2. A multi-scale RSM

Let $\varphi _{ij}$ be the spherical mean of the Fourier transform of the two-point correlation tensor depending on the position $\displaystyle \boldsymbol {x}$ and the wavenumber $\kappa =\Vert \boldsymbol {\kappa } \Vert$, such that the Reynolds stress $R_{ij}$ is given by its integral:

(2.1)\begin{equation} \displaystyle R_{ij}(\boldsymbol{x}) = \int\limits_{0}^{\infty} \varphi_{ij}(\boldsymbol{x},\kappa) \,{\rm d}\kappa. \end{equation}

The corresponding spectrum can be split into $n+1$ slices as illustrated in figure 1. Formally, a multi-scale RSM derived from the approach proposed by Schiestel (Reference Schiestel2007) may be written for an incompressible fluid as follows:

(2.2)\begin{equation} \displaystyle \frac{{\rm D}R^{(m)}_{ij}}{{\rm D}t} = P^{(m)}_{ij}+ F^{(m-1)}_{ij}-F^{(m)}_{ij}+\varPhi^{(m)}_{ij} + D^{(m)}_{ij} - \varepsilon^{(m)}_{ij}, \end{equation}

with

(2.3)\begin{equation} \displaystyle R_{ij}^{(m)}={\overline{u'_iu'_j}}^{(m)}=\int\limits_{\kappa_{m-1}}^{\kappa_m} \varphi_{ij} \,{\rm d}\kappa. \end{equation}

The overbar symbol is used for the Reynolds-averaged quantities whereas prime is used for fluctuating quantities. Here $P^{(m)}_{ij}=-R^{(m)}_{ik}({\partial \overline {u_j}}/\partial {x_k})-R^{(m)}_{jk}({\partial \overline {u_i}}/\partial {x_k})$ is the exact production term in slice $m$, $\varPhi ^{(m)}_{ij}$ is the pressure–strain correlation corresponding to the redistribution term in the slice, ${D}^{(m)}_{ij}$ the diffusion term including the viscous diffusion, the turbulent diffusion, denoted by ${T}^{(m)}_{ij}$ in the following, and the pressure diffusion and $\varepsilon ^{(m)}_{ij}$ the dissipation associated with slice $m$. The velocity–pressure gradient correlation is here decomposed into the pressure–strain correlation $\varPhi _{ij}$ and the pressure diffusion term. The flux term $F^{(m)}_{ij}$ denotes the rate of energy transfer leaving slice $m$. Therefore, in the last slice all the outgoing energy is dissipated and $F^{(n)}_{ij}=\varepsilon ^{(n+1)}$.

Figure 1. Spectrum partitioning of the two-point correlation tensor $\varphi _{ij}$. In each slice $m$ such that $\kappa \in [{\kappa }^{(m-1)},{\kappa }^{(m)}]$, ${P}^{(m)}_{ij}$ is the corresponding production term, ${D}^{(m)}_{ij}$ the diffusion term, ${\varPhi }^{(m)}_{ij}$ the redistribution term and ${F}^{(m)}_{ij}$ the rate of transfer tensor. The one-point velocity correlation ${R}^{(m)}_{ij}$ is obtained by integration of $\varphi _{ij}$ over slice $m$.

Implicitly, the above partitioning means that any flow quantity $q$ is averaged from an extended form of the Reynolds decomposition which reads

(2.4)\begin{equation} \displaystyle q = \bar{q} + \sum_{j=1}^{n} q^{'(j)}. \end{equation}

To model the energy flux, Schiestel (Reference Schiestel2007) assumed that the slice width is given by a length scale related to the turbulent kinetic energy flux $F^{(m)}=\tfrac {1}{2}F^{(m)}_{ii}$ such that

(2.5)\begin{equation} \kappa(m)-\kappa(m-1) \propto \frac{F^{(m)}}{k^{{(m)}^{3/2}}}. \end{equation}

When deriving the $\varphi _{ij}$ equation (Schiestel Reference Schiestel2007) and then applying (2.3), it is shown that the transfer rate ${F}^{(m)}$ is composed of an inertial transfer contribution ${\mathcal {F}_t}^{(m)}$ from the energy cascade, a rapid transfer contribution ${\mathcal {F}_u}^{(m)}$ due to straining of turbulence through the action of the mean velocity gradients and a contribution due to the variations in the splitting wavenumbers, which gives

(2.6)\begin{equation} {F}^{(m)} = {\mathcal{F}_t}^{(m)} + {\mathcal{F}_u}^{(m)} - K\left(\kappa\right)\frac{\partial \kappa}{\partial t}, \end{equation}

where $K$ is the turbulent kinetic energy density in the spectral space such that ${k}^{(m)}=\int _{\kappa (m-1)}^{\kappa (m)}K \,{\rm d}\kappa$. We remark that in developed turbulent channel flows, the contribution ${\mathcal {F}_u}^{(m)}$ is zero due to homogeneity in the horizontal ($x$ and $z$) directions. From the two previous relations (2.5) and (2.6) and after a little algebra, Schiestel (Reference Schiestel2007) proves that ${F}^{(m)}$ is ruled by a transport equation of the general form:

(2.7)\begin{equation} \frac{{\rm D}F^{(m)}}{{\rm D}t} = \frac{F^{(m)}}{k^{(m)}}\left( C_1^{(m)}P^{(m)}+ C_2^{(m)}F^{(m-1)}- C_3^{(m)}F^{(m)} - C_4^{(m)}\varepsilon^{(m)}\right). \end{equation}

Finally, a formulation inspired by Rotta hypothesis is retained to assess $F_{ij}^{(m)}$:

(2.8)\begin{equation} F_{ij}^{(m)}=\frac{F^{(m)}}{k^{(m)}}\left(A^{(m)}R_{ij}^{(m)}+\frac{2}{3}\left(1-A^{(m)}\right)k^{(m)}\delta_{ij}\right), \end{equation}

with $A^{(m)}$ a constant ranging from $0$ to $1$. For slices $m$ involving large wavenumbers where isotropy is expected, $A^{(m)}$ is assumed to take a value close to zero, whereas for slices $m$ in the region of low wavenumbers, $A^{(m)}$ may reach values close to unity. More complicated attempts were also proposed by Schiestel (Reference Schiestel2007) to model the energy transfer, in particular to account for the fast transfer contribution which also causes energy redistribution among the Reynolds stress tensor components.

All previous developments are valid for free flows, without consideration of wall effects. Although some proposals were formulated by Schiestel (Reference Schiestel2007) to incorporate near-wall behaviour with ad hoc wall functions, no complete multi-scale RSM was fully derived to compute wall-bounded flows. This is one of the most challenging aspects of developing a multi-scale model, and was a stumbling block to date. For the description of wall-bounded flows with emphasis on the similarities existing in the near-wall region, physical quantities are generally made dimensionless using the kinematic viscosity $\nu$ and the friction velocity $u_{\tau }$. The sign $+$ is used in the following to denote dimensionless quantities.

To revisit the approach developed by Schiestel and implement it to capture high-Reynolds-number effects in channel flows, the spectrum is split into three parts ($n=2$, see figure 1) in analogy with the scale separation proposed by Lee & Moser (Reference Lee and Moser2019) and, thus, forming a two-scale model. The EBRSM model of Manceau (Reference Manceau2015) is considered as the base model from which the closure relations for the pressure–strain correlation and the turbulent diffusion are taken. In order to distinguish dissipation from energy transfers between spectral bands, which are mixed up in single-scale models, we slightly modify Manceau's model by decomposing dissipation into a homogeneous part $\tilde {\varepsilon }$, corresponding to the energy flux, and a near-wall inhomogeneous part $\varepsilon _w$, as is conventionally done (Jones & Launder Reference Jones and Launder1972; Gleize et al. Reference Gleize, Schiestel and Couaillier1996). Numerical tests on the EBRSM model showed the best agreement with the original model for $\varepsilon = \tilde {\varepsilon } + \varepsilon _w$, where

(2.9)\begin{equation} \varepsilon_w=\frac{2\nu k}{y^{2}}\left[\frac{1}{y^{+^{0.8}}}\tanh \left(\frac{y^+}{2.5}\right) + \left(1-\tanh \left(\frac{y^+}{2.5}\right)\right)\right], \end{equation}

which guarantees correct theoretical behaviour at the wall. In Schiestel's approach, the transfer rate $F^{(m)}$ is directly related to the spectral width of the slice. By adjusting the level of these fluxes, we can adjust the location of the cut-off in spectral space. In the analysis proposed by Lee & Moser (Reference Lee and Moser2019), the energy transfers are found to take place in the logarithmic region, well beyond the buffer layer and the viscous sublayer. This reflects the fact that the large structures transfer energy to the small structures and that the large structures are essentially located further from the wall than the small ones (Hutchins & Marusic Reference Hutchins and Marusic2007b). Hence, in order to position the spectral cut-off for a wavelength $\lambda ^+=1000$ as Lee and Moser did, the damping functions associated with the two transported scales are judiciously adjusted. In the EBRSM model, the damping is controlled by a blending function derived from an elliptic equation. In our two-scale model, the coefficients governing each of the elliptical equations associated with the transported scales can be used to adjust the position of the spectral cut-off.

Lee & Moser (Reference Lee and Moser2019) showed that partitioning the spectral space into a LS contribution and a SS contribution with a cut-off wavenumber fixed at $\lambda ^+=1000$ is an appropriate choice for describing the near-wall behaviour of channel flows. This corroborates previous observations on wall flows where two distinct turbulent scales can be identified, one characterising the near-wall cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999) and a second characterising the ‘superstructures’ appearing at high Reynolds numbers (Hutchins & Marusic Reference Hutchins and Marusic2007a). To comply with these findings and to limit the number of transport equations, the multi-scale approach is applied by considering only two scales. As mentioned above, the original EBRSM model, which is implicitly based on the single-scale hypothesis, admits a unique solution in the inner region, irrespective of the Reynolds number. Lee and Moser have shown that the contributions of the SS are practically independent of the Reynolds number, which means that the EBRSM formulation can be reused almost identically to describe the $m=2$ (${\rm SS}= {}^{(2)}$) spectral slice. In addition, for this slice, the outgoing energy flux is equal to the viscous dissipation, which confirms the idea of reusing the equation for $\varepsilon$ from the EBRSM model with minor adjustments to take account of certain effects such as energy transfers from LS contributions of slice $m=1$ (${\rm LS}= {}^{(1)}$).

The pressure–strain correlation of the EBRSM model (see Appendix A) is composed of the Speziale–Sarkar–Gatski (SSG) model (Speziale, Sarkar & Gatski Reference Speziale, Sarkar and Gatski1991) for its homogeneous part and, in order to be consistent with the strong anisotropy encountered near the walls, includes an inhomogeneous contribution taken from the work of Manceau & Hanjalić (Reference Manceau and Hanjalić2002). This closure formulation is left unchanged in the two-scale RSM for both spectral slices to model ${\varPhi }^{(m)}_{ij}$ terms. This choice is questionable, particularly for the inhomogeneous part close to the wall in the slice $m=1$ (LS). The use of SSG model for the homogeneous part, which performs satisfactorily in the EBRSM model, can legitimately be applied to the slice $m=2$ (SS) and its use in the slice $m=1$ (LS), fairly far from the wall, seems reasonable, although Schiestel (Reference Schiestel2007) showed that increasing coefficients for the linear part should be considered as wavenumbers increase in the case of homogeneous anisotropic turbulence subjected to strain. The use of the inhomogeneous part defined by Manceau & Hanjalić (Reference Manceau and Hanjalić2002) for the slice $m=2$ also seems coherent. For $m=1$ there is no evidence to justify this choice, but the inhomogeneous contribution for the slice $m=1$ is almost zero, since by construction the slice $m=1$ is essentially acting far from the wall. Similarly, the Daly & Harlow (Reference Daly and Harlow1970) formulation used for turbulent diffusion is directly reused in the transport equations for partial Reynolds stresses ${R}^{(m)}_{ij}$.

Analysis of Lee and Moser's DNS clearly shows that the transfer rate term ${F}^{(1)}$ from LS to SS essentially acts from the logarithmic region onwards and that no transfer from LS to SS exists in the buffer layer, particularly around the peak of turbulence kinetic energy about $y^+=15$. However, as demonstrated by Marusic et al. (Reference Marusic, Mathis and Hutchins2010a), the LS acting in the outer region modulate the behaviour of the SS in the buffer layer, leading notably to an increase in the peak near the wall of $R_{11}$ as the Reynolds number increases. To take this modulation effect into account, a term denoted ${M}^{(1)}_{ij}$ is added to the transport equation of ${R}^{(1)}_{ij}$ to reproduce the action of the LS in the inner region.

In the present two-scale context, the extended Reynolds decomposition (2.4) reduces to $q=\bar {q}+{q'}^{(1)}+{q'}^{(2)}$ for any quantity $q$. Therefore, the Reynolds tensor is decomposed into two parts $R_{ij}={R}^{(1)}_{ij}+{R}^{(2)}_{ij}$. To comply with the usual notation used in RANS modelling, we note ${\tilde {\varepsilon }}^{(m)}={F}^{(m)}$, $m\in \{1,2\}$, in what follows. Partial energy dissipation ${\varepsilon }^{(m)}=\frac {1}{2}{\varepsilon }^{(m)}_{ii}$ can thus be broken down into a transfer component ${\tilde {\varepsilon }}^{(m)}$ and a near-wall viscous dissipation component ${\varepsilon }^{(m)}_w$. To complete the two-scale RSM, a transport equation must be defined for the flux ${F}^{(1)}={\tilde {\varepsilon }}^{(1)}$. The model equation (2.7) is completed with a diffusion term to resemble a standard transport equation. Ultimately, the two-scale RSM reads

(2.10) \begin{equation} \left.\begin{gathered} \frac{{\rm D}{R}^{(1)}_{ij}}{{\rm D}t} ={-}{R}^{(1)}_{ik}\frac{\partial \overline{u_j}}{\partial x_k}-{R}^{(1)}_{jk} \frac{\partial \overline{u_i}}{\partial x_k} + {\varPhi}^{(1)}_{ij} - {\varepsilon}^{(1)}_{ij} + {M}^{(1)}_{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)}_{ij}}{\partial x_m}\right] ,\\ \frac{{\rm D}{R}^{(2)}_{ij}}{{\rm D}t} ={-}{R}^{(2)}_{ik}\frac{\partial \overline{u_j}}{\partial x_k}-{R}^{(2)}_{jk}\frac{\partial \overline{u_i}}{\partial x_k} + {\tilde{\varepsilon}}^{(1)}_{ij} + {\varPhi}^{(2)}_{ij} - {\varepsilon}^{(2)}_{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)}_{ij}}{\partial x_m}\right] ,\\ \frac{{\rm D}{\tilde{\varepsilon}}^{(1)}}{{\rm D}t} = \frac{{C'^{(1)}_{\varepsilon_1}}{P_k}^{(1)}- {C^{(1)}_{\varepsilon_2}}{\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] ,\\ \frac{{\rm D}{\tilde{\varepsilon}}^{(2)}}{{\rm D}t} = \frac{{C'^{(2)}_{\varepsilon_1}}{P_k}^{(2)}- {C'^{(2)}_{\varepsilon_2}}{\tilde{\varepsilon}}^{(2)} +{C^{(2)}_{\varepsilon_3}} {\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] ,\\ {\alpha}^{(1)} - \displaystyle {{l}^{(1)}_t}^2\nabla^2 {\alpha}^{(1)} = 1;\quad{\alpha}^{(2)} - \displaystyle {{l}^{(2)}_t}^2\nabla^2 {\alpha}^{(2)} = 1. \end{gathered}\right\} \end{equation}

Concerning the partial dissipation tensors ${\varepsilon }^{(1)}_{ij}$ and ${\varepsilon }^{(2)}_{ij}$, the approach followed by Manceau (Reference Manceau2015) is modified slightly by generalising the formulation used far from the walls. Instead of considering an isotropic distribution of ${\varepsilon }^{(m)}_{ij}$, the possibility of taking into account some anisotropy far from the walls is introduced:

(2.11) \begin{align} \left.\begin{gathered} {\varepsilon}^{(m)}_{ij} = \displaystyle \left(1-{f_w}^{(m)}\right)\frac{{R}^{(m)}_{ij}}{{k}^{(m)}}{\varepsilon}^{(m)} + {f_w}^{(m)}\left(C_{\varepsilon}^{(m)}\frac{{R}^{(m)}_{ij}}{{k}^{(m)}}{\varepsilon}^{(m)}+\left(1-C_{\varepsilon}^{(m)}\right)\frac{2}{3}{\varepsilon}^{(m)}\delta_{ij}\right);\\ {\varepsilon}^{(m)} = \displaystyle {\tilde{\varepsilon}}^{(m)} + {\varepsilon}^{(m)}_w;\quad {\varepsilon}^{(m)}_w = \frac{2\nu{k}^{(m)}}{y^{2.8}}\tanh \left(\frac{y}{2.5}\right) + \frac{2\nu{k}^{(m)}}{y^2}\left(1-\tanh \left(\frac{y}{2.5}\right)\right). \end{gathered}\right\} \end{align}

If the original approach of Manceau (Reference Manceau2015) was followed, the coefficients would have been $C_{\varepsilon }^{(m)}=0$. In practice, for the LS contribution, $C_{\varepsilon }^{(1)}=0$ is used, but taking $C_{\varepsilon }^{(2)}=0.3$ slightly improves the results. The transfer rate tensor ${\tilde {\varepsilon }}^{(1)}_{ij}$, which enters the transport equation of ${R}^{(2)}_{ij}$ is also modelled using the same ideas:

(2.12) \begin{equation} {\tilde{\varepsilon}}^{(1)}_{ij} = \displaystyle \left(1-{f_w^{(2)}}\right)\frac{{R}^{(1)}_{ij}}{{k}^{(1)}}{\tilde{\varepsilon}}^{(1)} + {f_w^{(2)}}\left( C_{\tilde{\varepsilon}}\frac{{R}^{(1)}_{ij}}{{k}^{(1)}}{\tilde{\varepsilon}}^{(1)} + \left(1-C_{\tilde{\varepsilon}}\right)\frac{2}{3}{\tilde{\varepsilon}}^{(1)}\delta_{ij}\right). \end{equation}

Here $C_{\tilde {\varepsilon }}=0.3$ was chosen as the best compromise. The previous formulations for ${\varepsilon }^{(m)}_{ij}$ and ${\tilde {\varepsilon }}^{(1)}_{ij}$ are discussed in § 3 from the DNS results.

The damping functions ${f_w^{(m)}}$, blending the homogeneous and inhomogeneous parts of ${\varPhi }^{(m)}_{ij}$, ${\varepsilon }^{(m)}_{ij}$ and ${\tilde {\varepsilon }}^{(1)}_{ij}$, are calculated directly from ${\alpha }^{(m)}$ and are written as ${f_w^{(1)}}={{\alpha }^{(1)}}^2$ and ${f_w}^{(2)}={{\alpha }^{(2)}}^3$. They play a key role in the spatial separation of turbulent scales. To complete the model, characteristic turbulent timescale ${t}^{(m)}_t$ and length scale ${l}^{(m)}_t$ are taken identically from the relations used by Manceau (Reference Manceau2015) and originally introduced by Durbin (Reference Durbin1991):

(2.13)\begin{equation} \left.\begin{gathered} \displaystyle {t}^{(m)}_t = \displaystyle \max\left(\frac{{k}^{(m)}}{{\varepsilon}^{(m)}},{C_T}^{(m)} \left(\frac{\nu}{{\varepsilon}^{(m)}}\right)^{{1}/{2}}\right),\\ \displaystyle {l}^{(m)}_t = \displaystyle {C_L}^{(m)}\max\left(\frac{{{k}^{(m)}}^{{3}/{2}}}{{\varepsilon}^{(m)}}, {C_\eta}^{(m)}\left(\frac{\nu^{3}}{{\varepsilon}^{(m)}}\right)^{{1}/{4}}\right). \end{gathered}\right\} \end{equation}

The modulation term ${M}^{(1)}_{ij}$ was added to the transport equations of ${R}^{(1)}_{ij}$ to reproduce the increase in the near-wall peaks with Reynolds number only observed for ${R}^{(1)}_{11}$ and ${R}^{(2)}_{33}$ in the DNS calculations of Lee & Moser (Reference Lee and Moser2019). In wall-parallel bounded flow, with no transverse pressure gradient, ${M}^{(1)}_{ij}$ must thus have two non-zero components, and in this sense it can be written as aligned with $\delta _{ij}-n_in_j$. However, in the present study, no generalisable expression for this term was found and we simply use a formulation adapted to developed turbulent channel flows. The terms are made proportional to the production of the turbulent kinetic energy of slice $m=2$ and to the ratio of the Reynolds stress component to ${k}^{(2)}$:

(2.14) \begin{equation} \left.\begin{gathered} \displaystyle {M}^{(1)}_{11} = C_{m}\left(1-{{f_w^{(2)}}}\right)\frac{{P_k}^{(2)}}{{k}^{(2)}}{R}^{(1)}_{11},\\ \displaystyle {M}^{(1)}_{33} = C_{m}\left(1-{{f_w^{(1)}}}\right)\frac{{P_k}^{(2)}}{{k}^{(2)}}{R}^{(1)}_{33}. \end{gathered}\right\} \end{equation}

Although the model proposed in (2.14) is not rotationally invariant, it can nevertheless demonstrate the influence of the LS contribution in the inner region as the Reynolds number increases.

The coefficients involved in the two-scale RSM model are given in Appendix A.

3. High-Reynolds-number channel flows application

The two-scale RSM model described above is applied on the turbulent plane channel configuration for the four friction Reynolds, i.e. $Re_{\tau }=\{550;1000;2000;5200\}$, treated by Lee and Moser. A specific one-dimensional (1-D) channel code was developed for this purpose. Details on the numerical procedure are given in Appendix B. Figure 2 shows the mean velocity profiles on the left and the evolution of the cut-off wavelength $\lambda ^+$ in the boundary layer thickness on the right. The latter was assessed directly using the assumption proposed by Schiestel (Reference Schiestel2007) $\lambda ^+=2{\rm \pi} ({{{{k}^{(1)}}^+}^{{3}/{2}}}/{{\displaystyle {\tilde {\varepsilon }}^{(1)}}^+})$ given by (2.5) since ${\kappa }^{(0)}=0$. The mean velocity profiles obtained with the two-scale RSM model are very similar to those obtained with the EBRSM model and match the DNS profiles. We also noticed that the results between the two versions of the EBRSM model, with or without the decomposition $\varepsilon = \tilde {\varepsilon } +\varepsilon _w$, are almost identical and only the results from the original model are presented. Since in slice $m=1$ the turbulent kinetic energy ${k}^{(1)}=\frac {1}{2}{R}^{(1)}_{ii}$ and the transfer rate ${\varepsilon }^{(1)}=\frac {1}{2}{\varepsilon }^{(1)}_{ii}$ are dominated by their first diagonal component, i.e. ${R}^{(1)}_{11}={\overline {u^{\prime 2}}}^{(1)}$ and ${\varepsilon }^{(1)}_{11}$ (see figures 3 and 6), $\lambda$ can be considered as the characteristic scale of eddies carrying ${\overline {u^{\prime 2}}}^{(1)}$. These eddies are known to be very elongated in the longitudinal direction $x$ (Hutchins & Marusic Reference Hutchins and Marusic2007a) and therefore $\lambda$ can be equated with a longitudinal wavelength $\lambda _x$. This explains why $\lambda ^+$ can reach very large values, well above the dimensionless height of the channel $Re_{\tau }$. Figure 2 indicates that the evolution of $\lambda ^+$ is not very sensitive to $Re_{\tau }$. However, $\lambda ^+$ is not constant in $y^+$. Here $\lambda ^+$ is around $1000$ in the inner region and increases with $y^+$, which is consistent with the results of Lee and Moser. Indeed, the spectra show a shift in the LS contribution towards longer wavelengths as $y^+$ increases. The model therefore appears to be able to predict a separation between the SS and LS contributions as observed in DNS.

Figure 2. (a) Velocity profiles for the different Reynolds numbers, $Re_{\tau }\in [550,1000,2000,5200]$. Profiles are shifted up two units as $Re_{\tau }$ increases. Darker symbols or lines indicate higher $Re_{\tau }$ values. (b) Cut-off wavelength estimates (see (2.5)) obtained with the two-scale RSM for the four Reynolds numbers.

Figure 3. Reynolds stress diagonal component profiles. The Reynolds number value $Re_{\tau }$ increase from top to bottom. For the sake of clarity, the SS, LS and total contributions of the two-scale RSM results are all drawn with solid lines. The y-axis label is given on top for each column.

In figure 3, the SS and LS contributions of the diagonal Reynolds stresses are plotted for all $Re_{\tau }$ numbers. First, the figure shows that the EBRSM model behaves well even though it is unable to capture the effects of the Reynolds number, such as the evolution of the $\overline {u^{\prime 2}}$ peak. The two-scale model performs very well at both SS and LS, and is able to give a fairly accurate account of Reynolds-number effects. Some notable weaknesses can be pointed out in the model concerning the LS contribution on $\overline {v^{\prime 2}}$ and $\overline {w^{\prime 2}}$ or at low Reynolds numbers, but the overall agreement is very satisfactory and was hitherto unattainable with RANS models. The model predicts a SS contribution almost independent of $Re_{\tau }$ as found in the DNS. The LS contribution is twofold, with on the one hand a bump appearing on each of the diagonal stress in the logarithmic region and on the other the modulation effect occurring in the buffer region for $\overline {u^{\prime 2}}$ and $\overline {w^{\prime 2}}$. The LS contribution in the logarithmic region is slightly overpredicted when the Reynolds number diminished, which reflects the limits of this approach specifically designed to handle high-Reynolds-number effects where the scale separation is large enough. The LS contribution is even more overestimated for the $\overline {v^{\prime 2}}$ and $\overline {w^{\prime 2}}$ components. However, the modulation effects occurring in the buffer layer for $\overline {u^{\prime 2}}$ and $\overline {w^{\prime 2}}$ are reasonably captured for all $Re_{\tau }$ numbers. According to the DNS results of Lee & Moser (Reference Lee and Moser2019), there is no effect of modulation on the $\overline {v^{\prime 2}}$ component, which somewhat mitigates the view that the inner region ‘sees’ a modified mean outer condition (Marusic et al. Reference Marusic, Baars and Hutchins2017). If this was the case, all the Reynolds stresses would be affected by the modulation by the change in the inner scaling.

A further analysis of the results is presented in figure 4 where the SS and LS contributions to the production term, the transfer rate and the dissipation of the turbulent kinetic energy are presented. To emphasise the LS contributions all terms are multiplied by $y^+$. The SS and LS contributions to the production terms ${P_k}^{(1)}$ and ${P_k}^{(2)}$ are well-described by the present two-scale model. The independence of ${P_k}^{(2)}$ on $Re_{\tau }$ is satisfied and the growing contribution of the LS with $Re_{\tau }$ fairly captured, even though the agreement is not perfect, especially for low Reynolds numbers. From a modelling perspective, the recovery of the dissipation ${\varepsilon }^{(2)}$ and, more importantly, of the transfer rate ${\tilde {\varepsilon }}^{(1)}$ are remarkable. Concerning ${\varepsilon }^{(2)}$, the agreement with DNS results is very satisfactory and reflects the good ability of the transport equations for the SS $m=2$ to mimic the self-sustained near-wall cycle (Jiménez & Pinelli Reference Jiménez and Pinelli1999) and to maintain it independent of the outer-layer turbulence. However, the most noticeable agreement is that on the transfer rate ${\tilde {\varepsilon }}^{(1)}$. Discrepancies can be seen with the DNS results regarding the evolution with $Re_{\tau }$ or the amplitude of the term but the two-scale offers a very reasonable description of this rate of transfer. This is a key component of the two-scale model which is absent of single-scale-based RANS models since the integral of the transfer rate over the all wavenumber spectrum is zero.

Figure 4. Turbulent kinetic energy budget for production $({P_k}^{(m)})$, transfer rate $({\tilde {\varepsilon }}^{(1)})$ and dissipation terms $({\varepsilon }^{(2)})$, as log densities. Darker grey indicates higher Reynolds number $Re_{\tau }$.

In order to validate the assumptions made in the model (2.11) and (2.12) about the anisotropy of the dissipation tensor ${\varepsilon }^{(2)}_{ij}$ and the transfer rate tensor ${\tilde {\varepsilon }}^{(1)}_{ij}$, the evolution of the diagonal components of these terms are plotted on figures 5 and 6. Only the highest Reynolds number $Re_{\tau }=5200$ is presented since the SS terms are almost independent of $Re_{\tau }$. Similar results are obtained for the lower Reynolds numbers. The results for the EBRSM model are also shown in figure 5 since the vast majority of the energy is dissipated on SS and so we have $\varepsilon _{ij} \approx {\varepsilon }^{(2)}_{ij}$. The tensorial expression in (2.11), also used by Manceau (Reference Manceau2015) in the EBRSM with $C_{\varepsilon }=0$, turns out to provide convincing agreement with the DNS results allowing a good representation of the anisotropy. The strong anisotropy observed in the inner region is well-reproduced by this formulation. The use of ${C^{(2)}_{\varepsilon }}=0.3$ for the two-scale model delays the expected return to isotropy at large values of $y^+$ as the energy decreases on each of the components. However, slight improvements are observed in the inner region (up to $y^+=300$) with ${C^{(2)}_{\varepsilon }}=0.3$ in the two-scale RSM results where the anisotropy is enhanced. The benefit of using ${C^{(2)}_{\varepsilon }}=0.3$ is more tangible on the Reynolds stress profiles of figure 3, although the effect is quite moderate. A lesser match is obtained for the diagonal components of the transfer rate tensor ${\tilde {\varepsilon }}^{(1)}_{ij}$. The choice made for modelling ${\tilde {\varepsilon }}^{(1)}_{ij}$ turns out to be a good approximation of what is observed at high Reynolds numbers, i.e. for $Re_{\tau }=5200$ in figure 6. The introduction of constant $C_{\tilde {\varepsilon }}=0.3$ in (2.12) enables to recover a correct breakdown between the diagonal components of ${\tilde {\varepsilon }}^{(1)}_{ij}$. Even far from the wall, the transfer rate tensor is still strongly anisotropic. More surprising, contrasted effects are observed on the anisotropy as the Reynolds number changes. The ratios ${{\tilde {\varepsilon }}^{(1)}_{22}}/{{\tilde {\varepsilon }}^{(1)}_{11}}$ and ${{\tilde {\varepsilon }}^{(1)}_{33}}/{{\tilde {\varepsilon }}^{(1)}_{11}}$ increase with $Re_{\tau }$ while ${{\tilde {\varepsilon }}^{(1)}_{22}}/{{\tilde {\varepsilon }}^{(1)}_{33}}$ drops as $Re_{\tau }$ increases. These trends are poorly captured by the model. According to (2.12), $C_{\tilde {\varepsilon }}$ should be a function of $Re_{\tau }$, which is not desirable from a numerical point of view. This shows quite clearly the limitations of the model adopted for ${\tilde {\varepsilon }}^{(1)}_{ij}$. Despite these limitations, the agreement with the DNS data is reasonable and this partly explains the good behaviour of the two-scale model on the SS and LS contributions (figure 3).

Figure 5. Diagonal components of the dissipation tensors (a) ${\varepsilon }^{(2)}_{ij}$ and (b) $\varepsilon _{ij}$ at $Re_{\tau }=5200$.

Figure 6. Diagonal components of the transfer rate tensor ${\tilde {\varepsilon }}^{(1)}_{ij}$ at different $Re_{\tau }$ values. Darker grey indicates higher Reynolds number $Re_{\tau }$.

To better understand the origin of the modulation effect observed in the buffer region (figure 3) and whose contribution is due to the LS, figures 7 and 8 present the budgets for ${\overline {u^{\prime 2}}}^{(1)}$ and ${\overline {w^{\prime 2}}}^{(1)}$ for $y^+<1000$ at $Re_{\tau }=5200$, respectively. Similar trends but less pronounced are observed at lower Reynolds numbers. The budgets of ${\overline {v^{\prime 2}}}^{(1)}$ for all $Re_{\tau }$ values do not show any LS contribution in the buffer layer as it was expected from figure 3. The very near-wall balance in the viscous sublayer is governed by viscous diffusion and dissipation. Above, for $y^+\in [2,20]$, the budget for ${\overline {u^{\prime 2}}}^{(1)}$ is governed by a balance between the LS contributions of production, rate of transfer and turbulent diffusion on the one hand and viscous diffusion and dissipation on the other, the LS contribution of the pressure–strain correlation being almost zero. The dominant positive contribution is due to turbulent diffusion $T_{11}^{(1)}$ which must be seen as the transport by turbulence of the ${\overline {u^{\prime 2}}}^{(1)}$ in the wall-normal direction $y$ as pointed out by Lee & Moser (Reference Lee and Moser2019). A positive contribution is also due to $-\tilde {\varepsilon}_{11}^{(1)}$ involving energy transfer from the SS to the LS. This backscatter is not accessible with the current model since the transfer rate tensor ${\tilde {\varepsilon }_{ij}^{(1)}}$ has only positive values, i.e. only energy transfer from LS to SS are accounted for. From $y^+=5$ a small contribution of the production term $P_{11}^{(1)}$ manifests, the balance above $y^+ =20$ being mainly governed by the production and the rate of transfer. The two-scale RSM uses the term $M_{11}^{(1)}$ to represent the various contributions of LS in the buffer layer and figure 7 shows an overall good agreement of this term with the sum ${{T}^{(1)}_{11}}-{{\tilde {\varepsilon }}^{(1)}_{11}}+{{P}^{(1)}_{11}}+{{\varPhi }^{(1)}_{11}}$ despite a slight shift in $y^+$ and a small underestimation of the amplitude. The picture is slightly different regarding the budget of ${\overline {w^{\prime 2}}}^{(1)}$. Since the production is zero for ${\overline {w^{\prime 2}}}^{(1)}$, the rate transfer $-\tilde {\varepsilon }_{33}^{(1)}$ is balanced with the redistribution term $\varPhi _{33}^{(1)}$ for large $y^+$ values, i.e. $y^+>40$. In the buffer layer, the balance involves the turbulent transport $\displaystyle T_{33}^{(1)}$ and a backscatter flux from SS with negative values of $\tilde{\varepsilon }_{33}^{(1)}$ observed for $y^+\in [2,40]$. The model (2.14) for the $M_{33}^{(1)}$ term gives a reasonable agreement compared with the positive contribution ${{T}^{(1)}_{33}}-{{\tilde {\varepsilon }}^{(1)}_{33}}+{{\varPhi }^{(1)}_{33}}$ obtained in the DNS, still with a shift in $y^+$ and this time an overprediction of the amplitude. The LS contribution for $\overline {w^{\prime 2}}$ is more spread out than for $\overline {u^{\prime 2}}$, resulting in a wider spread of ${\overline {w^{\prime 2}}}^{(1)}$ and a more pronounced peak for ${\overline {u^{\prime 2}}}^{(1)}$ around $y^+=15$ (figure 3). To conclude on the so-called modulation effect, the DNS results show that, in fact, there are multiple contributions, involving in particular turbulent transport and backscattering, and affecting only the $R_{11}$ and $R_{33}$ components of the Reynolds tensor. It is difficult to judge from these data alone the precise origin of these transfers and the underlying mechanisms. The modelling proposed in the multi-scale model framework reflects the fact that backscatter cannot be considered, i.e. the rate of transfer components ${\tilde {\varepsilon }}^{(1)}_{ij}$ are strictly positive in the current approach. The modulation model (2.14) that gathers the main contributions is nevertheless consistent for the two contributions to the stresses $R_{11}$ and $R_{33}$.

Figure 7. Budget of ${R}^{(1)}_{11}={\overline {u^{\prime 2}}}^{(1)}$ for $Re_{\tau }=5200$. (a) DNS contributions. (b) Comparisons between sums of different DNS contributions and the modulation term (2.14) of the two-scale model.

Figure 8. Budget of ${R_{33}}^{(1)}={\overline {w^{\prime 2}}}^{(1)}$ for $Re_{\tau }=5200$. (a) DNS contributions. (b) Comparisons between sums of different DNS contributions and the modulation term (2.14) of the two-scale model.