4.1. Analysis of the leading edge

The abrupt change of boundary conditions taking place at $x/\delta = 0$ generates enduring non-equilibrium effects which, as will be seen more in detail in § 4.2, do not vanish completely even at the distance of $x/\delta = 18{\rm \pi}$. However, there is a portion of the channel where these effects are so strong that the flow has a behaviour of its own that needs to be carefully investigated. This region is the leading edge, where the temperature gradient is the most elevated, as a consequence of the sudden change of thermal wall conditions (see the evolution of $Nu$ and $B_q$ in figure 5). Since the temperature is treated as an active scalar in our simulation, an impact on the velocity fields and momentum balance is expected. On the other hand, in this region the thermal boundary layer is too thin for any appreciable phenomenon to be observed on the temperature.

Figure 9 shows the mean streamwise velocity for $x/\delta$ ranging between $[-0.06, 0.18]$ compared with the canonical profiles seen in § 3. Even if semi-local scaling is used, none of the profiles agree with the equilibrium ones, showing that, at these close distances from the leading edge, the perturbation cannot be ascribed to the only variation of mean fluid properties. This is particularly true for the profile at $x/\delta = 0.015$, for which the velocity boundary layer seems to be at equilibrium only for $y^* \lesssim 2$, i.e. in the viscous sublayer. The discrepancy of the profile at $x/\delta = - 0.06$ with respect to the equilibrium profile of § 3.1, instead, shows how the perturbation slightly propagates upstream.

Figure 9.

Mean streamwise velocity profile for different $x/\delta$ in semi-local scaling: ____ present results at $x/\delta = -0.06$, 0.015, 0.09, 0.18 (from lightest to darkest, respectively); $\cdots \cdots$ equilibrium profile of § 3.1; – $\cdot$ – $\cdot$ – equilibrium profile of § 3.2.

A similar impact on the streamwise velocity has been observed by Sanchez et al. (Reference Sanchez, Aulery, Toutant and Françoise2014) with a higher $B_q$, arguing that the perturbation should be ascribed to the increase of the wall-normal velocity induced by the temperature gradient. It is certainly true that the wall-normal velocity considerably increases near the leading edge, as it can be seen in figure 10. Yet, the wall-normal velocity is also important at $x/\delta = - 0.06$, and, above all, always greater than zero. If the wall-normal velocity were only responsible for the deviation of the streamwise velocity, the impact would be expected to be similar to the one observed downstream of the leading edge, where the normal velocity is also always positive and of the same order of magnitude. Instead, as can be seen in figure 9, it is not the case, since an upper and lower shift is respectively observed at $x/\delta = -0.06$ and, for example, $x/\delta = 0.18$.

Figure 10.

Wall-normal velocity profile for different $x/\delta$ in semi-local scaling (a) and normalised with the mean streamwise velocity (b): ____ present results at $x/\delta = 0.015$, 0.18, 0.73 (from lightest to darkest); – $\cdot$ – $\cdot$ – equilibrium profile.

Hence, we propose a different interpretation, i.e.  that the destabilisation of the boundary layer is due to the abrupt variation of the wall shear stress. Figure 11(a) shows the evolution of the skin friction coefficient $C_f = \overline {\tau _w} / ( \frac {1}{2} \rho _b u_b^2 )$, where $\rho _b$ and $u_b$ are respectively the bulk density and the bulk velocity. At $x/\delta = 0^+$, the skin friction coefficient is at its maximum, which can be explained by the abrupt increase of the fluid dynamic viscosity at the leading edge of the isothermal wall. Then, the following relaxation and adaptation of the velocity gradient at the wall to the increased viscosity, leads to a quickly decreasing $C_f$ until it attains a plateau. At $x/\delta = 0^-$, instead, the skin friction coefficient is at its minimum. This is due to the fact that the flow, being subsonic, perceives the upcoming isothermal wall, and adapts by decreasing the gradient of streamwise velocity at the wall, while the dynamic viscosity remains constant since the wall is adiabatic. This perturbation, limited to a very small portion of the channel flow between $x/\delta \in [-1/2{\rm \pi}, 1/2{\rm \pi} ]$, results in a local perturbation of the wall streamwise pressure gradient, as can be seen in figure 11(b). While $({\partial \bar {p}}/{\partial x})|_w / ({\overline {\tau _w}}/{\delta }) \approx -1$ far from the leading edge, a positive (and, thus, adverse) and strongly negative (and, thus, favourable) pressure gradient is observed at $x/\delta = 0^-$ and $x/\delta = 0^+$, respectively.

Figure 11.

Skin friction coefficient $C_f$ (a) and non-dimensionalised wall pressure gradient (b) along the channel flow.

In order to take the effect of the streamwise pressure gradient into account, we introduce, following Simpson (Reference Simpson1983), the velocity scale $u_p$ defined as

(4.1)\begin{equation} u_p = \left|\left.\frac{\mu}{\rho^2} \frac{\partial p}{\partial x}\right|_w \right|^{{1}/{3}}, \end{equation}

which, combined with the classic friction velocity $u_{\tau } = | \tau _w / \rho |^{{1}/{2}}$, gives as proposed by Manhart, Peller & Brun (Reference Manhart, Peller and Brun2008),

(4.2)\begin{equation} u_{\tau p} = \sqrt{u_{\tau}^2 + u_{p}^2} . \end{equation}

We propose here a slight modification of this definition, which distinguishes positive and negative pressure gradients,

(4.3)\begin{equation} u_{\tau p} = \sqrt{u_{\tau}^2 + \rm{sign} \left( \left.\frac{\partial p}{\partial x}\right|_w \right)u_{p}^2} . \end{equation}

In this manner, the decrease of $u_{\tau }$ in case of an adverse pressure gradient, and its increase in case of a favourable pressure gradient, are compensated by $u_p$, correcting the upward and downward deviation of the streamwise velocity at $x/\delta = 0^-$ and $x/\delta = 0^+$, respectively.

We can thus introduce the wall scaling

(4.4a,b)\begin{equation} y^*_{\tau p} = \frac{y u_{\tau p}}{\nu}, \quad u^*_{\tau p} = \frac{u}{u_{\tau p}} , \end{equation}

where, analogously to semi-local scaling, fluid properties depend on the wall-normal direction in order to take the effect of the temperature gradient into account.

Figure 12(a) shows the same velocity profiles of figure 9 in the newly introduced wall scaling, while figure 12(b) illustrates the r.m.s. velocity profiles. Very good agreement is obtained between the different profiles in every region of the boundary layer, with the only exception of the r.m.s. streamwise velocity, for which the upward shift in the log layer has in any case to be ascribed to what is explained in § 3.2.

Figure 12.

Mean streamwise velocity profile for different $x/\delta$ in the newly introduced wall scaling (a). See figure 9 for a reference to lines and colours. Profiles of r.m.s. streamwise, wall-normal and spanwise velocities, respectively (b): ____, – $\cdot$ – $\cdot$ – and - - - - present results at $x/\delta = -0.06$, 0.015, 0.09, 0.18 and equilibrium (see § 3).

The fact that the velocity profiles are corrected by adopting a different wall scaling, suggests that, in our case, the perturbation is confined in the near-wall region and does not propagate towards the core of the flow. In the most general case, however, the leading edge effects are expected to become more important as $B_q$ increases, with, in particular, larger wall-normal velocities and stronger variations of the pressure gradient. In these conditions, a modified wall scaling might not suffice to correct these effects.

In order to corroborate our interpretation of the leading edge perturbations, the evolution of the different contributions to the momentum balance is analysed. For every crosswise section, integrating along the wall-normal direction the streamwise momentum balance yields

(4.5)\begin{align} \bar{\tau}_w(x) & ={-} \underbrace{\int_0^y \left(\bar{\rho} \tilde{u}\frac{\partial \tilde{u}}{\partial x}\right) \mathrm{d} y}_{I_x} - \underbrace{\int_0^y \left(\bar{\rho} \tilde{v}\frac{\partial \tilde{u} }{\partial y}\right) \mathrm{d} y}_{I_y} - \underbrace{\int_0^y \left( \frac{\partial \bar{p}}{\partial x}\right) \mathrm{d} y}_{II} \nonumber\\ &\quad + \underbrace{\int_0^y \left ( \frac{\partial \overline{\boldsymbol{\tau}_{xx}}}{\partial x}\right) \mathrm{d} y}_{III_x} + \underbrace{\overline{\boldsymbol{\tau}_{xy}}(y)}_{III_y} - \underbrace{\int_0^y \left(\frac{\partial }{\partial x}(\bar{\rho} \widetilde{u''u''} )\right)\mathrm{d} y}_{IV_x} - \underbrace{\bar{\rho} \widetilde{u''v''}(y)}_{IV_y} , \end{align}

where on the right-hand side of the equation there are two mean convective terms ($I_x$ and $I_y$), the pressure gradient ($II$), two viscous terms ($III_x$ and $III_y$) and two terms associated with turbulent transport ($IV_x$ and $IV_y$). Figure 13 shows the different terms non-dimensionalised with respect to the average local wall shear stress $\overline {\tau _w}$ as a function of $y/\delta$ for two values of $x/\delta$, slightly upstream and downstream of $x/\delta = 0$.

Figure 13.

Momentum flux balance at $x/\delta = -0.06$ (a) and $0.18$ (b): ____ pressure gradient; - - - - streamwise (grey) and wall-normal (black) viscous terms; $\cdots \cdots$ streamwise (grey) and wall-normal (black) mean convective terms; – $\cdot$ – $\cdot$ – streamwise (grey) and wall-normal (black) turbulent terms.

Figure 13(a) is relative to $x/\delta = -0.06$. At this close distance from the leading edge, the terms of (4.5) are strongly perturbed. The pressure gradient flux does not follow a straight line, showing that $\partial \bar {p} / \partial x$ is not uniform along the wall-normal direction and that the pressure field is bidimensional; it is also evident how the streamwise pressure gradient is positive (and, thus, adverse) until $y/\delta \approx 0.05$. Both mean convective terms are important; note the positive slope of the streamwise convective term near the wall, indicating that $\partial \tilde {u} / \partial x$ is negative and resulting in the aforementioned decrease of $\tau _w$. The remaining terms seem not to be significantly modified, indicating that the perturbation of the pressure gradient is fully compensated by the two convective terms. This is in agreement with the hypothesis that the non-equilibrium contributions to the momentum balance tend to self-compensate (see Larsson et al. Reference Larsson, Kawai, Bodart and Bermejo-Moreno2016).

Figure 13(b) is relative to $x/\delta = 0.18$. The pressure gradient flux still does not follow a straight line and the pressure field is thus two dimensional; the positive slope of the pressure gradient term indicates that $\partial \bar {p} / \partial x$ is now negative for every $y / \delta$. Both mean convective terms are still important; near the wall, like at $x/\delta = -0.06$, $\partial \tilde {u} / \partial x < 0$, indicating that $\tau _w$ is decreasing. The perturbation of the momentum fluxes becomes negligible at $x/\delta \approx {\rm \pi}/2$, and completely disappears by $x/\delta \approx 3{\rm \pi}$, where the canonic momentum fluxes are finally retrieved, and the effects of the leading edge can be considered fully dissipated (not shown here).

Note that the results described in this section can only be observed if the temperature is handled as an active scalar, i.e. if the momentum and energy equations are coupled.

4.2. Identification of an equilibrium sublayer

Observe the behaviour of the instantaneous temperature isosurface illustrated in figure 7. Near the leading edge of the isothermal wall, the thermal layer is extremely thin, and the heated pockets are entirely located in the purely shear region of the flow. As the distance increases, these pockets thicken, breach through the viscous region and gradually become turbulent, until a fully developed state is attained at the outlet. The goal of this section is to describe the development of the thermal boundary layer from the leading edge to the outlet, and to identify, thanks to the mean energy balance, the different non-equilibrium regions of the flow.

The progressive development of the thermal boundary layer is shown in figure 14, where several mean temperature profiles are plotted at different $x/\delta$, and compared with the equilibrium profile of § 3.2. For every $x/\delta$, three distinct parts of the developing thermal boundary layer can be clearly identified. The first is the near-wall region where, for $y^* \in [0, h_{eq}^*]$, where $h_{eq}^*$ depends on and increases with $x/\delta$, the thermal boundary layer is fully developed and the mean temperature profile agrees with the one at equilibrium; note that this equilibrated behaviour is visible even at $x/\delta = 0.015$, where the leading edge effects described in § 4.1 are very much present. The second, for $y^* \in [h_{n-eq}^*, \delta ^{*}]$, where $h_{n-eq}^*$ also increases with $x/\delta$, is a region that is still not affected by the isothermal wall and the temperature profile is flat. The third, for $y^{*} \in [h_{eq}^{*}, h_{n-eq}^{*}]$, is instead the actual non-equilibrium developing portion of the thermal boundary layer, where the mean temperature profile is neither flat nor agreeing with the equilibrium one. As $x/\delta$ increases, both $h_{eq}^{*}$ and $h_{n-eq}^{*}$ tend towards the mid-height of the channel $\delta ^{*}$, and the non-equilibrium region disappears.

Figure 14.

Profiles of mean temperature for different $x/\delta$: ____ present results at $x/\delta = 0.015$, 0.21, $\frac {1}{2}{\rm \pi}$, $\frac {3}{2}{\rm \pi}$, $3.6{\rm \pi}$, $7.75{\rm \pi}$, $17.9{\rm \pi}$ (from lighter to darker); – $\cdot$ – $\cdot$ – equilibrium profile of § 3.2.

The existence of an equilibrium region in the development of the thermal boundary layer has been shown experimentally, for example, by Blom (Reference Blom1970) (who called this zone the adapted region) or Teitel & Antonia (Reference Teitel and Antonia1993). The objective is to analyse the evolution of the equilibrium layer (delimited by $h_{eq}^{*}$) and of the non-equilibrium region (delimited by $h_{n-eq}^{*}$), yet it is first necessary to quantitatively define the notions of equilibrium and non-equilibrium.

We propose to do so through the analysis of the different terms of the energy balance. The integration along the wall-normal direction of the local averaged energy equation gives a local heat budget for every crosswise section, i.e.

(4.6)\begin{align} \bar{q}_w(x) & = \underbrace{\int_0^y \left ( \bar{\rho} \tilde{u}\frac{\partial \tilde{h}}{\partial x} \right) \mathrm{d} y}_{I_x} + \underbrace{\int_0^y \left ( \bar{\rho} \tilde{v} \frac{\partial\tilde{h}}{\partial y} \right) \mathrm{d} y}_{I_y} + \underbrace{ \int_0^y \left( \frac{\partial \overline{q_x^{cd}}}{\partial x} \right) \mathrm{d} y}_{II_x} + \underbrace{ q_y^{cd}}_{II_y} \nonumber\\ &\quad + \underbrace{\int_0^y \left ( \frac{\partial }{\partial x} ( \bar{\rho} \widetilde{u''h''} ) \right) \mathrm{d} y}_{III_x} + \underbrace{\bar{\rho} \widetilde{v''h''}}_{III_y} - \underbrace{\int_0^y \left ( \frac{\mathrm{D} \bar{p}}{\mathrm{D} t} + \overline{\boldsymbol{\tau} : \nabla \boldsymbol{v} } \right)\mathrm{d} y}_{IV} - \underbrace{y S_{ener}}_{V} , \end{align}

where on the right-hand side of the equation there are two mean convective terms ($I_x$ and $I_y$), two mean conductive terms ($II_x$ and $II_y$), two terms associated with turbulent heat transport ($III_x$ and $III_y$), one flux combining compressibility effects ($IV$) and the source term contribution ($V$). The evolution of these terms, normalised by $\bar {q}_w$, is shown for several $x/\delta$ in figure 15.

Figure 15.

Energy flux balance at $x/\delta = 0.73$ (a), $2{\rm \pi}$ (b), $5.7{\rm \pi}$ (c) and $16.5{\rm \pi}$ (d): ____ source term; - - - - streamwise (grey) and wall-normal (black) conductive fluxes; $\cdots \cdots$ streamwise (grey) and wall-normal (black) convective fluxes; – $\cdot$ – $\cdot$ – streamwise (grey) and wall-normal (black) turbulent fluxes; – $\cdot$ $\cdot$ – compressibility effects.

Figures 15(a) and 15(b) are relative to $x/\delta = 0.73$ and $x/\delta = 2{\rm \pi}$, respectively. At these short distances from the leading edge, many contributions which do not appear in equilibrium channel flows, are important. It is the case, for example, of the wall-normal convective flux since, as illustrated in figure 10, the mean vertical velocity is considerably higher compared with equilibrium. It is also the case of the streamwise turbulent flux, indicating that the correlation $\widetilde {u''h''}$ is evolving and increasing (in module). Yet, the preponderant contribution is that of the streamwise convective flux. Its positive slope near the wall indicates that the thermal boundary layer is developing and, as can be seen, the wall-normal conductive and turbulent fluxes are active. Farther away from the wall, its slope is negative and linear; since all the other contributions are constant, it is evident that the only active phenomenon is the uniform cooling caused by the source term. It is thus a portion of the boundary layer which still has not perceived the presence of the heated wall.

Further away from the leading edge, the canonic heat fluxes are gradually retrieved. At $x/\delta = 5.7{\rm \pi}$ (see figure 15c), $\bar {\rho }\widetilde {v''h''} > 0$ everywhere but at $y/\delta = 1$, suggesting that the impact of the heated wall has by now reached the centre of the channel. At $x/\delta = 16.5{\rm \pi}$ (see figure 15d), the thermal boundary layer seems to be at equilibrium. However, notice how at $y/\delta \approx 1$ the streamwise convective flux is negative and decreasing, indicating that the temperature at the centre of the channel is still cooling under the action of the source term. This signals that even at the outlet the thermal boundary layer is not perfectly developed, given the limited extent of the simulated channel.

The evolution of the different contributions to the energy balance shows that the fluxes can be divided into two categories. The former is that of the equilibrium terms, which tend to become more and more important with $x/\delta$ until stabilisation. The contributions that belong to this category are the wall-normal conductive flux and the wall-normal turbulent flux, which, summed, form the mean equilibrium flux $q^{eq}$. The latter is that of the non-equilibrium terms which, even if potentially preponderant near the leading edge, tend to become negligible and disappear with $x/\delta$. The contributions belonging to this category are the streamwise and wall-normal convective terms as well as the streamwise turbulent flux, which form the mean non-equilibrium flux $q^{n-eq}$. The remaining contributions, at least in the present case, are always negligible.

Now, defining the average total heat flux $q^{tot}$ as

(4.7)\begin{equation} q^{tot} = \overline{q_w}(x) + y S_{ener} , \end{equation}

(4.6) can be rearranged in the form

(4.8)\begin{equation} q^{tot} = q^{eq} + q^{n-eq} , \end{equation}

where the compressibility effects and the streamwise conductive flux have been neglected. For every $x/\delta$ and $y/\delta$, it is thus possible to define the ratio

(4.9)\begin{equation} R^{n-eq}_{tot} = \frac{| q^{n-eq} |}{| q^{eq} | + | q^{n-eq}|}, \end{equation}

quantifying the importance of the non-equilibrium terms in the energy balance.

Figure 16(a) shows the evolution of $R^{n-eq}_{tot}$ along the wall-normal direction for different $x/\delta$. The ratio $R^{n-eq}_{tot}$ allows us to define a quantitative criterion for distinguishing the three aforementioned regions of the developing thermal boundary layer. Indeed, as can be seen in figure 16(a), for every $x/\delta$, there is a near-wall portion of the boundary layer where the equilibrium fluxes are predominant with respect to the non-equilibrium ones; consequently, $R^{n-eq}_{tot} \approx 0$ and the energy equation can be simply approximated as

(4.10)\begin{equation} \overline{q_w} + y S_{ener} \approx q_y^{cd} + \bar{\rho} \widetilde{v''h''} . \end{equation}

The equilibrium region of the boundary layer, delimited by $h^*_{eq}(x)$, can thus be defined as the region laying beneath an isoline of $R^{n-eq}_{tot}$ in the $x$–$y$ plane, for example, $R^{n-eq}_{tot} = 0.1$. For every $x/\delta$, there is also a portion of the boundary layer adjacent to the centre of the channel, where the non-equilibrium terms are predominant; consequently, ${R^{n-eq}_{tot} \approx 1}$, the only active physical phenomenon is the cooling caused by the source term and the mean temperature profile is flat along the wall-normal direction. This region of the boundary layer, delimited by $h^*_{n-eq}(x)$, can thus be defined as the region laying above an isoline of $R^{n-eq}_{tot}$, for example, $R^{n-eq}_{tot} = 0.9$. Finally, for every $x/\delta$ and for $y^* \in [h^{*}_{eq}, h^{*}_{n-eq}]$, the equilibrium and non-equilibrium fluxes are of the same order of magnitude, and the thermal boundary layer is in development.

Figure 16.

Evolution of $R^{n-eq}_{tot}$ with respect to $y/\delta$ for $x/\delta = 0.18$, $1.5{\rm \pi}$, $2.5{\rm \pi}$ and $16.5{\rm \pi}$ (from light grey to black) (a). Evolution of $h^*_{eq}$, $h^*_{n-eq}$ and $\delta ^*_{99\,\%}$ along the channel flow: $\cdots \cdots$ $h^*_{eq}$; – $\cdot$ – $\cdot$ – $h^{*}_{n-eq}$; ____ $\delta ^*_{99\,\%}$ (b).

Figure 16(b) shows the evolution of $h^*_{eq}$ (defined as the isoline $R^{n-eq}_{eq} = 0.1$) and $h^{*}_{n-eq}$ (defined as the isoline $R^{n-eq}_{eq} = 0.9$) compared with the wall-scaled thermal boundary layer thickness $\delta ^*_{99\,\%} := h^* \ : \ | T_w - \bar {T}(h^*) | / | T_w - \bar {T}_c(x) | = 0.99$ .

The isoline $h^{*}_{n-eq}$ is defined by a very loose criterion on $R^{n-eq}_{tot}$. Therefore, the advance of $h^{*}_{n-eq}$ towards the centre of the channel can benefit from the slightest expansion of the wall-normal heat flux, which takes place in regions of the flow where heat transport is efficiently carried out by turbulence. Consequently, the region beyond $h^{*}_{n-eq}$ quickly decays and disappears before $x/\delta = 4{\rm \pi}$. Downstream of this crosswise section, the effect of the heated wall has reached the centre of the channel and, as shown in figure 14, the mean temperature profile is nowhere flat any more.

The isoline $h^{*}_{eq}$, instead, is defined by a very stringent criterion on the development of the equilibrium terms. Therefore, its growth is initially driven by the slow development of the wall-normal conductive flux, which takes place in regions of the flow where heat transport is essentially carried out by diffusion. As $h^{*}_{eq}$ grows and reaches the most turbulent parts of the boundary layer, its expansion accelerates, and, as can be qualitatively seen in figure 16(b), at $x/\delta \approx 10{\rm \pi}$ its growth rate is similar to that of $h^{*}_{n-eq}$ at $x/\delta \approx 3{\rm \pi}$, where $h^{*}_{eq} \approx h^{*}_{n-eq} \approx 300$. However, $h^{*}_{eq}$ does not reach the centre of the channel, and attains a peak at $y^* \approx 350$ before slowly decreasing. The criterion $R^{n-eq}_{tot} = 0.1$ is thus so stringent that the slight cooling of the source term at the centre of the channel (recall figure 15d) becomes appreciable at a certain $x/\delta$. With a higher criterion for $h^{*}_{eq}$, the same behaviour of $h^{*}_{n-eq}$ would be observed. However, note that $h^{*}_{eq} > \delta ^*_{99\,\%}$, indicating that the equilibrium sublayer attains in any case an essentially temperature-homogeneous region.

The evolution of $h^{*}_{eq}$ and $h^{*}_{n-eq}$ is understandably case dependent. However, note that their definition, based on $R_{tot}^{n-eq}$, is general and can be used in all cases. By computing or estimating the size of $h^{*}_{eq}$ and comparing it with respect to a characteristic length of a given flow configuration, one can thus quantify the importance of the non-equilibrium effects in the thermal boundary layer.

Finally, figure 17(a) shows the evolution of $R_{tot}^{n-eq}$ as a function of $x/\delta$ for several wall distances. The intersection of every iso-$y^*$ with the isoline $R_{tot}^{n-eq} = 0.1$ can be seen as the non-equilibrium distance associated to that wall distance, i.e. the distance from the leading edge which is necessary for the thermal boundary layer to be at equilibrium between the wall and the specified $y^{*}$. Some contours of $T^*$ are given in figure 17(b) at $y^* = 12.5$ (above) and $y^* = 40$ (below). By comparing figure 17(a,b), one can appreciate the correspondence between the values of $R_{tot}^{n-eq}$ at these wall distances and the development of the heated turbulent structures as $x/\delta$ increases.

Figure 17.

Evolution of $R^{n-eq}_{tot}$ as a function of $x/\delta$ for several wall distances: ____ $R^{n-eq}_{tot}$ for $y^* = 5$, $12.5$, $30$ and $40$ (from light grey to black, respectively); – $\cdot$ – $\cdot$ – isoline $R^{n-eq}_{tot} = 0.1$ (a). Instantaneous contours of temperature at $y^* = 12.5$ (above) and $y^* = 40$ (below) (b).

4.3. Development of turbulent heat transfer

The previous section has allowed us to appreciate and quantitatively define an equilibrium sublayer, where both the mean temperature and the mean energy balance are equilibrated. In this section we investigate its relevance with respect to the evolution along the isothermal wall of several turbulent quantities.

Figure 18 shows the evolution of the temperature fluctuations and of the wall-normal turbulent heat flux for different crosswise sections compared with equilibrium. Observe how both quantities progressively grow with $x/\delta$, and how the peak shifts towards the higher $y^*$, until it stabilises at $y^* \approx 20$ for the r.m.s. temperature and $y^{*} \approx 45$ for the turbulent heat flux.

Figure 18.

Profiles of r.m.s. temperature (a) and wall-normal turbulent heat flux (b) for different $x/\delta$: ____ present results at $x/\delta = 0.015$, 0.21, $\frac {1}{2}{\rm \pi}$, $\frac {3}{2}{\rm \pi}$, $3.6{\rm \pi}$, $7.75{\rm \pi}$, $17.9{\rm \pi}$ (from lighter to darker); – $\cdot$ – $\cdot$ – equilibrium profiles of § 3.2.

The wall-normal turbulent heat flux (see figure 18b) clearly exhibits an equilibrated behaviour in the near-wall region for every $x/\delta$. Interestingly, the same cannot be said about the fluctuating temperature (see figure 18a), which starts agreeing with equilibrium only at a certain distance from the leading edge ($x/\delta \approx 3/2{\rm \pi}$). This illustrates how the r.m.s. temperature has a stronger inertia, and needs a larger distance to settle, an aspect which has also been observed experimentally (see Teitel & Antonia Reference Teitel and Antonia1993). Also observe, in figure 8, the scant fluctuating structures near the leading edge. Figure 18(b) also outlines the fast wall-normal expansion of the turbulent heat flux during the early boundary layer development. As suggested in § 4.2, this explains the fast decay of the region $y^*>h^*_{n-eq}$ seen in figure 16(b).

The turbulent budgets of the enthalpy variance are now analysed. The time-averaged transport equation of $\widetilde {h''^{2}}$ reads as (see, for a similar derivation, Vicquelin et al. Reference Vicquelin, Zhang, Gicquel and Taine2014)

(4.11)\begin{align} 0 & ={-}\underbrace{\frac{\partial }{\partial x} \left( \bar{\rho}\tilde{u} \frac{\widetilde{h''^2}}{2} \right)}_{I_x} -\underbrace{\frac{\partial }{\partial y} \left( \bar{\rho}\tilde{v} \frac{\widetilde{h''^2}}{2} \right)}_{I_y} -\underbrace{ \frac{\partial}{\partial x} ( \overline{q_x^{cd'}h'} )}_{II_x} -\underbrace{ \frac{\partial}{\partial y} ( \overline{q_y^{cd'}h'} )}_{II_y} \nonumber\\ &\quad + \underbrace{\overline{q_x^{cd'} \frac{\partial h'}{\partial x}} + \overline{q_y^{cd'} \frac{\partial h'}{\partial y}} + \overline{q_z^{cd'} \frac{\partial h'}{\partial z}}}_{III} + \underbrace{\overline{h'N'}}_{IV} -\underbrace{ \bar{\rho} \widetilde{u''h''} \frac{\partial \tilde{h}}{\partial x}}_{V_x} -\underbrace{\bar{\rho} \widetilde{v''h''} \frac{\partial \tilde{h}}{\partial y}}_{V_y} \nonumber\\ &\quad -\underbrace{\frac{1}{2} \frac{\partial}{\partial x} ( \bar{\rho} \widetilde{u''h''h''} )}_{VI_x} -\underbrace{\frac{1}{2} \frac{\partial}{\partial y} ( \bar{\rho} \widetilde{v''h''h''} )}_{VI_y} + \underbrace{\overline{h''} \left( - \frac{\partial \overline{q_x^{cd}}}{\partial x} - \frac{\partial \overline{q_y^{cd}}}{\partial y} + \overline{N} \right)}_{VII} , \end{align}

composed of the streamwise and wall-normal mean convective terms ($I_x$ and $I_y$) and molecular diffusion ($II_x$ and $II_y$), molecular dissipation ($III$), a term of correlation between the enthalpy and $N$ ($IV$), where $N= ({\mathrm {D} p}/{\mathrm {D} t}) + \boldsymbol {\tau }_{ij} ({\partial u_i}/{\partial x_j}) + S_{ener}$ regroups the compressibility effects and the source term, the streamwise and wall-normal production ($V_x$ and $V_y$) and turbulent diffusion terms ($VI_x$ and $VI_y$), and, finally, a term related to the enthalpy–density correlation ($VII$). Figure 19 shows the evolution of the different terms (scaled with respect to $\overline {q_w}(x)^2/\overline {\mu _w}(x)$) for different crosswise sections. For each $x/\delta$, the values of $h^*_{eq}$ and $h^*_{n-eq}$ defined in § 4.2 are put into evidence. Some terms ($IV$ and $VII$), being always small and negligible, are not plotted. Finally, we have observed non-negligible levels of spurious numerical dissipation in our simulation, which has been added to the molecular dissipation ($III$) to yield a total dissipation term and a closed balance. The justification of this approach is given in supplementary materials B.

Figure 19.

Budgets of enthalpy variance at $x/\delta = 0.73$ (a), $2{\rm \pi}$ (b), $5.7{\rm \pi}$ (c) and $16.5{\rm \pi}$ : ____ streamwise ($V_x$: grey) and wall-normal ($V_y$: black) production; - $\cdot$ $\cdot$ - $\cdot$ $\cdot$ - molecular dissipation $III$; - - - - - streamwise ($II_x$: grey) and wall-normal ($II_y$: black) molecular diffusion; – $\cdot$ – $\cdot$ – streamwise ($VI_x$: grey) and wall-normal ($VI_y$: black) turbulent diffusion; $\cdots \cdots$ streamwise ($I_x$: grey) and wall-normal ($I_y$: black) mean convective terms.

Figure 19(a) is relative to $x/\delta = 0.73$. At this close distance from the leading edge, the different terms of (4.11) are active only in a small portion (approximately $10\,\%$) of the boundary layer, delimited by $h^*_{n-eq}$, beyond which the flow is still not affected by the heated wall. Observe how it is between $h^*_{eq}$ and $h^*_{n-eq}$ that the non-equilibrium effects are located, with non-negligible streamwise convective and production terms. The classic contributions (wall-normal production, molecular dissipation and wall-normal turbulent and molecular diffusion), instead, appear to be under-developed. This is true even between the wall and $h^*_{eq}$, which is in agreement with the low level of r.m.s. temperature observed near the leading edge.

Figure 19(b) is relative to $x/\delta = 2{\rm \pi}$. It is interesting to notice how all the streamwise non-equilibrium terms in the temperature variance budget have vanished. In the region between $h^*_{eq}$ and $h^*_{n-eq}$, while the mean energy balance presents strong non-equilibrium terms as seen in § 4.2, the enthalpy variance budget seems to merely feature the gradual development of the classic terms. Observe how in the equilibrium sublayer, instead, the different contributions controlling the r.m.s. temperature have by now settled, as can be seen by comparing figure 19(b–d) for $y^*< h^*_{eq}$. Accordingly, the r.m.s. temperature exhibits an equilibrated behaviour for $y^*< h^*_{eq}$ at these crosswise sections (see figure 18a).

Concerning the wall-normal turbulent heat flux, the time-averaged transport equation reads as (see, for a similar derivation, Vicquelin et al. Reference Vicquelin, Zhang, Gicquel and Taine2014)

(4.12) \begin{align} 0 & ={-}\underbrace{ \frac{\partial }{\partial x} ( \bar{\rho} \tilde{u} \widetilde{v''h''} )}_{I_x} -\underbrace{ \frac{\partial }{\partial y} ( \bar{\rho} \tilde{v} \widetilde{v''h''})}_{I_y} + \underbrace{ \frac{\partial }{\partial x} ( \overline{\tau_{2,1}'h'} - \overline{q_x^{cd'}u'} )}_{II_x} + \underbrace{\frac{\partial }{\partial y} ( \overline{\tau_{2,2}'h'} - \overline{q_y^{cd'}v'})}_{II_y} \nonumber\\ & \quad -\underbrace{ \left ( \overline{\tau_{2,1}'\frac{\partial h'}{\partial x}} + \overline{\tau_{2,2}'\frac{\partial h'}{\partial y}} + \overline{\tau_{2,3}'\frac{\partial h'}{\partial z}} \right ) + \left ( \overline{q_x^{cd'} \frac{\partial v'}{\partial x}} + \overline{q_y^{cd'} \frac{\partial v'}{\partial y}} + \overline{q_z^{cd'} \frac{\partial v'}{\partial z}} \right )}_{III} -\underbrace{ \overline{h'\frac{\partial p'}{\partial y}}}_{IV} + \underbrace{\overline{v'N'}}_{V} \nonumber\\ &\quad - \underbrace{ \bar{\rho} \widetilde{u''h''} \frac{\partial \tilde{v}}{\partial x} - \bar{\rho} \widetilde{u''v''} \frac{\partial \tilde{h}}{\partial x}}_{VI_x} -\underbrace{ \bar{\rho} \widetilde{v''h''} \frac{\partial \tilde{v}}{\partial y} - \bar{\rho} \widetilde{v''v''} \frac{\partial \tilde{h}}{\partial y}}_{VI_y} -\underbrace{ \frac{\partial }{\partial x} ( \bar{\rho} \widetilde{u''v''h''} )}_{VII_x} -\underbrace{ \frac{\partial }{\partial y} ( \bar{\rho} \widetilde{v''v''h''} )}_{VII_y}\nonumber\\ &\quad + \underbrace{\overline{h''} \left( - \frac{\partial \bar{p}}{\partial y} + \frac{\partial \overline{\tau_{2,1}}}{\partial x} + \frac{\partial \overline{\tau_{2,2}}}{\partial y} \right) + \overline{v''} \left ( \overline{N} - \frac{\partial \overline{q_x^{cd'}}}{\partial x} - \frac{\partial \overline{q_y^{cd'}}}{\partial y} \right )}_{VIII} , \end{align}

composed of the mean streamwise and wall-normal convective terms ($I_x$ and $I_y$) and molecular diffusion ($II_x$ and $II_y$), the molecular dissipation ($III$), the enthalpy–pressure-gradient correlation ($IV$), the correlation between the wall-normal velocity and $N$ ($V$), the streamwise and wall-normal production ($VI_x$ and $VI_y$) and turbulent diffusion ($VII_x$ and $VII_y$), and, finally, two terms related to the enthalpy–density and wall-normal-velocity–density correlations ($VIII$). Figure 20 shows the evolution of the different terms (scaled with respect to $\overline {q_w}(x) \overline {\tau _w}(x)/\overline {\mu _w}(x)$) for different crosswise sections with the respective $h^*_{eq}$ and $h^*_{n-eq}$ defined in § 4.2. The terms $V$ and $VIII$ are always small and negligible, and are not plotted.

Figure 20.

Budgets of wall-normal turbulent heat flux at $x/\delta = 0.73$ (a), $2{\rm \pi}$ (b), $5.7{\rm \pi}$ (c) and $16.5{\rm \pi}$: ____ streamwise ($VI_x$: grey) and wall-normal ($VI_y$: black) production; - $\cdot$ $\cdot$ - $\cdot$ $\cdot$ - molecular dissipation $III$; - - - - - streamwise ($II_x$: grey) and wall-normal ($II_y$: black) molecular diffusion; – $\cdot$ – $\cdot$ – streamwise ($VII_x$: grey) and wall-normal ($VII_y$: black) turbulent diffusion; $\cdots \cdots$ streamwise ($I_x$: grey) and wall-normal ($I_y$: black) convective terms; –  –  –  – enthalpy–pressure-gradient correlation $IV$.

Most of the observations made concerning the budgets of enthalpy variance are still valid. The non-equilibrium region $h^*_{eq} < y^* < h^*_{n-eq}$ manifests itself with a gradual development of the classic terms (including, in this case, the enthalpy–pressure-gradient), rather than with the presence of strong streamwise terms as seen in § 4.2 for the mean energy balance. In this case, the only streamwise terms that are important near the leading edge are the streamwise production and convection, as illustrated in figure 20(a). Yet, even these terms become negligible at $x/\delta = 2{\rm \pi}$ (see figure 20b).

The only difference with the enthalpy variance budget is the behaviour for $y^*< h^*_{eq}$. For the wall-normal turbulent heat flux, the classic budgets have settled in the equilibrium sublayer even for the smallest $x/\delta$, which is in agreement with the equilibrated profiles observed for $y^*< h^*_{eq}$ in figure 18(b).

Figure 21 shows the details of the budgets of enthalpy variance and wall-normal turbulent heat flux for $y^* \approx h^*_{n-eq}$ at $x/\delta = 0.73$, i.e.  the same crosswise section of figures 19(a) and 20(a). In both figures 21(a) and 21(b), it can be seen how the turbulent diffusion plays an important role in the development of both $\widetilde {h''^{2}}$ and $\widetilde {v''h''}$. In the case of the enthalpy variance, turbulent diffusion is actually the only term inducing a gain of enthalpy fluctuations, mostly counterbalanced by the mean streamwise convective term. The presence of non-negligible turbulent diffusion around $y^* \approx h^*_{n-eq}$ corroborates the hypothesis formulated in § 4.2, i.e.  that the fast decay of the temperature-homogeneous region beyond $y^* > h^*_{n-eq}$ might be due to the efficient heat transport carried out by turbulence.

Figure 21.

Budgets of enthalpy variance (a) and wall-normal turbulent heat flux (b) at $x/\delta = 0.73$ zoomed at $y^* \sim h^*_{n-eq}$. See figures 19 and 20 for the notation.

Finally, the behaviour of the turbulent Prandtl number $Pr_t$ is investigated. The turbulent Prandtl number $Pr_t$ offers a simple way of relating the turbulent kinematic viscosity $\nu _t$ and conductivity $\lambda _t$ in Reynolds-averaged Navier–Stokes (RANS), LES or wall-modelled LES (WMLES) turbulence modelling, and, analogously to the molecular Prandtl number, $Pr_t$ is defined as

(4.13)\begin{equation} Pr_t = \frac{\bar{\rho} \nu_t \overline{c_p}}{\lambda_t} = \frac{-\bar{\rho}\widetilde{u''v''}}{-\bar{\rho}\widetilde{v''h''}/\overline{c_p}} \frac{\mathrm{d}\bar{T}/\mathrm{d} y}{\mathrm{d}\overline{u}/\mathrm{d} y} . \end{equation}

Experimentally, Blom (Reference Blom1970) analysed the evolution of the turbulent Prandtl number in a developing thermal boundary layer, and found that it varies in both wall-normal and streamwise directions, while suggesting that a universal distribution of $Pr_t$ can only be expected in the near-wall region. Antonia et al. (Reference Antonia, Danh and Prabhu1977) carried out similar investigations, and found that the turbulent Prandtl number is generally of the order of magnitude of one, thus showing the overall validity of the Reynolds analogy; according to their experimental results, values of $Pr_t$ strongly exceeding unity can only be found near the leading edge of the thermal boundary layer.

Figure 22 displays the evolution of $Pr_t$ for different $x/\delta$, compared with its equilibrium profile. The latter is represented by the dashed-dotted line. It can be seen that $Pr_t$ is slightly greater than unity at the wall, before decreasing and attaining $Pr_t \approx 0.7$ in the channel core flow. Its mild gradient along the wall-normal direction can justify the use of a constant average turbulent Prandtl number (usually $Pr_t \approx 0.9$), at least in equilibrium boundary layers (see, for instance, Kawai & Larsson Reference Kawai and Larsson2012).

Figure 22.

Evolution of turbulent Prandtl number in the wall-normal direction for different $x/\delta$: ____ present results at (from light to dark grey) $x/\delta = 0.18$, $0.73$ $0.5{\rm \pi}$ and $0.75{\rm \pi}$ (a), at $x/\delta = {\rm \pi}$, $1.25{\rm \pi}$, $1.5{\rm \pi}$ and $1.75{\rm \pi}$ (b); at $x/\delta = 5.67{\rm \pi}$, $8.25{\rm \pi}$, $10.8{\rm \pi}$ and $16.5{\rm \pi}$ (c); – $\cdot$ – $\cdot$ – equilibrium profile.

Figure 22(a) shows the evolution of $Pr_t$ for four crosswise sections which are relatively close to the leading edge. As can be seen, the turbulent Prandtl number does not exhibit an equilibrated behaviour in the near-wall region, where its value largely exceeds unity. Since the turbulent viscosity and conductivity, on their part, are at equilibrium in the wall vicinity (not shown here), this behaviour is explained by the abrupt increase of $c_p$ astride $x/\delta = 0$, and represents another leading edge effect. Further away from $x/\delta = 0$, $Pr_t$ remains for the most part greater than unity and, as $x/\delta$ increases, its perturbation expands in the wall-normal direction, while its intensity decreases. This tendency is particularly evident in figure 22(b) for $x/\delta$ ranging between $[{\rm \pi}, 1.75{\rm \pi} ]$. The profiles of figure 22(c), instead, illustrate how the turbulent Prandtl number essentially returns to equilibrium at $x/\delta \approx 5{\rm \pi}$.

The larger values of $Pr_t$ during the development of the thermal boundary layer are explained by the fact that its numerator, $\nu _t$, is already well established with the incoming fully developed adiabatic flow; its denominator $\lambda _t$, instead, is initially small, since heat transfer is essentially carried out by mean convection, as seen in § 4.2. The results are therefore in qualitative agreement with those of Antonia et al. (Reference Antonia, Danh and Prabhu1977). On the other hand, there is no evidence of the existence of a universal distribution in the near-wall region, as indicated in Blom (Reference Blom1970). It appears to be quite the opposite since the highest values of $Pr_t$ can be found near the leading edge in the very proximity of the wall.