3.1. Prediction of surface turbulent heat flux

In this section we present the results of the cGAN with a decomposition algorithm for predicting the surface heat flux $q_w$ from the streamwise and spanwise wall-shear stresses $\tau _{w,x}$ and $\tau _{w,z}$ in a turbulent channel flow. Before training our model, we investigated the fundamental behaviour of the surface heat flux and its relationship to the wall-shear stresses for the range of $Pr$ considered in this study using DNS data. Basic statistics such as the Nusselt number ($Nu=2\delta h/k=2\langle q_w \rangle$; where $h$ and $k$ are the heat transfer coefficient and thermal conductivity of fluid, respectively, and $\langle \rangle$ denotes an average operation), and r.m.s. of fluctuations $q_{w,rms}$ are presented in figure 2. The Nusselt number shows two distinct limiting behaviours, as shown in figure 2(a): it increases monotonically with $Pr$ for $Pr \geq 0.1$, while it converges to $1$ with decreasing $Pr$, indicating that the temperature field approaches a linear profile, which is a signature of pure conduction heat transfer across the channel. As $Pr$ decreases, the r.m.s. value of the surface heat flux decreases, as shown in figure 2(b), as $q_{w,rms}/\langle q_w \rangle \simeq 10.67 Pr$. The distribution becomes Gaussian, as shown in figure 24 in Appendix C. However, the r.m.s. value remained at 40 % of the mean value as $Pr$ increased.

Figure 2. Statistics obtained from DNS data. (a) Relation between Prandtl numbers and Nusselt numbers. (b) Root mean square of surface heat flux with $Pr$.

The correlation between the surface heat flux and wall-shear stress is presented in figure 3 in terms of the correlation coefficient $R(\equiv \langle \tau '_w q'_w\rangle / (\sigma (\tau _w) \sigma (q_w)) )$ and the scatter plot. Here, $\tau _{w}$ = $\sqrt {\tau _{w,x}^2+\tau _{w,z}^2}$, and superscript $'$ and $\sigma$ denote the fluctuation and standard deviation, respectively. The correlation coefficient shows a peak greater than 0.9 at $Pr =1$, and a strong correlation is seen in the scatter plot. These clearly support the Reynolds analogy. For the range $0.1 \leq Pr \leq 10$, a certain level of correlation is observable, whereas for $Pr < 0.1$, the two quantities are hardly correlated, although the correlation coefficient approaches the limiting value of 0.2, as $Pr$ approaches zero. These observations indicate that as $Pr$ approaches 0, the surface heat flux is mostly determined by the conduction process, and convection due to turbulence has little effect. Therefore, we focus on the range of $Pr=0.1\unicode{x2013}7$ and present the results for this range in the main text. The training and prediction results for $Pr=0.001\unicode{x2013}0.05$ are presented in Appendix C.

Figure 3. Relation between wall-shear stresses and surface heat flux for the Prandtl number obtained from DNS data. (a) Correlation coefficient. (b) Scatter plots.

The network was trained for $Pr = 0.2, 0.71, 2$ and $5$, and the trained network was tested for $Pr = 0.1, 0.2, 0.4, 0.71, 1, 2, 3, 5$ and $7$. As shown in figure 2, for the range of $Pr$ considered here, the mean surface heat flux exhibits two slightly different scaling behaviours in $Pr$ depending on whether $Pr$ is less than or greater than one, whereas the r.m.s. value shows almost the same behaviour as the mean value. When the surface heat flux fields for various $Pr$ were used together as the output in training, the training sometimes became unstable due to the different ranges of output fields. To alleviate this problem, the surface heat flux fields were normalized using empirical scaling between $Nu$ and $Pr$. Ignoring the difference between the two scaling relations in figure 2(a), we used the heat flux fields normalized by $Pr^{1/2}$, the empirical correlation indicating that the Nusselt number is a function of the Prandtl number, as the output. The wall-shear stresses and input data were normalized to have $\text {mean}=0$ and $\text {std}=1$. The training and validation data were 1000 and 100 in number, respectively, with $\Delta t^+=9$, which is an interval of data fields, for trained $Pr$; and the number of testing data was 1000 with $\Delta t^+=9$ for all $Pr$. The superscript ($+$) indicates that these parameters were normalized by $u_\tau$ and $\nu$ and made dimensionless. The testing data were sufficiently decorrelated from the training data. In the training process, a randomly sampled subregion ($64 \times 64$) in the $x$-$z$ plane was used for the input and output data. The subregion is of a sufficiently large size, over which the correlation decays to almost zero in two-dimensional two-point correlation between the wall-shear stresses and surface heat flux for all $Pr$. Furthermore, we double checked through an analysis of the trained model that the input information in a much smaller region than the subregion is mainly used for prediction. Before presenting the prediction results, we want to emphasize that our model does not overfit the training data based on the comparison of the training and validation errors of cGAN, presented in Appendix B.

To provide an idea of the structures in the surface heat flux for various Prandtl numbers, the surface heat flux fields for $Pr$ predicted for the same wall-shear stresses using cGAN in a domain smaller than the full domain are shown in figure 4. The wall-shear stresses and predicted $q_w$ values are presented in figures 4(a) and 4(b), respectively. The surface heat flux field for $Pr=0.71$ was very similar to the streamwise wall-shear stress, and a high local heat flux was observed at the location where the wall-shear stress was strong. When $Pr=0.1$, the surface heat flux distribution was smoother in the spanwise direction and wavier in the streamwise direction than that for $Pr=0.71$, whereas the thermal structures for $Pr=2$ were elongated in the streamwise direction and sharper in the spanwise direction than in the case of $Pr=0.71$. This trend strengthens as $Pr$ increases. Small-scale structures appeared in the predicted fields for $Pr=7$. In other words, the thermal structures tend to become streakier as $Pr$ increases.

Figure 4. Surface heat flux fields for various $Pr$ obtained from same input data using cGAN. (a) Streamwise and spanwise wall-shear stress used as input data. (b) Surface heat flux with $Pr$.

For the qualitative evaluation of the performance of the developed network, instantaneous surface heat flux fields in the whole domain predicted by cGAN compared with DNS data for trained Prandtl and untrained Prandtl numbers are shown in figures 5 and 6, respectively. As shown in figure 5, cGAN slightly underpredicted the local maximum values of $q_w$ for $Pr=0.2$ observed in DNS, whereas the predicted surface heat flux shows an overall similar distribution to that of DNS by capturing small-scale variations. In contrast, our model can generate streaky structures observed in $q_w$ for $Pr=0.71$ and $5$. As shown in figure 6 for the untrained Prandtl numbers, our model slightly overpredicts the surface heat flux field compared with DNS for $Pr=0.1$. The predicted surface heat flux fields for $Pr=1$ and $7$ were consistent with those of DNS. Therefore, we confirmed that our model could generate surface heat fluxes similar to those of DNS for untrained $Pr$ and trained $Pr$.

Figure 5. Instantaneous surface heat flux for trained $Pr$ obtained from wall-shear stresses using cGAN.

Figure 6. Instantaneous surface heat flux for untrained $Pr$ obtained from wall-shear stresses using cGAN.

To quantitatively assess the prediction accuracy of cGAN, we additionally considered CNN, ShallowCNN and multiple-linear models as comparative models. In table 2 we provide the correlation coefficient $R$ between surface heat flux of DNS data and that predicted by DL models, where $R ={\langle q^{'DNS}_{w} q^{'DL}_{w}\rangle / ({\sigma (q^{DNS}_{w})}{\sigma (q^{DL}_{w})}})$, demonstrating that CNN has a higher correlation coefficient for all untrained $Pr$ than those obtained from the linear model and cGAN. The correlation coefficient $R$ for cGAN is slightly lower than that for CNN, whereas ShallowCNN and the linear model have a relatively low correlation coefficient, particularly for $Pr = 0.2$ and 5. ShallowCNN predicts heat flux better than linear models, but not as accurate as CNN with more layers. This indicates that there is a complex nonlinear relation between the local heat flux and the wall-shear stresses, suggesting that a sufficiently large number of layers should be used to develop an integrated model for $Pr$. For the untrained $Pr$, the performance of all models is similar to that of the trained $Pr$. Commonly, the performance of all models is best for $Pr$ around 1, and the performance deteriorates as $Pr$ increases or decreases because the temperature is not dominantly determined by the near-wall transport and the dissimilarity becomes very strong. Furthermore, the slight inaccuracy might be caused by an improper normalization technique, which is needed for learning data of highly different scales. However, the prediction by CNN and cGAN is good for the tested range. Because CNN learns to minimize the pointwise error between DNS data and generated data, the correlation coefficient of CNN is naturally higher than that of cGAN. One might think that CNN is better than the GAN-based model by comparing the point-by-point error, but CNN is somewhat inaccurate in the prediction of statistics, as shown below. To improve the prediction performance of the model, we considered a cGAN model using wall pressure fluctuations and wall-shear stresses as input information. The predictive performance of the model was improved when additional pressure fluctuations were used, but only marginally, as shown in table 4 in Appendix D, where the performance of the model with pressure data as input is provided in detail. The pressure information was found to be auxiliary in the prediction of the surface heat flux. Consequently, only the wall-shear stresses were considered as input information.

Table 2. Correlation coefficient between target data (DNS data) and surface heat flux for trained and untrained $Pr$ predicted by various learning models.

To investigate the performance of the models in more detail, we present the basic statistics of the surface heat flux, such as the Nusselt number, r.m.s., skewness and flatness in figure 7. Because the heat flux is normalized by $Pr^{1/2}$ based on our observation, which is not accurate, the predicted mean such as the Nusselt number needs to be checked. As shown in figures 7(a) and 7(b), the Nusselt number and r.m.s. predicted by cGAN and CNN are very accurate compared with those by DNS for both trained and untrained $Pr$. On the other hand, the multiple-linear model predicts the Nusselt number relatively well because of the use of empirical correlation but underpredicts the r.m.s. values. As shown in figures 7(c) and 7(d), cGAN produces the smallest errors for all $Pr$, whereas the CNN and multiple-linear models underpredict the skewness and flatness factors for all $Pr$. These results confirm that the multiple-linear model is not suitable for prediction in the considered range of $Pr$.

Figure 7. Statistics of surface heat flux for trained $Pr$ (0.2, 0.71, 2, 5) and untrained $Pr$ (0.1, 0.4, 1, 3, 7) obtained using DL models; (a) $Nu$, (b) r.m.s., (c) skewness, (d) flatness.

Figure 8 compares the probability distribution function (PDF) of the surface heat flux predicted for the trained and untrained $Pr$ with the DNS data. In this investigation only the results of the cGAN and CNN are presented, except for the multiple-linear model, which showed the lowest accuracy of the previous statistics. As shown in figure 8(a), both cGAN and CNN produce a PDF for the trained $Pr$ that is similar to that of DNS, but cGAN outperforms CNN in capturing high values of the surface heat flux for the trained $Pr$. For the untrained $Pr$, the prediction performance of the two models is comparable to that of the trained $Pr$, as shown in figure 8(b); however, cGAN outperforms CNN. Given that this type of statistical information is not used in training, our model captures the asymmetric statistical nature remarkably well. As $Pr$ decreases, the PDF tends to recover symmetry, gradually becoming a Gaussian distribution, as shown in figure 24 in Appendix C.

Figure 8. Probability density function (p.d.f.) of surface heat flux for (a) trained $Pr$ ($=0.2, 0.71, 2, 5$) and (b) untrained $Pr$ ($=0.1,0.4,1,3,7$) obtained through DL models. Arrows indicate increasing $Pr$.

Additionally, we examined the energy spectrum of the surface heat flux for the reproducibility of scale behaviour. The streamwise and spanwise energy spectra of $q_w$ are defined as

(3.1a,b)\begin{equation} E(\kappa_x) = \frac{1}{{2{\rm \pi}}}\int_{-\infty}^{\infty}\,{{\rm e}^{-{\rm i}p\kappa_x}\phi(p)\, {\rm d} p},\quad E(\kappa_z) = \frac{1}{2{\rm \pi}}\int_{-\infty}^{\infty}\,{\rm e}^{-{\rm i}q\kappa_z}\psi(q)\, {\rm d} q, \end{equation}

with

(3.2a,b)\begin{equation} \phi(p) = \langle q'_w(x,z)q'_w(x+p,z)\rangle ,\quad \psi(q) = \langle q'_w(x,z)q'_w(x,z+q)\rangle ,\end{equation}

where $\phi (p)$ and $\psi (q)$ are the two-point correlations of the surface heat flux in the $x$ and $z$ directions, respectively. As shown in figures 9(a) and 9(b) for the trained $Pr$, cGAN produces a spectrum that matches well with that obtained from DNS for all ranges of both wavenumbers, whereas CNN underestimates the streamwise spectrum for all wavenumbers and the spanwise spectrum for low wavenumbers. For the untrained $Pr$, the performance of cGAN does not deteriorate in the prediction of the streamwise spectrum, except for $Pr=0.1$, whereas CNN tends to underestimate the spectrum except for $Pr=0.1$, which appears to be coincidental, as shown in figure 9(c). However, the prediction of the spanwise spectrum by both cGAN and CNN worsens, especially for high wavenumbers, as shown in figure 9(d). It is noteworthy that both cGAN and CNN do not perform well for $Pr=0.1$, because as $Pr$ decreases, the surface heat flux becomes less dependent on the wall-shear stresses, as discussed in Appendix C. Overall, cGAN shows better performance in capturing statistical characteristics than CNN because statistical consistency is considered in the training process of the cGAN through the discriminator network, in addition to the local loss based on pointwise errors.

Figure 9. One-dimensional energy spectra of surface heat flux for various $Pr$ obtained from wall-shear stresses through DL models. Arrows indicate increasing $Pr$. (a) Streamwise and (b) spanwise energy spectrum of surface heat flux with trained $Pr(=0.2, 0.71, 2, 5)$; (c) streamwise and (d) spanwise energy spectrum of surface heat flux with untrained $Pr(=0.1,0.4,1,3,7)$.

In order to provide a reliable interpretation of the trained model, which will be discussed in §§ 3.2 and 3.3, we investigated whether the model well reflects the spatial correlation between the wall-shear stresses and the surface heat flux. The two-dimensional two-point correlation between the surface heat flux at a point $(x,z)$ and the wall-shear stresses at a point $(x+r_x,z+r_z)$ is defined by

(3.3)\begin{equation} R_{V_{1} V_{2}}(r_{x},r_{z})= \frac{\langle V'_{1}(x,z)V'_{2}(x+r_{x},z+r_{z})\rangle}{\sigma(V_{1}) \sigma(V_{2})}, \end{equation}

where $V_{1}$ and $V_{2}$ represent $\tilde {q}_{w}$ and $\tilde {\tau }_{w}$, respectively. The two-point correlation between the wall-shear stresses and surface heat flux predicted through cGAN and CNN for the trained $Pr$ is presented in figure 10. In $R_{q_{w}\tau _{w,x}}(r_x,0)$, both cGAN and CNN well reflect the spatial shifting phenomenon of the correlation peak location depending on the Prandtl number, which is observed in DNS (figure 10a). However, for $Pr = 0.2$ and $5$, CNN highly overestimates the maximum values of DNS, whereas cGAN reproduces the correlation more accurately than CNN. These results indicate that cGAN, unlike CNN, generates a more input-dependent output by using input information as conditions in the learning process. Figure 10(b) shows $R_{q_{w}\tau _{w,z}}(r_{x,max}, r_z)$, where the streamwise location $r_{x,max}$ is the maximum location of $R_{q_{w}\tau _{w,z}}$ for each $Pr$. The cGAN and CNN follow DNS well overall, while cGAN presents slightly better correlation for the lowest $Pr$. In addition, the models reflected well the symmetric properties and the decorrelated tendency with an increase in $Pr$. Although the models present a relatively inaccurate two-point correlation for untrained $Pr$ (not shown here), it is obvious that cGAN predicts the spatial correlation between input and target better than CNN. Therefore, it seems reasonable to focus on the interpretation of cGAN rather than CNN.

Figure 10. Two-point correlations (a) $R_{q_{w}\tau _{w,x}}(r_x,0)$ along the streamwise direction and (b) $R_{q_{w}\tau _{w,z}}(r_{x,max}, r_z)$ along the streamwise direction for trained $Pr(=0.2,0.71,2,5)$.

From all the tests in this section, we confirm that cGAN is able to predict the surface heat flux for various $Pr$ values using only the wall-shear stress information. In addition to the prediction of pointwise distribution of the surface heat flux, cGAN captures statistical features very well compared with DNS for the trained $Pr$. Meanwhile, for the untrained $Pr$, our model showed the highest accuracy among other models, including CNN and multiple-linear models, exhibiting an accuracy comparable to that of the trained $Pr$. These results indicate that our model can distinguish the effect of the Prandtl number well and can express the relationship between the wall-shear stresses and the surface heat flux depending on $Pr$. The performance of CNN and cGAN is not extremely different in this application, but in a situation where the input information is insufficient, CNN can generate an output with non-physical characteristics and highly underestimate the target magnitude. On the other hand, a GAN-based network can generate an output that reflects physical or statistical properties of turbulence, although the pointwise error is slightly higher than CNN (Ledig et al. Reference Ledig2016; Deng et al. Reference Deng, He, Liu and Kim2019; Lee & You Reference Lee and You2019; Kim et al. Reference Kim, Kim, Won and Lee2021). It is highly probable that a GAN-based network tries to find a solution in the space that satisfies physical and statistical properties of turbulence data. Therefore, GAN is considered as a promising tool for the generation and modelling as well as prediction of turbulence. In addition, although we tested the Prandtl number effect only, we expect that our model could work well for a higher Reynolds number from previous studies (Kim & Lee Reference Kim and Lee2020b; Kim et al. Reference Kim, Kim, Won and Lee2021), where the DL model showed successful predictions for a higher Reynolds number than the trained number under the condition that grid resolution of input is the same as that of the trained one and input data are normalized by proper length and velocity scales (e.g. the viscous length scale and friction velocity in wall turbulence).

3.2. Interpretation of DL model

By observing the gradient maps obtained from the trained DL model, we attempt to analyse the physical relationship between the input and output of our prediction network. For the analysis, a cGAN was used, which is the most accurate model as discussed in § 3.1. We only investigated the surface heat flux for the trained $Pr$. Because the mean surface heat flux is very different for different Prandtl numbers, it is difficult to accurately analyse their effect. For the reliability of the analysis, cGAN was trained using data preprocessed by different methods. The surface heat flux for each $Pr$ was normalized so that its $\text {mean}=1$, and the normalized value is denoted by $\tilde {q}_{w}$. The wall-shear stresses were normalized to have $\text {mean}=0$ and $\text {std}=1$, and the normalized stress is denoted by $\tilde {\tau }_{w}$. Furthermore, we performed data augmentation by applying spanwise symmetry and a phase shift, considering the characteristics of the channel flow and boundary conditions.

Prior to interpreting the physical relationship between the surface heat flux and wall-shear stresses based on the gradient maps, we observed two-dimensional two-point correlation $R_{\tilde {q}_{w}\tilde {\tau }_{w}}$ for $Pr=0.2, 0.71$ and $5$ obtained from DNS data in figure 11, which is calculated by (3.3). As shown in figure 11(a), the peak correlation of the streamwise shear stress for $Pr=0.2$ and $0.71$ is observed in the upstream region at approximately $r_x^+=-60$, $r_z^+ \simeq 0$ and $r_x^+=-10, r_z^+\simeq 0$, respectively, whereas that for $Pr=5$ is found approximately at $r_x^+=80, r_z^+\simeq 0$. This shift in the location showing peak correlation can be understood using the convection velocity of temperature near the wall; the convection velocity of temperature for $Pr < 1$ is higher than that of the streamwise velocity and vice versa for $Pr >1$ (Kowalewski, Mosyak & Hetsroni Reference Kowalewski, Mosyak and Hetsroni2003; Abe et al. Reference Abe, Kawamura and Matsuo2004). Depending on $Pr$, the dominance of thermal diffusion and momentum diffusion enhances the convection velocity. Similar to the effect of the Prandtl number, it has been reported that the local concentration field response to velocity fluctuations in turbulent mass transfer has a significant time lag for high Schmidt numbers (Shaw & Hanratty Reference Shaw and Hanratty1977; Hasegawa & Kasagi Reference Hasegawa and Kasagi2009), explaining the spatial shifting phenomenon of surface heat flux well. On the other hand, for all $Pr$, the correlation between the surface heat flux and the spanwise shear stress shown in figure 11(b) exhibits positive and negative correlations in the first quadrant ($r_x^+>0$, $r_z^+>0$) and the fourth quadrant ($r_x^+>0$,$r_z^+<0$), respectively. As $Pr$ increased, the correlation peak gradually moved in the downstream direction. This is consistent with a previous observation that the maximum correlation between the spanwise shear stress and streamwise vortices near the wall occurs downstream (Kravchenko, Choi & Moin Reference Kravchenko, Choi and Moin1993). In addition, as $Pr$ increases, the correlation weakens, and its distribution is stretched in the streamwise direction.

Figure 11. Two-dimensional two-point correlation of (a) streamwise wall-shear stress ($\tilde {\tau }_{w,x}$) and (b) spanwise wall-shear stress ($\tilde {\tau }_{w,z}$) with respect to surface heat flux ($\tilde {q}_{w}$) for $Pr$ obtained from DNS data.

A two-point correlation between two variables indicates that there exists a relationship between them, but it does not reveal any information about the cause-and-effect relationship between them. To investigate the cause-and-effect relationship between the wall-shear stresses and surface heat flux, we performed a sensitivity analysis between the input and output of the trained DL model. Sensitivity analysis has been applied to image classification and regression problems, which enables the analysis of the relationship between the input and output (Simonyan, Vedaldi & Zisserman Reference Simonyan, Vedaldi and Zisserman2013; Kim & Lee Reference Kim and Lee2020b). From a sensitivity analysis, we can determine the reasons for the inaccuracy of the DL model, which can help improve its performance. In addition, in the case of a regression problem, sensitivity analysis can provide a guide on the input size, by taking into account the essential components required for efficient training of the DL. Sensitivity analysis was performed using the gradient of the output with respect to the input as

(3.4)\begin{equation} S^k (i,j) = \frac{\partial N(I) }{\partial I^k (i,j)},\end{equation}

where $S^k(i,j)$ is the gradient map of the input variables ($k$), such as the wall-shear stresses at $(x+r_x, z+r_z)$, with $r_x=i\Delta x$ and $r_z=j\Delta z$. Here $I^k(i,j)$ is an input variable at $(x+r_x, z+r_z)$ and $N(I)$ is an output variable, such as the surface heat flux, obtained from the output of DL at $(x,z)$; $i = -(m_{x}-1)/2 \sim (m_{x}-1)/2$ and $j = -(m_{z}-1)/2 \sim (m_{z}-1)/2$, where $m_x \times m_z$ is the input kernel size. The gradient map indicates the sensitivity of the change in the surface heat flux at a point with respect to the change in the wall-shear stress around that point. Due to the nonlinearity of the trained model, the gradient map typically varies with the spatial location of the predicted heat flux, unlike a linear model that applies the same operation regardless of spatial location (Kim & Lee Reference Kim and Lee2020b). Therefore, the significant difference between a DL model and a linear model could be understood by the input-dependent nature observed in the gradient map.

Through an investigation of the gradient map, we were able to analyse how the wall-shear stresses affected the surface heat flux. However, the recognition of meaningful patterns in the gradient map is not straightforward because of the noise generated in the complicated learning process of the non-unique relationship between the input and output. Therefore, we first investigated an average gradient map $\bar {S}^{k}$ (where $\bar {{\cdot }}$ is the average operation in space and time), which is an average of the instantaneous gradient maps of input variables $(k)$ obtained at various locations of output in space over a time span.

The average gradient maps $\bar {S}^{\tilde {\tau }_{w,x}}$ and $\bar {S}^{\tilde {\tau }_{w,z}}$ for the local heat flux $\tilde {q}_w$ with respect to the streamwise wall-shear stress $\tilde {\tau }_{w,x}$ and spanwise wall-shear stress $\tilde {\tau }_{w,z}$ for various $Pr$ are presented in figures 12(a) and 12(b), respectively. As shown in figure 12(a), non-trivial values of $\bar {S}^{\tilde {\tau }_{w,x}}$ are observed in a local region of the input fields for all $Pr$, indicating that DL primarily uses the local information of the input field $\tilde {\tau }_{w,x}$ to predict the heat flux at a point. The magnitude of the gradient is small for $Pr=0.2$ compared with the cases for $Pr=0.71$ and $5$, implying that the surface heat flux at $Pr=0.2$ is relatively less sensitive to the change in $\tilde {\tau }_{w,x}$ than that at $Pr=0.71$ and $5$. An important characteristic observed in $\bar {S}^{\tilde {\tau }_{w,x}}$ is that a positive peak is found in the upstream region for $Pr < 1$, whereas in the downstream region the peak is found for $Pr > 1$. The peaks for $Pr=0.2, 0.71$ and $5$ occur at $r_x^+=-20, 0$ and $30$, respectively. Although the specific peak locations are different, this behaviour is similar to that of the two-point correlation shown in figure 11(a). The shift of a peak observed in the gradient maps presents an important observation that the DL model predicts local heat flux by reflecting the physical features affected by large-scale motion as the Prandtl number decreases. Also, the fact that the region showing high two-point correlation for all $Pr$ is much longer in the streamwise direction than the region with a non-trivial gradient map indicates that the average gradient map pinpoints the region of input that influences the output on average.

Figure 12. Average gradient maps of surface heat flux ($\tilde {q}_{w}$) with respect to (a) streamwise wall-shear stress ($\tilde {\tau }_{w,x}$) and (b) spanwise wall-shear stress ($\tilde {\tau }_{w,z}$) for $Pr$ obtained through cGAN. The average gradient maps were obtained using a sufficiently large amount of instantaneous gradient maps, where the number of data are $81\ 920$ (all points in five fields with $N_x=N_z=128$ and $\Delta t^+=9$) for each $Pr$.

In figure 12(b) the average gradient map $\bar {S}^{\tilde {\tau }_{w,z}}$ shows a skew-symmetric pattern in the spanwise direction $\bar {S}^{\tilde {\tau }_{w,z}}(i,j)=-\bar {S}^{\tilde {\tau }_{w,z}}(i,-j)$ for all $Pr$. For $Pr=0.2$, the gradient map was pronounced in the first and fourth quadrants, and a high sensitivity for $Pr=0.71$ was observed in all quadrants. The gradient map for $Pr=0.71$ indicates that the local heat flux can be enhanced when a pair of vortices is located downstream, or when a vortex crosses diagonally at the centre point, even when the streamwise wall-shear stress is weak (Kim & Lee Reference Kim and Lee2020b). However, the gradient for $Pr=5$ is relatively weak everywhere, implying that the surface heat flux at $Pr = 5$ is hardly influenced by the spanwise wall-shear stress compared with other Prandtl numbers. These patterns in the gradient maps, which were not observed in the two-point correlation in figure 11(b), indicate that the model can capture intermittent physical phenomena through a combination of multi-point streamwise and spanwise wall-shear stresses. We also observed the average gradient map of the model predicting the surface heat flux from the wall-shear stresses and pressure fluctuations; the results are presented in detail in Appendix D.

Next, we investigate the instantaneous gradient map for specific flow situations to discuss the cause-and-effect relationship between turbulent variables depending on the Prandtl number in detail. As described above, noise removal in an instantaneous gradient map is essential because noise makes it difficult to recognize and analyse meaningful patterns. Smilkov et al. (Reference Smilkov, Thorat, Kim, Viégas and Wattenberg2017) used Gaussian noise as a method to remove noise in the salient map for classification problems. However, in our problem this method did not show a remarkable effect in removing noise, and thus, we took advantage of symmetry properties such as reflectional equivariance and invariance with respect to the phase shift in the channel geometry. The reflectional equivariance property can be implemented using mirror data in the spanwise direction. Therefore, a gradient map was obtained by averaging the original gradient map and the mirrored gradient map of the mirrored data. Additionally, phase shift invariance can be implemented by averaging the instantaneous gradient maps of several data points with a phase shift of less than $0.2\Delta x$, under the assumption that the turbulence structure does not significantly change over a small distance. To illustrate the effect of noise removal, we present instantaneous gradient maps ($S^{\tilde {\tau }_{w,z}}$) before and after noise removal in figure 13, which clearly demonstrate the effect of noise removal. Compared with the original gradient map shown in figure 13(a), the noise-removed gradient map in figure 13(b) exhibits a more pronounced pattern, similar to the average gradient map. The gradient in the region far from the origin is very small. The instantaneous gradient maps presented in this study were obtained by applying the noise-removal method.

Figure 13. Effect of the noise-removal method for the instantaneous gradient map ($S^{\tilde {\tau }_{w,z}}$) at $Pr=0.71$ obtained through cGAN. (a) Original gradient map and (b) gradient map applying both reflectional equivariance and phase shift method.

To investigate the detailed cause-and-effect mechanism based on the analysis of the average gradient map, we chose two specific examples of the flow field: one showing locally intense streamwise shear stress and the other exhibiting intense spanwise shear stress. Figure 14(a) illustrates an example with intense streamwise shear stress located at the origin. The nearby vortices are visualized using the $\lambda _2$ method (Jeong & Hussain Reference Jeong and Hussain1995). Figure 14(b) shows instantaneous gradient maps, relative to the peak location of heat flux marked by the black dot in figure 14(c), which are similar to the average gradient maps shown in figure 12. This instantaneous gradient map suggests that the peak heat flux is caused by different parts of the nearby streamwise shear stress distribution depending on $Pr$. The peak heat flux is observed at a slightly downstream location of the peak streamwise shear stress for $Pr=0.2$, while the peak heat flux is found at a slightly upstream location of the peak stress for $Pr=5$. Because this slight shift of peaks clearly indicates that the gradient of input shear stress may play a key role in determining heat flux, we investigated the spatial gradient of input to observe the Prandtl number effect, which is also connected to sweep and ejection events by the Taylor expansion of the near-wall wall-normal velocity. The streamwise gradient of the streamwise shear stress $\partial {\tilde {\tau }_{w,x}}/\partial {x}$ shown in coloured contours in figure 14(c) clearly confirms that the negative peak of $\partial {\tilde {\tau }_{w,x}}/\partial {x}$ is observed at almost the same location as the peak heat flux for $Pr=0.2$, whereas the positive peak $\partial {\tilde {\tau }_{w,x}}/\partial {x}$ coincides with the peak heat flux location for $Pr=5$. When $Pr$ is approximately 1, the heat flux and shear stress exhibit peaks at the same location. The relatively weak sensitivity shown in the gradient map for $Pr=0.2$ in figure 14(b) is due to relatively weak heat flux in this particular situation.

Figure 14. Representative example of the region where dissimilarity between $\tilde {\tau }_{w,x}$ and $\tilde {q}_{w}$ is weak and $\tilde {q}_{w}$ is high for $Pr=0.71$. (a) The right panel is the top view of the left panel. The colours and lines represent $\tilde {\tau }_{w,x}$ and $\tilde {\tau }_{w,z}$, respectively, and $\lambda _{2}^+=-0.02$. (b) The instantaneous gradient map of $\tilde {q}_{w}$ with respect to $\tilde {\tau }_{w,x}$ for $Pr$ obtained through cGAN. (c) Instantaneous contours of $\partial {\tilde {\tau }_{w,x}}/\partial {x}$ (colour) and $\tilde {q}_{w}$ for $Pr$ obtained using cGAN (lines are from 2.0 to 4.0 with increments of 0.2).

We now investigate the second example showing a strong spanwise shear stress around the peak heat flux, as demonstrated in figure 15(a). The strongest local heat flux is observed around the region with a strong positive spanwise shear stress for $Pr=0.2$. Figure 15(b) shows the instantaneous gradient map relative to the peak heat flux location marked by the black dot in figure 15(c) for $Pr=0.2$ and 0.71. For $Pr=5$, because there is no strong heat flux observed around the black dot, the same output location as that of $Pr=0.71$ was selected in the calculation of the gradient map. The instantaneous gradient map is similar to the average gradient map of figure 12, except that the peak streamwise location of the instantaneous gradient map is almost the same as that of the heat flux, whereas the peak of the average gradient map is located slightly downstream of the heat flux. This suggests that for $Pr=0.2$, the spanwise variation of the spanwise shear stress might cause a strong local heat flux. It naturally leads to the investigation of the spanwise gradient of the spanwise shear stress (coloured contours) with the corresponding heat flux distributions (line contours) in figure 15(c). The positive spanwise gradient of the spanwise shear stress $\partial {\tilde {\tau }_{w,z}}/\partial {z}$ clearly confirms a strong correlation with the peak heat flux. On the other hand, when $Pr=5$, the strong spanwise shear stress does not contribute to the local heat flux. However, when $Pr=0.71$, the instantaneous gradient map exhibits a skew-symmetric pattern in both the streamwise and spanwise directions: $S^{\tilde {\tau }_{w,z}}(x+r_x,z+r_z)=-S^{\tilde {\tau }_{w,z}}(x+r_x,z-r_z) =-S^{\tilde {\tau }_{w,z}}(x-r_x,z+r_z)$. The only situation allowing the local heat flux at $(x,z)$ to be caused by the nearby spanwise shear stress through such a skew-symmetric sensitivity map is that a streamwise vortex is slightly slanted from the streamwise direction, as shown in figure 15(a). In the two-point correlation (figure 11) it is difficult to identify which part is necessary for prediction because correlation exists in a wide region, indicating the limitation of the conventional statistical analysis. On the other hand, it should be noted that the gradient map for the high-performance DL model clearly reveals which part is actually essential for the prediction and which operation acts on the input.

Figure 15. Representative example of the region where dissimilarity between $\tilde {\tau }_{w,x}$ and $\tilde {q}_{w}$ occurs. (a) right panel is top view of left panel. The colours and lines represent $\tilde {\tau }_{w,x}$ and $\tilde {\tau }_{w,z}$, respectively, and $\lambda _{2}^+=-0.03$. (b) The instantaneous gradient map of $\tilde {q}_{w}$ with respect to $\tilde {\tau }_{w,z}$ for $Pr$ obtained through cGAN. (c) Instantaneous contours of $\partial {\tilde {\tau }_{w,z}}\partial {z}$ (colour) and $\tilde {q}_{w}$ for $Pr$ obtained using cGAN (lines are from 1.5 to 2.5 with increments of 0.1).

We discovered a strong correlation between the heat flux and gradient of the wall-shear stresses depending on $Pr$ after a detailed investigation of the gradient map and its relation to the gradient of the wall-shear stresses for two example fields demonstrated in figures 14 and 15. For $Pr=0.2$, a strong heat flux occurs in the region with a negative streamwise gradient of the streamwise wall-shear stress or a positive spanwise gradient of the spanwise wall-shear stress. However, for $Pr=5$, the region of high heat flux is highly correlated with the region with a positive gradient of the streamwise wall-shear stress. When $Pr = 0.71$, a strong heat flux was found in the region with a strong streamwise wall-shear stress. The scatter plots between the normalized heat flux and the normalized gradient of the wall-shear stresses obtained from DNS for $Pr=0.2, 1$ and $5$, shown in figure 16, confirm this behaviour. The locally high heat flux ($\tilde {q}'_w \geq \tilde {q}_{w,rms}$), marked by the black dot in figure 16(a), clearly indicates negative and positive correlations with the streamwise gradient of the streamwise wall-shear stress for $Pr=0.2$ and 5, respectively, while showing almost no correlation for $Pr=1$. A locally high heat flux, on the other hand, shows an overall positive correlation with the spanwise gradient of the spanwise wall-shear stress for $Pr \leq 1$, as shown in figure 16(b); however, the smaller the value of $Pr$, the stronger is the positive correlation. The correlation coefficients provided in figure 16 quantitatively support this behaviour, but less conspicuously, because all data points, including the data with negative fluctuations of heat flux, which do not have any correlation with the gradient of wall-shear stresses, were considered in the calculation of the correlation coefficient.

Figure 16. Scatterplots between surface heat flux with $Pr$ and gradient of (a) streamwise wall-shear stress $\partial {\tilde {\tau }_{w,x}}/\partial {x}$ and (b) spanwise wall-shear stress ${\partial {\tilde {\tau }_{w,z}}}/\partial {z}$. High heat flux data ($\tilde {q}'_w > \tilde {q}_{w,rms}$) are marked with darker points to highlight the correlation.

Recognizing that the gradient of the wall-shear stresses is associated with the near-wall flow structures, we propose plausible pictures by which the region with locally high heat flux is estimated from the flow behaviour, as presented in figure 17. A sweep and ejection event commonly observed in near-wall turbulence causes streamwise wall-shear stress distribution, as shown in figure 17(a). From our discussion of the correlation between the heat flux and streamwise wall-shear stress above, we can pinpoint the region of the local hotspot where high heat flux occurs, depending on $Pr$, as shown in figure 17(a). This indicates that sweeping flow causes a high heat flux for $Pr >1$, whereas ejection flow induces a high heat flux for $Pr <1$. When $Pr =1$, high heat flux and strong streamwise shear tend to occur at the same location. A streamwise vortex also causes the spanwise wall-shear stress distribution shown in figure 17(b), from which a local hotspot can be identified from the correlation between the heat flux and the spanwise wall stress for $Pr < 1$. However, when $Pr >1$, a streamwise vortex does not contribute to a high heat flux.

Figure 17. Schematic for physical relationship between near-wall transport and heat transfer with the Prandtl number.

In summary, we demonstrated the interpretability of DL through an investigation of the gradient map between the input wall-shear stresses and output heat flux. Depending on the Prandtl number, the diverse physics of near-wall turbulent heat transfer can be identified.

3.3. Decomposition of Prandtl number effect

For efficient learning, we introduced decomposition of the surface heat flux ($\tilde {q}_{w}=\tilde {q}_{w}^{C}+\tilde {q}_{w}^{P}$) into $Pr$-independent features ($\tilde {q}_{w}^{C}$) and $Pr$-dependent features ($\tilde {q}^{P}$) in the network, as discussed in § 2.2. In this section we investigate the behaviour of the features obtained using the trained model in § 3.2. The $Pr$-independent features contain common characteristics of the surface heat flux irrelevant to $Pr$, whereas the $Pr$-dependent features reflect the characteristics of the surface heat flux dependent on $Pr$. These decomposed features are automatically obtained through the training process by adopting GAN explained in § 2.2 without additional knowledge regarding the criterion of the data. Therefore, DL with a decomposition algorithm is a convenient and effective method for extracting decomposed physical features. A detailed analysis of the noticeable physical properties from the decomposed features aids in understanding the $Pr$ effect in turbulent heat transfer.

First, we investigate statistics such as the variance and correlation coefficient of the decomposed fields of the surface heat flux ($\tilde {q}_{w}^{C}$ and $\tilde {q}_{w}^{P}$), as listed in table 3. By definition, the mean of $\tilde {q}_w$ is 1, whereas the mean of the $Pr$-independent feature $\langle \tilde {q}_{w}^{C}\rangle$ is approximately $1$, and that of $Pr$-dependent features $\langle \tilde {q}_{w}^{P}\rangle$ for all $Pr$ is negligibly small when the training network is constructed. The variance of the $Pr$-dependent feature is relatively small compared with that of the $Pr$-independent feature, which is comparable to the total heat flux, suggesting that the $Pr$-independent feature captures the common behaviour of the heat flux for all $Pr$. The high correlation coefficient between $\tilde {q}_w$ and $\tilde {q}_w^C$ for all $Pr$ values also supports this argument. On the other hand, $\tilde {q}_w^P$ is mildly correlated with $\tilde {q}_w$ but hardly correlated with $\tilde {q}_{w}^{C}$, indicating that the $Pr$-independent feature extracts common characteristics regardless of $Pr$. Furthermore, the high correlation coefficients between $\tilde {q}_w$ and $\tilde {q}_{w}^{C}$, and $\tilde {q}_w$ and $\tilde {q}_w^P$ for $Pr=0.71$ and 2, strongly suggest that the $Pr$-independent feature is most pronounced in the heat flux field for $Pr$ around 1.

Table 3. Variance and correlation coefficient of the $Pr$-independent feature, $Pr$-dependent feature and surface heat flux with trained $Pr$.

Figure 18 shows example snapshots of $\tilde {q}_{w}^{C}$, $\tilde {q}_{w}^{P}$ and $\tilde {q}_w$ for the same input field for various values of $Pr$, with the contours of $\tilde {q}_{w}^{C}$ and $\tilde {q}_w$ displayed for values greater than 1. The strong correlation between $\tilde {q}_{w}^{C}$, $\tilde {q}_{w}^{P}$ and $\tilde {q}_w$, shown in table 3, suggests that the similarity between them is most pronounced for $Pr=0.71$ or 2. The only difference is that $\tilde {q}_w$ has higher peak values than $\tilde {q}_w^C$, as evidenced by the $\tilde {q}_w^P$ distribution, which has a positive correlation with $\tilde {q}_w^C$. When $Pr=0.2$ or 5, however, there are mismatches in the peak locations of $\tilde {q}_{w}^{C}$ and $\tilde {q}_w$ in the opposite sense, with $\tilde {q}_w$ peaks found slightly downstream and upstream of those of $\tilde {q}_w^C$ for $Pr=0.2$ and $Pr=5$, respectively. The opposite behaviour of the $\tilde {q}_w^P$ distributions for $Pr=0.2$ and 5 clearly confirms this. Another difference is that the high heat flux regions for $Pr=5$ are significantly elongated in the streamwise direction compared with those for $Pr=0.2$. This kind of shift in the peak locations is obviously caused by the different physical relationships between the surface heat flux and the flow field depending on $Pr$, as described in figure 17. The decomposition algorithm captured this difference.

Figure 18. Example fields of the $Pr$-independent feature $\tilde {q}_w^C$ (line contours with levels from 1.0 to 3.0 with an increment of 0.2 in the left panels), $Pr$-dependent feature (colour contours in the left panels) and the total heat flux $\tilde {q}_w$ (line contours with levels from 1.0 to 3.0 with an increment of 0.2 in the right panels) for various $Pr$ decomposed through cGAN.

Figure 19 shows the two-point correlations along the streamwise direction $R_{\tilde {q}_{w}\tilde {q}_{w}^C}(r_x)$ and $R_{\tilde {q}_{w}\tilde {q}_{w}^P}(r_x)$ for $Pr=0.2, 0.71, 2$ and 5, for a quantitative investigation of the spatial relationship between the local heat flux, $Pr$-independent feature and $Pr$-dependent feature. The surface heat flux for all $Pr$ has a high correlation with the $Pr$-independent feature, as confirmed by table 3, with peaks shifted at $-50< r_x^+<50$. The correlation peaks for $Pr=5$ and 2 are observed at approximately $r_x^+=35$ and 20, respectively, while those for $Pr=0.71$ and 0.2 are found at $r_x^+=-10$ and $-35$, indicating that the peak for $Pr \approx 1$ is located in the centre, and shifts downward/upward as $Pr$ increases/decreases. On the other hand, the surface heat flux for all $Pr$ shows a mild correlation with the $Pr$-dependent feature, as shown in figure 19(b). As $Pr$ varies, the correlation peaks shift in the opposite direction to that of the correlation between the surface heat flux and the $Pr$-independent feature shown in figure 19(a), but the amount of shift is smaller. Furthermore, the correlation is asymmetric and even becomes negative for large $r_x$ for $Pr > 1$ and for large $-r_x$ for $Pr < 1$. For a clear understanding of these correlations, we provide a schematic describing the plausible streamwise distribution of the typical normalized surface heat flux and the corresponding decomposed features for three ranges of $Pr$ in figure 20. Three distributions of locally high surface heat flux for each $Pr$ regime for the same flow field inferred from DNS data of the normalized surface heat flux shown in figure 18, which is consistent with the schematic in figure 17, are plotted in the upper panel of figure 20 along with the corresponding $Pr$-independent feature. The corresponding $Pr$-dependent features for each $Pr$ regime are displayed in the lower panel. This clearly explains the high correlation between the surface heat flux and the $Pr$-independent feature and the mild and asymmetric correlation between the surface heat flux and $Pr$-dependent feature. The shifts in the peak location of the two-point correlations shown in figure 19 are self-evident, and the negative correlation between the surface heat flux and the $Pr$-dependent feature for large distances can be explained by the schematic in figure 20. This also supports the necessity of the decomposition algorithm, because a major common part clearly exists among the locally high surface heat flux distribution, and the $Pr$-dependent components are clearly distinct.

Figure 19. Two-point correlations along the streamwise direction (a) $R_{\tilde {q}_{w}\tilde {q}_{w}^C}(r_x)$ between surface heat flux $\tilde {q}_{w}$ and the $Pr$-independent feature $\tilde {q}_{w}^C$, and (b) $R_{\tilde {q}_{w}\tilde {q}_{w}^P}(r_x)$ between surface heat flux $\tilde {q}_{w}$ and the $Pr$-dependent feature $\tilde{q}_{w}^{P}$).

Figure 20. Schematics describing typical distributions of $\tilde {q}_w$, $\tilde {q}_w^C$ and $\tilde {q}_w^P$ for the same flow field for three regimes of $Pr$. The vertical dashed lines indicate the different peak locations for $\tilde {q}_w$ and $\tilde {q}_w^P$.

The $Pr$-independent feature tends to capture the spanwise variation that is most similar to that for $Pr=0.71$, as shown in figure 18. The spanwise one-dimensional energy spectrum of the surface heat flux and the $Pr$-independent feature shown in figure 21(a) quantitatively confirms that the energy spectra of the $Pr$-independent feature and the surface heat flux for $Pr=0.71$ almost coincide in the high-wavenumber region, whereas the spectra in the low-wavenumber region for $Pr=5$ are similar. The $Pr$-dependent feature clearly has a lower energy level in the low-wavenumber region than the $Pr$-independent feature, except for $Pr = 0.2$, as shown in figure 21(b). It is noteworthy that the $Pr$-independent feature shows a peak at $\kappa _z = 7$, indicating the capture of the average spanwise spacing of the thermal streaks of 160 wall units for $Pr = 0.71, 2$ and 5, while the spacing for $Pr = 0.2$ and 360 wall units is not reflected. Correspondingly, the spectra of the $Pr$-dependent feature for $Pr = 0.71$, 2 and 5 monotonically decrease with the wavenumber, whereas the spectrum for $Pr=0.2$ shows a mild peak in the low-wavenumber range.

Figure 21. One-dimensional energy spectra for decomposed fields with $Pr$. (a) Spanwise energy spectra of the $Pr$-independent feature ($\tilde {q}_{w}^C$) and surface heat flux ($\tilde {q}_{w}$). (b) Spanwise energy spectra of the $Pr$-independent feature ($\tilde {q}_{w}^C$) and the $Pr$-dependent feature ($\tilde {q}_{w}^P$).

In this section we demonstrated how our algorithm decomposes the surface heat flux into a $Pr$-independent feature and $Pr$-dependent features during the training process. For the considered range of $Pr$, this decomposition clearly confirms that there exists a common feature in the high heat flux events, but the $Pr$-dependent features are quite distinct. By investigating the correlation between these features, we explored the effect of the Prandtl number in determining their physical relationship with the surface heat flux.