2. Interscale energy transfers

The DNS computations in turbulent channel flows at moderate to high Reynolds numbers, namely $\displaystyle \textit{Re}_{\tau }\in \{550,1000,2000,5200\}$ , conducted by Lee & Moser (Reference Lee and Moser2019) are thoroughly analysed in this section in order to shed light on the interscale energy transfer processes occurring in both the inner and outer regions of wall-bounded flows. For this purpose, energy spectra are first scrutinised. We denote $\varPsi _{\textit{ij}} (\kappa _x,y,\kappa _z )$ as the two-dimensional spectrum tensor corresponding to the Fourier transform of the two-point correlation tensor $R_{\textit{ij}} (r_x,y,r_z )$ , where $r_x$ and $r_z$ are the separations between the two points in the streamwise and spanwise directions, respectively, and $\kappa _x$ and $\kappa _z$ are the corresponding wavenumbers. By partitioning spectra $\varPsi _{\textit{ij}}$ in two parts for a given cut-off wavenumber $\kappa ^+=\sqrt {\kappa _x^{+^2}+\kappa _z^{+^2}}=\kappa _c^+$ , Lee & Moser (Reference Lee and Moser2019) demonstrated that the mean Reynolds stress components can be split into two distinct contributions with respect to the Reynolds number dependency. The first, mainly concentrated in the inner region, is independent of the Reynolds number and related to the near-wall cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999). This first contribution corresponds to the action of the small scales of turbulence, i.e. those located above the chosen cut-off wavenumber. The second contribution, which strongly depends on the Reynolds number, is twofold. In the outer region, LS structures promote the growth of the Reynolds stress component as the Reynolds number increases. In the meantime, a LS contribution also exists in the inner region for the streamwise and the spanwise diagonal Reynolds stress components. This specific LS contribution was proved to be correlated to the LS structures of the outer region via the superposition phenomenon, which ultimately leads to the increase of the near-wall peak of the turbulent kinetic energy as the Reynolds number is increased. In their analysis, Lee & Moser (Reference Lee and Moser2019) applied a scale separation to the two-dimensional spectra at a fixed wavelength $\displaystyle \lambda _c^+={\displaystyle 2\pi }/{\displaystyle \kappa _c^+}=1000$ . In figure 2 of Lee & Moser (Reference Lee and Moser2019), showing results obtained at $\textit{Re}_{\tau }=5200$ , there is a clear evidence that large scales imprint throughout the whole channel height for $R_{11}=\overline {{u^{\prime }}^2}$ and $R_{33}=\overline {{w^{\prime }}^2}$ components of the two-point velocity correlation tensor. The energy on $\overline {{u^{\prime }}^2}$ is mainly carried out by streamwise elongated structures (streak-like structures) with $\displaystyle \lambda _x^+ \gg \lambda _z^+$ , while the energy for $\overline {{v^{\prime }}^2}$ – and in a lesser manner for $\overline {{w^{\prime }}^2}$ – is more spread over a large range of wavelengths. But more interestingly, a remarkable behaviour is observed on the spanwise component $\overline {{w^{\prime }}^2}$ in the logarithmic layer, i.e. $y^+\in [100,1000]$ in the Lee & Moser (Reference Lee and Moser2019) figure, where the energy is being transferred between streamwise elongated structures at large scale and more isotropic structures ( $\displaystyle \lambda _x^+ \approx \lambda _z^+$ ) at lower scale, $\lambda ^+$ changing for approximately a decade. This behaviour is almost absent on the $R_{22}=\overline {{v^{\prime }}^2}$ component. This observation points out the specific role played by the spanwise component ${w^{\prime }}^2$ in the energy transfer from large to small scale acting in the logarithmic layer.

Figure 1. One-dimensional premultiplied spectra $\displaystyle \varPhi _{\textit{ij}}^z(\kappa _x,y)$ (red line contours) and $\displaystyle \varPhi _{\textit{ij}}^x(y,\kappa _z)$ (grey filled contours) of the two-point velocity correlations (a) $\overline {{u^{\prime }}^2}$ , (b) $\overline {{v^{\prime }}^2}$ and (c) $\overline {{w^{\prime }}^2}$ ) obtained from DNS of channel flows of Lee & Moser (Reference Lee and Moser2019). Each spectrum is normalised by its maximum value. Equidistant isocontour values are $\{1/6,1/3,1/2,2/3,5/6\}$ , and are represented with increasingly darker colours.

To further confirm these observations, we consider the one-dimensional premultiplied energy spectra $\displaystyle \varPhi _{\textit{ij}}^z(\kappa _x,y)=\smallint \kappa _z\varPsi _{\textit{ij}}\, {\textrm{d}}\kappa _z$ and $\displaystyle \varPhi _{\textit{ij}}^x(y,\kappa _z)=\smallint \kappa _x\varPsi _{\textit{ij}}\, {\textrm{d}}\kappa _x$ . Three of their components are plotted in figure 1 for the three diagonal components of the two-point velocity correlation tensor for the channel configuration at the highest friction Reynolds number $\textit{Re}_{\tau }=5200$ . The one-dimensional spectra for the streamwise velocity cover wavelength ranges separated by approximately an order of magnitude, indicating there again the elongated nature of the structures bearing the $\overline {{u^{\prime }}^2}$ component, whereas spectra for $\overline {{v^{\prime }}^2}$ and $\overline {{w^{\prime }}^2}$ show more comparable ranges of $\lambda _x$ and $\lambda _z$ values for a given isocontour value. The Reynolds number effects in the inner region, which lead to the failure of viscous scaling for wall-parallel Reynolds stress components (Monkewitz & Nagib Reference Monkewitz and Nagib2015), are related to the emergence at high Reynolds numbers of an outer peak, visible in figure 1(a). The location of the peak on the spectrum $\displaystyle \varPhi _{11}^z$ is $\textit{Re}_{\tau }$ -dependent (Mathis et al. Reference Mathis, Hutchins and Marusic2009), and at very high $\textit{Re}_{\tau }$ , the location of the outer peak marks the start of a broad plateau (Vallikivi, Hultmark & Smits Reference Vallikivi, Hultmark and Smits2015; Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). The high wavelength energy in the near-wall region in spectra $\displaystyle \varPhi _{11}^z$ and $\displaystyle \varPhi _{11}^x$ is the footprint of this outer energy site. The picture is different on spectra $\displaystyle \varPhi _{33}^z$ and $\displaystyle \varPhi _{33}^x$ , for which no clear outer peak can be identified. However, a Reynolds number dependence in the outer and inner regions is also reported (see e.g. Lee & Moser Reference Lee and Moser2019; Jiménez Reference Jiménez2024) for the spanwise fluctuations intensity $\overline {{w^{\prime }}^2}$ , while spectra $\displaystyle \varPhi _{22}^z$ and $\displaystyle \varPhi _{22}^x$ do not show any $\textit{Re}_{\tau }$ -dependence in the inner region (Lee & Moser Reference Lee and Moser2019). In figure 1(c) for spectra $\displaystyle \varPhi _{33}^z$ , energy around the inner peak is spread over a large range of wavelength, with amplitudes at large scales almost constant from $y^+=30$ up to several hundreds of wall units for the most elongated structures.

Figure 2. One-dimensional premultiplied spectra (a) $\displaystyle \varPhi _{11}^x(y,\kappa _z)$ and (b) $\displaystyle \varPhi _{33}^x(y,\kappa _z)$ for the four considered Reynolds numbers $\textit{Re}_{\tau }=550,1000,2000,5200$ in channel flow DNS from Lee & Moser (Reference Lee and Moser2019). Isocontour values are $1/6,1/3,1/2,2/3,5/6$ , and are represented with increasingly darker grey lines: dotted for $\textit{Re}_{\tau }=550$ , dashed for $\textit{Re}_{\tau }=1000$ , dash-dotted for $\textit{Re}_{\tau }=2000$ , and solid for $\textit{Re}_{\tau }=5200$ . Square symbols denote the maximum $y^+$ location reached by each contour. Darker red colour indicates increased $\textit{Re}_{\tau }$ values.

To illustrate the determining role of the $\overline {{w^{\prime }}^2}$ component in the Reynolds dependence of the inner region, figure 2 compares one-dimensional spectra $\displaystyle \varPhi ^x_{11}$ and $\displaystyle \varPhi ^x_{33}$ for the four considered Reynolds numbers. Below $\lambda _z^+=1000$ , spectra are almost independent of the Reynolds number. Above, Reynolds number effects on the $\overline {{u^{\prime }}^2}$ component tend to increase the LS contributions in the outer region following a linear evolution in the $ (y^+,\lambda _z^+ )$ plane, independent of the Reynolds number, marked in the figure by the locations of the maximum $y^+$ value reached by each isocontour. This line is very close to that found by Jiménez (Reference Jiménez2024) when plotting the maxima for each wavelength in the outer region, which approximately follows a line $\lambda _z^+=5y^+$ . In the inner region, the near-wall peak of $\overline {{u^{\prime }}^2}$ spreads over larger scales as $\textit{Re}_{\tau }$ is increased. The effect of the Reynolds number, and therefore of the increase of the LS contributions, on the $\overline {{u^{\prime }}^2}$ correlation appears to be localised in the near-wall peak region. Important differences can be noticed when looking at the $\varPhi ^x_{33}$ spectra. Below $\lambda _z^+=1000$ , if spectra are still independent of the Reynolds number, then the distribution of the increasing contribution of the large scale in the $ (y^+,\lambda _z^+ )$ plane is dependent on the Reynolds number, as indicated by the different lines drawn by the maximum locations of the isocontours. Isocontours of $\varPhi ^x_{33}$ are significantly modified in the region $y^+\in [100,400]$ as $\textit{Re}_{\tau }$ is increased. Unlike component $\overline {{u^{\prime }}^2}$ , the Reynolds number dependence is not restricted to the near-wall peak region, but spreads over a large range of $y^+$ values.

Figure 3. Opposite of the energy flux rates (a) $\displaystyle {\tilde {\varepsilon }_{11}}^{(1)}$ and (b) $\displaystyle {\tilde {\varepsilon }_{33}}^{(1)}$ obtained by DNS (Lee & Moser Reference Lee and Moser2019) for $\textit{Re}_{\tau }=550,1000,2000,5200$ . Darker symbols indicate increased $\textit{Re}_{\tau }$ values.

In order to see how the observations made on spectra $\varPhi ^x_{11}$ and $\varPhi ^x_{33}$ translate in terms of interscale energy transfers, the evolutions across the channel height of the energy flux rate components $\displaystyle {\tilde {\varepsilon }_{11}}^{(1)}$ and $\displaystyle {\tilde {\varepsilon }_{33}}^{(1)}$ for the four considered friction Reynolds numbers are plotted in figure 3. The energy flux rate tensor ${\tilde {\varepsilon }_{\textit{ij}}}^{(1)}$ , following notations introduced by Chedevergne et al. (Reference Chedevergne, Coroama, Gleize and Bézard2024) and recalled in § 3, is computed from the DNS data as the rest of the turbulent term due to triple correlations once the wall-normal turbulent transport is removed. Therefore, by definition, this term is symmetrical between the Reynolds stress budgets of the small and large scales, and is exactly zero if integrated over the whole spectra. Negative values of $\displaystyle -{\tilde {\varepsilon }_{11}}^{(1)}$ or $\displaystyle -{\tilde {\varepsilon }_{33}}^{(1)}$ in figure 3 indicate energy transfers from large to small scales. The energy flux rate tensor must be seen as a sink term in the budget of the LS contribution to the Reynolds stress tensor, and as a source term for the SS contribution respectively (see § 3). In figure 3, $\displaystyle -{\tilde {\varepsilon }_{11}}^{(1)}$ and $\displaystyle -{\tilde {\varepsilon }_{33}}^{(1)}$ are shown to have positive values in the inner region. In the framework of the scale separation performed by Lee & Moser (Reference Lee and Moser2019), these positive values indicate backscatter with energy transfer from small to large scale on each of the components $\overline {{u^{\prime }}^2}$ and $\overline {{w^{\prime }}^2}$ . Only negative values are observed for $\displaystyle -{\tilde {\varepsilon }_{22}}^{(1)}$ , confirming that backscattering only concerns the streamwise and spanwise components of the Reynolds stress tensor. Backscatter fluxes are very localised for the ${u^{\prime }}^2$ component, and centred about $y^+=6.5$ and vanishing at $y^+=15$ , while they cover a region ranging from $y^+=1$ up to a value at approximately $y^+=100$ that drops as $\textit{Re}_{\tau }$ increases. The locations of these energy transfers from small to large scales are in line with the energy distributions observed in figure 2. Transfers to large scales are more distributed across the inner region, up to the logarithmic region, for the $\overline {{w^{\prime }}^2}$ component. The LS contribution to $\overline {{w^{\prime }}^2}$ is thus increased with $\textit{Re}_{\tau }$ in the whole inner region, and due to pressure distribution, so is the LS contribution to $\overline {{u^{\prime }}^2}$ . The peak location of $\displaystyle {\tilde {\varepsilon }_{33}}^{(1)}$ tends to lower the $y^+$ value as $\textit{Re}_{\tau }$ increases, whereas it remains fixed for $\displaystyle {\tilde {\varepsilon }_{11}}^{(1)}$ . Interestingly, the amplitude of $\displaystyle -{\tilde {\varepsilon }_{11}}^{(1)}$ scales with $\textit{Re}_{\tau }^{3/5}$ , while $\displaystyle -{\tilde {\varepsilon }_{33}}^{(1)}$ scales with $\textit{Re}_{\tau }^{1/3}$ , showing a more rapid increase on $\overline {{u^{\prime }}^2}$ than on $\overline {{w^{\prime }}^2}$ of the LS contribution in the near-wall region. In the present study, we do not discuss the asymptotic behaviour of the near-wall intensity, whether it is of logarithmic type (Monkewitz & Nagib Reference Monkewitz and Nagib2015; Hwang Reference Hwang2024) or following a power law (Chen & Sreenivasan Reference Chen and Sreenivasan2020; Pirozzoli Reference Pirozzoli2024). A thorough discussion on this point is reported by Jiménez (Reference Jiménez2024). The observed scalings in figure 3 are limited to a small range of Reynolds numbers, and no conclusion can be drawn from that concerning the asymptotic behaviour of the near-wall peak. However, these observations will benefit the energy transfer modelling described in § 3, in particular to characterise evolutions with respect to $\textit{Re}_{\tau }$ for intermediate values, i.e. $\textit{Re}_{\tau } \in [500,4000]$ .

3. Modelling energy transfers from small to large scales

Before revising the two-scale RSM of Chedevergne et al. (Reference Chedevergne, Coroama, Gleize and Bézard2024), the foundations of the model are briefly recalled. A thorough explanation of the model’s principles can be found in Schiestel (Reference Schiestel1987). To model the LS and SS contributions resulting from the partitioning of spectra of two-point velocity correlations, Chedevergne et al. (Reference Chedevergne, Coroama, Gleize and Bézard2024), relying on ideas introduced by Schiestel (Reference Schiestel1974), developed a two-scale RSM whose underlying basis for each of the two transported scales is that of the RANS RSM proposed by Manceau (Reference Manceau2015). The spectral partitioning consists of two slices covering the wavenumber range $[0,\kappa _c]\times [\kappa _c,\kappa _\eta ]$ (see figure 1 in Chedevergne et al. Reference Chedevergne, Coroama, Gleize and Bézard2024). The first slice, $m=1$ , corresponding to large scale, is delimited by the cut-off wavenumber $\kappa _c$ . The second slice, $m=2$ , representing the SS contribution, ranges from $\kappa _c$ to $\kappa _\eta$ , the Kolmogorov scale at which energy is disspated. The cut-off wavenumber $\kappa _c$ is not imposed but results from the energy balance between the two spectral slices provided by the model. Practically, it was shown (Chedevergne et al. Reference Chedevergne, Coroama, Gleize and Bézard2024) to adequately discriminate the SS and LS contributions in a similar manner to that of Lee & Moser (Reference Lee and Moser2019), where $\lambda ^+=1000$ was imposed. The transport equations of partially integrated Reynolds stress tensors, over spectral slices $m=1$ and $m=2$ are considered, resulting in a set of $16$ equations, i.e. $2\times 6$ equations for the partial Reynolds stresses, $2$ equations for the associated outgoing energy fluxes from each spectral bandwidth, and $2$ equations to manage wall damping according to the elliptic blending approach of Manceau & Hanjalić (Reference Manceau and Hanjalić2002). The outgoing energy flux for the SS contribution, namely ${\tilde {\varepsilon }}^{(2)}$ , corresponds to dissipation at Kolmogorov scale, whereas ${\tilde {\varepsilon }}^{(1)}$ is the rate of energy transferred from large to small scale. The near-wall dissipations for each of the two spectral slices, denoted ${\varepsilon }^{(1)}_w$ and ${\varepsilon }^{(2)}_w$ , respectively, are treated separately using the algebraic expressions

(3.1) \begin{equation} {\varepsilon }^{(m)}_w = \frac {2\nu {k}^{(m)}}{y^{2.8}}\tanh \left (\frac {y}{y_0}\right ) + \frac {2\nu {k}^{(m)}}{y^2}\left (1-\tanh \left (\frac {y}{y_0}\right )\right )\!, \end{equation}

where ${k}^{(m)}$ is the turbulent kinetic energy in the spectral slice $m$ , and $y_0$ is a parameter controlling the extent of the dissipation due to the presence of the wall. On each spectral slice, the total dissipation ${\varepsilon }^{(m)}$ is therefore decomposed into a homogeneous part ${\tilde {\varepsilon }}^{(m)}$ corresponding to the outgoing energy flux, and an inhomogeneous part ${\varepsilon }^{(m)}_w$ due to wall. This is a key feature of the model, enabling us to isolate the energy flux ${\tilde {\varepsilon }}^{(1)}$ that characterises the energy transfers from large to small scales, and which can be then be treated directly as a transported variable.

The value of $y_0^+$ was slighly adjusted compared to that used in Chedevergne et al. (Reference Chedevergne, Coroama, Gleize and Bézard2024), where $y_0^+=2.5$ was retained to match the near-wall behaviour obtained by the reference model of Manceau (Reference Manceau2015). Here, we retain $y_0^+=4$ for both slices $m=1$ and $m=2$ . The validity of (3.1) is tested against the DNS data provided by Lee & Moser (Reference Lee and Moser2019) in figure 4. In the DNS, the dissipation is not separated into homogeneous and inhomogeneous parts; only the energy flux ${\tilde {\varepsilon }}^{(1)}$ has been explicitly computed. The matching with DNS data is then very good on the LS contribution insofar as the dissipation obtained in the DNS is only due to near-wall effects, the energy behind essentially transferred to small scale without viscous dissipation away from the walls. The matching is nonethless very satisfying on the SS contribution of the dissipation in the vicinity of the wall for all the considered $\textit{Re}_{\tau }$ numbers. A slightly better agreement can be obtained using the standard form $\displaystyle {\varepsilon }^{(2)}_w={2\nu {k}^{(2)}}/{y^2}$ , but for consistency reasons, and also for modelling reasons detailed below, we decided to keep the form given in (3.1).

Figure 4. Wall dissipations ${\varepsilon }^{(1)}$ and ${\varepsilon }^{(2)}$ for the LS and SS contributions, respectively. Darker grey colours show increasing $\textit{Re}_{\tau }$ values. Symbols are DNS data (Lee & Moser Reference Lee and Moser2019), and solid lines are the expressions given in (3.1). Dashed lines are obtained with the expression $\displaystyle {\varepsilon }^{(2)}_w={2\nu {k}^{(2)}}/{y^2}$ .

The two-scale approach relies on a generalised form of the Reynolds decomposition for each instantaneous velocity component $u_i=\overline {u_i} + {u^{\prime}_i}^{(1)}+{u^{\prime}_i}^{(2)}$ . To lighten notations, $R_{\textit{ij}}=\overline {u'_iu'\!_j}$ now designates the one-point velocity correlation, i.e. the $ij$ -component of the Reynolds stress tensor. The two-scale RSM is governed by the following system composed of sixteen equations:

(3.2) \begin{equation} \begin{array}{l} \kern-7pt \displaystyle \frac {{\textrm{D}}{R}^{(1)}_{\textit{ij}}}{{\textrm{D}}t} \!=\! - {R}^{(1)}_{\textit{ik}}\frac {\partial \overline {u\!_j}}{\partial x_k} - {R}^{(1)}_{jk}\frac {\partial \overline {u_i}}{\partial x_k} + {\varPhi }^{(1)}_{\textit{ij}} - {\varepsilon }^{(1)}_{\textit{ij}} + {M}^{(1)}_{\textit{ij}} + \frac {\partial }{\partial x_l} \! \left [\! \left (\nu +\frac {c_s}{{\sigma _k\!}^{(1)}}{R}^{(1)}_{lm}{t}^{(1)}_t\right )\frac {\partial\! {R}^{(1)}_{\textit{ij}}}{\partial x_m}\! \right ]\!, \\[0.5cm] \kern-7pt \displaystyle \frac {{\textrm{D}}{R}^{(2)}_{\textit{ij}}}{{\textrm{D}}t} \!=\! \displaystyle -{R}^{(2)}_{\textit{ik}}\frac {\partial \overline {u\!_j}}{\partial x_k}-{R}^{(2)}_{jk}\frac {\partial \overline {u_i}}{\partial x_k} + {\tilde {\varepsilon }}^{(1)}_{\textit{ij}} + {\varPhi }^{(2)}_{\textit{ij}} - {\varepsilon }^{(2)}_{\textit{ij}} + \frac {\partial }{\partial x_l}\left [\! \left (\nu +\frac {c_s}{{\sigma _k\!}^{(2)}}{R}^{(2)}_{lm}{t}^{(2)}_t\right )\frac {\partial {R}^{(2)}_{\textit{ij}}}{\partial x_m}\! \right ]\!, \\[0.5cm] \kern-7pt \displaystyle \frac {{\textrm{D}}{\tilde {\varepsilon }}^{(1)}}{{\textrm{D}}t} \!=\! \displaystyle \frac {{C'\!_{\varepsilon\! _1}}^{(1)}{P_k\!}^{(1)}-{C_{\varepsilon _2}\!}^{(1)}{\tilde {\varepsilon }}^{(1)}}{{t}^{(1)}_t} +\displaystyle \frac {\partial }{\partial x_l}\left [\! \left (\nu +\frac {c_s}{{\sigma _{\varepsilon }\!}^{(1)}}{R}^{(1)}_{lm}{t}^{(1)}_t \!\right )\frac {\partial {\tilde {\varepsilon }}^{(1)}}{\partial x_m}\right ]\!, \\[0.5cm] \kern-7pt \displaystyle \frac {{\textrm{D}}{\tilde {\varepsilon }}^{(2)}}{{\textrm{D}}t} \!=\! \displaystyle \frac {{C'_{\varepsilon _1}\!\!}^{(2)}{P_k\!}^{(2)}-{C'\!_{\varepsilon _2}\!\!\!}^{(2)}{\tilde {\varepsilon }}^{(2)} +{C_{\varepsilon _3}\!\!\!}^{(2)}{\tilde {\varepsilon }}^{(1)}}{{t}^{(2)}_t} +\frac {\partial }{\partial x_l}\left [\left (\nu +\frac {c_s}{{\sigma _{\varepsilon }\!}^{(2)}}{R}^{(2)}_{lm}{t}^{(2)}_t\right )\frac {\partial {\tilde {\varepsilon }}^{(2)}}{\partial x_m}\right ]\!, \\[0.3cm] \kern-5pt \displaystyle {\alpha }^{(1)} - \displaystyle {{{l}_t^{(1)}}^2}{\nabla} ^2 {\alpha }^{(1)} = 1,\quad{\alpha }^{(2)} - \displaystyle {{l}^{(2)}_t}^2{\nabla} ^2 {\alpha }^{(2)} = 1. \end{array} \end{equation}

Here, $\varPhi ^{(m)}_{\textit{ij}}$ is the pressure–strain correlation corresponding to the redistribution term in the slice. It is composed of a homogeneous part given by the Spezial–Sarkar–Gatski model (Speziale, Sarkar & Gatski Reference Speziale, Sarkar and Gatski1991) and an inhomogeneous contribution provided by Manceau & Hanjalić (Reference Manceau and Hanjalić2002). Also, $\displaystyle {t}^{(m)}_t$ is the turbulent time scale in slice $m$ , and is defined in (3.4) below following Durbin (Reference Durbin1991). Each partial dissipation tensor $\displaystyle {\varepsilon }^{(m)}_{\textit{ij}}$ is computed from its partial energy dissipation $\displaystyle {\varepsilon }^{(m)}=({1}/{2}){\varepsilon }^{(m)}_{ii}$ following the approach of Manceau (Reference Manceau2015). The same is applied to the partial rate of transfer ${\tilde {\varepsilon }}^{(1)}_{\textit{ij}}$ :

(3.3) \begin{eqnarray} && {\varepsilon }^{(m)}_{\textit{ij}} = \displaystyle \big (1-{f_w\!\!}^{(m)}\big )\frac {{R}^{(m)}_{\textit{ij}}}{{k}^{(m)}}{\varepsilon }^{(m)} + {f_w\!\!}^{(m)}\left (C_{\varepsilon }^{(m)}\frac {{R}^{(m)}_{\textit{ij}}}{{k}^{(m)}}{\varepsilon }^{(m)}+\big (1-C_{\varepsilon }^{(m)}\big )\frac {2}{3}{\tilde {\varepsilon }}^{(m)}\delta _{\textit{ij}}\right )\!, \nonumber\\ && {\tilde {\varepsilon }}^{(1)}_{\textit{ij}} = \displaystyle \big (1-{f_w\!\!}^{(2)}\big )\frac {{R}^{(1)}_{\textit{ij}}}{{k}^{(1)}}{\tilde {\varepsilon }}^{(1)} + {f_w\!\!}^{(2)}\left ( C_{\tilde {\varepsilon }}\frac {{R}^{(1)}_{\textit{ij}}}{{k}^{(1)}}{\tilde {\varepsilon }}^{(1)} +\left (1-C_{\tilde {\varepsilon }}\right )\frac {2}{3}{\tilde {\varepsilon }}^{(1)}\delta _{\textit{ij}}\right )\!. \end{eqnarray}

The blending function $\displaystyle {f_w\!\!}^{(m)}$ , evolving from $0$ at the wall to $1$ far from walls, and linking the homogeneous and inhomogeneous parts of ${\varPhi }^{(m)}_{\textit{ij}}$ , $\displaystyle {\varepsilon }^{(m)}_{\textit{ij}}$ and $\displaystyle {\tilde {\varepsilon }}^{(1)}_{\textit{ij}}$ , are calculated directly from functions $\displaystyle {\alpha }^{(m)}$ and are taken as $\displaystyle {f_w\!}^{(1)}=\displaystyle {{\alpha }^{(1)}}^2$ and $\displaystyle {f_w\!}^{(2)}=\displaystyle {{\alpha }^{(2)}}^3$ . Functions ${\alpha }^{(m)}$ are controlled by the turbulent length scale ${l}^{(m)}_t$ , given in

(3.4) \begin{equation} \begin{array}{l} \displaystyle {t}^{(m)}_t = \displaystyle \max \left (\frac {{k}^{(m)}}{{\varepsilon }^{(m)}},{C_T\!\!\kern1.5pt}^{(m)}\left (\frac {\nu }{{\varepsilon }^{(m)}}\right )^{\frac {1}{2}}\right ), \nonumber\\ \displaystyle {l}^{(m)}_t = \displaystyle {C_L\!\!\kern1.5pt}^{(m)}\max \left (\frac {{{k}^{(m)}}^{\frac {3}{2}}}{{\varepsilon }^{(m)}},{C_\eta\!\!}^{(m)}\left (\frac {\nu ^{3}}{{\varepsilon }^{(m)}}\right )^{\frac {1}{4}}\right ). \end{array} \end{equation}

By construction, the length scale ${l}^{(2)}_t$ associated with small scale is almost independent of $\textit{Re}_{\tau }$ , and so is the blending function $\displaystyle {f_w\!\!}^{(2)}$ . On the contrary, $\displaystyle {f_w\!\!}^{(1)}$ is shifted up towards higher $y^+$ values when $\textit{Re}_{\tau }$ is increased.

To account for the Reynolds number dependence of $\overline {{u^{\prime }}^2}$ and $\overline {{w^{\prime }}^2}$ in the inner region and due to LS contributions, the ${M}^{(1)}_{\textit{ij}}$ term was introduced into the model. Because ${\tilde {\varepsilon }}^{(1)}_{\textit{ij}}$ is given by (3.3), it is strictly positive, and instead of searching for a formulation allowing ${\tilde {\varepsilon }}^{(1)}_{\textit{ij}}$ to have negative values, it is more convenient to consider an additional term to balance the budget of ${R}^{(1)}_{\textit{ij}}$ in the inner region. In this regard, according to the decomposition introduced by Lee & Moser (Reference Lee and Moser2019), the rate of transfer ${\tilde {\varepsilon }}^{(1)}_{\textit{ij}}$ computed in the DNS results from multiple contributions, and there is no contradiction in modelling energy transfers between large and small scales from several terms. Term ${M}^{(1)}_{\textit{ij}}$ is thus the key term on which efforts are concentrated below, in order to reproduce the observations made in § 2.

The analysis presented in § 2 supports the idea of two separated contributions for interscale energy transfer in the near-wall region, i.e. a first one acting on energy transfers for $\overline {{u^{\prime }}^2}$ , and a second one for $\overline {{w^{\prime }}^2}$ , with different Reynolds number dependences. Let $\displaystyle n_i$ be the wall-normal vector, and let $t_i^x$ and $t_i^z$ be the streamwise and spanwise components of the wall-tangent vector. We begin with the contribution acting in the spanwise direction ${t_i^z}$ . According to figure 3, the contribution should be spread over the buffer layer and have a weak dependence on $\textit{Re}_{\tau }$ . We choose to make this latter be carried by the damping function $ {f_w\!}^{(1)}$ , and to use the $\textit{Re}_{\tau }$ -independent function $\displaystyle {{\tilde {\varepsilon }}^{(2)}{\overline {{w^{\prime }}^2}}^{(2)}\!}/{{k}^{(2)}}$ to spatially distribute the contribution. Concerning the streamwise contribution, we make use of the function $ (1-{f}^{(2)}_w )( {{\varepsilon }^{(2)}_w{\overline {{u^{\prime }}^2}}^{(2)}\!}/{{k}^{(2)}})$ to locate the action around $y^+=6.5$ . This choice was made in conjunction with that for the expression of ${\varepsilon }^{(2)}_w$ in (3.1). The dependence to $\textit{Re}_{\tau }^{3/5}$ is hardly reproducible from any of the transported variables. Instead, we introduce a function of $\textit{Re}_{\theta }$ , the Reynolds number based on the momentum thickness $\theta$ . The $\displaystyle {M}^{(1)}_{\textit{ij}}$ term finally reads

(3.5) \begin{equation} \displaystyle {M}^{(1)}_{\textit{ij}}= C_m^x \tanh \left (\frac {\textit{Re}_{\theta }}{\textit{Re}_{\theta _0}}\right ) \big (1-{f}^{(2)}_w\big )\frac {{\varepsilon }^{(2)}_w{\overline {{u^{\prime }}^2}}^{(2)}}{{k}^{(2)}} t_i^xt_j^x + C_m^z {f}^{(1)}_w \frac {{\tilde {\varepsilon }}^{(2)}{\overline {{w^{\prime }}^2}}^{(2)}}{{k}^{(2)}} t_i^zt_j^z, \end{equation}

with $\textit{Re}_{\theta _0}=16\,000$ . It must be noticed that the $\textit{Re}_{\theta }$ -dependence of the streamwise contribution in (3.5) suggests a finite value for the near-wall peak intensity $\overline {{u^{\prime }}^2}$ . However, as the contribution on $\overline {{w^{\prime }}^2}$ is not bounded with $\textit{Re}_{\theta }$ since it depends on ${l_t}^{(1)}$ through ${f_w}^{(1)}$ , the near-wall peak of $\overline {{u^{\prime }}^2}$ may still tend to infinity for large Reynolds number values because of the action of redistribution. As a reminder, the purpose of the work is not to discuss the theoretical behaviour of the near-wall peak intensity, but to focus on the practical modelling of the $\textit{Re}_{\tau }$ -dependence of the Reynolds tensor components, which will remain in this context limited to a restricted range of the Reynolds numbers. The introduction of the integral parameter $\textit{Re}_{\theta }$ is not suited to most actual computational fluid dynamics solvers – although some modern solvers are able to cope with integral parameters – since it makes the expression in (3.5) non-local, but the two-scale RSM is not intended to become an industrial RANS model due to its inherent complexity. Above all, this should be considered as proof of concept that will enable the development of industrialisable models at a later stage, particularly with a reduced number of transported variables of the first-order RANS model type.

To further enrich the original two-scale RSM, a Reynolds-number dependence was also applied to two other terms. First, the amplitude of ${\tilde {\varepsilon }}^{(1)}$ in the outer region was shown (Chedevergne et al. Reference Chedevergne, Coroama, Gleize and Bézard2024) to be overestimated for low $\textit{Re}_{\tau }$ values, leading to overpredicted Reynolds stress components of the LS contribution. Therefore, constant ${C_{\varepsilon _3}\!\!\!}^{(2)}$ was changed for $\displaystyle {{C_{\varepsilon _3}\!\!\!}^{(2)}}^1 + {{C_{\varepsilon _3}\!\!\!}^{(2)}}^2\tanh ( {\textit{Re}_{\theta }}/{\textit{Re}_{\theta _0}} )$ such that $\displaystyle {{C_{\varepsilon _3}\!\!\!}^{(2)}}^1 + {{C_{\varepsilon _3}\!\!\!}^{(2)}}^2={C_{\varepsilon _3}\!\!\!}^{(2)}$ and ${{C_{\varepsilon _3}\!\!\!}^{(2)}}^1 \approx 5{{C_{\varepsilon _3}\!\!\!}^{(2)}}^2$ . The second modification concerns the damping function ${f}^{(1)}_w$ , and more specifically ${\alpha }^{(1)}_w$ . At low to moderate Reynolds numbers, ${\alpha }^{(1)}$ could not reach its maximal value $1$ at $y^+=\textit{Re}_{\tau }$ , which also contributed to degraded predictions of the LS contribution at low Reynolds numbers. Instead of applying a Reynolds dependence to ${l}^{(1)}_t$ via the coefficient ${C_L\!}^{(1)}$ , we preferred to rearrange the elliptic equation in the form ${\alpha }^{(1)}- {{l}^{(1)}_t}^2{\nabla} ^2 {\alpha }^{(1)} = \min (0.5+0.5( {\textit{Re}_{\theta }}/{2\textit{Re}_{\theta _0}}),1 )$ and to normalise ${\alpha }^{(1)}$ by the factor $\displaystyle \min (0.5+0.5( {\textit{Re}_{\theta }}/{2\textit{Re}_{\theta _0}}),1 )$ so that it remains bounded between $0$ and $1$ . These two modifications have virtually no effect, or at least very limited effect, on results for $\textit{Re}_{\theta }\gt 16\,000$ , and only serve to accommodate some terms at low Reynolds number conditions, smoothing the transition towards the solution, independent of $\textit{Re}_{\tau }$ , as given by any standard turbulence models having a single scale.

Model constants are recapped in Appendix A. Minor modifications to constants were made to adapt to the introduction of the new formulation of term ${M}^{(1)}_{\textit{ij}}$ in (3.5), and to the use of $\textit{Re}_{\theta }$ -dependent functions.

4. High Reynolds number boundary layer applications

To go beyond the validation previously carried out on turbulent channel flows (Chedevergne et al. Reference Chedevergne, Coroama, Gleize and Bézard2024), which remains satisfied with modifications presented in § 3, we are turning to zero pressure gradient (ZPG) turbulent boundary layer configurations. The two-scale RSM was implemented in a RANS boundary layer code, developed at ONERA for several decades and called CLICET. The parabolic system of equations is solved at each station and marched in the streamwise direction. The code uses a second-order finite volume scheme together with adaptive grids to ensure convergence.

In order to challenge the model and reach very high Reynolds numbers, we are using data from the University of Melbourne and, notably, the measurements undertaken by Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018). But before considering these data, for which only streamwise mean and fluctuating velocity are available, DNS data provided by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2013) at $\textit{Re}_{\tau }\approx 2000$ ( $\textit{Re}_{\theta }\approx 6500$ ) are thoroughly examined. One-dimensional pre-multiplied spectra $\varPhi ^x_{\textit{ij}}$ are available in this dataset, but for a limited number of locations in the boundary layer thickness. Each spectrum is split into two parts with a cut-off wavelength $\lambda _z^+=1000$ , allowing an assessment of the SS and LS contributions, although it is not similar to the decomposition proposed by Lee & Moser (Reference Lee and Moser2019), and therefore not strictly coherent with respect to the model. Additionally, the power law used to discretise the spectra only leaves $15$ points above $\lambda _z^+=1000$ over the $1364$ available, introducing some uncertainty on the assessment of the LS contribution. Figure 5 shows the mean velocity profile and the three profiles of the diagonal stresses obtained with the two-scale RSM and compared to the DNS data. The agreement is satisfying given that the Reynolds number $\textit{Re}_{\tau }$ is still limited. The results support the choices made to include Reynolds number dependent functions to accommodate the model to low to moderate Reynolds number configurations. When plotting the SS and LS contributions (see figure 6), the agreement is all the more remarkable, with an energy splitting well distributed in the boundary layer thickness for all Reynolds stress components. A too abrupt separation between inner and outer contributions on the large scale can nevertheless be pointed out on $\overline {{u^{\prime }}^2}$ and $\overline {{w^{\prime }}^2}$ , without being able to say whether deviations are coming from the model or are due to the use of one-dimensional spectra. In any case, this has no major consequences on the complete profiles. Figure 6 also permits us to assess the weak impact of the spectra discretisation when comparing Reynolds stress profiles obtained from the integration over the whole spectra and those directly computed by Sillero et al. (Reference Sillero, Jiménez and Moser2013), proving the overall good estimation of the LS contributions, even though some errors may be observed in the outer region, particularly on $\overline {{w^{\prime }}^2}$ .

Figure 5. Profiles of the velocity (black) and the streamwise (red), wall-normal (blue) and spanwise (green) Reynolds stress components for a ZPG boundary layer at $\textit{Re}_{\tau }\approx 2000$ . Symbols are DNS data from Sillero et al. (Reference Sillero, Jiménez and Moser2013), and solid lines are results of the two-scale RSM.

Figure 6. The SS (triangle symbols) and LS (diamond symbols) contributions of the diagonal components of the Reynolds stress tensor for a ZPG boundary layer at $\textit{Re}_{\tau } \approx 2000$ obtained from partitioned one-dimensional spectra of Sillero et al. (Reference Sillero, Jiménez and Moser2013). Circle symbols correspond to diagonal components computed over the whole spectra. Dashed black lines are DNS results from Sillero et al. (Reference Sillero, Jiménez and Moser2013) as in figure 5. Solid lines are SS and LS contributions in the results of the two-scale model.

Experiments conducted by Samie et al. (Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) are simulated using the two-scale RSM to assess the performances of the model at high Reynolds number. These measurements, acquired using the nanoscale thermal anemometry probes (NSTAPs) manufactured at Princeton University (Vallikivi & Smits Reference Vallikivi and Smits2014), were obtained at four stations corresponding to dimensionless boundary layer thickness $\delta ^+=\textit{Re}_{\tau }\in \{6000,10\,000,14\,500,20\,000\}$ ( $\textit{Re}_{\theta }\in [14\,000,55\,000]$ ). Over this range of Reynolds numbers, high Reynolds number effects are clearly identifiable on the fluctuating velocity profiles. The inner and outer contributions of the large scale are indeed distinctly spatially separated at such high $\textit{Re}_{\tau }$ . In figure 7, the mean streamwise velocity profiles are plotted for the four considered cases.

Figure 7. Experimental (solid lines) mean velocity profiles in a ZPG boundary layer (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) at $\textit{Re}_{\tau }=6000,10\,000,14\,500,20\,000$ . Darker purple lines indicate increasing $\textit{Re}_{\tau }$ values. Corresponding purple dashed lines are results from the two-scale RSM. Mean velocity profiles are shifted up by $2$ units for each increasing value of $\textit{Re}_{\tau }$ .

Figure 8. Streamwise Reynolds stress component profiles (solid lines) from (Samie et al. Reference Samie, Marusic, Hutchins, Fu, Fan, Hultmark and Smits2018) at $\textit{Re}_{\tau }=6000,10\,000,14\,500,20\,000$ . Darker purple lines indicate increasing $\textit{Re}_{\tau }$ values. Corresponding (a) green and (b) purple dashed lines are results from the previous version of the two-scale RSM and the present one, respectively. Red dashed lines are results obtained with the EBRSM (Manceau Reference Manceau2015).

Mean profiles are well reproduced by the two-scale RSM although some differences are visible in the wake regions. The agreement reflects the good predictions of the SS and LS contributions to $\overline {u'v'}$ for all Reynolds numbers. In figure 8, fluctuating streamwise profiles are drawn for the previous version of the two-scale RSM (Chedevergne et al. Reference Chedevergne, Coroama, Gleize and Bézard2024) and for the one described above in order to highlight the improvements made by the new closure of the interscale energy fluxes. The most remarkable achievement of the present model is the agreement across all the $\overline {{u^{\prime }}^2}$ profiles. The first, trivial, observation that can be made from these results is that two-scale modelling enables behaviours in the outer region to be recovered, which is not possible with a single-scale RANS model. To exemplify this problem, results obtained with the elliptic blending RSM (EBRSM) of Manceau (Reference Manceau2015), which is a reference RANS model for calculating Reynolds stress profiles of wall-bounded flows, are plotted in figure 8. As the EBRSM uses only one length scale to characterise energy transfer, a single profile for $\overline {{u^{\prime }}^2}$ is obtained up to $y^+=1000$ , irrespective of the Reynolds number, which prevents the effects of high Reynolds numbers from being restored. Second, the $\textit{Re}_{\tau }$ -dependence at the $\overline {{u^{\prime }}^2}$ peak is fairly well captured by the new version of the two-scale RSM. However, there are a slight shift in the position of the peak, and some differences in amplitude. The dynamics with respect to the Reynolds number is nevertheless fairly faithful to the experimental data. Finally, a particularly noteworthy feature of the results obtained is the contribution of $\overline {{w^{\prime }}^2}$ in (3.5) on $\overline {{u^{\prime }}^2}$ profiles in the region covering values of $y^+$ ranging from $80$ to a few hundreds. Figure 8 clearly shows the improvements achieved by modifying the ${M}^{(1)}_{\textit{ij}}$ term (3.5) compared to the previous version of the two-scale RSM. The $\overline {{u^{\prime }}^2}$ profiles are strongly affected by the increase in Reynolds number in this region of the boundary layer. This effect on $\overline {{u^{\prime }}^2}$ is due to the redistribution term, since it is in fact the contribution on $\overline {{w^{\prime }}^2}$ that acts in this region. The modelling strategy, inspired by the analysis presented in § 2, therefore proves to be entirely relevant. Most interestingly, at these Reynolds numbers, the dependence functions introduced in the modelling (see § 3) have practically no effect. Thus the Reynolds evolution obtained is the result of the balance between the different terms of the $\overline {{u^{\prime }}^2}$ budget – more precisely, the budget of ${\overline {{u^{\prime }}^2}}^{(1)}$ – involving in particular interscale energy transfers within the inner and outer regions, and some pressure redistribution in the logarithmic zone due to the evolution of the Reynolds stress component $\overline {{w^{\prime }}^2}$ at large scales.