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 $D=|v|+|w|$; see Marxen & Zaki (Reference Marxen and Zaki2019) and Yao et al. (Reference Yao, Alves-Portela and Papadakis2020) 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. (i) 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. (ii) 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. (iii) 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. (iv) 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 upper-case letter denotes the state of point $x_i^+$ and the second the state of $x_i^-$. We 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. All points are located at the same horizontal plane, i.e. $r_2=0$.

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

(4.1a)$$\begin{gather} \gamma^{(TT)}{(X,Y;r_1,r_2,r_3)} \equiv \frac{1}{{\rm \Delta} T L_z} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.1b)$$\begin{gather}\gamma^{(TL)}{(X,Y;r_1,r_2,r_3)} \equiv \frac{1}{{\rm \Delta} T L_z } \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+(1-\tau^{-}) \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.1c)$$\begin{gather}\gamma^{(LT)}{(X,Y;r_1,r_2,r_3)} \equiv \frac{1}{{\rm \Delta} T L_z } \int_0^{{\rm \Delta} T} \int_0^{L_z} (1-\tau^+)\tau^{-} \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.1d)$$\begin{gather}\gamma^{(LL)} {(X,Y;r_1,r_2,r_3)} \equiv \frac{1}{{\rm \Delta} T L_z } \int_0^{{\rm \Delta} T} \int_0^{L_z} (1-\tau^+)(1-\tau^{-}) \, {\rm d} z \, {\rm d} t. \end{gather}$$

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. (Reference Yao, Alves-Portela and Papadakis2020), 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 increment ${\rm d} Q=Q(x_i^+)-Q(x_i^-)$ are defined as

(4.2a)$$\begin{gather} \, \overline{{\rm d} Q}^{(TT)}(X,Y;r_1,{r_2},r_3)\equiv \frac{1}{{\rm \Delta} T L_z \gamma^{(TT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \, {\rm d} Q \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.2b)$$\begin{gather}\overline{{\rm d} Q}^{(TL)}(X,Y;r_1,{r_2},r_3)\equiv \frac{1}{{\rm \Delta} T L_z \gamma^{(TL)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+(1-\tau^{-}) \, {\rm d} Q \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.2c)$$\begin{gather}\overline{{\rm d} Q}^{(LT)}(X,Y;r_1,{r_2},r_3)\equiv \frac{1}{{\rm \Delta} T L_z \gamma^{(LT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} (1-\tau^+)\tau^{-} \, {\rm d} Q \, {\rm d} z \, {\rm d} t, \end{gather}$$

(4.2d)$$\begin{gather}\overline{{\rm d} Q}^{(LL)}(X,Y;r_1,{r_2},r_3)\equiv \frac{1}{{\rm \Delta} T L_z \gamma^{(LL)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} (1-\tau^+)(1-\tau^{-}) \, {\rm d} Q \, {\rm d} z \, {\rm d} t. \end{gather}$$

This means that the standard time average can be decomposed as

(4.3)\begin{align} \overline{{\rm d} Q}(X,Y;r_1,r_2,r_3) & = \frac{1}{{\rm \Delta} T L_z} \int_0^{{\rm \Delta} T} \int_0^{L_z} \, {\rm d} Q \, {\rm d} z \, {\rm d} t \nonumber\\ & = \gamma^{(TT)}\, \overline{{\rm d} Q}^{(TT)}+\gamma^{(TL)}\, \overline{{\rm d} Q}^{(TL)}+ \gamma^{(LT)}\, \overline{{\rm d} Q}^{(LT)}+\gamma^{(LL)}\, \overline{{\rm d} Q}^{(LL)}. \end{align}

The definitions are similar for the conditionally averaged midpoint variable $Q^*=0.5[Q(x_i^+)+Q(x_i^-)]$. Thus,

(4.4)\begin{align} \overline{Q^*}(X,Y;r_1,r_2,r_3) & = \frac{1}{{\rm \Delta} T L_z} \int_0^{{\rm \Delta} T} \int_0^{L_z} Q^* \, {\rm d} z \, {\rm d} t \nonumber\\ & = \gamma^{(TT)} \overline{Q^*}^{(TT)}+\gamma^{(TL)}\overline{Q^*}^{(TL)}+ \gamma^{(LT)} \overline{Q^*}^{(LT)}+\gamma^{(LL)} \overline{Q^*}^{(LL)}. \end{align}

Referring back to figure 3, it is clear that $\overline {{\rm d} Q}^{(TL)}$ and $\overline {{\rm d} Q}^{(LT)}$ are two-point averages across the laminar/turbulent interface. For $r_1=0$ and $r_3\ne 0$ (case shown in figure 3), these two conditional averages are taken across the interface in the spanwise direction. For $r_1>0$ 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 § 6. Note that $\overline {{\rm d} Q}^{(TL)}(r_1,r_2,r_3)=\overline {{\rm d} Q}^{(LT)}(-r_1,-r_2,-r_3)$, but since we consider only positive separations, we need to retain both $\overline {{\rm d} Q}^{(TL)}$ and $\overline {{\rm d} Q}^{(LT)}$.

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 $x_i^-$, (3.1), and define the conditional velocity fluctuation difference as

(4.5)\begin{equation} {\rm d} u_i'^{(AA)} \equiv {\rm d} u_i-{\rm d} U_i^{(AA)},\end{equation}

where $AA =TT$ or $LL$ or $TL$ or $LT$ as mentioned earlier, ${\rm d} u_i=u_i^+-u_i^-$ and ${\rm d} U_i^{(AA)}=\overline {{\rm d} u_i}^{(AA)}$ (from definition (4.2)). It is straightforward to prove that $\overline {{\rm d} u_i'^{(AA)}}^{(AA)}=0$; this is the equivalent of $\overline {{\rm d} u_i'}=0$ in standard averaging.

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

(4.6)\begin{equation} {u'^*_i}^{(AA)} \equiv {u^*_i}-{U^*_i}^{(AA)},\end{equation}

where $u^*_{i}=0.5(u_i^++u_i^-)$ and ${U^*_i}^{(AA)}=\overline {u^*_{i}}^{(AA)}$, and again $\overline {{u'^*_i}^{(AA)}}^{(AA)}=0$.

Subtracting equation (3.1b) from (3.1a), multiplying each term by $2\, {\rm d} u_i'^{(AA)}$ and then applying the $(AA)$ averaging operation as defined in (4.2), we obtain

(4.7)\begin{align} &\underbrace{ \overline{2\, {\rm d} u_i'^{(AA)}\frac{\partial \, {\rm d} u_i}{\partial t}}^{(AA)}}_{\text{Transient term}} +\underbrace{ \overline{2\, {\rm d} u_i'^{(AA)}u^+_j \frac{\partial \, {\rm d} u_i}{\partial x^+_j} + 2\, {\rm d} u_i'^{(AA)}u^-_j \frac{\partial \, {\rm d} u_i}{\partial x^-_j}}^{(AA)}}_{\text{Nonlinear term}} \nonumber\\ &\quad =\underbrace{ \overline{-2\, {\rm d} u_i'^{(AA)}\left(\frac{\partial \, {\rm d} p}{\partial x^+_i} - \frac{\partial \, {\rm d} p}{\partial x^-_i}\right)}^{(AA)} }_{\text{Pressure--velocity correlation}} + \underbrace{ \overline{2 \nu \, {\rm d} u_i'^{(AA)}\frac{\partial^2 \, {\rm d} u_i}{\partial x_j^+ \partial x_j^+} + 2\nu \, {\rm d} u_i'^{(AA)}\frac{\partial^2 \, {\rm d} u_i}{\partial x_j^- \partial x_j^-}}^{(AA)} }_{\text{Viscous term}}, \end{align}

where we have used again ${\partial u^+_i}/{\partial x^-_j}=0$ and ${\partial u^-_i}/{\partial x^+_j}=0$. This is the equivalent of (3.5) for standard averaging.

We now define the two-point conditional energy as $\overline {{\rm d} q^{2^{(AA)}}}^{(AA)} =\overline {{\rm d} u_i'^{(AA)} \, {\rm d} u_i'^{(AA)}}^{(AA)}$ and seek its transport equation in the physical and scale spaces, similar to (3.6). Applying again the variable transformation $X_i=0.5(x^+_i + x^-_i)$ and $r_i = x^+_i - x^-_i$ and the definitions (4.5) and (4.6) into (4.7), after some algebra we get the following conditionally averaged KHMH equation for $\overline {{\rm d} q^{2^{(AA)}}}^{(AA)}$:

(4.8)\begin{align} & \underbrace{\overline{\frac{\partial \, {\rm d} q^{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'^*_j}^{(AA)}\frac{\partial \, {\rm d} q^{2^{(AA)}}}{\partial X_j} }^{(AA)}}_{\text{Turbulent advection}}+ \underbrace{\overline{{\rm d} U_j^{(AA)} \frac{\partial \, {\rm d} q^{2^{(AA)}}}{\partial r_j} }^{(AA)}}_{\text{Linear transfer}}+\underbrace{\overline{{\rm d} u_j'^{(AA)} \frac{\partial \, {\rm d} q^{2^{(AA)}}}{\partial r_j}}^{(AA)} }_{\text{Nonlinear transfer}}\nonumber\\ &\quad =\underbrace{\overline{-2\, {\rm d} u_i'^{(AA)} \frac{\partial \, {\rm d} p'^{(AA)}}{\partial X_i} }^{(AA)}}_{\text{Pressure--velocity correlation}} +\underbrace{\nu \frac{1}{2} \overline{ \frac{\partial^2 \, {\rm d} q^{2^{(AA)}}}{\partial X_j \partial X_j} }^{(AA)}}_{\text{Physical diffusion}}+ \underbrace{2\nu \overline{ \frac{\partial^2 \, {\rm d} q^{2^{(AA)}} }{\partial r_j \partial r_j}}^{(AA)}}_{\text{Scale diffusion}}\nonumber\\ &\qquad -\underbrace{\overline{2\, {\rm d} u_i'^{(AA)} {u'^*_j}^{(AA)}\frac{\partial \, {\rm d} U_i^{(AA)}}{\partial X_j}}^{(AA)} - \overline{2\, {\rm d} u_i'^{(AA)} \, {\rm d} u_j'^{(AA)} \frac{\partial \, {\rm d} U_i^{(AA)}}{\partial r_j}}^{(AA)} }_{\text{Production}}\nonumber\\ & \qquad - \underbrace{2\nu \overline{ \left( \frac{\partial \, {\rm d} u'^{(AA)}_i }{\partial x^+_j}\frac{\partial \, {\rm d} u'^{(AA)}_i }{\partial x^+_j} + \frac{\partial \, {\rm d} u'^{(AA)}_i }{\partial x^-_j}\frac{\partial \, {\rm d} u'^{(AA)}_i }{\partial x^-_j} \right ) }^{(AA)}}_{\text{Dissipation}}. \end{align}

The form of (4.8) is similar to the standard form (3.6). This is due to the appropriate definitions of the conditional fluctuating quantities in (4.5) and (4.6) that satisfy $\overline {{\rm d} u_i'^{(AA)}}^{(AA)}=0$ and $\overline {{u'^*_i}^{(AA)}}^{(AA)}=0$. There is, however, an important difference. Standard averaging commutes with the spatial differentiation operation, for example,

(4.9)\begin{equation} \frac{\partial \, \overline{{\rm d} Q}}{\partial r_j}= \frac{1}{{\rm \Delta} T L_z} \frac{\partial}{\partial r_j} \left( \int_0^{{\rm \Delta} T} \int_0^{L_z} \, {\rm d} Q \, {\rm d} z \, {\rm d} t \right)= \frac{1}{{\rm \Delta} T L_z} \int_0^{{\rm \Delta} T} \int_0^{L_z} \frac{\partial \, {\rm d} Q}{\partial r_j} \, {\rm d} z \, {\rm d} t =\overline{\frac{\partial \,{{\rm d} Q}}{\partial r_j}}. \end{equation}

However, this is not the case for conditionally averaged quantities, for example

(4.10)\begin{equation} \frac{\partial \, \overline{{\rm d} Q}^{(TT)}}{\partial r_j}= \frac{1}{{\rm \Delta} T L_z} \frac{\partial}{\partial r_j} \left( \frac{1}{\gamma^{(TT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \, {\rm d} Q \, {\rm d} z \, {\rm d} t \right), \end{equation}

while

(4.11)\begin{equation} \overline{\frac{\partial \,{{\rm d} Q}}{\partial r_j}}^{(TT)}= \frac{1}{{\rm \Delta} T L_z} \left( \frac{1}{\gamma^{(TT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \frac{\partial \,{{\rm d} Q}}{\partial r_j} \, {\rm d} z \, {\rm d} t \right) \ne \frac{\partial \, \overline{{\rm d} Q}^{(TT)}}{\partial r_j}, \end{equation}

because $\gamma ^{(TT)}$ depends on $r_j$. The same issue appears in the conditionally averaged turbulent kinetic energy equation (Marxen & Zaki Reference Marxen and Zaki2019).

This lack of commutation has two important implications. First, (4.8) cannot be written in conservative form. For example, for the nonlinear interscale energy transfer term, we have

(4.12)\begin{equation} \overline{{\rm d} u_j'^{(AA)} \frac{\partial \, {\rm d} q^{2^{(AA)}}} {\partial r_j}}^{(AA)} = \overline{ \frac{\partial \, {\rm d} q^{2^{(AA)}} \, {\rm d} u_j'^{(AA)}}{\partial r_j}}^{(AA)} \neq \frac{\partial (\overline{{\rm d} q^{2^{(AA)}} \, {\rm d} u_j'^{(AA)}}^{(AA)} )}{\partial r_j}. \end{equation}

The first equality in the above equation is because ${\partial {\, {\rm d} u_j'^{(AA)}}}/{\partial r_j}=0$ (this is equivalent to ${\partial \, \overline {{\rm d} u_j'}}/{\partial r_j}=0$ for standard averaging). It is easy to prove; for example if $AA=TT$, we have

(4.13)\begin{align} \frac{\partial \, {\rm d} u_j'^{(TT)}}{\partial r_j}&= \frac{\partial \, {\rm d} u_j}{\partial r_j}-\frac{\partial \, {\rm d} U_j^{(TT)}}{\partial r_j}=\frac{\partial \, {\rm d} u_j}{\partial r_j}- \frac{\partial}{\partial r_j} \left( \frac{1}{{{\rm \Delta} T L_z} \gamma^{(TT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \,{\rm d} U_j \, {\rm d} z \, {\rm d} t \right)\nonumber\\ & =\frac{\partial \, {\rm d} u_j}{\partial r_j}- \frac{\partial}{\partial r_j} \left( \frac{{\rm d} U_j }{{\rm \Delta} T L_z \gamma^{(TT)}} \int_0^{{\rm \Delta} T} \int_0^{L_z} \tau^+\tau^{-} \, {\rm d} z \, {\rm d} t \right)\nonumber\\ &= \frac{\partial \, {\rm d} u_j}{\partial r_j}-\frac{\partial \, {\rm d} U_j}{\partial r_j}=0-0=0, \end{align}

because ${\rm d} u_j$ and ${\rm d} U_j$ satisfy the continuity equation in the scale space. We also took into account that ${\rm d} U_j$ is constant with respect to $Z$ and $t$ and used (4.1a).

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

(4.14)\begin{align} \frac{\partial \, {\rm d} U_i }{\partial r_j}&= \frac{\partial \, \overline{{\rm d} u_i} }{\partial r_j} =\overline{\frac{\partial {\, {\rm d} u_i} }{\partial r_j}} =\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)\nonumber\\ &=\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). \end{align}

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

(4.15)\begin{align} \frac{\partial \, {\rm d} U_i^{(AA)} }{\partial r_j}&=\frac{\partial \, \overline{{\rm d} u_i}^{(AA)} }{\partial r_j} \ne \overline{\frac{\partial {\, {\rm d} u_i} }{\partial r_j}}^{(AA)} \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. \nonumber\\ & =\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]. \end{align}

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.

6.1. Fluxes on $(r_3,Y)$ plane

Figure 6 shows the conditionally averaged energy fluxes $( {{\phi ^F_{r_3}}^{(AA)}},{{\phi ^F_{s_2}}^{(AA)}} )$, where $AA = TT$ or $TL+LT$ or $LL$ at locations $TR1,TR2,TR3$. The last row (marked as ‘Total’) depicts $( {{\phi ^F_{r_3}}},{{\phi ^F_{s_2}}} )$. In all plots, $r_1=0$ and $r_2=0$.

Figure 6.

Conditionally averaged flux vectors $({{\phi ^F_{r_3}}^{(AA)}},{{\phi ^F_{s_2}}^{(AA)}} )$ (a–i) and standard-averaged flux vector $( {{\phi ^F_{r_3}}},{{\phi ^F_{s_2}}} )$ (j–l) at locations $TR1$ (a,d,g,j), $TR2$ (b,e,h,k) and $TR3$ (c, f,i,l). The contours represent the magnitude of the corresponding flux vectors and the purple horizontal lines indicate the local boundary layer thickness.

At the early stages of transition, at location $TR1$ (left column), the flux vector $( {{\phi ^F_{r_3}}^{(LL)}},{{\phi ^F_{s_2}}^{(LL)}} )$ 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 $( {{\phi ^F_{r_3}}^{(TT)}}, {{\phi ^F_{s_2}}^{(TT)}} )$ and $( {{\phi ^F_{r_3}}^{(TL)}}, {{\phi ^F_{s_2}}^{(TL)}} )+( {{\phi ^F_{r_3}}^{(LT)}},{{\phi ^F_{s_2}}^{(LT)}} )$ 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, $({{\phi ^F_{r_3}}^{(TT)}}, {{\phi ^F_{s_2}}^{(TT)}} )$, compared with 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 $TL$ and $LT$ 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 with 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 conditionally 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 $TL$ and $LT$ events again give rise to inverse cascade. Interestingly, as intermittency increases, $TT$ events show stronger inverse cascade. For comparison, the flux vector $( \phi ^F_{r_3}, \phi ^F_{s_2} )$ 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.

Figure 7.

Standard-averaged flux vector $( {{\phi ^F_{r_3}}},{{\phi ^F_{s_2}}} )$ at $TU$ location. The contour represents the magnitude of the flux vectors.

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 $TL$ and $LT$ events. Indeed, in the $TR2$ location their magnitude is 5–6 times larger compared with $LL$ events (that also contribute to inverse cascade), while in $TR3$ it is 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 $({\phi ^F_{r_1}}^{(AA)}, {\phi ^F_{r_3}}^{(AA)},{\phi ^F_{s_2}}^{(AA)})$. We select location $TR2$, which is in the middle of the transition region, and has single-point intermittency of about 0.4 (see figure 1).

Stream-tubes obtained from the total and the nonlinear flux vectors (standard- or conditionally averaged) are shown in figure 8; see caption for details. Figure 8(a) shows the standard-averaged total energy flux vector $(\phi _{r_1},\phi _{r_3},\phi _{s_2})$ that includes the nonlinear, linear, pressure and viscous components (see (3.9)). This plot is very similar to that of figure 8(b) that shows the nonlinear component $(\phi ^F_{r_1}, \phi ^F_{r_3}, \phi ^F_{s_2})$, the latter being the dominant component (see Yao et al. (Reference Yao, Mollicone and Papadakis2022) 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 ($r_1=0,r_3=0$).

Figure 8.

Stream-tubes of (a) the total flux vector $({{\phi _{r_1}}},{{\phi _{r_3}}},{{\phi _{s_2}}})$, (b) the standard-averaged nonlinear flux vector $({\phi ^F_{r_1}}, {\phi ^F_{r_3}}, {\phi ^F_{s_2}})$, and the conditionally averaged vectors (c) $({{\phi ^F_{r_1}}^{(LT)}}, {{\phi ^F_{r_3}}^{(LT)}},{{\phi ^F_{s_2}}^{(LT)}})$, (d) $({{\phi ^F_{r_1}}^{(TL)}}, {{\phi ^F_{r_3}}^{(TL)}},{{\phi ^F_{s_2}}^{(TL)}})$, (e) $({{\phi ^F_{r_1}}^{(TT)}}, {{\phi ^F_{r_3}}^{(TT)}},{{\phi ^F_{s_2}}^{(TT)}})$ and ( f) $({{\phi ^F_{r_1}}^{(LL)}}, {{\phi ^F_{r_3}}^{(LL)}},{{\phi ^F_{s_2}}^{(LL)}})$ 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 minimum/maximum values are the same to facilitate comparison).

The decomposition (5.3) 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 (see figure 4).

Figures 8(c) and 8(d) show the stream-tubes for the conditionally averaged flux vectors $({\phi ^F_{r_1}}^{(LT)}, {\phi ^F_{r_3}}^{(LT)},{\phi ^F_{s_2}}^{(LT)})$ and $({\phi ^F_{r_1}}^{(TL)}, {\phi ^F_{r_3}}^{(TL)},{\phi ^F_{s_2}}^{(TL)})$ 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 interscale energy transfer processes are different at the upstream and downstream interfaces of turbulent spots.

Figures 8(e) and 8( f) show the stream-tubes of the conditionally averaged flux vectors $({\phi ^F_{r_1}}^{(TT)}, {\phi ^F_{r_3}}^{(TT)},{\phi ^F_{s_2}}^{(TT)})$ and $({\phi ^F_{r_1}}^{(LL)}, {\phi ^F_{r_3}}^{(LL)},{\phi ^F_{s_2}}^{(LL)})$. 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.

Plots in the three-dimensional 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 nonlinear component (figures 9a and 9b 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 three-dimensional 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 figures 9(c) and 9(d) respectively. The flux component ${\phi ^F_{r_1}}^{(LL)}$ (figure 9e) clearly demonstrates forward cascade over the whole $r_3$ range examined, but ${\phi ^F_{r_1}}^{(TT)}$ (figure 9f) 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 9.

Flux vectors (a) $(\phi _{r_1},\phi _{r_3})$, (b) $(\phi ^F_{r_1}, \phi ^F_{r_3})$, (c) $({\phi ^F_{r_1}}^{(LT)}, {\phi ^F_{r_3}}^{(LT)})$, (d) $({\phi ^F_{r_1}}^{(TL)}, {\phi ^F_{r_3}}^{(TL)})$, (e) $({\phi ^F_{r_1}}^{(TT)}, {\phi ^F_{r_3}}^{(TT)})$ and ( f) $({\phi ^F_{r_1}}^{(LL)}, {\phi ^F_{r_3}}^{(LL)})$ 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.

The above figures have demonstrated the central role of the downstream laminar/turbulent interface in 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 in figure 10 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 (see figures 12 and 14 in Anthony, Jones & Lagraff (Reference Anthony, Jones and Lagraff2005)). 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 (Reference Marxen and Zaki2019). However, analysis of this simplified shape can provide significant physical insight, as is seen next.

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 circles represent the $x_i^-$, $X_i$ and $x_i^+$ points respectively. Only points with fixed $r_3={\rm \Delta} z$ separation are shown.

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.

A compilation of reported leading and trailing edge propagation speeds suggests typical values around $0.9 U_\infty$ and $0.5 U_\infty$ respectively (see Fransson Reference Fransson2010). The fact that the the leading edge propagates faster results in the spot growth in the streamwise direction and is consistent with the dominant role of the downstream laminar/turbulent interface in inverse cascade. This explains the direction of the nonlinear energy fluxes shown in figure 8, but does not explain the difference in the streamwise and spanwise separations over which TL and LT events are active.

The underlying cause for this difference can be elucidated with the aid of figure 10 that depicts snapshots of the propagating spot at three time instants $t_1, t_2$ and $t_3$. In all snapshots, we consider a fixed middle point (denoted with a yellow circle) located at the streamwise position, $X$. The blue and red circles represent the $x_i^+$ and $x_i^-$ points respectively. Only points with a fixed spanwise separation, $r_3={\rm \Delta} z$, are shown in the figure. At 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 ${\rm \Delta} z=0$, $r_1(t_1)$ attains a maximum value, equal to twice the spot length.

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 $r_3={\rm \Delta} z$, $TL$ and $LT$ events coexist, but it is clear that the $r_1(t_2)$ separation of an $LT$ event is shorter than 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 $t_3$ are more narrow compared with $LT$ events at $t=t_1$ and $t_2$. This explains the inverse cascade over a wider range of separations for $LT$ events shown in figure 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.

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 ${\phi ^F_{r_1}}^{(AA)}=\overline {({\rm d} u'^{(AA)}_i)^2\, {\rm d} u'^{(AA)}_1}^{(AA)}$. In figure 12, the probability density functions (PDFs) of $({\rm d} u'^{(AA)}_i)^2\, {\rm d} u'^{(AA)}_1$ 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 $({\rm d} u'^{(LT)}_i)^2\, {\rm d} u'^{(LT)}_1$ is asymmetric and skewed to positive values (implying more intense instantaneous inverse cascade events than forward cascade events). This is also the case but it is less evident for $({\rm d} u'^{(TL)}_i)^2\, {\rm d} u'^{(TL)}_1$, while $({\rm d} u'^{(TT)}_i)^2\, {\rm d} u'^{(TT)}_1$ is almost symmetric and $({\rm d} u'^{(LL)}_i)^2\, {\rm d} u'^{(LL)}_1$ is skewed to the left. Note the large positive and negative fluctuations of the instantaneous fluxes compared with 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.

Figure 12.

The PDFs of $({\rm d} u'^{(AA)}_i)^2\, {\rm d} u'^{(AA)}_1$ for $AA= LT$, $TL$, $TT$ and $LL$ for $r_1=30L_0$ and $r_3=10L_0$ at $Y=4.5L_0$.

7.1. Conditionally and standard-averaged production and fluxes

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 averaged 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 $Y\le 1.3L_0$) 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 13.

Contour plots of conditionally averaged production $\mathcal {P}_{r(1,2)}^{(TT)}$ at $TR2$ (a), standard-averaged production $\mathcal {P}_{r(1,2)}$ at $TR2$ (b) and $TU$ (c) 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.

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 $TR2$.

Figure 15(a) shows stream-tubes in the $(r_1,r_3,Y)$ hyperplane obtained from the conditionally averaged total fluxes $({\phi _{r_1}^{(TT)}}, {\phi _{r_3}^{(TT)}},{\phi _{s_2}^{(TT)}})$ at $TR2$ together with an isosurface of the conditionally averaged production. In figure 15(b) we plot $({{\phi _{r_1}}},{{\phi _{r_3}}},{{\phi _{s_2}}})$ 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. Reference Cimarelli, De Angelis and Casciola2013). A similar pattern can be discerned in figure 15(a), 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).

Figure 15.

Stream-tubes of the conditionally averaged total flux vector $({\phi _{r_1}^{(TT)}}, {\phi _{r_3}^{(TT)}},{\phi _{s_2}^{(TT)}})$ at $TR2$ (a) and of the standard-averaged vector $({{\phi _{r_1}}},{{\phi _{r_3}}},{{\phi _{s_2}}})$ at $TU$ (b) in the three-dimensional ($r_1,r_3,Y$) hyperplane. The stream-tubes are coloured according to the sign of $\phi _{r_3}$ (red for positive, blue for negative, thus indicating inverse and forward cascade respectively). The colour 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 $0.9\times \max (\mathcal {P}_{r(1,2)}^{(TT)})$ (a) and $0.85\times \max (\mathcal {P}_{r(1,2)})$ (b) 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 $0.95\times \max (\mathcal {P}_{r(1,2)}^{(TT)})$ (a) and $0.85\times \max (\mathcal {P}_{r(1,2)})$ (b) are shown in yellow.

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