3.1. Effects of input variables: a priori and a posteriori tests

Before conducting an actual LES, the effects of input variables on ANN-SGS models are investigated in an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } = 106$. Although consistent predictive capability in the a priori and a posteriori tests is not always guaranteed (Park, Lee & Choi Reference Park, Lee and Choi2005; Park & Choi Reference Park and Choi2021), a priori tests are still considered to be a useful step for evaluating SGS models (Piomelli, Moin & Ferziger Reference Piomelli, Moin and Ferziger1988; Salvetti & Banerjee Reference Salvetti and Banerjee1995). It is also worth noting that the performance of ANN-SGS models are actually assessed in a posteriori tests in the present study. While the use of $\bar {S}_{ij}$ is expected to be preferable for homogeneous isotropic turbulence in terms of the accuracy and stability (Park & Choi Reference Park and Choi2021), it is of interest to investigate how differently the predicted SGS stress aligns with respect to $\bar {S}_{ij}$. To examine the alignment, the correlation coefficients between $\bar {S}_{ij}$ and the predicted SGS stress ($\tau _{ij}^{ANN}$) are calculated and listed in table 3. The correlation coefficient between the components of arbitrary second-order tensors $\alpha _{ij}$ and $\beta _{ij}$ is defined as

(3.1)\begin{equation} Corr(\alpha_{ij},\beta_{ij}) = \frac{\left\langle \alpha_{ij} \beta_{ij} \right\rangle}{\left\langle \alpha_{ij}^2 \right\rangle^{1/2} \left\langle \beta_{ij}^2 \right\rangle^{1/2}}, \end{equation}

where $\langle {\cdot } \rangle$ denotes the ensemble averaging. As shown in table 3, S-106H has much higher correlation coefficients between $\bar {S}_{ij}$ and $\tau _{ij}^{ANN}$ than fDNS with the value exceeding $-0.8$, which indicates that the predicted SGS stress is aligned closer to the strain-rate tensor rather than the true SGS stress.

Table 3. Correlation coefficients ($Corr(\bar {S}_{ij}, \tau _{ij})$) between $\bar {S}_{ij}$ and the predicted SGS stress by ANN-SGS models from an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } = 106$.

Additionally, the p.d.f. of the SGS dissipation $\varepsilon _{SGS}$ ($= -\tau _{ij}\bar {S}_{ij}$) predicted by S-106H is shown in figure 4(a). Negative SGS dissipation corresponds to the backscatter, which is the kinetic energy transfer from the subgrid scale to the resolved scale. Interestingly, S-106H shows a similar characteristic to that of the eddy-viscosity models that are not capable of predicting backscatter. It can be considered that an ANN-SGS model that uses the resolved strain-rate tensor alone as the input only rescales the given input $\bar {S}_{ij}$ to predict the SGS stress while maintaining its alignment.

Figure 4. Results from an a priori test of forced homogeneous isotropic turbulence at $Re_\lambda = 106$. (a) P.d.f. of the SGS dissipation $\varepsilon _{SGS}$ ($= -\tau _{ij}\bar {S}_{ij}$); (b) p.d.f. of $\tau _{23}$. Here —— (thick black solid line), fDNS; —— (thick red solid line), SL-106H; – – - (thick green dashed line), S-106H.

Based on this observation, the performance of S-106H in actual LES is compared with that of DSM by conducting a posteriori tests of forced isotropic turbulence at $Re_\lambda = 106$. The energy spectrum and p.d.f.s of the SGS dissipation and the SGS stress from S-106H are compared with those from DSM and fDNS. The SGS stress of DSM (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991; Lilly Reference Lilly1992) is given as

(3.2)\begin{equation} \tau_{ij} - \tfrac{1}{3}\delta_{ij}\tau_{kk} ={-}2C\bar{\varDelta}^2|\bar{S}|\bar{S}_{ij}, \end{equation}

where $|\bar {S}|=\sqrt {2\bar {S}_{ij} \bar {S}_{ij}}$, $\bar {S}_{ij}=\frac {1}{2}(\partial \bar {u}_i/\partial x_j+\partial \bar {u}_j/\partial x_i)$, $C=\langle L_{ij}M_{ij}\rangle /\langle M_{ij}M_{ij}\rangle$, $L_{ij}=\widehat {\bar {u}_i\bar {u}_j}-\hat {\bar {u}}_i\hat {\bar {u}}_j$, $M_{ij}=-2\hat {\varDelta }^2|\hat {\bar {S}}|\hat {\bar {S}}_{ij}+2\bar {\varDelta }^2|\widehat {\bar {S}|\bar {S}_{ij}}$, $\bar {\varDelta }$ and $\hat {\varDelta } (= \sqrt {5}\bar {\varDelta })$ are the grid-filter and test-filter scales, respectively. $\overline {(\,{\cdot }\,)}$ denotes the grid-level filter at $\bar {\varDelta }$ scale, $\widehat {(\,{\cdot }\,)}$ denotes the test filter at $\hat {\varDelta }$ scale and $\langle {\cdot } \rangle$ denotes averaging over homogeneous directions (volume averaging for homogeneous isotropic turbulence). The LES are performed using a pseudospectral code HIT3D (Chumakov Reference Chumakov2007, Reference Chumakov2008) with the dealiasing method of the $2/3$ rule. The second-order Adams–Bashforth scheme is used for time integration and the time step size is set so that the Courant–Friedrichs–Lewy number of LES is the same as that of DNS (Wang et al. Reference Wang, Luo, Li, Tan and Fan2018). The present ANN-SGS models predict the SGS stress tensor at each local grid point using resolved flow variables at the corresponding grid point as inputs. Large-eddy simulations with ANN-SGS models are performed without any ad hoc stabilisation procedures.

Figure 5 shows energy spectra from LES of forced isotropic turbulence at $Re_\lambda = 106$ with a grid resolution of $48^3$, which is the same as that of the training data. The DSM is found to overestimate the energy spectrum in the range of $k \leqslant 5$. Interestingly, S-106H shows nearly identical performance to that of DSM in predicting the energy spectrum. In addition, from the p.d.f. of the SGS dissipation shown in figure 6(a), S-106H and DSM have the same characteristic that they are not capable of predicting backscatter. Note that the p.d.f. of the SGS stress ($\tau _{23}$) predicted by S-106H is almost identical to that of DSM as both p.d.f.s are narrower than that of fDNS (figure 6b). This indicates that S-106H produces the SGS stress similar to that of DSM.

Figure 5. Energy spectra from fDNS and LES of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with grid resolution of $48^3$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – – (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick green dashed line), S-106H.

Figure 6. Results from an a posteriori test of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with grid resolution of $48^3$. (a) P.d.f. of the SGS dissipation $\varepsilon _{SGS}$ ($= -\tau _{ij}\bar {S}_{ij}$); (b) p.d.f. of $\tau _{23}$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; —— (thick red solid line), SL-106H; – – - (thick green dashed line), S-106H.

From the above observation, it can be concluded that the ANN-SGS model with $\bar {S}_{ij}$ as an input predicts the SGS stress that is closely aligned with the given input tensor, and performs similarly to DSM. Consequently, it is expected that S-106H can be improved by the similar concept employed in dynamic mixed models (Liu et al. Reference Liu, Meneveau and Katz1994; Anderson & Meneveau Reference Anderson and Meneveau1999). It is reported that a linear combination of the eddy-viscosity model with the resolved stress $L_{ij} (=\widehat {\bar {u}_i\bar {u}_j}-\widehat {\bar {u}}_i\hat {\bar {u}}_j)$ improves the accuracy of the predicted SGS stress with better alignment to the true SGS stress.

Similarly to the algebraic dynamic mixed models, it is expected that an ANN-SGS model can achieve closer alignment to the true SGS stress and more accurate prediction of the magnitude of the SGS stress if the resolved stress tensor $L_{ij}$ is considered as an input in addition to the strain-rate tensor $\bar {S}_{ij}$ (i.e. ANN-SGS mixed model). A total of 12 components, six from $\bar {S}_{ij}$ and six from the resolved stress $L_{ij}$, are simultaneously provided as inputs for the ANN-SGS mixed model. To investigate the effect of the additional input variable, results of S-106H and SL-106H are compared in a priori and a posteriori tests of forced homogeneous isotropic turbulence at $Re_\lambda = 106$. Results of SL-106H are also compared with those of the algebraic dynamic mixed models in § 3.3.

The performance of SL-106H is compared with that of S-106H in an a priori test. The correlation coefficients between the true and the predicted SGS stress tensors are calculated following the definition of (3.1) and shown in table 4. The correlation coefficients of SL-106H are significantly improved compared with those of S-106H, which indicates that SL-106H reconstructs the instantaneous SGS stress closer to the true SGS stress. In table 3 the correlation coefficients between $\bar {S}_{ij}$ and the predicted SGS stress $\tau _{ij}$ ($Corr(\bar {S}_{ij}, \tau _{ij})$) from SL-106H are found to be closer to those of fDNS than those from S-106H, indicating that SL-106H is capable of aligning the principal axes of the predicted SGS stress closer to the true SGS stress. In addition, SL-106H provides excellent prediction of the p.d.f. of the SGS dissipation (figure 4a), which almost overlaps with that of fDNS. At the same time, SL-106H predicts the p.d.f. of the SGS stress ($\tau _{23}$) closer to that of fDNS, whereas S-106H predicts a narrower p.d.f. as shown in figure 4(b).

Table 4. Correlation coefficients ($Corr(\tau _{ij}^{fDNS}, \tau _{ij}^{ANN})$) between the traceless parts of the true SGS stress ($\tau _{ij}^{fDNS}$) and the predicted SGS stress by ANN-SGS models ($\tau _{ij}^{ANN}$) from an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } = 106$.

An a posteriori test of forced isotropic turbulence at $Re_\lambda = 106$ is also conducted using SL-106H. As shown in figure 5, SL-106H predicts the energy spectrum closer to that of fDNS than S-106H and DSM in the range of $k \leqslant 5$. Furthermore, SL-106H noticeably better predicts the p.d.f. of the SGS dissipation than S-106H (figure 6a). It is worth noting that SL-106H is capable of accurately predicting the backscatter even in the a posteriori test; consequently, LES becomes stable without any ad hoc procedures, which is a clear advantage over the algebraic dynamic SGS models. Model SL-106H is found to predict the p.d.f. of the SGS stress ($\tau _{23}$) more accurately, as shown in figure 6(b).

Figure 7 shows p.d.f.s of the resolved strain-rate tensor from fDNS and LES. Both SL-106H and DSM accurately predict the p.d.f. of $\bar {S}_{11}$, while S-106H shows slight overestimation from the right tail of the p.d.f. On the other hand, no significant differences in the p.d.f.s of $\bar {S}_{23}$ from DSM, SL-106H and S-106H are observed.

Figure 7. The p.d.f.s of the resolved strain-rate tensor (a) $\bar {S}_{11}$ and (b) $\bar {S}_{23}$ from an a posteriori test of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with grid resolution of $48^3$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; —— (thick red solid line), SL-106H; – – - (thick green dashed line), S-106H.

Contours of the SGS dissipation ($\varepsilon _{SGS}$) from fDNS and LES are shown in figure 8 for qualitative comparison. Snapshots are sampled after 21.5 large-eddy turnover times and at the centre of the domain in the $z$ direction. The SGS dissipation of SL-106H shows a relatively closer spatial distribution to that of fDNS than those of DSM and S-106H, since the backscatter (i.e. negative regions in the contours) is observed in the contours of SL-106H and fDNS, while DSM and S-106H are not capable of predicting the backscatter. This result is consistent with the p.d.f.s in figure 6(a).

Figure 8. Contours of the SGS dissipation $\varepsilon _{SGS}$ ($= -\tau _{ij}\bar {S}_{ij}$) from an a posteriori test of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with grid resolution of $48^3$. Results are shown for (a) fDNS, (b) DSM, (c) SL-106H, (d) S-106H.

The second-order longitudinal velocity structure functions from fDNS and LES of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with a grid resolution of $24^3$, $48^3$ and $96^3$ are compared in figure 9. Note that SL-106H and S-106H are trained only for fDNS data with the filter size for $48^3$ resolution. Generalization to untrained resolution will be discussed in § 3.2. The second-order longitudinal velocity structure function $S_2^L(r)$ is defined as

(3.3a,b)\begin{equation} S_2^L(r) \equiv \left\langle (\Delta u^L)^2 \right\rangle, \quad \Delta u^L \equiv \left[\boldsymbol{\bar{u}}(\boldsymbol{x}+\boldsymbol{r}) - \bar{\boldsymbol{u}}(\boldsymbol{x})\right]\boldsymbol{\cdot} \hat{\boldsymbol{r}}, \end{equation}

where $\bar {\boldsymbol {u}}(\boldsymbol {x})$ is the filtered velocity vector at $\boldsymbol {x}$, $\hat {\boldsymbol {r}} = \boldsymbol {r}/|\boldsymbol {r}|$ is a unit vector in the direction of the separation $\boldsymbol {r}$. No distinctive differences in the structure functions at small separations are observed except for those of the no-SGS case that shows notable errors at small separations, indicating an inaccurate prediction of small-scale fluctuations. For all tested grid resolutions, SL-106H shows a more accurate prediction of the structure functions at large separations than DSM and S-106H.

Figure 9. The second-order longitudinal velocity structure functions $S_2^L(r)$ from fDNS and LES of forced homogeneous isotropic turbulence at $Re_\lambda = 106$ with a grid resolution of (a) $24^3$, (b) $48^3$ and (c) $96^3$. The domain size $2{\rm \pi}$ is denoted by $L$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick green dashed line), S-106H.

Results of a priori and a posteriori tests using SL-106H clearly indicate that considering the resolved stress $L_{ij}$ as an input in addition to $\bar {S}_{ij}$ improves the performance of the ANN-SGS model. Unlike the SGS stress predicted by S-106H that is closely aligned with the given input $\bar {S}_{ij}$, the SGS stress predicted by SL-106H is closely aligned with the true SGS stress. Although the performance of S-106H is almost identical to that of DSM, both S-106H and SL-106H have advantages over DSM as they are free from the requirement of a stabilisation process such as averaging over statistically homogeneous directions or clipping of the negative model coefficients.

3.2. Generalization to untrained conditions: decaying homogeneous isotropic turbulence

In this section the application of ANN-SGS models trained with only forced homogeneous isotropic turbulence data to untrained decaying isotropic turbulence is discussed. For successful generalization to decaying isotropic turbulence, ANN-SGS models have to provide an accurate prediction of the SGS stress for various Reynolds numbers and grid resolution.

In this regard, normalisation of variables is important, as it plays a critical role for the consistent performance of an ANN under various conditions. As explained in § 2.4, the input and output tensors are normalised as $\bar {S}_{ij}^* = \bar {S}_{ij}/\langle | \bar {S} |\rangle$, $L_{ij}^* = L_{ij}/\langle | L |\rangle$ and $\tau _{ij}^* = \tau _{ij}/\langle | \tau |\rangle$, where the denominator $\langle | \tau |\rangle$ requires an approximation in actual LES. The input variables ($\bar {S}_{ij}$ and $L_{ij}$) are properly normalised by the magnitudes of the variables to have similar distributions for various conditions. However, normalising the SGS stress tensor to have similar distributions for various conditions is a challenging task because the approximation for $\langle | \tau |\rangle$ is not accurate enough. In other words, an inaccurate approximation of the normalisation factor of the output SGS stress results in a significantly different distribution of the normalised output for different conditions. In this situation, an ANN-based model could suffer from the prior probability shift issue, which occurs when the output variable distributions are different at training and test conditions (i.e. $P_{train}(y) \neq P_{test}(y)$, where $P(y)$ is a probability distribution of an output variable $y$) (Quiñonero-Candela et al. Reference Quiñonero-Candela, Sugiyama, Schwaighofer and Lawrence2009; Moreno-Torres et al. Reference Moreno-Torres, Raeder, Alaiz-Rodríguez, Chawla and Herrera2012). Dataset shift, including the prior probability shift, is not desirable for ANNs as it can cause significant changes in their performance during testing with untrained data (Gawlikowski et al. Reference Gawlikowski, Tassi, Ali, Lee, Humt, Feng, Kruspe, Triebel, Jung and Roscher2021). Consequently, the performance of an ANN-SGS model trained at a specific Reynolds number and on a certain grid resolution could be different from that at other flow conditions.

Therefore, it is suggested that a normalisation factor, which enables the distributions of the normalised inputs and outputs to remain unchanged for various Reynolds numbers and grid resolution, should be selected to avoid such performance inconsistency of an ANN-SGS model during the generalization. The modified Leonard term $\mathcal {L}_{ij}^m$ ($= \overline {\bar {u}_i \bar {u}_j} - \overline {\bar {u}}_i \overline {\bar {u}}_j)$, the resolved stress $L_{ij}$ ($= \widehat {\bar {u}_i \bar {u}_j} - \hat {\bar {u}}_i \hat {\bar {u}}_j)$ and the gradient model term $\tau _{ij}^{grad}$ ($= \frac {1}{12}\bar {\varDelta }^2({\partial \bar {u}_i}/{\partial x_k})({\partial \bar {u}_j}/{\partial x_k}))$ are considered to approximate the normalisation factor for the SGS stress. Among the three candidates, the term that has the most constant ratio to the true SGS stress $\langle | \tau |\rangle$ for various Reynolds numbers and grid resolution should be selected. Figure 10 shows ratios of $\langle | \tau ^{grad}|\rangle$, $\langle | \mathcal {L}^m |\rangle$ and $\langle | L |\rangle$ to $\langle | \tau ^{fDNS}|\rangle$, obtained at three different Reynolds numbers on a fixed grid resolution and three different grid resolutions at $Re_\lambda =106$. The gradient model term $\langle | \tau ^{grad}|\rangle$ shows the most constant ratio to the true SGS stress $\langle | \tau ^{fDNS}|\rangle$ for various Reynolds numbers and grid resolution, compared with the other two terms. Therefore, in the present study, the averaged $L_2$ norm of the gradient model term is selected to normalise and rescale the output SGS stress tensor as $\tau _{ij}^* = \tau _{ij}/\langle | \tau ^{grad}|\rangle$.

Figure 10. Ratios of $\left \langle | \tau ^{grad}|\right \rangle$, $\left \langle | \mathcal {L}^m |\right \rangle$ and $\left \langle | L |\right \rangle$ to $\left \langle | \tau ^{fDNS}|\right \rangle$ at (a) $Re_{\lambda }=106$, 168 and 286 for $\bar {\varDelta }_{train}/\bar {\varDelta }=1$, and for (b) $\bar {\varDelta }_{train}/\bar {\varDelta }=1/2$, 1, 2 at $Re_{\lambda }=106$. Here —— (thick red solid line), $\left \langle | \tau ^{grad}|\right \rangle /\left \langle | \tau ^{fDNS}|\right \rangle$; —— (thick blue solid line), $\left \langle | \mathcal {L}^m |\right \rangle /\left \langle | \tau ^{fDNS}|\right \rangle$; —— (thick green solid line), $\left \langle | L |\right \rangle /\left \langle | \tau ^{fDNS}|\right \rangle$.

Figure 11 shows the p.d.f. of the SGS stress from fDNS of forced isotropic turbulence with and without normalisation. Distributions of the SGS stress without normalisation vary significantly depending on the Reynolds number and grid resolution, while the normalised SGS stress tensors show similar distributions for various Reynolds numbers and grid resolution, which enables an ANN-SGS model trained at a single condition to be successfully generalized to other conditions. In figure 12 effectiveness of the normalisation is verified in LES of forced homogeneous isotropic turbulence at $Re_\lambda =106$ with a grid resolution of $24^3$ and $96^3$, which are coarser and finer by a factor of 2 than that of the training data, respectively. As shown in figure 12(a), SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ predict the energy spectra more accurately than DSM and S-106H at all wavenumbers on a coarser resolution ($24^3$). On a finer resolution of $96^3$, SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ predict the energy spectra more accurately than DSM and S-106H in the range of $k \leqslant 6$ (figure 12b). This indicates that a proper choice of the normalisation factor for the output SGS stress enables the present ANN-SGS mixed models to perform accurately on an untrained coarser and finer grid resolution.

Figure 11. The p.d.f.s of the SGS stress $\tau _{23}$ from fDNS of forced isotropic turbulence with various Reynolds numbers and grid resolution. (a) The p.d.f.s of $\tau _{23}$ without normalisation; (b) the p.d.f.s of normalised $\tau _{23}$ with $\langle | \tau ^{grad}|\rangle$. —— (thick black solid line), $Re_{\lambda }=106$, $\bar {\varDelta }_{train}/\bar {\varDelta }=1$; —— (thick red solid line), $Re_{\lambda }=106$, $\bar {\varDelta }_{train}/\bar {\varDelta }=1/2$; —— (thick blue solid line), $Re_{\lambda }=106$, $\bar {\varDelta }_{train}/\bar {\varDelta }=2$; —— (thick green solid line), $Re_{\lambda }=286$, $\bar {\varDelta }_{train}/\bar {\varDelta }=1$.

Figure 12. Energy spectra from fDNS and LES of forced homogeneous isotropic turbulence at $Re_\lambda =106$ with a grid resolution of (a) $24^3$ and (b) $96^3$ (two-times coarser and finer resolution than that of training data, respectively). Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick blue dashed line), SL-286H; – - – (thick yellow orange dashed-dot line), $\text {SL-106}+286{\rm H}$; – – – (thick green dashed line), S-106H.

Using $\langle | \tau ^{grad}|\rangle$ as the normalisation factor, ANN-SGS models trained with only forced homogeneous isotropic turbulence data are applied to LES for decaying homogeneous isotropic turbulence to assess the generalizability to untrained transient flow. Table 5 shows the initial Reynolds numbers and grid resolution of test LES cases for decaying isotropic turbulence. The numerical methods are the same as those used for LES of forced isotropic turbulence. A fully converged instantaneous flow field of fDNS of forced homogeneous isotropic turbulence is selected as the initial field of decaying homogeneous isotropic turbulence. The robustness of ANN-SGS models to a random-phase initial condition is assessed in Appendix D.

Table 5. Test LES cases for decaying homogeneous isotropic turbulence with ANN-SGS models and DSM. Here $N$ is the number of grid points in each direction. Effects of grid resolution and the initial Reynolds number on the performance of ANN-SGS models are considered.

Figure 13 shows energy spectra from LES of decaying homogeneous isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$. Energy spectra are obtained at $t/T_{e,0} = 1.1$, $3.3$ and $6.6$, where time $t$ is normalised by the initial eddy turnover time $T_{e,0}$. Energy spectra from S-106H and DSM are found to be almost identical. This is consistent with the a posteriori test of forced isotropic turbulence where S-106H performs similar to DSM. Both S-106H and DSM overestimate the energy spectra in the range of $k < 5$, and errors increase as flow evolves. In contrast, energy spectra from SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ show better agreement with those of fDNS than DSM at all time steps.

Figure 13. Energy spectra from fDNS and LES of decaying homogeneous isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$ (DHIT106 case). Results are shown for (a) $t/T_{e,0} = 1.1$; (b) $t/T_{e,0} = 3.3$; (c) $t/T_{e,0} = 6.6$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick blue dashed line), SL-286H; – - – (thick yellow orange dashed-dot line), $\text {SL-106}+286\text {H}$; – – - (thick green dashed line), S-106H.

Additionally, temporal evolution of the resolved kinetic energy and the mean SGS dissipation are investigated from the same simulations and presented in figure 14. The DSM and S-106H predict smaller SGS dissipation than that of fDNS, which leads to a slower decaying rate of the resolved kinetic energy. Compared with DSM, SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ significantly better predict temporal evolutions of the resolved kinetic energy and the SGS dissipation, which are in good agreement with those of fDNS.

Figure 14. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$ (DHIT106 case). Temporal evolution of (a) the resolved kinetic energy and (b) the mean SGS dissipation $\langle \varepsilon _{SGS}\rangle$ ($= \langle -\tau _{ij}\bar {S}_{ij}\rangle$) are shown. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick blue dashed line), SL-286H; – - – (thick yellow orange dashed-dot line), $\text {SL-106}+286\text {H}$; – – - (thick green dashed line), S-106H.

Similarly to the results of forced isotropic turbulence, SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ perform significantly better than DSM for LES of decaying isotropic turbulence, which clearly indicates that the additional input variable $L_{ij}$ is beneficial. In addition, SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ provide stable solutions without any ad hoc stabilisation procedures unlike DSM during the temporal evolution of the flow.

Note that SL-106H, SL-286H and $\text {SL-106}+286\text {H}$ show almost identical performance, although they are trained at different Reynolds numbers. This is because the present normalisation of the input and output variables enables the mapping of an ANN between the input resolved variables and the output SGS stress to remain unchanged for various flow conditions. Therefore, in the present study, training an ANN-SGS model at a single Reynolds number and on a single grid resolution is found to be sufficient for generalization to untrained flow conditions.

The performance of ANN-SGS models is further investigated through LES of decaying isotropic turbulence with different grid resolutions. Coarser ($24^3$) and finer ($96^3$) grid resolutions by a factor of 2 than that of the training dataset ($48^3$) are considered. These are challenging cases, as the magnitudes of the SGS stress and the resolved kinetic energy differ from those of the training data as the turbulence decays. Nevertheless, SL-106H and SL-286H perform consistently better than DSM.

The LES of decaying homogeneous isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $24^3$ is performed and results are shown in figure 15. The DSM and S-106H show similar performance; they overestimate the energy spectra at all wavenumbers, and the resolved kinetic energy decays slower than that of fDNS. On the other hand, SL-106H and SL-286H provide more accurate prediction of the energy spectra than that of DSM, especially with smaller errors at $k \leqslant 2$. Furthermore, SL-106H and SL-286H predict the temporal evolution of the resolved kinetic energy more accurately than DSM and S-106H.

Figure 15. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $24^3$ (DHIT106c case). (a) Energy spectra at $t/T_{e,0} = 1.1$, $3.3$ and $6.6$; (b) temporal evolution of the resolved kinetic energy. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – –  (thick black dashed line), no-SGS; —— (thick red dashed line), SL-106H; – – - (thick blue dashed line), SL-286H; – – - (thick green dashed line), S-106H.

Figure 16 shows results from LES of decaying homogeneous isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $96^3$. As the grid resolution becomes finer, the energy spectra and the temporal evolution of the resolved kinetic energy obtained by SL-106H and SL-286H become more accurate and almost overlap with those of fDNS. However, DSM and S-106H overestimate energy spectra in the range of $k \leqslant 6$ and predict slower decaying rates of the resolved kinetic energy.

Figure 16. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $96^3$ (DHIT106f case). (a) Energy spectra at $t/T_{e,0} = 1.1$, $3.3$ and $6.6$; (b) temporal evolution of the resolved kinetic energy. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick blue dashed line), SL-286H; – – - (thick green dashed line), S-106H.

The performance of ANN-SGS models is tested through LES at a higher initial Reynolds number $Re_\lambda$ of 286. Figure 17 shows results from LES of decaying homogeneous isotropic turbulence with a grid resolution of $128^3$, in which the cutoff wavenumber is located in the inertial range, same as in the training data. Notably, SL-106H is successfully generalized to the higher Reynolds number, and it predicts the energy spectra and the temporal evolution of the resolved kinetic energy more accurately than DSM. In figure 17(a), SL-106H and SL-286H accurately predict the energy spectra, which are in good agreement with those of fDNS. In contrast, DSM and S-106H slightly underestimate the energy at low wavenumbers ($k \leqslant 2$), while they overestimated the energy at $4 \leqslant k \leqslant 10$. In figure 17(b) temporal evolutions of the resolved kinetic energy of SL-106H and SL-286H are well matched with those of fDNS at all time steps, whereas DSM and S-106H show slower decaying rates of the resolved kinetic energy until $t/T_{e,0} \approx 5$.

Figure 17. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 286 with a grid resolution of $128^3$ (DHIT286 case). (a) Energy spectra at $t/T_{e,0} = 2.0$, $4.9$, $7.9$; (b) temporal evolution of the resolved kinetic energy. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; —— (thick red solid line), SL-106H; – – - (thick blue dashed line), SL-286H; – – - (thick green dashed line), S-106H.

Despite the fact that the present ANN-SGS models are not trained for the transient characteristics of decaying isotropic turbulence, they can be successfully utilised for LES of transient decaying isotropic turbulence with various initial Reynolds numbers and grid resolution. This is again possible due to the selection of proper normalisation factors for the input and output variables such that the normalisation enables the mapping of an ANN between the input resolved variables and the output SGS stress to remain unchanged during the temporal decay of turbulence.

It is worth noting that, for wall-bounded turbulent flow, normalisation of the SGS stress using the gradient model term may not be appropriate since it is known to have an incorrect near-wall scaling (Liu et al. Reference Liu, Meneveau and Katz1994). For turbulent channel flow and also for homogeneous isotropic turbulence, the magnitude of the resolved stress tensor is found to be the best alternative to the magnitude of the gradient model term. The advantage of using the resolved stress tensor for normalisation of the SGS stress over other normalisation options is discussed in Appendix E in detail.

3.3. Comparison with algebraic dynamic mixed models

In § 3.1 the predictive performance of SL-106H is found to meet the expectation that the use of both $\bar {S}_{ij}$ and $L_{ij}$ as inputs could perform better than DSM due to better prediction of the SGS stress in terms of the magnitude and alignment to the true SGS stress. Consequently, it is of interest to investigate how well the ANN-SGS mixed model performs compared with the algebraic dynamic mixed models. Therefore, the performance of the developed ANN-SGS mixed model is compared with one-parameter (Zang et al. Reference Zang, Street and Koseff1993; Vreman, Geurts & Kuerten Reference Vreman, Geurts and Kuerten1994) and two-parameter (Anderson & Meneveau Reference Anderson and Meneveau1999) algebraic dynamic mixed models through LES of both forced and decaying isotropic turbulence.

The one-parameter dynamic mixed model that combines the eddy-viscosity model with the modified Leonard term $(\mathcal {L}_{ij}^m = \overline {\bar {u}_i \bar {u}_j} - \bar {\bar {u}}_i \bar {\bar {u}}_j)$, which was formulated by Zang et al. (Reference Zang, Street and Koseff1993) and later modified by Vreman et al. (Reference Vreman, Geurts and Kuerten1994), is denoted as DMM. The model coefficient of DMM is dynamically determined to close the SGS stress as

(3.4)\begin{equation} \tau_{ij} - \tfrac{1}{3}\delta_{ij}\tau_{kk} ={-}2C\bar{\varDelta}^2|\bar{S}|\bar{S}_{ij} + \mathcal{L}_{ij}^m - \tfrac{1}{3}\delta_{ij}\mathcal{L}_{kk}^m, \end{equation}

where $C=\left \langle M_{ij}(L_{ij}-H_{ij})\right \rangle /\left \langle M_{ij}M_{ij}\right \rangle$, $M_{ij}=-2\hat {\varDelta }^2|\hat {\bar {S}}|\hat {\bar {S}}_{ij}+2\bar {\varDelta }^2|\widehat {\bar {S}|\bar {S}_{ij}}$, $L_{ij}=\widehat {\bar {u}_i\bar {u}_j}-\hat {\bar {u}}_i\hat {\bar {u}}_j$, $H_{ij}=\widehat {\overline {\hat {\bar {u}}_i\hat {\bar {u}}_j}}-\hat{\bar{\hat{\bar{u}}_i}}\hat{\bar{\hat{\bar{u}}}_{j}}-(\widehat {\overline {\bar {u}_i\bar {u}_j}}- \widehat {\bar {\bar {u}}_i\bar {\bar {u}}_j})$, $\mathcal {L}_{ij}^m = \overline {\bar {u}_i \bar {u}_j} - \bar {\bar {u}}_i \bar {\bar {u}}_j$.

Anderson & Meneveau (Reference Anderson and Meneveau1999) formulated the dynamic two-parameter mixed model ($DTM_{sim}$), which combines the eddy-viscosity model with the resolved stress and calculates two model coefficients for eddy-viscosity and mixed terms using the dynamic procedure as

(3.5)$$\begin{gather} \tau_{ij} ={-}2C_{1}\bar{\varDelta}^2|\bar{S}|\bar{S}_{ij} + C_{2}L_{ij}, \end{gather}$$

(3.6)$$\begin{gather}C_{1} = \frac{\left\langle L_{ij}M_{ij}\right\rangle \left\langle N_{ij}N_{ij}\right\rangle - \left\langle L_{ij}N_{ij}\right\rangle \left\langle M_{ij}N_{ij}\right\rangle }{\left\langle M_{ij}M_{ij}\right\rangle \left\langle N_{ij}N_{ij}\right\rangle - \left\langle N_{ij}M_{ij}\right\rangle ^2}, \end{gather}$$

(3.7)$$\begin{gather}C_{2} = \frac{\left\langle L_{ij}N_{ij}\right\rangle \left\langle M_{ij}M_{ij}\right\rangle - \left\langle L_{ij}M_{ij}\right\rangle \left\langle M_{ij}N_{ij}\right\rangle }{\left\langle M_{ij}M_{ij}\right\rangle \left\langle N_{ij}N_{ij}\right\rangle - \left\langle N_{ij}M_{ij}\right\rangle ^2}, \end{gather}$$

where $L_{ij}=\widehat {\bar {u}_i\bar {u}_j}-\hat {\bar {u}}_i\hat {\bar {u}}_j$, $M_{ij}=-2\hat {\varDelta }^2|\hat {\bar {S}}|\hat {\bar {S}}_{ij}+2 \bar {\varDelta }^2|\widehat {\bar {S}|\bar {S}_{ij}}$, $N_{ij}=(\widetilde {\hat {\bar {u}}_i\hat {\bar {u}}_j}-\widetilde {\hat {\bar {u}}_i} \widetilde {\hat {\bar {u}}_j}) - (\widehat {\widehat {\bar {u}_i\bar {u}_j}}-\widehat {\hat {\bar {u}}_i\hat {\bar {u}}_j})$ and $\widetilde {(\,{\cdot }\,)}$ denotes the filtering at the $5\bar {\varDelta }$ scale.

Anderson & Meneveau (Reference Anderson and Meneveau1999) proposed another dynamic two-parameter mixed model ($DTM_{nl}$), which combines the eddy-viscosity model with the Clark model as

(3.8)\begin{equation} \tau_{ij} ={-}2C_{1}\bar{\varDelta}^2|\bar{S}|\bar{S}_{ij} + C_{2}\bar{\varDelta}^2\frac{\partial \bar{u}_i}{\partial x_k}\frac{\partial \bar{u}_j}{\partial x_k}, \end{equation}

where $C_{1}$ and $C_{2}$ are determined by (3.6) and (3.7), respectively, with $L_{ij}=\widehat {\bar {u}_i\bar {u}_j}-\hat {\bar {u}}_i\hat {\bar {u}}_j$, $M_{ij}=-2\hat {\varDelta }^2|\hat {\bar {S}}|\hat {\bar {S}}_{ij}+2\bar {\varDelta }^2| \widehat {\bar {S}|\bar {S}_{ij}}$, $N_{ij}=\hat {\varDelta }^2({\partial \hat {\bar {u}}_i}/{\partial x_k})({\partial \hat {\bar {u}}_j}/{\partial x_k}) - \bar {\Delta }^2\widehat { ( {\partial \bar {u}_i}/{\partial x_k} ) ({\partial \bar {u}_j}/{\partial x_k}) }$.

Table 6 shows correlation coefficients between the true and predicted SGS stresses by ANN-SGS models and algebraic SGS models from an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } = 106$. Model $DTM_{nl}$ shows the best correlation coefficients, while DSM shows the worst correlation coefficients among tested models with similar values to those reported by Xie et al. (Reference Xie, Yuan and Wang2020b). Correlation coefficients from $DTM_{sim}$ are better than those from DSM and worse than those from DMM (Salvetti & Banerjee Reference Salvetti and Banerjee1995; Xie et al. Reference Xie, Yuan and Wang2020b). Model S-106H shows poor correlation coefficients similar to DSM, while SL-106H shows comparable correlation coefficients to those of DMM. The p.d.f.s of the SGS stress $\tau _{23}$ in figure 18(b) show a similar trend. The DSM shows the most inaccurate p.d.f. of $\tau _{23}$, while $DTM_{sim}$ and SL-106H show similarly improved prediction. The DMM and $DTM_{nl}$ show more accurate p.d.f.s of the SGS stress compared with SL-106H. However, it is found from the following a posteriori test that higher correlation coefficients of the SGS stress in an a priori test do not guarantee more accurate LES results as also reported by Park et al. (Reference Park, Lee and Choi2005) and Park & Choi (Reference Park and Choi2021).

Figure 18. Results from an a priori test of forced homogeneous isotropic turbulence at $Re_\lambda = 106$. (a) The p.d.f. of the SGS dissipation $\varepsilon _{SGS}$ ($= -\tau _{ij}\bar {S}_{ij}$); (b) p.d.f. of $\tau _{23}$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – - – (thick black dashed-dot line), DMM; – - – (thick blue dashed-dot line), $DTM_{nl}$; – - – (thick green dashed-dot line), $DTM_{sim}$; —— (thick red solid line), SL-106H.

Table 6. Correlation coefficients ($Corr(\tau _{ij}^{fDNS}, \tau _{ij}^{model})$) between the traceless parts of the true SGS stress ($\tau _{ij}^{fDNS}$) and the predicted SGS stress ($\tau _{ij}^{model}$) from an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } =106$.

Table 7 shows correlation coefficients between $\bar {S}_{ij}$ and the predicted SGS stress $\tau _{ij}^{model}$. Model S-106H shows high absolute magnitudes of $Corr(\bar {S}_{ij}, \tau _{ij})$ similar to those of DSM, as discussed in § 3.1. Both DMM and $DTM_{nl}$ show closer values of $Corr(\bar {S}_{ij}, \tau _{ij})$ to those of fDNS compared with DSM. On the other hand, SL-106H predicts $Corr(\bar {S}_{ij}, \tau _{ij})$ most closely to those of fDNS. In addition, SL-106H shows the most accurate prediction of the p.d.f. of the SGS dissipation, while the dynamic mixed models overestimate the forward scatter (figure 18a). The DSM is not capable of predicting the backscatter.

Table 7. Correlation coefficients ($Corr(\bar {S}_{ij}, \tau _{ij})$) between $\bar {S}_{ij}$ and the predicted SGS stress from an a priori test of forced homogeneous isotropic turbulence at $Re_{\lambda } =106$.

Figure 19 shows energy spectra from LES of forced isotropic turbulence at $Re_\lambda =106$ with a grid resolution of $48^3$, including those from the algebraic dynamic mixed models. Except for DSM, all models well predict the energy spectrum in the range of $k < 5$. However, energy spectra of DMM and $DTM_{sim}$ deviate from that of fDNS in the range of $k \geqslant 5$, showing large errors near the cutoff wavenumber. Model $DTM_{nl}$ accurately predicts the energy spectrum at all wavenumbers. Model SL-106H produces the energy spectrum nearly identical to that of $DTM_{nl}$. Both SL-106H and $DTM_{sim}$ accurately predict the SGS dissipation, whereas DSM and DMM underestimate the backscatter and $DTM_{nl}$ overestimates the forward scatter as shown in figure 20(b).

Figure 19. Energy spectra from fDNS and LES of forced homogeneous isotropic turbulence at $Re_\lambda =106$ with a grid resolution of $48^3$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – - – (thick black dashed-dot line), DMM; – - – (thick blue dashed-dot line), $DTM_{nl}$; – - – (thick green dashed-dot line), $DTM_{sim}$; —— (thick red solid line), SL-106H.

Figure 20. The p.d.f. of the SGS dissipation $\varepsilon _{SGS}$ from LES of forced homogeneous isotropic turbulence at $Re_\lambda =106$ with a grid resolution of $48^3$. (a) Without ad hoc stabilisation; (b) with ad hoc stabilisation including averaging of the model coefficients in statistically homogeneous directions and clipping of the negative model coefficients. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – - – (thick black dashed-dot line), DMM; – - – (thick blue dashed-dot line), $DTM_{nl}$; – - – (thick green dashed-dot line), $DTM_{sim}$; —— (thick red solid line), SL-106H.

Note that the algebraic dynamic SGS models tested in the present study are stable only after applying ad hoc procedures (i.e. averaging of the model coefficients in statistically homogeneous directions and clipping of the negative model coefficients). A possible reason for this problem can be found from the p.d.f. of the SGS dissipation without ad hoc procedures, as shown in figure 20(a). The dynamic Smagorinsky and algebraic dynamic mixed models significantly overestimate both backscatter and forward scatter when the ad hoc procedures are not applied. The excessive backscatter causes LES to diverge within tens of time steps. In contrast, LES with SL-106H is stable without any ad hoc procedures, since SL-106H accurately predicts the p.d.f. of the SGS dissipation, which is a notable advantage of the present ANN-SGS mixed model.

Additionally, LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$ is performed, and the results are shown in figures 21 and 22. Similarly to the results from LES of forced isotropic turbulence, DMM accurately predicts energy spectra at $k < 5$ but underpredicts the energy at $k \geqslant 5$. On the other hand, energy spectra of $DTM_{nl}$ are accurate at $k \geqslant 5$ but the energy of large scale at $k \leqslant 2$ is slightly overestimated at $t/T_{e,0} = 3.3$ and $6.6$. Model $DTM_{sim}$ provides similar energy spectra, resolved kinetic energy and SGS dissipation to those of DSM after $t/T_{e,0} \geqslant 1.62$ since the model coefficient $C_2$ of (3.5) becomes negative and is clipped as zero after that time. The same issue in decaying isotropic turbulence is reported by Anderson & Meneveau (Reference Anderson and Meneveau1999); the similarity coefficient of $DTM_{sim}$ becomes negative when the grid resolution is too coarse so that the test-filter size exceeds the integral scale. Compared with the algebraic dynamic mixed models, SL-106H predicts most accurately the energy spectra at all time steps; especially, it shows the smallest error in the range of $k < 5$. In addition, SL-106H most accurately predicts the temporal evolution of the resolved kinetic energy and the mean SGS dissipation, as shown in figure 22. This indicates that the present ANN-SGS mixed model with input tensors of $\bar {S}_{ij}$ and $L_{ij}$ is able to predict the decay of isotropic turbulence more accurately than the algebraic dynamic mixed models.

Figure 21. Energy spectra from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda =106$ with a grid resolution of $48^3$ (DHIT106 case). Results are shown for (a) $t/T_{e,0} = 1.1$; (b) $t/T_{e,0} = 3.3$; (c) $t/T_{e,0} = 6.6$. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – - – (thick black dashed-dot line), DMM; – - – (thick blue dashed-dot line), $DTM_{nl}$; – - – (thick green dashed-dot line), $DTM_{sim}$; —— (thick red solid line), SL-106H.

Figure 22. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$ (DHIT106 case). Temporal evolution of (a) the resolved kinetic energy and (b) the mean SGS dissipation $\langle \varepsilon _{SGS}\rangle$ ($= \langle -\tau _{ij}\bar {S}_{ij}\rangle$) are shown. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – - – (thick black dashed-dot line), DMM; – - – (thick blue dashed-dot line), $DTM_{nl}$; – - – (thick green dashed-dot line), $DTM_{sim}$; —— (thick red solid line), SL-106H.

3.4. Computational cost of ANN-SGS models

Computational cost is an important factor of ANN-SGS models, as it is one of the main obstacles to the practical application of the models. Previous studies (Wang et al. Reference Wang, Luo, Li, Tan and Fan2018; Xie & Wang Reference Xie and Wang2019; Xie et al. Reference Xie, Wang and Weinan2020a; Yuan et al. Reference Yuan, Xie and Wang2020; Park & Choi Reference Park and Choi2021) reported that ANN-SGS models are computationally more expansive than DSM, which can be considered a major disadvantage. Therefore, a parameter study of the present ANN-SGS mixed model with the number of neurons per hidden layer is conducted to find an optimal size of the ANN, as the computational time for forward propagation of an ANN depends on the size of the ANN.

Five different ANNs with 12 to 48 neurons and 2 to 4 hidden layers are considered. The ANN-SGS mixed models with the five different ANNs are trained with the fDNS dataset of forced isotropic turbulence at $Re_\lambda = 106$. Figure 23 shows results from LES of decaying homogeneous isotropic turbulence with a grid resolution of $48^3$ using ANN-SGS mixed models with ANNs of five different sizes. It is found that 12 neurons and 2 hidden layers are sufficient, and additional neurons or hidden layers do not improve the performance of the ANN-SGS mixed model. However, the computational time to evaluate the SGS stress increases with the number of neurons as ratios of ${\rm DSM} : 12n : 24n : 48n = 1 : 0.71 : 0.89 : 1.42$, where $n$ denotes the number of neurons per hidden layer. Therefore, in the present study an ANN with 12 neurons at each of the two hidden layers is selected.

Figure 23. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$. (a) Energy spectra at $t/T_{e,0} = 1.1$, $3.3$ and $6.6$; (b) temporal evolution of the resolved kinetic energy. Here $\blacksquare$, fDNS; ANN-SGS mixed model with —— (thick red solid line), 12 neurons and 2 hidden layers; – - – (thick yellow orange dashed-dot line), 24 neurons and 2 hidden layers; – - – (thick blue dashed-dot line), 48 neurons and 2 hidden layers; – – - (thick green dashed line), 12 neurons and 3 hidden layers; – – - (thick magenta dashed line), 12 neurons and 4 hidden layers.

The computational cost of the developed ANN-SGS mixed model is compared with those of the algebraic dynamic SGS models. For a fair comparison, the matrix operations of the ANN are implemented on the LES solver. The weight matrices and the bias vectors of the trained ANN are saved and transferred to the solver to perform a forward propagation of the ANN. All simulations are conducted using 16 CPU cores of Intel(R) Xeon(R) E5-2650 v2, and the time consumption for evaluating the SGS stress tensors during the same 100 time steps are compared. Figure 24 shows ratios of the computational time of the tested SGS models for evaluating the SGS stress. The algebraic dynamic mixed models require 1.6 to 2.2 times greater computational time than DSM to calculate the SGS stress. It is noteworthy that the developed ANN-SGS mixed model is computationally cheaper than the algebraic dynamic mixed models and even than DSM.

Figure 24. Ratio of computational time of tested SGS models for evaluating the SGS stress. Computational time is normalised to that of DSM.

3.5. Training and testing for turbulent channel flow

In this section application of the developed ANN-SGS mixed model to a wall-bounded flow is discussed. Turbulent channel flow at $Re_{\tau }=180$ and 395 is considered. The predictive capabilities of the ANN-SGS mixed model that uses both $\bar {S}_{ij}$ and $L_{ij}$ as inputs and the ANN-SGS model with $\bar {S}_{ij}$ as the only input are investigated through a posteriori tests for a turbulent channel flow with various untrained Reynolds number and grid resolution. Additionally, it is further investigated whether the ANN-SGS mixed model trained with both channel flow and homogeneous isotropic turbulence at a certain condition is capable of providing accurate solutions for both types of flow at untrained conditions. Lastly, results from LES of turbulent channel flow with the ANN-SGS mixed model that is trained only with homogeneous isotropic turbulence are discussed.

Direct numerical simulations are performed with parameters summarized in table 8 to construct training datasets and reference solutions for LES. A fourth-order finite-difference code by Bose, Moin & You (Reference Bose, Moin and You2010) is utilised to conduct DNS and LES. The convection terms are discretized in a skew-symmetric form (Morinishi et al. Reference Morinishi, Lund, Vasilyev and Moin1998). A semi-implicit method is used for time marching. The viscous diffusion terms in the wall-normal direction are integrated implicitly with the Crank–Nicolson method, and the other terms are explicitly advanced with the third-order Runge–Kutta method (Morinishi et al. Reference Morinishi, Lund, Vasilyev and Moin1998; Bose et al. Reference Bose, Moin and You2010; Kang & You Reference Kang and You2021).

Table 8. Parameters for DNS of turbulent channel flow. Here $L_x$ and $L_z$ are the streamwise and spanwise domain sizes, respectively; $\delta$ is the channel half-width, and $N_x$, $N_y$ and $N_z$ are numbers of grid points in the streamwise, wall-normal, and spanwise directions, respectively; $\Delta x^+$ and $\Delta z^+$ are the streamwise and the spanwise grid sizes in wall units, respectively; $\Delta y_{min}^+$ and $\Delta y_{c}^+$ are the wall-normal grid sizes at the wall and the centreline, respectively.

The fDNS dataset of a turbulent channel flow at $Re_{\tau } = 180$ is constructed by applying the grid and test filters with $\bar {\varDelta } = 4\varDelta _{DNS}$ and $\hat {\bar {\varDelta }} = 8\varDelta _{DNS}$ to instantaneous DNS fields, where $\varDelta _{DNS}$ is the grid spacing of DNS. The filter width $\bar {\varDelta }$ of the fDNS dataset is the same as the grid size of LES with $N_x\times N_y \times N_z = 48\times 48\times 48$ cells at $Re_{\tau }=180$. Fourth-order commutative discrete filters by Vasilyev, Lund & Moin (Reference Vasilyev, Lund and Moin1998) are utilised to construct the fDNS dataset and to conduct test filtering of LES. A total of $3.5\times 10^7$ data points are sampled from five instantaneous velocity fields of fDNS after flow becomes fully developed, and $2.8\times 10^7$ data points are used for training, whereas the rest are used for testing. It has been found that training with more than $2.8\times 10^7$ data points (four instantaneous fDNS fields) does not show meaningful alteration of the ANN performance.

The ANN-SGS models are trained from scratch using the fDNS dataset of a turbulent channel flow by the same method in § 2.4, except for normalisation of the SGS stress. The normalisation factor of the SGS stress is estimated using the resolved stress tensor $L_{ij}$ instead of the gradient model term since it is reported that the gradient model term has incorrect near-wall scaling (Liu et al. Reference Liu, Meneveau and Katz1994) in contrast to the resolved stress tensor.

Each ANN-SGS model is named following the same rules as in § 2.4. The characters C and H at the end of the model names denote channel flow and homogeneous isotropic turbulence, respectively, and represent flow with which each ANN-SGS model is trained. For example, S-180C uses $\bar {S}_{ij}$ as the input, whereas SL-180C uses $\bar {S}_{ij}$ and $L_{ij}$ as inputs, and both models are trained with turbulent channel flow at $Re_{\tau } = 180$. Model SL-106H180C is trained with both channel flow at $Re_{\tau } = 180$ and homogeneous isotropic turbulence at $Re_{\lambda } = 106$; SL-106H is only trained with homogeneous isotropic turbulence at $Re_{\lambda } = 106$. For training of SL-106H180C, the same amount of fDNS data points ($2.8\times 10^7$) from both homogeneous isotropic turbulence and turbulent channel flow are used to prevent the model from being biased. In addition, the same number of training samples are provided with the mini-batch size of 128 by extracting 64 fDNS data from homogeneous isotropic turbulence and turbulent channel flow, respectively.

Trained ANN-SGS models are initially evaluated in an a priori test of a turbulent channel flow at $Re_{\tau } = 180$ (table 9). The DSM shows poor correlation coefficients of the SGS stress and the SGS dissipation as reported by Park & Choi (Reference Park and Choi2021). The SGS stress from DMM shows the best correlation coefficient of 0.88, a similar value reported by Salvetti & Banerjee (Reference Salvetti and Banerjee1995). Model SL-106H shows higher correlation coefficients than those of DSM even though the model is not trained with turbulent channel flow. Model SL-180C is more accurate than S-180C in predicting the SGS stress and the SGS dissipation, indicating that the use of $L_{ij}$ in addition to $\bar {S}_{ij}$ as an input is also beneficial to the prediction of the SGS stress for channel flow.

Table 9. Correlation coefficients ($Corr(\tau _{xy}^{fDNS},\tau _{xy}^{model})$, $Corr(\varepsilon _{SGS}^{fDNS},\varepsilon _{SGS}^{model})$) from an a priori test of turbulent channel flow at $Re_\tau =180$.

In order to investigate the generalizability of the ANN-SGS models, the models are tested for LES of turbulent channel flow at various conditions listed in table 10. The Reynolds number and the grid size of LES180 case are the same as those of the training data (i.e. $Re_{\tau } = 180$, $N_x\times N_y \times N_z = 48\times 48\times 48$), and the remaining four cases correspond to untrained Reynolds number and grid resolution conditions.

Table 10. Parameters for LES of turbulent channel flow with ANN-SGS models and DSM. The effects of grid resolution and the Reynolds number on the performance of ANN-SGS models are considered. Here $L_x$ and $L_z$ are the streamwise and spanwise domain sizes, respectively; $\delta$ is the channel half-width, and $N_x$, $N_y$ and $N_z$ are numbers of grid points in the streamwise, wall-normal, and spanwise directions, respectively; $\Delta x^+$ and $\Delta z^+$ are the streamwise and spanwise grid sizes in wall units, respectively; $\Delta y_{min}^+$ and $\Delta y_{c}^+$ are the wall-normal grid sizes at the wall and the centreline, respectively.

Figure 25 shows results of the LES180 case. The DSM overpredicts the mean streamwise velocity and the streamwise velocity fluctuations $\bar {u}'_{rms}$, while underpredicts the wall-normal velocity fluctuations $\bar {v}'_{rms}$ and the spanwise velocity fluctuations $\bar {w}'_{rms}$. The overprediction of the mean streamwise velocity by DSM is consistent with the results of the previous studies (Horiuti Reference Horiuti1997; Park & Choi Reference Park and Choi2021; Kim et al. Reference Kim, Kim, Kim and Lee2022). In addition, it has been reported that LES with DSM exhibits overprediction of $\bar {u}'_{rms}$ and underpredictions of $\bar {v}'_{rms}$ and $\bar {w}'_{rms}$ (Park & Mahesh Reference Park and Mahesh2009; You & Moin Reference You and Moin2007, Reference You and Moin2009). The larger errors of LES with DSM than those of the no-SGS case in the prediction of velocity fluctuations have also been reported by Morinishi & Vasilyev (Reference Morinishi and Vasilyev2001) and Cabot (Reference Cabot1994). The DSM also shows slight underprediction of the mean Reynolds shear stress at $y^+<30$. Large-eddy simulation without an SGS model (no-SGS) underpredicts the mean streamwise velocity and overpredicts the Reynolds shear stress. However, SL-180C more accurately predicts the mean velocity, root-mean-squared (r.m.s.) velocity fluctuations, and the mean Reynolds shear stress than DSM and no-SGS. Model S-180C performs similar to SL-180C in terms of the velocity fluctuations and the Reynolds shear stress. However, S-180C shows slight underprediction of the mean streamwise velocity, compared with SL-180C. Model SL-106H180C shows almost identical performance to SL-180C, while showing slightly better prediction of the mean Reynolds shear stress at $y^+>40$. Interestingly, SL-106H is stable and accurately predicts the velocity fluctuations, despite the fact that it was trained only for homogeneous isotropic turbulence. The Reynolds shear stress of SL-106H shows a similar error near $y^+=25$ and smaller errors at $y^+>45$, compared with that of DSM. However, SL-106H underpredicts the mean streamwise velocity, which is similar to that of no-SGS. Model SL-106H may inaccurately predict the SGS stress at wall-bounded flow since the mean of the SGS shear stress (off-diagonal components) vanishes (Kang et al. Reference Kang, Chester and Meneveau2003) in homogeneous isotropic turbulence. Therefore, training an ANN-SGS model for a turbulent channel flow dataset seems to be essential to accurately predict the mean streamwise velocity.

Figure 25. Results from fDNS and LES of turbulent channel flow at $Re_\tau =180$ with a grid resolution of $48\times 48\times 48$ (LES180). (a) The mean streamwise velocity; (b) root-mean-squared (r.m.s.) velocity fluctuations; (c) the mean Reynolds shear stress $\langle \bar {u}'\bar {v}'\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x\unicode{x2013}z$ plane and time. The three sets of curves in (b) represent $\bar {u}_{rms}/u_{\tau }$, $\bar {w}_{rms}/u_{\tau }$ and $\bar {v}_{rms}/u_{\tau }$ from top to bottom, respectively. Here $\bigcirc$, fDNS; $+$, DSM; ${\nabla }$, no-SGS; —— (thick red solid line), SL-180C; – – - (thick green dashed line), S-180C; – - – (thick blue dashed-dot line), SL-106H180C; – – (thick yellow orange dashed line), SL-106H; – – - (thick black dashed line), the law of the wall $\langle \bar {u}\rangle /u_{\tau } = 0.41^{-1}\log {y^+}+5.2$.

Table 11 shows the averaged wall shear stress and the skin-friction coefficients ($C_f = 2\langle \tau _w\rangle /\rho U_b^2$, where $\tau _w$ is the wall shear stress and the bulk mean velocity $U_b = 1/2\delta \int _{-\delta }^{+\delta }\langle \bar {u}\rangle {\rm d}y$) of fDNS and LES for the LES180 case. Models SL-180C, S-180C and SL-106H180C predict the wall shear stress more accurately than DSM, while SL-106H shows a slightly larger deviation from that of DSM. On the other hand, the no-SGS case overestimates $C_f$, while LES with DSM underestimates $C_f$. The overestimation of $C_f$ by the no-SGS case is also reported by Vreman (Reference Vreman2004), although the computed $Re_{\tau }$ by the no-SGS case is in good agreement with that of DNS. Model SL-180C shows the most accurate prediction of $C_f$ among tested models.

Table 11. Averaged wall shear stress $\langle \tau _w\rangle /\rho$ and the skin-friction coefficient $C_f$ from fDNS and LES of a turbulent channel flow at $Re_\tau =180$ with a grid resolution of $48\times 48\times 48$ (LES180), where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. Here $C_f = 2\langle \tau _w\rangle /\rho U_b^2$, where $\tau _w$ is the wall shear stress and the bulk mean velocity $U_b = 1/2\delta \int _{-\delta }^{+\delta }\langle \bar {u}\rangle {{\rm d}y}$.

Figure 26 shows results in LES180c and LES180f cases that test the performance of ANN-SGS models on an untrained grid resolution. On a coarser grid resolution (LES180c), DSM shows large errors in the prediction of the mean velocity and velocity fluctuations. Although DSM more accurately predicts the Reynolds shear stress than SL-180C and SL-106H180C (figure 26e), the mean streamwise velocity and the velocity fluctuations predicted by SL-180C and SL-106H180C are significantly better than those of DSM (figure 26a,c).

Figure 26. Results from fDNS and LES of turbulent channel flow at $Re_\tau =180$ with a grid resolution of $32\times 48\times 32$ (a,c,e; LES180c) and $64\times 64\times 64$ (b,d,f; LES180f). (a,b) The mean streamwise velocity; (c,d) r.m.s. velocity fluctuations; (e,f) the mean Reynolds shear stress $\langle \bar {u}'\bar {v}'\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. The three sets of curves in (c,d) represent $\bar {u}_{rms}/u_{\tau }$, $\bar {w}_{rms}/u_{\tau }$ and $\bar {v}_{rms}/u_{\tau }$ from top to bottom, respectively. Here $\bigcirc$, fDNS; $+$, DSM; $\nabla$, no-SGS; —— (thick red solid line), SL-180C; – - – (thick blue dashed-dot line), SL-106H180C; – – - (thick black dashed line), the law of the wall $\langle \bar {u}\rangle /u_{\tau } = 0.41^{-1}\log y^++5.2$.

On a finer grid resolution (LES180f), DSM slightly overpredicts the mean streamwise velocity. The DSM is less accurate than SL-180C and SL-106H180C in the prediction of the velocity fluctuations and the Reynolds shear stress. In contrast, all results of SL-180C and SL-106H180C are in excellent agreement with those of fDNS (figure 26b,d,f). Models SL-180C and SL-106H180C are found to perform better than DSM regardless of grid resolution.

The developed ANN-SGS models are tested for LES at a higher Reynolds number $Re_{\tau }$ of 395, and results are shown in figure 27. The DSM overpredicts the mean streamwise velocity in the LES395 and LES395f cases as shown in figures 27(a) and 27(b). Model S-180C underpredicts the mean streamwise velocity in both LES395 and LES395f cases, and the error becomes larger on a finer grid resolution (figure 27a,b). On the other hand, the mean velocity profiles of SL-180C and SL-106H180C are found to be closer to those of fDNS and less sensitive to grid resolution than DSM and S-180C.

Figure 27. Results from fDNS and LES of a turbulent channel flow at $Re_\tau =395$ with a grid resolution of $48\times 48\times 48$ (a,c,e; LES395) and $64\times 64\times 64$ (b,d,f; LES395f). (a,b) The mean streamwise velocity; (c,d) r.m.s. velocity fluctuations; (e,f) the mean Reynolds shear stress $\langle \bar {u}'\bar {v}'\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. The three sets of curves in (c,d) represent $\bar {u}_{rms}/u_{\tau }$, $\bar {w}_{rms}/u_{\tau }$ and $\bar {v}_{rms}/u_{\tau }$ from top to bottom, respectively. Here $\bigcirc$, fDNS; $+$, DSM; $\nabla$, no-SGS; —— (thick red solid line), SL-180C; – – - (thick green dashed line), S-180C; – - – (thick blue dashed-dot line), SL-106H180C; – – - (thick black dashed line), the law of the wall $\langle \bar {u}\rangle /u_{\tau } = 0.41^{-1}\log y^++5.2$.

In figures 27(c) and 27(d), DSM overpredicts $\bar {u}'_{rms}$ while it underpredicts $\bar {v}'_{rms}$ and $\bar {w}'_{rms}$. In contrast, S-180C, SL-180C and SL-106H180C show similar predictions of $\bar {u}'_{rms}$, $\bar {v}'_{rms}$ and $\bar {w}'_{rms}$ that are more accurate than those of DSM. In figures 27(e) and 27(f), DSM and S-180C similarly overpredict the Reynolds shear stress, while the Reynolds shear stress of SL-180C shows good agreement with that of fDNS. As SL-180C more accurately predicts the mean streamwise velocity and the Reynolds shear stress than S-180C, it can be concluded that the use of $L_{ij}$ in addition to the $\bar {S}_{ij}$ as an input improves the performance of the ANN-SGS model for a turbulent channel flow, similarly to the case of homogeneous isotropic turbulence. In LES of a turbulent channel flow (figures 25, 26 and 27), SL-106H180C performs similarly to SL-180C, especially in the prediction of the mean velocity and the velocity fluctuations.

In figure 28 a qualitative comparison of vortical structures is shown by isosurfaces of Q that is defined as the second invariant of the velocity gradient tensor from fDNS and LES for the LES395f case. Similar scales and distributions of flow structures to those of fDNS are observed by SL-180C, S-180C and DSM, while significantly more vortical structures are found in the flow field of no-SGS due to the lack of SGS dissipation (Park & Choi Reference Park and Choi2021).

Figure 28. Isosurfaces of $Q=0.0025 u_\tau ^4/\nu ^2$ from fDNS and LES of a turbulent channel flow at $Re_\tau =395$ with a grid resolution of $64\times 64\times 64$ (LES395f). Results are shown for (a) fDNS, (b) DSM, (c) no-SGS, (d) SL-180C, (e) S-180C.

Additionally, it is tested whether SL-106H180C provides accurate solutions for LES of decaying isotropic turbulence, and the results are shown in figure 29. Model SL-106H180C shows almost identical performance to SL-106H in the prediction of energy spectra and the resolved kinetic energy. Therefore, the present ANN-SGS mixed model that is trained with isotropic turbulence and turbulent channel flow at a given Reynolds number and grid resolution condition (SL-106H180C) is found to be capable of accurately and stably predicting both turbulent channel flow and isotropic turbulence flow at untrained Reynolds number and grid resolution conditions.

Figure 29. Results from fDNS and LES of decaying isotropic turbulence at the initial Reynolds number $Re_\lambda$ of 106 with a grid resolution of $48^3$. (a) Energy spectra at $t/T_{e,0} = 1.1$, $3.3$ and $6.6$; (b) temporal evolution of the resolved kinetic energy. Here $\blacksquare$, fDNS; —— (thick black solid line), DSM; – – - (thick black dashed line), no-SGS; – - – (thick blue dashed-dot line), SL-106H180C; —— (thick red solid line), SL-106H.

3.6. Normalisation of input and output variables for turbulent channel flow

In § 3.5 the input and output variables of ANN-SGS models are normalised with volume-averaged quantities as done in homogeneous isotropic turbulence. However, as flow variables are inhomogeneous in the wall-normal direction in the case of a turbulent channel flow, normalisation with plane-averaged quantities can also be considered. Thus, two different normalisations for ANN-SGS models using plane-averaged and volume-averaged quantities (denoted as plane-wise normalisation and volume-wise normalisation hereafter, respectively) are compared in this section. For plane-wise normalisation, variables are normalised in every wall-parallel plane with plane-averaged quantities. The additional character V or P is added at the end of the model names to denote volume-wise normalisation or plane-wise normalisation, respectively.

Although, as shown in figures 30(d) and 30(e), SL-180C-P is found to be more accurate in the prediction of the mean SGS shear stress, SL-180C-V provides a more accurate prediction of the backscatter. However, as shown in figures 30(a), 30(b) and 30(c), no significant differences in the mean streamwise velocity, r.m.s. velocity fluctuations and the Reynolds shear stress in SL-180C-P and SL-180C-V are found. The same trends are observed in the comparison of results from S-180C-P and S-180C-V. The volume-wise averaging is not taken on the output SGS stress itself but only on the normalisation factor that is used for regularising the input and output data range. In addition, important physical characteristics of SL-180C-V, such as the near-wall behaviour of the SGS stress (figure 30d) and the backscatter (figure 30e), remain accurate even when volume-wise normalisation is used. However, it is worth noting that an ANN-SGS model trained with plane-wise normalisation is applicable only when statistically homogeneous directions exist.

Figure 30. Results from fDNS and LES of a turbulent channel flow at $Re_\tau =180$ with a grid resolution of $48\times 48\times 48$ (LES180). (a) The mean streamwise velocity; (b) r.m.s. velocity fluctuations; (c) the mean Reynolds shear stress $\langle \bar {u}'\bar {v}'\rangle$; (d) the mean SGS shear stress $\langle \tau _{xy}\rangle$; (e) the mean backscatter $\langle \varepsilon _{SGS}^{-}\rangle =\frac {1}{2}\langle \varepsilon _{SGS}-|\varepsilon _{SGS}|\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. The three sets of curves in (b) represent $\bar {u}_{rms}/u_{\tau }$, $\bar {w}_{rms}/u_{\tau }$ and $\bar {v}_{rms}/u_{\tau }$ from top to bottom, respectively. Here $\bigcirc$, fDNS; $+$, DSM; $\nabla$, no-SGS; —— (thick red solid line), SL-180C-V; – – - (thick green dashed line), S-180C-V; – – (thick yellow orange dashed line), SL-106H-V; —— (thick magenta solid line), SL-180C-P; – - – (thick cyan dashed-dot line), S-180C-P; – – (thick blue dashed line), SL-106H-P; – – - (thick black dashed line), the law of the wall $\langle \bar {u}\rangle /u_{\tau } = 0.41^{-1}\log y^++5.2$.

On the other hand, as shown in figure 30(a), it is found that the plane-wise normalisation in SL-106H-P produces the mean streamwise velocity more accurately than the volume-wise normalisation in SL-106H-V. The mean SGS shear stress by SL-106H-V does not vanish at the wall since SL-106H-V was not trained with the SGS stress distribution near the wall (figure 30d). Compared with SL-180C-V that is trained with channel flow, the mean SGS shear stress is overall overestimated by SL-106H-V, leading to an underestimated Reynolds shear stress (figure 30c). Interestingly, the mean SGS shear stress by SL-106H-P vanishes at the wall, which seems to improve the prediction of the mean streamwise velocity, regardless of the inaccurate prediction of the mean SGS shear stress. Therefore, plane-wise normalisation is found to improve the applicability of the ANN-based mixed SGS model that is trained only with the homogeneous isotropic turbulence to an untrained turbulent channel flow.

Regardless of the normalisation method, providing $L_{ij}$ in addition to the $\bar {S}_{ij}$ as an input is beneficial to the prediction of the backscatter and the mean SGS shear stress (figure 30d,e). Although the mean streamwise velocity and r.m.s.velocity fluctuations from SL-180C-V and S-180C-V are similar to each other, S-180C-V significantly underestimates the backscatter, which is consistent with the results from LES of homogeneous isotropic turbulence. In addition, SL-180C-V more accurately predicts the mean SGS shear stress than S-180C-V. The similar trend is observed in the comparison of SL-180C-P and S-180C-P. Model SL-180C-P is found to more accurately predict the mean SGS shear stress and the backscatter. Owing to the better prediction of the SGS shear stress and backscatter, SL-180C is considered to better predict channel flow at a higher Reynolds number $Re_\tau =395$ than S-180C as shown in figure 27.

3.7. Application of ANN-SGS models trained with homogeneous isotropic turbulence to turbulent channel flow

In this section it is discussed how SL-106H that is trained only with homogeneous isotropic turbulence is partially successful in the prediction of the SGS stress for a turbulent channel flow. As discussed in § 3.1, the ANN-SGS mixed model combines two input tensors ($\bar {S}_{ij}$ and $L_{ij}$) to produce the SGS stress. Therefore, the investigation of statistical relations between each input tensor and the true SGS stress for homogeneous isotropic turbulence and a turbulent channel flow is important.

Correlation coefficients between each input tensor and the true SGS stress for both flows from a priori tests are shown in tables 12 and 13. Because $Corr(\bar {S}_{ij},\tau _{ij})$ and $Corr(L_{ij},\tau _{ij})$ from fDNS are different in the two flows, it can be concluded that $\bar {S}_{ij}$ and $L_{ij}$ contribute differently to the prediction of the SGS stress. In particular, the absolute value of $Corr(\bar {S}_{ij},\tau _{ij})$ from fDNS is close to zero in a turbulent channel flow (see table 13) while it is close to $0.3$ in homogeneous isotropic turbulence (see table 12), which implies that the relative importance of $\bar {S}_{ij}$ for the prediction of $\tau _{ij}$ should be considerably different. Such difference in relative importance of $\bar {S}_{ij}$ is particularly noticeable in the observation of the near-wall behaviours of both input tensors and the SGS stress. As shown in figure 31(a), in fDNS, $\langle \bar {S}_{xy}\rangle$ increases as $y^+$ approaches $0$ whereas $\langle \tau _{xy}\rangle$ decreases from the peak value near the wall as shown in figure 31(d). On the other hand, $\langle L_{xy}\rangle$ shows a similar distribution to $\langle \tau _{xy}\rangle$ as shown in figures 31(b) and 31(d).

Figure 31. Results from an a priori test of a turbulent channel flow at $Re_\tau =180$. (a) Strain-rate tensor $\langle \bar {S}_{xy}\rangle$, (b) resolved stress tensor $\langle L_{xy}\rangle$, (c) $L_2$ norm of resolved stress tensor $\langle |L|\rangle$, (d) SGS shear stress $\langle \tau _{xy}\rangle$, (e) SGS dissipation $\langle \varepsilon _{SGS}\rangle$ and (f) backscatter $\langle \varepsilon _{SGS}^{-}\rangle =\frac {1}{2}\langle \varepsilon _{SGS}-|\varepsilon _{SGS}|\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. Here $\bigcirc$, fDNS; – – (thick yellow orange dashed line), SL-106H-V; – - – (thick yellow orange dashed-dot line), S-106H-V; – – (thick blue dashed line), SL-106H-P.

Table 12. Correlation coefficients ($Corr(\bar {S}_{ij}, \tau _{ij})$, $Corr(L_{ij}, \tau _{ij})$) between the input variables and the predicted SGS stress by ANN-SGS models from an a priori test of forced homogeneous isotropic turbulence at ${Re_{\lambda } = 106}$.

Table 13. Correlation coefficients ($Corr(\tau _{xy}^{fDNS},\tau _{xy}^{ANN})$, $Corr(\bar {S}_{xy},\tau _{xy})$, $Corr(L_{xy},\tau _{xy})$, $Corr(\varepsilon _{SGS}^{fDNS},\varepsilon _{SGS}^{ANN})$) from an a priori test of a turbulent channel flow at $Re_\tau =180$.

These observations explain the incorrect prediction of $\langle \tau _{xy}\rangle$ near the wall by S-106H-V in figure 31(d) since S-106H-V only uses $\bar {S}_{xy}$ to produce $\tau _{xy}$. The absolute value of $Corr(\bar {S}_{xy},\tau _{xy})$ from S-106H-V in a turbulent channel flow is close to 1 indicating that S-106H-V predicts the SGS stress nearly aligned with $\bar {S}_{ij}$. This leads to the prediction of $\langle \tau _{xy}\rangle$ by S-106H-V of which the near-wall behaviour mimics that of $\langle \bar {S}_{xy}\rangle$. Consequently, S-106H-V significantly overestimates the mean SGS dissipation $\langle \varepsilon _{SGS}\rangle$ near the wall (figure 31e).

Model SL-106H-V, on the other hand, predicts $\langle \tau _{xy}\rangle$ near the wall more accurately than S-106H-V. This is because (i) SL-106H-V highly relies on $L_{ij}$ to predict the tensor alignment of $\tau _{ij}$ that can be confirmed from tables 12 and 13 where $Corr(L_{ij},\tau _{ij})$ values of SL-106H are higher than $0.8$ for both types of flow, and (ii) $L_{ij}$ has high similarity to the SGS stress in terms of the correlation coefficient to the true SGS stress (table 13) and the near-wall behaviour (figure 31). It is noteworthy that Zang et al. (Reference Zang, Street and Koseff1993) reported that the scale-similarity part of the dynamic mixed model (the modified Leonard term) shows significant contribution to the Reynolds stress near the wall, while the contribution of the eddy-viscosity part is negligible.

Consequently, SL-106H-V shows an improved near-wall behaviour of the mean SGS shear stress $\langle \tau _{xy}\rangle$ as it is influenced by the near-wall behaviour of the mean resolved stress $\langle L_{xy}\rangle$. Model SL-106H-P shows the best near-wall behaviour among the tested models since the $L_2$ norm of $L_{ij}$, which is multiplied to the output of SL-106H-P for rescaling, shows a correct near-wall behaviour at the wall (figure 31c).

However, SL-106H-V and SL-106H-P show higher absolute values of $Corr(\bar {S}_{xy},\tau _{xy})$ than those of SL-180C-V and fDNS in channel flow (table 13), which indicates the contribution of $\bar {S}_{ij}$ for $\tau _{ij}$ is overestimated. Since $Corr(\bar {S}_{xy},\tau _{xy})$ of fDNS has a small absolute value of 0.08 in channel flow, alignment of $\tau _{ij}$ is inaccurately predicted when the contribution of $\bar {S}_{ij}$ is overestimated. This explains the overprediction of the SGS stress and the SGS dissipation in figures 31(d) and 31(e).

It is also investigated whether SL-106H-V and SL-106H-P can reflect the anisotropy of $L_{ij}$ to produce $\tau _{ij}$ in a turbulent channel flow. In order to quantitatively compare the anisotropy of the predicted SGS stress by ANN-SGS models, the anisotropy invariant maps (Lumley & Newman Reference Lumley and Newman1977) of the SGS anisotropy tensor are examined in figure 32. The SGS anisotropy tensor is defined as (Rasam et al. Reference Rasam, Pouransari, Vervisch and Johansson2016; Inagaki & Kobayashi Reference Inagaki and Kobayashi2020)

(3.9)\begin{equation} a_{ij} = \frac{\left\langle \tau_{ij}\right\rangle }{\left\langle \tau_{kk}\right\rangle } - \frac{1}{3}\delta_{ij}, \end{equation}

where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. The anisotropy invariant maps are constructed based on the two invariants of the anisotropy tensor $a_{ij}$,

(3.10a,b)\begin{equation} II = a_{ij}a_{ij}, \quad III = a_{ik}a_{kj}a_{ji}, \end{equation}

where $II$ and $III$ are the second and third invariants of $a_{ij}$, respectively. The invariant map lies in the region bounded by the limits of the two-component turbulence state ($II = 2/9 + 2III$) and the axisymmetric turbulence state ($II = 3/2(4/3|III|)^{2/3}$). The triangular region defined by the limits represents the realizability condition for the invariants of the SGS tensor.

Figure 32. Anisotropy invariant maps of the SGS stress anisotropy tensor from an a priori test of a turbulent channel flow at $Re_\tau =180$. Here $\bigcirc$, fDNS; —— (thick green solid line), $L_{ij}$; —— (thick red solid line), SL-180C-V; —— (thick yellow orange solid line), SL-106H-V; —— (thick blue solid line), SL-106H-P; $\blacktriangledown$ (yellow orange triangle), S-106H-V.

In the invariant map, the two-component turbulence ($II = 2/9 + 2III$), which is the condition in which the wall-normal component is negligible relative to the other two components, occurs at the wall (Inagaki & Kobayashi Reference Inagaki and Kobayashi2020). The right curved edgy is the axisymmetric turbulence in which the streamwise component is larger than the other two components (Rasam et al. Reference Rasam, Pouransari, Vervisch and Johansson2016). The origin corresponds to the isotropic turbulence state that occurs at the channel centreline.

The invariant map for SL-180C-V approaches the top right corner and ends at the two-component limit as it approaches the wall, indicating that SL-180C-V represents the anisotropy of the SGS stress similar to the true SGS stress. The invariant map of $L_{ij}$ indicates that $L_{ij}$ itself has similar anisotropy to that of the true SGS stress. Since the invariant map of S-106H-V remains near the origin, it is concluded that the SGS stress predicted by S-106H-V is isotropic. On the other hand, the SGS stress from SL-106H-V and SL-106H-P show larger anisotropy than that of S-106H-V as the invariant maps follow the axisymmetric limit, which represents the streamwise component becoming larger than the other two components. Due to the large contribution of $L_{ij}$ for predicting $\tau _{ij}$ by SL-106H-V and SL-106H-P, they represent a limited magnitude of anisotropy by indirectly accounting for the anisotropy of $L_{ij}$. However, the anisotropy magnitude is significantly underestimated and the invariant map does not approach the two-component turbulence state at the wall. This indicates that SL-106H-V and SL-106H-P are not fully capable of representing the asymptotic behaviour of the SGS stress near the wall since they are not trained with channel flow.

In figure 33 results from an a posteriori test of a turbulent channel flow are presented to compare the ANN-SGS models trained only with homogeneous isotropic turbulence. Model S-106H-V shows a significant underprediction of the mean streamwise velocity due to the inaccurate prediction of the SGS shear stress and the SGS dissipation, although the model employs the same normalisation of variables with SL-106H-V. Model S-106H-V also shows poor prediction of the r.m.s.velocity fluctuations and the mean Reynolds shear stress. Compared with SL-106H-V, SL-106H-P that employs plane-wise normalisation further improves the near-wall behaviour of the SGS shear stress and the prediction of the mean streamwise velocity since the rescaling factor ($\langle |L|\rangle _{x-z}$) becomes zero at the wall (figure 31c).

Figure 33. Results from fDNS and LES of a turbulent channel flow at $Re_\tau =180$ with a grid resolution of $48\times 48\times 48$ (LES180). (a) The mean streamwise velocity; (b) r.m.s. velocity fluctuations; (c) the mean Reynolds shear stress $\langle \bar {u}'\bar {v}'\rangle$, where $\langle {\cdot } \rangle$ denotes averaging over the $x$–$z$ plane and time. The three sets of curves in (b) represent $\bar {u}_{rms}/u_{\tau }$, $\bar {w}_{rms}/u_{\tau }$ and $\bar {v}_{rms}/u_{\tau }$ from top to bottom, respectively. Here $\bigcirc$, fDNS; $+$, DSM; $\nabla$, no-SGS; – – (thick yellow orange dashed line), SL-106H-V; – - – (thick yellow orange dashed-dot line), S-106H-V; – – (thick blue dashed line), SL-106H-P; – – - (thick black dashed line), the law of the wall $\langle \bar {u}\rangle /u_{\tau } = 0.41^{-1}\log y^++5.2$.

In summary, since the strain-rate tensor that is almost aligned with the SGS stress in S-106H-V is poorly correlated to the true SGS stress and has an inaccurate near-wall behaviour, proper normalisation of variables is not enough for accurate prediction of a turbulent channel flow. Improved results by SL-106H-V and SL-106H-P are explained as they predict the SGS stress using $L_{ij}$ that has high similarity to the true SGS stress in terms of the near-wall behaviour and anisotropy. Therefore, the most important factor for generalization to an untrained turbulent channel flow is considered to be the use of $L_{ij}$.