# On the role of laminar/turbulent interface on energy transfer between scales
in bypass transition

H. Yao1  G. Papadakis1<EMAIL_ADDRESS>1Department of Aeronautics,
Imperial College London, London SW7 2AZ, UK

###### Abstract

We investigate the role of laminar/turbulent interface in the interscale
energy transfer in a boundary layer undergoing bypass transition, with the aid
of the Karman-Howarth-Monin-Hill (KHMH) equation. A local binary indicator
function is used to detect the interface and employed subsequently to define
two-point intermittencies. These are used to decompose the standard-averaged
interscale and interspace energy fluxes into conditionally-averaged
components. We find that the inverse cascade in the streamwise direction
reported in an earlier work arises due to events across the downstream or
upstream interfaces (head or tail respectively) of a turbulent spot. However,
the three-dimensional energy flux maps reveal significant differences between
these two regions: in the downstream interface, inverse cascade is stronger
and dominant over a larger range of streamwise and spanwise separations. We
explain this finding by considering a propagating spot of simplified shape as
it crosses a fixed streamwise location. We derive also the conditionally-
averaged KHMH equation, thus generalising similar equations for single-point
statistics to two-point statistics. We compare the three-dimensional maps of
the conditionally-averaged production and total energy flux within turbulent
spots against the maps of standard-averaged quantities within the fully
turbulent region. The results indicate remarkable dynamical similarities
between turbulent spots and the fully turbulent region for two-point
statistics. This has been known only for single-point quantities, and we show
here that the similarity extends to two-point quantities as well.

###### keywords:

##  1 Introduction

###  1.1 Bypass transition

Transition to turbulence that does not involve linear instability paths, such
as Tollmien–Schlichting waves, is called bypass transition (Morkovin, 1969).
This type of transition can be triggered by high levels of free-stream
turbulence, surface roughness, etc. In the case of free-stream turbulence,
which is the triggering mechanism considered in this paper, bypass transition
comprises three stages. In the first stage, low-frequency fluctuations from
the free-stream penetrate inside the boundary layer, forming high and low-
speed streaks, while high-frequency fluctuations remain in the free-stream due
to shear sheltering (Leib et al., 1999; Hunt & Durbin, 1999; Zaki & Saha,
2009). In the second stage, the streaks breakdown to intermittent turbulent
patches (or spots) due to secondary instability (Andersson et al., 2001;
Vaughan & Zaki, 2011), while in the final stage the spots propagate and merge,
forming a fully turbulent region. More details can be found in the review
papers of Durbin & Wu (2007); Zaki (2013); Durbin (2017).

Most previous investigations of the structural details of turbulent spots,
such as shape, propagation speed, growth rate etc. have employed analysis of
single point statistics, see Emmons (1951); Wygnanski et al. (1976); Cantwell
et al. (1978); Perry et al. (1981); Singer (1996); Nolan & Zaki (2013) and
more recently Wang et al. (2021, 2022). This type of analysis however cannot
capture the underlying physical mechanisms that explain the spot growth and
the amalgamation process of smaller spots to form larger turbulent patches as
transition progresses. In order to study this process in more detail, an
analysis of two-point statistics is required. The second order structure
function at point $X_{i}$ is defined as the second moment of the fluctuating
velocity difference at points $x^{\pm}_{i}=X_{i}\pm\frac{1}{2}r_{i}$, i.e.
$\overline{dq^{2}}(X_{i},r_{i})=\overline{(u^{\prime+}_{i}-u^{\prime-}_{i})^{2}}$,
where the overbar denotes time-averaging. The volume integral of
$\overline{dq^{2}}$ over a sphere of radius $r=|r_{i}|$ (divided by the volume
of the sphere) represents physically the energy of eddies located at $X_{i}$
that have size (or scale) less than $r$; this is also known as scale energy,
for details see Davidson (2015). This is the appropriate quantity to study in
order to better understand the process of spots growth and merging.

The transport equation of $\overline{{dq^{2}}}(X_{i},r_{i})$ is known as the
Karman-Howarth-Monin-Hill (KHMH) equation. It contains all the physical
mechanisms that determine the energy contained within eddies of scale less
than $r$, such as transfer of energy in scale space (i.e. from smaller or
larger scales), transfer of energy in physical space, production (due to mean
shear), dissipation (due to viscosity), etc. It was first derived by Karman &
Howarth (1938) for homogeneous isotropic turbulence (HIT) using the two-point
velocity correlation tensor, $\overline{u^{\prime+}_{i}u^{\prime-}_{j}}$, and
later reformulated in terms of structure function $\overline{dq^{2}}(r_{i})$
by Kolmogorov (1941). The equation was used to prove the famous -4/5’th law,
that links the interscale energy flux, the separation between the two points,
and the dissipation rate. For homogeneous, isotropic turbulence the interscale
flux is always negative, i.e. energy is transferred from large to small
scales; this is known as forward cascade. The most general form of the KHMH
equation, applicable to inhomogeneous and anisotropic flows, was derived
directly from Navier-Stokes equations by Hill (2002). The equation was applied
recently to transitional boundary layers and demonstrated strong inverse
cascade in the transition region, especially in the streamwise direction, see
Yao et al. (2022). Analysis of instantaneous velocity fields and flux vectors
revealed that the inverse cascade was related to the growth of turbulent
spots.

The KHMH equation has been applied to several other flow settings, such as
homogeneous shear flow (Casciola et al., 2003), channel flow (Marati et al.,
2004; Cimarelli et al., 2013, 2015, 2016), Von Karman flow (Dubrulle, 2019;
Knutsen et al., 2020), temporal planar jet (Cimarelli et al., 2021), wake
behind a square cylinder (Alves Portela et al., 2017, 2020) and has revealed
convoluted paths of interscale energy fluxes with a mixture of forward and
inverse cascade. In all these investigations, the points $x^{\pm}_{i}$ used to
define $\overline{dq^{2}}$ are immersed within a turbulent flow. This is not
the case however in transitional flows, where for a fixed streamwise location
$X_{i}$ and separation $r_{i}$, the two points will experience different flow
conditions as a spot propagates, for example $x^{\pm}_{i}$ may be within the
laminar region, straddle the laminar/turbulent interface, or be within a
turbulent patch. It is therefore difficult to explain the origin of the
inverse cascade found in Yao et al. (2022). For example, does it arise from
the turbulent conditions within the spot? What is the role of the
laminar/turbulent interface? Is there competition between the different flow
conditions, i.e. do some lead to forward and others to inverse cascade?

To answer these questions in the present paper we perform conditional
averaging of the interscale energy fluxes based on the state of the two points
i.e. whether they experience laminar or turbulent flow conditions. This
process clearly elucidates the effect of different flow states and the role of
the laminar/turbulent interface. As will be seen latter, it even characterises
the separate roles of the downstream and upstream interfaces (head or tail
respectively) of the spot. We also examine the production term of the KHMH
equation conditioned on turbulent events within a turbulent patch, and compare
it with the production term when the two points are located within the fully
turbulent region. Similar comparisons have been made for single-point
quantities, such as turbulent kinetic energy, see Marxen & Zaki (2019). We
derive also the conditionally-averaged form of the KHMH equation, which is
analogous to the conditionally-averaged turbulent kinetic energy equation.

The work of Zhou & Vassilicos (2020) has some similarities, but also
significant differences, with the present work. The authors studied the energy
cascade across the turbulent/non-turbulent (TNTI) interface at one axial
position of an axisymmetric turbulent wake. They found that the interscale
energy transfer at the vicinity of the interface is from small to large scales
(inverse cascade) in directions close to the interface’s tangent plane where
motions are predominantly stretching, but from large to small scales (forward
cascade) in the other directions where motions are predominantly compressive.
This reflects the fundamental mechanism that sustains the TNTI, i.e. fluid is
entrained from the irrotational region and the wake grows due to turbulent
diffusion, see schematic 3(d) in Zhou & Vassilicos (2020). This mechanism
however is different compared to the one that determines the growth of spots
in a transitional boundary layer. The authors also did not perform conditional
analysis, because the midpoint $X_{i}$ was located at the interface (thus was
not fixed in the cross-stream direction), and the two points $x^{\pm}_{i}$
always straddled the TNTI.

The paper is organised as follows; in section §2 the bypass transition case is
briefly presented, in §3 we summarise the derivation of the standard time-
averaged form of the KHMH equation, while in §4 we derive the conditionally-
averaged form; this is followed by the conditional decomposition of the energy
fluxes in §5. The next two sections present the results; in §6 maps of the
conditionally-averaged non-linear energy fluxes are shown (focusing on the
flux across the laminar/turbulent interface), while in §7 we compare the two-
point energy production and total flux (both conditioned on turbulent events)
to the corresponding quantities in the fully turbulent region. We conclude in
§8.

##  2 Details of the test case examined

We consider the transition of a boundary layer developing on a flat plate due
to free-stream turbulence. At the inlet of the computational domain, a random
velocity field is superimposed on the Blasius velocity profile. In the free-
stream, the random field follows a von-Karman spectrum with turbulence
intensity $3.4\%$ and integral length scale $L_{11}=5L_{0}$, where
$L_{0}=\sqrt{\nu X_{0}/U_{\infty}}$ is the Blasius similarity variable,
$X_{0}$ is the distance of the inlet of the domain from the leading edge of
the plate, $\nu$ is the kinematic viscosity and $U_{\infty}$ is the free-
stream velocity. The inlet Reynolds number is $Re_{L_{0}}=160$ (or
$Re_{\theta}=110$ based on momentum thickness).

The size of the computational domain is $(3000\times 200\times 150)L_{0}$,
with the number of cells $2049\times 192\times 169$ in the streamwise ($X$),
wall-normal ($Y$), and spanwise ($Z$) directions. Velocities are denoted as
$u$, $v$, $w$ in the $X,Y,Z$ directions respectively. This notation is used
interchangeably with the indexed notation $X_{i}$ and $u_{i}$ (with
$i=1,2,3$), for example $X_{2}=Y$ and $u_{2}=v$. The spacing is uniform in the
streamwise and spanwise directions, with $\Delta x^{+}_{max}\approx 11.78$ and
$\Delta z^{+}_{max}\approx 7.14$, where ${max}$ represents the maximum value
(located in the fully turbulent region). In the wall-normal direction, grid
spacing increases gradually; $y^{+}$ at the centroid of the first cell close
to the wall is around 0.24. The generated DNS database contains 350
uncorrelated snapshots. The results have been validated against the T3A
experimental data (Roach, 1990). More details about the computational method
and comparison of velocity profiles (mean and RMS) against experiments can be
found in Yao et al. (2020).

For future reference, the normalised skin friction coefficient,
$C_{f}/\max(C_{f})$, and the maximum value of the time- and spanwise-averaged
intermittency are plotted in figure 1. For the methodology used to compute the
intermittency refer to the aforementioned paper and also to section 4.1.
Vertical lines indicate the streamwise locations in the laminar ($LA$),
transitional ($TR1,TR2,TR3$) and fully turbulent regions ($TU$) where velocity
data are extracted in order to compute the interscale fluxes.

Figure 1: Normalised skin friction coefficient, $C_{f}/\max(C_{f})$ (blue
solid line), maximum intermittency $\gamma_{max}$ (red solid line), and
$\gamma(\xi)$ from the formula of Narasimha (1985), where
$\xi=(X-X_{s})/(X_{\gamma=0.75}-X_{\gamma=0.25})$ and $X_{s}-X_{0}=1100L_{0}$
is the location where transition starts (black dash line) against streamwise
distance $(X-X_{0})/L_{0}$. The purple vertical lines are located in the
laminar ($LA=540L_{0}+X_{0}$), transitional
($TR1,TR2,TR3=1215,1515,1815L_{0}+X_{0}$) and fully turbulent
($TU=2415L_{0}+X_{0}$) regions.

##  3 Standard time-averaged KHMH equation

In this section, the form of the standard time-averaged KHMH equation is
presented. The basic steps of the derivation are sketched below; more details
can be found in Hill (2002). Similarities and differences with the
conditionally-averaged form are presented and discussed in section 4.

We start with the Navier-Stokes equations at two points $x_{i}^{+}$ and
$x_{i}^{-}$ (see sketch 2),

$\displaystyle\frac{\partial u^{+}_{i}}{\partial t}+u^{+}_{j}\frac{\partial
u^{+}_{i}}{\partial x^{+}_{j}}$ $\displaystyle=-\frac{\partial p^{+}}{\partial
x^{+}_{i}}+\nu\frac{\partial^{2}u^{+}_{i}}{\partial x_{j}^{+}\partial
x_{j}^{+}},$ (1a) $\displaystyle\frac{\partial u^{-}_{i}}{\partial
t}+u^{-}_{j}\frac{\partial u^{-}_{i}}{\partial x^{-}_{j}}$
$\displaystyle=-\frac{\partial p^{-}}{\partial
x^{-}_{i}}+\nu\frac{\partial^{2}u^{-}_{i}}{\partial x_{j}^{-}\partial
x_{j}^{-}}$ (1b)

and define the time- and spanwise-averaged velocities as usual,

$\displaystyle U^{+}_{i}(X,Y)=\overline{u^{+}_{i}}=\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}u_{i}^{+}\>dzdt,$ (2a)
$\displaystyle U^{-}_{i}(X,Y)=\overline{u^{-}_{i}}=\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}u_{i}^{-}\>dzdt.$ (2b)

In the following, we use an overbar $\overline{(\quad)}$ to denote the
standard averaging operation in the time and Z-direction, as defined by (2).
Velocity fluctuations around $U^{+}_{i}$ and $U^{+}_{i}$ are denoted using
primes $(\quad)^{\prime}$, i.e.

$u^{\prime+}_{i}=u^{+}_{i}-U^{+}_{i},\quad
u^{\prime-}_{i}=u^{-}_{i}-U^{-}_{i},$ (3)

and fluctuating velocity differences are defined as,

$du_{i}^{\prime}\equiv du_{i}-dU_{i},$ (4)

where $du_{i}=u_{i}^{+}-u_{i}^{-}$ and
$dU_{i}=\overline{du_{i}}=U^{+}_{i}-U^{+}_{i}$. It is straightforward to prove
that $\overline{du_{i}^{\prime}}=0$.

Subtracting equation (1b) from (1a), multiplying each term by
$2du_{i}^{\prime}$, and applying the time- and spanwise- averaging operation
defined in (2) we obtain,

$\displaystyle\underbrace{\overline{{2du_{i}^{\prime}}\frac{\partial
du_{i}}{\partial t}}}_{\text{Transient
term}}+\underbrace{\overline{2du_{i}^{\prime}u^{+}_{j}\frac{\partial
du_{i}}{\partial x^{+}_{j}}+2du_{i}^{\prime}u^{-}_{j}\frac{\partial
du_{i}}{\partial x^{-}_{j}}}}_{\text{Non-linear term}}=$ (5)
$\displaystyle\underbrace{\overline{{-2du_{i}^{\prime}}\left(\frac{\partial
dp}{\partial x^{+}_{i}}-\frac{\partial dp}{\partial
x^{-}_{i}}\right)}}_{\text{Pressure-velocity
correlation}}+\underbrace{\overline{{2\nu
du_{i}^{\prime}}\frac{\partial^{2}du_{i}}{\partial x_{j}^{+}\partial
x_{j}^{+}}+2\nu du_{i}^{\prime}\frac{\partial^{2}du_{i}}{\partial
x_{j}^{-}\partial x_{j}^{-}}}}_{\text{Viscous term}},$

where we have used $\dfrac{\partial u^{+}_{i}}{\partial x^{-}_{j}}=0$ and
$\dfrac{\partial u^{-}_{i}}{\partial x^{+}_{j}}=0$ (because $x^{+}_{j}$ and
$x^{-}_{j}$ are independent variables).

We now define the second order structure function as
$\overline{dq^{2}}=\overline{(du^{\prime}_{i})^{2}}=\overline{(u^{\prime+}_{i}-u^{\prime-}_{i})^{2}}$.
This function has six dimensions, three in physical space ($X_{i}$) and three
in scale space ($r_{i}$). In the particular case examine in this paper, due to
the homogeneity in the spanwise direction, there are only two dimensions in
physical space. The integral of $\overline{dq^{2}}$ in a sphere of radius
$r=|r_{i}|$ (divided with the volume of the sphere) represents the energy of
eddies with size smaller than $r=|r_{i}|$, see Davidson (2015); thus
$\overline{dq^{2}}$ is usually referred to as scale energy.

Figure 2: Sketch that shows the position vector $\vv{X}=(X_{1},X_{2},X_{3})$
of the mid point, and the separation vector $\vv{r}=(r_{1},r_{2},r_{3})$
between the two points $\vv{x^{+}}=(x^{+}_{1},x^{+}_{2},x^{+}_{3})$ and
$\vv{x^{-}}=(x^{-}_{1},x^{-}_{2},x^{-}_{3})$.

We seek the transport equation of $\overline{dq^{2}}$ in the physical and
scale spaces. Applying the variable transformation
$X_{i}=0.5\left(x^{+}_{i}+x^{-}_{i}\right)$ and $r_{i}=x^{+}_{i}-x^{-}_{i}$
and the definitions (3) and (4) into (5), after some algebra we get the
following standard KHMH equation for $\overline{dq^{2}}$,

$\displaystyle\underbrace{\overline{\frac{\partial dq^{2}}{\partial
t}}}_{\text{Transient term}}+\underbrace{\overline{U^{*}_{j}\frac{\partial
dq^{2}}{\partial X_{j}}}}_{\text{Mean flow
advection}}+\underbrace{\overline{u^{\prime*}_{j}\frac{\partial
dq^{2}}{\partial X_{j}}}}_{\text{Turbulent
advection}}+\underbrace{\overline{dU_{j}\frac{\partial dq^{2}}{\partial
r_{j}}}}_{\text{Linear interscale transfer}}+$ (6)
$\displaystyle\underbrace{\overline{du_{j}^{\prime}\frac{\partial
dq^{2}}{\partial r_{j}}}}_{\text{Nonlinear interscale
transfer}}=\underbrace{\overline{-2du_{i}^{\prime}\frac{\partial
dp^{\prime}}{\partial X_{i}}}}_{\text{Pressure-velocity
correlation}}-\underbrace{\overline{2du_{i}^{\prime}u^{\prime*}_{j}\frac{\partial
dU_{i}}{\partial
X_{j}}}-\overline{2du_{i}^{\prime}du_{j}^{\prime}\frac{\partial
dU_{i}}{\partial r_{j}}}}_{\text{Production by mean flow }(=\mathcal{P})}$
$\displaystyle+\underbrace{\overline{\nu\frac{1}{2}\frac{\partial^{2}dq^{2}}{\partial
X_{j}\partial X_{j}}}}_{\text{Physical
diffusion}}+\underbrace{\overline{2\nu\frac{\partial^{2}dq^{2}}{\partial
r_{j}\partial r_{j}}}}_{\text{Scale
diffusion}}-\underbrace{4\nu\left(\frac{1}{4}\overline{\frac{\partial
du^{\prime}_{i}}{\partial X_{j}}\frac{\partial du^{\prime}_{i}}{\partial
X_{j}}}+\overline{\frac{\partial du^{\prime}_{i}}{\partial
r_{j}}\frac{\partial du^{\prime}_{i}}{\partial
r_{j}}}\right)}_{\text{Dissipation }(=\epsilon)}$

Note that $U^{*}_{j}$ and $u^{\prime*}_{j}$ denote the midpoint values of the
time-average and fluctuating velocities respectively, i.e.
$U^{*}_{j}=\left(U^{+}_{j}+U^{-}_{j}\right)/2$ and
$u^{\prime*}_{j}=\left(u^{\prime+}_{j}+u^{\prime-}_{j}\right)/2$. The physical
meaning of each term is also provided; the production by mean flow and
dissipation are denoted by $\mathcal{P}$ and $\epsilon$ respectively. Assuming
that the transient term is 0, the above equation can be written in divergence
form as

$\frac{\partial\boldsymbol{\phi_{s}}_{i}}{\partial
X_{i}}+\frac{\partial\boldsymbol{\phi_{r}}_{i}}{\partial
r_{i}}=\mathcal{P}-\epsilon,$ (7)

where

$\boldsymbol{\phi_{s}}_{i}=\underbrace{\overline{U_{i}^{*}\delta
q^{2}}}_{=\phi_{s_{i}}^{M}}+\underbrace{\overline{u^{\prime*}_{i}\delta
q^{2}}}_{=\phi_{s_{i}}^{F}}+\underbrace{\overline{2\delta u^{\prime}_{i}\delta
p^{\prime}}}_{=\phi_{s_{i}}^{P}}\underbrace{-\frac{1}{2}\nu\frac{\partial\overline{\delta
q^{2}}}{\partial X_{i}}}_{=\phi_{s_{i}}^{V}},$ (8)

is the total flux vector in physical space and

$\boldsymbol{\phi_{r}}_{i}=\underbrace{\overline{\delta u^{\prime}_{i}\delta
q^{2}}}_{=\phi_{r_{i}}^{F}}+\underbrace{\overline{\delta U_{i}\delta
q^{2}}}_{=\phi_{r_{i}}^{M}}\underbrace{-2\nu\frac{\partial\overline{\delta
q^{2}}}{\partial r_{i}}}_{=\phi_{r_{i}}^{V}},$ (9)

is the total flux vector in scale space. We use the superscript ’F’ to denote
the non-linear components, $\phi_{s_{i}}^{F}$ and $\phi_{r_{i}}^{F}$, of these
vectors respectively. The conditional decomposition of the non-linear fluxes
$\phi_{s_{i}}^{F}$ and $\phi_{r_{i}}^{F}$ will be examined in section 5. We
are now ready to proceed with the derivation of the conditionally-averaged
KHMH equation.

##  4 Conditionally-averaged KHMH equation

###  4.1 Definitions

In order to derive the conditionally-averaged form of the KHMH equation, we
need first to define the conditions under which the averaging is performed. To
this end, we employ the local binary function $\tau(X_{i},t)$ to distinguish
between instantaneous laminar and turbulent states at point $X_{i}$ at time
$t$. More specifically, $\tau(X_{i},t)$ takes the value of $0$ for the former
(i.e. laminar), and $1$ for the latter (i.e. turbulent) state. This binary
function is computed using the standard deviation of $D=|v|+|w|$; refer to
Marxen & Zaki (2019) and Yao et al. (2020) for more details.

Since the KHMH equation involves two points, the conditions for the averaging
operation should be defined using the states at both points. Four combinations
are possible,

  1. 1.

If both $x_{i}^{+}$ and $x_{i}^{-}$ are located within a turbulent patch, this
is denoted as a Turbulent-Turbulent (or $TT$) event, and it is defined by the
condition $\tau^{+}\tau^{-}=1$, where $\tau^{+}=\tau(x_{i}^{+},t)$ and
$\tau^{-}=\tau(x_{i}^{-},t)$.

  2. 2.

If both points are within the laminar region, this is a Laminar-Laminar (or
$LL$) event, and it is defined by the condition $(1-\tau^{+})(1-\tau^{-})=1$.

  3. 3.

If $x_{i}^{+}$, $x_{i}^{-}$ are within a turbulent and a laminar patch
respectively, this is a Turbulent-Laminar (or $TL$) event, defined by
$\tau^{+}(1-\tau^{-})=1$.

  4. 4.

If $x_{i}^{+}$, $x_{i}^{-}$ are within a laminar and turbulent region
respectively, this is a Laminar-Turbulent (or $LT$) event, defined by
$(1-\tau^{+})\tau^{-}=1$.

In our notation, the first capital letter denotes the state of point
$x_{i}^{+}$ and the second the state of $x_{i}^{-}$. We will also use the
generic notation $AA$ to refer to a general event, i.e. $AA=TT$ or $LL$ or
$TL$ or $LT$. The four events are shown schematically in figure 3.

Figure 3: Contour plot of wall-normal velocity fluctuations in the
transitional region. A turbulent spot is clearly visualised on the right half
of the figure. Within the spot $\tau=1$, while outside (grey region) $\tau=0$.
The four different two-point event types, $LL$, $TT$, $TL$ and $LT$, are
shown.

We can now define the time- and spanwise-average two-point intermittencies as,

$\displaystyle\gamma^{(TT)}$ $\displaystyle\equiv\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}dzdt,$ (10a)
$\displaystyle\gamma^{(TL)}$ $\displaystyle\equiv\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}\tau^{+}(1-\tau^{-})dzdt,$ (10b)
$\displaystyle\gamma^{(LT)}$ $\displaystyle\equiv\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}(1-\tau^{+})\tau^{-}dzdt,$ (10c)
$\displaystyle\gamma^{(LL)}$ $\displaystyle\equiv\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}(1-\tau^{+})(1-\tau^{-})dzdt.$ (10d)

Since
$\tau^{+}\tau^{-}+(1-\tau^{+})(1-\tau^{-})+\tau^{+}(1-\tau^{-})+(1-\tau^{+})\tau^{-}=1$,
we have $\gamma^{(TT)}+\gamma^{(TL)}+\gamma^{(LT)}+\gamma^{(LL)}=1$. Two point
intermittencies were also defined in Yao et al. (2020), where $TL$ and $TL$
events were amalgamated as a combined $TL$ event. Here we consider the two
event types separately for reasons that will become clear shortly.

The conditional time-averages of the general two-point variable,
$dQ=Q(x_{i}^{+})-Q(x_{i}^{-})$, are defined as,

$\displaystyle\overline{dQ}^{(TT)}(X,Y;r_{1},r_{3})$
$\displaystyle\equiv\frac{1}{\Delta TL_{z}\gamma^{(TT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}dQ\>dzdt,$ (11a)
$\displaystyle\overline{dQ}^{(TL)}(X,Y;r_{1},r_{3})$
$\displaystyle\equiv\frac{1}{\Delta TL_{z}\gamma^{(TL)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}(1-\tau^{-})dQ\>dzdt,$ (11b)
$\displaystyle\overline{dQ}^{(LT)}(X,Y;r_{1},r_{3})$
$\displaystyle\equiv\frac{1}{\Delta TL_{z}\gamma^{(LT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}(1-\tau^{+})\tau^{-}dQ\>dzdt,$ (11c)
$\displaystyle\overline{dQ}^{(LL)}(X,Y;r_{1},r_{3})$
$\displaystyle\equiv\frac{1}{\Delta TL_{z}\gamma^{(LL)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}(1-\tau^{+})(1-\tau^{-})dQ\>dzdt.$ (11d)

This means that the standard time-average can be decomposed as

$\displaystyle\overline{dQ}(X,Y;r_{1},r_{3})$ $\displaystyle=\frac{1}{\Delta
TL_{z}}\int_{0}^{\Delta T}\int_{0}^{L_{z}}dQ\>dzdt$ (12)
$\displaystyle=\gamma^{(TT)}\overline{dQ}^{(TT)}+\gamma^{(TL)}\overline{dQ}^{(TL)}+\gamma^{(LT)}\overline{dQ}^{(LT)}+\gamma^{(LL)}\overline{dQ}^{(LL)}$

The definitions are similar for the conditionally average midpoint variable
$Q^{*}=0.5\left[Q(x_{i}^{+})+Q(x_{i}^{-})\right]$ thus,

$\displaystyle\overline{Q^{*}}(X,Y;r_{1},r_{3})$
$\displaystyle=\frac{1}{\Delta TL_{z}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}Q^{*}\>dzdt$ (13)
$\displaystyle=\gamma^{(TT)}\overline{Q^{*}}^{(TT)}+\gamma^{(TL)}\overline{Q^{*}}^{(TL)}+\gamma^{(LT)}\overline{Q^{*}}^{(LT)}+\gamma^{(LL)}\overline{Q^{*}}^{(LL)}$

Referring back to figure 3, it is clear that $\overline{dQ}^{(TL)}$ and
$\overline{dQ}^{(LT)}$ are two-point averages across the laminar/turbulent
interface. For $r_{1}=0$ and $r_{3}\neq 0$ (case shown in figure 3), these two
conditional averages are taken across the interface in the spanwise direction.
If ${dQ}$ is a quadratic function (for example $dQ=\overline{dq^{2}}$) due to
homogeneity in $Z$ we have
$\overline{dq^{2}}^{(TL)}(X,Y;0,r_{3})=\overline{dq^{2}}^{(LT)}(X,Y;0,r_{3})$.
For $r_{1}\neq 0$ and $r_{3}=0$ (case shown in figure 4), $LT$ averaging is
taken across the tail (i.e. the downstream end) of a turbulent spot, while
$TL$ is taken across the head of the spot (i.e. the upstream end). It is
therefore possible to distinguish the different properties of the head or tail
of a spot using the appropriate conditionally averaged quantity. This is an
important observation and facilitates the physical interpretation of the
results presented in section 6.

Figure 4: Contour plot of instantaneous streamwise velocity fluctuations.
Yellow ovals demarcate two turbulent spots. The purple vertical dotted line
represents a fixed streamwise location, X. Two event types, TL and LT, are
shown with $r_{1}\neq 0$ and $r_{3}=0$.

We can now proceed to derive the conditionally-averaged KHMH equation.

###  4.2 Derivation of the conditionally-averaged KHMH equation

We start again with the Navier-Stokes equations at two points $x_{i}^{+}$ and
$x_{i}^{-}$, equations (1), and define the conditional velocity fluctuation
difference as

$du_{i}^{\prime(AA)}\equiv du_{i}-dU_{i}^{(AA)},$ (14)

where $AA=TT$ or $LL$ or $TL$ or $LT$ as mentioned earlier,
$du_{i}=u_{i}^{+}-u_{i}^{-}$ and $dU_{i}^{(AA)}=\overline{du_{i}}^{(AA)}$
(from definition (11)). It is straightforward to prove that
$\overline{du_{i}^{\prime(AA)}}^{(AA)}=0$; this is the equivalent of
$\overline{du_{i}^{\prime}}=0$ in standard averaging.

Similarly, the conditional fluctuation velocity at the midpoint is defined as

${u^{\prime*}_{i}}^{(AA)}\equiv{u^{*}_{i}}-{U^{*}_{i}}^{(AA)},$ (15)

where $u^{*}_{i}=0.5\left(u_{i}^{+}+u_{i}^{-}\right)$ and
${U^{*}_{i}}^{(AA)}=\overline{u^{*}_{i}}^{(AA)}$, and again
$\overline{{u^{\prime*}_{i}}^{(AA)}}^{(AA)}=0$.

Subtracting equation (1b) from (1a), multiplying each term by
$2du_{i}^{\prime(AA)}$, and then applying the $(AA)$ averaging operation as
defined in equation (11) we obtain,

$\displaystyle\underbrace{\overline{2du_{i}^{\prime(AA)}\frac{\partial
du_{i}}{\partial t}}^{(AA)}}_{\text{Transient
term}}+\underbrace{\overline{2du_{i}^{\prime(AA)}u^{+}_{j}\frac{\partial
du_{i}}{\partial x^{+}_{j}}+2du_{i}^{\prime(AA)}u^{-}_{j}\frac{\partial
du_{i}}{\partial x^{-}_{j}}}^{(AA)}}_{\text{Non-linear term}}=$ (16)
$\displaystyle\underbrace{\overline{-2du_{i}^{\prime(AA)}(\frac{\partial
dp}{\partial x^{+}_{i}}-\frac{\partial dp}{\partial
x^{-}_{i}})}^{(AA)}}_{\text{Pressure-velocity
correlation}}+\underbrace{\overline{2\nu
du_{i}^{\prime(AA)}\frac{\partial^{2}du_{i}}{\partial x_{j}^{+}\partial
x_{j}^{+}}+2\nu du_{i}^{\prime(AA)}\frac{\partial^{2}du_{i}}{\partial
x_{j}^{-}\partial x_{j}^{-}}}^{(AA)}}_{\text{Viscous term}},$

where we have used again $\dfrac{\partial u^{+}_{i}}{\partial x^{-}_{j}}=0$
and $\dfrac{\partial u^{-}_{i}}{\partial x^{+}_{j}}=0$. This is the equivalent
of (5) for standard-averaging.

We now define the two-point conditional energy as
$\overline{dq^{2^{(AA)}}}^{(AA)}=\overline{du_{i}^{\prime(AA)}du_{i}^{\prime(AA)}}^{(AA)}$
and seek its transport equation in the physical and scale spaces, similar to
(6). Applying again the variable transformation
$X_{i}=0.5\left(x^{+}_{i}+x^{-}_{i}\right)$ and $r_{i}=x^{+}_{i}-x^{-}_{i}$
and the definitions (14) and (15) into (16), after some algebra we get the
following conditionally-averaged KHMH equation for
$\overline{dq^{2^{(AA)}}}^{(AA)}$,

$\displaystyle\underbrace{\overline{\frac{\partial dq^{2^{(AA)}}}{\partial
t}}^{(AA)}}_{\text{Transient
term}}+\underbrace{\overline{{U^{*}_{j}}^{(AA)}\frac{\partial
q^{2^{(AA)}}}{\partial X_{j}}}^{(AA)}}_{\text{Mean
advection}}+\underbrace{\overline{{u^{\prime*}_{j}}^{(AA)}\frac{\partial
dq^{2^{(AA)}}}{\partial X_{j}}}^{(AA)}}_{\text{Turbulent
advection}}+\underbrace{\overline{dU_{j}^{(AA)}\frac{\partial
dq^{2^{(AA)}}}{\partial r_{j}}}^{(AA)}}_{\text{Linear transfer}}$ (17)
$\displaystyle+\underbrace{\overline{du_{j}^{\prime(AA)}\frac{\partial
dq^{2^{(AA)}}}{\partial r_{j}}}^{(AA)}}_{\text{Nonlinear
transfer}}=\underbrace{\overline{-2du_{i}^{\prime(AA)}\frac{\partial
dp^{\prime(AA)}}{\partial X_{i}}}^{(AA)}}_{\text{Pressure-velocity
correlation}}+\underbrace{\nu\frac{1}{2}\overline{\frac{\partial^{2}dq^{2^{(AA)}}}{\partial
X_{j}\partial X_{j}}}^{(AA)}}_{\text{Physical
diffusion}}+\underbrace{2\nu\overline{\frac{\partial^{2}dq^{2^{(AA)}}}{\partial
r_{j}\partial r_{j}}}^{(AA)}}_{\text{Scale diffusion}}$
$\displaystyle\underbrace{-\overline{2du_{i}^{\prime(AA)}{u^{\prime*}_{j}}^{(AA)}\frac{\partial
dU_{i}^{(AA)}}{\partial
X_{j}}}^{(AA)}-\overline{2du_{i}^{\prime(AA)}du_{j}^{\prime(AA)}\frac{\partial
dU_{i}^{(AA)}}{\partial r_{j}}}^{(AA)}}_{\text{Production}}$
$\displaystyle-\underbrace{2\nu\overline{\left(\frac{\partial
du^{\prime(AA)}_{i}}{\partial x^{+}_{j}}\frac{\partial
du^{\prime(AA)}_{i}}{\partial x^{+}_{j}}+\frac{\partial
du^{\prime(AA)}_{i}}{\partial x^{-}_{j}}\frac{\partial
du^{\prime(AA)}_{i}}{\partial x^{-}_{j}}\right)}^{(AA)}}_{\text{Dissipation}}$

The form of equation (17) is similar to the standard form (6). This is due to
the appropriate definitions of the conditional fluctuating quantities in (14)
and (15) that satisfy $\overline{du_{i}^{\prime(AA)}}^{(AA)}=0$ and
$\overline{{u^{\prime*}_{i}}^{(AA)}}^{(AA)}=0$. There is however an important
difference. Standard averaging commutes with the spatial differentiation
operation, for example,

$\frac{\partial\overline{dQ}}{\partial r_{j}}=\frac{1}{\Delta
TL_{z}}\frac{\partial}{\partial r_{j}}\left(\int_{0}^{\Delta
T}\int_{0}^{L_{z}}dQdzdt\right)=\frac{1}{\Delta TL_{z}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\frac{\partial dQ}{\partial
r_{j}}dzdt=\overline{\frac{\partial{dQ}}{\partial r_{j}}}$ (18)

This is however not the case for conditionally-averaged quantities, for
example

$\frac{\partial\overline{dQ}^{(TT)}}{\partial r_{j}}=\frac{1}{\Delta
TL_{z}}\frac{\partial}{\partial
r_{j}}\left(\frac{1}{\gamma^{(TT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}dQdzdt\right),$ (19)

while

$\overline{\frac{\partial{dQ}}{\partial r_{j}}}^{(TT)}=\frac{1}{\Delta
TL_{z}}\left(\frac{1}{\gamma^{(TT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}\frac{\partial{dQ}}{\partial
r_{j}}dzdt\right)\neq\frac{\partial\overline{dQ}^{(TT)}}{\partial r_{j}},$
(20)

because $\gamma^{(TT)}$ depends on $r_{j}$. The same issue appears in the
conditionally-averaged TKE equation (Marxen & Zaki, 2019).

This lack of commutation has two important implications. First, equation (17)
cannot be written in conservative form. For example, for the non-linear inter-
scale energy transfer term, we have

$\overline{du_{j}^{\prime(AA)}\frac{\partial dq^{2^{(AA)}}}{\partial
r_{j}}}^{(AA)}=\overline{\frac{\partial
dq^{2^{(AA)}}du_{j}^{\prime(AA)}}{\partial
r_{j}}}^{(AA)}\neq\frac{\partial\left(\overline{dq^{2^{(AA)}}du_{j}^{\prime(AA)}}^{(AA)}\right)}{\partial
r_{j}}$ (21)

The first equality in the above equation is because
$\dfrac{\partial{du_{j}^{\prime(AA)}}}{\partial r_{j}}=0$ (this is equivalent
to $\dfrac{\partial\overline{du_{j}^{\prime}}}{\partial r_{j}}=0$ for standard
averaging). It is easy to prove; for example if $AA=TT$, we have

$\displaystyle\frac{\partial du_{j}^{\prime(TT)}}{\partial
r_{j}}=\frac{\partial du_{j}}{\partial r_{j}}-\frac{\partial
dU_{j}^{(TT)}}{\partial r_{j}}=\frac{\partial du_{j}}{\partial
r_{j}}-\frac{\partial}{\partial r_{j}}\left(\frac{1}{{\Delta
TL_{z}}\gamma^{(TT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}dU_{j}dzdt\right)=$ (22)
$\displaystyle\frac{\partial du_{j}}{\partial r_{j}}-\frac{\partial}{\partial
r_{j}}\left(\frac{dU_{j}}{\Delta TL_{z}\gamma^{(TT)}}\int_{0}^{\Delta
T}\int_{0}^{L_{z}}\tau^{+}\tau^{-}dzdt\right)=\frac{\partial du_{j}}{\partial
r_{j}}-\frac{\partial dU_{j}}{\partial r_{j}}=0-0=0,$

because $du_{j}$ and $dU_{j}$ satisfy the continuity equation in the scale
space. We also took into account that $dU_{j}$ is constant with respect to $Z$
and $t$ and used (10a).

The second implication is computational. For standard averaging, all the terms
in the KHMH equation can be evaluated numerically by differentiating locally
at points $x_{i}^{+}$ and $x_{i}^{-}$, for example

$\displaystyle\frac{\partial dU_{i}}{\partial
r_{j}}=\frac{\partial\overline{du_{i}}}{\partial
r_{j}}=\overline{\frac{\partial{du_{i}}}{\partial r_{j}}}$
$\displaystyle=\frac{1}{2}\left(\overline{\frac{\partial{(u_{i}^{+}-u_{i}^{-})}}{\partial
x^{+}}}-\overline{\frac{\partial{(u_{i}^{+}-u_{i}^{-})}}{\partial
x_{j}^{-}}}\right)=\frac{1}{2}\left(\overline{\frac{\partial{u_{i}^{+}}}{\partial
x_{j}^{+}}}+\overline{\frac{\partial{u_{i}^{-}}}{\partial x_{j}^{-}}}\right)=$
(23)
$\displaystyle\frac{1}{2}\left(\frac{\partial\overline{u_{i}^{+}}}{\partial
x_{j}^{+}}+\frac{\partial\overline{u_{i}^{-}}}{\partial
x_{j}^{-}}\right)=\frac{1}{2}\left({\frac{\partial U_{i}^{+}}{\partial
x_{j}^{+}}}+{\frac{\partial U_{i}^{-}}{\partial x_{j}^{-}}}\right)$

However, this is not the case for the conditionally-averaged velocity
difference,

$\displaystyle\frac{\partial dU_{i}^{(AA)}}{\partial
r_{j}}=\frac{\partial\overline{du_{i}}^{(AA)}}{\partial
r_{j}}\neq\overline{\frac{\partial{du_{i}}}{\partial r_{j}}}^{(AA)}$
$\displaystyle\left[=\frac{1}{2}\left(\overline{\frac{\partial{(u_{i}^{+}-u_{i}^{-})}}{\partial
x_{j}^{+}}}^{(AA)}-\overline{\frac{\partial{(u_{i}^{+}-u_{i}^{-})}}{\partial
x_{j}^{-}}}^{(AA)}\right)\right.$ (24)
$\displaystyle=\left.\frac{1}{2}\left(\overline{\frac{{\partial
u_{i}^{+}}}{\partial x_{j}^{+}}}^{(AA)}+\overline{\frac{{\partial
u_{i}^{-}}}{\partial x_{j}^{-}}}^{(AA)}\right)\right]$

This means that the derivatives of the conditionally-averaged two-point
variables must be calculated directly in scale space. The process and
validation against standard averaging are presented in appendix A.

##  5 Conditional decomposition of non-linear energy fluxes

In this section we decompose the non-linear energy fluxes in scale and
physical spaces,
$\phi_{r_{j}}^{F}=\overline{dq^{2}du^{\prime}_{j}}=\overline{(du^{\prime}_{i})^{2}du^{\prime}_{j}}$
and ${\phi_{s_{j}}^{F}}=\overline{u^{\prime*}_{j}\delta q^{2}}$ respectively,
see equations (9) and (8), into conditionally-averaged components, i.e. we
seek to derive expressions similar to (12) and (13). The decomposition for the
second-order structure function (quadratic quantity) was derived in Yao et al.
(2020). Here we extend the method to energy fluxes (cubic quantities).

For the non-linear energy flux in scale space
$\phi_{r_{j}}^{F}=\overline{(du^{\prime}_{i})^{2}du^{\prime}_{j}}$ we have,

$\displaystyle\phi_{r_{j}}^{F}$
$\displaystyle=\overline{(du^{\prime}_{i})^{2}du^{\prime}_{j}}=\overline{(du_{i}-dU_{i})^{2}(du_{j}-dU_{j})}$
(25)
$\displaystyle=\overline{(du_{i})^{2}du_{j}}-\overline{(du_{i})^{2}}dU_{j}-\overline{2du_{i}du_{j}}dU_{i}+2dU_{i}dU_{i}dU_{j}$
$\displaystyle=\gamma^{(TT)}\overline{(du_{i})^{2}du_{j}}^{(TT)}+\gamma^{(TL)}\overline{(du_{i})^{2}du_{j}}^{(TL)}+\gamma^{(LT)}\overline{(du_{i})^{2}du_{j}}^{(LT)}+$
$\displaystyle\gamma^{(LL)}\overline{(du_{i})^{2}du_{j}}^{(LL)}\quad-\overline{(du_{i})^{2}}dU_{j}-\overline{2du_{i}du_{j}}dU_{i}+2dU_{i}dU_{i}dU_{j},$

where (12) was applied to $\overline{(du_{i})^{2}du_{j}}$.

We now write $\overline{(du_{i})^{2}du_{j}}^{(AA)}$ ($AA=TT$ or $LL$ or $TL$
or $LT$) in terms of conditional fluctuations. To do this, we use the
definition $du_{i}^{\prime(AA)}\equiv du_{i}-dU_{i}^{(AA)}$, see equation
(14), and express the conditional non-linear energy flux
${\phi^{F}_{r_{j}}}^{(AA)}=\overline{\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{j}}^{(AA)}$
as

$\displaystyle{\phi^{F}_{r_{j}}}^{(AA)}=\overline{\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{j}}^{(AA)}$
$\displaystyle=\overline{(du_{i})^{2}du_{j}}^{(AA)}-\overline{(du_{i})^{2}}^{(AA)}dU_{j}^{(AA)}$
(26)
$\displaystyle-\overline{2du_{i}du_{j}}^{(AA)}dU_{i}^{(AA)}-2dU_{i}^{(AA)}dU_{i}^{(AA)}dU_{j}^{(AA)}$

Solving for $\overline{(du_{i})^{2}du_{j}}^{(AA)}$ and substituting into (25),
we obtain the desired decomposition

$\displaystyle\phi^{F}_{r_{j}}=\gamma^{(TT)}{\phi^{F}_{r_{j}}}^{(TT)}+\gamma^{(TL)}{\phi^{F}_{r_{j}}}^{(TL)}+\gamma^{(LT)}{\phi^{F}_{r_{j}}}^{(LT)}+\gamma^{(LL)}{\phi^{F}_{r_{j}}}^{(LL)}+\phi_{r_{j}},$
(27)

where the additional term $\phi_{r_{j}}$ is given by,

$\displaystyle\phi_{r_{j}}$
$\displaystyle=\gamma^{(TT)}\left[\overline{(du_{i})^{2}}^{(TT)}dU_{j}^{(TT)}+\overline{2du_{i}du_{j}}^{(TT)}dU_{i}^{(TT)}-2dU_{i}^{(TT)}dU_{i}^{(TT)}dU_{j}^{(TT)}\right]$
(28)
$\displaystyle+\gamma^{(TL)}\left[\overline{(du_{i})^{2}}^{(TL)}dU_{j}^{(TL)}+\overline{2du_{i}du_{j}}^{(TL)}dU_{i}^{(TL)}-2dU_{i}^{(TL)}dU_{i}^{(TL)}dU_{j}^{(TL)}\right]$
$\displaystyle+\gamma^{(LT)}\left[\overline{(du_{i})^{2}}^{(LT)}dU_{j}^{(LT)}+\overline{2du_{i}du_{j}}^{(LT)}dU_{i}^{(LT)}-2dU_{i}^{(LT)}dU_{i}^{(LT)}dU_{j}^{(LT)}\right]$
$\displaystyle+\gamma^{(LL)}\left[\overline{(du_{i})^{2}}^{(LL)}dU_{j}^{(LL)}+\overline{2du_{i}du_{j}}^{(LL)}dU_{i}^{(LL)}-2dU_{i}^{(LL)}dU_{i}^{(LL)}dU_{j}^{(LL)}\right]$
$\displaystyle-\overline{(du_{i})^{2}}dU_{j}-\overline{2du_{i}du_{j}}dU_{i}+2dU_{i}dU_{i}dU_{j}$

Figure 5: Non-linear interscale energy flux vector
$\left(\phi^{F}_{r_{3}},\phi^{F}_{s_{2}}\right)$ (LHS) and the constituent
components
$\left(\gamma^{(AA)}{\phi^{F}_{r_{3}}}^{(AA)},\gamma^{(AA)}{\phi^{F}_{s_{2}}}^{(AA)}\right)$
(where $AA$=$TT$,$TL+LT$,$LL$) in the $(r_{3},Y)$ plane at location $TR2$. RHS
denotes the sum of the terms on the right-hand side of equation (27). Contours
represent the magnitude of the flux vectors and the purple horizontal lines
indicate the local boundary layer thickness.

Following the same process, the following decomposition can be obtained for
the non-linear energy flux in physical space,
${\phi_{s_{j}}^{F}}=\overline{u^{\prime*}_{j}\delta
q^{2}}=\overline{u^{\prime*}_{j}(du^{\prime}_{i})^{2}}$,

$\displaystyle\phi^{F}_{s_{j}}=\gamma^{(TT)}{\phi^{F}_{s_{j}}}^{(TT)}+\gamma^{(TL)}{\phi^{F}_{s_{j}}}^{(TL)}+\gamma^{(LT)}{\phi^{F}_{s_{j}}}^{(LT)}+\gamma^{(LL)}{\phi^{F}_{s_{j}}}^{(LL)}+\phi_{s_{j}},$
(29)

where
${\phi^{F}_{s_{j}}}^{(AA)}=\overline{{u^{\prime*}_{j}}^{(AA)}\left(du^{\prime(AA)}_{i}\right)^{2}}^{(AA)}$
and the additional term $\phi_{s_{j}}$ is given by,

$\displaystyle\phi_{s_{j}}$
$\displaystyle=\gamma^{(TT)}\left[\overline{(du_{i})^{2}}^{(TT)}{U^{*}_{j}}^{(TT)}+\overline{2du_{i}u^{*}_{j}}^{(TT)}dU_{i}^{(TT)}-2dU_{i}^{(TT)}dU_{i}^{(TT)}{U^{*}_{j}}^{(TT)}\right]$
(30)
$\displaystyle+\gamma^{(TL)}\left[]\overline{(du_{i})^{2}}^{(TL)}{U^{*}_{j}}^{(TL)}+\overline{2du_{i}u^{*}_{j}}^{(TL)}dU_{i}^{(TL)}-2dU_{i}^{(TL)}dU_{i}^{(TL)}{U^{*}_{j}}^{(TL)}\right]$
$\displaystyle+\gamma^{(LT)}\left[\overline{(du_{i})^{2}}^{(LT)}{U^{*}_{j}}^{(LT)}+\overline{2du_{i}u^{*}_{j}}^{(LT)}dU_{i}^{(LT)}-2dU_{i}^{(LT)}dU_{i}^{(LT)}{U^{*}_{j}}^{(LT)}\right]$
$\displaystyle+\gamma^{(LL)}\left[\overline{(du_{i})^{2}}^{(LL)}{U^{*}_{j}}^{(LL)}+\overline{2du_{i}u^{*}_{j}}^{(LL)}dU_{i}^{(LL)}-2dU_{i}^{(LL)}dU_{i}^{(LL)}{U^{*}_{j}}^{(LL)}\right]$
$\displaystyle-\overline{(du_{i})^{2}}U^{*}_{j}-\overline{2du_{i}u^{*}_{j}}dU_{i}+2dU_{i}dU_{i}U^{*}_{j}$

Figure 5 presents contours of all the terms appearing in (27) for the
decomposition of the energy flux vector $(\phi^{F}_{r_{3}},\phi^{F}_{s_{2}})$
at location $TR2$ on $(r_{3},Y)$ plane (the separations in the other two
directions are equal to 0, i.e. $r_{1}=0,r_{2}=0$). In the figure, we combine
the results of $TL$ and $LT$ together due to homogeneity in the spanwise
direction. The upper left panel, denoted as LHS, depicts the results from the
direct calculation of the flux vector $(\phi^{F}_{r_{3}},\phi^{F}_{s_{2}})$
using standard time-average, i.e.
$\left(\overline{\left(du^{\prime}_{i}\right)^{2}du^{\prime}_{3}},\overline{\left(du^{\prime}_{i}\right)^{2}u^{\prime*}_{2}}\right)$.
The lower right panel, denoted as RHS, presents the sum of all terms in the
(27). It is clear that the results in the two panels are essentially
identical, confirming the validity of the derived decomposition. The flux
vector originates from the focal point $Y/L_{0}\approx 2.5$ and
$r_{3}/L_{0}\approx 10$ and transfers energy radially to different directions.
The strongest flux is located in the region $0<Y/L_{0}<10,r_{3}/L_{0}>10$ and
is found to be positive, indicating strong inverse cascade. This pattern was
observed in Yao et al. (2022).

Apart from verifying the decomposition (27), figure 5 provides important
insight into the origin of the aforementioned energy flux pattern. It is
interesting to observe that the inverse cascade arises from the $TL+LT$, $LL$,
and $\phi$ components; all contribute to the strong positive flux in the
region $0<Y/L_{0}<10$, $r_{3}/L_{0}>10$ with values that are of the same order
of magnitude. Outside this region, they have small values. On the other hand,
the $TT$ component contributes to the forward cascade in the region left of
the focal point. The shape therefore of the energy flux vectors on the
$(r_{3},Y)$ plane arises from the superposition of the contribution of $TT$
term which is responsible for forward cascade, and all the other terms which
are responsible for inverse cascade.

The constituent terms plotted in figure 5 are weighted by the corresponding
two-point intermittencies, $\gamma^{(AA)}$, which can have very small values
depending on the location examined, refer to figure 1. In the following
section, the evolution of the vectors
$\left({{\phi^{F}_{r_{3}}}^{(AA)}},{{\phi^{F}_{s_{2}}}^{(AA)}}\right)=\left(\overline{\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{3}}^{(AA)},\overline{\left(du^{\prime(AA)}_{i}\right)^{2}{u^{\prime*}_{2}}^{(AA)}}^{(AA)}\right)$
at different streamwise locations is examined.

##  6 Evolution of conditionally-averaged non-linear energy fluxes

###  6.1 Fluxes on $(r_{3},Y)$ plane

Figure 6: Conditionally-averaged flux vectors
$\left({{\phi^{F}_{r_{3}}}^{(AA)}},{{\phi^{F}_{s_{2}}}^{(AA)}}\right)$ (first,
second and third row respectively starting from the top) and standard-averaged
flux vector $\left({{\phi^{F}_{r_{3}}}},{{\phi^{F}_{s_{2}}}}\right)$ (bottom
row) at locations $TR1$ (left column), $TR2$ (middle column) and $TR3$ (right
column). The contours represent the magnitude of the corresponding flux
vectors and the purple horizontal lines indicate the local boundary layer
thickness.  Figure 7: Standard-averaged flux vector
$\left({{\phi^{F}_{r_{3}}}},{{\phi^{F}_{s_{2}}}}\right)$ at $TU$ location. The
contour represents the magnitude of the flux vectors.

Figure 6 shows the conditionally averaged energy fluxes
$\left({{\phi^{F}_{r_{3}}}^{(AA)}},{{\phi^{F}_{s_{2}}}^{(AA)}}\right)$ where
$AA$$=$ $TT$ or $TL+LT$ or $LL$ at locations $TR1,TR2,TR3$. The last row
(marked as ’Total’) depicts
$\left({{\phi^{F}_{r_{3}}}},{{\phi^{F}_{s_{2}}}}\right)$. In all plots,
$r_{1}=0$ and $r_{2}=0$.

At the early stages of transition, at location $TR1$ (left column), the flux
vector $\left({{\phi^{F}_{r_{3}}}^{(LL)}},{{\phi^{F}_{s_{2}}}^{(LL)}}\right)$
is dominant and is almost identical to the total flux. We can clearly see
strong inverse cascade occurring to the right of the focal point, which is
mainly due to $LL$ events. The vectors
$\left({{\phi^{F}_{r_{3}}}^{(TT)}},{{\phi^{F}_{s_{2}}}^{(TT)}}\right)$ and
$\left({{\phi^{F}_{r_{3}}}^{(TL)}},{{\phi^{F}_{s_{2}}}^{(TL)}}\right)+\left({{\phi^{F}_{r_{3}}}^{(LT)}},{{\phi^{F}_{s_{2}}}^{(LT)}}\right)$
are very localised and their behaviour is difficult to interpret. This may be
due to the small number of $TT$ and $TL/LT$ events at this early transition
location. Notice however the high values of the magnitude of the $TT$ vector,
$\left({{\phi^{F}_{r_{3}}}^{(TT)}},{{\phi^{F}_{s_{2}}}^{(TT)}}\right)$,
compared to all the other vectors. However, its overall contribution to the
total flux, $\gamma^{(TT)}\boldsymbol{\phi^{F}}^{(TT)}$, is small because the
values of intermittency $\gamma^{(TT)}$ are negligible at this location (see
figure 1).

At location $TR2$ (middle column), $TT$ and $TL/LT$ events start to play a
more significant role as expected, but they act in different areas of the map.
Events of the $LL$ type maintain their dominant contribution to the inverse
cascade, while $TL$ & $LT$ events also amplify and contribute to the inverse
cascade in the area $10L_{0}<r_{3}<20L_{0}$ above $Y\approx 5L_{0}$. Notice
again that $TT$-averaged flux vectors have a much higher magnitude compared to
the total flux, and show a mixture of the forward and inverse cascade to the
left and right of the focal point respectively (with the forward cascade being
slightly stronger). This picture is consistent with figure 5 where the
conditional-averaged fluxes are weighted by the two-point intermittency.

At location $TR3$ (right column), $TT$ events now assume the dominant role
because $\gamma^{(TT)}$ approaches 1, $LL$ events are localised (in the same
way that $TT$ events were localised at $TR1$ location), while $TL$ & $LT$
events again give rise to inverse cascade. Interestingly, as intermittency
increases, $TT$ events show stronger inverse cascade. For comparison, the flux
vector $\left(\phi^{F}_{r_{3}},\phi^{F}_{s_{2}}\right)$ at the fully turbulent
location $TU$ is plotted in figure 7. At this location, the inverse cascade is
weakened (but it is still visible) and energy flows in the wall-normal
direction before looping back to small spanwise length scales.

The strong inverse cascade found at $TR2$ and $TR3$ locations is clearly not
observed in the fully turbulent region. The origin of the inverse cascade
arises mainly due to $TL$ & $LT$ events. Indeed, in the $TR2$ location their
magnitude is 5-6 times larger compared to $LL$ events (that also contribute to
inverse cascade), while in $TR3$ about 2-3 times larger. The laminar/turbulent
interface, therefore, plays a crucial role in the inverse cascade process. In
the next section, we focus on the $TR2$ location and explore in more depth the
cascade process in the three-dimensional $(r_{1},r_{3},Y)$ hyperplane.

###  6.2 Fluxes on the $(r_{1},r_{3},Y)$ hyperplane and $(r_{1},r_{3})$ plane

We project the conditionally-averaged fluxes ${\phi^{F}}^{(AA)}$ and
${\phi^{S}}^{(AA)}$ on the $(r_{1},r_{3},Y)$ hyperplane, i.e. plot three-
dimensional maps of the vector
$\left({\phi^{F}_{r_{1}}}^{(AA)},{\phi^{F}_{r_{3}}}^{(AA)},{\phi^{F}_{s_{2}}}^{(AA)}\right)$.
We select location $TR2$, which is in the middle of the transition region, and
has single-point intermittency about 0.4 (refer to figure 1).

Figure 8: Stream-tubes of (a) the total flux vector
$\left({{\phi_{r_{1}}}},{{\phi_{r_{3}}}},{{\phi_{s_{2}}}}\right)$, (b) the
standard-averaged non-linear flux vector
$\left({\phi^{F}_{r_{1}}},{\phi^{F}_{r_{3}}},{\phi^{F}_{s_{2}}}\right)$, and
the conditionally-averaged vectors (c)
$\left({{\phi^{F}_{r_{1}}}^{(LT)}},{{\phi^{F}_{r_{3}}}^{(LT)}},{{\phi^{F}_{s_{2}}}^{(LT)}}\right)$,
(d)
$\left({{\phi^{F}_{r_{1}}}^{(TL)}},{{\phi^{F}_{r_{3}}}^{(TL)}},{{\phi^{F}_{s_{2}}}^{(TL)}}\right)$,
(e)
$\left({{\phi^{F}_{r_{1}}}^{(TT)}},{{\phi^{F}_{r_{3}}}^{(TT)}},{{\phi^{F}_{s_{2}}}^{(TT)}}\right)$
and (f)
$\left({{\phi^{F}_{r_{1}}}^{(LL)}},{{\phi^{F}_{r_{3}}}^{(LL)}},{{\phi^{F}_{s_{2}}}^{(LL)}}\right)$
at $TR2$. The plots are generated by placing a sphere of radius $5L_{0}$ at
point $(r_{1},r_{3},Y)=(5L_{0},10L_{0},3L_{0})$ and tracing the stream-tubes
crossing the sphere. The stream-tubes are coloured according to the sign of
the first component i.e. ${\phi_{r_{1}}}$, ${\phi^{F}_{r_{1}}}$,
${\phi^{F}_{r_{1}}}^{(LT)}$, ${\phi^{F}_{r_{1}}}^{(TL)}$,
${\phi^{F}_{r_{1}}}^{(TT)}$ and ${\phi^{F}_{r_{1}}}^{(LL)}$ (red for positive,
blue for negative, thus indicating inverse or forward cascade in the $r_{1}$
direction respectively). The colour bars also refer to the value of the first
component (the min/max values are the same to facilitate comparison)

Stream-tubes obtained from the total and the non-linear flux vectors (standard
or conditionally-averaged) are shown in figure 8; see caption for details.
Panel (a) shows the standard-averaged total energy flux vector
$\left(\phi_{r_{1}},\phi_{r_{3}},\phi_{s_{2}}\right)$ that includes the non-
linear, linear, pressure, and viscous components, refer to equation (9). This
plot is very similar to that of panel (b) that shows the non-linear component
$\left(\phi^{F}_{r_{1}},\phi^{F}_{r_{3}},\phi^{F}_{s_{2}}\right)$, the latter
being the dominant component (see Yao et al. (2022) for a detailed discussion
on the other components). Both plots depict a dense cluster of stream-tubes
with energy flowing to larger $r_{1}$ scales (inverse cascade) with energy
originating at $Y\approx 5L_{0}$ and $r_{3}\approx 10L_{0}$. There is milder
inverse cascade in larger $r_{3}$ scales. In another (smaller) cluster, energy
flux vectors rotate and bend towards the Y-axis ($r_{1}=0,r_{3}=0$).

The decomposition (27) allows us to probe in more detail the origin of the
strong inverse cascade in the $r_{1}$ direction and identify the flow events
that determine it. In the homogeneous spanwise direction $TL$ and $LT$ events
can be combined together as $TL+LT$ (see figure 3). However, the streamwise
direction is inhomogeneous and these events have to be considered separately.
As mentioned earlier, for a fixed $(X,Y)$ spatial location and $r_{3}$
separation, a $TL$ event with $r_{1}>0$ takes place across the upstream
laminar/turbulent interface of a spot. On the other hand, an $LT$ event is
taken across the downstream interface, refer to figure 4.

Panels (c) and (d) of figure 8 show the stream-tubes for the conditionally-
averaged flux vectors
$\left({\phi^{F}_{r_{1}}}^{(LT)},{\phi^{F}_{r_{3}}}^{(LT)},{\phi^{F}_{s_{2}}}^{(LT)}\right)$
and
$\left({\phi^{F}_{r_{1}}}^{(TL)},{\phi^{F}_{r_{3}}}^{(TL)},{\phi^{F}_{s_{2}}}^{(TL)}\right)$
respectively. It is very clear that the $LT$ flux vector contributes most
strongly to the inverse cascade; this corresponds to the downstream
laminar/turbulent interface of a spot. Indeed there is a cluster of stream-
tubes whose direction indicates the transfer of energy to larger streamwise
scales, up to $r_{1}\approx 50L_{0}$. The stream-tubes then bend towards
smaller scales, $r_{1},r_{3}\to 0$. On the other hand, $TL$ events that
correspond to energy flux across the upstream interface of a turbulent spot,
also contribute to inverse cascade, but over a shorter streamwise range,
$0<r_{1}<20L_{0}$. This clearly indicates that inter-scale energy transfer
processes are different at the upstream and downstream interfaces of turbulent
spots.

Panels (e) and (f) of figure 8 show the stream-tubes of the conditionally-
averaged flux vectors
$\left({\phi^{F}_{r_{1}}}^{(TT)},{\phi^{F}_{r_{3}}}^{(TT)},{\phi^{F}_{s_{2}}}^{(TT)}\right)$
and
$\left({\phi^{F}_{r_{1}}}^{(LL)},{\phi^{F}_{r_{3}}}^{(LL)},{\phi^{F}_{s_{2}}}^{(LL)}\right)$.
The former corresponds to interscale transfer within spots; there is weak
inverse cascade and the stream-tubes bend towards smaller scales. The latter
corresponds to time instants where both points are located within laminar
regions. There is strong forward cascade to small scales and then bending and
energy transfer to larger scales away from the wall. This is a quite
complicated energy flux pattern, which is difficult to interpret physically.

Figure 9: Flux vectors (a) $\left(\phi_{r_{1}},\phi_{r_{3}}\right)$, (b)
$\left(\phi^{F}_{r_{1}},\phi^{F}_{r_{3}}\right)$ ,(c)
$\left({\phi^{F}_{r_{1}}}^{(LT)},{\phi^{F}_{r_{3}}}^{(LT)}\right)$, (d)
$\left({\phi^{F}_{r_{1}}}^{(TL)},{\phi^{F}_{r_{3}}}^{(TL)}\right)$, (e)
$\left({\phi^{F}_{r_{1}}}^{(TT)},{\phi^{F}_{r_{3}}}^{(TT)}\right)$ and (f)
$\left({\phi^{F}_{r_{1}}}^{(LL)},{\phi^{F}_{r_{3}}}^{(LL)}\right)$ on the
$(r_{1},r_{3})$ plane at wall-normal height $Y=4.5L_{0}$ and location $TR2$.
Contours represent the magnitude of the energy flux vectors in the plane.

Plots in the 3D hyperplane $(r_{1},r_{3},Y)$ visualise the main features of
the energy flux paths, but can hide important detail. To uncover this detail,
in figure 9 we plot the flux vectors in the $(r_{1},r_{3})$ plane at the
specific height $Y=4.5L_{0}$. The total flux vector and the standard-averaged
non-linear component (panels (a) and (b) respectively) are very similar and
show a recirculating pattern with inverse cascade for $r_{3}>10$ over the
range of $r_{1}$ examined and forward cascade for $r_{3}<10L_{0}$. It is very
interesting to see that around $r_{3}=10L_{0}$ the energy flux is negligible;
this cannot be easily observed from figure 8. The conditionally-averaged
fluxes also show detail that cannot be discerned from the 3D plots. For
example, the strong inverse cascade of ${\phi^{F}_{r_{1}}}^{(LT)}$ extends
over the whole range of $r_{3}$ and $r_{1}$, while for
${\phi^{F}_{r_{1}}}^{(TL)}$ it extends only in a specific range of
separations, $r_{3}\approx(5-15)L_{0}$, depending on $r_{1}$, as can be seen
from panels (c) and (d) respectively. The flux component
${\phi^{F}_{r_{1}}}^{(LL)}$ (panel (e)) clearly demonstrates forward cascade
over the whole $r_{3}$ range examined, but ${\phi^{F}_{r_{1}}}^{(TT)}$ (panel
(f)) is very small around $r_{3}\approx 10$ and increases at the boundaries of
the domain. These plots confirm that the strong inverse cascade in the
standard-averaged flux is due to $LT$ events at the downstream interface of
the turbulent spots. Interestingly, the energy fluxes due to $TL$ and $LL$
events almost cancel out around $r_{3}\approx 10L_{0}$, and this explains the
very small fluxes in this area for ${\phi^{F}_{r_{1}}}$. Notice also that the
$TT$ events account for the forward cascade observed in ${\phi^{F}_{r_{1}}}$
and ${\phi_{r_{1}}}$ for small $r_{3}$ separations.

Figure 10: A propagating turbulent spot of diamond shape crossing a fixed $X$
location; the view shown is in the $(X,Z)$ plane. Red, yellow and blue dots
represent the $x_{i}^{-}$, $X_{i}$, and $x_{i}^{+}$ points respectively. Only
points with fixed $r_{3}=\Delta z$ separation are shown.

The above figures have demonstrated the central role of the downstream
laminar/turbulent interface on the interscale transfer and in particular the
inverse cascade over a large range of scales. Additionally, $TL$ events were
localised in a smaller range of spanwise and streamwise separations. We now
try to explain physically this behaviour with the aid of the cartoons shown in
figure 10. More specifically, we consider a fixed $X$ location (for figures 8
and 9, $X=X_{TR2}$) and follow a turbulent spot of diamond shape as it
propagates to the right and crosses this location. The arrowhead shape at the
upstream and downstream ends is a simplified, but rather realistic,
approximation. This can be seen from figure 11 where we demarcate the
boundaries of two turbulent spots. It is also consistent with experimental
spot observations, refer to figures 12 and 14 in Anthony et al. (2005). The
sharp corners of the spot around the maximum thickness are less realistic; the
shape is more rounded in this region as can be seen from figure 18 of Marxen &
Zaki (2019). However, analysis of this simplified shape can provide
significant physical insight, as will be seen next.

In figure 10, snapshots of the propagating spot at three time instants
$t_{1},t_{2}$ and $t_{3}$ are shown. In all snapshots, we consider a fixed
middle point (denoted with a yellow dot) located at the streamwise position,
$X$. The blue and red dots represent the $x_{i}^{+}$ and $x_{i}^{-}$ points
respectively. Only points with a fixed spanwise separation, $r_{3}=\Delta z$,
are shown in the figure. At the time $t=t_{1}$, the downstream apex of the
spot lies exactly at the fixed $X$ location. It can be seen that for an $LT$
event (the only type of event possible at this time instant), the streamwise
separation $r_{1}(t_{1})$ is very long, of the order of the spot length. For
$\Delta z=0$, $r_{1}(t_{1})$ attains a maximum value, equal to twice the spot
length.

Figure 11: Contour plot of instantaneous streamwise velocity fluctuations. The
boundaries of two turbulent spots in the transition region (marked by dashed
yellow lines) indicate spots of approximately diamond shape.

At time instant $t=t_{2}$, approximately half of the spot has crossed $X$. The
valid spanwise locations of the middle point are determined by the spreading
angle of the front apex. Note that for fixed $r_{3}=\Delta z$, $TL$ and $LT$
events co-exist, but it is clear that the $r_{1}(t_{2})$ separation of an $LT$
event is shorter to the one at $t=t_{1}$, i.e. $r_{1}(t_{2})<r_{1}(t_{1})$. At
$t=t_{3}$, the whole spot has crossed the considered location, thus the rear
apex is at $X$. At this time instant, only $TL$ events are possible. It can be
seen that only a narrow range of $r_{1}$ separations is admissible for a given
$r_{3}$. The actual range depends on the spreading angle of the rear apex.
Thus on average, the valid $r_{1}$ separations corresponding to $TL$ events at
$t_{2}$ and $t_{3}$ is more narrow compared to $LT$ events at $t=t_{1}$ and
$t_{2}$. This explains the inverse cascade over a wider range of separations
for $LT$ events shown in panel 9(c). We also conjecture that the largest
admissible $r_{1}$ value mentioned earlier explains why the stream-tubes shown
in figure 8(c) reach up to a maximum $r_{1}$ (the exact value depends on point
of origin of the stream-tubes in the $(r_{3},Y)$ plane) and then bend backward
towards small scales.

Figure 12: PDFs of $\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{1}$
for $AA=$ $LT$, $TL$, $TT$ and $LL$ for $r_{1}=30L_{0}$ and $r_{3}=10L_{0}$ at
$Y=4.5L_{0}$.

To provide further insight into the observed behaviour of the conditionally-
averaged fluxes, we examine in more detail the flux vector component in the
$r_{1}$ direction
${\phi^{F}_{r_{1}}}^{(AA)}=\overline{\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{1}}^{(AA)}$.
In figure 12, the probability density functions (PDFs) of
$\left(du^{\prime(AA)}_{i}\right)^{2}du^{\prime(AA)}_{1}$ for $AA=$ $LT$,
$TL$, $TT$ and $LL$ are plotted for separations $r_{1}=30L_{0}$ and
$r_{3}=10L_{0}$ at plane $Y=4.5L_{0}$. The PDF of the instantaneous flux
$\left(du^{\prime(LT)}_{i}\right)^{2}du^{\prime(LT)}_{1}$ is asymmetric and
skewed to positive values (implying inverse cascade after time-averaging).
This is also the case but it is less evident for
$\left(du^{\prime(TL)}_{i}\right)^{2}du^{\prime(TL)}_{1}$, while
$\left(du^{\prime(TT)}_{i}\right)^{2}du^{\prime(TT)}_{1}$ is almost symmetric,
and $\left(du^{\prime(LL)}_{i}\right)^{2}du^{\prime(LL)}_{1}$ is skewed to the
left. Note the large positive and negative fluctuations of the instantaneous
fluxes compared to the time-average values reported in figure 9. This means
that instantaneously energy flows in either direction, and intense fluxes of
relatively low probability tip the balance in one direction or another after
time-averaging.

##  7 Conditionally-averaged scale energy production and transfer within a
turbulent spot and comparison with fully developed turbulence

The conditional averaging operations defined in section 4.1 allow us to
compute the scale energy production and interscale transfer within a turbulent
spot and compare them with the corresponding quantities in the fully turbulent
region. Similar work has been done for single-point statistics, for example,
the turbulent kinetic energy by Park et al. (2012); Nolan & Zaki (2013);
Marxen & Zaki (2019). To the best of our knowledge, this is the first time
this type of analysis is extended to two-point statistics.

We start by deriving a decomposition similar to (27) for the production term
$\mathcal{P}$ of the KHMH equation (6). This term consists of two components,
one due to inhomogeneity of the mean flow in scale space,
$\mathcal{P}_{r}=-2\overline{du^{\prime}_{i}du^{\prime}_{j}}\frac{\partial
dU_{i}}{\partial r_{j}}$, and the other due to inhomogeneity in physical
space,
$\mathcal{P}_{s}=-2\overline{du^{\prime}_{i}u^{\prime*}_{j}}\frac{\partial
dU_{i}}{\partial X_{j}}$. Here we decompose the former component,
$\mathcal{P}_{r}$, which is the dominant one; this is also discussed below.
Applying a process similar to that described in section 5, we obtain

$\displaystyle\mathcal{P}_{r}=-2\overline{du^{\prime}_{i}du^{\prime}_{j}}\frac{\partial
dU_{i}}{\partial r_{j}}=$ (31)
$\displaystyle-2\left(\gamma^{(TT)}\right)^{2}\overline{du^{\prime(TT)}_{i}du^{\prime(TT)}_{j}}^{(TT)}\frac{\partial
dU_{i}^{(TT)}}{\partial
r_{j}}-2\left(\gamma^{(TL)}\right)^{2}\overline{du^{\prime(TL)}_{i}du^{\prime(TL)}_{j}}^{(TL)}\frac{\partial
dU_{i}^{(TL)}}{\partial r_{j}}$
$\displaystyle-2\left(\gamma^{(LT)}\right)^{2}\overline{du^{\prime(LT)}_{i}du^{\prime(LT)}_{j}}^{(LT)}\frac{\partial
dU_{i}^{(LT)}}{\partial
r_{j}}-2\left(\gamma^{(LL)}\right)^{2}\overline{du^{\prime(LL)}_{i}du^{\prime(LL)}_{j}}^{(LL)}\frac{\partial
dU_{i}^{(LL)}}{\partial r_{j}}$ $\displaystyle-\phi,$

where the additional term $\phi$ is given by

$\displaystyle\phi=$ (32) $\displaystyle
2\left(\gamma^{(TT)}\right)^{2}dU_{i}^{(TT)}dU_{j}^{(TT)}\frac{\partial
dU_{i}^{(TT)}}{\partial
r_{j}}+2\left(\gamma^{(TL)}\right)^{2}dU_{i}^{(TL)}dU_{j}^{(TL)}\frac{\partial
dU_{i}^{(TL)}}{\partial r_{j}}$
$\displaystyle+2\left(\gamma^{(LT)}\right)^{2}dU_{i}^{(LT)}dU_{j}^{(LT)}\frac{\partial
dU_{i}^{(LT)}}{\partial
r_{j}}+2\left(\gamma^{(LL)}\right)^{2}dU_{i}^{(LL)}dU_{j}^{(LL)}\frac{\partial
dU_{i}^{(LL)}}{\partial r_{j}}$
$\displaystyle+2\gamma^{(TT)}\overline{du_{i}du_{j}}^{(TT)}\left[dU_{i}^{(TT)}\frac{\partial\gamma^{(TT)}}{\partial
r_{j}}+\frac{\partial}{\partial
r_{j}}\left(\gamma^{(LL)}dU_{i}^{(LL)}+\gamma^{(TL)}dU_{i}^{(TL)}+\gamma^{(LT)}dU_{i}^{(LT)}\right)\right]$
$\displaystyle+2\gamma^{(TL)}\overline{du_{i}du_{j}}^{(TL)}\left[dU_{i}^{(TL)}\frac{\partial\gamma^{(TL)}}{\partial
r_{j}}+\frac{\partial}{\partial
r_{j}}\left(\gamma^{(LL)}dU_{i}^{(LL)}+\gamma^{(TT)}dU_{i}^{(TT)}+\gamma^{(LT)}dU_{i}^{(LT)}\right)\right]$
$\displaystyle+2\gamma^{(LT)}\overline{du_{i}du_{j}}^{(LT)}\left[dU_{i}^{(LT)}\frac{\partial\gamma^{(LT)}}{\partial
r_{j}}+\frac{\partial}{\partial
r_{j}}\left(\gamma^{(LL)}dU_{i}^{(LL)}+\gamma^{(TL)}dU_{i}^{(TL)}+\gamma^{(TT)}dU_{i}^{(TT)}\right)\right]$
$\displaystyle+2\gamma^{(LL)}\overline{du_{i}du_{j}}^{(LL)}\left[dU_{i}^{(LL)}\frac{\partial\gamma^{(LL)}}{\partial
r_{j}}+\frac{\partial}{\partial
r_{j}}\left(\gamma^{(TT)}dU_{i}^{(TT)}+\gamma^{(TL)}dU_{i}^{(TL)}+\gamma^{(LT)}dU_{i}^{(LT)}\right)\right]$
$\displaystyle-2dU_{i}dU_{j}\frac{\partial dU_{i}}{\partial r_{j}}$

Note that the terms
$\mathcal{P}_{r}^{(AA)}=-2\overline{du^{\prime(AA)}_{i}du^{\prime(AA)}_{j}}^{(AA)}\frac{\partial
dU_{i}^{(AA)}}{\partial r_{j}}$ that appear in (31) are the production terms
of the conditionally-averaged KHMH equation (17). Due to the strong shear in
the wall-normal direction, the dominant component of
$\mathcal{P}_{r}=-2\overline{du^{\prime}_{i}du^{\prime}_{j}}\frac{\partial
dU_{i}}{\partial r_{j}}$ is
$\mathcal{P}_{r(1,2)}=-2\overline{du^{\prime}_{1}du^{\prime}_{2}}\left.\frac{\partial
dU_{1}}{\partial r_{2}}\right|_{r_{2}=0}$. It can be easily proved (see Yao et
al. (2022)) that for $r_{1}=0$ this component is equal to
$-2\overline{du^{\prime}_{1}du^{\prime}_{2}}\frac{\partial U_{1}}{\partial
x_{2}}$. The corresponding production component due to inhomogeneity of the
mean flow in physical space,
$\mathcal{P}_{s}=-2\overline{du^{\prime}_{1}u^{\prime*}_{2}}\frac{\partial
dU_{1}}{\partial X_{2}}$, is much smaller. Therefore in this section we
consider only
$\mathcal{P}_{r(1,2)}=-2\overline{du^{\prime}_{1}du^{\prime}_{2}}\left.\frac{\partial
dU_{1}}{\partial r_{2}}\right|_{r_{2}=0}$ and the corresponding conditionally-
averaged $(TT)$ component
$\mathcal{P}_{r(1,2)}^{(TT)}=-2\overline{du^{\prime(TT)}_{1}du^{\prime(TT)}_{2}}^{(TT)}\left.\frac{\partial
dU_{1}^{(TT)}}{\partial r_{2}}\right|_{r_{2}=0}$ within a turbulent spot. We
focus at location $TR2$ and compare the aforementioned component with
$\mathcal{P}_{r(1,2)}$ evaluated at the fully turbulent region, $TU$.

We also extend the decomposition (27) to the total fluxes in physical and
scales spaces $\boldsymbol{\phi_{s}}$ and $\boldsymbol{\phi_{r}}$, defined in
equations (8) and (9) respectively. The resulting expressions are

$\boldsymbol{\phi_{s}}_{j}=\gamma^{(TT)}\boldsymbol{\phi_{s}}^{(TT)}_{j}+\gamma^{(TL)}\boldsymbol{\phi_{s}}^{(TL)}_{j}+\gamma^{(LT)}\boldsymbol{\phi_{s}}^{(LT)}_{j}+\gamma^{(LL)}\boldsymbol{\phi_{s}}^{(LL)}_{j}+\phi,$
(33)

where

$\displaystyle\boldsymbol{\phi_{s}}_{j}^{(AA)}=\overline{{U^{*}_{j}}^{(AA)}dq^{2^{(AA)}}}^{(AA)}+\overline{{u^{\prime*}_{j}}^{(AA)}dq^{2^{(AA)}}}^{(AA)}+\overline{2du^{\prime(AA)}_{j}dp^{\prime(AA)}}^{(AA)}-\frac{1}{2}\nu\overline{\frac{\partial
dq^{2^{(AA)}}}{\partial X_{j}}}^{(AA)},$ (34)

and

$\boldsymbol{\phi_{r}}_{j}=\gamma^{(TT)}\boldsymbol{\phi_{r}}^{(TT)}_{j}+\gamma^{(TL)}\boldsymbol{\phi_{r}}^{(TL)}_{j}+\gamma^{(LT)}\boldsymbol{\phi_{r}}^{(LT)}_{j}+\gamma^{(LL)}\boldsymbol{\phi_{r}}^{(LL)}_{j}+\phi,$
(35)

where

$\boldsymbol{\phi_{r}}_{j}^{(AA)}=\overline{du^{\prime(AA)}_{j}dq^{2^{(AA)}}}^{(AA)}+\overline{dU_{j}^{(AA)}dq^{2^{(AA)}}}^{(AA)}-2\nu\overline{\frac{\partial
dq^{2^{(AA)}}}{\partial r_{j}}}^{(AA)}.$ (36)

The full expressions, including the remainder terms, are provided in Appendix
B.

###  7.1 Conditionally- and standard-averaged production and fluxes

Figure 13: Contour plots of conditionally-averaged production
$\mathcal{P}_{r(1,2)}^{(TT)}$ at $TR2$ (left), standard-averaged production
$\mathcal{P}_{r(1,2)}$ at $TR2$ (middle) and $TU$ (right) on the $(r_{3},Y)$
plane for $r_{1}=0$. The red and black vertical dotted lines are placed at
$r_{3}=5L_{0}$ and $7.5L_{0}$ respectively. The horizontal purple line
indicates the local boundary layer thickness.

In figure 13, contours of $\mathcal{P}_{r(1,2)}^{(TT)}$ at $TR2$ and of
$\mathcal{P}_{r(1,2)}$ at $TR2$ and $TU$ are plotted in the $(r_{3},Y)$ plane
for $r_{1}=0$. It can be seen that the production peaks within the turbulent
spot and the fully turbulent region are located at approximately the same
spanwise separation and wall-normal height, $r_{3}\approx 5L_{0}$ and
$Y\approx 1.3L_{0}$ respectively. On the other hand, the peak of
$\mathcal{P}_{r(1,2)}$ at $TR2$ is found to be at larger $r_{3}$ separation
and further away from the wall, $r_{3}\approx 7.5L_{0}$ and $Y\approx
4.5L_{0}$. We mark the spanwise scales where the peaks appear, $r_{3}=5L_{0}$
and $7.5L_{0}$, with vertical dotted lines in figure 13, and plot the
variation of the three production terms along these lines in figure 14. Notice
the very close matching of the conditionally-average production
$\mathcal{P}_{r(1,2)}^{(TT)}$ at $TR2$ (dashed line) and the standard-averaged
production $\mathcal{P}_{r(1,2)}$ at $TU$ (solid line) close to the wall (for
$Y\leq 1.3L_{0}$) while further away the two sets deviate. On the other hand,
$\mathcal{P}_{r(1,2)}$ at $TR2$ shows significantly different behaviour even
close to the wall, and of course peaks at a different distance.

Figure 14: Variation of production terms along the wall-normal distance at
$r_{3}/L_{0}=5.0$ and $7.5$. The variation is along the dotted vertical lines
shown in figure 13 that pass through the corresponding production peaks. The
plots are normalised by the value at $r_{3}/L_{0}=5.0$. The solid lines
represent $\mathcal{P}_{r(1,2)}$ at $TU$. The dashed lines denote
$\mathcal{P}_{r(1,2)}^{(TT)}$ and the circles $\mathcal{P}_{r(1,2)}$, both at
location $TR2$.

The left panel of figure 15 shows stream-tubes in the $(r_{1},r_{3},Y)$
hyperplane obtained from the conditionally-averaged total fluxes
$\left({\phi_{r_{1}}^{(TT)}},{\phi_{r_{3}}^{(TT)}},{\phi_{s_{2}}^{(TT)}}\right)$
at $TR2$ together with an isosurface of the conditionally-averaged production.
On the right panel we plot
$\left({{\phi_{r_{1}}}},{{\phi_{r_{3}}}},{{\phi_{s_{2}}}}\right)$ and
production in the fully turbulent region. The latter figure reflects the
dynamics of near-wall turbulence; energy is extracted from the mean flow at
the buffer layer where the production peak is located, then it is transferred
away from the wall and towards larger $r_{1}$ scales before bending back to
smaller scales (dissipation region). This behaviour is related to the self-
sustained turbulence mechanism near the wall, see Cimarelli et al. (2013). A
similar pattern can be discerned in the left panel, but the inverse cascade
and flow of energy away from the wall is over a smaller range of $r_{1}$
separations (up to $r_{1}\approx 10$); the stream-tubes again bend towards
small scales. There are also some deviations between the two plots for larger
$r_{3}$ separations. If the centre of the sphere (used for identifying which
stream-tubes to trace) is placed at smaller $r_{3}$ and the radius is reduced,
the similarity between the two panels is more evident, see figure 16 and
caption for details.

The shorter range of inverse cascade in the $TR2$ location compared to the
$TU$ is probably because the spots are still developing, the merging is not
yet complete, thus they have a smaller footprint in the streamwise direction.
Note also the similarities of the left panels of figures 15 and 16 with the
bottom left panel of figure 8 that shows only the non-linear component of the
flux vector. This similarity confirms that this is the most important
component that determines the overall behaviour.

Figure 15: Stream-tubes of the conditionally-averaged total flux vector
$\left({\phi_{r_{1}}^{(TT)}},{\phi_{r_{3}}^{(TT)}},{\phi_{s_{2}}^{(TT)}}\right)$
at $TR2$ (left) and of the standard-averaged vector
$\left({{\phi_{r_{1}}}},{{\phi_{r_{3}}}},{{\phi_{s_{2}}}}\right)$ at $TU$ in
the 3D ($r_{1},r_{3},Y$) hyper-plane. The stream-tubes are colored according
to the sign of $\phi_{r_{3}}$ (red for positive, blue for negative, thus
indicating inverse and forward cascade respectively). The color bar refers to
the value of $\phi_{r_{3}}$. The plots were generated by placing a sphere of
radius $15L_{0}$ at $(r_{1}=5L_{0},r_{3}=5L_{0},Y=1.3L_{0})$ and tracing the
stream-tubes crossing the sphere. Isosurfaces of the production term with
values $0.9\times\max\left(\mathcal{P}_{r(1,2)}^{(TT)}\right)$(left) and
$0.85\times\max\left(\mathcal{P}_{r(1,2)}\right)$(right) are shown in yellow.
Figure 16: Same as figure 15, but the sphere is now placed at
$(r_{1}=5L_{0},r_{3}=2.5L_{0},Y=1.3L_{0})$ and has smaller radius, $4L_{0}$.
Isosurfaces of the production term with values
$0.95\times\max\left(\mathcal{P}_{r(1,2)}^{(TT)}\right)$(left) and
$0.85\times\max\left(\mathcal{P}_{r(1,2)}\right)$(right) are shown in yellow.

##  8 Conclusions

We apply conditional averaging to study the interscale energy transfer process
during bypass transition. To this end, we define two-point intermittencies and
apply them to decompose the energy fluxes into different components that
depend on the local conditions at the two points used to define the flux; the
points are both within a laminar region or a turbulent spot or straddle the
laminar/turbulent interface. The flux terms are evaluated numerically directly
in the scale space because conditional averaging does not commute with the
spatial derivative operator.

In the $(r_{1},r_{3},Y)$ hyper-plane, strong inverse cascade is found in the
$r_{1}$ direction, due to the non-linear fluxes across the downstream and
upstream boundaries of a spot (head and tail respectively). For the former
boundary, the inverse cascade extends over a larger range of $r_{1}$
separations compared to the latter boundary. We explain this finding by
considering a propagating spot as it passes across a fixed streamwise
location.

We derive also the conditionally-averaged KHMH equation and consider the
production term and the total energy fluxes when both points are located
within a turbulent spot ($TT$ events). We compare with the corresponding terms
in the fully turbulent region and find significant similarities, but also some
differences. In both plots, a cluster of stream-tubes originates from the
production peak and transfers energy to larger scales before bending back to
small scales and the near-wall region. This spiral shape is similar to that
found in the fully turbulent region and in channel flow. However, the extent
of the spiralling motion is confined to smaller separations, probably because
the spots have not fully merged yet. Also, a smaller cluster of stream-tubes
transfers energy in the $r_{1}$ and $r_{3}$ directions, which is not found in
the fully turbulent region.

The conditional averaging approach for two-point statistics developed in the
paper can be applied to other flow configurations that exhibit sharp
interfaces, such as wakes and jets, where a turbulent/non-turbulent interface
separates the irrotational and vortical regions. Important questions remain to
be answered, for example, do the conditionally-averaged statistics exhibit
self-similarity? How does this develop as the jet/wake expands? Research in
this direction is left as future work.

## Acknowledgements

H. Yao acknowledges financial support from the Dept. of Aeronautics, Imperial
College London and the Imperial College-CSC scholarship. The authors also wish
to acknowledge the UK Turbulence Consortium (UKTC) for providing access to the
ARCHER high performance computing facility through EPSRC grant EP/R029326/1.

## Declaration of Interests.

The authors report no conflict of interest.

##  Appendix A Calculation of derivatives of two-point quantities directly in
scale space

In this Appendix, we provide the steps for the numerical evaluation of
derivatives of two-point quantities directly in scale space.

###  A.1 Calculation of $\left.\frac{\partial dU_{i}^{(AA)}}{\partial
r_{2}}\right|_{r_{2}=0}$

  1. 1.

Consider a cell with centroid at distance $Y=Y_{0}$ from the wall. The
distances of the centroids of the cells located above and below are $dy_{1}$
and $dy_{2}$ respectively, refer to figure 17. Recall that the grid is non-
uniform in the wall-normal direction, so $dy_{1}\neq dy_{2}$. Calculate
$dU_{i}^{(AA)}(r_{2}=dy_{1}+dy_{2})=U_{i}^{(AA)}\left(Y_{0}+dy_{1}\right)-U_{i}^{(AA)}\left(Y_{0}-dy_{2}\right)$
i.e. the velocity difference between the red $+/-$ markers in figure 17.

  2. 2.

Calculate
$dU_{i}^{(AA)}\left(r_{2}=-(dy_{1}+dy_{2})\right)=-dU_{i}^{(AA)}(r_{2}=dy_{1}+dy_{2})$
i.e. the velocity difference between the blue $+/-$ markers.

  3. 3.

Use central difference scheme to compute

$\left.\frac{\partial dU_{i}^{(AA)}}{\partial
r_{2}}\right|_{r_{2}=0}=\frac{dU_{i}^{(AA)}(r_{2}=dy_{1}+dy_{2})-dU_{i}^{(AA)}(r_{2}=-(dy_{1}+dy_{2}))}{2(dy_{1}+dy_{2})}$
(37)

and store the value at the midpoint (denoted with a green dot in figure 17).

  4. 4.

Repeat steps (i)-(iii) for cell centroids at different heights $Y_{0}$.

  5. 5.

Because the mesh is non-uniform, the middle point is not located at $Y_{0}$,
so interpolate values of $\frac{\partial dU_{i}^{(AA)}}{\partial
r_{2}}|_{r_{2}=0}$ at midpoints to obtain value at $Y_{0}$.

Figure 17: Sketch for the calculation of $\left.\frac{\partial
dU_{i}^{(AA)}}{\partial r_{2}}\right|_{r_{2}=0}$.

Comparison with evaluation at points $x_{i}^{+}$ and $x_{i}^{-}$ for the
standard-averaged streamwise velocity shows that the results are identical,
refer to figure 18.

Figure 18: Contour plot of $\left.\frac{\partial dU_{1}}{\partial
r_{2}}\right|_{r_{2}=0}$ in the $(r_{3},Y)$ plane. Evaluation directly in
scale space (left) and from $\frac{1}{2}\left({\frac{\partial
U_{1}^{+}}{\partial x_{2}^{+}}}+{\frac{\partial U_{1}^{-}}{\partial
x_{2}^{-}}}\right)$ (right)

###  A.2 Calculation of $\left.\frac{\partial dU_{i}^{(AA)}}{\partial
r_{1}}\right|_{r_{1}=0}$

Figure 19: Sketch for the calculation of $\left.\frac{\partial
dU_{i}^{(AA)}}{\partial r_{1}}\right|_{r_{1}=0}$.

  1. 1.

Calculate $dU_{i}^{(AA)}(r_{3},r_{1}=2\Delta x)$, i.e. the velocity difference
between the red $+/-$ points shown in the left panel of figure 19, at a fixed
height $y=Y_{0}$.

  2. 2.

Calculate $dU_{i}^{(AA)}(r_{3},r_{1}=-2\Delta x)$, i.e. the velocity
difference between the blue $+/-$ points shown in right panel of figure 19, at
a fixed height $y=Y_{0}$.

  3. 3.

Use central difference scheme to compute

$\left.\frac{\partial dU_{i}^{(AA)}}{\partial
r_{1}}\right|_{r_{1}=0}=\frac{dU_{i}^{(AA)}(r_{3},r_{1}=2\Delta
x)-dU_{i}^{(AA)}(r_{3},r_{1}=-2\Delta x)}{2(2\Delta x)}$ (38)

and store the value at the middle point (marked with a green dot in both
panels of figure 19).

  4. 4.

Repeat steps (i)-(iii) at different heights $Y_{0}$.

Schematic 19 shows the four points involved for $\left.\frac{\partial
dU_{i}^{(AA)}}{\partial r_{1}}\right|_{r_{1}=0}$. The process is similar for
the evaluation of $\frac{\partial dU_{i}^{(AA)}}{\partial r_{1}}$ for
different $r_{1}$ values. Comparison with evaluation at points $x_{i}^{+}$ and
$x_{i}^{-}$ for the standard-averaged streamwise velocity shows identical
results, see figure 20.

Figure 20: Contour plot of $\left.\frac{\partial dU_{1}}{\partial
r_{1}}\right|_{r_{1}=0}$ in the $(r_{3},Y)$ plane. Evaluation directly in
scale space (left) and from $\frac{1}{2}\left({\frac{\partial
U_{1}^{+}}{\partial x_{1}^{+}}}+{\frac{\partial U_{1}^{-}}{\partial
x_{1}^{-}}}\right)$ (right).

##  Appendix B Conditional decomposition of total fluxes in scale and physical
spaces

Decomposition of the total flux in scale space

$\boldsymbol{\phi_{r}}_{j}=\gamma^{(TT)}\boldsymbol{\phi_{r}}^{(TT)}_{j}+\gamma^{(TL)}\boldsymbol{\phi_{r}}^{(TL)}_{j}+\gamma^{(LT)}\boldsymbol{\phi_{r}}^{(LT)}_{j}+\gamma^{(LL)}\boldsymbol{\phi_{r}}^{(LL)}_{j}+\phi,$
(39)

where

$\boldsymbol{\phi_{r}}_{j}^{(AA)}=\overline{du^{\prime(AA)}_{j}dq^{2^{(AA)}}}^{(AA)}+\overline{dU_{j}^{(AA)}dq^{2^{(AA)}}}^{(AA)}-2\nu\overline{\frac{\partial
dq^{2^{(AA)}}}{\partial r_{j}}}^{(AA)}$ (40)

$\displaystyle\phi$
$\displaystyle=\gamma^{(TT)}\left(\overline{(du_{i})^{2}}^{(TT)}dU_{j}^{(TT)}+\overline{2du_{i}du_{j}}^{(TT)}dU_{i}^{(TT)}-2dU_{i}^{(TT)}dU_{i}^{(TT)}dU_{j}^{(TT)}\right)$
(41)
$\displaystyle+\gamma^{(TL)}\left(\overline{(du_{i})^{2}}^{(TL)}dU_{j}^{(TL)}+\overline{2du_{i}du_{j}}^{(TL)}dU_{i}^{(TL)}-2dU_{i}^{(TL)}dU_{i}^{(TL)}dU_{j}^{(TL)}\right)$
$\displaystyle+\gamma^{(LT)}\left(\overline{(du_{i})^{2}}^{(LT)}dU_{j}^{(LT)}+\overline{2du_{i}du_{j}}^{(LT)}dU_{i}^{(LT)}-2dU_{i}^{(LT)}dU_{i}^{(LT)}dU_{j}^{(LT)}\right)$
$\displaystyle+\gamma^{(LL)}\left(\overline{(du_{i})^{2}}^{(LL)}dU_{j}^{(LL)}+\overline{2du_{i}du_{j}}^{(LL)}dU_{i}^{(LL)}-2dU_{i}^{(LL)}dU_{i}^{(LL)}dU_{j}^{(LL)}\right)$
$\displaystyle+\gamma^{(TT)}dU_{i}^{(TT)}dU_{i}^{(TT)}dU_{j}+\gamma^{(TL)}dU_{i}^{(TL)}dU_{i}^{(TL)}dU_{j}$
$\displaystyle+\gamma^{(LT)})dU_{i}^{(LT)}dU_{i}^{(LT)}dU_{j}+\gamma^{(LL)}dU_{i}^{(LL)}dU_{i}^{(LL)}dU_{j}$
$\displaystyle-2\nu\gamma^{(TT)}\left(2dU_{i}^{(TT)}\frac{\partial
dU_{i}^{(TT)}}{\partial r_{j}}+2dU_{i}^{(TT)}\overline{\frac{\partial
du_{i}}{\partial r_{j}}}^{(TT)}-\frac{\partial(dU_{i}^{(TT)})^{2}}{\partial
r_{j}}\right)$
$\displaystyle-2\nu\gamma^{(TL)}\left(2dU_{i}^{(TL)}\frac{\partial
dU_{i}^{(TL)}}{\partial r_{j}}+2dU_{i}^{(TL)}\overline{\frac{\partial
du_{i}}{\partial r_{j}}}^{(TL)}-\frac{\partial(dU_{i}^{(TL)})^{2}}{\partial
r_{j}}\right)$
$\displaystyle-2\nu\gamma^{(LT)}\left(2dU_{i}^{(LT)}\frac{\partial
dU_{i}^{(LT)}}{\partial r_{j}}+2dU_{i}^{(LT)}\overline{\frac{\partial
du_{i}}{\partial r_{j}}}^{(LT)}-\frac{\partial(dU_{i}^{(LT)})^{2}}{\partial
r_{j}}\right)$
$\displaystyle-2\nu\gamma^{(LL)}\left(2dU_{i}^{(LL)}\frac{\partial
dU_{i}^{(LL)}}{\partial r_{j}}+2dU_{i}^{(LL)}\overline{\frac{\partial
du_{i}}{\partial r_{j}}}^{(LL)}-\frac{\partial(dU_{i}^{(LL)})^{2}}{\partial
r_{j}}\right)$
$\displaystyle-\overline{(du_{i})^{2}}dU_{j}-\overline{2du_{i}du_{j}}dU_{i}+dU_{i}dU_{i}dU_{j}-2\nu\left(-2dU_{i}\frac{\partial
dU_{i}}{\partial r_{j}}-2dU_{i}\overline{\frac{\partial du_{i}}{\partial
r_{j}}}+\frac{\partial(dU_{i})^{2}}{\partial r_{j}}\right)$

Decomposition of the total flux in physical space

$\boldsymbol{\phi_{s}}_{j}=\gamma^{(TT)}\boldsymbol{\phi_{s}}^{(TT)}_{j}+\gamma^{(TL)}\boldsymbol{\phi_{s}}^{(TL)}_{j}+\gamma^{(LT)}\boldsymbol{\phi_{s}}^{(LT)}_{j}+\gamma^{(LL)}\boldsymbol{\phi_{s}}^{(LL)}_{j}+\phi,$
(42)

where

$\displaystyle\boldsymbol{\phi_{s}}_{j}^{(AA)}=$ (43)
$\displaystyle\overline{{U^{*}_{j}}^{(AA)}dq^{2^{(AA)}}}^{(AA)}+\overline{{u^{\prime*}_{j}}^{(AA)}dq^{2^{(AA)}}}^{(AA)}+\overline{2du^{\prime(AA)}_{j}dp^{\prime(AA)}}^{(AA)}-\frac{1}{2}\nu\overline{\frac{\partial
dq^{2^{(AA)}}}{\partial X_{j}}}^{(AA)}$

$\displaystyle\phi$
$\displaystyle=\gamma^{(TT)}\left(\overline{(du_{i})^{2}}^{(TT)}{U^{*}_{j}}^{(TT)}+\overline{2du_{i}{U^{*}_{j}}}^{(TT)}dU_{i}^{(TT)}-2dU_{i}^{(TT)}dU_{i}^{(TT)}{U^{*}_{j}}^{(TT)}\right)$
(44)
$\displaystyle+\gamma^{(TL)}\left(\overline{(du_{i})^{2}}^{(TL)}{U^{*}_{j}}^{(TL)}+\overline{2du_{i}{U^{*}_{j}}}^{(TL)}dU_{i}^{(TL)}-2dU_{i}^{(TL)}dU_{i}^{(TL)}{U^{*}_{j}}^{(TL)}\right)$
$\displaystyle+\gamma^{(LT)}\left(\overline{(du_{i})^{2}}^{(LT)}{U^{*}_{j}}^{(LT)}+\overline{2du_{i}{U^{*}_{j}}}^{(LT)}dU_{i}^{(LT)}-2dU_{i}^{(LT)}dU_{i}^{(LT)}{U^{*}_{j}}^{(LT)}\right)$
$\displaystyle+\gamma^{(LL)}\left(\overline{(du_{i})^{2}}^{(LL)}{U^{*}_{j}}^{(LL)}+\overline{2du_{i}{U^{*}_{j}}}^{(LL)}dU_{i}^{(LL)}-2dU_{i}^{(LL)}dU_{i}^{(LL)}{U^{*}_{j}}^{(LL)}\right)$
$\displaystyle+\gamma^{(TT)}dU_{i}^{(TT)}dU_{i}^{(TT)}U^{*}_{j}+\gamma^{(TL)}dU_{i}^{(TL)}dU_{i}^{(TL)}U^{*}_{j}$
$\displaystyle+\gamma^{(LT)}dU_{i}^{(LT)}dU_{i}^{(LT)}U^{*}_{j}+\gamma^{(LL)}dU_{i}^{(LL)}dU_{i}^{(LL)}U^{*}_{j}$
$\displaystyle-\frac{1}{2}\nu\gamma^{(TT)}\left(2dU_{i}^{(TT)}\frac{\partial
dU_{i}^{(TT)}}{\partial X_{j}}+2dU_{i}^{(TT)}\overline{\frac{\partial
du_{i}}{\partial X_{j}}}^{(TT)}-\frac{\partial(dU_{i}^{(TT)})^{2}}{\partial
X_{j}}\right)$
$\displaystyle-\frac{1}{2}\nu\gamma^{(TL)}\left(2dU_{i}^{(TL)}\frac{\partial
dU_{i}^{(TL)}}{\partial X_{j}}+2dU_{i}^{(TL)}\overline{\frac{\partial
du_{i}}{\partial X_{j}}}^{(TL)}-\frac{\partial(dU_{i}^{(TL)})^{2}}{\partial
X_{j}}\right)$
$\displaystyle-\frac{1}{2}\nu\gamma^{(LT)}\left(2dU_{i}^{(LT)}\frac{\partial
dU_{i}^{(LT)}}{\partial X_{j}}+2dU_{i}^{(LT)}\overline{\frac{\partial
du_{i}}{\partial X_{j}}}^{(LT)}-\frac{\partial(dU_{i}^{(LT)})^{2}}{\partial
X_{j}}\right)$
$\displaystyle-\frac{1}{2}\nu\gamma^{(LL)}\left(2dU_{i}^{(LL)}\frac{\partial
dU_{i}^{(LL)}}{\partial X_{j}}+2dU_{i}^{(LL)}\overline{\frac{\partial
du_{i}}{\partial X_{j}}}^{(LL)}-\frac{\partial(dU_{i}^{(LL)})^{2}}{\partial
X_{j}}\right)$
$\displaystyle+2\left(\gamma^{(TT)}dU_{i}^{(TT)}dP^{(TT)}+\gamma^{(TL)}dU_{i}^{(TL)}dP^{(TL)}+\gamma^{(LT)}dU_{i}^{(LT)}dP^{(LT)}+\right.$
$\displaystyle\qquad\left.\gamma^{(LL)}dU_{i}^{(LL)}dP^{(LL)}\right)-\frac{1}{2}\nu\left(-2dU_{i}\frac{\partial
dU_{i}}{\partial X_{j}}-2dU_{i}\overline{\frac{\partial du_{i}}{\partial
X_{j}}}+\frac{\partial(dU_{i})^{2}}{\partial X_{j}}\right)$
$\displaystyle-\overline{(du_{i})^{2}}U^{*}_{j}-\overline{2du_{i}U^{*}_{j}}dU_{i}+dU_{i}dU_{i}U^{*}_{j}-2dU_{i}dP$

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