the mid-point
$\unicode[STIX]{x1D74C}$
and the separation vector
$\unicode[STIX]{x1D746}$
(here
$\unicode[STIX]{x1D74C}$
and
$\unicode[STIX]{x1D746}$
are used instead of
$\boldsymbol{X}$
and
$\boldsymbol{r}$
when considering Lagrangian variables) and advected by the instantaneous velocity
$\boldsymbol{u}_{T}$
, is governed by

From the Navier–Stokes equations for the fluctuating velocity, the velocity difference
$\unicode[STIX]{x1D6FF}\boldsymbol{u}$
transported along the path characteristics defined by (2.8) evolves according to

Scalar multiplication by the velocity increment and few manipulations yield

After averaging, taking into account that

the equation

follows. Since the field is solenoidal, the left-hand side of this equation can be rearranged to read

Transport velocities of the structure function
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
in separation space,
$\boldsymbol{w}_{\boldsymbol{r}}$
, and position space,
$\boldsymbol{w}_{\boldsymbol{X}}$
, can be conveniently defined as

The resulting velocity in the six-dimensional space will be denoted by

Equation (2.13) can then be recast as

where

Equation (2.16) is the Lagrangian version of the GKE and its interpretation is revealing. The velocity structure function is transported in space by the field
$\boldsymbol{w}_{\boldsymbol{X}}$
and moved to a different separation and orientation by the field
$\boldsymbol{w}_{\boldsymbol{r}}$
. Along the convection, the structure function changes due to pressure–velocity correlation, diffusion and dissipation. The physical process of turbulent kinetic energy production corresponds to the two terms
$\unicode[STIX]{x1D6F1}_{\boldsymbol{X}}$
and
$\unicode[STIX]{x1D6F1}_{\boldsymbol{r}}$
. Two additional source terms occur due to the definition of the transport velocity,
$\boldsymbol{w}_{\boldsymbol{X}}$
and
$\boldsymbol{w}_{\boldsymbol{r}}$
, which are generally not solenoidal.

One expects that, at large separations, far from boundaries and from strong shear layers such as those occurring at the edge of separation bubbles, diffusion is negligible. On the other hand, diffusion is expected to become dominant at small scale and near solid walls and not negligible in strong shear layers. It may be stressed that the notion of a turbulent cascade is explicitly captured by this formulation. Denoting the length of
$\unicode[STIX]{x1D73B}$
as
$\ell$
and
$\hat{\boldsymbol{e}}$
its unit vector,
$\unicode[STIX]{x1D73B}=\ell \hat{\boldsymbol{e}}$
, the second line of (2.14) can be expressed as
$\dot{\ell }\hat{\boldsymbol{e}}+\ell \dot{\hat{\boldsymbol{e}}}=\boldsymbol{w}_{\boldsymbol{r}}$
, where
$\dot{\ell }=\boldsymbol{w}_{\boldsymbol{r}}\boldsymbol{\cdot }\hat{\boldsymbol{e}}$
and
$\dot{\hat{\boldsymbol{e}}}=(\hat{\boldsymbol{e}}\times \boldsymbol{w}_{\boldsymbol{r}})\times \hat{\boldsymbol{e}}/\ell$
. A forward cascade is implied by
$\dot{\ell }<0$
, i.e. the structure function is advected (in separation space) towards smaller scales. On the other hand,
$\dot{\ell }>0$
indicates that locally, a backward cascade is occurring.
$\dot{\hat{\boldsymbol{e}}}$
accounts for the reorientation of the separation vector
$\unicode[STIX]{x1D73B}$
. While these two processes take place, the position where the structure function is evaluated, as described by the mid-point
$\unicode[STIX]{x1D743}$
, changes with velocity
$\boldsymbol{w}_{\boldsymbol{X}}$
. In addition to transport, the intensity of the structure function is modified according to the right-hand side of (2.16).

For the flow over a bulge in a channel, statistics are independent of the mid-point coordinate
$Z$
, hence we shall be dealing with a five-dimensional phase space that is quite difficult to grasp. To provide a visualisation of the results and make things as clear as possible, we shall employ different two-dimensional sections of the phase space. In doing so, the subscript
$\Vert$
will be used to denote projections of five-dimensional vectors on a given planar section. The (three-dimensional) orthogonal component of the vector will be denoted by the subscript
$\bot$
. For example, when selecting the plane
$(Y,r_{x})$
, the parallel component of the flux will be
$\unicode[STIX]{x1D731}_{\Vert }=(\unicode[STIX]{x1D6F7}_{Y},\unicode[STIX]{x1D6F7}_{r_{x}})$
. The orthogonal component of, e.g. the transport velocity will be
$\boldsymbol{w}_{\bot }=(w_{X},w_{r_{y}},w_{r_{z}})$
. When plotting data in the selected plane, the following conventions are used: if both directions belong to either the mid-point position or the separation space, the plane will be plotted according to the rules of three-dimensional vector spaces. Otherwise
$X$
and/or
$Y$
will be placed on abscissas and ordinates, respectively.

## 4 Results

Figure 3. A point in the
$(Y,r_{x})$
-plane at constant
$X=X^{0}$
,
$Z=Z^{0}$
,
$r_{y}=r_{y}^{0}$
,
$r_{z}=r_{z}^{0}$
(*a*) corresponds to two points,
$\tilde{\boldsymbol{x}},\,\boldsymbol{x}$
, of the physical space such that the mid-point
$\boldsymbol{X}$
is constrained at
$X=X^{0}$
,
$Z=Z^{0}$
and two components of the separation vector
$\boldsymbol{r}$
are fixed,
$r_{y}=r_{y}^{0},\;r_{z}=r_{z}^{0}$
(*b*). The red lines sketch the lower wall of the channel and the bump. By moving from
$(Y,r_{x})$
to
$(Y+\text{d}Y,r_{x}+\text{d}r_{x})$
the separation vector
$\boldsymbol{r}$
(*b*) changes from the blue to the green one.

In order to visualise the second-order velocity structure function, it is helpful to consider two-dimensional sections of the five-dimensional phase space
$(X,Y,r_{x},r_{y},r_{z})$
introduced in § 2. The sketch in figure 3 addresses a typical
$(Y,r_{x})$
-plane (section at constant
$X=X^{0}$
,
$Z=Z^{0}$
,
$r_{y}=r_{y}^{0}$
and
$r_{z}=r_{z}^{0}$
). It shows the physical space points,
$\tilde{\boldsymbol{x}}$
and
$\boldsymbol{x}$
, across which the velocity difference is evaluated and the corresponding separation vector
$\boldsymbol{r}$
for two different phase space points in the section. By analogy, the reader may easily figure out the picture for the other planar sections.

The isolines of the second-order structure function
$\langle |\unicode[STIX]{x1D6FF}\boldsymbol{u}|^{2}\rangle$
in four selected phase space planes are plotted in figure 4. We shall focus near the maximum of the effective production of the second-order structure function intensity
$\unicode[STIX]{x1D6F1}_{6}-2\langle \unicode[STIX]{x1D700}^{\ast }\rangle$
. Based on the physical intuition provided by the single-point energy balance discussed in § 3 on the one hand and on the formal theory developed in § 2 on the other, that region (i.e. that range of mid-point positions and separations) is expected to be the source of turbulent fluxes that redistribute turbulent activity across the phase space to different positions
$\boldsymbol{X}$
and scales
$\boldsymbol{r}$
. The planes addressed in the four panels actually go through the production maximum. Although not difficult to interpret, the plots are not immediately readable. They will be illustrated in detail in the following discussions regarding the in-plane components of the flux
$\unicode[STIX]{x1D731}_{\Vert }$
, after a few more details are given on how the four planes shown in figure 4 were selected.

Figure 4.
$Re=2500$
. The second-order structure function in different planes. (*a*)
$(Y,r_{x})$
-plane at
$X^{0}=5.8$
,
$r_{y}^{0}=0$
,
$r_{z}^{0}=0.4$
. (*b*)
$(X,Y)$
-plane at
$r_{x}^{0}=0.45$
,
$r_{y}^{0}=0$
,
$r_{z}^{0}=0.4$
. (*c*)
$(Y,r_{y})$
-plane at
$X^{0}=5.8$
,
$r_{x}^{0}=0.45$
,
$r_{z}^{0}=0.4$
. (*d*)
$(X,r_{x})$
-plane at
$Y^{0}=0.45$
,
$r_{y}^{0}=0$
,
$r_{z}^{0}=0.4$
. See figure 7 and the related discussion for a description of the yellow symbols.

### 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,
$y_{sl}=0.45$
.

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

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

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).

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

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

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

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

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

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.

## 5 An intuitive link with coherent structure dynamics

An intuitive illustration of the processes described so far is provided by associating the phase space dynamics described by the GKE with effects directly visualised in physical space. The phase space processes consist of advection of a given turbulent scale
$\ell$
by a suitably defined vector field and intensification or depletion of the structure function intensity along the advection. The shear layer and its interaction with the recirculation region have been identified as features of interest. In proposing a meaningful association between phase space and physical space dynamics one should first identify an observable in physical space that could be related to the structure function. In the context of wall-bounded shear flows, scientists are familiar with the notion of the so-called coherent structures and the effects that these intermittent features have on the flow statistics. A well-known example are the quasi-streamwise vortices and low/high speed streaks that populate the near-wall region of canonical wall-bounded flows. These objects affect correlation functions, e.g. minimum of spanwise correlations and average distance between streamwise vortices. Correlations and second-order structure functions are very close relatives. It could be envisaged that production and advection of structure function intensity are related to generation and convection of some sort of coherent structures which can be identified in the regions (mid-point positions) and scales (separation vectors) that were analysed by the GKE. Relying on the widely accepted Q-criterion for coherent structure visualisation, the picture that emerges is shown in figure 13. A forest of hairpin-like vortical structures appear out of nowhere just behind the bump, at the level of the shear layer. Such structures are indeed expected there, given the observed maximum of net production.

The vortical structures shown in the figure are coloured with the local streamwise velocity intensity, red meaning strong downstream convection, blue meaning upstream convection. It is apparent that the structures materialise at an
$x$
-station that approximately corresponds, for this Reynolds number, to the end of the bump at
$x=5.8$
, matching the
$X^{0}$
corresponding to the maximum net production of structure function intensity. As the structures in the top part of the layer are convected downstream, they are tilted and their characteristic
$r_{y}$
scale decreases and are stretched, i.e. their
$r_{x}$
characteristic scale increases. Part of the structures get trapped in the recirculation bubble and are then convected upstream, remaining close to the wall. Their intensity weakens and they eventually vanish as they move upstream climbing up the bump. The structures form again at the top edge of the recirculating bubble. Figure 14 provides a short sequence of successive snapshots where certain specific structures have been isolated from the surroundings to provide a better illustration. The downstream convection, streamwise stretching and clockwise rotation is apparent. The bottom part of the plots displays blue coloured structures that move upstream, shrink and disappear. The process of extracting a few selected structures may introduce a possible bias. To remove this potential bias, two animations are provided in the supplementary material in Movie1 and Movie2, available at https://doi.org/10.1017/jfm.2018.114. The first animation concerns the whole forest of hairpins shown in figure 13; the second one corresponds to the sequence extracted in figure 14. By looking at the animations the reader may be convinced of the strong link between the phase space interpretation and the coherent structure evolution.

Figure 14. Isolated selection of isosurfaces of
$Q$
coloured by instantaneous streamwise velocity in sequential snapshots for case
$Re=2500$
. The threshold chosen is the same as in figure 13.

Finally, it is worth commenting on the effect of the Reynolds number on the dynamics described so far. At
$Re=10\,000$
, the recirculation region becomes smaller, see Mollicone *et al.* (Reference Mollicone, Battista, Gualtieri and Casciola2017). Figure 15 shows spanwise and wall-parallel views of the instantaneous streamwise velocity in panel (*a*) and (*b*) respectively. Despite flow structures becoming smaller and more attached to the wall, the gross features are similar to those described for the smaller Reynolds number case. The maximum net production is now found at the phase space position
$X^{0}=5.1$
,
$Y^{0}=0.45$
,
$r_{x}^{0}=0.17$
,
$r_{y}^{0}=0$
,
$r_{z}^{0}=0.13$
.

Without repeating the whole analysis, figure 16 shows in-plane fluxes, advection velocities and advective flux normal to the separatrix in the
$(X,Y)_{r_{x}=0.17,r_{y}=0,r_{z}=0.13}$
- plane, to be compared with figure 12 at the smaller Reynolds number. The qualitative features are analogous, once the shorter extension of the high intensity region of the shear layer and the shorter and narrower recirculation regions are accounted for. A corresponding visualisation of the hairpin forest at
$Re=10\,000$
is provided in figure 17, while an animation is provided in the supplementary material in Movie3.