2.2.1. Conditional GAN

Conditional GAN (cGAN) (Mirza & Osindero Reference Mirza and Osindero2014) learns to generate data (output) from a given condition (input) from the annotated set of data. For the purposes of this study, cGAN can be considered as a mapping function that transforms the input data to the output. Numerous studies have extended cGAN for use in super-resolution (cf. the review by Tian et al. (Reference Tian, Zhang, Lin, Zuo, Zhang and Lin2022)), and is used in this study as the supervised machine learning model for super-resolution from fDNS data to DNS-quality flows. In the following, the employed formulation is described.

A schematic of the employed cGAN architecture is shown in figure 1. In cGAN used in this study, the generator network takes the low-resolution flow fields as input and tries to output the corresponding high-resolution flow fields. The discriminator takes either the output of the generator or the high-resolution (e.g. DNS) flow fields and tries to discern between the ‘real’ data in the training dataset (DNS data) and the generated ‘fake’ data (DNS-quality data generated by the generator). Because of the need to provide the corresponding high-resolution flow data for each low-resolution data as the paired training dataset, cGAN is classified as a supervised learning model.

To make the training procedure more stable, we employ Wasserstein GAN with gradient penalty (Gulrajani et al. Reference Gulrajani, Ahmed, Arjovsky, Dumoulin and Courville2017). Therefore, the loss function used in this study is as follows:

(2.16) \begin{align} \mathcal{L}_{\textit{cGAN}}(G,D) &= \mathbb{E}_{x^{\textit{HR}} \sim P_{X}} \left [ D \big( x^{\textit{HR}} | x^{\textit{LR}} \big) - D \big( G \big( x^{\textit{LR}}\big) | x^{\textit{LR}} \big) \right ] \nonumber \\ &\quad - \lambda _{\textit{GP}} \mathbb{E}_{\hat {x} \sim P_{\hat {X}}} \left [ \left ( \| \nabla _{\hat {x}}D(\hat{x}) \|_2 - 1 \right ) ^2 \right ]. \end{align}

Here $G$ and $D$ are the generator and discriminator represented as a mapping function. Here $D ( x^{\textit{HR}} | x^{\textit{LR}} )$ represents the output of the discriminator for the input $x^{\textit{HR}}$ , provided the condition $x^{\textit{LR}}$ , where the superscripts ‘HR’ and ‘LR’ represent ‘high resolution’ and ‘low resolution’ respectively. Here $\mathbb{E}_{x\sim P_X} [ f(x) ]$ denotes the expected value of an arbitrary function $f(x)$ when $x$ is sampled from distribution $X$ . In this study, $X$ represents the distribution of high-resolution data (DNS data), and $x^{\textit{LR}}$ (filtered DNS data) is obtained by applying a top-hat filter to $x^{\textit{HR}}$ . The first term is known as the adversarial loss. This loss encourages the discriminator to differentiate between real data and generated data more accurately, while the generator is led to be more able to deceive the discriminator. Here $\hat {x} \sim P_{\hat {X}}$ represents random samples from a random distribution $\hat {X}$ . Here $\nabla _z f(z)$ is the partial derivative of $f(z)$ with respect to $z$ , and $\|\cdot \|_2$ is the L2 norm. Here $\lambda _{\textit{GP}}$ is the constant weight coefficient for the last term, the gradient penalty loss, which constrains the discriminator to be a 1-Lipschitz function as is required in the Wasserstein GAN formulation. We found through preliminary experiments that for $\lambda _{\textit{GP}} = 1, 10, 100$ , the results show no significant sensitivities, in which $\lambda _{\textit{GP}} = 10$ (recommended in the original paper (Gulrajani et al. Reference Gulrajani, Ahmed, Arjovsky, Dumoulin and Courville2017)) performs slightly better. Therefore, in this paper, $\lambda _{\textit{GP}}$ is set to $10$ .

The process of learning the super-resolution mapping can be expressed as the following optimisation problem:

(2.17) \begin{equation} \arg \min _G \max _D \mathcal{L}_{\textit{cGAN}}(G,D). \end{equation}

The goal of this process is to obtain the super-resolution mapping $G$ that, given low-resolution data $x^{\textit{LR}}$ , predicts the high-resolution data, i.e. $G (x^{\textit{LR}} ) \approx x^{\textit{HR}}$ . This is solved by iteratively updating the parameters of the networks according to the used optimiser algorithm.

2.2.2. Cycle-consistency GAN

In this study, CycleGAN (Zhu et al. Reference Zhu, Park, Isola and Efros2020) is employed for the unsupervised learning of mappings between the vLES flows and fDNS flows. Figure 2 shows the schematic of a CycleGAN architecture. Here $X$ and $Y$ indicate the domains of data. In this study, they represent the vLES flow fields and fDNS flow fields, respectively. Unlike regular GAN architectures, a CycleGAN architecture consists of two generator networks ( $F, G$ ) and two discriminator networks ( $D_X, D_Y$ ). Each generator is encouraged to learn a mapping that conserves the common characteristics between the two domains through the use of the cycle consistency loss. Mathematically, this process is understood as an optimisation problem of mapping functions $F$ and $G$ between data distributions $X$ and $Y$ .

Figure 2. Schematic of unsupervised CycleGAN architecture: $F$ and $G$ , generator models; $D_X$ and $D_Y$ , discriminator models; $X$ and $Y$ , training datasets.

The loss function used in this study is expressed as follows:

(2.18) \begin{equation} \mathcal{L}(F,G,D_X,D_Y)=\mathcal{L}_{\textit{GAN}}(F,D_X)+\mathcal{L}_{\textit{GAN}}(G,D_Y)+\lambda _{\textit{cyc}} \mathcal{L}_{\textit{cyc}}(F,G). \end{equation}

The first two terms of (2.18) are the adversarial losses defined as follows:

(2.19) \begin{align} \mathcal{L}_{\textit{GAN}}(F,D_X) =& \mathbb{E}_{x \sim P_X} \left [ D_X(x) \right ] -\mathbb{E}_{y\sim P_Y} \left [ D_X(F(y)) \right ] \nonumber \\ &- \lambda _{\textit{GP}} \mathbb{E}_{\hat {x} \sim P_{\hat {x}}} \left [ \left ( \| \nabla _{\hat {x}} D_X(\hat {x}) \| _2 - 1 \right )^2 \right ],\\[-10pt]\nonumber \end{align}

(2.20) \begin{align} \mathcal{L}_{\textit{GAN}}(G,D_Y) =& \mathbb{E}_{y \sim P_Y} \left [ D_Y(y) \right ] -\mathbb{E}_{x \sim P_X} \left [ D_Y(G(x)) \right ] \nonumber \\ &- \lambda _{\textit{GP}} \mathbb{E}_{\hat {y} \sim P_{\hat {y}}} \left [ \left ( \| \nabla _{\hat {y}} D_Y(\hat {y}) \| _2 - 1 \right )^2 \right ]. \end{align}

The adversarial losses are similar to the loss function of cGAN (2.16) except that they are not conditioned; i.e. the discriminators have only one input as opposed to two (input and condition) in cGAN. The third term of (2.18) is the cycle consistency loss, with a constant weight coefficient of $\lambda _{\textit{cyc}}=1$ . The cycle consistency loss is defined as

(2.21) \begin{equation} \mathcal{L}_{\textit{cyc}}(F,G)=\mathbb{E}_{x \sim P_X}[\| F(G(x))-x\|_2]\\+\mathbb{E}_{y \sim P_Y}[\|G(F(y))-y\|_2)]. \end{equation}

This loss ensures that after data passes through the two generators, the original data are recovered; that is, $F(G(x)) \approx x$ and $G(F(y)) \approx y$ . This incentivises each mapping function $F$ and $G$ to retain the shared characteristics of the two datasets. The value of the weight coefficient $\lambda _{\textit{cyc}}$ was determined based on the preliminary experiments in which $\lambda _{\textit{cyc}} = 1$ performed better than the originally recommended value of $\lambda _{\textit{cyc}} = 10$ (Zhu et al. Reference Zhu, Park, Isola and Efros2020). The objective of the training phase is to find the model parameters that meet the following condition:

(2.22) \begin{equation} \arg \underset {F,G}{\min } \underset {D_X, D_Y}{\max } \mathcal{L}(F,G,D_X,D_Y). \end{equation}

5.1.1. Instantaneous flow fields

Figure 6 shows the instantaneous velocity distributions of the input vLES flow fields, the predicted fDNS-quality flow fields obtained through the unsupervised $G_{\textit{vLES}-\textit{fDNS}}$ in figure 3, and the reference fDNS flow fields at $y^+ \approx 15$ and $100$ . Here, because the CycleGAN model is trained unsupervisedly without paired instantaneous flow fields, there does not exist the exactly same instantaneous fDNS flow field that corresponds to the input instantaneous vLES flow field. Thus, the reference fDNS data are shown for qualitative comparisons only, and quantitative comparisons are in the discussion of the turbulence statistics in § 5.1.2.

Figure 6. Instantaneous velocity distributions in wall-parallel ( $x$ – $z$ ) plane for vLES-to-fDNS model $G_{\textit{vLES}-\textit{fDNS}}$ at $Re_\tau \approx 1000$ . The region corresponds to $(L_x, L_z) = (1.152 \delta , 0.576 \delta )$ . Velocity components are non-dimensionalised by the bulk velocity $u_b$ . Here (i,iv,vii) streamwise component ( $u$ ); (ii,v,viii) wall-normal component ( $v$ ); (iii,vi,ix) spanwise component ( $w$ ); (i–iii) input vLES; (iv–vi) predicted flow; (vii–ix) reference fDNS.

In the predicted flow fields, the high streamwise velocity in the middle of the domain in figure 6(a) and throughout the domain in figure 6(b) obtained by the vLES (figure 6(a)i–iii and figure 6(b)i–iii) decreases to match the velocity magnitude observed in the fDNS flow field. The difference between the highest and lowest velocity magnitude is also lowered, which signifies that the variances of the predicted velocities are appropriately decreased from the vLES input. The small-scale structures are missing in the wall-normal velocity distributions of vLES at $y^+ \approx 15$ , unlike its fDNS counterpart. This difference is alleviated by the machine learning model creating fine structures similar to the ones obtained by the fDNS. These observations will be quantitatively assessed in the following § 5.1.2. It is important to point out that these changes are performed while the large resolved turbulent structures in the vLES data remain unchanged. In other words, the information of the original vLES remains although the proper adjustments are made by the unsupervised machine learning model $G_{\textit{vLES}-\textit{fDNS}}$ . This characteristic is crucial as the SGS stress represents the interaction between the large resolved scales and small unresolved scales. That is, if the resolved scales are significantly altered by the unsupervised machine learning model $G_{\textit{vLES}-\textit{fDNS}}$ , the predicted SGS stress components predicted using the changed resolved scales (as in § 5.3.2) would not be able to appropriately represent the effect on the original flow field.

Figure 7 shows a streamwise ( $y$ – $z$ ) cross-sectional plane of the instantaneous velocity distributions. As discussed in § 3, the present machine learning model processes each wall-normal plane independently. This method may cause spurious discontinuities between each wall-normal plane, however, the results suggest that there are no clear discontinuous velocity distributions across the wall-normal direction. This may be understood by considering that the machine learning model consists of convolution layers which are continuous functions with respect to the change in input. Therefore, because the flow fields at adjacent wall-normal planes are continuous, the output flow fields are also continuous in the wall-normal direction.

We stress again that the fDNS flow fields shown in figures 6 and 7 do not correspond on a one-to-one basis to the flow fields predicted by the CycleGAN model. Because the input vLES flow fields are not created by filtering the DNS flow fields, there does not exist an exactly-the-same instantaneous reference flow field corresponding to the input vLES flow field. The fDNS flow fields shown in the figure as ‘reference’ are taken from a simulation that is independent from the vLES simulation. As a result, the instantaneous flow fields shown in the figures should only be compared qualitatively in terms of the exhibited flow features, not quantitatively in terms of their congruence. Quantitative comparisons based on turbulence statistics are presented in § 5.1.2.

Figure 7. Instantaneous velocity distributions in streamwise ( $y$ – $z$ ) cross-sectional plane for vLES-to-fDNS model $G_{\textit{vLES}-\textit{fDNS}}$ at $Re_\tau \approx 1000$ . The region corresponds to $(L_y, L_z) = (\delta , 2 \delta )$ . Velocity components are non-dimensionalised by the bulk velocity $u_b$ . Here (i,iv,vii) streamwise component ( $u$ ); (ii,v,viii) wall-normal component ( $v$ ); (iii,vi,ix) spanwise component ( $w$ ); (i–iii) input vLES; (iv–vi) predicted flow; (vii–ix) reference fDNS.

Figure 8. (a,c) Streamwise and (b,d) spanwise energy spectra of predicted velocity for vLES-to-fDNS model $G_{\textit{vLES}-\textit{fDNS}}$ at $Re_\tau \approx 1000$ . Here (a,b) $y^+ \approx 15$ ; (c,d) $y^+ \approx 100$ ; blue, streamwise component; orange, wall-normal component; green, spanwise component; solid lines, predicted flow; circles, reference fDNS; pluses, input vLES.

Figure 9. (a) Mean streamwise velocity and (b) resolved reynolds stresses of predicted flow at $Re_\tau \approx 1000$ . Solid lines, predicted flow; circles, reference fDNS; pluses, input vLES. For (b) blue, streamwise normal component ( $\overline {u^{\prime}u^{\prime}}$ ); orange, wall-normal normal component ( $\overline {v^{\prime}v^{\prime}}$ ); green, spanwise normal component ( $\overline {w^{\prime}w^{\prime}}$ ); red, shear component ( $\overline {u^{\prime}v^{\prime}}$ ).

5.1.2. Turbulence spectra and statistics

Figure 8 shows the streamwise and spanwise energy spectra for the three components of velocity at $y^+ \approx 15$ and $y^+ \approx 100$ . The input vLES spectra show significant deviations from the fDNS spectra, which show the effects of the insufficient grid resolution of the very coarse computational grid. On the other hand, the predicted output spectra show good agreements with those of the reference fDNS for both off-wall locations shown.

The mean streamwise velocity and the resolved Reynolds stresses of the predicted flows are shown in figure 9. The turbulence statistics show good agreements with those of the reference fDNS. A slight underprediction is observed for the streamwise normal Reynolds stress profiles at $0.1 \lesssim y/\delta \lesssim 0.7$ . Although not shown here, we have confirmed that this discrepancy originates from the underprediction of the low-wavenumber components of turbulence. However, as shown in the later subsections, this discrepancy does not degrade the machine learning pipeline’s ability to predict the correct SGS stresses.

The results indicate that the unsupervised CycleGAN model is highly effective at converting the non-physical vLES flow fields to physically correct fDNS-quality flow fields. This kind of conversion is impossible without the use of an unsupervised learning method, where paired training data of inputs and expected outputs are not required.

Figure 10. Instantaneous velocity distributions in wall-parallel ( $x$ – $z$ ) plane for super-resolution model $G_{\textit{SR}}$ at $Re_\tau \approx 1000$ . The region corresponds to $(L_x, L_z) = (1.152 \delta , 0.576 \delta )$ . Velocity components are non-dimensionalised by the bulk velocity $u_b$ . (i,iv,vii) Streamwise component ( $u$ ); (ii,v,viii) wall-normal component ( $v$ ); (iii,vi,ix) spanwise component ( $w$ ); (i–iii) input fDNS; (iv–vi) predicted super-resolved flow; (vii–ix) reference DNS.

5.2.1. Instantaneous flow fields

Figure 10 shows the instantaneous velocity distributions of the super-resolved flows from the cGAN model $G_{\textit{SR}}$ . As with figure 6, the wall-normal planes at $y^+ \approx 15$ and $100$ are shown. Unlike the vLES-to-fDNS model $G_{\textit{vLES}-\textit{fDNS}}$ which is trained using unsupervised machine learning, this super-resolution model is supervisedly trained since the paired training data of fDNS and DNS can easily be obtained by applying a filter to DNS flow fields. Thus, the reference DNS flow field is the expected instantaneous output for the input fDNS flow field. The figure shows that the super-resolution model can reconstruct the high-wavenumber components that are not present in the input fDNS. Figure 11 shows the streamwise ( $y$ – $z$ ) cross-sectional planes of the velocity fields. As with figure 7, there are no clear discontinuous velocity distributions across the wall-normal direction owing to the similarities between the adjacent wall-normal planes.

Figure 11. Instantaneous velocity distributions in streamwise ( $y$ – $z$ ) cross-sectional plane for super-resolution model $G_{\textit{SR}}$ at $Re_\tau \approx 1000$ . The region corresponds to $(L_y, L_z) = (\delta , 2 \delta )$ . Velocity components are non-dimensionalised by the bulk velocity $u_b$ . (i,iv,vii) Streamwise component ( $u$ ); (ii,v,viii) wall-normal component ( $v$ ); (iii,vi,ix) spanwise component ( $w$ ); (i–iii) input fDNS; iv–vi) predicted super-resolved flow; (vii–ix) reference DNS.

Figure 12. (a,c) Streamwise and (b,d) spanwise energy spectra of predicted super-resolved velocity fields at $Re_\tau \approx 1000$ . Here (a,b) $y^+ \approx 15$ ; (c,d) $y^+ \approx 100$ . Blue, streamwise component; orange, wall-normal component; green, spanwise component. Solid lines, predicted super-resolved flow; circles, reference DNS; pluses, input fDNS. Black vertical dashed lines indicate maximum wavenumber resolved by the input fDNS (cutoff wavenumber).

Figure 13. (a) Mean streamwise velocity and (b) resolved Reynolds stresses of predicted super-resolved flows at $Re_\tau \approx 1000$ . Solid lines, predicted super-resolved flow; circles, reference DNS; pluses, input fDNS. For (b) blue, streamwise normal component ( $\overline {u^{\prime}u^{\prime}}$ ); orange, wall-normal normal component ( $\overline {v^{\prime}v^{\prime}}$ ); green, spanwise normal component ( $\overline {w^{\prime}w^{\prime}}$ ); red, shear component ( $\overline {u^{\prime}v^{\prime}}$ ).

5.2.2. Turbulence spectra and statistics

In figure 12, the energy spectra of the super-resolved flows are shown. The components of the wavenumbers that are lower than the vertical dashed lines in the figure are resolved by the fDNS, while the higher wavenumber components must be newly generated. In other words, the higher wavenumber components are not included in the input, and they are predicted by the super-resolution model. The predicted flows of the super-resolution model show good agreements in the high-wavenumber components of the flow. While discrepancies are observed in the highest-wavenumber components, they are insignificantly small by three to four orders of magnitude compared with the most energetic eddies. In fact, the discrepancies are indeed negligible to the mean streamwise velocity and the resolved Reynolds stresses, as shown in figure 13. Furthermore, in the resolved wavenumbers (left of the vertical dashed lines), the discrepancies between the input fDNS and the reference DNS are corrected by the super-resolution, most evidently in the wall-normal component. As the low-wavenumber components carry much of the turbulent kinetic energy, this correction is an important part of the super-resolution process in terms of the recovery of the energy. As shown in figure 13, the mean streamwise velocity and the Reynolds stresses of the super-resolved flows show good agreements with those of the reference DNS. We note that the spikes of the spectra occurring at some of the high wavenumbers are known as the chequerboard artefact (Odena, Dumoulin & Olah Reference Odena, Dumoulin and Olah2016). These artefacts are known in the field of computer vision to be avoidable by employing more sophisticated upsampling layers, while the current study uses nearest-neighbour interpolation as shown in Appendix A. However, as the spikes only exist in few wavenumber components and contribute little to the total energy, they pose little problem to the SGS modelling in this study.

5.2.3. Extraction of SGS stresses

The super-resolution model’s ability to predict the SGS stresses is tested in an a priori manner. Figure 14 shows the SGS stresses obtained from the super-resolved DNS-quality flows through the process described in § 3.2. The predicted super-resolved flows agree well with the reference SGS stresses extracted from the original DNS data. Also, the discrepancies in the highest-wavenumbers of the predicted spectra observed in figure 12 do not negatively affect the predicted SGS stresses. The largest energy contained in the unresolved wavenumbers (the energy at the near-cutoff wavenumbers which constitute the majority of the extracted SGS stresses) and the prediction errors near the highest-wavenumbers are separated by around two orders of magnitude, leading to the negligible contribution to the total SGS stress components. The results indicate that the proposed turbulence super-resolution can be used to predict accurate SGS stresses.

Figure 14. The SGS stresses extracted from flows predicted by the super-resolution model $G_{\textit{SR}}$ at ${Re}_\tau \approx 1000$ . Solid lines, predicted super-resolved flow; circles, reference DNS. Blue, streamwise normal component ( $\widehat {u^* u^*}$ ); orange, wall-normal normal component ( $\widehat {v^* v^*}$ ); green, spanwise normal component ( $\widehat {w^* w^*}$ ); red, shear component ( $\widehat {u^* v^*}$ ).

5.3.1. Instantaneous flow fields

The instantaneous velocity fields of the flows super-resolved by the proposed pipeline are shown in figure 16. We note that the proposed super-resolution machine learning pipeline performs unsupervised super-resolution, and therefore the reference DNS is not exactly the same instantaneous reference solution to the particular instantaneous vLES input as with § 5.1.1. The predicted instantaneous flow fields are expected to show qualitative agreement, such as the turbulent structures and the peak velocity magnitude, and the quantitative assessments will be assessed using turbulence statistics in § 5.3.2. It can be seen from figure 16 that the proposed pipeline successfully decreases the high streamwise velocity at the centre of the domain and successfully generates the stronger fluctuations in the wall-normal velocity, which are the characteristics of the vLES-to-fDNS model discussed in § 5.1.1. Additionally, the predicted distributions show the small-scale turbulent structures similar to the reference DNS that are not present in the input vLES, which is a characteristic of the super-resolution model discussed in § 5.2.1. The predicted velocity distributions suggest that the constructed pipeline exhibits the characteristics of each of the constituents: the vLES-to-fDNS model $G_{\textit{vLES}-\textit{fDNS}}$ and the super-resolution model $G_{\textit{SR}}$ . It should be noted that this kind of super-resolution is impossible without the use of an unsupervised method such as CycleGAN because it is impossible to obtain the paired flow fields of vLES and DNS required by supervised learning methods. The streamwise ( $y$ – $z$ ) cross-sectional planes of the velocity fields are shown in figure 17. One may observe that the proposed pipeline predicts the high-wavenumber components qualitatively well, especially near the wall. It can also be seen that the velocity distributions do not present clear discontinuities in the wall-normal direction, similar to figures 7 and 11. As shown in § 5.3.2, the predictions made by the proposed pipeline show quantitative agreements in terms of turbulence spectra and statistics. We also note that the resolved large eddies in the original vLES remain after the super-resolution. This shows that the proposed pipeline successfully creates the DNS-quality flow fields which still contain the large eddies resolved by vLES.

5.3.2. Turbulence spectra and statistics

Figure 18 shows the energy spectra of the flows super-resolved by the proposed unsupervised–supervised pipeline. The proposed pipeline newly predicts the high-wavenumber components (right of the vertical dashed lines) and corrects the spectra differences in the lower wavenumber range (left of the vertical dashed lines) accurately. As also observed in § 5.2.2, some discrepancies are seen in the highest wavenumber components. These differences, however, are insignificant in the turbulence statistics and the extracted SGS stresses as will be shown in § 5.3.3. The dashed lines show that using only the supervised learning model (as typical flow super-resolution) to super-resolve vLES data does not result in accurate super-resolution. The discrepancy occurs because fDNS flow fields do not have the same turbulence statistics as vLES as discussed in § 1. Therefore, typical supervised learning using fDNS flows as the training data does not learn to properly super-resolve the non-physical vLES flow fields, and thus is not appropriate for application to vLES flow fields. Specifically, in the spanwise premultiplied spectra of the streamwise velocity at $y^+\approx 15$ (although not shown here), the proposed unsupervised–supervised pipeline shows a peak at $k_z \delta \approx 60$ , whereas the typical supervised model shows a peak at $k_z \delta \approx 40$ . Converted to wavelengths in wall units, the length scales correspond to $\lambda _z^+ \approx 100$ and $\lambda _z^+ \approx 150$ , respectively. The proposed unsupervised–supervised methodology reproduces the peak at the length scale that corresponds to the well-known streak structures $\lambda _z^+ \approx 100$ (Smith & Metzler Reference Smith and Metzler1983), while the typical supervised model produces the peak at the length scale that is approximately 1.5 times longer. These results show the superiority of the proposed unsupervised pipeline method for the super-resolution of the vLES flow fields.

Figure 18. (a,c) Streamwise and (b,d) spanwise energy spectra of predicted super-resolved velocity field with the proposed unsupervised–supervised pipeline at ${Re}_\tau \approx 1000$ . Here (a,b) $y^+ \approx 15$ ; (c,d) $y^+ \approx 100$ . Blue, streamwise component; orange, wall-normal component; green, spanwise component. Solid lines, predicted super-resolved flow using proposed unsupervised pipeline model (figure 15 a); dashed lines, typical supervised model (figure 15 b); circles, reference DNS; pluses, input vLES. Black vertical dashed lines indicate maximum wavenumber resolved by the input vLES (cutoff wavenumber).

Figure 19. Premultiplied spanwise energy spectra of streamwise velocity for input vLES (a), predicted super-resolved flow using proposed unsupervised–supervised pipeline (b) and reference DNS (c).

Figure 20. Mean streamwise velocity (a) and resolved Reynolds stresses (b) of predicted super-resolved flows at $Re_\tau \approx 1000$ . Solid lines, predicted super-resolved flow using proposed unsupervised–supervised pipeline (figure 15 a); dashed lines, typical supervised model (figure 15 b); circles, reference DNS; pluses, input vLES. For (b): blue, streamwise normal component ( $\overline {u^{\prime}u^{\prime}}$ ); orange, wall-normal normal component ( $\overline {v^{\prime}v^{\prime}}$ ); green, spanwise normal component ( $\overline {w^{\prime}w^{\prime}}$ ); red, shear component ( $\overline {u^{\prime}v^{\prime}}$ ).

Figure 19 shows the premultiplied spanwise energy spectra of the streamwise velocity for the input vLES, the flow field super-resolved by the proposed pipeline, and the reference DNS. As can be seen from the figure, while the reference DNS shows an increase in energy in the low wavenumber range ( $2 \lesssim k_z \delta \lesssim 10$ ) at the outer layer ( $y / \delta \gtrsim 0.1$ ), the spectra are missing in the input vLES. This energy corresponds to the so-called outer peak (or very large-scale motions) of the turbulent boundary layer in high-Reynolds number flows (Hutchins & Marusic Reference Hutchins and Marusic2007; Marusic et al. Reference Marusic, Mathis and Hutchins2010a , b; Lee & Moser Reference Lee and Moser2015). The outer peak typically manifests in the spanwise wavelengths of $\lambda _z / \delta \approx 1$ ( $k_z \delta = 2 \pi \delta / \lambda _z \approx 6$ ), as is observed in the present DNS. Furthermore, the outer peak is reported to be missing for wall-modelled LES (Maeyama & Kawai Reference Maeyama and Kawai2023), where the grid resolution with respect to the wall-unit is significantly coarse as is in vLES. On the other hand, the super-resolved flow field shows the presence of the energy in the outer layer, which does not exist in the input vLES data. It can also be seen that the energy peak in the inner-layer ( $y / \delta \approx 10^{-2}$ ) predicted by the proposed pipeline shows better agreement with the reference DNS compared with the vLES. This shows that the proposed unsupervised–supervised machine-learning pipeline not only learns to predict the high-frequency component of turbulence, but also to amend the errors in the low-frequency components that occur in coarse computational grids.

Figure 20 shows the turbulence statistics of the reconstructed super-resolved flows. The mean streamwise velocity obtained by the proposed pipeline accurately predicts the mean velocity of the reference DNS, whereas the typical supervised super-resolution model, which was supervisedly trained using fDNS flows, yields distributions that do not differ from the input vLES. Similarly, the proposed unsupervised machine learning pipeline can correct the resolved Reynolds stresses and show good agreements with the reference DNS. The super-resolution model alone overestimates the streamwise and shear stresses because of the incorrect mapping learned from fDNS. This likely occurs because the super-resolution model $G_{\textit{SR}}$ is trained merely to predict the small-scale structures missing from the fDNS flow fields, and does not learn to correct the low-frequency components of turbulence. Therefore, the typical supervised model is unable to correct the flow fields that differ from the fDNS flow fields, such as the vLES flow fields. It thus follows that an unsupervisedly trained model is required to perform accurate super-resolution from vLES flow fields, as is proposed in this study.

5.3.3. Extraction of SGS stresses

Figure 21 shows the SGS stress components extracted from the super-resolved flows with the proposed pipeline using the method described in § 3.2. The SGS stress distributions obtained by the proposed unsupervised–supervised pipeline show good agreements with those from the reference DNS. On the other hand, the super-resolution model based on the typical supervised method overpredicts the magnitude of the SGS stress in the streamwise and shear components. The discrepancies in the predicted SGS stresses are caused by the inappropriate super-resolution performed by the typical supervised model. Additionally, the reference SGS stress components show that the streamwise component is stronger compared with the spanwise and the wall-normal components in the near-wall region, and the proposed pipeline shows good predictions of this relationship. It is also notable that the shear component is well predicted by the proposed pipeline. As the shear stress is directly related to the mean velocity profile through the total shear stress balance within the boundary layer, its accurate prediction is important for the accurate prediction of the mean velocity with vLES.

Figure 21. The SGS stresses extracted from predicted super-resolved flows with the proposed unsupervised–supervised pipeline at $Re_\tau \approx 1000$ . Solid lines, predicted super-resolved flow using proposed unsupervised pipeline model (figure 15 a); dashed lines, typical supervised model (figure 15 b); circles, reference DNS. Blue, streamwise normal component ( $\widehat {u^* u^*}$ ); orange, wall-normal normal component ( $\widehat {v^* v^*}$ ); green, spanwise normal component ( $\widehat {w^* w^*}$ ); red, shear component ( $\widehat {u^* v^*}$ ).

Figure 22. Mean streamwise velocity (a) and Reynolds shear stress (b,c ) of a posteriori tests at $Re_\tau \approx 1000$ . Panel (c) is a near-wall zoomed view of (b). Blue, vLES with proposed unsupervised–supervised-pipeline SGS model (figure 15 a); green, vLES with typical supervised SGS model (figure 15 b); red, vLES with typical SGS model; black circles, reference DNS; grey symbols, reference fDNS. For (b,c): dashed lines and squares, resolved components; dotted lines and triangles, SGS components; solid lines, resolved + SGS components.

6.2.1. Effect of SGS stresses on near-wall Reynolds stresses

As discussed in § 1, this study aims to obtain the accurate prediction of the mean velocity (first-order statistics) and the closely related Reynolds shear stress (second-order statistics) from the vLES. Therefore, the agreement of the Reynolds stress budgets (third-order statistics) between the proposed SGS model and the reference DNS is not the goal of this study. Rather, we focus on understanding the mechanism that lead to the accurate predictions of the mean streamwise velocity and the Reynolds shear stress obtained by the proposed unsupervised–supervised-pipeline SGS model. We also note that the following analyses reveal that the DNS-obtained Reynolds normal stresses and the budgets should not be expected in the vLES to obtain the correct mean velocity and Reynolds shear stress.

The budget equation of the resolved Reynolds stresses is given as follows:

(6.2) \begin{align} \frac {\partial }{\partial t} \left ( \overline {\rho u^{\prime}_i u^{\prime}_j} \right ) & = C_{ij} + P_{ij} + T_{t,ij} + T_{p,ij} + T_{v,ij} + T_{\textit{SGS},\textit{ij}} \nonumber \\ & \quad + D_{p,ij} + D_{v,ij} + D_{\textit{SGS},\textit{ij}} \nonumber \\ &= 0. \end{align}

Here, the subscripts $i,j = (1,2,3)$ denote the streamwise, wall-normal and spanwise directions respectively, and

(6.3) \begin{align}&\qquad\qquad C_{ij} = - \frac {\partial }{\partial x_k} \left ( \overline {\rho u^{\prime}_i u^{\prime}_j} \overline {u_k} \right ), \end{align}

(6.4) \begin{align}&\qquad P_{ij} = - \overline {\rho u^{\prime}_i u^{\prime}_k} \frac {\partial \overline {u_j}}{\partial x_k} - \overline {\rho u^{\prime}_j u^{\prime}_k} \frac {\partial \overline {u_i}}{\partial x_k}, \end{align}

(6.5) \begin{align}&\qquad\qquad T_{t,ij} = - \frac {\partial }{\partial x_k} \left ( \overline {\rho u^{\prime}_i u^{\prime}_j u^{\prime}_k} \right ), \end{align}

(6.6) \begin{align}&\quad T_{p,ij} = - \frac {\partial }{\partial x_k} \left ( \overline {p^{\prime} \delta _{ik} u^{\prime}_j} \right ) - \frac {\partial }{\partial x_k} \left ( \overline {p^{\prime} \delta _{jk} u^{\prime}_i} \right ), \end{align}

(6.7) \begin{align}&\qquad T_{v,ij} = \frac {\partial }{\partial x_k} \left ( \overline {\tau ^{\prime}_{ik} u^{\prime}_j} \right ) + \frac {\partial }{\partial x_k} \left ( \overline {\tau ^{\prime}_{jk} u^{\prime}_i} \right ), \end{align}

(6.8) \begin{align}& T_{\textit{SGS},\textit{ij}} = \frac {\partial }{\partial x_k} \left ( \overline {\tau^{\prime}_{ik, {\textit{SGS}}} u^{\prime}_j} \right ) + \frac {\partial }{\partial x_k} \left ( \overline {\tau^{\prime}_{jk, {\textit{SGS}}} u^{\prime}_i} \right ), \end{align}

(6.9) \begin{align}&\qquad\quad D_{p,ij} = \overline {p^{\prime} \delta _{ik} \frac {\partial u^{\prime}_j}{\partial x_k}} + \overline {p^{\prime} \delta _{jk} \frac {\partial u^{\prime}_i}{\partial x_k}}, \end{align}

(6.10) \begin{align}&\qquad\quad D_{v,ij} = - \overline {\tau ^{\prime}_{ik} \frac {\partial u^{\prime}_j}{\partial x_k}} - \overline {\tau ^{\prime}_{jk} \frac {\partial u^{\prime}_i}{\partial x_k}}, \end{align}

(6.11) \begin{align}&\quad\,\, D_{\textit{SGS},\textit{ij}} = - \overline {\tau^{\prime}_{ik, {\textit{SGS}}} \frac {\partial u^{\prime}_j}{\partial x_k}} - \overline {\tau^{\prime}_{jk, {\textit{SGS}}} \frac {\partial u^{\prime}_i}{\partial x_k}}. \end{align}

Here $C_{ij}, P_{ij}, T_{t,ij}, T_{p,ij}, T_{v,ij}, T_{\textit{SGS},\textit{ij}}, D_{p,ij}, D_{v,ij}$ and $D_{\textit{SGS},\textit{ij}}$ denote convection, production, turbulent transport, pressure transport, viscous transport, SGS transport, pressure redistribution, viscous dissipation and SGS dissipation terms, respectively. As can be seen from (6.3), (6.5)–(6.8), the convection and transport terms are written in conservative forms; that is, they satisfy

(6.12) \begin{equation} \int _V F_{ij} dV = 0, \: F_{ij} \in \left ( C_{ij}, T_{t,ij}, T_{p,ij}, T_{v,ij}, T_{\textit{SGS},\textit{ij}} \right ) \end{equation}

for the entire computational domain $V$ . Thus, they do not result in the net increase/decrease in the Reynolds stresses in the flow field. The other terms do not satisfy the above relation, therefore, they act as the source/sink terms of the Reynolds stresses and increase/decrease the Reynolds stresses of the computational domain. The SGS-related terms $T_{\textit{SGS}, \textit{ij}}$ and $D_{\textit{SGS}, \textit{ij}}$ represent the direct effects of the employed SGS model on the Reynolds stresses. The other terms are also affected by the change in the SGS-related terms to satisfy the relation $({\partial }/{\partial t}) ( \overline {\rho u^{\prime}_i u^{\prime}_j} ) = 0$ . That is, as the SGS terms increase, the other terms should decrease and vice versa.

Figure 25 shows the budgets of the resolved Reynolds normal stresses for the vLES with the proposed unsupervised–supervised-pipeline SGS model, with the typical SGS model, and the reference DNS by taking $j=i$ . It can be observed that the overall shape of the profiles are similar between the three cases with differences in the magnitude. Focusing on the SGS dissipation term $D_{\textit{SGS}, \textit{ii}}$ (dashed blue lines in figure 25), discrepancies are observed between the two SGS models, most evidently for the spanwise component. Furthermore, comparatively large differences in the spanwise viscous dissipation $D_{v, 33}$ (solid blue line in figure) and the wall-normal and spanwise pressure redistribution terms $D_{p, 22}, D_{p,33}$ (solid yellow lines in figure) are observed. Here, based on the discussion in § 6.1, we will focus on the effects of the predicted SGS stresses and the resulting changes to the near-wall wall-normal Reynolds stress, which was found to have significant effects on the correctly predicted near-wall Reynolds shear stress and the mean streamwise velocity in the vLES. Therefore, in the following analyses, the mechanism behind the SGS dissipation and the pressure redistribution are discussed in detail to clarify their roles in predictions of the near-wall wall-normal Reynolds stress.

Figure 25. Budgets of Reynolds normal stresses in a posteriori tests at $Re_\tau \approx 1000$ . (a–c) Streamwise component; (d–f) wall-normal component; (g–i) spanwise component; (a,d,g) vLES with proposed unsupervised–supervised-pipeline SGS model; (b,e,h) vLES with typical SGS model; (c,f,i) reference DNS. Solid red, production $P_{ij}$ ; solid green, turbulent diffusion $T_{t,ij}$ ; solid orange, pressure diffusion $T_{p,ij}$ ; solid purple, viscous diffusion $T_{v,ij}$ ; dashed purple, SGS diffusion $T_{\textit{SGS},\textit{ij}}$ ; solid yellow, pressure redistribution $D_{p,ij}$ ; solid blue, viscous dissipation $D_{v,ij}$ ; dashed blue, SGS dissipation $D_{\textit{SGS},\textit{ij}}$ .

First, the effects of the SGS dissipation term ( $D_{\textit{SGS}, \textit{ii}}$ , dashed blue lines in figure) are discussed. In the wall-normal component of the SGS dissipation term $D_{\textit{SGS}, 22}$ , there is little difference between the SGS models (and also $D_{\textit{SGS}, 22}$ is negligibly small), suggesting that the near-wall differences in the wall-normal Reynolds stress are not directly caused by the SGS dissipation term $D_{\textit{SGS}, 22}$ . On the other hand, the spanwise SGS dissipation term $D_{\textit{SGS},33}$ shows opposite tendencies between the proposed unsupervised–supervised-pipeline SGS model and the conventional SGS model. The conventional SGS model predicts negative values of the SGS dissipation, whereas the proposed unsupervised–supervised-pipeline SGS model predicts positive values in the near-wall region ( $y^+ \lesssim 100$ ). In general, positive values in the SGS dissipation term (i.e. the net increase in the resolved Reynolds stress) are attributed to the occurrences of SGS backscatter events; that is, the resolved Reynolds stress is transported from the unresolved SGS components. The results suggest that the increased near-wall spanwise Reynolds stress predicted by the proposed unsupervised–supervised-pipeline SGS model seen in figure 23 is primarily caused by the SGS backscatter, which induces the near-wall spanwise velocity fluctuations.

Next, we focus on the pressure redistribution term $D_{p,ii}$ (yellow lines in figure). In the incompressible limit, this term satisfies $\sum _{i} D_{p,ii} = 0$ . Therefore, this term represents the exchange of Reynolds normal stress components between each other. Note that the vertical axis ranges vary among the three components depicted in figure 25. Particularly, the wall-normal component (figure 25 d–f) exhibits the smallest range among the three directions. In the near-wall region ( $20 \lesssim y^+ \lesssim 100$ ) of the proposed unsupervised–supervised-pipeline SGS model, the spanwise component shows a negative value while the wall-normal component shows a positive value. The decrease of the spanwise stress and the increase of the wall-normal stress corresponds to the transfer of the Reynolds stress from the spanwise component to the wall-normal component through the pressure redistribution term. In fact, this process is known to be the dominant energy transfer mechanism towards the wall-normal component (Lee & Moser Reference Lee and Moser2019), and will be discussed in more detail in § 6.2.2. In comparison, the conventional SGS model predicts a smaller positive peak in the wall-normal component slightly farther from the wall ( $y^+ \approx 70$ ) compared with the proposed unsupervised–supervised-pipeline SGS model. This shows that the redistribution of the Reynolds stress to the wall-normal component is weaker and occurs farther away from the wall using the conventional SGS model. It can be considered that these differences between the SGS models in the pressure redistribution term are the causes of the differences in the near-wall wall-normal and the shear Reynolds stresses.

Considering that the terms in (6.2) balance each other out to satisfy ${(\partial }/{\partial t}) ( \overline {\rho u^{\prime}_i u^{\prime}_i} ) = 0$ , the prediction of SGS backscatter by the proposed SGS model in the near-wall spanwise Reynolds stress leads to a net decrease in the other terms. In the present vLES, the decrease was observed for the pressure redistribution term, which represents the outflow of Reynolds stress from the spanwise component. The decrease then leads to the increase of the pressure redistribution term in the near-wall wall-normal component to satisfy $\sum _{i} D_{p,ii} = 0$ , acting as the source term for the increasing wall-normal velocity fluctuations. Therefore, it can be considered that the increase in the near-wall wall-normal Reynolds stress is caused by the SGS backscatter in the near-wall spanwise Reynolds stress. The increase of the near-wall spanwise Reynolds stress leads to the increase of the near-wall Reynolds shear stress and the resultant correct mean velocity. On the other hand, the typical SGS model does not predict the SGS backscatter in the spanwise Reynolds stress and resultantly does not cause the redistribution of near-wall Reynolds stresses from the spanwise to the wall-normal component. Therefore, the conventional SGS model leads to a smaller wall-normal Reynolds stress in the near-wall region, leading to the underprediction of the near-wall Reynolds shear stress and the overprediction of the mean streamwise velocity.

6.2.2. Mechanisms of redistribution of Reynolds stresses

In § 6.2.1, the SGS dissipation and the pressure redistribution terms are shown to have a significant effect on the prediction of the mean velocity and the Reynolds shear stress by the proposed unsupervised–supervised-pipeline SGS model. The redistribution of the stress from the spanwise to the wall-normal component is to be expected from streamwise vortices near the wall (Lee & Moser Reference Lee and Moser2019), as shown in figure 26. For example, at locations with $({\partial v^{\prime}}/{\partial y})\gt 0$ (red regions in figure), the spanwise velocity satisfies $({\partial w^{\prime}}/{\partial z})\lt 0$ . Conversely, at blue regions in the figure, $({\partial v^{\prime}}/{\partial y})\gt 0$ and $({\partial w^{\prime}}/{\partial z})\lt 0$ are satisfied. Therefore, $p^{\prime} ({\partial v^{\prime}}/{\partial y})$ and $p^{\prime} ({\partial w^{\prime}}/{\partial z)}$ have opposite signs at each of the red and blue regions which represent the redistribution of the Reynolds normal stress as discussed earlier. However, because the grid resolutions between the DNS and vLES are different by a factor of eight, the resolved structures of the near-wall turbulence are also expected to be different. Here, we perform the spectral analyses of the pressure redistribution terms to quantify the differences between the resolved turbulence in vLES using the proposed unsupervised–supervised-pipeline SGS model and the conventional SGS. Specifically, we aim to elucidate the reason of the differences in the pressure redistribution terms by examining the spanwise wavelengths (denoted by $\lambda _z$ in figure) associated with these structures.

Figure 26. Schematic of near-wall streaky streamwise vortices in wall turbulence.

As with § 6.2.1, we disregard the density variations because of the low Mach number condition at $M_b \approx 0.1$ . Considering the channel flow as in this study in which the flow is homogeneous in the streamwise and the spanwise directions, the spectral Reynolds stress budget equation (Mizuno Reference Mizuno2016; Lee & Moser Reference Lee and Moser2019) can be written as

(6.13) \begin{align} \textrm{Re} \left [ \frac {\partial }{\partial t} \left (\overline {\rho \widehat {u^{\prime}_i} \widehat {u^{\prime}_j}^* } \right ) \right ] = & \check C_{ij} + \check P_{ij} + \check T_{t,ij} + \check T_{p,ij} + \check T_{v,ij} + \check T_{\textit{SGS},\textit{ij}} \nonumber \\ & + \check D_{p,ij} + \check D_{v,ij} + \check D_{\textit{SGS},\textit{ij}}. \end{align}

Here, $\widehat {(\cdot )}$ denotes the Fourier transformed quantity, $(\cdot )^*$ the complex conjugate and $\textrm{Re} [\cdot ]$ the real part of the complex number. For each term on the right-hand side, the following equality holds:

(6.14) \begin{equation} \int _{0}^{\infty } \check F_{ij} (k,y) {\textrm d}k = F_{ij}(y). \end{equation}

This equation shows that the terms in the spectral budget equation represents the contribution of each wavenumber to the budget term in the total budget equation (6.2). Note that the previous studies (Mizuno Reference Mizuno2016; Lee & Moser Reference Lee and Moser2019) further split the turbulent transport term $\check T_{t,ij}$ into the spatial and interscale transfers, which is not employed here for consistency with (6.2). Following the discussion in § 6.2.1, we focus on the spectral pressure redistribution term $\check D_{p,ij}$ which is written as

(6.15) \begin{equation} \check D_{p,ij} = \textrm{Re} \left [ \overline {\widehat {p^{\prime}} \delta _{ik} \widehat {\frac {\partial u^{\prime}_j}{\partial x_k}}^*} + \overline {\widehat {p^{\prime}} \delta _{jk} \widehat {\frac {\partial u^{\prime}_i}{\partial x_k}}^*} \right ]. \end{equation}

Figure 27. Premultiplied spanwise-spectral components of pressure redistribution in a posteriori tests at $y^+ \approx 15$ at $Re_\tau \approx 1000$ . (a) Streamwise component; (b) wall-normal component; (c) spanwise component. Blue lines, proposed unsupervised–supervised-pipeline SGS model; red lines, typical SGS model, circles, reference DNS. Horizontal dotted lines denote values of $0$ in the vertical axes.

Figure 27 shows the premultiplied spanwise spectral components of the pressure redistribution term for the proposed unsupervised–supervised-pipeline SGS model, the conventional SGS model and the reference DNS at the $y^+ \approx 15$ plane. In the DNS, the wall-normal component of the pressure redistribution term takes a positive peak in the high-wavenumber range of $k_z^+ \gtrsim 0.1$ . There also exists a small negative peak in the spanwise component in this high-wavenumber range. As also discussed by Lee & Moser (Reference Lee and Moser2019), these peaks show that there is the redistribution of the Reynolds stress from the spanwise direction to the wall-normal direction, and the near-wall streaky streamwise vortices discussed above occur in the spanwise wavelength of $\lambda _z^+ \lesssim 2 \pi / k_z^+ \approx 60$ . Here, it is notable that the spanwise grid spacings for the vLES are set at $\Delta z ^+ \approx 36$ and therefore cannot resolve the structures smaller than the Nyquist wavelength $2 \Delta z^+ \approx 72$ . This is shown in the figure by the fact that the solid lines do not exist in the high-wavenumber range $k^+ \gtrsim 0.09$ . Instead, the proposed unsupervised–supervised-pipeline SGS model shows a clear positive peak of the spanwise pressure redistribution term at the smaller wavenumber of $k_z^+ \approx 0.04$ , for which the corresponding wavelength at $\lambda _z^+ = 2 \pi / k_z^+ \approx 160$ is larger than the DNS. There is also a negative peak in the spanwise pressure redistribution term at the similar wavenumber range, suggesting that there exist dominant near-wall streaky streamwise vortical structures at this wavenumber range. These observations can be understood as the mechanism by which the near-wall wall-normal Reynolds stress increases for the proposed unsupervised–supervised-pipeline SGS model; that is, the pressure redistribution of near-wall Reynolds normal stress from the spanwise to wall-normal component is driven by the larger vortical structures. As discussed in § 6.2.1, the redistribution leads to the increase of the near-wall wall-normal Reynolds stress and the Reynolds shear stress, resulting in the correct prediction of the mean streamwise velocity. On the other hand, it can be seen from the wall-normal budget of the conventional SGS model that the positive peak of the pressure redistribution term does not exist in the wall-normal component. The lack of redistribution towards the wall-normal component causes the discrepancies in the near-wall wall-normal Reynolds stress and Reynolds shear stress, leading to the overprediction of the mean streamwise velocity.

6.2.3. Discussions on the near-wall turbulence mechanisms

Based on the above analyses of the Reynolds stress budget, we discuss the differences in the resolved phenomena that leads to the differences in the near-wall Reynolds shear stress between the high-fidelity DNS and vLES with the proposed unsupervised–supervised-pipeline SGS model and the conventional SGS model.

We first discuss the DNS, which is a high-fidelity representation of real turbulence. As also shown by Lee & Moser (Reference Lee and Moser2019), near-wall wall-normal Reynolds stress is generated primarily by the pressure redistribution from the spanwise Reynolds stress. The redistribution occurs in the spanwise wavenumber of $k_z^+ \approx 60$ . The resulting increase in the wall-normal stress gives rise to the Reynolds shear stress in the near-wall buffer layer of the turbulent boundary layer.

In the vLES using the proposed unsupervised–supervised-pipeline SGS model, the SGS backscatter represented by the positive SGS dissipation term (SGS backscatter) introduces the additional production of the resolved Reynolds stress in the spanwise direction, as shown in figure 25. The effect of the predicted SGS backscatter can also be seen in the instantaneous flow fields in figure 24, where the non-physical large elongated structures in the near-wall region such as those observed by the conventional SGS model are disturbed to generate the smaller structures. To satisfy the balance of the Reynolds stress budget (6.2), the increased near-wall spanwise stress then gives rise to the increased pressure redistribution towards the wall-normal stress which results in the increased near-wall wall-normal Reynolds stress and Reynolds shear stress. Here, the spectral budgets (figure 27) show that the redistribution mainly occurs in the spanwise wavenumbers of $0.03 \lesssim k_z^+ \lesssim 0.05$ , which correspond to the resolved wavelengths of $3.5 \Delta z \lesssim \lambda _z \lesssim 5.8 \Delta z$ ( $130 \lesssim \lambda _z^+ \lesssim 210$ ). While these length scales are larger than those observed in the DNS because of the very coarse grid employed in the vLES, the resolved turbulent structures serve the similar roles in the redistribution of the Reynolds normal stresses. Here we repeat that the spanwise wavenumber for DNS ( $k_z^+ \approx 60$ ) cannot be resolved by the vLES grid. Considering that the redistribution process is induced by the predicted SGS backscatter, the increased wall-normal stress can thus be interpreted as compensation for the unresolved turbulence dynamics by the proposed SGS model. As a result, although the resolved flow field of vLES cannot reproduce the near-wall turbulent physics of DNS (because of the very coarse grid), the vLES with the proposed unsupervised–supervised-pipeline SGS model leads to the accurate near-wall Reynolds shear stress and the resultant mean streamwise velocity profile.

In the vLES using the conventional SGS model, where the turbulence physics in the unresolved wavenumbers $k_z^+ \gtrsim 0.09$ that are crucial for the reproduction of the near-wall wall-normal Reynolds stress are also not resolved, the redistribution of Reynolds stress from the spanwise to the wall-normal component does not occur in the near-wall region. As the conventional SGS model fails to reproduce the near-wall energy redistribution that is crucial for the accurate prediction of the Reynolds shear stress, the mean streamwise velocity resultantly shifts to satisfy the total shear stress balance, leading to the overprediction of velocity.

We also note that the identified mechanisms of the proposed unsupervised–supervised-pipeline SGS model show similarities with the SGS dynamics described by Hamba (Reference Hamba2019) and Inagaki & Kobayashi (Reference Inagaki and Kobayashi2023). By analysing the DNS data of turbulent channel flows, it was shown that SGS backscatter in the spanwise velocity fluctuations occur in the near-wall region. It is also shown that the physics-based stabilised mixed SGS model proposed by Abe (Reference Abe2013) enables appropriate prediction of the redistribution term for the near-wall wall-normal Reynolds stress in coarse-grid LES, which leads to the enhanced near-wall Reynolds shear stress (Inagaki & Kobayashi Reference Inagaki and Kobayashi2020). As these observations are consistent with the identified near-wall dynamics of the proposed unsupervised–supervised-pipeline SGS model in this study, we believe that the spanwise SGS backscatter and the pressure redistribution terms may provide insight into further development of SGS models for coarse-grid LES.